International Mathematics Research Notices

Frantzikinakis,, Nikos;Host,, Bernard

doi: 10.1093/imrn/rnz037pmid: N/A

Abstract We prove a structural result for measure preserving systems naturally associated with any finite collection of multiplicative functions that take values on the complex unit disc. We show that these systems have no irrational spectrum and their building blocks are Bernoulli systems and infinite-step nilsystems. One consequence of our structural result is that strongly aperiodic multiplicative functions satisfy the logarithmically averaged variant of the disjointness conjecture of Sarnak for a wide class of zero entropy topological dynamical systems, which includes all uniquely ergodic ones. We deduce that aperiodic multiplicative functions with values plus or minus one have super-linear block growth. Another consequence of our structural result is that products of shifts of arbitrary multiplicative functions with values on the unit disc do not correlate with any totally ergodic deterministic sequence of zero mean. Our methodology is based primarily on techniques developed in a previous article of the authors where analogous results were proved for the Möbius and the Liouville function. A new ingredient needed is a result obtained recently by Tao and Teräväinen related to the odd order cases of the Chowla conjecture. 1 Introduction and Main Results 1.1 Main results A function |$f\colon \mathbb{N}\to \mathbb{C}$| is called multiplicative if \begin{equation*} f(mn)=f(m)f(n) \ \textrm{whenever} \ (m,n)=1. \end{equation*} Perhaps the most well-known example of a bounded multiplicative function is the Möbius function, which is defined to be |$0$| on integers divisible by a square, |$-1$| on square-free integers with an odd number of prime factors, and |$1$| elsewhere. Its non-zero values are expected to fluctuate between |$-1$| and |$1$| in a random way and many famous conjectures have been formulated based on this belief. One example that has received a lot of attention in recent years is the Möbius disjointness conjecture of Sarnak [27, 28]. It asserts that the Möbius function does not correlate with any bounded deterministic sequence, meaning, any sequence that is produced by a continuous function evaluated along the orbit of a point in a zero entropy topological dynamical system. In [14] we verified the logarithmically averaged variant of the conjecture of Sarnak for a wide class of deterministic sequences. Our approach was to study measure preserving systems (which we call Furstenberg systems) naturally associated with the Möbius function; in particular, we studied structural properties that allow to deduce disjointness from a wide class of zero entropy systems. Various interesting results, establishing non-correlation of deterministic sequences with the Möbius function and products of its shifts, are natural consequences of these disjointness results. The main purpose of this article is to extend the approach from [14] in order to cover multiplicative functions that take values on the complex unit disc |$\mathbb{U}:=\{z\in{{\mathbb{C}}}\colon |z|\leq 1\}$|⁠. Our 1st main result concerns a class of multiplicative functions that are expected to satisfy similar disjointness properties as the Möbius function. These are the strongly aperiodic multiplicative functions (see Definition 2.9), and we verify that they do not correlate with a wide class of deterministic sequences. Throughout the paper, we denote by |$(Y,R)$| a topological dynamical system, meaning, a compact metric space |$Y$| together with a continuous homeomorphism |$R\colon Y\to Y$|⁠. Theorem 1.1. Let |$f\colon{{\mathbb{N}}}\to \mathbb{U}$| be a strongly aperiodic multiplicative function. Let |$(Y,R)$| be a topological dynamical system with zero topological entropy and at most countably many ergodic invariant measures. Then for every |$y\in Y$| and every |$g\in C(Y)$| we have \begin{equation*} \lim_{N\to\infty} \frac 1{\log N}\sum_{n=1}^N\frac{g(R^ny)\, f(n)}{n} =0. \end{equation*} Furthermore, the convergence is uniform in |$y\in Y$|⁠. Remarks. |$\bullet$| Using rotations on finite cyclic groups, one deduces that non-correlation (using logarithmic averages) of |$f$| with all periodic sequences (which implies strong aperiodicity in the real valued case) is a necessary assumption for the conclusion to hold. |$\bullet$| We believe that the countability assumption on the number of ergodic invariant measures of |$(Y,R)$| can be dropped. In the case where |$f$| is the Möbius function, this is equivalent to the logarithmically averaged variant of the Sarnak conjecture. An interesting consequence of the previous result is a statement about the block complexity of multiplicative functions |$f\colon{{\mathbb{N}}}\to \mathbb{U}$| that have finite range. In the next statement we denote by |$P_f(n)$| the number of patterns of size |$n$| that are taken by consecutive values of |$f$| (see Section 5.3 for a more formal definition). Theorem 1.2. If the multiplicative function |$f\colon{{\mathbb{N}}}\to \mathbb{U}$| has finite range, is strongly aperiodic, and does not converge to zero in logarithmic density, then |$\lim _{n\to \infty }\frac{P_f(n)}{n}=\infty$|⁠. Remarks. |$\bullet$| In fact, we establish a stronger statement, if |$a\colon{{\mathbb{N}}}\to{{\mathbb{C}}}$| has finite range and linear block growth, then |$\lim _{N\to \infty }\frac{1}{\log{N}}\sum _{n=1}^{N} \frac{a(n) \, f(n)}{n}=0$| for every strongly aperiodic multiplicative function |$f\colon{{\mathbb{N}}}\to \mathbb{U}$|⁠. Thus, even if we modify the values of |$f$| on a set of logarithmic density |$0$|⁠, using values taken from a finite set of complex numbers, the new sequence still has super-linear block growth. |$\bullet$| The assumptions are satisfied if |$f$| takes only the values |$\pm 1$| and is aperiodic, meaning, it does not correlate with any periodic sequence. Previously, it was not even known that for such multiplicative functions we have |$\lim _{n\to \infty }(P_f(n)-n)=\infty$|⁠. On the other hand, a conjecture of Elliott [9, 10] predicts if |$f\colon{{\mathbb{N}}}\to \{-1,1\}$| is aperiodic, then |$P_f(n)=2^n$| for every |$n\in{{\mathbb{N}}}$|⁠, and if |$f\colon{{\mathbb{N}}}\to \mathbb{U}$| has finite range, is strongly aperiodic, and does not converge to zero in logarithmic density, then |$P_f(n)$| grows exponentially. Henceforth, whenever needed, we assume that a multiplicative function |$f\colon{{\mathbb{N}}}\to \mathbb{U}$| is extended to the integers in an arbitrary way. In the next result if |$(Y,R)$| is a topological dynamical system, we say that a point |$y\in Y$| is generic for logarithmic averages for a Borel probability measure |$\nu$| on |$Y$| if for every |$g\in C(Y)$| we have |$\lim _{N\to \infty } \frac{1}{\log{N}}\sum _{n=1}^N \frac{g(R^ny)}{n}=\int g\, \textrm{d}\nu .$| Our methods also allow us to prove non-correlation between products of shifts of arbitrary multiplicative functions with values on the unit disc and totally ergodic deterministic sequences of zero mean. Theorem 1.3. Let |$f_1,\ldots ,f_\ell \colon{{\mathbb{N}}}\to \mathbb{U}$| be multiplicative functions. Let |$(Y,R)$| be a topological dynamical system and let |$y\in Y$| be generic for logarithmic averages for a measure |$\nu$| with zero entropy and at most countably many ergodic components, all of which are totally ergodic. Then for every |$g\in C(Y)$| that is orthogonal in |$L^2(\nu )$| to all |$R$|-invariant functions we have \begin{equation} \lim_{N\to\infty} \frac 1{\log N}\sum_{n=1}^N\frac{g(R^ny)\, \prod_{j=1}^{\ell} f_j(n+h_j)}{n}=0 \end{equation} (1) for all |$h_1,\dots ,h_\ell \in{{\mathbb{Z}}}$|⁠. Remarks. |$\bullet$| The unweighted version of (1) (take |$g:=1$|⁠) is expected to hold if the shifts are distinct and at least one of the multiplicative functions is strongly aperiodic. This is the logarithmically averaged variant of a conjecture of Elliott [9, 10] (see [26, Theorem B.1] for a corrected version and the need to assume strong aperiodicity). |$\bullet$| If |$(Y,R)$| has zero topological entropy and is uniquely and totally ergodic, then it is easy to deduce from Theorem 1.3 that (1) holds for all |$g\in C(Y)$| and |$y\in Y$| such that |$\lim _{N\to \infty }\frac 1{\log N}\sum _{n=1}^N\frac{g(R^ny)}{n}=0$|⁠. |$\bullet$| If |$(Y,\nu ,R)$| is totally ergodic, then using an approximation argument one can also conclude that (1) holds for every |$g\colon Y\to{{\mathbb{C}}}$| that is Riemann integrable with respect to the measure |$\nu$| and |$\int g\, \textrm{d}\nu =0$|⁠. We say that a function |$g\colon Y\to{{\mathbb{C}}}$| is Riemann integrable with respect to the measure |$\nu$| if for every |$\varepsilon>0$| there exist |$g_1,g_2\in C(Y)$| such that |$g_1(y)\leq g(y)\leq g_2(y)$| for every |$y\in Y$| and |$\int (g_2-g_1)\, d\nu \leq \varepsilon$|⁠. Theorem 1.3 is new even in the very special case where |$R$| is given by an irrational rotation on |${{\mathbb{T}}}$| and |$g(t):=\textrm{e}^{2\pi i t}$|⁠, |$t\in{{\mathbb{T}}}$|⁠. In this case we have |$g(R^n0)=\textrm{e}^{2\pi i n\alpha }$|⁠, |$n\in{{\mathbb{N}}}$|⁠, for some irrational |$\alpha$|⁠, and we get the following result as a consequence: Corollary 1.4. Let |$f_1,\ldots ,f_\ell \colon{{\mathbb{N}}}\to \mathbb{U}$| be multiplicative functions and let |$\alpha \in{{\mathbb{R}}}$| be irrational. Then \begin{equation} \lim_{N\to\infty} \frac 1{\log N}\sum_{n=1}^N\frac{\textrm{e}^{2\pi\, in\, \alpha}\, \prod_{j=1}^{\ell} f_j(n+h_j)}{n}=0 \end{equation} (2) for all |$h_1,\dots ,h_\ell \in{{\mathbb{Z}}}$|⁠. Remarks. |$\bullet$| For |$\ell =1$| the result is the logarithmically averaged variant of a classical result of Daboussi [5–7]. But even for |$\ell =2$| the result is new. |$\bullet$| More generally, if we apply Theorem 1.3 for |$R$| given by appropriate totally ergodic affine transformations on a torus with the Haar measure (as in [16, Section 3.3]), we get that (2) holds with |$(\textrm{e}^{2\pi i n\alpha })_{n\in{{\mathbb{N}}}}$| replaced by any sequence of the form |$(\textrm{e}^{2\pi i P(n)})_{n\in{{\mathbb{N}}}}$|⁠, where |$P\in{{\mathbb{R}}}[t]$| has at least one non-constant coefficient irrational. Moreover, one could use as weights zero mean sequences arising from more general totally ergodic nilsystems, giving rise to generalized polynomial sequences. One such example is the sequence |$(e^{2\pi i [n\alpha ]n\beta })_{n\in{{\mathbb{N}}}}$|⁠, where |$\alpha ,\beta \in{{\mathbb{R}}}$| are rationally independent. In order to establish this variant, one has to use Theorem 1.3 (see the last remark following this result) for a totally ergodic nilsystem |$(Y,R)$| defined on the Heisenberg nilmanifold and an appropriate Riemann integrable function |$g$| with respect to the Haar measure on |$Y$| with zero integral (see [2, Section 0.14] for details). A key step in the proof of the previous results is a structural result for measure preserving systems naturally associated with any collection of multiplicative functions that take values on the complex unit disc. We call such systems Furstenberg systems, and they are defined as follows: For convenience, let |$f$| be a multiplicative function that takes values on a finite subset |$A$| of |$\mathbb{U}$| and admits correlations for logarithmic averages on a sequence of intervals |${\mathbf N}=([N_k])_{k\in{{\mathbb{N}}}}$| with |$N_k\to \infty$| (see Definition 2.1). Then the Furstenberg system associated with |$f$| and |${\mathbf N}$| is defined on the sequence space |$X:=A^{{\mathbb{Z}}}$| with the shift transformation, by a measure that assigns to each cylinder set |$\{x\in X\colon x(j)=a_j,j=-m,\ldots ,m\}$| value equal to the logarithmic density, taken along the sequence |${\mathbf N}$|⁠, of the set |$\{n\in{{\mathbb{N}}}\colon f(n+j)=a_j,j=-m,\ldots ,m\}$|⁠, where |$a_{-m},\ldots , a_m\in A$| and |$m\in{{\mathbb{N}}}$|⁠. Similarly, one defines Furstenberg systems associated with any finite collection of multiplicative functions |$f_1,\ldots , f_\ell \colon{{\mathbb{N}}}\to \mathbb{U}$| and a sequence of intervals |${\mathbf N}$| on which |$f_1,\ldots , f_\ell$| admit correlations for logarithmic averages; we call these measure preserving systems joint Furstenberg systems. The precise constructions are given in Section 2.3 and are motivated by analogous constructions made by Furstenberg in [16, 17] in order to restate Szemerédi’s theorem on arithmetic progressions in ergodic terms. We prove the following structural result for joint Furstenberg systems of multiplicative functions: Theorem 1.5. A joint Furstenberg system of the multiplicative functions |$f_1,\ldots , f_\ell \colon{{\mathbb{N}}}\to \mathbb{U}$| is a factor of a system that (i) has no irrational spectrum; (ii) has ergodic components isomorphic to direct products of infinite-step nilsystems and Bernoulli systems. Remarks. |$\bullet$| We refer the reader to Section 2 and Appendix A of [14] for the definition of the ergodic notions used in the previous statement. |$\bullet$| The product decomposition depends on the ergodic component, in particular, the infinite-step nilsystems depend on the ergodic component. See Section 1.3 for more refined conjectural statements regarding the structure of joint Furstenberg systems of multiplicative functions with values on the unit disc. 1.2 Proof strategy Our general strategy in the proofs of Theorems 1.1–1.3 is similar to the one used in [14] to cover the case of the Möbius and the Liouville functions, but there are also some serious additional difficulties that we have to overcome. Our main focus is to prove the structural result stated in Theorem 1.5; then Theorems 1.1–1.3 are consequences of this result and the deduction is carried out using joining arguments in Section 5 (Theorem 1.1 also uses additional number theory input provided by Theorem 2.10). The 1st step in the proof of Theorem 1.5 is to apply the identity of Theorem 3.1 that allows to express an arbitrary joint correlation of multiplicative functions as a weighted average of their dilated joint correlations taken over all prime dilates (this step necessitates the use of logarithmic averages). This leads, via the correspondence principle of Furstenberg (see Proposition 2.3), to certain ergodic identities that any joint Furstenberg system |$(X,\mu ,T)$| of these multiplicative functions satisfies. The next goal is to utilize the ergodic identities in order to prove the structural properties of Theorem 1.5. Unfortunately, the presence of some unwanted weights, which appear because the multiplicative functions are not constant on primes, creates serious problems that do not allow us to continue as in [14], especially when the multiplicative functions take infinitely many distinct values on primes. The way to overcome this obstacle is to first utilize a recent result of Tao and Teräväinen, which proves that joint correlations of multiplicative functions vanish if the product of the multiplicative functions is far from being periodic. This result enables us to obtain a variant of the identity in Theorem 3.1, which has the additional property that all the weights are equal to |$1$| (see Corollary 3.7). As a consequence, we get an ergodic identity, stated in Theorem 3.8, which allows to show that the system |$(X,\mu ,T)$| is a factor of some system of arithmetic progressions with steps given by all primes in an appropriate congruence class (see Definition 4.1). Finally, this system can be easily linked to a system of arithmetic progressions with prime steps (see Lemma 4.4). The structure of these systems was studied in [14] and they were shown to satisfy the structural properties of Theorem 1.5. Combining the above facts we get a proof of Theorem 1.5. A simpler and more elementary way to link Furstenberg systems of multiplicative functions to systems of arithmetic progressions with primes steps is explained in Section 4.2, but this simpler approach only works if the range of the multiplicative functions on the primes is a subset of the unit interval or a finite subset of the complex unit disc. 1.3 Further remarks and conjectures The structural result of Theorem 1.5 is not expected to be optimal and we give below some more refined conjectural structural statements. In what follows, unless explicitly specified, a Bernoulli system is allowed to be the trivial one point system. Moreover, an ergodic procyclic system (often referred to as an odometer) is an ergodic inverse limit of periodic systems, or equivalently, an ergodic system |$(X,\mu ,T)$| for which the rational eigenfunctions span a dense subspace of |$L^2(\mu )$|⁠. Conjecture 1. If the multiplicative functions |$f_1,\ldots , f_\ell$| take values in |$[-1,1]$| or in a finite subset of |$\mathbb{U}$|⁠, then they have a unique joint Furstenberg system (This is equivalent to the statement that all sequences of the form |$(\prod _{j=1}^m g_j(n+h_j))_{n\in{{\mathbb{N}}}}$| have logarithmic averages, where |$g_1,\ldots , g_m \in \{\,f_1,\ldots , f_\ell , \overline{f_1}, \ldots , \overline{f_\ell }\}$| and |$m, h_1,\ldots , h_m\in{{\mathbb{N}}}$| are arbitrary.), which is ergodic and isomorphic to the direct product of a procyclic system and a Bernoulli system. This generalizes [14, Conjecture 1], which concerned Furstenberg systems of a single multiplicative function |$f\colon{{\mathbb{N}}}\to [-1,1]$|⁠. If we further restrict to the case where |$f$| takes values in |$\{-1,1\}$|⁠, then we conjectured in [14, Conjecture 2] that |$f$| should have a unique Furstenberg system, which is either an ergodic procyclic system or a Bernoulli system. Combining [1, Theorem 1.7] and [7, Theorem 6] we get that the 1st alternative holds if |$f$| is not aperiodic (this happens if and only if |${{\mathbb{D}}}(\,f,\chi )<\infty$| for some Dirichlet character |$\chi$|⁠, see terminology in Section 2.5). We expect that the 2nd alternative holds exactly when |$f$| is aperiodic. This is known to be the case conditionally to the assumption that all Furstenberg systems of |$f$| are ergodic [12, Corollary 1.5]. Unconditionally, this is not even known for the Liouville function; it is equivalent to the logarithmically averaged variant of the Chowla conjecture. Perhaps surprisingly, multiplicative functions with values on the unit circle may have non-ergodic Furstenberg systems, in fact, with uncountably many ergodic components. Consider for instance the multiplicative function |$f(n):=n^{it}$|⁠, |$n\in{{\mathbb{N}}}$|⁠, for some non-zero |$t\in{{\mathbb{R}}}$|⁠. We claim that it has a unique Furstenberg system |$(X,\mu ,T)$|⁠, which is isomorphic to the system |$({{\mathbb{T}}},m_{{\mathbb{T}}},R)$|⁠, where |$R$| is the identity transformation on |${{\mathbb{T}}}$|⁠. Indeed, let |$G_0(y):=e^{2\pi i y}$|⁠, |$y\in{{\mathbb{T}}}$|⁠, and |$X:=\mathbb{U}^{{\mathbb{Z}}}$|⁠, |$T$| be the shift transformation on |$X$|⁠, |$F_0(x):=x(0)$|⁠, |$x\in X$|⁠, and |$\mu :=\lim _{N\to \infty }\frac 1{\log N}\sum _{n=1}^N\frac{ \delta _{f(n)}}{n}$| (we show below that the weak-star limit exists). We claim that |$\Phi \colon{{\mathbb{T}}}\to X$|⁠, defined by |$\Phi (y):=(e^{2\pi i y})_{n\in{{\mathbb{Z}}}}$|⁠, |$y\in{{\mathbb{T}}}$|⁠, is an isomorphism between the systems |$({{\mathbb{T}}},m_{{\mathbb{T}}},R)$| and |$(X,\mu ,T)$|⁠. The map |$\Phi$| is clearly one to one and satisfies |$T\circ \Phi =\Phi \circ R$|⁠. It remains to show that |$\mu =m_{{\mathbb{T}}}\circ \Phi ^{-1}$|⁠. Notice first, that due to the slowly varying nature of |$n^{it}$|⁠, for fixed |$h\in{{\mathbb{Z}}}$| we have |$(n+h)^{it}-n^{it}\to 0$| as |$n\to \infty$|⁠. Using this and the fact that |$\lim _{N\to \infty }\frac 1{\log N}\sum _{n=1}^{N} \frac{n^{it}}{n}=0$| for |$t\neq 0$|⁠, we get that for every |$m\in{{\mathbb{N}}}$| and |$k_{-m},\ldots , k_m\in{{\mathbb{Z}}}$|⁠, we have \begin{equation*} \int_X\prod_{j={-m}}^m T^{h_j}F_0^{k_j}\, \textrm{d}\mu=\lim_{N\to\infty}\frac 1{\log N}\sum_{n=1}^N\frac{\prod_{j={-m}}^m f^{k_j}(n+h_j)}{n}=\int_{{{\mathbb{T}}}} \prod_{j={-m}}^m T^{h_j}G_0^{k_j}\, \textrm{d}m_{{\mathbb{T}}}, \end{equation*} since the 2nd and 3rd terms are either simultaneously |$0$| or |$1$| depending on whether |$\sum _{j=-m}^mk_j\neq 0$| or |$\sum _{j=-m}^mk_j= 0$|⁠. Using this identity and the fact that |$G_0=F_0\circ \Phi$| we get that |$\mu =m_{{\mathbb{T}}}\circ \Phi ^{-1}$|⁠, completing the proof that |$\Phi$| is an isomorphism. Similarly, if |$f(n):=n^{it}{\boldsymbol{\mu }}(n)$|⁠, |$n\in{{\mathbb{N}}}$|⁠, where |$t\neq 0$| and |${\boldsymbol{\mu }}$| is the Möbius function, then we expect (but cannot prove) that |$f$| has a unique Furstenberg system with uncountably many ergodic components, all of them isomorphic to a direct product of a non-trivial procyclic system and a non-trivial Bernoulli system. It seems likely that a similar structural result holds for general multiplicative functions with values on the unit disc: Conjecture 2. A joint Furstenberg system of any multiplicative functions |$f_1,\ldots , f_\ell \colon\! {{\mathbb{N}}}\!\to \mathbb{U}$| has ergodic components isomorphic to direct products of procyclic systems and Bernoulli systems. 1.4 Notation For readers’ convenience, we gather here some notation that we use frequently throughout the article. We write |${{\mathbb{T}}}$| for the unit circle, which we often identify with |${{\mathbb{R}}}/{{\mathbb{Z}}}$|⁠, and we write |$\mathbb{U}$| for the complex unit disc. We denote by |${{\mathbb{N}}}$| the set of positive integers, by |${{\mathbb{P}}}$| the set of prime numbers, and for |$d\in{{\mathbb{N}}}$| we denote by |${{\mathbb{P}}}_d$| the set |${{\mathbb{P}}}\cap (d{{\mathbb{N}}}+1)$|⁠. For |$N\in{{\mathbb{N}}}$| we denote by |$[N]$| the set |$\{1,\ldots , N\}$|⁠. Whenever we write |${\mathbf N}$| we mean a sequence of intervals of integers |$([N_k])_{k\in{{\mathbb{N}}}}$| with |$N_k\to \infty$|⁠. 2 Background in Ergodic Theory and Number Theory 2.1 Notation regarding averages If |$A$| is a non-empty finite subset of |${{\mathbb{N}}}$| we let \begin{equation*} {{\mathbb{E}}}_{n\in A}\,a(n):=\frac{1}{|A|}\sum_{n\in A}\, a(n), \quad{\mathbb{E}}^{\log}_{n\in A}\,a(n):=\frac{1}{\sum_{n\in A}\frac{1}{n}}\sum_{n\in A}\frac{a(n)}{n}. \end{equation*} If |$A$| is an infinite subset of |${{\mathbb{N}}}$| we let \begin{equation*} {{\mathbb{E}}}_{n\in A}\, a(n):=\lim_{N\to\infty} {{\mathbb{E}}}_{n\in A\,\cap\, [N]}\, a(n), \quad{\mathbb{E}}^{\log}_{n\in A}\, a(n):=\lim_{N\to\infty} {\mathbb{E}}^{\log}_{n\in A\,\cap\, [N]}\, a(n) \end{equation*} if the limits exist. Let |${\mathbf N}= ([N_k])_{k\in{{\mathbb{N}}}}$| be a sequence of intervals with |$N_k\to \infty$|⁠. We let \begin{equation*} {{\mathbb{E}}}_{n\in{\mathbf N}}\, a(n):=\lim_{k\to\infty}{{\mathbb{E}}}_{n\in[N_k]} \, a(n), \quad{\mathbb{E}}^{\log}_{n\in{\mathbf N}}\, a(n):=\lim_{k\to\infty}{\mathbb{E}}^{\log}_{n\in[N_k]} \, a(n) \end{equation*} if the limits exist. Using partial summation one sees that if |${{\mathbb{E}}}_{n\in{{\mathbb{N}}}}\,a(n)=0$|⁠, then also |${\mathbb{E}}^{\log }_{n\in{{\mathbb{N}}}}\,a(n)=0$|⁠. The converse does not hold in general, and the forward implication fails if one replaces |${{\mathbb{N}}}$| with general sequences |${\mathbf N}= ([N_k])_{k\in{{\mathbb{N}}}}$| with |$N_k\to \infty$|⁠. 2.2 Measure preserving systems Throughout the article, we make the standard assumption that all probability spaces |$(X,{{\mathcal{X}}},\mu )$| considered are Lebesgue, meaning, |$X$| can be given the structure of a compact metric space and |${{\mathcal{X}}}$| is its Borel |$\sigma$|-algebra. A measure-preserving system, or simply a system, is a quadruple |$(X,{{\mathcal{X}}},\mu ,T)$| where |$(X,{{\mathcal{X}}},\mu )$| is a probability space and |$T\colon X\to X$| is an invertible, measurable, measure preserving transformation. We typically omit the |$\sigma$|-algebra |${{\mathcal{X}}}$| and write |$(X,\mu ,T)$|⁠. Throughout, for |$n\in{{\mathbb{N}}}$| we denote by |$T^n$| the composition |$T\circ \cdots \circ T$| (⁠|$n$| times) and let |$T^{-n}:=(T^n)^{-1}$| and |$T^0:=\operatorname{id}_X$|⁠. Also, for |$f\in L^1(\mu )$| and |$n\in{{\mathbb{Z}}}$| we denote by |$T^nf$| the function |$f\circ T^n$|⁠. In order to avoid unnecessary repetition, we refer the reader to the article [14] for some other standard notions from ergodic theory. In particular, the reader will find in Section 2 and in Appendix A of [14] the definition of the terms factor, Kronecker factor, isomorphism, inverse limit, spectrum, rational and irrational spectrum, ergodicity, ergodic components, total ergodicity, nilsystem, infinite-step nilsystem, Bernoulli system, joining, and disjoint systems; all these notions are used subsequently. 2.3 Furstenberg systems associated with several sequences To each finite collection of sequences |$a_1,\ldots , a_\ell \colon{{\mathbb{N}}}\to \mathbb{U}$| that are distributed “regularly” along a sequence of intervals, we associate a measure preserving system defined using the joint distribution of the sequences |$a_1,\ldots , a_\ell$|⁠. For the purposes of this article, all averages in the definition of joint Furstenberg systems are taken to be logarithmic. Definition 2.1. Let |${\mathbf N}:=([N_k])_{k\in{{\mathbb{N}}}}$| be a sequence of intervals with |$N_k\to \infty$|⁠. We say that the sequences |$a_1,\ldots , a_\ell \colon{{\mathbb{Z}}} \to \mathbb{U}$| admit log correlations on |${\mathbf N}$|⁠, if the limits \begin{equation} \lim_{k\to\infty} {\mathbb{E}}^{\log}_{n\in [N_k]}\, \prod_{j=1}^m b_j(n+h_j) \end{equation} (3) exist for all |$m \in{{\mathbb{N}}}$|⁠, all |$h_1,\ldots , h_m\in{{\mathbb{Z}}}$|⁠, and all |$b_1,\ldots , b_m\in \{a_1,\ldots , a_\ell , \overline{a_1}, \ldots , \overline{a_\ell }\}$|⁠. Remarks. |$\bullet$| Given |$a_1,\ldots , a_\ell \colon{{\mathbb{Z}}} \to \mathbb{U}$|⁠, using a diagonal argument, we get that every sequence of intervals |${\mathbf N}=([N_k])_{k\in{{\mathbb{N}}}}$| has a subsequence |${\mathbf N}^{\prime}=([N_k^{\prime}])_{k\in{{\mathbb{N}}}}$|⁠, such that the sequences |$a_1,\ldots , a_\ell$| admit log correlations on |${\mathbf N}^{\prime}$|⁠. |$\bullet$| If the sequences |$a_1, \ldots , a_\ell$| are only defined on |${{\mathbb{N}}}$|⁠, then we extend them in an arbitrary way to |${{\mathbb{Z}}}$| and give analogous definitions. Then all the limits in (3) do not depend on the choice of the extension. Definition 2.2. Let |$(X,T)$| be a topological dynamical system. We say that the collection of functions |$F_1,\ldots , F_\ell \in C(X)$| is |$T$|-generating if the functions |$T^nF_1,\ldots , T^nF_\ell$|⁠, |$n\in{{\mathbb{Z}}}$|⁠, separate points of |$X$|⁠. Remark. By the Stone–Weierstrass theorem, the functions |$F_1,\ldots , F_\ell \in C(X)$| are |$T$|-generating if and only if the |$T$|⁠, |$T^{-1}$|-invariant subalgebra generated by |$F_1,\ldots , F_\ell$| and |$\overline{F_1},\ldots ,\overline{F_\ell }$| is dense in |$C(X)$| with the uniform topology. We use the following variant of the correspondence principle of Furstenberg [16, 17] that applies to finite collections of bounded sequences of complex numbers: Proposition 2.3. Let |$a_1,\ldots , a_\ell \colon{{\mathbb{Z}}}\to \mathbb{U}$| be sequences that admit log correlations on |${\mathbf N}:=([N_k])_{k\in{{\mathbb{N}}}}$|⁠. Then there exist a topological dynamical system |$(X,T)$|⁠, a |$T$|-invariant Borel probability measure |$\mu$|⁠, and a |$T$|-generating collection of functions |$F_{0,1}\ldots , F_{0,\ell }\in C(X)$|⁠, such that \begin{equation} {\mathbb{E}}^{\log}_{n\in{{\mathbf N}}} \, \prod_{j=1}^m b_j(n+h_j) =\int_X \prod_{j=1}^m T^{h_j}F_j \, \textrm{d}\mu \end{equation} (4) for all |$m\in{{\mathbb{N}}}$|⁠, all |$h_1, \ldots , h_m\in{{\mathbb{Z}}}$|⁠, and all |$b_j,\ldots , b_m\in \{a_1,\ldots , a_\ell , \overline{a_1},\ldots , \overline{a_\ell }\}$|⁠, where for |$j=1,\ldots , m,$| if the sequence |$b_j$| is |$a_k$| or |$\overline{a_k}$| for some |$k\in \{1,\ldots , \ell \}$|⁠, then |$F_j$| is |$F_{0,k}$| or |$\overline{F_{0,k}}$| respectively. Remark. In the arguments that follow we often use the explicit choice of |$X$| and |$T$| made in the proof below, namely, we take |$X=(\mathbb{U}^{\ell })^{{\mathbb{Z}}}$| and let |$T$| be the shift transformation on |$X$|⁠. We also often assume that the functions |$F_{0,1}, \ldots , F_{0,\ell }$| are defined by (5) below. Proof. Let |$X:=(\mathbb{U}^{\ell })^{{\mathbb{Z}}}$| and |$T$| be the shift transformation on |$X$|⁠, namely, |$T$| maps an element |$((x_1(n),\ldots , x_\ell (n)))_{n\in{{\mathbb{Z}}}}$| of |$X$| to |$((x_1(n+1),\ldots , x_\ell (n+1)))_{n\in{{\mathbb{Z}}}}$|⁠. For |$j=0,\ldots , \ell$| we define the functions |$F_{0,j}\in C(X)$| as follows \begin{equation} F_{0,j}(x):=x_j(0), \quad \textrm{for} \ x=((x_1(n),\ldots, x_\ell(n)))_{n\in{{\mathbb{Z}}}}\in X. \end{equation} (5) Finally, the measure |$\mu$| is defined to be the weak-star limit of the sequence of measures |${\mathbb{E}}^{\log }_{n\in [N_k]}\delta _{T^na}$|⁠, |$k\in{{\mathbb{N}}}$|⁠, where |$a:=((a_1(n),\ldots , a_\ell (n)))_{n\in{{\mathbb{Z}}}}\in X$|⁠. Then |$\mu$| is |$T$|-invariant and we have |$F_{0,j}(T^na)=a_j(n)$|⁠, |$n\in{{\mathbb{Z}}}$|⁠, for |$j=1,\ldots , \ell$|⁠. It follows that (4) holds and the proof is complete. Definition 2.4. Let |$a_1,\ldots , a_\ell \colon{{\mathbb{Z}}}\to \mathbb{U}$| be sequences that admit log correlations on |${\mathbf N}:=(N_k)_{k\in{{\mathbb{N}}}}$|⁠. We call the system (or the measure |$\mu$|⁠) defined in Proposition 2.3 the joint Furstenberg system (or measure) associated with |$a_1,\ldots , a_\ell$| and |${\mathbf N}$|⁠. Remarks. |$\bullet$| Given |$a_1,\ldots ,a_\ell \colon{{\mathbb{Z}}}\to \mathbb{U}$| and |${\mathbf N}$|⁠, the measure |$\mu$| is uniquely determined by (4) since this identity determines the values of |$\int f\, \textrm{d}\mu$| for all real valued |$f\in C(X)$|⁠. |$\bullet$| If two or more sequences coincide, say for example that |$a_m=\cdots =a_\ell$| for some |$m\in \{1,\ldots , \ell -1\}$|⁠, then it is not hard to see that the joint Furstenberg system associated with |$a_1,\ldots , a_\ell$| and |${\mathbf N}$| is isomorphic with the one associated with |$a_1,\ldots , a_m$| and |${\mathbf N}$|⁠. |$\bullet$| A collection of sequences |$a_1,\ldots , a_\ell \colon{{\mathbb{Z}}}\to \mathbb{U}$| may have several non-isomorphic joint Furstenberg systems depending on which sequence of intervals |${\mathbf N}$| we use in the evaluation of their joint correlations. When we write that a joint Furstenberg measure or system of the sequences |$a_1,\ldots , a_\ell$| has a certain property, we mean that any of these measures or systems has the asserted property. 2.4 Convergence results Henceforth, we use the following notation: Definition 2.5. If |$d\in{{\mathbb{N}}}$| we let |${{\mathbb{P}}}_d:={{\mathbb{P}}}\cap (d{{\mathbb{N}}}+1)$|⁠. We will use the following convergence result that was proved in [32] and also in [15] conditional to some conjectures obtained later in [21, 22]: Theorem 2.6. Let |$(X,\mu ,T)$| be a system and |$d\in{{\mathbb{N}}}$|⁠. Then for every |$\ell \in{{\mathbb{N}}}$| and |$F_1,\ldots ,F_\ell \in L^{\infty }(\mu )$| the following limit exists in |$L^2(\mu )$| \begin{equation*} {{\mathbb{E}}}_{p\in{{\mathbb{P}}}_d}\prod_{j=1}^{\ell} T^{pj}F_j. \end{equation*} Remark. Convergence is proved in [32] and [15] for |$d=1$|⁠. The more general statement follows by using the |$d=1$| case for product systems of the form |$T\times R$| acting on |$X\times{{\mathbb{Z}}}/d{{\mathbb{Z}}}$| with the product measure, where |$R$| is the translation by |$1$| on |${{\mathbb{Z}}}/d{{\mathbb{Z}}}$|⁠, and for the functions |$F_1\otimes{\textbf 1}_{d{{\mathbb{Z}}}+1}, F_2,\ldots , F_\ell$|⁠; of course, one also uses the fact that the relative density of the set |${{\mathbb{P}}}_d$| in the primes exists. We will make use of the following consequence of Theorem 2.6: Proposition 2.7. Suppose that the sequences |$a_1,\ldots , a_\ell \colon{{\mathbb{Z}}}\to \mathbb{U}$| admit log correlations on the sequence of intervals |${\mathbf N}$|⁠. Then for every |$d\in{{\mathbb{N}}}$| the limit \begin{equation*} {{\mathbb{E}}}_{p\in{{\mathbb{P}}}_d}\left({\mathbb{E}}^{\log}_{n\in{\mathbf N}}\prod_{j=1}^m b_j(n+ph_j)\right) \end{equation*} exists for all |$m \in{{\mathbb{N}}}$|⁠, all |$h_1,\ldots , h_m\in{{\mathbb{Z}}}$|⁠, and all |$b_1,\ldots ,b_{m}\in \{a_1,\ldots , a_\ell , \overline{a_1},\ldots , \overline{a_\ell }\}$|⁠. Proof. Let |$(X,{{\mathcal{X}}}, \mu ,T)$| be the joint Furstenberg system associated with |$a_1,\ldots , a_\ell$| and |${\mathbf N}$|⁠, and let also |$F_{0,1},\ldots , F_{0,\ell }\in L^{\infty }(\mu )$| be as in Proposition 2.3. Using Theorem 2.6 we get that the limit \begin{equation*} {{\mathbb{E}}}_{p\in{{\mathbb{P}}}_d}\int_X\prod_{j=1}^m T^{ph_j}F_j\,\textrm{d}\mu \end{equation*} exists for all |$m\in{{\mathbb{N}}}$|⁠, all |$h_1,\ldots , h_m\in{{\mathbb{Z}}}$|⁠, and all |$F_1,\ldots ,F_{m}\in L^{\infty }(\mu )$|⁠. Combining this with identity (4) we get the asserted convergence. 2.5 Aperiodic and strongly aperiodic multiplicative functions We denote by |$\mathcal{M}$| the set of all multiplicative functions |$f\colon{{\mathbb{N}}}\to \mathbb{U}$|⁠, where |$\mathbb{U}$| is the complex unit disc. A Dirichlet character is a periodic completely multiplicative function |$\chi$| with |$\chi (1)=1$|⁠. We say that |$f\in \mathcal{M}$| is aperiodic (or non-pretentious using terminology from [18]) if it averages to |$0$| on every infinite arithmetic progression, meaning, if \begin{equation*} {{\mathbb{E}}}_{n\in{{\mathbb{N}}}}\, f(an+b)=0, \quad \ \textrm{for all} \ a,b\in \mathbb{N}. \end{equation*} This is equivalent to asserting that |${{\mathbb{E}}}_{n\in{{\mathbb{N}}}}\, f(n)\, d(n)=0$| for every periodic sequence |$d\colon{{\mathbb{N}}}\to{{\mathbb{C}}}$|⁠, or that |${{\mathbb{E}}}_{n\in{{\mathbb{N}}}}\, f(n)\, \chi (n)=0$| for every Dirichlet character |$\chi$|⁠. In order to give easier to verify necessary conditions for aperiodicity, we use a notion of distance between two multiplicative functions defined as in [18]: Definition 2.8. We let |${{\mathbb{D}}}\colon \mathcal{M}\times \mathcal{M}\to [0,\infty ]$| be given by \begin{equation*} {{\mathbb{D}}}(\,f,g)^2:=\sum_{p\in{{\mathbb{P}}}} \frac{1}{p}\,\big(1-\Re\big(\,f(p) \overline{g(p)}\big)\big) \end{equation*} where |$\Re (z)$| denotes the real part of a complex number |$z$|⁠. It is shown in [8, Theorem 1] that |$f\in \mathcal{M}$| is aperiodic if and only if |${{\mathbb{D}}}(\,f, \chi \cdot n^{it})=\infty$| for every |$t\in{{\mathbb{R}}}$| and every Dirichlet character |$\chi$|⁠. Moreover, if |$f$| takes real values, then |$f$| is aperiodic if and only if |${{\mathbb{D}}}(\,f, \chi )=\infty$| for every Dirichlet character |$\chi$|⁠. In particular, the Möbius and the Liouville functions are aperiodic. For our purposes we also need a notion introduced in [26] that is somewhat stronger than aperiodicity. Definition 2.9. Let |${{\mathbb{D}}}\colon \mathcal{M}\times \mathcal{M}\times \mathbb{N} \to [0,\infty ]$| be given by \begin{equation*} {{\mathbb{D}}}(\,f,g;N)^2:=\sum_{p\in{{\mathbb{P}}}\cap [N]} \frac{1}{p}\,\bigl(1-\Re\bigl(\,f(p) \overline{g(p)}\bigr)\bigr) \end{equation*} and |$M\colon \mathcal{M}\times \mathbb{N} \to [0,\infty )$| be given by \begin{equation*} M(\,f;N):=\min_{|t|\leq N} {{\mathbb{D}}}(\,f, n^{it};N)^2. \end{equation*} The multiplicative function |$f\in \mathcal{M}$| is strongly aperiodic if |$M(\,f\cdot \chi ;N)\to \infty$| as |$N \to \infty$| for every Dirichlet character |$\chi$|⁠. Note that strong aperiodicity implies aperiodicity. The converse is not in general true (see [26, Theorem B.1]), but it is true for real-valued multiplicative functions (see [26, Appendix C]). In particular, the Möbius and the Liouville functions are strongly aperiodic. Furthermore, if |$f\in \mathcal{M}$| is aperiodic and |$f(p)$| is a |$d$|-th root of unity for all |$p\in{{\mathbb{P}}}$|⁠, then |$f$| is strongly aperiodic (see [13, Proposition 6.1]). In particular, if |$f(p)$| is a nontrivial |$d$|-th root of unity for all |$p\in{{\mathbb{P}}}$|⁠, then |$f$| is strongly aperiodic (see [13, Corollary 6.2]). The hypothesis of strong aperiodicity is useful for our purposes because it enables us to use the following result of Tao [30, Corollary 1.5]: Theorem 2.10. Let |$f\in \mathcal{M}$| be a strongly aperiodic multiplicative function. Then we have \begin{equation*} {\mathbb{E}}^{\log}_{n\in{{\mathbb{N}}}}\, f(n)\, \overline{f(n+h)}=0 \end{equation*} for every |$h\in{{\mathbb{N}}}$|⁠. Remark. By adjusting the example in [26, Theorem B.1], it follows that strong aperiodicity cannot be replaced by aperiodicity; in particular, there exist an aperiodic multiplicative function |$f\in \mathcal{M}$|⁠, a positive constant |$c$|⁠, and a sequence of intervals |${\textbf N}:=([N_k])_{k\in \mathbb{N}}$| with |$N_k\to \infty$|⁠, such that \begin{equation*} \Big|{\mathbb{E}}^{\log}_{n\in{\textbf N}} \, f(n)\cdot \overline{f(n+h)} \Big|\geq c, \quad \textrm{ for every}\ h\in \mathbb{N}. \end{equation*} 3 Correlation Identities and Ergodic Consequences 3.1 Correlation identities If |$a\colon{{\mathbb{P}}}\to \mathbb{U}$| is a sequence and |$A$| is a non-empty finite or infinite subset of the primes we define |${\mathbb{E}}^{\log }_{p\in A}$| as in Section 2.1. The following identity of Tao and Teräväinen from [31, Theorem 3.6] is key for our purposes: Theorem 3.1. Suppose that the multiplicative functions |$f_1,\ldots , f_\ell \colon{{\mathbb{Z}}}\to \mathbb{U}$| admit log correlations on the sequence of intervals |${\mathbf N}$|⁠. Then we have \begin{equation} {\mathbb{E}}^{\log}_{p\in{{\mathbb{P}}}}\left|c_{p,m}\, {\mathbb{E}}^{\log}_{n\in{\mathbf N}}\, \prod_{j=1}^m g_j(n+h_j)- \, {\mathbb{E}}^{\log}_{n\in{\mathbf N}}\, \prod_{j=1}^{m} g_j(n+ph_j)\right|=0 \end{equation} (6) for all |$m \in{{\mathbb{N}}}$|⁠, all |$h_1,\ldots , h_m\in{{\mathbb{Z}}}$|⁠, and all |$g_1,\ldots ,g_{m}\in \{\,f_1,\ldots , f_\ell , \overline{f_1},\ldots , \overline{f_\ell }\}$|⁠, where |$c_{p,m}:=\prod _{j=1}^m g_j(p)$|⁠, |$p\in{{\mathbb{P}}}$|⁠. Remarks. |$\bullet$| A variant of this result is also implicit in the article of Tao [30] and was also used in [14] for |$f_1=\cdots =f_\ell =\mu$| or |$\lambda$| using a different averaging scheme. The current version is more suitable for our purposes. |$\bullet$| In [31] the result is proved for a class of generalized limit functionals in place of |${\mathbb{E}}^{\log }_{n\in{\mathbf N}}$|⁠. Assuming that |${\mathbf N}=([N_k])_{k\in{{\mathbb{N}}}}$|⁠, the asserted version follows if one uses a generalized limit functional of the form |$\widetilde \lim _{k\to \infty }{\mathbb{E}}^{\log }_{n\in{[N_k]}}$| since it coincides with the standard limit |$\lim _{k\to \infty }{\mathbb{E}}^{\log }_{n\in{[N_k]}}={\mathbb{E}}^{\log }_{n\in{\mathbf N}}$| whenever this limit exists. For the record, we mention the following identity for general sequences, which follows from the proof of [31,Theorem 3.6] without any essential change; Theorem 3.1 is an easy consequence of this identity: Theorem 3.2. Let |${\mathbf N}$| be a sequence of intervals, |$a_1,\ldots , a_\ell \colon{{\mathbb{Z}}}\to \mathbb{U}$| be sequences, and |$h_1,\ldots , h_\ell \in{{\mathbb{Z}}}$|⁠. Then, assuming that for every |$p\in{{\mathbb{P}}}$| the limits |${\mathbb{E}}^{\log }_{n\in{\mathbf N}}$| below exist, we have the identity \begin{equation*} {\mathbb{E}}^{\log}_{p\in{{\mathbb{P}}}} \left|{\mathbb{E}}^{\log}_{n\in{\mathbf N}}\,\prod_{j=1}^{\ell} a_j(pn+ph_j)- {\mathbb{E}}^{\log}_{n\in{\mathbf N}}\, \prod_{j=1}^{\ell} a_j(n+ph_j)\right|=0. \end{equation*} 3.2 A consequence of the correlation identities We are going to combine Theorem 3.1 with Theorem 3.5 stated below in order to prove a variant of the identity (6) in which the weights |$c_{p,m}$| are all equal to |$1$|⁠. For convenience we introduce the following notation: Definition 3.3. Let |$a,b\colon{{\mathbb{P}}}\to \mathbb{U}$| be sequences. We write |$a\sim b$| if \begin{equation*} {\mathbb{E}}^{\log}_{p\in{{\mathbb{P}}}}\Big(1-\Re(a(p)\cdot \overline{b(p)})\Big)=0. \end{equation*} Remarks. |$\bullet$| If we restrict to sequences that take values on the unit circle, then |$\sim$| is an equivalence relation and |$a\sim b$| is equivalent to |${\mathbb{E}}^{\log }_{p\in{{\mathbb{P}}}}|a(p)-b(p)|=0$|⁠. |$\bullet$| Using terminology from [31] we have that two multiplicative functions |$f,g\colon{{\mathbb{Z}}}\to \mathbb{U}$| satisfy |$f\sim g$| exactly when “|$f$| weakly pretends to be |$g$|⁠.” We will use the following basic properties: Lemma 3.4. If |$a,b,c,d\colon{{\mathbb{P}}} \to \mathbb{U}$| are sequences, then the following properties hold: (i) If |$a\sim b$|⁠, then |$\overline{a}\sim \overline{b}$|⁠. (ii) If |$a\sim b$| and |$b\sim c$|⁠, then |$a \sim c$|⁠. (iii) If |$a\sim b$| and |$c\sim d$|⁠, then |$a c\sim b d$|⁠. (iv) If |$a\sim b$|⁠, then |${\mathbb{E}}^{\log }_{p\in{{\mathbb{P}}}}|a(p)-b(p)|=0$|⁠. Proof. Property |$\text{(i)}$| is obvious. Properties |$\text{(ii)}$| and |$\text{(iii)}$| follow from the estimate \begin{equation} 1-\Re(uv)\leq 2\big(1-\Re(u)+1-\Re(v)\big), \end{equation} (7) which holds for all |$u,v\in \mathbb{U}$|⁠. One way to prove this is to first consider the case where |$|u|=|v|=1$|⁠; in this case the estimate is equivalent to |$|u-v|^2\leq 2(|1-u|^2+|1-v|^2)$|⁠, which follows from the Cauchy–Schwarz inequality. One then deduces from this the general case by expressing arbitrary |$u,v\in \mathbb{U}$| as a convex combination of two elements on the unit circle and taking advantage of the linearity features of (7). Property |$\text{(iv)}$| follows from the estimate \begin{equation*} |u-v|^2\leq 2\big(1-\Re(u\overline{v})\big), \end{equation*} which holds for all |$u,v\in \mathbb{U}$|⁠. We will use the next result of Tao and Teräväinen [31, Theorem 1.2]: Theorem 3.5. Let |$f_1,\ldots , f_\ell \colon{{\mathbb{Z}}}\to \mathbb{U}$| be multiplicative functions. Suppose that for every Dirichlet character |$\chi$| we have |$f_1\cdots f_\ell \nsim \chi$|⁠. Then \begin{equation*} {\mathbb{E}}^{\log}_{n\in{{\mathbb{N}}}}\, \prod_{j=1}^{\ell} f_j(n+h_j)=0 \end{equation*} for all |$h_1,\ldots , h_\ell \in{{\mathbb{Z}}}$|⁠. Remarks. |$\bullet$| The use of logarithmic averages is essential for the statement to hold. For example, take |$\ell =1$| and let |$f_1(n):=n^{it}$|⁠, |$n\in{{\mathbb{N}}}$|⁠, for some non-zero |$t\in{{\mathbb{R}}}$|⁠. Then |$f_1 \nsim \chi$| for every Dirichlet character |$\chi$| but the limit |$\lim _{N\to \infty }{{\mathbb{E}}}_{n\in [N]} \, n^{it}$| does not exist (since |${{\mathbb{E}}}_{n\in [N]}\, n^{it}=\frac{N^{it}}{it+1}+o(1)$|⁠). On the other hand we have that |${\mathbb{E}}^{\log }_{n\in{{\mathbb{N}}}}\, n^{it}=0$|⁠. |$\bullet$| The proof of Theorem 3.5 depends crucially on deep results from ergodic theory such as [23–25] and analytic number theory [19, 20]. The next result is a key ingredient in our argument (recall that |${{\mathbb{P}}}_d={{\mathbb{P}}}\cap (d{{\mathbb{N}}}+1)$|⁠): Proposition 3.6. Let |$f_1,\ldots , f_\ell \colon{{\mathbb{Z}}}\to \mathbb{U}$| be multiplicative functions. There exists |$d\in{{\mathbb{N}}}$| such that the following holds: If |$f_1,\ldots , f_\ell$| admit log correlations on the sequence of intervals |${\mathbf N}$|⁠, then \begin{equation} {\mathbb{E}}^{\log}_{p\in{{\mathbb{P}}}_d}\left|{\mathbb{E}}^{\log}_{n\in{\mathbf N}}\, \prod_{j=1}^{m} g_j(n+h_j)- \, {\mathbb{E}}^{\log}_{n\in{\mathbf N}}\, \prod_{j=1}^{m}g_j(n+ph_j)\right|=0 \end{equation} (8) for all |$m \in{{\mathbb{N}}}$|⁠, all |$h_1, \ldots , h_{m} \in{{\mathbb{Z}}}$|⁠, and all |$g_1,\ldots ,g_{m}\in \{\,f_1,\ldots , f_\ell , \overline{f_1},\ldots , \overline{f_\ell }\}$|⁠. Proof. Suppose first that for some |$j\in \{1,\ldots , \ell \}$| we have |${\mathbb{E}}^{\log }_{n\in{\mathbf N}}|\,f_j(n)|=0$|⁠. Then whenever one of the functions |$g_1, \ldots , g_m$| is equal to |$f_j$| or |$\overline{f_j}$|⁠, all logarithmic averages in (8) vanish and the identity holds trivially for |$d=1$|⁠. Thus, without loss of generality, we can assume that for |$j=1,\ldots , \ell$| we have |${\mathbb{E}}^{\log }_{n\in{\mathbf N}}|\,f_j(n)|> 0$| (note that by our assumptions the average exists). Using Theorem 3.1 for |$\ell =1$|⁠, |$g_1=f_j$|⁠, and |$h_1=0$|⁠, we deduce that |${\mathbb{E}}^{\log }_{p\in{{\mathbb{P}}}}(1-|\,f_j(p)|)=0$|⁠. (We note that the |$\ell =1$| case of Theorem 3.1 admits a simple elementary proof via the Turàn-Kubilius inequality.) Hence, we can work under the assumption that \begin{equation} |\,f_j|\sim 1, \quad \textrm{for} \ \ j=1,\ldots, \ell. \end{equation} (9) Next, for |$k\in{{\mathbb{N}}}$| and |$j=1,\ldots , \ell$|⁠, we denote by |$f_j^{-k}$| the function |$\overline{f_j^k}$| and let |$K=K_{f_1,\ldots , f_\ell }$| be the subset of |${{\mathbb{Z}}}^{\ell }$| defined as follows: \begin{equation*} K:=\left\{(k_1,\ldots, k_\ell)\in{{\mathbb{Z}}}^{\ell}\colon \prod_{j=1}^{\ell} f_j^{k_j}\sim \chi \textrm{ for some Dirichlet character}\ \chi \right\}. \end{equation*} Using (9) and properties |$\text{(i)}$|–|$\text{(iii)}$| of Lemma 3.4, and since products and complex conjugates of Dirichlet characters are Dirichlet characters, we get that |$K$| is a subgroup of |${{\mathbb{Z}}}^{\ell }$|⁠. Since every subgroup of |${{\mathbb{Z}}}^{\ell }$| is finitely generated, |$K$| is finitely generated. We let |$F_K=F_{K,f_1,\ldots , f_\ell }$| be the following set of multiplicative functions \begin{equation*} F_K:=\left\{\prod_{j=1}^{\ell} f_j^{k_j}\colon (k_1,\ldots, k_\ell)\in K\right\}. \end{equation*} We have that |$F_K$| is finitely generated under multiplication. Let |$\{\,f_{0,1},\ldots , f_{0,r}\}$|⁠, for some |$r\in{{\mathbb{N}}}$|⁠, be a set of generators for |$F_K$|⁠. Then for |$j=1,\ldots , r$| there exist Dirichlet characters |$\chi _j$| such that |$f_{0,j}\sim \chi _j$|⁠. If |$d\in{{\mathbb{N}}}$| is a common period of all these Dirichlet characters, then for |$j=1,\ldots ,r$| we have |$\chi _j(dn+1)=1$| for every |$n\in{{\mathbb{N}}}$|⁠. Let |$f\in F_K$|⁠, then |$f=\prod _{j=1}^rf_{0,j}^{k_j}$| for some |$k_1,\ldots , k_r\in{{\mathbb{Z}}}$|⁠. Since |$f_{0,j}\sim \chi _j$| for |$\,j=1,\ldots , r,$| we get from property |$\text{(iii)}$| of Lemma 3.4 that |$f\sim \prod _{j=1}^r\chi _j^{k_j}$|⁠, and since |$\chi _j(p)=1$| for all |$j\in \{1,\ldots , r\}$| and all |$p\in{{\mathbb{P}}}_d$|⁠, we deduce from property |$\text{(iv)}$| of Lemma 3.4 that |${\mathbb{E}}^{\log }_{p\in{{\mathbb{P}}}_d}|\,f(p)-1|=0$| (we also used that |${{\mathbb{P}}}_d$| has positive relative density in |${{\mathbb{P}}}$|⁠). Hence, \begin{equation} {\mathbb{E}}^{\log}_{p\in{{\mathbb{P}}}_d}\big|\,f(p)-1\big|=0, \quad \textrm{ for every}\ \ f\in F_K. \end{equation} (10) We now show that (8) holds. Let |$\widetilde g:=\prod _{j=1}^mg_j$|⁠. Since |$g_j\in \{\,f_1,\ldots , f_\ell , \overline{f_1},\ldots , \overline{f_\ell }\}$| for |$j=1,\ldots , r$|⁠, using (9) and properties |$\text{(ii)}$| and |$\text{(iii)}$| of Lemma 3.4, we get that |$\widetilde g\sim \prod _{j=1}^{\ell } f_j^{k_j}$| for some |$k_1,\ldots , k_\ell \in{{\mathbb{Z}}}$|⁠, where we continue to use the notation |$f^k$| for |$\overline{f^{-k}}$| if |$k$| is a negative integer. We consider two cases. If |$\widetilde g \notin F_K$|⁠, then |$\widetilde g\nsim \chi$| for all Dirichlet characters |$\chi$|⁠, in which case (8) holds because by Theorem 3.5 we have |${\mathbb{E}}^{\log }_{n\in{\mathbf N}}\prod _{j=1}^{m} g_j(n+h_j)=0$| for all |$h_1,\ldots , h_{m}\in{{\mathbb{Z}}}$|⁠. On the other hand, if |$\widetilde g\in F_K$|⁠, we see that (8) holds by combining Theorem 3.1 (with |${{\mathbb{P}}}_d$| in place of |${{\mathbb{P}}}$|⁠) and (10). This completes the proof. We are going to use the following consequence of Proposition 3.6, which is better suited for our purposes: Corollary 3.7. Let |$f_1,\ldots , f_\ell \colon{{\mathbb{Z}}}\to \mathbb{U}$| be multiplicative functions. There exists |$d\in{{\mathbb{N}}}$| such that the following holds: If |$f_1,\ldots , f_\ell$| admit log correlations on the sequence of intervals |${\mathbf N}$|⁠, then the limit on the right-hand side below exists and we have \begin{equation} {\mathbb{E}}^{\log}_{n\in{\mathbf N}}\, \prod_{j=1}^m g_j(n+h_j)= {{\mathbb{E}}}_{p\in{{\mathbb{P}}}_d} {\mathbb{E}}^{\log}_{n\in{\mathbf N}}\, \prod_{j=1}^{m}g_j(n+ph_j) \end{equation} (11) for all |$m \in{{\mathbb{N}}}$|⁠, all |$h_1, \ldots , h_{m} \in{{\mathbb{Z}}}$|⁠, and all |$g_1,\ldots ,g_{m}\in \{\,f_1,\ldots , f_\ell , \overline{f_1},\ldots , \overline{f_\ell }\}$|⁠. Proof. This follows immediately from Proposition 3.6 and the fact that by Proposition 2.7 the limit |${{\mathbb{E}}}_{p\in{{\mathbb{P}}}_d}$| on the right-hand side exists and as a consequence the limit |${{\mathbb{E}}}^*_{p\in{{\mathbb{P}}}_d}$| is equal to the limit |${{\mathbb{E}}}_{p\in{{\mathbb{P}}}_d}$|⁠. 3.3 An ergodic consequence Using Proposition 2.3 we deduce from Corollary 3.7 the following ergodic result concerning joint Furstenberg systems of multiplicative functions: Theorem 3.8. Let |$f_1,\ldots , f_\ell \colon{{\mathbb{Z}}}\to \mathbb{U}$| be multiplicative functions. There exists |$d\in{{\mathbb{N}}}$| such that the following holds: If |$(X,\mu ,T)$| is a joint Furstenberg system of |$f_1,\ldots , f_\ell$|⁠, and if |$F_{0,1}, \ldots , F_{0,\ell }$| are as in Proposition 2.3, then we have \begin{equation} \int_X \prod_{j=1}^{m} T^{h_j} F_j \, d\mu= {{\mathbb{E}}}_{p\in{{\mathbb{P}}}_d} \int_X \prod_{j=1}^{m} T^{ph_j}F_j\, \textrm{d}\mu \end{equation} (12) for all |$m \in{{\mathbb{N}}}$|⁠, all |$h_1,\ldots , h_{m}\in{{\mathbb{Z}}}$|⁠, and all |$F_1,\ldots , F_{m}\in \{F_{0,1},\ldots F_{0,\ell }, \ \overline{F_{0,1}},\ldots , \overline{F_{0,\ell }} \}$|⁠. 4 The Structure of Furstenberg Systems of Multiplicative Functions The goal of this section is to prove Theorem 1.5. In the next section we use this structural result to prove Theorems 1.1–1.3. 4.1 Proof of Theorem 1.5 Given a system |$(X,\mu ,T)$| and |$d\in{{\mathbb{N}}}$| we define the system of arithmetic progressions with steps in |${{\mathbb{P}}}_d$| as follows: Definition 4.1. Let |$(X, \mu ,T)$| be a system and let |$X^{{\mathbb{Z}}}$| be endowed with the product |$\sigma$|-algebra. For |$d\in{{\mathbb{N}}}$| we write |$\widetilde \mu _d$| for the probability measure on |$X^{{\mathbb{Z}}}$| defined as follows: for every |$m\in{{\mathbb{N}}}$| and all |$F_{-m},\ldots ,F_m\in L^{\infty }(\mu )$|⁠, we let \begin{equation} \int_{X^{{\mathbb{Z}}}}\prod_{j=-m}^m F_j(x_j)\,\textrm{d}\widetilde\mu_{\textrm{d}}(\underline x):= {{\mathbb{E}}}_{p\in{{\mathbb{P}}}_d}\int_X\prod_{j=-m}^m T^{pj}F_j\,\textrm{d}\mu, \end{equation} (13) where |$\underline x:=(x_j)_{j\in{{\mathbb{Z}}}}$| and the limit on the right-hand side exists by Theorem 2.6. The measure |$\widetilde \mu _d$| is invariant under the shift transformation |$S$| on |$X^{{\mathbb{Z}}}$| and induces a system |$(X^{{\mathbb{Z}}}, \widetilde \mu _d,S)$|⁠, which we call the system of arithmetic progressions with steps in |${{\mathbb{P}}}_d$| associated with the system |$(X,\mu ,T)$|⁠. Remark. For |$d=1$| the system |$(X^{{\mathbb{Z}}}, \widetilde \mu _1,S)$| coincides with the system of arithmetic progressions with prime steps introduced in [14, Definition 3.8]. The relevance of the systems |$(X^{{\mathbb{Z}}}, \widetilde \mu _d,S)$| to our problem is demonstrated by the following result: Proposition 4.2. Let |$f_1,\ldots , f_\ell \colon{{\mathbb{Z}}}\to \mathbb{U}$| be multiplicative functions. Then there exists |$d\in{{\mathbb{N}}}$| such that any joint Furstenberg system |$(X,\mu ,T)$| of the multiplicative functions |$f_1,\ldots , f_\ell$| is a factor of the system |$(X^{{\mathbb{Z}}},\widetilde \mu _d,S)$|⁠. Proof. We can assume that the joint Furstenberg system is defined on the space |$X:=(\mathbb{U}^{\ell })^{{\mathbb{Z}}}$| and |$T$| is the shift transformation on |$X$|⁠. We denote elements of |$X$| by |$x=(x_1(k), \ldots , x_\ell (k))_{k\in{{\mathbb{Z}}}}$|⁠, where |$x_1(k), \ldots , x_\ell (k)\in U$| for |$k\in{{\mathbb{Z}}}$|⁠, and elements of |$X^{{\mathbb{Z}}}$| with |$\underline x=(x_n)_{n\in{{\mathbb{Z}}}}$|⁠, where |$x_n\in X$| for |$n\in{{\mathbb{Z}}}$|⁠. Hence, |$\underline x=(x_{n,1}(k), \ldots , x_{n,\ell }(k))_{k,n \in{{\mathbb{Z}}}}$| can be identified with |$(x_{n,1},\ldots , x_{n,\ell })_{n\in{{\mathbb{Z}}}}$|⁠, where |$x_{n,j}=(x_{n,j}(k))_{k\in{{\mathbb{Z}}}}$| for |$j=1,\ldots , \ell$|⁠. We define the map |$\pi \colon X^{{\mathbb{Z}}}\to X$| as follows: For |$\underline x=(x_{n, 1},\ldots , x_{n,\ell })_{n\in{{\mathbb{Z}}}}\in X^{{\mathbb{Z}}}$| let \begin{equation*} (\pi(\underline x))(n):=( x_{n,1}(0),\ldots, x_{n, \ell}(0))=(F_{0,1}(x_{n,1}),\ldots, F_{0,\ell}(x_{n, \ell})), \quad n\in{{\mathbb{Z}}}, \end{equation*} where \begin{equation*} F_{h, j}(x):=x_j(h),\quad x\in X,\ h\in{{\mathbb{Z}}},\ j\in \{1,\ldots, \ell\}. \end{equation*} For |$n\in{{\mathbb{Z}}}$| we have \begin{align*} (\pi(S\underline x))(n)=&\ (F_{0,1}((S\underline x)_n), \ldots,F_{0,\ell}((S\underline x)_n)) =(F_{0,1}(x_{n+1, 1}), \ldots,F_{0,\ell}(x_{n+1, \ell}))=\\ (\pi(\underline x))(n+1)=&\ (T\pi(\underline x))(n). \end{align*} Thus \begin{equation*} \pi\circ S=T\circ\pi. \end{equation*} Next, we claim that |$\widetilde \mu _d\circ \pi ^{-1}=\mu$|⁠. Indeed, for all |$m \in{{\mathbb{N}}}$|⁠, all |$h_1,\dots ,h_m\in{{\mathbb{Z}}}$|⁠, and all |$k_1,\ldots , k_m\in \{\pm 1,\ldots , \pm \ell \}$|⁠, using identity (12) in Theorem 3.8 and the definition of |$\widetilde \mu _d$| given in (13), we have \begin{align*} \int_X\prod_{j=1}^m F_{h_j, k_j}(x)\, \textrm{d}\mu(x)=&\ \int_X\prod_{j=1}^m F_{0, k_j}(T^{h_j}x)\,\textrm{d}\mu(x) = {{\mathbb{E}}}_{p\in{{\mathbb{P}}}_d} \int_X\prod_{j=1}^{\ell} F_{0, k_j}(T^{ph_j}x)\,\textrm{d}\mu(x) =\\ \int_{X^{{\mathbb{Z}}}}\prod_{j=1}^m F_{0,k_j}(x_{h_j})\,\textrm{d}\widetilde\mu_{\textrm{d}}(\underline x) =&\ \int_{X^{{\mathbb{Z}}}}\prod_{j=1}^m (F_{h_j, k_j}\circ\pi)(\underline x)\,\textrm{d}\widetilde\mu_{\textrm{d}}(\underline x), \end{align*} where we let |$F_{h,-k}:=\overline{F_{h,k}}$| for |$h\in{{\mathbb{Z}}}$| and |$k\in \{1, \ldots , \ell \}$|⁠. Since the algebra generated by the functions |$F_{h,1}, \ldots , F_{h,\ell }, \overline{F_{h,1}}, \ldots , \overline{F_{h,\ell }}$|⁠, |$h\in{{\mathbb{Z}}}$|⁠, is dense in |$C(X)$| with the uniform topology, the claim follows. Therefore, |$\pi \colon (X^{{\mathbb{Z}}},\widetilde \mu _d,S)\to (X,\mu ,T)$| is a factor map and the proof is complete. Our next task is to obtain structural results for the systems |$(X^{{\mathbb{Z}}},\widetilde \mu _d,S)$|⁠. This crucially depends on the following result from [14] that deals with the case where |$d=1$|⁠: Theorem 4.3. Let |$(X,\mu ,T)$| be a system. Then the system |$(X^{{\mathbb{Z}}},\widetilde \mu _1,S)$| (i) has no irrational spectrum; (ii) has ergodic components isomorphic to direct products of infinite-step nilsystems and Bernoulli systems. Remark. The infinite-step nilsystems and the Bernoulli systems are allowed to be trivial. The proof of Theorem 4.3 uses some deep ergodic machinery such as the main result from [23] (or [33]) regarding characteristic factors of Furstenberg averages, results about arithmetic progressions on nilmanifolds, and properties of partially strongly stationary systems. It also uses indirectly (via the use of variants of limit formulas obtained in [15]) some deep number theoretic input such as the Gowers uniformity of the |$W$|-tricked von Mangoldt function from [20–22]. Luckily, we do not have to modify the argument from [14] in order to get a similar result for the measures |$\widetilde \mu _d$|⁠; instead, we make use of the following simple observation, which allows us to use Theorem 4.3 as a “black box”: Lemma 4.4. Let |$(X,\mu ,T)$| be a system and |$\widetilde \mu _d$|⁠, |$d\in{{\mathbb{N}}}$|⁠, be the measures on |$X^{{\mathbb{Z}}}$| defined by (13). Then |$\widetilde \mu _d\leq \phi (d) \, \widetilde \mu _1$| for every |$d\in{{\mathbb{N}}}$|⁠. Proof. It suffices to show that for all |$m\in{{\mathbb{N}}}$| and all non-negative |$F_{-m},\ldots , F_m\in L^{\infty }(\mu )$| we have \begin{equation*} \int_{X^{{\mathbb{Z}}}}\prod_{j=-m}^m F_j(x_j)\,\textrm{d}\widetilde\mu_{\textrm{d}}(\underline x) \leq \phi(\textrm{d})\, \int_{X^{{\mathbb{Z}}}}\prod_{j=-m}^m F_j(x_j)\,\textrm{d}\widetilde\mu_1(\underline x). \end{equation*} This follows immediately from (13), the fact that the relative density |$d_{{\mathbb{P}}}({{\mathbb{P}}}_d)$| of the set |${{\mathbb{P}}}_d$| in the primes is |$1/\phi (d)$|⁠, and the estimate \begin{equation*} {{\mathbb{E}}}_{p\in{{\mathbb{P}}}_d}\, a(p)\leq (d_{{\mathbb{P}}}({{\mathbb{P}}}_d))^{-1}\, {{\mathbb{E}}}_{p\in{{\mathbb{P}}}} \, a(p), \end{equation*} which holds for all sequences |$a\colon{{\mathbb{P}}}\to{{\mathbb{R}}}^+$| assuming that the limits on the left- and right-hand sides exist. Combining Theorem 4.3 and Lemma 4.4 we deduce the following: Theorem 4.5. Let |$(X,\mu ,T)$| be a system. Then for every |$d\in{{\mathbb{N}}}$| the system |$(X^{{\mathbb{Z}}},\widetilde \mu _d,S)$| (i) has no irrational spectrum; (ii) has ergodic components isomorphic to direct products of infinite-step nilsystems and Bernoulli systems. Proof. By Lemma 4.4 we have |$\widetilde \mu _d\leq \phi (d)\, \widetilde \mu _1$|⁠; hence the measure |$\widetilde \mu _d$| is absolutely continuous with respect to the measure |$\widetilde \mu _1$|⁠. This implies that the spectrum of the system |$(X^{{\mathbb{Z}}},\widetilde \mu _d,S)$| is a subset of the spectrum of the system |$(X^{{\mathbb{Z}}},\widetilde \mu _1,S)$|⁠, so the former has no irrational spectrum since the same holds for the latter by Theorem 4.3. Furthermore, if |$\widetilde \mu _1=\int _\Omega \widetilde \mu _{1,\omega }\, \textrm{d}P(\omega )$| is the ergodic decomposition of the measure |$\widetilde \mu _1$|⁠, then the ergodic decomposition of the measure |$\widetilde \mu _d$| is |$\widetilde \mu _d=\int _\Omega \widetilde \mu _{1,\omega }\, \textrm{d}P_d(\omega )$| for some probability measure |$P_d$| that is absolutely continuous with respect to |$P$|⁠. This implies that property |$\text{(ii)}$| holds for the ergodic components of the measure |$\widetilde \mu _d$| since it holds for the ergodic components of the measure |$\widetilde \mu _1$| by Theorem 4.3. Theorem 1.5 now follows by combining Proposition 4.2 and Theorem 4.5. 4.2 An alternative proof of Theorem 1.5 for some special cases In some interesting special cases we can prove Theorem 1.5 (and hence Theorems 1.1–1.3) using an alternative approach that avoids the use of Theorem 3.5. We present the details below, let us emphasize though, that this alternative approach breaks down when |$f_j({{\mathbb{P}}})$| is an infinite subset of the unit circle for some |$j\in \{1,\ldots , \ell \}$|⁠, and we do not see how to avoid the use of Theorem 3.5 in order to cover such cases. 4.2.1 The case where |$f_1({{\mathbb{P}}}),\ldots , f_\ell ({{\mathbb{P}}})$| are finite subsets of |$\,{{\mathbb{T}}}$| Suppose first that the multiplicative functions |$f_1,\ldots , f_\ell$| are such that |$f_j({{\mathbb{P}}})$| is a finite subset of |${{\mathbb{T}}}$| for |$j=1,\ldots , \ell$|⁠. Let |$(X,\mu ,T)$| be a joint Furstenberg system associated with these multiplicative functions and a sequence of intervals |${\mathbf N}$|⁠. Then there exist |$c_1,\ldots , c_\ell \in \mathbb{U}$|⁠, a subset |$A$| of |${{\mathbb{P}}}$|⁠, and a sequence of intervals |${\mathbf{M}}=([M_k])_{k\in{{\mathbb{N}}}}$| with |$M_k\to \infty$|⁠, such that (i) |$f_j(p)=c_j$|⁠, |$j=1\ldots , \ell$|⁠, for all |$p\in A$|⁠; (ii) |$d^{\log }_{{\mathbf{M}},{{\mathbb{P}}}}(A):={\mathbb{E}}^{\log }_{{\mathbf{M}},p\in{{\mathbb{P}}}}\, {\textbf 1}_A(p)$| exists and is positive; (iii) the averages \begin{equation*} {\mathbb{E}}^{\log}_{{\mathbf{M}}, p\in A}\int_X\prod_{j=-m}^m T^{pj}F_j\,\textrm{d}\mu \end{equation*} exist for all |$m\in{{\mathbb{N}}}$| and |$F_{-m},\ldots , F_m\in L^{\infty }(\mu )$|⁠, where we used the notation \begin{equation*} {\mathbb{E}}^{\log}_{{\mathbf{M}}, p\in A}\, a(p):=\lim_{k\to\infty}{\mathbb{E}}^{\log}_{p\in A\cap [M_k]}\, a(p) \end{equation*} for |$a\colon{{\mathbb{P}}}\to{{\mathbb{C}}}$|⁠, if the limit exists. Using Theorem 3.1 and property |$\text{(i)}$|⁠, we get as in the proof of Proposition 4.2, using the factor map |$\pi \colon X^{{\mathbb{Z}}}\to X$| defined by |$(\pi (\underline x))(n):=(\overline{c_1} \, x_{n,1}(0),\ldots , \overline{c_\ell }\, x_{n, \ell }(0))$|⁠, that the system |$(X,\mu ,T)$| is a factor of the system |$(X^{{\mathbb{Z}}},\mu ^*,S)$| where the measure |$\mu ^*$| is defined as follows: for every |$m\in{{\mathbb{N}}}$| and all |$F_{-m},\ldots ,F_m\in L^{\infty }(\mu )$|⁠, we let \begin{equation} \int_{X^{{\mathbb{Z}}}}\prod_{j=-m}^m F_j(x_j)\,\textrm{d}\mu^*(\underline x):= {{\mathbb{E}}}^*_{{\mathbf{M}},p\in A}\int_X\prod_{j=-m}^m T^{pj}F_j\,\textrm{d}\mu, \end{equation} (14) where the limit on the right-hand side exists by property |$\text{(iii)}$|⁠. Since |$d^*_{{\mathbf{M}},{{\mathbb{P}}}}(A)>0$| (by property |$\text{(ii)}$|⁠), we get that for every sequence |$a\colon{{\mathbb{P}}}\to{{\mathbb{R}}}^+$| for which the limits below exist that \begin{equation*} {\mathbb{E}}^{\log}_{{\mathbf{M}},p\in A}a(p) \leq C\, \lim_{k\to\infty}{\mathbb{E}}^{\log}_{p\in{{\mathbb{P}}}\cap [M_k]}\, a(p)= C\, {{\mathbb{E}}}_{p\in{{\mathbb{P}}}}\, a(p) \end{equation*} where |$C:=(d^{\log }_{{\mathbf{M}},{{\mathbb{P}}}}(A))^{-1}$| (the last identity holds because |$\lim _{mkto\infty }{\mathbb{E}}^{\log }_{p\in{{\mathbb{P}}}\cap [M_k]}\, a(p)= {{\mathbb{E}}}_{p\in{{\mathbb{P}}}}\, a(p)$| since the last limit is assumed to exist). Using this estimate in the case where |$a(p):=\int _X\prod _{j=-m}^m T^{pj}F_j\,\textrm{d}\mu$|⁠, |$p\in{{\mathbb{P}}}$|⁠, is non-negative, we get that the measures |$\widetilde \mu _1$| and |$\mu ^*$|⁠, defined by (13) and (14), respectively, satisfy the estimate \begin{equation*} \int_{X^{{\mathbb{Z}}}}\prod_{j=-m}^m F_j(x_j)\,\textrm{d}\mu^*(\underline x)\leq C\, {{\mathbb{E}}}_{p\in{{\mathbb{P}}}}\int_X\prod_{j=-m}^m T^{pj}F_j\,\textrm{d}\mu= C \int_{X^{{\mathbb{Z}}}}\prod_{j=-m}^m F_j(x_j)\,\textrm{d}\widetilde\mu_1(\underline x) \end{equation*} for all |$m\in{{\mathbb{N}}}$| and all non-negative |$F_{-m},\ldots ,F_m\in L^{\infty }(\mu )$|⁠. Hence, |$\mu ^*\leq C\, \widetilde \mu _1$|⁠, and as in the proof of Theorem 4.5 we conclude that the system |$(X^{{\mathbb{Z}}},\mu ^*,S)$| satisfies properties |$\text{(i)}$| and |$\text{(ii)}$| of Theorem 4.5. Since |$(X,\mu ,T)$| is a factor of the system |$(X^{{\mathbb{Z}}},\mu ^*,S)$|⁠, it also satisfies these two properties. 4.2.2 The case of real valued multiplicative functions Let |$(X,\mu ,T)$| be a joint Furstenberg system associated with the multiplicative functions |$f_1,\ldots , f_\ell \colon{{\mathbb{Z}}}\to [-1,1]$| and a sequence of intervals |${\mathbf N}$|⁠. Suppose first that for some |$j\in \{1, \ldots , \ell \}$| we have |${{\mathbb{E}}}_{n\in{{\mathbb{N}}}}|f_j(n)|=0$|⁠, say for |$j=\ell$|⁠. Then all correlations involving the function |$f_\ell$| are trivial. As a consequence, the joint Furstenberg system associated with the functions |$f_1,\ldots , f_\ell$| and |${\mathbf N}$| is isomorphic (in the measure theoretic sense) to the joint Furstenberg system associated with the functions |$f_1,\ldots , f_{\ell -1}$| and |${\mathbf N}$|⁠. Hence, it suffices to prove Theorem 1.5 in the case where |${{\mathbb{E}}}_{n\in{{\mathbb{N}}}}|\,f_j(n)|\neq 0$| for |$j=1,\ldots , \ell$|⁠. As in the 1st part of the proof of Proposition 3.6 we get that |${\mathbb{E}}^{\log }_{p\in{{\mathbb{P}}}}(1-|\,f_j(p)|)=0$| for |$j=1,\ldots , \ell$|⁠. Hence, |$f_j\sim f_j^{\prime}$| for some |$f_j^{\prime}\colon{{\mathbb{P}}}\to \{-1,1\}$| for |$j=1,\ldots , \ell$|⁠. As a consequence, in the identity of Theorem 3.1 we can replace the weights |$c_{p,m}=\prod _{j=1}^m g_j(p)$| with the weights |$c^{\prime}_{p,m}:=\prod _{j=1}^m g^{\prime}_j(p)$|⁠, where for |$j=1,\ldots , m$|⁠, if |$g_j(p)$| is |$f_k(p)$| or |$\overline{f_k(p)}$| for some |$k\in \{1,\ldots , \ell \}$|⁠, then |$g_j^{\prime}(p)$| is |$f^{\prime}_k(p)$| or |$\overline{f^{\prime}_k(p)}$|⁠, respectively. Using this new identity, we deduce Theorem 1.5 as in the case treated above where |$f_j({{\mathbb{P}}})$| is finite for |$j=1,\ldots , \ell$|⁠. 5 Proof of Theorems 1.1–1.3 We will use the following disjointness result, proved in [14, Corollary 3.13]: Proposition 5.1. Let |$(X,\mu ,T)$| be a system with ergodic components isomorphic to direct products of infinite-step nilsystems and Bernoulli systems. Let |$(Y,\nu ,R)$| be a zero entropy system with at most countably many ergodic components. (i) If the two systems have disjoint irrational spectrum, then for every joining |$\sigma$| of the two systems and function |$F\in L^{\infty }(\mu )$| that is orthogonal to |${{\mathcal{K}}}_{\textrm{rat}}(T)$|⁠, we have \begin{equation*} \int_{X\times Y} F(x)\, G(y)\, \textrm{d}\sigma(x,y)=0 \end{equation*} for every |$G\in L^{\infty }(\nu )$|⁠. (ii) If the two systems have no common eigenvalue except |$1$|⁠, then for every joining |$\sigma$| of the two systems and function |$G\in L^{\infty }(\nu )$| that is orthogonal in |$L^2(\nu )$| to all |$R$|-invariant functions, we have \begin{equation*} \int_{X\times Y} F(x)\, G(y)\, \textrm{d}\sigma(x,y)=0 \end{equation*} for every |$F\in L^{\infty }(\mu )$|⁠. 5.1 Proof of Theorem 1.1 We follow the argument used in [14, Section 3.9]. Arguing by contradiction, suppose that under the assumptions of Theorem 1.1 we do not have uniform convergence to |$0$| of the related averages. Then there exist a strongly aperiodic multiplicative function |$f\colon{{\mathbb{N}}}\to \mathbb{U}$|⁠, which we extend to |${{\mathbb{Z}}}$| in an arbitrary way, a topological dynamical system |$(Y,R)$|⁠, positive integers |$N_k\to \infty$|⁠, points |$y_k\in Y$|⁠, |$k\in{{\mathbb{N}}}$|⁠, and a function |$g_0\in C(Y)$| such that the averages \begin{equation*} {\mathbb{E}}^{\log}_{n\in [N_k]}\, g_0(R^ny_k)\, f(n) \end{equation*} converge to a non-zero number as |$k\to \infty$|⁠. After passing to a subsequence that we denote again by |$([N_k])_{k\in{{\mathbb{N}}}}$|⁠, we can further assume that the averages |${\mathbb{E}}^{\log }_{n\in [N_k]}\delta _{R^ny_k}$| converge (as |$k\to \infty$|⁠) weak-star to an |$R$|-invariant probability measure |$\nu$| and the limit \begin{equation} \lim_{k\to\infty} {\mathbb{E}}^{\log}_{n\in[N_k]}\, g(R^ny_k)\prod_{j=1}^m\, f_j(n+h_j) \end{equation} (15) exists for all |$m \in{{\mathbb{N}}}$|⁠, |$h_1,\dots ,h_m\in{{\mathbb{Z}}}$|⁠, |$f_1,\ldots , f_m\in \{\,f,\overline{f}\}$|⁠, and |$g\in C(Y)$|⁠. Note that, by our assumptions, the system |$(Y,\nu ,R)$| has zero entropy and at most countably many ergodic components. Let |$X:=\mathbb{U}^{{\mathbb{Z}}}$|⁠, |$T\colon X\to X$| be the shift transformation, and |$x_0\in X$| be defined by \begin{equation*} x_0(n):=f(n), \quad n\in{{\mathbb{Z}}}. \end{equation*} Then the convergence (15) implies that the limit \begin{equation*} \lim_{k\to\infty}{\mathbb{E}}^{\log}_{n\in[N_k]}\,g(R^ny_k)\, \left(\,\prod_{j=1}^m G_{h_j}\right)(T^nx_0) \end{equation*} exists for all |$m \in{{\mathbb{N}}}$|⁠, |$h_1,\dots ,h_m\in{{\mathbb{Z}}}$|⁠, |$g\in C(Y)$|⁠, and |$G_h\in \{F_h,\overline{F_h}\}$|⁠, |$h\in{{\mathbb{Z}}}$|⁠, where |$F_h(x)=x(h)$|⁠, |$x\in X$|⁠, |$h\in{{\mathbb{Z}}}$|⁠. Since the algebra generated by the functions |$F_h, \overline{F_h}$|⁠, and |$h\in{{\mathbb{Z}}}$| is dense in |$C(X)$| with the uniform topology, we deduce that the sequence of measures \begin{equation*} {\mathbb{E}}^{\log}_{n\in [N_k]}\delta_{(T^nx_0,R^ny_k)}, \quad k\in{{\mathbb{N}}}, \end{equation*} converges weak-star to some probability measure |$\sigma$| on |$X\times Y$| that satisfies \begin{equation} \lim_{k\to\infty} {\mathbb{E}}^{\log}_{n\in[N_k]}\,g(R^ny_k)\, \prod_{j=1}^m f_j(n+h_j)=\int_{X\times Y} \prod_{j=1}^m G_{h_j}(x)\,g(y)\,\textrm{d}\sigma(x,y) \end{equation} (16) for all |$m \in{{\mathbb{N}}}$|⁠, |$h_1,\dots ,h_m\in{{\mathbb{Z}}}$|⁠, |$f_1,\ldots , f_m\in \{\,f,\overline{f}\}$|⁠, and |$g\in C(Y)$|⁠, where |$G_h$| is |$F_h$| or |$\overline{F_h}$| according to whether |$f_j$| is |$f$| or |$\overline{f}$|⁠. By construction, |$\sigma$| is invariant under |$T\times R$|⁠. The projection of |$\sigma$| on |$Y$| is the weak-star limit of the sequence of measures |${\mathbb{E}}^{\log }_{n\in [N_k]}\delta _{R^ny_k}$|⁠, |$k\in{{\mathbb{N}}}$|⁠, which is the measure |$\nu$|⁠, and thus the corresponding measure preserving system has zero entropy and at most countably many ergodic components. The projection of |$\sigma$| on |$X$| is the weak-star limit of the sequence of measures |${\mathbb{E}}^{\log }_{n\in [N_k]}\delta _{T^nx_0}$|⁠, |$k\in{{\mathbb{N}}}$|⁠. It is thus a |$T$|-invariant measure |$\mu$| that is the Furstenberg measure associated with |$f$| and |${\mathbf N}=([N_k])_{k\in{{\mathbb{N}}}}$| by Proposition 2.3. Hence, |$\sigma$| is a joining of the systems |$(X,\mu ,T)$| and |$(Y,\nu ,R)$|⁠. By the |$\ell =1$| case of Proposition 4.2 and its proof, there exists |$d\in{{\mathbb{N}}}$| such that |$(X,\mu ,T)$| is a factor of the system |$(X^{{\mathbb{Z}}},\widetilde \mu _d, S)$|⁠, with factor map |$\pi \colon X^{{\mathbb{Z}}}\to X$| given by \begin{equation*} (\pi(\underline x))(k):=x_k(0), \quad \underline x=(x_n)_{n\in{{\mathbb{Z}}}}\in X^{{\mathbb{Z}}}, \ k\in{{\mathbb{Z}}}. \end{equation*} We define the joining |$\widetilde \sigma$| of the systems |$(X^{{\mathbb{Z}}}, \widetilde \mu _d, S)$| and |$(Y,\nu ,R)$| by \begin{equation*} \int_{X^{{\mathbb{Z}}}\times Y} H(\underline x)\cdot g(y) \, \textrm{d}\widetilde\sigma(\underline x,y):= \int_{X\times Y} {{\mathbb{E}}}_{\widetilde\mu_d}(H\mid X)(x) \cdot g(y) \, \textrm{d}\sigma(x,y) \end{equation*} for every |$H\in L^{\infty }(\widetilde \mu _d)$| and |$g\in L^{\infty }(\nu )$|⁠, where |${{\mathbb{E}}}_{\widetilde \mu _d}(H\mid X)$| in |$L^1(\nu )$| is determined by the property |$\int _{A}{{\mathbb{E}}}_{\widetilde \mu _d}(H\mid X)\, \textrm{d}\mu =\int _{\pi ^{-1}(A)} H\, \textrm{d}\widetilde \mu _d$| for every |$A\in{{\mathcal{X}}}$|⁠. We show now that the systems |$(X^{{\mathbb{Z}}},\widetilde \mu _d, S)$| and |$(Y,\nu ,R)$| verify the assumptions of part |$\text{(i)}$| of Proposition 5.1. By Theorem 4.5, the system |$(X^{{\mathbb{Z}}},\widetilde \mu _d,S)$| has no irrational spectrum and its ergodic components are isomorphic to direct products of infinite-step nilsystems and Bernoulli systems. We show next that the function |$F^{\prime}_0:=F_0\circ \pi$| is orthogonal to the rational Kronecker factor of the system |$(X^{{\mathbb{Z}}},\widetilde \mu _d, S)$|⁠, in fact, we establish the stronger property that |$F^{\prime}_0$| is orthogonal to the Kronecker factor of this system. By a well-known consequence of the spectral theorem for unitary operators, this is equivalent to establishing that \begin{equation} {{\mathbb{E}}}_{n\in{{\mathbb{N}}}} \left|\int_{X^{{\mathbb{Z}}}} F^{\prime}_0\cdot S^n \overline{ F^{\prime}_0}\, \textrm{d}\widetilde \mu_d\right|=0. \end{equation} (17) By the Definition 4.1 of the measure |$\widetilde \mu _d$| and since for |$n\in{{\mathbb{N}}}$| we have |$F^{\prime}_0(\underline x) \, \overline{ F^{\prime}_0}(S^n\underline x)=F_0(x_0) \, \overline{F_0}(x_n)$|⁠, we get for every |$n\in{{\mathbb{N}}}$| that \begin{equation*} \int_{X^{{\mathbb{Z}}}} F^{\prime}_0\cdot S^n \overline{F^{\prime}_0}\, \textrm{d}\widetilde \mu_d={{\mathbb{E}}}_{p\in{{\mathbb{P}}}_d}\int_X F_0\cdot T^{pn} \overline{F_0}\, \textrm{d} \mu. \end{equation*} By (4), for every |$h\in{{\mathbb{N}}}$| we have \begin{equation*} \int_X F_0\cdot T^h\overline{F_0}\, \textrm{d} \mu= {\mathbb{E}}^{\log}_{n\in{{\mathbf N}}} \, f(n)\, \overline{f(n+h)}=0 \end{equation*} where the vanishing of the average follows from Theorem 2.10 and our assumption that |$f$| is strongly aperiodic. Combining the above identities we get (17). By part |$\text{(i)}$| of Proposition 5.1, we have \begin{equation*} 0=\int_{X^{{\mathbb{Z}}}\times Y} F^{\prime}_0(\underline x) \cdot g_0(y)\,\textrm{d}\widetilde\sigma(\underline x,y)= \int_{X\times Y} F_0(x)\cdot g_0(y)\,\textrm{d}\sigma(x,y)=\lim_{k\to\infty} {\mathbb{E}}^{\log}_{n\in[N_k]}\,g_0(R^ny_k)\, f(n) \end{equation*} where the last identity follows by (16). This contradicts our initial assumption that |$\lim _{k\to \infty }{\mathbb{E}}^{\log }_{n\in [N_k]}\, g_0(R^ny_k)\, f(n)\neq 0$| and completes the proof. 5.2 Proof of Theorem 1.3 We follow the argument used in [14, Section 3.11]. Arguing by contradiction, suppose that the conclusion of Theorem 1.3 fails. Then there exist |$\ell \in{{\mathbb{N}}}$|⁠, multiplicative functions |$f_1,\ldots , f_{\ell }\colon{{\mathbb{N}}}\to \mathbb{U}$|⁠, which we extend to |${{\mathbb{Z}}}$| in an arbitrary way, a topological dynamical system |$(Y,R)$|⁠, a point |$y_0\in Y$| that is generic for a measure |$\nu$| such that the system |$(Y,\nu ,R)$| has zero entropy and at most countably many ergodic components all of which are totally ergodic, and a function |$g_0\in C(Y)$| orthogonal in |$L^2(\nu )$| to all |$R$|-invariant functions, such that for some |$h_{0,1},\dots ,h_{0,\ell }\in{{\mathbb{Z}}}$| the identity (1) fails, namely, the averages \begin{equation} {\mathbb{E}}^{\log}_{n\in [N]} \,g_0(R^ny_0)\, \prod_{j=1}^{\ell}f_j(n+h_{0,j}) \end{equation} (18) do not converge to |$0$| as |$N\to \infty$|⁠. Let |$X:=(\mathbb{U}^{\ell })^{{\mathbb{Z}}}$|⁠, |$T\colon X\to X$| be the shift transformation, and |$x_0\in X$| be defined by \begin{equation*} x_0(n):=(f_1(n),\ldots, f_{\ell}(n)), \quad n\in{{\mathbb{Z}}}. \end{equation*} If |$x=(x_1(n), \ldots , x_\ell (n))_{n\in{{\mathbb{Z}}}}\in X$|⁠, where |$x_j(n)\in \mathbb{U}$| for |$j=1,\ldots , \ell$|⁠, |$n\in{{\mathbb{Z}}}$|⁠, we let \begin{equation*} F_{h, j}(x):=x_j(h),\quad \ h\in{{\mathbb{Z}}},\ j\in \{1,\ldots, \ell\}. \end{equation*} As in the proof of Theorem 1.1 in the previous subsection, we define a sequence of intervals |${\mathbf N}=(N_k)_{k\in{{\mathbb{N}}}}$|⁠, with |$N_k\to \infty$|⁠, such that the averages (18), taken along |${\mathbf N}$|⁠, converge to some non-zero number, and a measure |$\sigma$| on |$X\times Y$| that is the weak-star limit of the sequence of measures \begin{equation*} {\mathbb{E}}^{\log}_{n\in [N_k]}\delta_{(T^nx_0,R^ny_0)}, \quad k\in{{\mathbb{N}}}. \end{equation*} In particular, the identity \begin{equation} {\mathbb{E}}^{\log}_{n\in{\mathbf N}}\,g(R^ny_0)\, \prod_{j=1}^\ell f_j(n+h_j)=\int_{X\times Y} \prod_{j=1}^\ell F_{h_j, j}(x)\,g(y)\,\textrm{d}\sigma(x,y) \end{equation} (19) holds for all |$h_1,\ldots, h_\ell\in \mathbb{Z}$|⁠, and |$g\in C(Y)$|⁠. By construction, |$\sigma$| is invariant under |$T\times R$|⁠. By assumption and the definition of genericity, the projection of |$\sigma$| on |$Y$| is the measure |$\nu$|⁠, and thus the system |$(Y,\nu ,R)$| has zero entropy, at most countably many ergodic components, and no rational eigenvalue except |$1$|⁠. Moreover, the projection of |$\sigma$| on |$X$| is the weak-star limit of the sequence of measures |${\mathbb{E}}^{\log }_{n\in [N_k]}\delta _{T^nx_0}$|⁠, |$k\in{{\mathbb{N}}}$|⁠. It is thus a |$T$|-invariant measure |$\mu$|⁠, which is the joint Furstenberg measure associated with the multiplicative functions |$f_1,\ldots , f_\ell$| and |${\mathbf N}$| by Proposition 2.3. Hence, by Proposition 4.2, for some |$d\in{{\mathbb{N}}}$| the system |$(X,\mu ,T)$| is a factor of the system |$(X^{{\mathbb{Z}}},\widetilde \mu _d, S)$|⁠. By Theorem 1.5, the system |$(X^{{\mathbb{Z}}},\widetilde \mu _d,S)$| has no irrational spectrum and its ergodic components are isomorphic to direct products of infinite-step nilsystems and Bernoulli systems. From the previous discussion it follows that the function |$g_0$| and the systems |$(X^{{\mathbb{Z}}},\widetilde \mu _d, S)$| and |$(Y,\nu ,R)$| satisfy the hypothesis of the second part of Proposition 5.1. Hence, for every joining |$\widetilde{\sigma }$| of these systems and |$\tilde{f}\in L^{\infty }(\widetilde \mu _d)$|⁠, we have |$\int \tilde{f}(\underline x)\, g_0(y)\, \textrm{d}\widetilde{\sigma }(\underline x,y)=0$|⁠. Since |$\sigma$| is a joining of the systems |$(X,\mu ,T)$| and |$(Y,\nu ,R)$|⁠, and the system |$(X,\mu ,T)$| is a factor of |$(X^{{\mathbb{Z}}},\widetilde \mu _d, S)$|⁠, the measure |$\sigma$| can be lifted to a joining |$\widetilde \sigma$| of |$(X^{{\mathbb{Z}}},\widetilde \mu _d, S)$| and |$(Y,\nu ,R)$|⁠. It follows that for every |$f\in L^{\infty }(\mu )$| we have |$\int f(x)\, g_0(y)\, \textrm{d}\sigma (x,y)=0$|⁠. We deduce that \begin{equation*} {\mathbb{E}}^{\log}_{n\in{\mathbf N}} \,g_0(R^ny_0)\, \prod_{j=1}^{\ell}f_j(n+h_{0,j}) =\int_{X\times Y} \prod_{j=1}^{\ell}F_{h_{0,\,j},j}(x)\cdot g_0(y)\,\textrm{d}\sigma(x,y)=0. \end{equation*} This contradicts our assumption that |${\mathbb{E}}^{\log }_{n\in{\mathbf N}} \,g_0(R^ny_0)\, \prod _{j=1}^{\ell } f_j(n+h_{0,j})\neq 0$| and completes the proof of Theorem 1.3. 5.3 Block complexity and proof of Theorem 1.2 We start with some definitions. Let |$A$| be a non-empty finite set. The set |$A$| is endowed with the discrete topology and |$A^{{\mathbb{Z}}}$| with the product topology and with the shift |$T$|⁠. For |$n\in{{\mathbb{N}}}$|⁠, a word of length |$n$| is a sequence |$u=u_1\dots u_n$| of |$n$| letters where |$u_1,\ldots , u_n\in A$|⁠, and we write |$[u]=\{x\in A^{{\mathbb{Z}}}\colon x_1\dots x_n=u_1\dots u_n\}$|⁠. A subshift is a closed non-empty |$T$|-invariant subset |$X$| of |$A^{{\mathbb{Z}}}$|⁠. It is transitive if it has at least one dense orbit under |$T$|⁠. Let |$(X,T)$| be a transitive subshift that is equal to the closed orbit of some point |$\omega \in A^{{\mathbb{Z}}}$|⁠. For every |$n\in{{\mathbb{N}}}$| we let |$L_n(X)$| denote the set of words |$u$| of length |$n$| such that |$[u]\cap X\neq \emptyset$|⁠. Then |$L_n(X)$| is also the set of words of length |$n$| that occur (as consecutive values) in |$\omega$|⁠. Note that the set |$L(X):=\bigcup _{n\in{{\mathbb{N}}}}L_n(X)$| determines |$X$|⁠. The block complexity of |$X$| or of |$\omega$| is defined by |$p_X(n)=|L_n(X)|$| for |$n\in{{\mathbb{N}}}$|⁠. We say that the subshift |$(X,T)$| (or the sequence |$\omega$|⁠) has linear block growth if |$\liminf _{n\to \infty } p_X(n)/n<\infty$|⁠. We are going to use the following consequence of a result from [4] (or [11, Theorem 7.3.7]) that was obtained in [14, Section 7.1]: Proposition 5.2. Let |$(X,T)$| be a transitive subshift with linear block growth. Then |$(X,T)$| admits only finitely many ergodic invariant measures. This result was proved in [3] under the stronger hypothesis that |$(X,T)$| is minimal. Proof of Theorem 1.2. We argue as in [14, Section 7.2] where a similar result was proved for the Liouville function. Let |$A$| be the range of |$f$|⁠, which we have assumed to be a finite subset of |$\mathbb{U}$|⁠. Suppose that |$f$| has linear block growth. We extend |$f$| to a two-sided sequence, which we denote by |$y_0\in A^{{\mathbb{Z}}}$|⁠, by letting |$y_0(n):=1$| for non-positive |$n\in{{\mathbb{Z}}}$|⁠; then the extended sequence still has linear block growth. Let |$Y$| be the closed orbit of |$y_0$| in |$A^{{\mathbb{Z}}}$| and let |$R$| be the shift on |$Y$|⁠. Then |$(Y,R)$| is a transitive subshift, and since it has linear block growth it has zero topological entropy. Moreover, by Proposition 5.2 this system admits only finitely many ergodic invariant measures. Note that for every |$n\in{{\mathbb{N}}}$| we have |$f(n)=F_0(R^ny_0)$|⁠, where |$F_0\colon A^{{\mathbb{Z}}}\to \mathbb{U}$| is the map defined by |$F_0(y):= y(0)$| for |$y=(y(n))_{n\in{{\mathbb{Z}}}}\in Y$|⁠. By Theorem 1.1 we get \begin{equation*} 0= {\mathbb{E}}^{\log}_{n\in{{\mathbb{N}}}} \, \overline{F_0}(R^ny_0)\, f(n) ={\mathbb{E}}^{\log}_{n\in{{\mathbb{N}}}}\, |\,f(n)|^2\neq 0, \end{equation*} where we used our assumption that |$f$| does not converge to zero in logarithmic density. We have thus established a contradiction and the proof is complete. Acknowledgments We would like to thank M. Lemańczyk for the observation that the convergence in Theorem 1.1 is uniform. We also thank M. Lemańczyk and T. de la Rue for pointing out a correction in Theorem 1.4 and Part (ii) of Proposition 5.1. References [1] V. Bergelson , J. Kułaga-Przymus , M. Lemańczyk , F. 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Buryak,, Alexandr;Dubrovin,, Boris;Guéré,, Jérémy;Rossi,, Paolo

doi: 10.1093/imrn/rnz029pmid: N/A

Abstract In this paper we study various aspects of the double ramification (DR) hierarchy, introduced by the 1st author, and its quantization. We extend the notion of tau-symmetry to quantum integrable hierarchies and prove that the quantum DR hierarchy enjoys this property. We determine explicitly the genus |$1$| quantum correction and, as an application, compute completely the quantization of the |$3$|- and |$4$|-KdV hierarchies (the DR hierarchies for Witten’s |$3$|- and |$4$|-spin theories). We then focus on the recursion relation satisfied by the DR Hamiltonian densities and, abstracting from its geometric origin, we use it to characterize and construct a new family of quantum and classical integrable systems that we call of DR type, as they satisfy all of the main properties of the DR hierarchy. In the 2nd part, we obtain new insight towards the Miura equivalence conjecture between the DR and Dubrovin-Zhang (DZ) hierarchies, via a geometric interpretation of the correlators forming the DR tau-function. We then show that the candidate Miura transformation between the DR and DZ hierarchies (which we uniquely identified in our previous paper) indeed turns the DZ Poisson structure into the standard form. Eventually, we focus on integrable hierarchies associated with rank-|$1$| cohomological field theories and their deformations, and we prove the DR/DZ equivalence conjecture up to genus |$5$| in this context. 1 Introduction The Dubrovin–Zhang (DZ) hierarchy [11] is an integrable system of Hamiltonian PDEs: partial differential equations associated with any given semisimple cohomological field theory (CohFT). As an important property, it is tau-symmetric and we can then define its partition function as the tau-function of its topological solution. The DZ hierarchy plays a central role in generalizing to any semisimple CohFT the notion, underlying the Witten–Kontsevich theorem [19, 29], which states that the partition function of the CohFT should correspond to the topological tau-function of some integrable Hamiltonian tau-symmetric hierarchy of evolutionary PDEs. The double ramification (DR) hierarchy has been introduced in [1] by the 1st author and is another integrable system of Hamiltonian PDEs, associated with any given CohFT. It does not require any semisimplicity condition and it is also defined for partial CohFTs, satisfying weaker axioms, see [2]. At the heart of its construction lies the DR cycle |$\textrm{DR}_g(a_1,\ldots ,a_n)$|⁠, which is the push-forward to the moduli space of stable curves |${\overline{{\mathcal{M}}}}_{g,n}$| of the virtual fundamental cycle of the moduli space of rubber stable maps to |$\mathbb{P}^1$| relative to |$0$| and |$\infty $|⁠, with ramification profile (orders of poles and zeros) given by |$(a_1,\ldots ,a_n)\in \mathbb{Z}^n$|⁠. We prove in [2] that the DR hierarchy is also tau-symmetric and we define its partition function as the tau-function of its string solution. The DR/DZ equivalence conjecture [2] predicts the existence (and unicity) of a normal Miura transformation under which the partition function of a given CohFT equals the associated DR partition function. As a consequence, we recover in the semisimple case the original conjecture from [1] that the DR and DZ hierarchies are Miura equivalent. One application of the DR/DZ equivalence conjecture, when proved true, is to give a quantization of any DZ hierarchy. Indeed, the DR hierarchy has a natural quantization, constructed in [7], and recalled in Section 2.4. In this paper, we prove that the quantum DR hierarchy is also tau-symmetric and we define a quantum tau-function. In the limit when the quantum parameter |$\hbar $| tends to zero, we recover results from [2]. We also study the 1st quantum correction in genus |$1$| and, as an application, we completely determine the quantum DR hierarchies associated with the Witten’s |$3$|- and |$4$|-spin theories. One of the most striking property of the quantum DR hierarchy is that it can be recovered recursively from the knowledge of one Hamiltonian, usually denoted |$\overline{G}_{1,1}$|⁠, via the recursion equations of Theorem 2.2, proved in [7]. Conversely, any Hamiltonian |$\overline{H}$| compatible with these recursion equations in the sense of Theorem 5.1 produces a unique quantum integrable tau-symmetric hierarchy. An integrable hierarchy obtained in this way is said to be of DR type. As an example, we study the dispersionless quantum deformations of DR type of the Riemann hierarchy and suggest they are in one-to-one correspondence with the DR hierarchies associated with CohFTs of rank |$1$|⁠. Starting from Section 6, we go back to the classical DR hierarchy and to the DR/DZ equivalence conjecture. In Theorem 6.1 we give a very explicit and geometric formula for the coefficients of the DR partition function, called the DR correlators. This formula is used in Section 7 towards the DR/DZ equivalence conjecture. More precisely, we prove in Theorem 7.2 that the candidate Miura transformation between the two, which we uniquely identified in [2], indeed transforms the Hamiltonian operator |$K^{\textrm{DZ}}$| of the DZ hierarchy to the standard operator |$\eta \partial _x$| used in the DR hierarchy, giving a new evidence for the conjecture. To conclude, we give various results about the DR and DZ hierarchies associated with CohFTs of rank |$1$|⁠. In particular, we show that the DR hierarchy is a standard deformation of the Riemann hierarchy in the sense of [10] and we prove the existence of a normal Miura transformation that reduces the DZ hierarchy to its unique standard form, proving one of the conjectures from [10] about tau-symmetric deformations of the Riemann hierarchy. Lastly, we prove that the DR/DZ equivalence conjecture holds for rank-|$1$| CohFTs at the approximation up to genus |$5$|⁠. 2 DR Hierarchy In this section we recall the main definitions and results from [1, 6, 7]. The classical DR hierarchy is a system of commuting Hamiltonians on an infinite dimensional phase space that can be heuristically thought of as the loop space of a fixed vector space. The entry datum for this construction is a CohFT in the sense of Kontsevich and Manin [20] or, more in general, a partial CohFT in the sense of [21] (the definition of a partial CohFT is the same as the one for a CohFT apart from the loop axiom, which is not required in the first). For actual CohFTs (not just partial), in [7] a quantization was constructed for the classical DR hierarchy, dubbed quantum double ramification (qDR) hierarchy. 2.1 Formal loop space Let |$V$| be an |$N$|-dimensional vector space and |$\eta $| a symmetric bilinear form on it. The loop space of |$V$| will be defined somewhat formally by describing its ring of functions. Following [11] (see also [26]), let us consider formal variables |$u^\alpha _i$|⁠, |$\alpha =1,\ldots ,N$|⁠, |$i=0,1,\ldots $|⁠, associated with a basis |$e_1,\ldots ,e_N$| of |$V$|⁠. Always just at a heuristic level, the variable |$u^\alpha :=u^\alpha _0$| can be thought of as the component |$u^\alpha (x)$| along |$e_\alpha $| of a formal loop |$u\colon S^1\to V$|⁠, where |$x$| is the coordinate on |$S^1$|⁠, and the variables |$u^\alpha _{x}:=u^\alpha _1, u^\alpha _{xx}:=u^\alpha _2,\ldots $| as its |$x$|-derivatives. We then define the ring |${\mathcal{A}}_N$| of differential polynomials as the ring of polynomials |$f(u^*;u^*_x,u^*_{xx},\ldots )$| in the variables |$u^\alpha _i, i>0$|⁠, with coefficients in the ring of formal power series in the variables |$u^\alpha =u^\alpha _0$| (when it does not give rise to confusion, we will use the symbol * to indicate any value, in the appropriate range, of the sub or superscript). We can differentiate a differential polynomial with respect to |$x$| by applying the operator |$\partial _x:= \sum _{i\geq 0} u^\alpha _{i+1} \frac{\partial }{\partial u^\alpha _i}$| (in general, we use the convention of sum over repeated Greek indices, but not over repeated Latin indices). Finally, we consider the quotient |$\Lambda _N$| of the ring of differential polynomials first by constants and then by the image of |$\partial _x$|⁠, and we call its elements local functionals. A local functional, which is the equivalence class of a differential polynomial |$f=f(u^*;u^*_x,u^*_{xx},\ldots )$|⁠, will be denoted by |$\overline{f}=\int f \text{d}x$|⁠. Let us introduce a grading |$\deg u^\alpha _i = i$| and define |${\mathcal{A}}^{[k]}_N$| and |$\Lambda ^{[k]}_N$| as the subspaces of degree |$k$| of |${\mathcal{A}}_N$| and of |$\Lambda _N,$| respectively. Differential polynomials and local functionals can also be described using another set of formal variables, corresponding heuristically to the Fourier components |$p^\alpha _k$|⁠, |$k\in{\mathbb{Z}}$|⁠, of the functions |$u^\alpha =u^\alpha (x)$|⁠. Let us, hence, define a change of variables \begin{equation} u^\alpha_j = \sum_{k\in{\mathbb{Z}}} (i k)^j p^\alpha_k e^{i k x}, \end{equation} (2.1) which allows us to express a differential polynomial |$f(u;u_x,u_{xx},\ldots )$| as a formal Fourier series in |$x$| where the coefficient of |$e^{i k x}$| is a power series in the variables |$p^\alpha _j$| (where the sum of the subscripts in each monomial in |$p^\alpha _j$| equals |$k$|⁠). Moreover, the local functional |$\overline{f}$| corresponds to the constant term of the Fourier series of |$f$|⁠. Let us describe a natural class of Poisson brackets on the space of local functionals. Given an |$N\times N$| matrix |$K=(K^{\mu \nu })$| of differential operators of the form |$K^{\mu \nu } = \sum _{j\geq 0} K^{\mu \nu }_j \partial _x^j$|⁠, where the coefficients |$K^{\mu \nu }_j$| are differential polynomials and the sum is finite, we define \begin{equation*} \{\overline{f},\overline{g}\}_{K}:=\int\left(\frac{\delta \overline{f}}{\delta u^\mu}K^{\mu \nu}\frac{\delta \overline{g}}{\delta u^\nu}\right)\mathrm{\,d}x, \end{equation*} where we have used the variational derivative |$\frac{\delta \overline{f}}{\delta u^\mu }:=\sum _{i\geq 0} (-\partial _x)^i \frac{\partial f}{\partial u^\mu _i}$|⁠. Imposing that such bracket satisfies the anti-symmetry and the Jacobi identity will translate, of course, into conditions for the coefficients |$K^{\mu \nu }_j$|⁠. An operator that satisfies such conditions will be called Hamiltonian. A standard example of a Hamiltonian operator is given by |$\eta \partial _x$|⁠. The corresponding Poisson bracket |$\{\cdot ,\cdot \}_{\eta{\partial }_x}$| will sometimes be denoted just by |$\{\cdot ,\cdot \}$| when no confusion arises. Such Poisson bracket also has a nice expression in terms of the variables |$p^\alpha _k$|⁠, \begin{equation} \big\{p^\alpha_k, p^\beta_j\big\}_{\eta \partial_x} = i k \eta^{\alpha \beta} \delta_{k+j,0}. \end{equation} (2.2) Finally, we will need to consider extensions |$\widehat{{\mathcal{A}}}_N$| and |$\widehat \Lambda _N$| of the spaces of differential polynomials and local functionals. Introduce a new variable |$\varepsilon $| with |$\deg \varepsilon = -1$|⁠. Then |$\widehat{{\mathcal{A}}}^{[k]}_N$| and |$\widehat \Lambda ^{[k]}_N$| are defined, respectively, as the subspaces of degree |$k$| of |$\widehat{{\mathcal{A}}}_N:={\mathcal{A}}_N[[\varepsilon ]]$| and of |$\widehat \Lambda _N:=\Lambda _N[[\varepsilon ]]$|⁠. Their elements will still be called differential polynomials and local functionals. We can also define Poisson brackets as above, starting from a Hamiltonian operator |$K=(K^{\mu \nu })$|⁠, |$K^{\mu \nu } = \sum _{i,j\geq 0} (K^{[i]}_j)^{\mu \nu } \varepsilon ^i \partial _x^j$|⁠, where |$(K^{[i]}_j)^{\mu \nu }\in{\mathcal{A}}_N$| and |$\deg (K^{[i]}_j)^{\mu \nu }=i-j+1$|⁠. The corresponding Poisson bracket will then have degree |$1$|⁠. In the sequel only such Hamiltonian operators will be considered. A Hamiltonian hierarchy of PDEs is a family of systems of the form \begin{equation} \frac{\partial u^\alpha}{\partial \tau_i} = K^{\alpha\mu} \frac{\delta\overline{h}_i}{\delta u^\mu}, \ \alpha=1,\ldots,N,\ i=1,2,\ldots, \end{equation} (2.3) where |$\overline h_i\in \widehat \Lambda ^{[0]}_N$| are local functionals with the compatibility condition |$\{\overline h_i,\overline h_j\}_K=0$|⁠, for |$i,j\geq 1$|⁠. The local functionals |$\overline h_i$| are called the Hamiltonians of the systems (2.3). 2.2 Classical DR hierarchy Let |$c_{g,n}\colon V^{\otimes n} \to H^{\textrm{even}}({\overline{{\mathcal{M}}}}_{g,n},{\mathbb{C}})$| be the system of linear maps defining a (possibly partial, in the sense of [21]) CohFT, |$V$| its underlying |$N$|-dimensional vector space, |$\eta $| its metric tensor, and |$e_1\in V$| the unit vector. Let |$\psi _i$| be the 1st Chern class of the line bundle over |${\overline{{\mathcal{M}}}}_{g,n}$| formed by the cotangent lines at the |$i$|-th marked point. Denote by |$\mathbb E$| the rank |$g$| Hodge vector bundle over |${\overline{{\mathcal{M}}}}_{g,n}$| whose fibers are the spaces of holomorphic one-forms. Let |$\lambda _j:=c_j(\mathbb E)\in H^{2j}({\overline{{\mathcal{M}}}}_{g,n},{{\mathbb{Q}}})$|⁠. The Hamiltonians of the DR hierarchy are defined as follows: \begin{equation} \overline g_{\alpha,d}:=\sum_{\substack{g\ge 0\\n\ge 2}}\frac{(-\varepsilon^2)^g}{n!}\sum_{\substack{a_1,\ldots,a_n\in{\mathbb{Z}}\\\sum a_i=0}}\left(\int_{{\overline{{\mathcal{M}}}}_{g,n+1}}\textrm{DR}_g(0,a_1,\ldots,a_n)\lambda_g\psi_1^d c_{g,n+1}\big(e_\alpha\otimes \otimes_{i=1}^n e_{\alpha_i}\big)\right)\prod_{i=1}^n p^{\alpha_i}_{a_i}, \end{equation} (2.4) for |$\alpha =1,\ldots ,N$| and |$d=0,1,2,\ldots $|⁠. Here |$\textrm{DR}_g(a_1,\ldots ,a_n) \in H^{2g}({\overline{{\mathcal{M}}}}_{g,n},{{\mathbb{Q}}})$| is the DR cycle. If not all of |$a_i$|’s are equal to zero, then the restriction |$\left .\textrm{DR}_g(a_1,\ldots ,a_n)\right |_{{{\mathcal{M}}}_{g,n}}$| can be defined as the Poincaré dual to the locus of pointed smooth curves |$[C,p_1,\ldots ,p_n]$| satisfying |$\mathcal O_C\left (\sum _{i=1}^n a_ip_i\right )\cong \mathcal O_C$|⁠, and we refer the reader, for example, to [8] for the definition of the DR cycle on the whole moduli space |${\overline{{\mathcal{M}}}}_{g,n}$|⁠. We will often consider the Poincaré dual to the DR cycle |$\textrm{DR}_g(a_1,\ldots ,a_n)$|⁠. It is an element of |$H_{2(2g-3+n)}({\overline{{\mathcal{M}}}}_{g,n},{{\mathbb{Q}}})$| and, abusing our notations a little bit, it will also be denoted by |$\textrm{DR}_g(a_1,\ldots ,a_n)$|⁠. In particular, the integral in (2.4) will often be written in the following way: \begin{equation} \int_{\textrm{DR}_g(0,a_1,\ldots,a_n)}\lambda_g\psi_1^{\mathrm{\,d}} c_{g,n+1}\big(e_\alpha\otimes\otimes_{i=1}^n e_{\alpha_i}\big). \end{equation} (2.5) The expression on the right-hand side of (2.4) can be uniquely written as a local functional from |$\widehat \Lambda _N^{[0]}$| using the change of variables (2.1). Concretely it can be done in the following way. The integral (2.5) is a polynomial in |$a_1,\ldots ,a_n$| homogeneous of degree |$2g$|⁠. It follows from Hain’s formula [17], the results of [22], and the fact that |$\lambda _g$| vanishes on |${\overline{{\mathcal{M}}}}_{g,n}\setminus{{\mathcal{M}}}_{g,n}^{\textrm{ct}}$|⁠, where |${{\mathcal{M}}}_{g,n}^{\textrm{ct}}$| is the moduli space of stable curves of compact type [13, 23]. Thus, the integral (2.5) can be written as a polynomial \begin{equation*} P_{\alpha,d,g;\alpha_1,\ldots,\alpha_n}(a_1,\ldots,a_n)=\sum_{\substack{b_1,\ldots,b_n\ge 0\\b_1+\ldots+b_n=2g}}P_{\alpha,d,g;\alpha_1,\ldots,\alpha_n}^{b_1,\ldots,b_n}a_1^{b_1}\ldots a_n^{b_n}. \end{equation*} Then we have \begin{equation*} \overline g_{\alpha,d}=\int\sum_{\substack{g\ge 0\\n\ge 2}}\frac{\varepsilon^{2g}}{n!}\sum_{\substack{b_1,\ldots,b_n\ge 0\\b_1+\ldots+b_n=2g}}P_{\alpha,d,g;\alpha_1,\ldots,\alpha_n}^{b_1,\ldots,b_n} u^{\alpha_1}_{b_1}\ldots u^{\alpha_n}_{b_n}\mathrm{\,d}x. \end{equation*} Note that the integral (2.5) is defined only when |$a_1+\ldots +a_n=0$|⁠. Therefore, the polynomial |$P_{\alpha ,d,g;\alpha _1,\ldots ,\alpha _n}$| is actually not unique. However, the resulting local functional |$\overline g_{\alpha ,d}\in \widehat \Lambda _N^{[0]}$| doesn’t depend on this ambiguity (see [1]). In fact, in [6], a special choice of differential polynomial densities |$g_{\alpha ,d} \in \widehat{{\mathcal{A}}}^{[0]}_N$| for |$\overline g_{\alpha ,d} = \int g_{\alpha ,d} \ \mathrm{\,d}x$| is selected. They are defined in terms of |$p$|-variables as \begin{equation*} g_{\alpha,d}:=\sum_{\substack{g\ge 0,\,n\ge 1\\2g-1+n>0}}\frac{(-\varepsilon^2)^g}{n!}\sum_{\substack{a_0,\ldots,a_n\in{\mathbb{Z}}\\\sum a_i=0}}\left(\int_{\textrm{DR}_g(a_0,a_1,\ldots,a_n)}\lambda_g\psi_1^{\mathrm{\,d}} c_{g,n+1}\big(e_\alpha\otimes \otimes_{i=1}^n e_{\alpha_i}\big)\right)\prod_{i=1}^n p^{\alpha_i}_{a_i} e^{-i a_0 x}, \end{equation*} and converted univocally to differential polynomials using again the change of variables (2.1). The fact that the local functionals |$\overline g_{\alpha ,d}$| mutually commute with respect to the standard bracket |$\eta{\partial }_x$| was proved in [1] for CohFTs and in [2] for partial CohFTs. The system of local functionals |$\overline g_{\alpha ,d}$|⁠, for |$\alpha =1,\ldots ,N$|⁠, |$d=0,1,2,\ldots $|⁠, and the corresponding system of Hamiltonian PDEs with respect to the standard Poisson bracket |$\{\cdot ,\cdot \}_{\eta \partial _x}$|⁠, \begin{equation*} \frac{{\partial} u^\alpha}{{\partial} t^\beta_q}=\eta^{\alpha\mu}{\partial}_x\frac{\delta\overline g_{\beta,q}}{\delta u^\mu}, \end{equation*} is called the DR hierarchy. 2.3 Quantum Hamiltonian systems We will need, first, to extend the space of differential polynomials to allow for dependence on the quantization formal parameter |$\hbar $|⁠. A quantum differential polynomial|$f=f(u^*,u^*_x,u^*_{xx,}\ldots ;\varepsilon ,\hbar )$| is a formal power series in |$\hbar $| and |$\epsilon $| whose coefficients are polynomials in |$u^\alpha _k$|⁠, for |$k>0$|⁠, and power series in |$u^\alpha _0$|⁠, where |$\alpha =1,\ldots ,N$|⁠. The quantization parameter has degree |$\deg \hbar =-2$| and all other formal variables retain the same degree as in the classical case. The space of quantum differential polynomials will be denoted by |$\widehat{{\mathcal{A}}}^\hbar _N$|⁠. The space of quantum local functionals|$\widehat \Lambda ^\hbar _N$| is given, as in the classical case, by taking the quotient of |$\widehat{{\mathcal{A}}}_N^\hbar $| with respect to formal power series in |$\varepsilon $| and |$\hbar $| and the image of the |$\partial _x$|-operator. As in the classical case, the change of variables \begin{equation*} u^\alpha_j=\sum_{k\in{\mathbb{Z}}}(ik)^jp^\alpha_k e^{ikx}, \end{equation*} allows one to express any quantum differential polynomial |$f=f(u^*_*;\varepsilon ,\hbar )$| as a formal Fourier series in |$x$| with coefficients that are (power series in |$\varepsilon $| with coefficients) in the Weyl algebra |${\mathbb{C}}[p^1_{k>0},\ldots ,p^N_{k>0}][[p^1_{k\leq 0},\ldots ,p^N_{k\leq 0},\hbar ]]$| endowed with the “normal ordering” |$\star $|-product \begin{equation*} f \star g =f \left( e^{\sum_{k>0} i \hbar k \eta^{\alpha \beta} \overleftarrow{\frac{\partial} {\partial p^\alpha_{k}}} \overrightarrow{\frac{\partial} {\partial p^\beta_{-k}}}}\right) g \end{equation*} and the commutator |$[f,g]:=f\star g - g \star f$|⁠. These structures can then be translated to the language of differential polynomials and local functionals. In [7] it was proved that, for any two differential polynomials |$f(x)=f(u^*,u^*_x,u^*_{xx},\ldots ;\varepsilon ,\hbar )$| and |$g(y)=g(u^*,u^*_y,u^*_{yy},\ldots ;\varepsilon ,\hbar )$|⁠, we have \begin{equation*} f(x)\star g(y) =\sum_{\substack{n\geq 0\\ r_1,\ldots,r_n\geq 0\\ s_1,\ldots, s_n\geq 0}} \frac{\hbar^{n}}{n!} \frac{\partial^n f}{\partial u^{\alpha_1}_{s_1}\ldots \partial u^{\alpha_n}_{s_n}}(x)\left( \prod_{k=1}^n (-1)^{r_k} \eta^{\alpha_k\beta_k} \delta_+^{(r_k + s_k +1)}(x-y) \right) \frac{\partial^n g}{\partial u^{\beta_1}_{r_1}\ldots \partial u^{\beta_n}_{r_n}}(y), \end{equation*} where |$\delta _+^{(s)}(x-y):= \sum _{k\geq 0} (ik)^s e^{i k (x-y)}$|⁠, |$s\geq 0$|⁠, is the positive frequency part of the |$s$|-th derivative of the Dirac delta distribution |$\delta (x-y)= \sum _{k\in{\mathbb{Z}}} e^{i k (x-y)}$| and \begin{equation} \begin{split} [f(x),g(y)]=\sum_{\substack{n\geq 1\\ r_1,\ldots,r_n\geq 0\\ s_1,\ldots,s_n\geq 0}}& \frac{(-i)^{n-1} \hbar^{n}}{n!} \frac{\partial^n f}{\partial u^{\alpha_1}_{s_1}\ldots \partial u^{\alpha_n}_{s_n}}(x) (-1)^{\sum_{k=1}^n r_k} \left( \prod_{k=1}^n \eta^{\alpha_k \beta_k}\right) \times\\ & \times \sum_{j=1}^{2n-1+\sum_{k=1}^n (s_k+r_k)} C_j^{s_1+r_1+1,\ldots,s_n+r_n+1}\delta^{(j)}(x-y)\frac{\partial^n g}{\partial u^{\beta_1}_{r_1}\ldots \partial u^{\beta_n}_{r_n}}(y), \end{split} \end{equation} (2.6) where \begin{equation} C_j^{a_1,\ldots,a_n}= \begin{cases} (-1)^{\frac{n-1+\sum a_i-j}{2}}\widetilde C_j^{a_1,\ldots,a_n},&\textrm{if}\ j=n-1+\sum_{i=1}^n a_i\ (\textrm{mod}\ 2),\\ 0,& \textrm{otherwise} \end{cases} \end{equation} (2.7) and \begin{equation} \prod_{i=1}^k\textrm{Li}_{-d_i}(z)=\sum_{j=1}^{k-1+\sum d_i}\widetilde C^{d_1,\ldots,d_k}_j\textrm{Li}_{-j}(z), \qquad \textrm{Li}_{-d}(z):=\sum_{k\ge 0}k^d z^k. \end{equation} (2.8) In particular, for |$f\in \widehat{{\mathcal{A}}}_N^\hbar $| and |$\overline g \in \widehat \Lambda _N^\hbar $|⁠, we get \begin{equation} \begin{split} [f,\overline g]=\sum_{\substack{n\geq 1\\ r_1,\ldots,r_n\geq 0\\ s_1,\ldots,s_n\geq 0}} \frac{(-i)^{n-1} \hbar^{n}}{n!} &\frac{\partial^n f}{\partial u^{\alpha_1}_{s_1}\ldots \partial u^{\alpha_n}_{s_n}} (-1)^{\sum_{k=1}^n r_k} \left( \prod_{k=1}^n \eta^{\alpha_k \beta_k}\right) \times\\ & \times \sum_{j=1}^{2n-1+\sum_{k=1}^n (s_k+r_k)} C_j^{s_1+r_1+1,\ldots,s_n+r_n+1} \partial_x^j \frac{\partial^n g}{\partial u^{\beta_1}_{r_1}\ldots \partial u^{\beta_n}_{r_n}}. \end{split} \end{equation} (2.9) If |$f$| and |$\overline g$| are homogeneous, |$[f,\overline g]$| is a nonhomogeneous element of |$\widehat{{\mathcal{A}}}_N^\hbar $| of top degree equal to |$\deg f + \deg \overline g - 1$|⁠. Taking the classical limit of this expression one obtains |$\left (\frac{1}{\hbar }[\overline{f},\overline g]\right )|_{\hbar =0}=\{\overline{f}|_{\hbar =0},\overline g|_{\hbar =0}\}$|⁠, that is, the standard hydrodynamic Poisson bracket on the classical limit of the local functionals. Notice that, given |$\overline g \in \widehat \Lambda _N^\hbar $|⁠, the morphism |$[\cdot ,\overline g]:\widehat{{\mathcal{A}}}_N^\hbar \to \widehat{{\mathcal{A}}}_N^\hbar $| is not a derivation of the commutative ring |$\widehat{{\mathcal{A}}}_N^\hbar $| (while it is if we consider the noncommutative |$\star $|-product instead). This means that, while it makes sense to describe the simultaneous evolution along different time parameters |$\tau _i$| (in the Heisenberg picture, to use the physical language) of a quantum differential polynomial |$f \in \widehat{{\mathcal{A}}}_N^\hbar $| by a system of the form \begin{equation} \frac{\partial f}{\partial \tau_i} = \frac{1}{\hbar}[f,\overline h_i], \ \alpha=1,\ldots,N,\ i=1,2,\ldots, \end{equation} (2.10) where |$\overline h_i\in \widehat \Lambda ^{[\leq 0]}_N$| are quantum local functionals with the compatibility condition |$[\overline h_i,\overline h_j]=0$|⁠, for |$i,j\geq 1$|⁠, one should refrain from interpreting it as the evolution induced by composition with |$\frac{{\partial } u^\alpha }{{\partial } \tau _i}=\frac{1}{\hbar } [u^\alpha ,\overline h_i]$|⁠, as the corresponding chain rule does not hold: |$\frac{{\partial } f}{\partial \tau _i} \neq \sum _{k\geq 0}\frac{{\partial } f}{{\partial } u^\alpha _k}{\partial }_x^k \left (\frac{{\partial } u^\alpha }{{\partial } \tau _i}\right )$|⁠. This corresponds to the familiar concept that in quantum mechanics there are no trajectories in the phase space along which observables evolve. A formal solution to the system (2.10) can be written in the form of an element in |$\widehat{{\mathcal{A}}}_N^\hbar [[\tau _*]]$|⁠, \begin{equation} f^{\tau_*}\big(u^*_*;\varepsilon,\hbar\big):= \exp\left(\sum_{i\geq 1} \frac{\tau_i}{\hbar}[\cdot,\overline h_i]\right) f\big(u^*_*;\varepsilon,\hbar\big) = \left(\prod_{i\geq 1} \exp \left( \frac{\tau_i}{\hbar}[\cdot,\overline h_i]\right) \right) f\big(u^*_*;\varepsilon,\hbar\big), \end{equation} (2.11) where \begin{equation} \exp\left(\frac{\tau_i}{\hbar}[\cdot,\overline h_i]\right):= \sum_{k\geq 0} \frac{\tau_i^k}{\hbar^k k!}[[\ldots [\cdot,\overline h_i],\ldots,\overline h_i],\overline h_i] \end{equation} (2.12) and |$f\in \widehat{{\mathcal{A}}}_N^\hbar $| in the right-hand side of (2.11) is interpreted as the initial datum. Lifting the quantum commutator |$[\cdot ,\cdot ]$| to |$\widehat{{\mathcal{A}}}^\hbar _N[[\tau _*]]$|⁠, it is easy to check that |$f^{\tau _*}$| satisfies equation (2.10). We do insist that |$f^{\tau _*}(u^*_*;\varepsilon ,\hbar ) \neq f((u^*_*)^{\tau _*},\varepsilon ,\hbar )$|⁠. 2.4 qDR hierarchy Given a CohFT |$c_{g,n}\colon V^{\otimes n} \to H^{\textrm{even}}({\overline{{\mathcal{M}}}}_{g,n};{\mathbb{C}})$|⁠, we define the Hamiltonian densities of the qDR hierarchy as the following generating series: \begin{equation} \begin{split} G_{\alpha,d}:=&\sum_{\substack{g\ge 0,n\ge 0\\2g-1+n>0}}\frac{(i \hbar)^g}{n!}\times\\ &\times\sum_{\substack{a_1,\ldots,a_n\in{\mathbb{Z}}\\ \alpha_1,\ldots,\alpha_n}}\left(\int_{\textrm{DR}_g\left(-\sum a_i,a_1,\ldots,a_n\right)}\Lambda\left(\frac{-\varepsilon^2}{i \hbar}\right) \psi_1^{\mathrm{\,d}} c_{g,n+1}\big(e_\alpha\otimes\otimes_{i=1}^n e_{\alpha_i}\big)\right)p^{\alpha_1}_{a_1}\ldots p^{\alpha_n}_{a_n}e^{ix\sum a_i}, \end{split} \end{equation} (2.13) for |$\alpha =1,\ldots ,N$| and |$d=0,1,2,\ldots $|⁠. Here |$\Lambda \left (\frac{-\varepsilon ^2}{i \hbar }\right ):=\left (1+ \left ( \frac{-\varepsilon ^2}{i \hbar }\right ) \lambda _1+\ldots + \left (\frac{-\epsilon ^2}{i\hbar }\right )^g \lambda _g \right )$|⁠, with |$\lambda _i$| the |$i$|-th Chern class of the Hodge bundle. Notice also that, since |$\Lambda (s)$| is itself a CohFT depending on the formal parameter |$s$|⁠, we could absorb such factor into |$c_{g,n+1}\big (e_\alpha \otimes \otimes _{i=1}^n e_{\alpha _i}\big )$| obtaining densities for a CohFT analogue of the Symplectic Field Theory Hamiltonians of [12, 15]. As for the “classical” Hamiltonian densities |$g_{\alpha ,p}=G_{\alpha ,p}|_{\hbar =0}$|⁠, we would like to rewrite the above expression in terms of formal jet variables |$u^\alpha _s = \sum _{k\in{\mathbb{Z}}} (ik)^s p^\alpha _k e^{ikx}$|⁠, |$\alpha =1,\ldots ,N$|⁠, |$s=0,1,2,\ldots $|⁠. Since the DR cycle |$\textrm{DR}_{g}(a_1,\ldots ,a_n)$| is a nonhomogeneous polynomial of degree at most |$2g$| in the variables |$a_1,\ldots ,a_n$| (as apparent from Pixton’s formula [18]), we actually obtain that each |$G_{\alpha ,p}$| can be uniquely written as a quantum differential polynomial of degree |$\deg G_{\alpha ,p} \leq 0$| and such that |$\deg G_{\alpha ,p}\big |_{\hbar =0} = 0$|⁠, that is, |$G_{\alpha ,p} \in (\widehat{{\mathcal{A}}}^\hbar _N)^{[\leq 0]}$| and |$G_{\alpha ,p}\big |_{\hbar =0} \in \widehat{{\mathcal{A}}}_N^{[0]}$|⁠. This means that the number of |$x$|-derivatives that can appear in the coefficient of |$\varepsilon ^k\hbar ^j$| is at most |$k+2j$|⁠, and exactly |$k$| in the coefficient of |$\varepsilon ^k \hbar ^0$|⁠. We finally add manually |$N$| extra densities |$G_{\alpha ,-1}:=\eta _{\alpha \mu } u^\mu $|⁠. Recall that by |${\overline G}_{\alpha ,p}= \int G_{\alpha ,p} \mathrm{\,d}x$| we denote the coefficient of |$e^{i0x}$| in |$G_{\alpha ,p}$| considered also up to a constant, for all |$\alpha =1,\ldots ,N$|⁠, |$p=-1,0,1,\ldots $|⁠. The fact that the local functionals |${\overline G}_{\alpha ,d}$| mutually commute with respect to the above commutator, |$[{\overline G}_{\alpha ,p},{\overline G}_{\beta ,q}]=0$|⁠, was proved in [7] together with the fact that |${\overline G}_{1,0}=\int \left (\frac{1}{2} \eta _{\mu \nu }u^\mu u^\nu \right )\mathrm{\,d}x$|⁠, so that, for any |$f\in \widehat{{\mathcal{A}}}_N^\hbar $|⁠, |${\partial }_{t^1_0} f= {\partial }_x f$|⁠. 2.5 Recursion for the qDR Hamiltonian densities We recall some of the properties of the DR hierarchies, in particular a recursion equation, proven in [6] for the classical case and in [7] for the quantum case, allowing to recover all the Hamiltonian densities |$G_{\alpha ,p}$|⁠, |$\alpha =1,\ldots ,N$|⁠, |$p\geq 0$|⁠, recursively from |$G_{\alpha ,-1}=\eta _{\alpha \mu } u^\mu $| starting from the knowledge of the functional |${\overline G}_{1,1}$| only. Let us define the following two-point potential for intersection numbers with the DR cycle: \begin{equation*} \begin{split} G_{\alpha,p;\beta,q}(x,y):=\sum_{\substack{g\ge 0,n\ge 0\\2g+n>0}}\frac{(i\hbar)^g}{n!} \sum_{\substack{a_0,\ldots,a_{n+1}\in{\mathbb{Z}}\\\sum a_i=0\\ \alpha_1,\ldots,\alpha_n}}&\left(\int_{\textrm{DR}_g\left(a_0,a_1,\ldots,a_n,a_{n+1}\right)}\right. \Lambda\left(\frac{-\varepsilon^2}{i \hbar}\right) \psi_0^p \psi_{n+1}^q \times \\ &\left.\vphantom{\int_{\textrm{DR}_g\left(a_0,a_1,\ldots,a_n,a_{n+1}\right)}}\times c_{g,n+2}\left(e_\alpha\otimes\otimes_{i=1}^n e_{\alpha_i}\otimes e_\beta\right)\right) p^{\alpha_1}_{a_1}\ldots p^{\alpha_n}_{a_n}e^{-i a_0 x-i a_{n+1} y}, \end{split} \end{equation*} for |$\alpha ,\beta =1,\ldots ,N$| and |$p,q=0,1,2,\ldots $|⁠. In [7] the following result was proven. Lemma 2.1 ([7]). For all |$\alpha ,\beta =1,\ldots ,N$| and |$p,q=0,1,2,\ldots $|⁠, we have \begin{equation} \partial_x G_{\alpha,p+1;\beta,q}(x,y) - \partial_y G_{\alpha,p;\beta,q+1}(x,y) =\frac{1}{\hbar} \left[ G_{\alpha,p}(x), G_{\beta,q}(y)\right]. \end{equation} (2.14) From this lemma the following theorem can be deduced. Theorem 2.2 ([7]). For all |$\alpha =1,\ldots ,N$| and |$p=-1,0,1,\ldots $|⁠, we have \begin{equation} \partial_x (D-1) G_{\alpha,p+1} =\frac{1}{\hbar} \left[ G_{\alpha,p}, {\overline G}_{1,1} \right], \end{equation} (2.15) \begin{equation} \partial_x \frac{\partial G_{\alpha,p+1}}{\partial u^\beta} =\frac{1}{\hbar} \left[G_{\alpha,p}, {\overline G}_{\beta,0} \right], \end{equation} (2.16) where |$D:=\varepsilon \frac{\partial }{\partial \varepsilon } + 2\hbar \frac{\partial }{\partial \hbar } + \sum _{s\ge 0} u^\alpha _s\frac{\partial }{\partial u^\alpha _s}$|⁠. Notice how equation (2.15) can be used to recover recursively (up to a constant) |$G_{\alpha ,p}$|⁠, |$\alpha =1,\ldots ,N$|⁠, |$p\geq 0$| from |$G_{\alpha ,-1} = \eta _{\alpha ,\mu } u^\mu $| and of course the knowledge of |${\overline G}_{1,1}$|⁠. From equation (2.16) we can instead deduce the string equation (always up to a constant, actually) \begin{equation} \frac{{\partial} G_{\alpha,p+1}}{{\partial} u^1} = G_{\alpha,p}. \end{equation} (2.17) Since we can prove such string equation separately from geometric considerations [7], the constant terms of the densities |$G_{\alpha ,p}$|⁠, which are left undetermined by the recursion (2.15), can then be chosen uniquely as those that verify equation (2.17). 3 qDR Hierarchy in Genus |$1$| In [2] we computed the genus |$1$| term of the classical DR hierarchy for any CohFT in terms of genus |$0$| data. In this section we compute the quantum correction, always in genus |$1$| and in terms of genus |$0$| data plus the genus |$1$||$\textsf{G}$|-function of the CohFT. As an application we compute the full qDR hierarchies for Witten’s |$3$|- and |$4$|-spin classes. 3.1 Genus-|$1$| quantum correction Let |$c_{g,n}:V^{\otimes n} \to H^*({\overline{{\mathcal{M}}}}_{g,n},{\mathbb{C}})$| be a CohFT with |$V$| an |$N$|-dimensional vector space endowed with a nondegenerate metric |$\eta $| and basis |$e_1,\ldots ,e_N$|⁠, where |$e_1$| is the unit of the CohFT. Let |${\overline G}_{\alpha ,d}$|⁠, |$1\leq \alpha \leq N$|⁠, |$d\geq -1$| be the corresponding qDR Hamiltonians and let |${\overline G} =(D-2)^{-1} {\overline G}_{1,1}$|⁠, with |$D$| as in Theorem 2.2. Let |$\overline g_{\alpha ,d}$| and |$\overline g$| be their classical counterparts and |$g_{\alpha ,d}^{[0]}=g_{\alpha ,d}^{[0]}(u^1,\ldots ,u^N)$| the genus |$0$| Hamiltonian densities. Theorem 3.1. Let |$F=F(u^1,\ldots ,u^N)$| be the Frobenius potential (genus |$0$| potential with no descendants) and |$\textsf{G}=\textsf{G}(u^1,\ldots ,u^N)$| the |$\textsf{G}$|-function (genus |$1$| potential with no descendants) of the CohFT. Let |$c_{\alpha \beta } = \frac{{\partial }^2 F}{{\partial } u^\alpha{\partial } u^\beta }$|⁠, |$c_{\alpha \beta \gamma } = \frac{{\partial } ^3 F}{{\partial } u^\alpha{\partial } u^\beta{\partial } u^\gamma }$|⁠, |$c_{\alpha \beta \gamma \delta } = \frac{{\partial } ^4 F}{{\partial } u^\alpha{\partial } u^\beta{\partial } u^\gamma{\partial } u^\delta }$|⁠, and indices be raised and lowered by the metric |$\eta $|⁠. Then we have \begin{equation} {\overline G} = \overline g + i \hbar \int \left[\left(\frac{1}{48} c_{\alpha \beta \mu}^\mu + \frac{1}{2} c_{\alpha \beta}^\mu \frac{{\partial} \textsf{G}}{{\partial} u^\mu}\right) u^\alpha_x u^\beta_x - \frac{1}{24}c^\mu_\mu \right] \mathrm{\,d}x+O(\hbar^2)+O(\hbar \varepsilon^2), \end{equation} (3.1) \begin{equation} \begin{split} {\overline G}_{\alpha,d} = \overline g_{\alpha,d} + i \hbar \int &\left[\left( \frac{1}{48} \frac{{\partial}^4 g_{\alpha,d}^{[0]}}{{\partial} u^\gamma{\partial} u^\beta{\partial} u^\mu{\partial} u^\nu} \eta^{\mu \nu} + \frac{1}{2} \frac{{\partial}^3 g_{\alpha,d}^{[0]}}{{\partial} u^\gamma{\partial} u^\beta{\partial} u^\mu} \eta^{\mu \nu} \frac{{\partial} \textsf{G}}{{\partial} u^\nu} \right. \right.\\ &\left.\left. + \frac{1}{2} c_{\gamma \beta}^\mu \frac{{\partial}}{{\partial} u^\mu} \left( \frac{1}{24}\frac{{\partial}^2 g_{\alpha,d-1}^{[0]}}{{\partial} u^\epsilon{\partial} u^\delta} \eta^{\epsilon \delta}+ \frac{{\partial} g_{\alpha,d-1}^{[0]}}{{\partial} u^\epsilon} \eta^{\epsilon \delta} \frac{{\partial} \textsf{G}}{{\partial} u^\delta}\right)\right) u^\gamma_x u^\beta_x\right. \\ & \left. -\frac{1}{24} \frac{{\partial}^2 g_{\alpha,d}^{[0]}}{{\partial} u^\mu{\partial} u^\nu} \eta^{\mu \nu}\right] \mathrm{\,d}x + O(\hbar^2)+O(\hbar\varepsilon^2). \end{split} \end{equation} (3.2) Proof. Let us prove equation (3.1). Recall that \begin{equation*} {\overline G}:=\sum_{\substack{g\ge 0,n\ge 1\\2g-2+n>0}}\frac{(i \hbar)^g}{n!}\sum_{\substack{a_1,\ldots,a_n\in{\mathbb{Z}}\\ \alpha_1,\ldots,\alpha_n}}\left(\int_{\textrm{DR}_g\left(a_1,\ldots,a_n\right)}\Lambda\left(\frac{-\varepsilon^2}{i \hbar}\right) c_{g,n}\left(\otimes_{i=1}^n e_{\alpha_i}\right)\right)p^{\alpha_1}_{a_1}\ldots p^{\alpha_n}_{a_n}, \end{equation*} so the relevant intersection numbers for the genus |$1$| quantum corrections are \begin{equation*}\int_{\textrm{DR}_1(a_1,\ldots,a_n)} c_{1,n}\left(\otimes_{i=1}^n e_{\alpha_i}\right).\end{equation*} To compute them we use the following formulae for the DR cycle (see [17]), psi, and lambda classes on |${\overline{{\mathcal{M}}}}_{1,n}$|⁠: \begin{equation*}\textrm{DR}_1(a_1,\ldots,a_n) = \sum_{i=1}^n \frac{\psi_i}{2} a_i^2 - \frac{1}{2} \left(\sum_{\substack{J\subset \{1,\ldots,n\}\\ |J|\geq 2}} \left(\sum_{j \in J}a_j\right)^2 \delta^J_0\right) -\lambda_1,\end{equation*} \begin{equation*}\psi_i = \frac{1}{24} \delta_{\textrm{irr}}+\sum_{\substack{J\subset \{1,\ldots,n\}\\ |J|\geq 2, i\in J}} \delta^J_0, \qquad \lambda_1 = \frac{1}{24} \delta_{\textrm{irr}},\end{equation*} where |$\delta _{\textrm{irr}}$| and |$\delta ^J_0$| denote the divisor in |${\overline{{\mathcal{M}}}}_{1,n}$| of singular curves with a non-separating node and of curves with a separating node whose rational component carries exactly the marked points labeled by |$J$| (the points labeled by the complement |$J^c$| belonging to the elliptic component), respectively. In particular we get \begin{equation*} \begin{split} &\int_{\textrm{DR}_1(a_1,\ldots,a_n)} c_{1,n}\big(\otimes_{k=1}^n e_{\alpha_k}\big)=\sum_{i=1}^n \frac{a_i^2}{48} \int_{{\overline{{\mathcal{M}}}}_{0,n+2}} c_{0,n+2}\big(\otimes_{k=1}^n e_{\alpha_k}\otimes e_\mu \otimes e_\nu\big)\eta^{\mu\nu}\\ & + \frac{1}{2} \sum_{\substack{J\subset \{1,\ldots,n\}\\ |J|\geq 2, i\in J}} a_i^2 \int_{{\overline{{\mathcal{M}}}}_{0,|J|+1}} c_{0,|J|+1}\big(\otimes_{k\in J} e_{\alpha_{k}}\otimes e_\mu\big) \eta^{\mu \nu} \int_{{\overline{{\mathcal{M}}}}_{1,n-|J|+1}}c_{1,n-|J|+1}(\otimes_{k\in J^c} e_{\alpha_k} \otimes e_\nu) \\ &-\frac{1}{2} \sum_{\substack{J\subset \{1,\ldots,n\}\\ |J|\geq 2}} \big(\sum_{j\in J} a_j\big)^2 \int_{{\overline{{\mathcal{M}}}}_{0,|J|+1}} c_{0,|J|+1}\big(\otimes_{k\in J} e_{\alpha_{k}}\otimes e_\mu\big) \eta^{\mu \nu} \int_{{\overline{{\mathcal{M}}}}_{1,n-|J|+1}}c_{1,n-|J|+1}(\otimes_{k\in J^c} e_{\alpha_k} \otimes e_\nu)\\ & -\frac{1}{24}\int_{{\overline{{\mathcal{M}}}}_{0,n+2}} c_{0,n+2}\big(\otimes_{k=1}^n e_{\alpha_k}\otimes e_\mu \otimes e_\nu\big)\eta^{\mu\nu}. \end{split} \end{equation*} In terms of generating functions, this becomes \begin{equation*} \begin{split} {\overline G} = \overline g + i\hbar \int \left[-\frac{1}{48} u^\alpha_{xx} c_{\alpha\mu}^\mu -\frac{1}{2}u^\alpha_{xx} c_\alpha^\mu \frac{{\partial} \textsf{G}}{{\partial} u^\mu} + \frac{1}{2} \left({\partial}_x^2 \frac{{\partial} F}{{\partial} u^\mu}\right) \eta^{\mu \nu} \frac{{\partial} \textsf{G}}{{\partial} u^\nu} - \frac{1}{24} c_\mu^\mu \right] \mathrm{\,d}x + O(\hbar^2), \end{split} \end{equation*} which can be brought to the form of equation (3.1) by integrating by parts. The proof of equation (3.2) is completely analogous, the only difference being the insertion of a psi class to the power |$d$| at an extra marked point, which makes it necessary to use genus |$1$| topological recursion relations (see [29]) \begin{equation*}\frac{{\partial} F_1(t^*_*)}{{\partial} t^\alpha_d} = \frac{1}{24} \frac{{\partial}^3 F_0(t^*_*)}{{\partial} t^\alpha_{d-1} {\partial} t^\epsilon_0 {\partial} t^\delta_0} \eta^{\epsilon \delta}+ \frac{{\partial}^2 F_0(t^*_*)}{{\partial} t^\alpha_{d-1} {\partial} t^\epsilon_0} \eta^{\epsilon \delta} \frac{{\partial} F_1(t^*_*)}{{\partial} t^\delta_0},\end{equation*} where |$F_g(t^*_*)$| is the genus |$g$| potential of the CohFT and whose right-hand side, when restricted to |$t^\alpha _0=u^\alpha $| and |$t^*_p = 0$| for |$p>0$|⁠, becomes the term \begin{equation*} \frac{1}{24}\frac{{\partial}^2 g_{\alpha,d-1}^{[0]}}{{\partial} u^\epsilon{\partial} u^\delta} \eta^{\epsilon \delta}+ \frac{{\partial} g_{\alpha,d-1}^{[0]}}{{\partial} u^\epsilon} \eta^{\epsilon \delta} \frac{{\partial} \textsf{G}}{{\partial} u^\delta}\end{equation*} in equation (3.2). 3.2 |$3$|- and |$4$|-spin qDR hierarchies As an application of the genus-|$1$| computation of the previous section we compute the qDR hierarchy of Witten’s |$r$|-spin class, for |$r=3,4$|⁠. In light of the results of [2, 3], which establish that the DR hierarchy in these cases coincides with the DZ hierarchy once we pass to the normal coordinates |$\widetilde{u}^\alpha = \eta ^{\alpha \mu }\frac{\delta \overline g_{\mu ,0}}{\delta u^1}$| (which, for |$r=4,$| also changes the form of the Hamiltonian operator, see [2]), and the fact that the DZ hierarchies for the |$3$|- and |$4$|-spin theories correspond in turn to the |$3$|- and |$4$|-KdV Gelfand–Dickey hierarchies [9, 11], we obtain this way a quantization for such two well-known integrable systems. Recall from [25, 30] that, fixing |$r\geq 2$| and an |$(r-1)$|-dimensional vector space |$V$| with a basis |$e_1,\ldots ,e_{r-1}$|⁠, Witten’s |$r$|-spin CohFT |$W_g(e_{a_1+1},\ldots ,e_{a_n+1})=W_g(a_1,\ldots ,a_n) \in H^*({\overline{{\mathcal{M}}}}_{g,n};{{\mathbb{Q}}})$| is a class of degree |$\deg W_g(a_1,\ldots ,a_n)=\frac{(r-2)(g-1)+\sum _{i=1}^n a_i}{r}$| if |$a_i\in \{0,\ldots ,r-2\}$| are such that this degree is a nonnegative integer, and vanishes otherwise. By [24], this CohFT is completely determined, thanks to generic semisimplicity, by the initial conditions |$W_0(a_1,a_2,a_3)=1$| if |$a_1+a_2+a_3=r-2$| (and zero otherwise) and |$W_0(1,1,r-2,r-2)=\frac{1}{r}[\textrm{pt}]$| for |$r\geq 3$| (while it vanishes for |$r=2$|⁠). In particular, the metric |$\eta $| takes the form |$\eta _{\alpha \beta }=\delta _{\alpha +\beta ,r}$|⁠. Theorem 3.2. For |$r=3,4$|⁠, the qDR hierarchies for Witten’s |$r$|-spin classes are uniquely determined by \begin{align*} {\overline G}_{1,1}^{3\text{-spin}} =&\int\left[\left(\frac{1}{2} \big(u^1\big)^2 u^2+\frac{\big(u^2\big)^4}{36}\right)\!+\!\left(-\frac{1}{12} \big(u_1^1\big){}^2\!-\!\frac{1}{24} u^2 \big(u_1^2\big){}^2\right) \varepsilon ^2+\!\frac{1}{432} \big(u_2^2\big){}^2 \varepsilon ^4 \!-\! \frac{i\hbar}{12} u^1\right] \mathrm{d}x,\\{\overline G}_{1,1}^{4\text{-spin}} =&\int \left[\left(\frac{ u^1 \big(u^2\big)^2}{2}+\frac{\left(u^1\right)^2 u^3}{2} +\frac{\left(u^2\right)^2 \left(u^3\right)^2}{8} +\frac{\left(u^3\right)^5}{320}\right)\right. \\ &\left.+\left(-\frac{\left(u_1^1\right){}^2}{8} -\frac{u^3 \left(u_1^2\right){}^2}{16} -\frac{u^3 u_1^1 u_1^3}{32} +\frac{3}{64} \big(u^2\big)^2 u_2^3+\frac{1}{192} \big(u^3\big)^3 u_2^3\right) \varepsilon ^2 \right. \\ & \left. +\left(\frac{1}{160} \big(u_2^2\big){}^2+\frac{3}{640} u_2^1 u_2^3+\frac{5 \left(u^3\right)^2 u_4^3}{4096}\right) \varepsilon ^4-\frac{\left(u_3^3\right){}^2 \varepsilon ^6}{8192}+ \right. \\ &\left.\vphantom{\left(\frac{ u^1 \left(u^2\right)^2}{2}+\frac{\left(u^1\right)^2 u^3}{2} +\frac{\left(u^2\right)^2 \left(u^3\right)^2}{8} +\frac{\left(u^3\right)^5}{320}\right)} \!+\left( \frac{1}{96} \big(u^3_1\big)^2 - \frac{1}{96} \big(u^3\big)^2 -\frac{1}{8} u^1 \right)i\hbar - \frac{1}{1280} u^3 i\hbar \varepsilon^2 \right]\mathrm{\,d}x. \end{align*} Proof. The classical parts of the above formulae are copied from [6]. Moreover, from dimension counting, we obtain that |${\overline G}^{r\text{-spin}}_{1,1}$| is a homogeneous local functional of degree |$2r+2$| with respect to the grading |$|u^{a+1}_k|=r-a$|⁠, |$|\varepsilon |=1$|⁠, |$|\hbar |=r+2$|⁠. This means that the quantum correction in |${\overline G}_{1,1}^{3\text{-spin}}$| is entirely in genus |$1$| and hence determined by Theorem 3.1 (recall that the |$\textsf{G}$|-function for the |$r$|-spin theory vanishes identically, see e.g. [27]). The quantum correction in |${\overline G}_{1,1}^{4\text{-spin}}$|⁠, instead, has a part in genus |$1$| (to be determined again using Theorem 3.1) but also the genus |$2$| term |$\int a u^3 i\hbar \varepsilon ^2\mathrm{\,d}x$|⁠, with |$a \in{{\mathbb{Q}}}$|⁠. The constant |$a$| corresponds to the intersection number |$a=-\int _{\textrm{DR}_2(0,0)}\lambda _1 \psi _1 W_2(e_1,e_3) = - 3\int _{\textrm{DR}_2(0)} \lambda _1 W_2(e_3)= -3\int _{{\overline{{\mathcal{M}}}}_{2,1}} \lambda _2 \lambda _1 W_2(e_3)$|⁠. Using the fact that, on |${\overline{{\mathcal{M}}}}_{2,0}$|⁠, |$\lambda _2\lambda _1 = \frac{1}{5760} [\textrm{pt}]$| and that the class of a fiber of |$\pi :{\overline{{\mathcal{M}}}}_{2,1}\to{\overline{{\mathcal{M}}}}_{2,0}$| is represented by the closure of the locus of singular genus |$2$| curves with |$3$| nodes (one separating, two non-separating) and a marked point on either of the two irreducible components we obtain |$a=-3 \times \frac{1}{5760} \times 2 \int _{{\overline{{\mathcal{M}}}}_{0,3}} W_0(e_\mu ,e_\nu ,e_\epsilon ) \eta ^{\mu \nu } \eta ^{\epsilon \delta } \int _{{\overline{{\mathcal{M}}}}_{0,4}} W_0(e_\delta ,e_\alpha ,e_\beta ,e_3) \eta ^{\alpha \beta }= -\frac{1}{1280}$|⁠. 4 Tau-symmetry and Tau-functions for Quantum Integrable Systems In this section we introduce a quantum version of the notions of tau-structure and tau-functions for a Hamiltonian hierarchy. Remark 4.1. We note here that, for the time being, we will restrict our definitions to the case (relevant for the qDR hierarchy) of quantum Hamiltonian systems whose commutator |$[\cdot ,\cdot ]$| is the one defined in Section 2.3. This means in particular that the semiclassical limit has the Poisson structure in standard form |$\{\cdot ,\cdot \}_{\eta{\partial }_x}$|⁠. A more general theory of quantum tau-structures will require a study and classification of star-products and commutators on the space of quantum differential polynomials and local functionals. We plan to study this subject in a future work. 4.1 Tau-symmetric quantum Hamiltonian hierarchies Consider a quantum Hamiltonian system defined by a family of pairwise commuting quantum local functionals |${\overline H}_{\beta ,q}\in (\widehat \Lambda ^{\hbar }_N)^{[\leq 0]}$|⁠, parameterized by two indices |$1\le \beta \le N$| and |$q\ge 0$|⁠, |$[{\overline H}_{\beta ,q},{\overline H}_{\gamma ,p}]=0$|⁠, with respect to the quantum commutator introduced in Section 2.3: \begin{equation*}\frac{{\partial} u^\alpha}{{\partial} {t^\beta_q}} = [u^\alpha,{\overline H}_{\beta,q}].\end{equation*} Let us assume that |${\overline H}_{1,0}=\frac{1}{2}\int \eta _{\mu \nu } u^\mu u^\nu $|⁠. Notice that, in this case, |$\frac{1}{\hbar } [f,{\overline H}_{1,0}] =\{f,{\overline H}_{1,0}\} =\sum _{k\geq 0} \frac{{\partial } f}{{\partial } u^\alpha _k} u^\alpha _{k+1}={\partial }_x f $| for any |$f \in \widehat{{\mathcal{A}}}^\hbar _N$|⁠. A tau-structure for such hierarchy is a collection of quantum differential polynomials |$H_{\beta ,q}\in (\widehat{{\mathcal{A}}}^\hbar _N)^{[\leq 0]}$|⁠, |$1\le \beta \le N$|⁠, |$q\ge -1$|⁠, such that the following conditions hold: (1) |${\overline H}_{\beta ,-1}:= \int H_{\beta ,-1} \mathrm{\,d}x =\int \eta _{\beta \mu } u^\mu \mathrm{\,d}x$|⁠, (2) For |$q\ge 0$|⁠, the quantum differential polynomials |$H_{\beta ,q}$| are densities for the Hamiltonians |${\overline H}_{\beta ,q}$|⁠, \begin{equation} {\overline H}_{\beta,q}=\int H_{\beta,q}\mathrm{\,d}x. \end{equation} (4.1) (3) Tau-symmetry: \begin{equation} \big[H_{\alpha,p-1},{\overline H}_{\beta,q}\big]=\big[H_{\beta,q-1},{\overline H}_{\alpha,p}\big],\quad1\le\alpha,\beta\le N,\quad p,q\ge 0. \end{equation} (4.2) Existence of a tau-structure imposes nontrivial constraints on a quantum Hamiltonian hierarchy. A quantum Hamiltonian hierarchy with a fixed tau-structure will be called tau-symmetric. 4.2 Sufficient condition for the existence of a tau-structure Consider again a quantum Hamiltonian hierarchy defined by a family of pairwise commuting quantum local functionals |${\overline H}_{\beta ,q}\in (\widehat \Lambda ^{\hbar }_N)^{[\leq 0]}$|⁠, parameterized by two indices |$1\le \beta \le N$| and |$q\ge 0$|⁠. In the same way, as in the previous section, we assume that |${\overline H}_{1,0}=\frac{1}{2}\int \eta _{\mu \nu } u^\mu u^\nu $|⁠. We have the following quantum analogue of a result from [2]. Proposition 4.2. Suppose that \begin{equation*} \frac{{\partial}{\overline H}_{\beta,q}}{{\partial} u^1}= \begin{cases} {\overline H}_{\beta,q-1},&\text{if {$q\ge 1$}},\\ \int\eta_{\beta\mu}u^\mu \mathrm{\,d}x,&\text{if {$q=0$}}. \end{cases} \end{equation*} Then the differential polynomials \begin{equation*} H_{\beta,q}:=\frac{\delta{\overline H}_{\beta,q+1}}{\delta u^1},\quad q\ge -1, \end{equation*} define a tau-structure for the quantum hierarchy. Proof. We have |${\overline H}_{\beta ,-1}=\int \eta _{\beta \mu }u^\mu \mathrm{\,d}x$|⁠. Condition (4.1) is clear, since for |$q\ge 0$| we have \begin{equation*} \int H_{\beta,q}\mathrm{\,d}x=\int\frac{\delta{\overline H}_{\beta,q+1}}{\delta u^1}\mathrm{\,d}x=\frac{{\partial}}{{\partial} u^1}{\overline H}_{\beta,q+1}={\overline H}_{\beta,q}. \end{equation*} Let us check the tau-symmetry condition (4.2). We have the commutativity |$[{\overline H}_{\alpha ,p},{\overline H}_{\beta ,q}]=0$|⁠. Let us apply the variational derivative |$\frac{\delta }{\delta u^1}$| to this equation. It is much easier to do it in the |$p$|-variables (2.1). We have |$[{\overline H}_{\alpha ,p},{\overline H}_{\beta ,q}] ={\overline H}_{\alpha ,p} \left ( e^{\sum _{k>0} i \hbar k \eta ^{\mu \nu } \overleftarrow{\frac{\partial } {\partial p^\mu _{k}}} \overrightarrow{\frac{\partial } {\partial p^\nu _{-k}}}} -e^{\sum _{k>0} i \hbar k \eta ^{\mu \nu } \overleftarrow{\frac{\partial } {\partial p^\mu _{-k}}} \overrightarrow{\frac{\partial }{\partial p^\nu _{k}}}}\right )\\{\overline H}_{\beta ,q}$|⁠. For the variational derivative we have |$\frac{\delta{\overline H}}{\delta u^\gamma }=\sum _{n\in{\mathbb{Z}}}e^{-inx}\frac{{\partial }{\overline H}}{{\partial } p^\gamma _n}$| for any |${\overline H}\in (\widehat \Lambda ^\hbar _N)^{[\leq 0]}$|⁠. Therefore, we get \begin{align*} 0&=\frac{\delta}{\delta u^1}\big[{\overline H}_{\alpha,p},{\overline H}_{\beta,q}\big]=\\ &=\sum_{n\in{\mathbb{Z}}}e^{-inx}\frac{{\partial}}{{\partial} p^1_n}\left({\overline H}_{\alpha,p} \left( e^{\sum_{k>0} i \hbar k \eta^{\mu \nu} \overleftarrow{\frac{\partial }{\partial p^\mu_{k}}} \overrightarrow{\frac{\partial }{\partial p^\nu_{-k}}}} -e^{\sum_{k>0} i \hbar k \eta^{\mu \nu} \overleftarrow{\frac{\partial }{\partial p^\mu_{-k}}} \overrightarrow{\frac{\partial }{\partial p^\nu_{k}}}}\right) {\overline H}_{\beta,q}\right)=\\ &=\left(\sum_{n\in{\mathbb{Z}}}e^{-inx}\frac{{\partial}{\overline H}_{\alpha,p}}{{\partial} p^1_n}\right)\left( e^{\sum_{k>0} i \hbar k \eta^{\mu \nu} \overleftarrow{\frac{\partial }{\partial p^\mu_{k}}} \overrightarrow{\frac{\partial }{\partial p^\nu_{-k}}}} -e^{\sum_{k>0} i \hbar k \eta^{\mu \nu} \overleftarrow{\frac{\partial }{\partial p^\mu_{-k}}} \overrightarrow{\frac{\partial }{\partial p^\nu_{k}}}}\right) {\overline H}_{\beta,q}\\ &\phantom{=}+{\overline H}_{\alpha,p}\left( e^{\sum_{k>0} i \hbar k \eta^{\mu \nu} \overleftarrow{\frac{\partial }{\partial p^\mu_{k}}} \overrightarrow{\frac{\partial }{\partial p^\nu_{-k}}}} -e^{\sum_{k>0} i \hbar k \eta^{\mu \nu} \overleftarrow{\frac{\partial }{\partial p^\mu_{-k}}} \overrightarrow{\frac{\partial }{\partial p^\nu_{k}}}}\right) \left(\sum_{n\in{\mathbb{Z}}}e^{-inx}\frac{{\partial}{\overline H}_{\beta,q}}{{\partial} p^1_n}\right)=\\ &=\big[H_{\alpha,p-1},{\overline H}_{\beta,q}\big]-\big[H_{\beta,q-1},{\overline H}_{\alpha,p}\big]. \end{align*} The proposition is proved. Corollary 4.3. The qDR hierarchy |$\{{\overline G}_{\alpha ,d}\}_{\substack{1 \leq \alpha \leq N, d\geq -1}}$|⁠, with |$G_{\alpha ,d}$| given by (2.13), is tau-symmetric. A tau-structure is given by the densities |$H_{\alpha ,d} = \frac{\delta{\overline G}_{\alpha ,d+1}}{\delta u^1}$|⁠. 4.3 Quantum tau-functions We consider again a quantum Hamiltonian hierarchy generated by Hamiltonians |${\overline H}_{\alpha ,p}$|⁠, |$1\leq \alpha \leq N$|⁠, |$p\geq -1,$| where |${\overline H}_{1,0}=\frac{1}{2}\int \eta _{\mu \nu } u^\mu u^\nu \mathrm{\,d}x$|⁠. Suppose that the quantum differential polynomials |$H_{\beta ,q}$|⁠, |$1\le \beta \le N$|⁠, |$q\ge -1$| define a tau-structure for such hierarchy. From commutativity of the Hamiltonians we have \begin{equation} \int \big[ H_{\alpha,p-1}, {\overline H}_{\beta,q}\big]\mathrm{\,d}x=0. \end{equation} (4.3) The quantum differential polynomial |$\big[ H_{\alpha ,p-1}, {\overline H}_{\beta ,q}\big]$| has no constant term (because of the form of the quantum commutator); hence, there exists a unique differential polynomial |$\Omega ^\hbar _{\alpha ,p;\beta ,q}\in (\widehat{{\mathcal{A}}}^\hbar _N)^{[\leq 0]}$| such that \begin{equation} {\partial}_x\Omega^\hbar_{\alpha,p;\beta,q}= \big[ H_{\alpha,p-1}, {\overline H}_{\beta,q}\big]\quad\textrm{and}\quad\left.\Omega^\hbar_{\alpha,p;\beta,q}\right|_{u^*_*=0}=0. \end{equation} (4.4) The differential polynomial |$\Omega ^\hbar _{\alpha ,p;\beta ,q}$| is called the two-point function of the given tau-structure of the hierarchy. From condition (4.2) it follows that \begin{equation} \Omega^\hbar_{\alpha,p;\beta,q}=\Omega^\hbar_{\beta,q;\alpha,p} \end{equation} (4.5) and, moreover, it implies that the differential polynomial \begin{equation} \big[\Omega^\hbar_{\alpha,p;\beta,q}, {\overline H}_{\gamma,r}\big] \end{equation} (4.6) is symmetric with respect to all permutations of the pairs |$(\alpha ,p)$|⁠, |$(\beta ,q)$|⁠, |$(\gamma ,r)$|⁠. Since the Hamiltonian |${\overline H}_{1,0}$| generates the spatial translations, equation (4.4) implies that |${\partial }_x\Omega ^\hbar _{\alpha ,p;1,0}={\partial }_x H_{\alpha ,p-1}$|⁠, |$p\ge 0$|⁠. Therefore, \begin{equation} \Omega^\hbar_{\alpha,p;1,0}-H_{\alpha,p-1}=C,\quad p\ge 0, \end{equation} (4.7) where |$C=C(\varepsilon ,\hbar )$| is a formal power series in |$\varepsilon $| and |$\hbar $|⁠. Consider also the evolved Hamiltonians \begin{equation} H^{t^*_*}_{\alpha,p} = \exp\left(\sum_{q\geq 0} \frac{t^\beta_q}{\hbar}\big[\cdot,{\overline H}_{\beta,q}\big]\right) H_{\alpha,p} \in \widehat{{\mathcal{A}}}_N^\hbar\big[\big[t^*_*\big]\big], \end{equation} (4.8) and the evolved two-point functions \begin{equation} \Omega^{\hbar,t^*_*}_{\alpha,p;\gamma,r} = \exp\left(\sum_{q\geq 0} \frac{t^\beta_q}{\hbar}\big[\cdot,{\overline H}_{\beta,q}\big]\right) \Omega^{\hbar}_{\alpha,p;\gamma,r} \in \widehat{{\mathcal{A}}}_N^\hbar\big[\big[t^*_*\big]\big]. \end{equation} (4.9) They satisfy, respectively, \begin{equation*}\frac{{\partial} H^{t^*_*}_{\alpha,p}}{{\partial} t^\beta_q}=\frac{1}{\hbar} \big[H^{t^*_*}_{\alpha,p},{\overline H}_{\beta,q}\big],\qquad \left.H^{t^*_*}_{\alpha,p}\right|_{t^*_*=0} = H_{\alpha,p} \in \widehat{{\mathcal{A}}}^\hbar_N\end{equation*} and \begin{equation*}\frac{{\partial} \Omega^{\hbar,t^*_*}_{\alpha,p;\gamma,r}}{{\partial} t^\beta_q}=\frac{1}{\hbar} \big[\Omega^{\hbar,t^*_*}_{\alpha,p;\gamma,r},{\overline H}_{\beta,q}\big],\qquad \left.\Omega^{\hbar,t^*_*}_{\alpha,p;\gamma,r}\right|_{t^*_*=0} = \Omega^\hbar_{\alpha,p;\gamma,r} \in \widehat{{\mathcal{A}}}^\hbar_N\end{equation*} together with \begin{equation} \frac{{\partial} H^{t^*_*}_{\alpha,p-1}}{{\partial} t^\beta_q} =\Omega^{\hbar,t^*_*}_{\alpha,p;\beta,q} =\Omega^{\hbar,t^*_*}_{\beta,q;\alpha,p} = \frac{{\partial} H^{t^*_*}_{\beta,q-1}}{{\partial} t^\alpha_p}. \end{equation} (4.10) Moreover, \begin{equation} \frac{{\partial} \Omega^{\hbar,t^*_*}_{\alpha,p;\beta,q}}{{\partial} t^\gamma_r} \end{equation} (4.11) is symmetric with respect to all permutations of the pairs |$(\alpha ,p)$|⁠, |$(\beta ,q),$| and |$(\gamma ,r)$|⁠. Then equation (4.10) and the symmetry of (4.11) imply that there exists a function |$P\in \widehat{{\mathcal{A}}}_N^\hbar [[t^*_*]]$| such that \begin{equation*} \Omega^{\hbar,t^*_*}_{\alpha,p;\beta,q}= \frac{{\partial}^2 P}{{\partial} t^\alpha_p{\partial} t^\beta_q},\quad\textrm{for any}\ 1\le\alpha,\beta\le N,\ \textrm{and}\ p,q\ge 0. \end{equation*} To each initial condition |$u^\alpha _k|_{x=t^*_*=0} = c^\alpha _k(\varepsilon ,\hbar )\in{\mathbb{C}}[[\varepsilon ,\hbar ]]$| with |$c^\alpha _k(0,0) = 0$| we can associate the restriction |$\left . P \right |_{u^\alpha _k=c^\alpha _k(\varepsilon ,\hbar )}\in{\mathbb{C}}[[t^*_*,\varepsilon ,\hbar ]]$| that is called (the logarithm of) the tau-function of the given solution. 5 Hierarchies of DR Type In this section we interpret the recursion (2.15) as a system of functional derivative equations for |${\overline G}_{1,1}$| and elevate it to the main axiom in the definition of a class of (quantum or classical) Hamiltonians producing integrable, tau-symmetric Hamiltonian systems. 5.1 An integrability condition for Hamiltonian systems Let us consider the quantum Hamiltonian system defined by a Hamiltonian |${\overline H}\in (\widehat \Lambda _N^\hbar )^{[\leq 0]}$| with respect to the standard quantum commutator introduced in Section 2.3. We give a sufficient condition for |${\overline H}$| to be part of an integrable hierarchy. Consider the operator |${{\mathcal{D}}}^\hbar _{\overline H}:\widehat{{\mathcal{A}}}_N^\hbar [[z]]\to \widehat{{\mathcal{A}}}_N^\hbar [[z]]$| defined by \begin{equation*} \begin{split} & {{\mathcal{D}}}^\hbar_{\overline H} f(z)= {\partial}_x (D-1) f(z) - \frac{z}{\hbar}\big[f(z),{\overline H}\big],\\ & f(z)= f\big(u^*_*;\varepsilon,\hbar;z\big) = \sum_{k\geq 0} f_{k-1}\big(u^*_*;\varepsilon,\hbar\big) z^k, \qquad f_{k-1}\big(u^*_*;\varepsilon,\hbar\big) \in \big(\widehat{{\mathcal{A}}}_N^\hbar\big)^{[\leq 0]}.\\ \end{split} \end{equation*} Suppose there exist |$N$| solutions |$G_{\alpha }(z) \in (\widehat{{\mathcal{A}}}_N^\hbar )^{[\leq 0]}[[z]]$|⁠, |$\alpha =1,\ldots ,N$|⁠, to |${{\mathcal{D}}}^\hbar _{\overline H} G_\alpha (z)=0$| with the initial conditions |$G_{\alpha }(z=0)=\eta _{\alpha \mu }u^\mu $|⁠. Then a new vector of solutions can be found by the following transformation: \begin{equation} G_{\alpha}(z) \mapsto A^\mu_\alpha(z) G_\mu(z) + B_\alpha(z), \end{equation} (5.1) where |$A^\mu _\alpha (z)=\delta ^\mu _\alpha + \sum _{i> 0} A^\mu _{\alpha ,i} z^i \in{\mathbb{C}}[[z]]$| and |$B_\alpha (z)=\sum _{i>0} B_{\alpha ,i}(\varepsilon ,\hbar ) z^i \in{\mathbb{C}}[[\varepsilon ,\hbar ,z]]$|⁠. Theorem 5.1. Assume that |${\overline H} \in (\widehat \Lambda _N^\hbar )^{[\leq 0]}$| has the following properties: (a) there exist |$N$| independent solutions |$G_{\alpha }(z) = \sum _{p\geq 0} G_{\alpha ,p-1} z^p\in (\widehat{{\mathcal{A}}}_N^\hbar )^{[\leq 0]}[[z]]$|⁠, |$\alpha =1,\ldots ,N$|⁠, to the equation \begin{equation} {{\mathcal{D}}}^\hbar_{\overline H} G_\alpha(z) = 0 \end{equation} (5.2) with the initial conditions |$ G_{\alpha }(z=0)=\eta _{\alpha \mu }u^\mu $|⁠, (b) |$\frac{\delta{\overline H}}{\delta u^1} = \frac{1}{2}\eta _{\mu \nu } u^\mu u^\nu + {\partial }_x R + c(\varepsilon ,\hbar ), \qquad R\in (\widehat{{\mathcal{A}}}^\hbar _N)^{[\le -1]}, \quad c(\varepsilon ,\hbar ) \in{\mathbb{C}}[[\varepsilon ,\hbar ]]$|⁠, (c) |${\overline G}_{1,1} = {\overline H}$|⁠. Then, up to a transformation of type (5.1), we have (i) |${\overline G}_{1,0} = \int \left ( \frac{1}{2}\eta _{\mu \nu } u^\mu u^\nu \right )\mathrm{\,d}x$|⁠, (ii) |$[{\overline G}_{\alpha ,p},{\overline G}_{\beta ,q}] = 0, \qquad \alpha ,\beta =1,\ldots ,N,\quad p,q\geq -1$|⁠, (iii) |$\frac{1}{\hbar }[G_{\alpha ,p},{\overline G}_{\beta ,0}] = {\partial }_x \frac{{\partial } G_{\alpha ,p+1}}{{\partial } u^\beta }, \qquad \beta =1,\ldots ,N, \quad p\geq -1,$| (iv) |$\frac{{\partial } G_{\alpha ,p}}{{\partial } u^1} = G_{\alpha ,p-1}, \qquad \alpha =1,\ldots ,N, \quad p\geq -1$|⁠; hence, in particular |${\overline H}$| is part of a quantum integrable tau-symmetric hierarchy. Proof. Equation (5.2) implies in particular that |$[{\overline G}_{\alpha ,p},{\overline H}] = 0$| for every |$\alpha =1,\ldots ,N$|⁠, |$p\geq -1$|⁠. Moreover, we have \begin{equation*}{\partial}_x (D-1) G_{1,0} = \frac{1}{\hbar}[G_{1,-1},{\overline H}] = {\partial}_x \frac{\delta{\overline H}}{\delta u^1} ={\partial}_x\left( \frac{1}{2} \eta_{\mu\nu} u^\mu u^\nu + {\partial}_x R + c(\varepsilon,\hbar)\right),\end{equation*} which proves |$\mathrm{(i)}$|⁠. We write equation (5.2) as |${\partial }_x (D-1) G_{\alpha ,p} = \frac{1}{\hbar } [G_{\alpha ,p-1},{\overline G}_{1,1}]$|⁠. To prove |$\mathrm{(ii)}$| we will show that such recursion implies \begin{equation*} \frac{1}{\hbar} \big[ G_{\alpha,p}(x), G_{\beta,q}(y)\big] = \partial_x G_{\alpha,p+1;\beta,q}(x,y) - \partial_y G_{\alpha,p;\beta,q+1}(x,y), \end{equation*} for |$\alpha ,\beta =1,\ldots ,N$|⁠, |$p,q\geq 0$| (which is equation (2.14)), for some opportunely defined |$G_{\alpha ,p;\beta ,q}(x,y)$|⁠, symmetric with respect to simultaneous exchange of the indices |$(\alpha ,p,x)$| and |$(\beta ,q,y)$|⁠. We proceed by recursion starting from the fact that, for |$p\geq 0$| \begin{equation*} \frac{1}{\hbar} \big[ G_{\alpha,p}(x), G_{\beta,-1}(y)\big] = \sum_{l\geq 0} \frac{{\partial} G_{\alpha,p}}{{\partial} u^\beta_l} \delta^{(l+1)}(x-y) = -{\partial}_y \left( \sum_{l\geq 0} \frac{{\partial} G_{\alpha,p}}{{\partial} u^\beta_l} \delta^{(l)}(x-y) \right), \end{equation*} so that we can pos e \begin{equation*} G_{\alpha,p;\beta,0}(x,y):= \sum_{l\geq 0} \frac{{\partial} G_{\alpha,p}}{{\partial} u^\beta_l} \delta^{(l)}(x-y)=: G_{\beta,0;\alpha,p}(y,x),\qquad G_{\alpha,p;\beta,-1}(x,y) = G_{\beta,-1;\alpha,p}(y,x) = 0, \end{equation*} and have \begin{equation*}\frac{1}{\hbar} \big[ G_{\alpha,p}(x), G_{\beta,-1}(y)\big] = \partial_x G_{\alpha,p+1;\beta,-1}(x,y) - \partial_y G_{\alpha,p;\beta,0}(x,y),\end{equation*} \begin{equation*}\frac{1}{\hbar} \big[ G_{\alpha,-1}(x), G_{\beta,q}(y)\big] = \partial_x G_{\alpha,0;\beta,q}(x,y) - \partial_y G_{\alpha,-1;\beta,q+1}(x,y).\end{equation*} Now we assume \begin{equation*}\frac{1}{\hbar} \big[ G_{\alpha,p}(x), G_{\beta,q-1}(y)\big] = \partial_x G_{\alpha,p+1;\beta,q-1}(x,y) - \partial_y G_{\alpha,p;\beta,q}(x,y),\end{equation*} \begin{equation*}\frac{1}{\hbar} \big[ G_{\alpha,p-1}(x), G_{\beta,q}(y)\big] = \partial_x G_{\alpha,p;\beta,q}(x,y) - \partial_y G_{\alpha,p-1;\beta,q+1}(x,y),\end{equation*} and obtain \begin{equation*} \begin{split} D \frac{1}{\hbar} & \big[ G_{\alpha,p}(x), G_{\beta,q}(y)\big] \\ &=\frac{1}{\hbar} \big[(D-1) G_{\alpha,p}(x),G_{\beta,q}(y)\big] + \frac{1}{\hbar} \big[ G_{\alpha,p}(x), (D-1) G_{\beta,q}(y)\big]\\ & = \frac{1}{\hbar} \left[{\partial}_x^{-1} \frac{1}{\hbar}\big[G_{\alpha,p-1}(x),{\overline G}_{1,1}\big],G_{\beta,q}(y)\right] + \frac{1}{\hbar} \left[G_{\alpha,p}(x),{\partial}_y^{-1} \frac{1}{\hbar}\big[G_{\beta,q-1}(y),{\overline G}_{1,1}\big]\right]\\ & = {\partial}_x^{-1}\frac{1}{\hbar} \left[\frac{1}{\hbar}\big[G_{\alpha,p-1}(x),{\overline G}_{1,1}\big],G_{\beta,q}(y)\right] + {\partial}_y^{-1}\frac{1}{\hbar} \left[G_{\alpha,p}(x), \frac{1}{\hbar}\big[G_{\beta,q-1}(y),{\overline G}_{1,1}\big]\right]\\ &={\partial}_x^{-1}\left(-\frac{1}{\hbar}\left[\frac{1}{\hbar}\big[G_{\beta,q}(y),G_{\alpha,p-1}(x)\big],{\overline G}_{1,1}\right]-\frac{1}{\hbar}\left[\frac{1}{\hbar}\big[{\overline G}_{1,1},G_{\beta,q}(y)\big],G_{\alpha,p-1}(x)\right]\right)\\ & \quad+{\partial}_y^{-1}\left(-\frac{1}{\hbar}\left[{\overline G}_{1,1},\frac{1}{\hbar}\big[G_{\alpha,p}(x),G_{\beta,q-1}(y)\big]\right]-\frac{1}{\hbar}\left[G_{\beta,q-1}(y),\frac{1}{\hbar}\big[{\overline G}_{1,1},G_{\alpha,p}(x)\big]\right]\right)\\ &={\partial}_x^{-1}\left(\frac{1}{\hbar}\left[\partial_x G_{\alpha,p;\beta,q}(x,y) - \partial_y G_{\alpha,p-1;\beta,q+1}(x,y),{\overline G}_{1,1}\right]\right. \\ & \quad\left.+\frac{1}{\hbar}\big[{\partial}_y (D-1) G_{\beta,q+1}(y),G_{\alpha,p-1}(x)\big]\right)\\ \end{split} \end{equation*} \begin{equation*} \begin{split} & \quad+{\partial}_y^{-1}\left(\frac{1}{\hbar}\big[\partial_x G_{\alpha,p+1;\beta,q-1}(x,y) - \partial_y G_{\alpha,p;\beta,q}(x,y),{\overline G}_{1,1}\big]\right.\\ & \quad\left.+\frac{1}{\hbar}\big[G_{\beta,q-1}(y),{\partial}_x (D-1) G_{\alpha,p+1}(x)\big]\right)\\ &=-{\partial}_y {\partial}_x^{-1}\left(\frac{1}{\hbar}\big[G_{\alpha,p-1;\beta,q+1}(x,y),{\overline G}_{1,1}\big]-\frac{1}{\hbar}\big[ (D-1) G_{\beta,q+1}(y),G_{\alpha,p-1}(x)\big]\right)\\ & \quad\,+{\partial}_x {\partial}_y^{-1}\left(\frac{1}{\hbar}\big[ G_{\alpha,p+1;\beta,q-1}(x,y),{\overline G}_{1,1}\big]+\frac{1}{\hbar}\big[G_{\beta,q-1}(y), (D-1) G_{\alpha,p+1}(x)\big]\right). \end{split} \end{equation*} Hence, we can define \begin{equation*}G_{\alpha,p+1;\beta,q}(x,y) = D^{-1}{\partial}_y^{-1}\left(\frac{1}{\hbar}\big[ G_{\alpha,p+1;\beta,q-1}(x,y),{\overline G}_{1,1}\big]+\frac{1}{\hbar}\big[G_{\beta,q-1}(y), (D-1) G_{\alpha,p+1}(x)\big]\right),\end{equation*} \begin{equation*}G_{\alpha,p;\beta,q+1}(x,y) = D^{-1} {\partial}_x^{-1}\left(\frac{1}{\hbar}\big[G_{\alpha,p-1;\beta,q+1}(x,y),{\overline G}_{1,1}\big]-\frac{1}{\hbar}\big[ (D-1) G_{\beta,q+1}(y),G_{\alpha,p-1}(x)\big]\right), \end{equation*} which enjoy the correct symmetry property with respect to exchange of indices and variables. By induction we arrive then to the proof of |$\mathrm{(ii)}$|⁠. From the last equation we can deduce in particular that |$\int G_{\alpha ,p+1;\beta ,0}(x,y) \mathrm{\,d}y = \frac{{\partial } G_{\alpha ,p+1}(x)}{{\partial } u^\beta }$|⁠. We also have \begin{equation*} \frac{1}{\hbar} \big[ G_{\alpha,p}(x), G_{\beta,0}(y)\big] = \partial_x G_{\alpha,p+1;\beta,0}(x,y) - \partial_y G_{\alpha,p;\beta,1}(x,y) \end{equation*} which, upon integration with respect to |$y$|⁠, gives \begin{equation*}\frac{1}{\hbar} \big[ G_{\alpha,p}(x), {\overline G}_{\beta,0}\big]=\int{\partial}_x G_{\alpha,p+1;\beta,0}(x,y) \mathrm{\,d}y={\partial}_x \frac{{\partial} G_{\alpha,p+1}(x)}{{\partial} u^\beta}\end{equation*} and proves |$\mathrm{(iii)}$|⁠. Point |$\mathrm{(iv)}$| follows from point |$\mathrm{(iii)}$| in the case |$\beta = 1$|⁠, which gives |${\partial }_x \frac{{\partial } G_{\alpha ,p+1}}{{\partial } u^1} = {\partial }_x G_{\alpha ,p}$|⁠. We also have the following theorem, which is slightly stronger than the classical version of the above one. For a local functional |$\overline h\in \widehat \Lambda ^{[0]}_N$| consider the operator |${{\mathcal{D}}}_{\overline h}:\widehat{{\mathcal{A}}}_N[[z]]\to \widehat{{\mathcal{A}}}_N[[z]]$| defined by \begin{equation*} \begin{split} & {{\mathcal{D}}}_{\overline h} f(z)= {\partial}_x (D-1) f(z) - z\{f(z),\overline h\},\\ & f(z)= f(u^*_*;\varepsilon;z) = \sum_{k\geq 0} f_{k-1}\big(u^*_*;\varepsilon\big) z^k, \qquad f_{k-1}\big(u^*_*;\varepsilon\big) \in \widehat{{\mathcal{A}}}_N^{[0]}.\\ \end{split} \end{equation*} Suppose there exist |$N$| solutions |$g_{\alpha }(z) \in \widehat{{\mathcal{A}}}_N^{[0]}[[z]]$|⁠, |$\alpha =1,\ldots ,N$|⁠, to |${{\mathcal{D}}}_{\overline h} g_\alpha (z)=0$| with the initial conditions |$g_{\alpha }(z=0)=\eta _{\alpha \mu }u^\mu $|⁠. Then a new vector of solutions can be found by the following transformation: \begin{equation} g_{\alpha}(z) \mapsto a^\mu_\alpha(z) g_\mu(z) + b_\alpha(z), \end{equation} (5.3) where |$a^\mu _\alpha (z)=\delta ^\mu _\alpha + \sum _{i> 0} a^\mu _{\alpha ,i} z^i \in{\mathbb{C}}[[z]]$| and |$b_\alpha (z)=\sum _{i>0} b_{\alpha ,i} z^i \in{\mathbb{C}}[[z]]$|⁠. Theorem 5.2. Assume that |$\overline h \in \widehat \Lambda _N^{[0]}$| has the following properties: (a) there exist |$N$| independent solutions |$g_{\alpha }(z) = \sum _{p\geq 0} g_{\alpha ,p-1} z^p\in \widehat{{\mathcal{A}}}_N^{[0]}[[z]]$|⁠, |$\alpha =1,\ldots ,N$|⁠, to the equation \begin{equation} {{\mathcal{D}}}_{\overline h} g_\alpha(z) = 0 \end{equation} (5.4) with the initial conditions |$ g_{\alpha }(z=0)=\eta _{\alpha \mu }u^\mu $|⁠, (b) |$\frac{\delta \overline h}{\delta u^1} = \frac{1}{2}\eta _{\mu \nu } u^\mu u^\nu + {\partial }_x^2 r, \qquad r\in \widehat{{\mathcal{A}}}_N^{[-2]}$|⁠. Then, up to a transformation of type (5.3), we have (i) |$g_{1,0} = \frac{1}{2}\eta _{\mu \nu } u^\mu u^\nu + {\partial }_x^2 (D-1)^{-1} r$|⁠, (ii) |$\overline g_{1,1} = \overline h$|⁠, (iii) |$\{\overline g_{\alpha ,p},\overline g_{\beta ,q}\} = 0, \qquad \alpha ,\beta =1,\ldots ,N,\quad p,q\geq -1$|⁠, (iv) |$\{g_{\alpha ,p},\overline g_{\beta ,0}\} = {\partial }_x \frac{{\partial } g_{\alpha ,p+1}}{{\partial } u^\beta }, \qquad \beta =1,\ldots ,N, \quad p\geq -1,$| (v) |$\frac{{\partial } g_{\alpha ,p}}{{\partial } u^1} = g_{\alpha ,p-1}, \qquad \alpha =1,\ldots ,N, \quad p\geq -1$|⁠; hence, in particular |$\overline h$| is part of an integrable tau-symmetric hierarchy. Proof. The only differences in the statement of this theorem from the classical limit of Theorem 5.1 are that hypothesis (b) has become stronger together with claim (i) and that hypothesis (c) of Theorem 5.1 has now become claim (ii) and so it needs to be proved. The proof of (i) follows from equation (5.4): \begin{equation*}{\partial}_x (D-1) g_{1,0} = \{g_{1,-1},\overline h\} = {\partial}_x \frac{\delta \overline h}{\delta u^1} ={\partial}_x\left( \frac{1}{2} \eta_{\mu\nu} u^\mu u^\nu + {\partial}^2_x r\right).\end{equation*} Also from equation (5.4) we obtain that |$g_{1,1} = (D-1)^{-1} {\partial }_x^{-1} \{g_{1,0},\overline h\}$|⁠. A direct computation shows that |$\{\frac{1}{2}\eta _{\mu \nu } u^\mu u^\nu , \overline h\} = {\partial }_x (D-1) h+ {\partial }_x^2 s$|⁠, where |$h\in \widehat{{\mathcal{A}}}^{[0]}_N$|⁠, |$s\in \widehat{{\mathcal{A}}}^{[-1]}_N$| with |$\overline h = \int h \mathrm{\,d}x$|⁠, so we deduce |$\{g_{1,0},\overline h\} = {\partial }_x (D-1) h+ {\partial }_x^2 s + {\partial }^2_x \{ (D-1)^{-1} r,\overline h\}$|⁠, where we used that |$D h = \left (\sum _{k\geq 0}(k+1)u^\alpha _k \frac{{\partial }}{{\partial } u^\alpha _k}\right )h$|⁠. This implies, always up to (5.3), \begin{equation*} \overline g_{1,1}= \int \left[(D-1)^{-1} {\partial}_x^{-1} \left({\partial}_x (D-1) h+ {\partial}_x^2 s + {\partial}^2_x \{(D-1)^{-1} r,\overline h\}\right)\right] \mathrm{\,d}x= \overline h. \end{equation*} Remark 5.3. When we restrict to |$\hbar =\varepsilon =0$|⁠, a par1ticular Hamiltonian satisfying conditions (a) and (b) of Theorem 5.1 is given by |$\left .{\overline H}\right |_{\hbar =\varepsilon =0} =(D-2)\int F(u^1,\ldots ,u^N) \mathrm{\,d}x $|⁠, where the function |$F=F(u^1,\ldots ,u^N)$| is a solution to the WDVV: Witten-Dijkgraaf-Verlinde-Verlinde equations \begin{equation*}\frac{{\partial}^3 F}{{\partial} u^\alpha{\partial} u^\beta{\partial} u^\mu} \eta^{\mu \nu}\frac{{\partial}^3 F}{{\partial} u^\nu{\partial} u^\gamma{\partial} u^\delta} = \frac{{\partial}^3 F}{{\partial} u^\alpha{\partial} u^\delta{\partial} u^\mu} \eta^{\mu \nu}\frac{{\partial}^3 F}{{\partial} u^\nu{\partial} u^\gamma{\partial} u^\beta},\end{equation*} \begin{equation*}\frac{{\partial}^3 F}{{\partial} u^1 {\partial} u^\alpha{\partial} u^\beta} = \eta_{\alpha \beta}, \end{equation*} for |$\alpha ,\beta ,\gamma ,\delta = 1,\ldots ,N$|⁠. This is because, at |$\hbar =\varepsilon =0$|⁠, equation (5.2) promptly reduces to an averaged (and hence weaker) form of genus |$0$| topological recursion relations, and the WDVV equations are equivalent to the existence of |$N$| independent solutions to such equations. At that point, such |$N$| solutions to (5.2) correspond to the |$N$| generating functions of the classical (⁠|$\hbar =0$|⁠) dispersionless (⁠|$\varepsilon =0$|⁠) Hamiltonian densities |$g^{[0]}_{\alpha }(z):= \left . G_{\alpha }(z)\right |_{\hbar =\varepsilon =0}$| of the principal hierarchy of the resulting (formal) Frobenius manifold, which is the |$N$| flat coordinates of its deformed flat connection (see [11] for details). In such classical dispersionless context, Theorem 5.1 is hence a generalization of results proved for instance in [11]. Definition 5.4. Let |${\overline H} \in (\widehat \Lambda _N^\hbar )^{[\leq 0]}$| (resp. |$\overline h \in \widehat \Lambda _N^{[0]}$|⁠) satisfy the hypothesis of Theorem 5.1 (resp. Theorem 5.2). Then we say that |${\overline H}$| (resp. |$\overline h$|⁠) and the induced quantum (resp. classical) integrable tau-symmetric hierarchy are of DR type. Theorem 5.5. The qDR hierarchy (2.13) with |${\overline H}={\overline G}_{1,1}$| and its classical limit are hierarchies of DR type. Proof. Hypothesis (a) of Theorem 5.1 is satisfied thanks to recursion (2.15). Hypothesis (b) follows for instance from the string equation (2.17) together with the fact that |${\overline G}_{1,0} = \int \big ( \frac{1}{2}\eta _{\mu \nu } u^\mu u^\nu \big ) \mathrm{\,d}x$| and hypothesis (c) holds by definition of DR hierarchy. For the classical counterpart, hypothesis (b) of Theorem 5.2 is a consequence of the divibility, for |$g,n\geq 1$|⁠, of |$\pi _*\big (\lambda _g\textrm{DR}_g(-\sum _{i=1}^n a_i,a_1,\ldots ,a_n)\big )$| by |$a_n^2$|⁠, where |$\pi :{\overline{{\mathcal{M}}}}_{g,n+1}\to{\overline{{\mathcal{M}}}}_{g,n}$| forgets the last marked point, which was proved in [2], and which implies the possibility of finding a density for |$\overline g_{1,1}$|⁠, which is independent of |$u^1_x$|⁠. 5.2 Classification of rank |$1$| quantum integrable hierarchies of DR type In this section we study quantum deformations of DR type of the Riemann hierarchy, which is the genus |$0$| DR hierarchy associated with the trivial CohFT with |$V={\mathbb{C}}\ni e_1$| and |$c_{g,n}(e_1^{\otimes n}) = 1 \in H^0({\overline{{\mathcal{M}}}}_{g,n},{{\mathbb{Q}}})$|⁠. At first we concentrate on purely quantum deformations of the Riemann hierarchy, which means that, in this classification problem, the variable |$\varepsilon $| will not appear. This amounts to classifying quantum Hamiltonians of the form \begin{equation*} {\overline G}_1 = \int \frac{u^3}{6} \mathrm{\,d}x + \sum_{k\geq 1} {\overline G}_1^k \hbar^k, \qquad{\overline G}_1^k \in \Lambda_1^{[\le 2k]}, \end{equation*} satisfying the hypothesis of Theorem 5.1, with |$\varepsilon =0$|⁠. An explicit computation gives, modulo terms proportional to the Casimir |$\int u \mathrm{\,d}x$|⁠, the following classification up to order |$3$| in |$\hbar $|⁠: \begin{align} {\overline G}_1 = \int \left[ \frac{u^3}{6}+\big( a u_1{}^2\big) i \hbar +\big(b u_2{}^2\big) (i\hbar) ^2+\left(c u_2{}^3+\frac{10 b^2 - c} {7 a}u_3{}^2\right) (i\hbar)^3+O\big(\hbar ^4\big)\right] \mathrm{\,d}x. \end{align} (5.5) In the above formula we assume |$a\neq 0$|⁠. In case |$a=0$|⁠, the computation gives |$b=c=0$| too. Let us compare this formula with the Hamiltonian |${\overline G}_1$| of the dispersionless (i.e., |$\varepsilon =0$|⁠) quantum DR hierarchy for a rank |$1$| CohFT with |$\eta _{1,1}=1$|⁠. According to [28] such CohFTs are parameterized by numbers |$r_1,r_2,\ldots $| in the following way: \begin{equation} c_{g,n}\big(e_1^{\otimes n}\big)=e^{-\sum_{i\ge 1}\frac{(2i)!}{B_{2i}}r_i\textrm{Ch}_{2i-1}(\mathbb E)}. \end{equation} (5.6) Here |$\textrm{Ch}_{2i-1}$| denotes the |$(2i-1)$|-th component of the Chern character and the |$B_{2i}$| are Bernoulli numbers (see also Section 8). A direct computation along the line of Section 8.2 gives, up to the term |$-\frac{i \hbar }{24}\int u \mathrm{\,d}x$|⁠, exactly equation (5.5) with \begin{equation*}a=-\frac{1}{2}r_1, \qquad b=-\frac{1}{12} r_2-\frac{2}{5}r_1^3,\qquad c=-\frac{7}{480} r_3 r_1+\frac{5}{72}r_2^2-\frac{1}{3} r_2 r_1^3-\frac{8}{25} r_1^6,\end{equation*} suggesting that dispersionless quantum deformations of the Riemann hierarchy are in one-to-one correspondence with rank |$1$| CohFTs with |$\eta _{1,1}=1$|⁠. Assuming this correspondence, it is possible to recover dispersive deformations too by defining new parameters |$s_i$| as follows: \begin{equation*}e^{-\sum_{i\ge 1}\frac{(2i)!}{B_{2i}}r_i\textrm{Ch}_{2i-1}(\mathbb E)} = \Lambda\left(\frac{-\varepsilon^2}{i \hbar}\right) e^{-\sum_{i\ge 1}\frac{(2i)!}{B_{2i}}s_i\textrm{Ch}_{2i-1}(\mathbb E)}.\end{equation*} This amounts to \begin{equation*}r_i = s_i +\frac{B_{2i}}{2i(2i-1)}\left(\frac{\varepsilon^2}{i \hbar}\right)^{2i-1},\end{equation*} which gives \begin{align*} a=&\frac{1}{i\hbar}\left(-\frac{\varepsilon ^2}{24}-\frac{1}{2} s_1 i\hbar\right),\\ b=&\frac{1}{(i\hbar)^2}\left(-\frac{1}{120} s_1 \varepsilon ^4-\frac{1}{10} s_1^2 \ i\hbar\varepsilon ^2 -\left(\frac{2}{5} s_1^3 +\frac{1}{12} s_2\right) (i\hbar) ^2\right),\\ c =&\frac{1}{(i\hbar)^3}\left( \left(-\frac{1}{360} s_1^3-\frac{s_2}{1728}\right)\varepsilon^6- \frac{24 s_1^4 +5 s_1 s_2} {720}i \hbar \varepsilon ^4-\frac{4608 s_1^5 +2400 s_2 s_1^2 +35 s_3 }{28800}(i\hbar)^2\varepsilon ^2\right.\\ &\left.-\frac{ 2304 s_1^6+2400 s_2 s_1^3+105 s_3 s_1-500 s_2^2 }{7200}(i\hbar)^3\right).\end{align*} Once plugged into (5.5), this parametrization provides the quantum correction to the density (8.2) or (8.23) up to genus |$3$|⁠. Rescaling |$\varepsilon ^2\to \varepsilon ^2 \gamma $| and |$\hbar \to \hbar \gamma $| to keep track of the genus, we obtain \begin{equation*} \begin{split} {\overline G}_1\!=\!\int &\left[\frac{u^3}{6}+\left(\left(-\frac{\varepsilon ^2}{24}-\frac{s_1}{2} i\hbar\right)u_1^2-\frac{i \hbar}{24} u\right)\gamma\vphantom{\frac{u^3_{1}}{6}}\right.\\ &\left.+\left(\left(-\frac{s_1}{120} \varepsilon ^4-\frac{s_1^2}{10} i\hbar\varepsilon ^2 -\frac{24 s_1^3 +5 s_2}{60} (i\hbar) ^2\right)u_2^2\right)\gamma^2\right.\\ &\left.+\left(\left( -\frac{s_1^3}{360} \varepsilon^6-\frac{s_2}{1728}\varepsilon^6- \frac{24 s_1^4 +5 s_1 s_2} {720}i \hbar \varepsilon^4-\frac{4608 s_1^5 +2400 s_2 s_1^2 +35 s_3 }{28800}(i\hbar)^2\varepsilon ^2\right.\right.\right.\\ &\left.\left.\left.-\frac{2304 s_1^6+2400 s_2 s_1^3+105 s_3 s_1-500 s_2^2 }{7200}(i\hbar)^3\right)\!u_2^3\! +\!\left(-\frac{s_1^2}{420} \varepsilon ^6-\frac{96 s_1^3+5 s_2}{2520}i \hbar \varepsilon^4\right.\right.\right.\\ &\left.\left.\left.\left.- \frac{24 s_1^4+5 s_2 s_1}{105}(i\hbar)^2\varepsilon^2-\frac{4608 s_1^5+2400 s_2 s_1^2+35 s_3 }{8400}(i\hbar ^3)\right)u_3^2\right)\gamma^3\right.\!+\!\!\ O\left(\gamma^4\right)\right]\mathrm{d}x. \end{split} \end{equation*} 6 Geometric Formula for the DR Correlators The goal of this section is to prove a geometric formula for the DR correlators. In Section 6.1 we recall the construction of these correlators from [2]. In Section 6.2 we introduce certain cohomology classes in |${\overline{{\mathcal{M}}}}_{g,n}$|⁠. They are used in the formulation of the geometric formula for the DR correlators in Section 6.3. In Section 6.4 we collect main formulas with the DR cycles and then use them in Section 6.5 for the proof of the geometric formula. Let |$c_{g,n}\colon V^{\otimes n}\to H^{\textrm{even}}({\overline{{\mathcal{M}}}}_{g,n},{\mathbb{C}})$| be an arbitrary CohFT, where |$V$| is an |$N$|-dimensional vector space, |$\eta $| is its metric tensor, and |$e_1,\ldots ,e_N$| is a basis in |$V$| such that |$e_1$| is the unit. 6.1 DR correlators Here we briefly recall the construction of the DR correlators from [2]. Define differential polynomials |$h^{\textrm{DR}}_{\alpha ,d}\in \widehat{{\mathcal{A}}}^{[0]}_N$|⁠, |$d\ge -1$|⁠, by \begin{equation*} h^{\textrm{DR}}_{\alpha,d}:=\frac{\delta\overline g_{\alpha,d+1}}{\delta u^1}. \end{equation*} For |$1\le \alpha ,\beta \le N$|⁠, and |$p,q\ge 0$| there exists a unique differential polynomial |$\Omega _{\alpha ,p;\beta ,q}^{\textrm{DR}}\in \widehat{{\mathcal{A}}}^{[0]}_N$| such that \begin{equation*} {\partial}_x\Omega_{\alpha,p;\beta,q}^{\textrm{DR}}=\frac{{\partial} h^{\textrm{DR}}_{\alpha,p-1}}{{\partial} t^\beta_q}=\left\{h^{\textrm{DR}}_{\alpha,p-1},\overline g_{\beta,q}\right\}_{\eta{\partial}_x}\quad\textrm{and}\quad\left.\Omega^{\textrm{DR}}_{\alpha,p;\beta,q}\right|_{u^*_*=0}=0. \end{equation*} The string solution |$(u^{\textrm{str}})^\alpha (x,t^*_*,\varepsilon )$| of the DR hierarchy is specified by the initial condition \begin{equation*} \left.(u^{\textrm{str}})^\alpha\right|_{t^*_*=0}=\delta^{\alpha,1}x. \end{equation*} Let |$(u^{\textrm{str}})^\gamma _n:={\partial }_x^n(u^{\textrm{str}})^\gamma $|⁠. Then there exists a unique power series |$F^{\textrm{DR}}(t^*_*,\varepsilon )\in{\mathbb{C}}[[t^*_*,\varepsilon ^2]]$| such that \begin{align} \frac{{\partial}^2 F^{\textrm{DR}}}{{\partial} t^\alpha_p{\partial} t^\beta_q}&=\left.\left(\left.\Omega^{\textrm{DR}}_{\alpha,p;\beta,q}\right|_{u^\gamma_n=(u^{\textrm{str}})^\gamma_n}\right)\right|_{x=0},\notag \\ \frac{{\partial} F^{\textrm{DR}}}{{\partial} t^1_0}&=\sum_{n\ge 0}t^\alpha_{n+1}\frac{{\partial} F^{\textrm{DR}}}{{\partial} t^\alpha_n}+\frac{1}{2}\eta_{\alpha\beta}t^\alpha_0 t^\beta_0, \end{align} (6.1) \begin{align} &\frac{{\partial} F^{\textrm{DR}}}{{\partial} t^1_1}=\sum_{n\ge 0}t^\alpha_n\frac{{\partial} F^{\textrm{DR}}}{{\partial} t^\alpha_n}+\varepsilon\frac{{\partial} F^{\textrm{DR}}}{{\partial}\varepsilon}-2F^{\textrm{DR}}+\varepsilon^2\frac{N}{24},\\& \left.\textrm{Coef}_{\varepsilon^2}F^{\textrm{DR}}\right|_{t^*_*=0}=0.\notag \end{align} (6.2) We see that the 1st equation here determines |$F^{\textrm{DR}}$| uniquely up to constant and linear terms in the variables |$t^\alpha _p$|⁠. The other equations fix this ambiguity. The power series |$F^{\textrm{DR}}$| is called the DR potential. Let \begin{equation*} F^{\textrm{DR}}\big(t^*_*,\varepsilon\big)=\sum_{g\ge 0}\varepsilon^{2g}F^{\textrm{DR}}_g(t^*_*). \end{equation*} The DR correlators |$\left <\tau _{d_1}(e_{\alpha _1})\ldots \tau _{d_n}(e_{\alpha _n})\right>_g^{\textrm{DR}}$| are defined as the coefficients of the expansion of |$F^{\textrm{DR}}_g$|⁠: \begin{equation*} F^{\textrm{DR}}_g=\sum_{n\ge 0}\sum_{d_1,\ldots,d_n\ge 0}\left<\tau_{d_1}(e_{\alpha_1})\ldots\tau_{d_n}(e_{\alpha_n})\right>_g^{\textrm{DR}}\frac{t^{\alpha_1}_{d_1}\ldots t^{\alpha_n}_{d_n}}{n!}. \end{equation*} In [2, Sections 6.6 and 6.7] we proved that a DR correlator |$\left <\tau _{d_1}(e_{\alpha _1})\ldots \tau _{d_n}(e_{\alpha _n})\right>_g^{\textrm{DR}}$| vanishes unless \begin{equation*} 2g-2+n>0\quad\textrm{and}\quad 2g-1\le\sum d_i\le 3g-3+n. \end{equation*} 6.2 Stable trees and cohomology classes in |${\overline{{\mathcal{M}}}}_{g,n}$| In this section we collect notations and definitions related to stable graphs that will be needed for the formulation of our geometric formula for the DR correlators. We will use the notations from [24, Sections 0.2 and 0.3]. By stable tree we mean a stable graph \begin{equation*} \Gamma=(V,H,L,g\colon V\to{\mathbb{Z}}_{\ge 0},v\colon H\to V, \iota\colon H\to H), \end{equation*} which is a tree. Let |$H^e(\Gamma ):=H(\Gamma )\backslash L(\Gamma )$|⁠. A path in |$\Gamma $| is a sequence of pairwise distinct vertices |$v_1,v_2,\ldots ,v_k\in V$|⁠, |$v_i\ne v_j$| for |$i\ne j$|⁠, such that for any |$1\le i\le k-1$| the vertices |$v_i$| and |$v_{i+1}$| are connected by an edge. For a vertex |$v\in V(\Gamma )$| define a number |$r(v)$| by \begin{equation*} r(v):=2g(v)-2+n(v). \end{equation*} A stable rooted tree is a pair |$(\Gamma ,v_0)$|⁠, where |$\Gamma $| is a stable tree and |$v_0\in V(\Gamma )$|⁠. The vertex |$v_0$| is called the root. Denote by |$H_+(\Gamma )$| the set of half-edges of |$\Gamma $| that are directed away from the root |$v_0$|⁠. Clearly, |$L(\Gamma )\subset H_+(\Gamma )$|⁠. Let |$H^e_+(\Gamma ):=H_+(\Gamma )\backslash L(\Gamma )$|⁠. A vertex |$w$| is called a descendant of a vertex |$v$| if |$v$| is on the unique path from the root |$v_0$| to |$w$|⁠. Note that according to our definition the vertex |$v$| is a descendant of itself. Denote by |$\textrm{Desc}[v]$| the set of all descendants of |$v$|⁠. Let |$g\ge 0$| and |$m,n\ge 1$|⁠. Denote by |$\textrm{ST}^m_{g,n+1}$| the set of stable trees of genus |$g$| with |$m$| vertices and with |$n+1$| legs marked by numbers |$0,1,\ldots ,n$|⁠. For a stable tree |$\Gamma \in \textrm{ST}^m_{g,n+1}$| denote by |$l_i(\Gamma )$| the leg in |$\Gamma $| that is marked by |$i$|⁠. We will always choose the vertex |$v(l_0(\Gamma ))$| as a root of |$\Gamma $|⁠. In this way a stable tree from |$\textrm{ST}^m_{g,n+1}$| automatically becomes a stable rooted tree. For a leg |$l\in L(\Gamma )$| denote by |$0\le i(l)\le n$| its marking. Consider a stable tree |$\Gamma \in \textrm{ST}^m_{g,n+1}$|⁠. We have the associated moduli space \begin{equation*} {\overline{{\mathcal{M}}}}_{\Gamma}:=\prod_{v\in V}{\overline{{\mathcal{M}}}}_{g(v),n(v)} \end{equation*} and the canonical morphism \begin{equation*} \xi_\Gamma\colon{\overline{{\mathcal{M}}}}_{\Gamma}\to{\overline{{\mathcal{M}}}}_{g(\Gamma),|L(\Gamma)|}. \end{equation*} Consider integers |$a_0,a_1,\ldots ,a_n$| such that |$a_0+a_1+\ldots +a_n=0$|⁠. To each half-edge |$h\in H(\Gamma )$| we assign an integer |$a(h)$| in such a way that the following conditions hold: a) If |$h\in L(\Gamma )$|⁠, then |$a(h)=a_{i(l)}$|⁠; b) If |$h\in H^e(\Gamma )$|⁠, then |$a(h)+a(\iota (h))=0$|⁠; c) For any vertex |$v\in V(\Gamma )$|⁠, we have |$\sum _{h\in H[v]}a(h)=0$|⁠. Since the graph |$\Gamma $| is a tree, it is easy to see that such a function |$a\colon H(\Gamma )\to{\mathbb{Z}}$| exists and is uniquely determined by the numbers |$a_0,a_1,\ldots ,a_n$|⁠. For each moduli space |${\overline{{\mathcal{M}}}}_{g(v),n(v)}$|⁠, |$v\in V(\Gamma )$|⁠, the numbers |$a(h)$|⁠, |$h\in H[v]$|⁠, define the DR cycle \begin{equation*} \textrm{DR}_{g(v)}\left((a(h))_{h\in H[v]}\right)\in H^{2g(v)}({\overline{{\mathcal{M}}}}_{g(v),n(v)},{{\mathbb{Q}}}). \end{equation*} If we multiply all these cycles, we get the class \begin{equation*} \prod_{v\in V(\Gamma)}\textrm{DR}_{g(v)}\left((a(h))_{h\in H[v]}\right)\in H^{2g}({\overline{{\mathcal{M}}}}_\Gamma,{{\mathbb{Q}}}). \end{equation*} We define a class |$\textrm{DR}_\Gamma (a_0,a_1,\ldots ,a_n)\in H^{2(g+m-1)}({\overline{{\mathcal{M}}}}_{g,n+1},{{\mathbb{Q}}})$| by \begin{equation*} \textrm{DR}_\Gamma(a_0,a_1,\ldots,a_n):=\left(\prod_{h\in H^e_+(\Gamma)}a(h)\right)\cdot \xi_{\Gamma*}\left(\prod_{v\in V(\Gamma)}\textrm{DR}_{g(v)}\left((a(h))_{h\in H[v]}\right)\right). \end{equation*} Note that in the case when the valency of some vertex |$v$| in |$\Gamma $| is equal to one, the class |$\textrm{DR}_\Gamma (a_0,a_1,\ldots ,a_n)$| is equal to zero. This happens because, if |$h$| is the half-edge incident to |$v$|⁠, then, obviously, |$a(h)=0$|⁠. From Hain’s formula [17] it follows that for an arbitrary stable tree |$\Gamma \in \textrm{ST}^m_{g,n+1}$| the class \begin{equation*} \lambda_g\textrm{DR}_\Gamma\left(-\sum_{i=1}^n a_i,a_1,\ldots,a_n\right)\in H^{2(2g+m-1)}({\overline{{\mathcal{M}}}}_{g,n+1},{{\mathbb{Q}}}) \end{equation*} is a polynomial in |$a_1,\ldots ,a_n$| homogeneous of degree |$2g+m-1$|⁠. For a stable tree |$\Gamma \in \textrm{ST}_{g,n+1}^m$| define a combinatorial coefficient |$C(\Gamma )$| by \begin{equation*} C(\Gamma):=\prod_{v\in V(\Gamma)}\frac{r(v)}{\sum_{\widetilde v\in\textrm{Desc}[v]}r(\widetilde v)}. \end{equation*} 6.3 Geometric formula for the correlators Recall that a DR correlator |$\left <\tau _{d_1}(e_{\alpha _1})\ldots \tau _{d_n}(e_{\alpha _n})\right>^{\textrm{DR}}_g$| vanishes unless |$\sum d_i\ge 2g-1$| (see [2, Section 6.7]). Theorem 6.1. Suppose |$g\ge 0$|⁠, |$n\ge 1$|⁠, and |$2g-2+n>0$|⁠. Let |$d\ge 2g-1$| and |$1\le \alpha _1,\ldots ,\alpha _n\le N$|⁠. Then we have the following equality of polynomials in |$a_1,\ldots ,a_n$| of degree |$d$|⁠: \begin{align} &\sum_{\substack{d_1,\ldots,d_n\ge 0\\\sum d_i=d}}\left<\tau_{d_1}(e_{\alpha_1})\ldots\tau_{d_n}(e_{\alpha_n})\right>^{\textrm{DR}}_ga_1^{d_1}\ldots a_n^{d_n}=\nonumber\\ &\qquad=\frac{1}{\sum a_i}\sum_{\Gamma\in\textrm{ST}^{d-2g+2}_{g,n+1}}C(\Gamma)\int_{{\overline{{\mathcal{M}}}}_{g,n+1}}\textrm{DR}_\Gamma\left(-\sum a_i,a_1,\ldots,a_n\right)\lambda_g c_{g,n+1}\big(e_1\otimes\otimes_{i=1}^n e_{\alpha_i}\big). \end{align} (6.3) Note that in the case |$d=2g-1$| formula (3) becomes particularly simple: \begin{multline} \sum_{\substack{d_1,\ldots,d_n\ge 0\\\sum d_i=2g-1}}\left<\tau_{d_1}(e_{\alpha_1})\ldots\tau_{d_n}(e_{\alpha_n})\right>^{\textrm{DR}}_ga_1^{d_1}\ldots a_n^{d_n}=\\ =\frac{1}{\sum a_i}\int_{{\overline{{\mathcal{M}}}}_{g,n+1}}\textrm{DR}_g\left(-\sum a_i,a_1,\ldots,a_n\right)\lambda_g c_{g,n+1}\big(e_1\otimes\otimes_{i=1}^n e_{\alpha_i}\big). \end{multline} (6.4) We will prove Theorem 6.1 in Section 6.5. 6.4 Main formulas with the DR cycles Here we collect main formulas with the DR cycles that we will use later. 6.4.1 Double ramification cycle and fundamental class Suppose |$\pi \colon{\overline{{\mathcal{M}}}}_{g,n+g}\to{\overline{{\mathcal{M}}}}_{g,n}$| is the forgetful map that forgets the last |$g$| marked points. Then we have [8, Example 3.7] \begin{equation} \pi_*\textrm{DR}_g(a_1,\ldots,a_{n+g})=g!a_{n+1}^2\ldots a_{n+g}^2\big[{\overline{{\mathcal{M}}}}_{g,n}\big]. \end{equation} (6.5) 6.4.2 Divisibility properties Let |$g,n\ge 1$|⁠. Suppose |$\pi \colon{\overline{{\mathcal{M}}}}_{g,n+1}\to{\overline{{\mathcal{M}}}}_{g,n}$| is the forgetful map that forgets the last marked point. Then the polynomial class \begin{equation*} \left.\pi_*\textrm{DR}_g\left(-\sum a_i,a_1,a_2,\ldots,a_n\right)\right|_{{{\mathcal{M}}}_{g,n}^{\textrm{ct}}}\in H^{2g-2}\big({{\mathcal{M}}}^{\textrm{ct}}_{g,n},{{\mathbb{Q}}}\big) \end{equation*} is divisible by |$a_n^2$|⁠. Suppose |$g,n,m\ge 1$|⁠. Then we have [2, Section 5.1] \begin{align} &\int_{\textrm{DR}_g(-\sum a_i-\sum b_j,a_1,\ldots,a_n,b_1,\ldots,b_m)}\lambda_g\psi_2^{\mathrm{\,d}} c_{g,n+m+1}\big(\otimes_{i=1}^{n+1} e_{\alpha_i}\otimes e_1^m\big)=\nonumber\\ =\!&\begin{cases} \int_{\textrm{DR}_g(-\sum a_i-\sum b_j,a_1+\sum b_j,a_2,\ldots,a_n)}\lambda_g\psi_2^{\mathrm{\,d}-m}c_{g,n+1}(\otimes_{i=1}^{n+1}e_{\alpha_i}) +O(b_1^2)+\ldots+O(b_m^2),&\!\textrm{if}\ \mathrm{d}\ge m;\\ O(b_1^2)+\ldots+O\big(b_m^2\big),&\!\textrm{if}\ \mathrm{d}<m. \end{cases} \end{align} (6.6) 6.4.3 DR cycle times a |$\psi $|-class Here we recall the formula from [8] for the product of the DR cycle with a |$\psi $|-class. Denote by \begin{equation*} \textrm{gl}_k\colon{\overline{{\mathcal{M}}}}_{g_1,n_1+k}\times{\overline{{\mathcal{M}}}}_{g_2,n_2+k}\to{\overline{{\mathcal{M}}}}_{g_1+g_2+k-1,n_1+n_2} \end{equation*} the gluing map that corresponds to gluing a curve from |${\overline{{\mathcal{M}}}}_{g_1,n_1+k}$| to a curve from |${\overline{{\mathcal{M}}}}_{g_2,n_2+k}$| along the last |$k$| marked points on the 1st curve and the last |$k$| marked points on the 2nd curve. Suppose |$n,m\ge k\ge 1$|⁠, and |$a_1,\ldots ,a_n$| and |$b_1,\ldots ,b_m$| are lists of integers with vanishing sums. Let \begin{align*} &\textrm{DR}_{g_1}(a_1,\ldots,a_n)\boxtimes_k\textrm{DR}_{g_2}(b_1,\ldots,b_m):=\\ =&\textrm{gl}_{k*}\big(\textrm{DR}_{g_1}(a_1,\ldots,a_n)\times\textrm{DR}_{g_2}(b_1,\ldots,b_m)\big)\in H^{2(g_1+g_2+k)}({\overline{{\mathcal{M}}}}_{g_1+g_2+k-1,n+m-2k},{{\mathbb{Q}}}). \end{align*} Let |$a_1,\ldots ,a_n$| be a list of integers with vanishing sum. Assume that |$a_s \ne 0$| for some |$1\le s\le n$|⁠. Then we have [8, Theorem 4] \begin{align} a_s\psi_s \textrm{DR}_g(a_1, \dots, a_n)&=\nonumber \\ &=\sum_{\substack{I\sqcup J=\{1,\ldots,n\}\\sum_{i\in I}a_i>0}}\sum_{p\ge 1}\sum_{\substack{g_1,g_2\ge 0\\g_1+g_2+p-1=g}}\sum_{\substack{k_1,\ldots,k_p\ge 1\\ \sum k_j=\sum_{i\in I}a_i}}\\&\quad\frac{\rho}{r}\frac{\prod_{i=1}^p k_i}{p!}\textrm{DR}_{g_1}(a_I,-k_1,\ldots,-k_p)\boxtimes_p\textrm{DR}_{g_2}(a_J,k_1,\ldots,k_p),\notag \end{align} (6.7) where |$a_I$| denotes the list |$(a_i)_{i\in I}$|⁠, |$r=2g-2+n$|⁠, and \begin{equation*} \rho= \begin{cases} 2g_2-2+|J|+p, &\text{if {$s\in I$}},\\ -(2g_1-2+|I|+p), &\text{if {$s\in J$}}. \end{cases} \end{equation*} 6.5 Proof of the geometric formula In this section we prove Theorem 6.1. The plan is the following. In Section 6.5.1 we put combinatorial definitions and constructions that we will need for the proof. In Section 6.5.2 we show how to use the combinatorial map |$\phi $| defined in Section 6.5.1 in order to simplify the geometric formula for the DR correlators from our previous work [2]. From this simplification we will see that Theorem 6.1 follows from a certain relation in the cohomology of the moduli space of curves. This relation is proved in Section 6.5.3. 6.5.1 More about stable trees In this section we collect a combinatorial material related to stable trees that we will need in the proof of the geometric formula. We will partly repeat the material from [2, Section 6.6.2]. Let |$\Gamma \in \textrm{ST}^m_{g,n+1}$|⁠. Introduce the following notations: \begin{equation*} L^{\prime}(\Gamma):=L(\Gamma)\backslash\{l_0(\Gamma)\},\qquad H_+^{\prime}(\Gamma):=H_+(\Gamma)\backslash\{l_0(\Gamma)\}. \end{equation*} Clearly, for any vertex |$v\in V(\Gamma )$| the set |$H[v]\backslash H_+^{\prime}[v]$| consists of exactly one element. The stable tree |$\Gamma $| will be called admissible if the following two conditions are satisfied: a) For any vertex |$v\in V(\Gamma )$| we have |$|L^{\prime}[v]|\ge 1$|⁠; b) For any two distinct vertices |$v_1,v_2\in V(\Gamma )$| such that |$v_2$| is a descendant of |$v_1$| we have \begin{equation*} \min_{l\in L^{\prime}[v_1]}i(l)<\min_{l\in L^{\prime}[v_2]}i(l). \end{equation*} The set of all admissible stable trees will be denoted by |$\textrm{AST}^m_{g,n+1}\subset \textrm{ST}^m_{g,n+1}$|⁠. Consider a stable tree |$\Gamma \in \textrm{ST}^m_{g,n+1}$| a vertex |$v\in V(\Gamma )$| and a half-edge |$h\in H^e_+[v]$|⁠. Denote by |$\Gamma _h$| the stable rooted tree formed by the descendants of |$v(\iota (h))$| and all half-edges incident to them together with the vertex |$v(\iota (h))$| as a root (see Figure 1). Fig. 1. View largeDownload slide Fig. 1. View largeDownload slide Let us define splitting and contracting operations on stable trees. Consider a stable tree |$\Gamma \in \textrm{ST}^m_{g,n+1}$|⁠, a vertex |$v\in V(\Gamma )$|⁠, a subset |$I\subset H^{\prime}_+[v],$| and an integer |$0\le g_1\le g(v)$| such that |$2g_1+|I|>0$| and |$2g_2+|I^c|-1>0$|⁠, where |$I^c:=H^{\prime}_+[v]\backslash I$| and |$g_2:=g(v)-g_1$|⁠. We define a stable tree |$\textrm{Spl}(\Gamma ,v,g_1,I)\in \textrm{ST}^{m+1}_{g,n+1}$| in the following way. We split the vertex |$v$| in two vertices of genera |$g_1$| and |$g_2$|⁠, respectively, connect them by an edge, attach the half-edge from |$H[v]\backslash H_+^{\prime}[v]$| to the 1st vertex, and then attach the half-edges from the set |$I$| to the 1st vertex and the half-edges from the set |$I^c$| to the 2nd vertex (see Figure 2). This is the splitting operation. Fig. 2. View largeDownload slide Splitting operation Fig. 2. View largeDownload slide Splitting operation Let us define a contracting operation. Suppose |$m\ge 2$|⁠. Let |$\Gamma \in \textrm{ST}^m_{g,n+1}$|⁠, |$v\in V(\Gamma ),$| and |$h\in H^e[v]$|⁠. A stable tree |$\textrm{Con}(\Gamma ,v,h)\in \textrm{ST}^{m-1}_{g,n+1}$| is defined simply by contracting the edge corresponding to the half-edges |$h$| and |$\iota (h)$|⁠. A modified stable tree is a stable tree |$\Gamma $| where we split the set of legs in two subsets: the set of legs of the 1st type and the set of legs of the 2nd type, where we require that each vertex of the tree is incident to exactly one leg of the 2nd type. The set of legs of the 1st type will be denoted by |$L_1(\Gamma )$| and the set of legs of the 2nd type will be denoted by |$L_2(\Gamma )$|⁠. For |$g\ge 0$| and |$m,n\ge 1$| denote by |$\textrm{MST}^m_{g,n+1}$| the set of modified stable trees of genus |$g$| with |$m$| vertices and with |$(m+n+1)$| legs. We mark the legs of 1st type by numbers |$0,1,\ldots ,n$| and the legs of the 2nd type by numbers |$n+1,\ldots ,n+m$|⁠. In the same way, as for usual stable trees, for a modified stable tree |$\Gamma \in \textrm{MST}^m_{g,n+1}$| we use the notation |$l_i(\Gamma )$| for the leg marked by |$i$| and the notation |$i(l)$| for the marking of a leg |$l\in L(\Gamma )$|⁠. We will always choose the vertex |$v(l_0(\Gamma ))$| as a root of |$\Gamma $|⁠. In this way a modified stable tree from |$\textrm{MST}^m_{g,n+1}$| automatically becomes a stable rooted tree. An example of a modified stable tree from |$\textrm{MST}^m_{g,n+1}$| is shown on the left-hand side of Figure 3. The legs of the 2nd type are drawn by double lines. The reader can see that in our example |$n=8$| and |$m=4$|⁠. Fig. 3. View largeDownload slide Map |$\phi \colon \textrm{MST}^{m,e}_{g,n+1}\to \textrm{ST}^{m-e}_{g,m+1}$| Fig. 3. View largeDownload slide Map |$\phi \colon \textrm{MST}^{m,e}_{g,n+1}\to \textrm{ST}^{m-e}_{g,m+1}$| Consider a modified stable tree |$\Gamma \in \textrm{MST}^m_{g,n+1}$|⁠. Define a function |$p\colon V(\Gamma )\to \{1,\ldots ,m\}$| by |$p(v):=i-n$|⁠, where |$i$| is the marking of a unique leg of the 2nd type incident to |$v$|⁠. The modified stable tree |$\Gamma $| is called admissible, if for any two distinct vertices |$v_1,v_2\in V(\Gamma )$| such that |$v_2$| is a descendant of |$v_1$|⁠, we have |$p(v_2)>p(v_1)$|⁠. The subset of admissible modified stable trees will be denoted by |$\textrm{AMST}^m_{g,n+1}\subset \textrm{MST}^m_{g,n+1}$|⁠. Note that the modified stable tree on the left-hand side of Figure 3 is admissible. Consider a modified stable tree |$\Gamma \in \textrm{MST}^m_{g,n+1}$| and integers |$a_0,a_1,\ldots ,a_n$| with vanishing sum. Define a function |$a\colon H(\Gamma )\to{\mathbb{Z}}$| by the properties a) If |$h\in L_1(\Gamma )$|⁠, then |$a(h)=a_{i(h)}$|⁠; b) If |$h\in L_2(\Gamma )$|⁠, then |$a(h)=0$|⁠; c) If |$h\in H^e(\Gamma )$|⁠, then |$a(h)+a(\iota (h))=0$|⁠; d) For any vertex |$v\in V(\Gamma )$|⁠, we have |$\sum _{h\in H[v]}a(h)=0$|⁠. In the same way, as in Section 6.2, we define the class |$\textrm{DR}_\Gamma (a_0,\ldots ,a_n)\in H^{2(g+m-1)}$||$({\overline{{\mathcal{M}}}}_{g,m+n+1},{{\mathbb{Q}}})$|⁠. Suppose |$\Gamma \in \textrm{MST}^m_{g,n+1}$|⁠. It is useful to introduce the notations \begin{align} &L^{\prime}_1(\Gamma):=L_1(\Gamma)\backslash\{l_0(\Gamma)\},\notag\\ &H^{\prime}(\Gamma):=H^e(\Gamma)\cup\{l_0(\Gamma)\}. \end{align} (6.8) Clearly, for any vertex |$v\in V(\Gamma )$| we have |$|H[v]\backslash L_1^{\prime}[v]|\ge 2$|⁠. A vertex |$v\in V(\Gamma )$| will be called exceptional, if |$g(v)=0$| and |$|H[v]\backslash L_1^{\prime}[v]|=2$|⁠. Otherwise, it will be called regular. The reader can see that one vertex in the graph on the left-hand side of Figure 3 is exceptional. Denote by |$V^{\textrm{exc}}(\Gamma )$| and |$V^{\textrm{reg}}(\Gamma )$| the sets of exceptional and regular vertices in |$\Gamma ,$| respectively. An edge in |$\Gamma $| that is incident to an exceptional vertex will be called exceptional. The set of modified stable graphs with |$e$| exceptional vertices will be denoted by |$\textrm{MST}^{m,e}_{g,n+1}\subset \textrm{MST}^m_{g,n+1}$|⁠. Consider |$g\ge 0$| and |$m,n\ge 1$| such that |$2g+m-1>0$|⁠. Note that for any modified stable tree |$\Gamma \in \textrm{MST}^m_{g,n+1}$| the root is regular. So we have |$|V^{\textrm{exc}}(\Gamma )|\le m-1$|⁠. Let |$0\le e\le m-1$|⁠. Let us define a map \begin{equation*} \phi\colon\textrm{MST}^{m,e}_{g,n+1}\to\textrm{ST}^{m-e}_{g,m+1} \end{equation*} in the following way. Suppose |$\Gamma \in \textrm{MST}^{m,e}_{g,n+1}$|⁠. We construct the graph |$\phi (\Gamma )$| by contracting all exceptional edges and then by throwing away all legs from |$L_1^{\prime}(\Gamma )$|⁠. It is easy to see that the graph |$\phi (\Gamma )$| has |$m-e$| vertices and |$m+1$| legs. We only have to specify how we mark them. A leg |$l$| in |$\phi (\Gamma )$| corresponds to some leg |$l_i(\Gamma )$| in |$\Gamma $|⁠, where |$i=0$| or |$n+1\le i\le m+n$|⁠. If |$i=0$|⁠, then we mark |$l$| by |$0$| and if |$n+1\le i\le m+n$|⁠, then we mark |$l$| by |$i-n$|⁠. An example of the action of the map |$\phi $| is shown in Figure 3. It is easy to see that for any |$\Gamma \in \textrm{AMST}^{m,e}_{g,n+1}$| we have |$\phi (\Gamma )\in \textrm{AST}^{m-e}_{g,m+1}$|⁠. So we have the map \begin{equation*} \phi\colon\textrm{AMST}^{m,e}_{g,n+1}\to\textrm{AST}^{m-e}_{g,m+1}. \end{equation*} 6.5.2 Map |$\phi $| and integrals over DR cycles The string equation (6.1) for the DR correlators implies that \begin{equation*} \sum_{\substack{d_1,\ldots,d_n\ge 0\\\sum d_i=d}}\left<\tau_{d_1}(e_{\alpha_1})\ldots\tau_{d_n}(e_{\alpha_n})\right>^{\textrm{DR}}_g\prod_{i=1}^n a_i^{d_i}\!=\frac{1}{\sum a_i}\sum_{\substack{d_1,\ldots,d_n\ge 0\\\sum d_i=d+1}}\left<\tau_0(e_1)\tau_{d_1}(e_{\alpha_1})\ldots\tau_{d_n}(e_{\alpha_n})\right>^{\textrm{DR}}_g\prod_{i=1}^n a_i^{d_i}. \end{equation*} Therefore, formula (6.3) is equivalent to \begin{multline} \sum_{\substack{d_1,\ldots,d_n\ge 0\\\sum d_i=d+1}}\left<\tau_0(e_1)\tau_{d_1}(e_{\alpha_1})\ldots\tau_{d_n}(e_{\alpha_n})\right>^{\textrm{DR}}_ga_1^{d_1}\ldots a_n^{d_n}=\\ =\sum_{\Gamma\in\textrm{ST}^{d-2g+2}_{g,n+1}}C(\Gamma)\int_{{\overline{{\mathcal{M}}}}_{g,n+1}}\textrm{DR}_\Gamma\left(-\sum a_i,a_1,\ldots,a_n\right)\lambda_g c_{g,n+1}\big(e_1\otimes\otimes_{i=1}^n e_{\alpha_i}\big). \end{multline} (6.9) In [2, Section 6.6.3] we proved that the correlator |$\left <\tau _0(e_1)\tau _{d_1}(e_{\alpha _1})\ldots \tau _{d_n}(e_{\alpha _n})\right>^{\textrm{DR}}_g$| is equal to the coefficient of |$b_1 b_2\ldots b_{2g+n-1}$| in the polynomial \begin{equation*} \frac{1}{(2g+n-1)!}\sum_{\Gamma\in\textrm{AMST}^n_{g,2g+n}}\int_{\textrm{DR}_\Gamma(-\sum b_i,b_1,\ldots,b_{2g+n-1})}\lambda_g c_{g,2g+2n}\big(e_1^{2g+n}\otimes\otimes_{i=1}^n e_{\alpha_i}\big)\prod_{i=1}^n\psi_{2g+n-1+i}^{\mathrm{\,d}_i}. \end{equation*} On the left-hand side of Figure 4 we schematically represent an example of an integral from this formula. Note that the modified stable tree |$\Gamma $| in this example coincides with the modified stable tree on the left-hand side of Figure 3.We have \begin{align} &\int_{\textrm{DR}_\Gamma(-\sum b_i,b_1,\ldots,b_{2g+n-1})}\lambda_g c_{g,2g+2n}\big(e_1^{2g+n}\otimes\otimes_{j=1}^n e_{\alpha_j}\big)\prod_{j=1}^n\psi_{2g+n-1+j}^{\mathrm{\,d}_j}=\\ =&\prod_{h\in H^e_+(\Gamma)}b(h)\sum_{\nu\colon H^e(\Gamma)\to\{1,\ldots,N\}}\eta^{\nu(h)\nu(\iota(h))}\times\notag\\ &\times\prod_{v\in V(\Gamma)}\int_{\textrm{DR}_{g(v)}\left(0,(b(l))_{l\in L_1[v]},(b(h))_{h\in H^e[v]}\right)}\lambda_{g(v)}\psi_0^{\mathrm{\,d}_{p(v)}}c_{g,|H[v]|}\big(e_{\alpha_{p(v)}}\otimes e_1^{|L_1[v]|}\otimes\otimes_{h\in H^e[v]}e_{\nu(h)}\big).\notag \end{align} (6.10) In order to simplify our formulas a little bit, it is convenient to use the notation (6.8) and set |$\nu (l_0(\Gamma )):=1$|⁠. Then we can rewrite formula (6.10) in the following way: \begin{align} &\int_{\textrm{DR}_\Gamma(-\sum b_i,b_1,\ldots,b_{2g+n-1})}\lambda_g c_{g,2g+2n}\big(e_1^{2g+n}\otimes\otimes_{j=1}^n e_{\alpha_j}\big)\prod_{j=1}^n\psi_{2g+n-1+j}^{\mathrm{\,d}_j}=\\ =&\prod_{h\in H^e_+(\Gamma)}b(h)\sum_{\nu\colon H^e(\Gamma)\to\{1,\ldots,N\}}\eta^{\nu(h)\nu(\iota(h))}\times\notag\\ &\times\prod_{v\in V(\Gamma)}\int_{\textrm{DR}_{g(v)}\left(0,(b(l))_{l\in L^{\prime}_1[v]},(b(h))_{h\in H^{\prime}[v]}\right)}\lambda_{g(v)}\psi_0^{\mathrm{\,d}_{p(v)}}c_{g,|H[v]|}\big(e_{\alpha_{p(v)}}\otimes e_1^{|L^{\prime}_1[v]|}\otimes\otimes_{h\in H^{\prime}[v]}e_{\nu(h)}\big).\notag \end{align} (6.11) Suppose that |$v\in V^{\textrm{reg}}(\Gamma )$|⁠. Then equation (6.6) implies that the integral \begin{equation} \int_{\textrm{DR}_{g(v)}\left(0,(b(l))_{l\in L^{\prime}_1[v]},(b(h))_{h\in H^{\prime}[v]}\right)}\lambda_{g(v)}\psi_0^{\mathrm{\,d}_{p(v)}}c_{g,|H[v]|}\big(e_{\alpha_{p(v)}}\otimes e_1^{|L^{\prime}_1[v]|}\otimes\otimes_{h\in H^{\prime}[v]}e_{\nu(h)}\big) \end{equation} (6.12) is equal to \begin{equation*} \int_{\textrm{DR}_{g(v)}\left(\sum_{l\in L_1^{\prime}[v]}b(l),(b(h))_{h\in H^{\prime}[v]}\right)}\lambda_{g(v)}\psi_0^{\mathrm{\,d}_{p(v)}-|L_1^{\prime}[v]|}c_{g,|H[v]|-|L_1^{\prime}[v]|}\big(e_{\alpha_{p(v)}}\otimes\otimes_{h\in H^{\prime}[v]}e_{\nu(h)}\big)+\sum_{i=1}^{2g+n-1}O\big(b_i^2\big) \end{equation*} in the case |$d_{p(v)}\ge |L_1^{\prime}[v]|$| and is equal to |$\sum _{i=1}^{2g+n-1}O(b_i^2)$|⁠, if |$d_{p(v)}<|L_1^{\prime}[v]|$|⁠. Suppose that |$v\in V^{\textrm{exc}}(\Gamma )$|⁠. Then the set |$H^{\prime}[v]$| consists of only one element, |$H^{\prime}[v]=\{l\}$|⁠. The integral (6.12) is equal to |$\eta _{\alpha _{p(v)}\nu (l)}$|⁠, if |$|L^{\prime}_1[v]|=d_{p(v)}+1$|⁠, and is equal to zero otherwise. Fig. 4. View largeDownload slide Map |$\phi $| and integrals over DR cycles Fig. 4. View largeDownload slide Map |$\phi $| and integrals over DR cycles We say that an admissible modified stable tree |$\Gamma \in \textrm{AMST}^n_{g,2g+n}$| is compatible with an |$n$|-tuple of nonnegative integers |$(d_1,\ldots ,d_n)$| if the following two conditions are satisfied: a) For any |$v\in V^{\textrm{reg}}(\Gamma )$| we have |$d_{p(v)}\ge |L_1^{\prime}[v]|$|⁠. b) For any |$v\in V^{\textrm{exc}}(\Gamma )$| we have |$d_{p(v)}+1=|L_1^{\prime}[v]|$|⁠. We obtain that the coefficient of |$b_1b_2\ldots b_{2g+n-1}$| in (6.10) can be nonzero only if |$\Gamma $| is compatible with |$(d_1,\ldots ,d_n)$|⁠. Suppose that an admissible modified stable tree |$\Gamma \in \textrm{AMST}^n_{g,2g+n}$| is compatible with an |$n$|-tuple |$(d_1,\ldots ,d_n)$|⁠, where |$\sum d_i=d+1$|⁠. Then from the computations above it follows that the coefficient of |$b_1b_2\ldots b_{2g+n-1}$| in (6.10) is equal to the coefficient of |$b_1b_2\ldots b_{2g+n-1}$| in \begin{align*} &\left(\prod_{v\in V^{\textrm{exc}}(\Gamma)}\sum_{l\in L_1^{\prime}[v]}b(l)\right)\times\\ &\times\int_{\textrm{DR}_{\phi(\Gamma)}\left(-\sum b_i,\big(\sum_{l\in L_1^{\prime}[v(l_{i+2g+n-1}(\Gamma))]}b(l)\big)_{1\le i\le n}\right)}\lambda_g c_{g,n+1}\big(e_1\otimes\otimes_{i=1}^n e_{\alpha_i}\big)\prod_{v\in V^{\textrm{reg}}(\Gamma)}\psi_{p(v)}^{\mathrm{\,d}_{p(v)}-|L_1^{\prime}[v]|}. \end{align*} An example of an integral from this formula is illustrated on the right-hand side of Figure 4. It is easy to see that the coefficient of |$b_1b_2\ldots b_{2g+n-1}$| in the last expression is equal to the coefficient of |$a_1^{d_1}\ldots a_n^{d_n}$| in \begin{equation*} \left(\prod_{v\in V(\Gamma)}|L_1^{\prime}[v]|!\right)\int_{\textrm{DR}_{\phi(\Gamma)}(-\sum a_i,a_1,\ldots,a_n)}\lambda_g c_{g,n+1}\big(e_1\otimes\otimes_{i=1}^n e_{\alpha_i}\big)\prod_{v\in V^{\textrm{reg}}(\Gamma)}\big(a_{p(v)}\psi_{p(v)}\big)^{\mathrm{\,d}_{p(v)}-|L_1^{\prime}[v]|}. \end{equation*} Let |$e:=|V^{\textrm{exc}}(\Gamma )|$|⁠. Note that \begin{equation*} \sum_{p\in V^{\textrm{reg}}(\Gamma)}\big(d_{p(v)}-|L_1^{\prime}[v]|\big)=d+1-(2g+n-1-e). \end{equation*} Note also that for any |$v\in V^{\textrm{reg}}(\Gamma )$| the leg |$l_{p(v)}(\phi (\Gamma ))\in L(\phi (\Gamma ))$| satisfies the following property: \begin{equation*} p(v)=\min_{l^{\prime}\in L^{\prime}[v(l_{p(v)}(\phi(\Gamma)))]}i(l^{\prime}). \end{equation*} This motivates the following definition. For |$0\le e\le n-1$| and an admissible stable tree |$\Gamma \in \textrm{AST}^{n-e}_{g,n+1}$| define a set |$S_{\Gamma ,d}\subset{\mathbb{Z}}^n_{\ge 0}$| by \begin{equation*} S_{\Gamma,d}:=\left\{(c_1,\ldots,c_n)\in{\mathbb{Z}}_{\ge 0}^n\left| \begin{smallmatrix} \text{{$c_i=0$}, if {$i\notin\{\min_{l\in L^{\prime}[v]}i(l)\}_{v\in V(\Gamma)}$}}\\ \sum c_i=d+1-(2g+n-1-e) \end{smallmatrix}\right.\right\}. \end{equation*} We obtain the following equation: \begin{align*} &\sum_{\substack{d_1,\ldots,d_n\ge 0\\\sum d_i=d+1}}\left<\tau_0(e_1)\tau_{d_1}(e_{\alpha_1})\ldots\tau_{d_n}(e_{\alpha_n})\right>^{\textrm{DR}}_g\prod_{i=1}^n a_i^{d_i}=\\ =&\sum_{e=0}^{n-1}\sum_{\Gamma\in\textrm{AST}^{n-e}_{g,n+1}}\sum_{(c_1,\ldots,c_n)\in S_{\Gamma,d}}\int_{\textrm{DR}_\Gamma(-\sum a_i,a_1,\ldots,a_n)}\lambda_g c_{g,n+1}\big(e_1\otimes\otimes_{i=1}^n e_{\alpha_i}\big)\prod_{i=1}^n(a_i\psi_i)^{c_i}.\notag \end{align*} We can now see that the following relation in the cohomology of |${\overline{{\mathcal{M}}}}_{g,n+1}$| implies formula (6.9): \begin{multline} \sum_{e=0}^{n-1}\sum_{\Gamma\in\textrm{AST}^{n-e}_{g,n+1}}\sum_{(c_1,\ldots,c_n)\in S_{\Gamma,d}}\lambda_g\textrm{DR}_\Gamma(a_0,a_1,\ldots,a_n)\prod_{i=1}^n(a_i\psi_i)^{c_i}=\\ =\sum_{\Gamma\in\textrm{ST}^{d-2g+2}_{g,n+1}}C(\Gamma)\lambda_g\textrm{DR}_\Gamma(a_0,a_1,\ldots,a_n), \end{multline} (6.13) where |$a_0:=-\sum _{i=1}^n a_i$|⁠. This relation will be proved in the next section. 6.5.3 Relation in the cohomology of |${\overline{{\mathcal{M}}}}_{g,n}$| We prove relation (13) by the double induction on |$d$| and on |$n$|⁠. The base cases are when |$d=2g-1$| or |$n=1$|⁠. If |$d=2g-1$|⁠, then the condition |$\sum c_i=d+1-(2g+n-1-e)$| in the definition of the set |$S_{\Gamma ,d}$| immediately implies that |$e=n-1$| and that the left-hand side of (13) is equal to |$\textrm{DR}_g(a_0,\ldots ,a_n)$|⁠. The right-hand side of (13) is clearly the same. Suppose |$n=1$|⁠. Then the left-hand side of (13) is equal to \begin{equation} \lambda_g\textrm{DR}_g(-a_1,a_1)(a_1\psi_1)^{d+1-2g}, \end{equation} (6.14) while the right-hand side of (13) is equal to \begin{equation} \sum_{\Gamma\in\textrm{ST}^{d-2g+2}_{g,2}}C(\Gamma)\lambda_g\textrm{DR}_\Gamma(-a_1,a_1). \end{equation} (6.15) The class |$\textrm{DR}_\Gamma (-a_1,a_1)$| is zero unless |$\Gamma $| is a chain. Therefore, applying formula (6.7) to (6.14) |$d+1-2g$| times, we get (6.15). So, the base cases for our induction are proved. Suppose now that |$d\ge 2g$| and |$n\ge 2$|⁠. We rewrite the left-hand side of (13) in the following way: \begin{align} \sum_{e=0}^{n-1}\sum_{\Gamma\in\textrm{AST}^{n-e}_{g,n+1}}\sum_{(c_1,\ldots,c_n)\in S_{\Gamma,d}}&\lambda_g\textrm{DR}_\Gamma(a_0,\ldots,a_n)\prod_{i=1}^n(a_i\psi_i)^{c_i}=\notag\\ =&a_1\psi_1\sum_{e=0}^{n-1}\sum_{\Gamma\in\textrm{AST}^{n-e}_{g,n+1}}\sum_{(c_1,\ldots,c_n)\in S_{\Gamma,d-1}}\lambda_g\textrm{DR}_\Gamma(a_0,\ldots,a_n)\prod_{i=1}^n(a_i\psi_i)^{c_i}+ \end{align} (6.16) \begin{align} &+\sum_{e=0}^{n-1}\sum_{\Gamma\in\textrm{AST}^{n-e}_{g,n+1}}\sum_{(0,c_2,\ldots,c_n)\in S_{\Gamma,d}}\lambda_g\textrm{DR}_\Gamma(a_0,\ldots,a_n)\prod_{i=2}^n(a_i\psi_i)^{c_i}. \end{align} (6.17) By the induction assumption, expression (6.16) is equal to \begin{align} &a_1\psi_1\sum_{\Gamma\in\textrm{ST}^{d-2g+1}_{g,n+1}}C(\Gamma)\lambda_g\textrm{DR}_\Gamma(a_0,\ldots,a_n)\stackrel{\text{by (6.7)}}{=}\notag\\ =&\sum_{\Gamma\in\textrm{ST}^{d-2g+1}_{g,n+1}}C(\Gamma)\lambda_g\sum_{\substack{g_1,g_2\ge 0\\g_1+g_2=g(v(l_1(\Gamma)))}}\sum_{\substack{I\sqcup J=H_+^{\prime}[v(l_1(\Gamma))]\\l_1(\Gamma)\in J\\2g_1+|I|>0\\2g_2+|J|-1}}\frac{2g_1+|I|}{r(v(l_1(\Gamma)))}\textrm{DR}_{\textrm{Spl}(\Gamma,v(l_1(\Gamma)),g_1,I)}(a_0,\ldots,a_n) \end{align} (6.18) \begin{align} &-\sum_{\Gamma\in\textrm{ST}^{d-2g+1}_{g,n+1}}C(\Gamma)\lambda_g\sum_{\substack{g_1,g_2\ge 0\\g_1+g_2=g(v(l_1(\Gamma)))}}\sum_{\substack{I\sqcup J=H_+^{\prime}[v(l_1(\Gamma))]\\l_1(\Gamma)\in I\\2g_2+|J|-1>0}}\frac{2g_2+|J|-1}{r(v(l_1(\Gamma)))}\textrm{DR}_{\textrm{Spl}(\Gamma,v(l_1(\Gamma)),g_1,I)}(a_0,\ldots,a_n). \end{align} (6.19) Let us now analyze expression (6.17). From the definition of an admissible stable tree it immediately follows that for any |$\Gamma \in \textrm{AST}^{n-e}_{g,n+1}$| the leg |$l_1(\Gamma )$| is incident to the root of |$\Gamma $|⁠. The stable rooted tree |$\Gamma $| is obtained by attaching the stable rooted trees |$\Gamma _h$|⁠, |$h\in H^e[v(l_0(\Gamma ))]$| together with the legs from |$L[v(l_0(\Gamma ))]$| to the vertex |$v(l_0(\Gamma ))$|⁠. Note that the number of legs in each tree |$\Gamma _h$| is strictly less than |$n+1$|⁠. Therefore, the induction assumption implies that expression (6.17) is equal to \begin{equation} \sum_{\substack{\Gamma\in\textrm{ST}^{d-2g+2}_{g,n+1}\\v(l_1(\Gamma))=v(l_0(\Gamma))}}\widetilde C(\Gamma)\lambda_g\textrm{DR}_\Gamma(a_0,\ldots,a_n), \end{equation} (6.20) where \begin{equation*} \widetilde C(\Gamma):=\prod_{v\in V(\Gamma)\backslash\{v(l_0(\Gamma))\}}\frac{r(v)}{\sum_{\widetilde v\in\textrm{Desc}[v]}r(\widetilde v)}. \end{equation*} It remains to prove that the sum of (6.18), (6.19), and (6.20) is equal to the right-hand side of (13). We see that all expressions (6.18), (6.19), (6.20), and the right-hand side of (13) are sums of classes \begin{equation} \lambda_g\textrm{DR}_\Gamma(a_0,\ldots,a_n),\quad \Gamma\in\textrm{ST}^{d-2g+2}_{g,n+1}, \end{equation} (6.21) with some rational coefficients. Consider a stable tree |$\Gamma \in \textrm{ST}^{d-2g+2}_{g,n+1}$|⁠. It remains to check that the coefficients of the class (6.21) in the sum of (6.18), (6.19), (6.20), and in the right-hand side of (13) are equal. Let |$v:=v(l_1(\Gamma ))$|⁠. Introduce the notations \begin{align*} R:=&\sum_{v^{\prime}\in\textrm{Desc}[v]}r(v^{\prime}),\\ R_h:=&\sum_{v^{\prime}\in\textrm{Desc}[v(\iota(h))]}r(v^{\prime}),\quad\text{for {$h\in H^e_+[v]$}}. \end{align*} There are two cases. Case 1. Suppose |$v\ne v(l_0(\Gamma ))$|⁠. Clearly, the set |$H^e[v]\backslash H^e_+[v]$| consists of a unique element. Let us denote it by |$h_-$| and let |$\widetilde v:=v(\iota (h_-))$| (see Figure 5). Fig. 5. View largeDownload slide Fig. 5. View largeDownload slide Let \begin{align*} \widetilde{R}:=&\sum_{v^{\prime}\in\textrm{Desc}[\widetilde v]}r(v^{\prime}),\\ B:=&\prod_{v^{\prime}\in V(\Gamma)\backslash(\{v,\widetilde v\}\cup\cup_{h\in H^e_+[v]}v(\iota(h)))}\frac{r(v^{\prime})}{\sum_{v^{\prime\prime}\in\textrm{Desc}[v^{\prime}]}r(v^{\prime\prime})}. \end{align*} So the constant |$C(\Gamma )$| can be written as \begin{equation*} C(\Gamma)=B\cdot\frac{r(\widetilde v)}{\widetilde{R}}\frac{r(v)}{R}\prod_{h\in H^e_+[v]}\frac{r(v(\iota(h)))}{R_h}. \end{equation*} Clearly, the stable tree |$\Gamma $| can be obtained by splitting of the tree |$\textrm{Con}(\Gamma ,v,h_-)$|⁠. Therefore, the coefficient of the class (6.21) in (6.18) is equal to \begin{equation} \frac{r(\widetilde v)}{r(\widetilde v)+r(v)}\cdot B\cdot\frac{r(\widetilde v)+r(v)}{\widetilde{R}}\prod_{h\in H^e_+[v]}\frac{r(v(\iota(h)))}{R_h}=B\cdot\frac{r(\widetilde v)}{\widetilde{R}}\prod_{h\in H^e_+[v]}\frac{r(v(\iota(h)))}{R_h}. \end{equation} (6.22) On the other hand, for any |$h\in H^e_+[v]$| the stable tree |$\Gamma $| can be obtained by splitting of the tree |$\textrm{Con}(\Gamma ,v,h)$|⁠. Therefore, the coefficient of the class (6.21) in (6.19) is equal to \begin{align} &-\sum_{h\in H^e_+[v]}\frac{r(v(\iota(h)))}{r(v(\iota(h)))+r(v)}\cdot B\cdot\frac{r(\widetilde v)}{\widetilde{R}}\frac{r(v)+r(v(\iota(h)))}{R}\prod_{h^{\prime}\in H^e_+[v]\backslash\{h\}}\frac{r(v(\iota(h^{\prime})))}{R_{h^{\prime}}}=\notag\\ =&-B\cdot\sum_{h\in H^e_+[v]}\frac{r(\widetilde v)}{\widetilde{R}}\frac{r(v(\iota(h)))}{R}\prod_{h^{\prime}\in H^e_+[v]\backslash\{h\}}\frac{r(v(\iota(h^{\prime})))}{R_{h^{\prime}}}. \end{align} (6.23) Obviously, the class (6.21) does not appear in (6.20). Summing (6.22) and (6.23), we get \begin{equation*} B\cdot\frac{r(\widetilde v)}{\widetilde{R}}\left(\prod_{h\in H^e_+[v]}\frac{r(v(\iota(h)))}{R_h}\right)\left(1-\sum_{h\in H^e_+[v]}\frac{R_h}{R}\right)=C(\Gamma). \end{equation*} So, this case is done. Case 2. Suppose |$v=v(l_0(\Gamma ))$|⁠. Let \begin{equation*} B:=\prod_{v^{\prime}\in V(\Gamma)\backslash(\{v\}\cup\cup_{h\in H^e_+[v]}v(\iota(h)))}\frac{r(v^{\prime})}{\sum_{v^{\prime\prime}\in\textrm{Desc}[v^{\prime}]}r(v^{\prime\prime})}. \end{equation*} Therefore, \begin{equation*} C(\Gamma)=B\cdot\frac{r(v)}{R}\prod_{h\in H^e_+[v]}\frac{r(v(\iota(h)))}{R_h}. \end{equation*} It is easy to see that the class (6.21) does not appear in (6.18). By the same arguments, as in the 1st case, the class (6.21) appears in (6.19) with the coefficient \begin{align} &-\sum_{h\in H^e_+[v]}\frac{r(v(\iota(h)))}{r(v(\iota(h)))+r(v)}\cdot B\cdot\frac{r(v(\iota(h)))+r(v)}{R}\prod_{h^{\prime}\in H^e_+[v]\backslash\{h\}}\frac{r(v(\iota(h^{\prime})))}{R_{h^{\prime}}}=\notag\\ =&-B\cdot\sum_{h\in H^e_+[v]}\frac{r(v(\iota(h)))}{R}\prod_{h^{\prime}\in H^e_+[v]\backslash\{h\}}\frac{r(v(\iota(h^{\prime})))}{R_{h^{\prime}}}. \end{align} (6.24) One can easily see that the coefficient of the class (6.21) in (6.20) is equal to \begin{equation} \widetilde C(\Gamma)=B\cdot\prod_{h\in H^e[v]}\frac{r(v(\iota(h)))}{R_{h}}. \end{equation} (6.25) Summing (6.24) and (6.25), we obtain \begin{equation*} B\cdot\left(\prod_{h\in H^e_+[v]}\frac{r(v(\iota(h)))}{R_h}\right)\left(1-\sum_{h\in H^e_+[v]}\frac{R_h}{R}\right)=C(\Gamma). \end{equation*} Case 2 is also done. Relation (13) is proved and, hence, Theorem 6.1 is also proved. 7 Miura Transformation for the DZ Operator In this section we show that our strong DR/DZ equivalence conjecture [2, Section 7.3] together with formula (4) gives a simple description of a Miura transformation that should reduce the Hamiltonian operator of the DZ hierarchy to the standard form. Remarkably, the description is given purely in terms of the potential of the CohFT. The main goal of this section is to prove that this Miura transformation indeed reduces the DZ operator to the standard form. This gives a new evidence for the strong DR/DZ equivalence conjecture. In Sections 7.1 and 7.2 we briefly recall the theory of the DZ hierarchies and our strong DR/DZ equivalence conjecture from [2]. The Miura transformation for the DZ operator is given at the end of Section 7.2. The main result is proved in Section 7.3. Throughout this section we fix a semisimple CohFT |$c_{g,n}\colon V^{\otimes n}\to H^{\textrm{even}}({\overline{{\mathcal{M}}}}_{g,n},{\mathbb{C}})$| with |$\dim V=N$|⁠. 7.1 Brief recall of the DZ theory Here we recall the construction of the DZ hierarchy. We follow the approach from [5] (see also [4]). The potential of the CohFT is defined by \begin{align*} F(t^*_*,\varepsilon):=&\sum_{g\ge 0}F_g(t^*_*)\varepsilon^{2g},\\ F_g(t^*_*):=&\sum_{\substack{n\ge 0\\2g-2+n>0}}\sum_{d_1,\ldots,d_n\ge 0}\left<\tau_{d_1}(e_{\alpha_1})\ldots\tau_{d_n}(e_{\alpha_n})\right>_g \frac{t^{\alpha_1}_{d_1}\ldots t^{\alpha_n}_{d_n}}{n!}, \end{align*} where \begin{equation*} \left<\tau_{d_1}(e_{\alpha_1})\ldots\tau_{d_n}(e_{\alpha_n})\right>_g:=\int_{{\overline{{\mathcal{M}}}}_{g,n}}c_{g,n}\big(\otimes_{i=1}^n e_{\alpha_1}\big)\psi_1^{\mathrm{\,d}_1}\ldots\psi_n^{\mathrm{\,d}_n}. \end{equation*} Recall the string and the dilaton equations for |$F$|⁠: \begin{align*} &\frac{{\partial} F}{{\partial} t^1_0}=\sum_{n\ge 0}t^\alpha_{n+1}\frac{{\partial} F}{{\partial} t^\alpha_n}+\frac{1}{2}\eta_{\alpha\beta}t^\alpha_0t^\beta_0+\varepsilon^2\left<\tau_0(e_1)\right>_1,\\ &\frac{{\partial} F}{{\partial} t^1_1}=\sum_{n\ge 0}t^\alpha_n\frac{{\partial} F}{{\partial} t^\alpha_n}+\varepsilon\frac{{\partial} F}{{\partial}\varepsilon}-2F+\varepsilon^2\frac{N}{24}. \end{align*} We will use rings of differential polynomials in different variables and Miura transformations between them. We refer the reader to [2, Section 3.4] for the corresponding notations. We also refer the reader to [2, Section 3.1] for a brief review of the theory of tau-symmetric Hamiltonian hierarchies. Introduce power series |$(w^{\textrm{top}})^\alpha \in{\mathbb{C}}[[x,t^*_*,\varepsilon ]]$| by \begin{equation*} (w^{\textrm{top}})^\alpha:=\left.\eta^{\alpha\mu}\frac{{\partial}^2 F}{{\partial} t^\mu_0{\partial} t^1_0}\right|_{t^1_0\mapsto t^1_0+x}. \end{equation*} Let |$(w^{\textrm{top}})^\alpha _n:={\partial }_x^n(w^{\textrm{top}})^\alpha $|⁠. For |$k\ge 0$| denote by |${\mathbb{C}}[[t^*_*]]^{(k)}$| the vector subspace of |${\mathbb{C}}[[t^*_*]]$| spanned by monomials |$t^{\alpha _1}_{d_1}\ldots t^{\alpha _n}_{d_n}$| with |$\sum d_i\ge k$|⁠. From the string equation for |$F$| it follows that \begin{equation} \left.(w^{\textrm{top}})^\alpha_n\right|_{\varepsilon=x=0}-t^\alpha_n-\delta_{n,1}\delta^{\alpha,1}\in{\mathbb{C}}\big[\big[t^*_*\big]\big]^{(n+1)}. \end{equation} (7.1) Therefore, any power series in |$t^\alpha _n$| and |$\varepsilon $| can be expressed as a power series in |$\left (\left .(w^{\textrm{top}})^\alpha _n\right |_{x=0}-\delta _{n,1}\delta ^{\alpha ,1}\right )$| and |$\varepsilon $| in a unique way. Consider formal variables |$w^1,\ldots ,w^N$|⁠. In [5] the authors proved that for any |$1\le \alpha ,\beta \le N,$| and |$p,q\ge 0$| there exists a unique differential polynomial |$\Omega ^{\textrm{DZ}}_{\alpha ,p;\beta ,q}\in \widehat{{\mathcal{A}}}^{[0]}_{w^1,\ldots ,w^N}$| such that \begin{equation*} \Omega^{\textrm{DZ}}_{\alpha,p;\beta,q}(w^{\textrm{top}},w^{\textrm{top}}_x,\ldots;\varepsilon)=\left.\frac{{\partial}^2 F}{{\partial} t^\alpha_p{\partial} t^\beta_q}\right|_{t^1_0\mapsto t^1_0+x}. \end{equation*} In particular, |$\Omega ^{\textrm{DZ}}_{\alpha ,0;1,0}=\eta _{\alpha \mu }w^\mu $|⁠. The equations of the DZ hierarchy are given by \begin{equation} \frac{{\partial} w^\alpha}{{\partial} t^\beta_q}=\eta^{\alpha\mu}{\partial}_x\Omega^{\textrm{DZ}}_{\mu,0;\beta,q},\quad 1\le\alpha,\beta\le N,\quad q\ge 0. \end{equation} (7.2) Clearly, the series |$(w^{\textrm{top}})^\alpha $| is a solution of these equations. It is called the topological solution. The system (7.2) has a Hamiltonian structure. The Hamiltonians are given by \begin{equation} \overline h^{\textrm{DZ}}_{\alpha,p}=\int\Omega^{\textrm{DZ}}_{\alpha,p+1;1,0}\mathrm{\,d}x,\quad p\ge 0. \end{equation} (7.3) The construction of the Hamiltonian operator is more complicated. Let \begin{equation*} (v^{\textrm{top}})^\alpha:=\left.(w^{\textrm{top}})^\alpha\right|_{\varepsilon=0}. \end{equation*} Then any power series in |$t^\alpha _n$| and |$\varepsilon $| can be expressed as a power series in |$\left (\left .(v^{\textrm{top}})^\alpha _n\right |_{x=0}-\delta _{n,1}\delta ^{\alpha ,1}\right )$| and |$\varepsilon $| in a unique way. In particular, for |$g\ge 1$| we can express the function |$F_g$| as a function of |$\left .(v^{\textrm{top}})^\alpha _n\right |_{x=0}$|⁠. Then |$F_g$| depends only on |$\left .(v^{\textrm{top}})^\alpha _n\right |_{x=0}$| with |$n\le 3g-2$| (see, e.g., [5, Proposition 4]). This property is called the |$3g-2$| property. Consider formal variables |$v^1,\ldots ,v^N$|⁠. Let |${\mathcal{A}}^{\textrm{wk}}_{v^1,\ldots ,v^N}$| be the ring of formal power series in |$(v^\alpha _n-\delta ^{\alpha ,1}\delta _{n,1})$| with complex coefficients. We have a natural inclusion \begin{equation*} {\mathcal{A}}_{v^1,\ldots,v^N}\subset{\mathcal{A}}^{\textrm{wk}}_{v^1,\ldots,v^N}. \end{equation*} Let |$\widehat{{\mathcal{A}}}^{\textrm{wk}}_{v^1,\ldots ,v^N}:={\mathcal{A}}^{\textrm{wk}}_{v^1,\ldots ,v^N}\otimes{\mathbb{C}}[[\varepsilon ]]$|⁠. Clearly, there exists a unique element |$w^\alpha (v^*_*;\varepsilon )\in \widehat{{\mathcal{A}}}^{\textrm{wk}}_{v^1,\ldots ,v^N}$| such that \begin{equation*} w^\alpha\big(v^{\textrm{top}},v^{\textrm{top}}_x,\ldots;\varepsilon\big)=(w^{\textrm{top}})^\alpha. \end{equation*} We have \begin{equation} w^\alpha\big(v^*_*;\varepsilon\big)=v^\alpha+\sum_{g\ge 1}\varepsilon^{2g}f^\alpha_g\big(v^*_*\big). \end{equation} (7.4) The |$3g-2$| property implies that the function |$f^\alpha _g(v^*_*)\in{\mathcal{A}}^{\textrm{wk}}_{v^1,\ldots ,v^N}$| depends only on |$v^\gamma _n$| with |$n\le 3g$|⁠. Then formula (7.4) can be considered as a change of variables between |$v^\gamma $| and |$w^\gamma $|⁠. Define an operator |$K^{\textrm{DZ}}(v^*_*;\varepsilon )=\left ((K^{\textrm{DZ}})^{\alpha \beta }(v^*_*;\varepsilon )\right )$| by \begin{equation} (K^{\textrm{DZ}})^{\alpha\beta}\big(v^*_*;\varepsilon\big)=\sum_{p,q\ge 0}\frac{{\partial} w^\alpha(v^*_*;\varepsilon)}{{\partial} v^\mu_p}{\partial}_x^p\circ\eta^{\mu\nu}{\partial}_x\circ(-{\partial}_x)^q\circ\frac{{\partial} w^\beta(v^*_*;\varepsilon)}{{\partial} v^\nu_q}. \end{equation} (7.5) Since |$f^\alpha _g(v^*_*)$| depends only on |$v^\gamma _n$| with |$n\le 3g$|⁠, the expression on the right-hand side of (7.5) is well defined. We have \begin{equation*} (K^{\textrm{DZ}})^{\alpha\beta}\big(v^*_*;\varepsilon\big)=\sum_{i\ge 0}(K^{\textrm{DZ}})^{\alpha\beta}_i\big(v^*_*;\varepsilon\big){\partial}_x^i. \end{equation*} Let |$(K^{\textrm{DZ}})^{\alpha \beta }_i(w^*_*;\varepsilon )$| be the function |$(K^{\textrm{DZ}})^{\alpha \beta }_i(v^*_*;\varepsilon )$| expressed in the variables |$w^\gamma $| using the change of variables (7.4). We have |$(K^{\textrm{DZ}})^{\alpha \beta }_i(w^*_*;\varepsilon )\in \widehat{{\mathcal{A}}}^{\textrm{wk}}_{w^1,\ldots ,w^N}$|⁠. In [5] the authors proved that we actually have \begin{equation*} (K^{\textrm{DZ}})^{\alpha\beta}_i\big(w^*_*;\varepsilon\big)\in\widehat{{\mathcal{A}}}^{[-i+1]}_{w^1,\ldots,w^N}. \end{equation*} The operator |$K^{\textrm{DZ}}=\sum _{i\ge 0}K^{\textrm{DZ}}_i(w^*_*;\varepsilon ){\partial }_x^i$| is Hamiltonian. Together with the local functionals (7.3) it defines the Hamiltonian structure for the DZ system (7.2). Finally, the tau-structure for the DZ hierarchy is given by the differential polynomials \begin{equation*} h^{\textrm{DZ}}_{\alpha,p}=\Omega^{\textrm{DZ}}_{\alpha,p+1;1,0},\quad p\ge -1. \end{equation*} Since |$h^{\textrm{DZ}}_{\alpha ,-1}=\eta _{\alpha \mu }w^\mu $|⁠, we see that the coordinates |$w^\alpha $| are normal. 7.2 Strong DR/DZ equivalence conjecture In [2, Section 7.3] we proved that there exists a unique differential polynomial |${\mathcal{P}}\in \widehat{{\mathcal{A}}}^{[-2]}_{w^1,\ldots ,w^N}$| such that the power series |$F^{\textrm{red}}\in{\mathbb{C}}[[t^*_*,\varepsilon ]]$|⁠, defined by \begin{equation} F^{\textrm{red}}:=F+\left.{\mathcal{P}}\big(w^{\textrm{top}},w^{\textrm{top}}_x,w^{\textrm{top}}_{xx},\ldots;\varepsilon\big)\right|_{x=0}, \end{equation} (7.6) satisfies the following vanishing property: \begin{equation} \left<\tau_{d_1}(e_{\alpha_1})\ldots\tau_{d_n}(e_{\alpha_n})\right>^{\textrm{red}}_g=0,\quad\textrm{if}\quad \sum d_i\le 2g-2, \end{equation} (7.7) where |$\left <\tau _{d_1}(e_{\alpha _1})\ldots \tau _{d_n}(e_{\alpha _n})\right>^{\textrm{red}}_g$| are the coefficients of the expansion of |$F^{\textrm{red}}$|⁠: \begin{equation*} F^{\textrm{red}}(t^*_*,\varepsilon):=\sum_{g,n\ge 0}\frac{\varepsilon^{2g}}{n!}\sum_{d_1,\ldots,d_n\ge 0}\left<\tau_{d_1}(e_{\alpha_1})\ldots\tau_{d_n}(e_{\alpha_n})\right>^{\textrm{red}}_g \frac{t^{\alpha_1}_{d_1}\ldots t^{\alpha_n}_{d_n}}{n!}. \end{equation*} We called the power series |$F^{\textrm{red}}$| the reduced potential of the CohFT. We proved that the reduced potential |$F^{\textrm{red}}$| satisfies the string and the dilaton equations \begin{align*} &\frac{{\partial} F^{\textrm{red}}}{{\partial} t^1_0}=\sum_{n\ge 0}t^\alpha_{n+1}\frac{{\partial} F^{\textrm{red}}}{{\partial} t^\alpha_n}+\frac{1}{2}\eta_{\alpha\beta}t^\alpha_0 t^\beta_0,\\ &\frac{{\partial} F^{\textrm{red}}}{{\partial} t^1_1}=\varepsilon\frac{{\partial} F^{\textrm{red}}}{{\partial}\varepsilon}+\sum_{n\ge 0}t^\alpha_n\frac{{\partial} F^{\textrm{red}}}{{\partial} t^\alpha_n}-2F^{\textrm{red}}+\varepsilon^2\frac{N}{24}. \end{align*} Recall (see [2, Section 4]) that the tau-structure for the DR hierarchy is given by the differential polynomials |$h^{\textrm{DR}}_{\alpha ,p}=\frac{\delta \overline g_{\alpha ,p+1}}{\delta u^1}$|⁠. The normal coordinates for this tau-structure are \begin{equation} {\widetilde u}^\alpha\big(u^*_*;\varepsilon\big)=\eta^{\alpha\mu}h^{\textrm{DR}}_{\mu,-1}=\eta^{\alpha\mu}\frac{\delta\overline g_{\mu,0}}{\delta u^1}. \end{equation} (7.8) In [2, Section 7.3] we proposed the following conjecture. Conjecture 7.1. The normal Miura transformation defined by the differential polynomial |${\mathcal{P}}$| transforms the DZ hierarchy to the DR hierarchy written in the normal coordinates |${\widetilde u}^\alpha $|⁠. We called this conjecture the strong DR/DZ equivalence conjecture. In [2, Section 7.3] we proved that the strong DR/DZ equivalence conjecture is true if and only if |$F^{\textrm{DR}}=F^{\textrm{red}}$|⁠. Note that formulas (7.8) and (4) together with the string equation for |$F^{\textrm{DR}}$| imply that the normal coordinates |${\widetilde u}^\alpha (u^*_*;\varepsilon )$| can be described using the DR correlators \begin{equation} {\widetilde u}^\alpha(u^*_*;\varepsilon)=u^\alpha+\sum_{g,n\ge 1}\frac{\varepsilon^{2g}}{n!}\sum_{d_1+\ldots+d_n=2g}\eta^{\alpha\mu}\left<\tau_0(e_1)\tau_0(e_\mu)\prod\tau_{d_i}(e_{\alpha_i})\right>^{\textrm{DR}}_g\prod u^{\alpha_i}_{d_i}. \end{equation} (7.9) If Conjecture 7.1 is true, then |$\left <\tau _0(e_1)\tau _0(e_\mu )\prod \tau _{d_i}(e_{\alpha _i})\right>^{\textrm{DR}}_g=\left <\tau _0(e_1)\tau _0(e_\mu )\prod \tau _{d_i}(e_{\alpha _i})\right>^{\textrm{red}}_g$|⁠. Together with equation (7.9), it motivates the following theorem. Theorem 7.2. Define Miura transformations |$w^\alpha \mapsto{\widetilde u}^\alpha (w^*_*;\varepsilon )$| and |$u^\alpha \mapsto{\widetilde u}^\alpha (u^*_*;\varepsilon )$| by \begin{align*} {\widetilde u}^\alpha(w^*_*;\varepsilon)=&w^\alpha+\eta^{\alpha\mu}{\partial}_x\left\{{\mathcal{P}},\overline h^{\textrm{DZ}}_{\mu,0}\right\}_{K^{\textrm{DZ}}},\\{\widetilde u}^\alpha(u^*_*;\varepsilon)=&u^\alpha+\sum_{g,n\ge 1}\frac{\varepsilon^{2g}}{n!}\sum_{d_1+\ldots+d_n=2g}\eta^{\alpha\mu}\left<\tau_0(e_1)\tau_0(e_\mu)\prod\tau_{d_i}(e_{\alpha_i})\right>^{\textrm{red}}_g\prod u^{\alpha_i}_{d_i}. \end{align*} Then the Miura transformation |$w^\alpha \mapsto u^\alpha (w^*_*;\varepsilon )$| transforms the operator |$K^{\textrm{DZ}}$| to |$\eta{\partial }_x$|⁠. We will prove this theorem in the next section. 7.3 Proof of Theorem 7.2 We split the proof into three steps. In Section 7.3.1 we introduce rational Miura transformations and discuss their properties. In Section 7.3.2 we prove that the change of variables |$v^\alpha \mapsto w^\alpha (v^*_*;\varepsilon )$| from Section 7.1 is a rational Miura transformation. Finally, in Section 7.3.3 we prove Theorem 7.2. 7.3.1 Rational Miura transformations For |$d\in{\mathbb{Z}}$| let |${\mathcal{A}}^{\textrm{rt},[d]}_{v^1,\ldots ,v^N}$| be the vector space spanned by expressions of the form \begin{equation} \sum_{i\ge m}\frac{P_i(v^*_*)}{(v^1_x)^i}, \end{equation} (7.10) where |$m\in{\mathbb{Z}}$|⁠, |$P_i\in{\mathcal{A}}^{[d+i]}_{v^1,\ldots ,v^N}$| and |$\frac{{\partial } P_i}{{\partial } v^1_x}=0$|⁠. Let |${\mathcal{A}}^{\textrm{rt}}_{v^1,\ldots ,v^N}:=\bigoplus _{d\in{\mathbb{Z}}}{\mathcal{A}}^{\textrm{rt},[d]}_{v^1,\ldots ,v^N}$|⁠. Since \begin{equation*} \frac{1}{(v^1_x)^i}=(1+(v^1_x-1))^{-i}=\sum_{k\ge 0}{-i\choose k}(v^1_x-1)^k, \end{equation*} we have a natural inclusion \begin{equation*} {\mathcal{A}}^{\textrm{rt}}_{v^1,\ldots,v^N}\subset{\mathcal{A}}^{\textrm{wk}}_{v^1,\ldots,v^N}. \end{equation*} In the same way, as for differential polynomials, we introduce a grading by |$\deg v^\alpha _i=i$|⁠. Then the subspace |${\mathcal{A}}^{\textrm{rt},[d]}_{v^1,\ldots ,v^N}\subset{\mathcal{A}}^{\textrm{rt}}_{v^1,\ldots ,v^N}$| consists precisely of elements of degree |$d$|⁠. For an element |$f(v^*_*)=\sum _{i\ge m}\frac{P_i(v^*_*)}{(v^1_x)^i}\in{\mathcal{A}}^{\textrm{rt}}_{v^1,\ldots ,v^N}$| define the polynomial part by \begin{equation*} f(v^*_*)^{\textrm{pol}}:=\sum_{i=m}^0\frac{P_i(v^*_*)}{(v^1_x)^i}\in{\mathcal{A}}_{v^1,\ldots,v^N}. \end{equation*} Define the extended space |$\widehat{{\mathcal{A}}}^{\textrm{rt}}_{v^1,\ldots ,v^N}:={\mathcal{A}}^{\textrm{rt}}_{v^1,\ldots ,v^N}[[\varepsilon ]]$|⁠. Denote by \begin{equation*} \widehat{{\mathcal{A}}}^{\textrm{rt},[d]}_{v^1,\ldots,v^N}\subset\widehat{{\mathcal{A}}}^{\textrm{rt}}_{v^1,\ldots,v^N} \end{equation*} the subspace of elements of degree |$d$|⁠, where we, as usual, set |$\deg \varepsilon =-1$|⁠. A rational function (7.10) is called tame if there exists a nonnegative integer |$C$| such that |$\frac{{\partial } P_i}{{\partial } v^\alpha _k}=0$| for |$k>C$|⁠. The subspace of tame elements in |${\mathcal{A}}^{\textrm{rt}}_{v^1,\ldots ,v^N}$| will be denoted by \begin{equation*} {\mathcal{A}}^{\textrm{rt},\mathrm{t}}_{v^1,\ldots,v^N}\subset{\mathcal{A}}^{\textrm{rt}}_{v^1,\ldots,v^N}. \end{equation*} An element |$f(v^*_*;\varepsilon )=\sum _{g\ge 0}\varepsilon ^g f_g(v^*_*)\in \widehat{{\mathcal{A}}}^{\textrm{rt}}_{v^1,\ldots ,v^N}$| will be called tame if all functions |$f_g\in{\mathcal{A}}^{\textrm{rt},\mathrm{t}}_{v^1,\ldots ,v^N}$| are tame. The subspace of tame elements in |$\widehat{{\mathcal{A}}}^{\textrm{rt}}_{v^1,\ldots ,v^N}$| will be denoted by \begin{equation*} \widehat{{\mathcal{A}}}^{\textrm{rt},\mathrm{t}}_{v^1,\ldots,v^N}\subset\widehat{{\mathcal{A}}}^{\textrm{rt}}_{v^1,\ldots,v^N}. \end{equation*} Consider changes of variables of the form \begin{equation} v^\alpha\mapsto w^\alpha\big(v^*_*;\varepsilon\big)=v^\alpha+\varepsilon f^\alpha\big(v^*_*;\varepsilon\big),\quad \alpha=1,\ldots,N,\quad f^\alpha\in\widehat{{\mathcal{A}}}^{\textrm{rt},\mathrm{t},[1]}_{v^1,\ldots,v^N}. \end{equation} (7.11) We will call them rational Miura transformations. These transformations form a group. Any tame rational function |$f(v^*_*;\varepsilon )\in \widehat{{\mathcal{A}}}^{\textrm{rt},\mathrm{t}}_{v^1,\ldots ,v^N}$| can be rewritten as a tame rational function in the new variables |$w^\alpha $|⁠. The resulting tame rational function will be denoted by |$f(w^*_*;\varepsilon )$|⁠. Clearly, the polynomial part |$w^\alpha (v^*_*;\varepsilon )^{\textrm{pol}}$| of a rational Miura transformation (7.11) is a usual Miura transformation. Define a subspace |$S_{v^1,\ldots ,v^N}\subset \widehat{{\mathcal{A}}}^{\textrm{rt},\mathrm{t}}_{v^1,\ldots ,v^N}$| by \begin{equation*} S_{v^1,\ldots,v^N}:=\left\{\left.f\in\widehat{{\mathcal{A}}}^{\textrm{rt},\mathrm{t}}_{v^1,\ldots,v^N}\right|f^{\textrm{pol}}=0,\frac{{\partial} f}{{\partial} v^1}=0\right\}. \end{equation*} It is easy to see that the subspace |$S_{v^1,\ldots ,v^N}$| is closed under multiplication and also under the derivations |${\partial }_x$| and |$\frac{{\partial }}{{\partial } v^\gamma _n}$|⁠. Lemma 7.3. Let |$v^\alpha \mapsto w^\alpha (v^*_*;\varepsilon )$| be a rational Miura transformation such that \begin{equation*} (w^\alpha)^{\textrm{pol}}(v^*_*;\varepsilon)=v^\alpha\quad\textrm{and}\quad \frac{{\partial} w^\alpha(v^*_*;\varepsilon)}{{\partial} v^1}=\delta^{\alpha,1}. \end{equation*} Consider an operator |$K=(K^{\alpha \beta })$| defined by \begin{equation} K^{\alpha\beta}:=\sum_{p,q\ge 0}\frac{{\partial} w^\alpha(v^*_*;\varepsilon)}{{\partial} v^\mu_p}{\partial}_x^p\circ\eta^{\mu\nu}{\partial}_x\circ(-{\partial}_x)^q\circ\frac{{\partial} w^\beta(v^*_*;\varepsilon)}{{\partial} v^\nu_q}=\sum_{i\ge 0}K^{\alpha\beta}_i\big(v^*_*;\varepsilon\big){\partial}_x^i. \end{equation} (7.12) Suppose that |$K^{\alpha \beta }_i(w^*_*;\varepsilon )\in \widehat{{\mathcal{A}}}_{w^1,\ldots ,w^N}$|⁠. Then |$K^{\alpha \beta }=\eta ^{\alpha \beta }{\partial }_x$|⁠. Proof. From formula (7.12) one can easily see that \begin{equation*} K^{\alpha\beta}_i\big(v^*_*;\varepsilon\big)-\delta_{i,1}\eta^{\alpha\beta}\in S_{v^1,\ldots,v^N}. \end{equation*} Observe that if |$f(v^*_*;\varepsilon )\in S_{v^1,\ldots ,v^N}$|⁠, then |$f(w^*_*;\varepsilon )\in S_{w^1,\ldots ,w^N}$|⁠. Since |$S_{w_1,\ldots ,w^N}\cap \widehat{{\mathcal{A}}}_{w_1,\ldots ,w^N}=0$|⁠, we get |$K^{\alpha \beta }_i(w^*_*,\varepsilon )-\delta _{i,1}\eta ^{\alpha \beta }=0$|⁠. The lemma is proved. Lemma 7.4. Consider three sets of variables |$v^\alpha $|⁠, |$u^\alpha $|⁠, and |$w^\alpha $|⁠. Suppose that we have rational Miura transformations |$v^\alpha \mapsto u^\alpha (v^*_*;\varepsilon )$| and |$u^\alpha \mapsto w^\alpha (u^*_*;\varepsilon )$| such that \begin{align*} &\frac{{\partial} u^\alpha(v^*_*;\varepsilon)}{{\partial} v^1}=\delta^{\alpha,1},&& \frac{{\partial} u^\alpha(v^*_*;\varepsilon)^{\textrm{pol}}}{{\partial} v^1_x}=0,\\ &\frac{{\partial} w^\alpha(u^*_*;\varepsilon)}{{\partial} u^1}=\delta^{\alpha,1},&& \frac{{\partial} w^\alpha(u^*_*;\varepsilon)^{\textrm{pol}}}{{\partial} u^1_x}=0. \end{align*} Then the polynomial part of the composition of these rational Miura transformations is equal to the composition of their polynomial parts. Proof. The proof is straightforward. One should just notice that the singularities of |$w^\alpha (u^*_*;\varepsilon )$| and |$u^\alpha (v^*_*;\varepsilon )$| cannot give a nontrivial contribution in the polynomial part of the composition of these rational Miura transformations. Let us formulate one more technical statement in this section. Lemma 7.5. Consider variables |$u^\alpha $| and |$w^\alpha $|⁠. Suppose we have a Miura transformation |$u^\alpha \mapsto w^\alpha (u^*_*;\varepsilon )$| such that |$\frac{{\partial } w^\alpha (u^*_*;\varepsilon )}{{\partial } u^1}=\delta ^{\alpha ,1}$| and |$\frac{{\partial } w^\alpha (u^*_*;\varepsilon )}{{\partial } u^1_x}=0$|⁠. Then the inverse Miura transformation |$w^\alpha \mapsto u^\alpha (w^*_*;\varepsilon )$| satisfies the following same properties: |$\frac{{\partial } u^\alpha (w^*_*;\varepsilon )}{{\partial } w^1}=\delta ^{\alpha ,1}$| and |$\frac{{\partial } u^\alpha (w^*_*;\varepsilon )}{{\partial } w^1_x}=0$|⁠. Proof. This is a direct computation based on the chain rule. 7.3.2 Rationality of the function |$w^\alpha (v^*_*,\varepsilon )$| Consider the function |$w^\alpha (v^*_*;\varepsilon )$| from Section 7.1. Proposition 7.6. We have |$w^\alpha (v^*_*;\varepsilon )\in \widehat{{\mathcal{A}}}^{\textrm{rt},\mathrm{t},[0]}_{v^1,\ldots ,v^N}$| and, moreover, |$\frac{{\partial } w^\alpha (v^*_*;\varepsilon )}{{\partial } v^1}=\delta ^{\alpha ,1}$|⁠. Remark 7.7. While this work was under preparation, we were informed that this proposition was independently proved by Shadrin, Lewanski, and Popolitov. Proof. of Proposition 7.6 The proof is very similar to the construction of the differential polynomial |${\mathcal{P}}$| from [2, Section 7.3]. Consider the |$\varepsilon $|-expansion of the topological solution |$(w^{\textrm{top}})^\alpha $|⁠: \begin{equation*} (w^{\textrm{top}})^\alpha\big(x,t^*_*,\varepsilon\big)=\sum_{g\ge 0}\varepsilon^{2g}(w^{\textrm{top}})^{\alpha,[g]}(x,t^*_*). \end{equation*} Define a linear differential operator |$O_{\textrm{dil}}$| by \begin{equation*} O_{\textrm{dil}}:=\frac{{\partial}}{{\partial} t^1_1}-x\frac{{\partial}}{{\partial} x}-\sum_{n\ge 0}t^\gamma_n\frac{{\partial}}{{\partial} t^\gamma_n}. \end{equation*} For |$g\ge 1$| let us construct a sequence of functions |$w^{\alpha ,[g,k]}\in{\mathcal{A}}^{\textrm{rt},[2g]}_{v^1,\ldots ,v^N}$|⁠, |$k\ge -1$|⁠, such that \begin{align} &w^{\alpha,[g,k]}=w^{\alpha,[g,k-1]}+\big(v^1_x\big)^{2g-k}P^{\alpha,[g,k]},\quad k\ge 0,\quad P^{\alpha,[g,k]}\in{\mathcal{A}}^{[k]}_{v^1,\ldots,v^N},\quad\frac{{\partial} P^{\alpha,[g,k]}}{{\partial} v^1_x}=0, \end{align} (7.13) \begin{align} &\left.\left((w^{\textrm{top}})^{\alpha,[g]}-w^{\alpha,[g,k]}\big(v^{\textrm{top}},v^{\textrm{top}}_x,\ldots\big)\right)\right|_{x=0}\in{\mathbb{C}}\big[\big[t^*_*\big]\big]^{(k+1)}, \end{align} (7.14) \begin{align} &O_{\textrm{dil}}w^{\alpha,[g,k]}\big(v^{\textrm{top}},v^{\textrm{top}}_x,\ldots\big)=2g\cdot w^{\alpha,[g,k]}\big(v^{\textrm{top}},v^{\textrm{top}}_x,\ldots\big). \end{align} (7.15) Let |$w^{\alpha ,[g,-1]}:=0$|⁠. Suppose that |$k\ge 0$| and that |$w^{\alpha ,[g,k-1]}$| is already constructed. Let \begin{equation*} \left<\tau_{d_1}(e_{\alpha_1})\ldots\tau_{d_n}(e_{\alpha_n})\right>^{\alpha,[g,k-1]}:=\left.\frac{{\partial}^n w^{\alpha,[g,k-1]}(v^{\textrm{top}},v^{\textrm{top}}_x,\ldots)}{{\partial} t^{\alpha_1}_{d_1}\ldots{\partial} t^{\alpha_n}_{d_n}}\right|_{x=t^*_*=0}. \end{equation*} Define \begin{align} &w^{\alpha,[g,k]}:=w^{\alpha,[g,k-1]}+\notag\\ &+\sum_{n\ge 0}\frac{\varepsilon^{2g}}{n!}\!\sum_{d_1+\ldots+d_n=k}\!\underline{\left(\eta^{\alpha\mu}\left<\tau_0(e_1)\tau_0(e_\mu)\prod_{i=1}^n\tau_{d_i}(e_{\alpha_i})\right>_g-\left<\prod_{i=1}^n\tau_{d_i}(e_{\alpha_i})\right>^{\alpha,[g,k-1]}\right)}\!\big(v^1_x\big)^{2g-k}\prod_{i=1}^n v^{\alpha_i}_{d_i}. \end{align} (7.16) Let us prove properties (7.13)–(7.15). We have \begin{equation*} O_{\textrm{dil}}\left((w^{\textrm{top}})^{\alpha,[g]}-w^{\alpha,[g,k-1]}\big(v^{\textrm{top}},v^{\textrm{top}}_x,\ldots\big)\right)=2g\left((w^{\textrm{top}})^{\alpha,[g]}-w^{\alpha,[g,k-1]}\big(v^{\textrm{top}},v^{\textrm{top}}_x,\ldots\big)\right). \end{equation*} Using (7.14) for |$w^{\alpha ,[g,k-1]}$|⁠, we see that the underlined expression in (7.16) is equal to zero, if |$\alpha _i=d_i=1$| for some |$i$|⁠. Therefore, formula (7.13) is clear. Equation (7.15) follows from the fact that |$O_{\textrm{dil}}(v^{\textrm{top}})^\alpha _n=n(v^{\textrm{top}})^\alpha _n$|⁠. Property (7.14) follows from (7.1). From (7.13) it follows that the limit |$w^{\alpha ,[g]}:=\lim _{k\to \infty }w^{\alpha ,[g,k]}\in{\mathcal{A}}^{\textrm{rt},[2g]}_{v^1,\ldots ,v^N}$| is well defined. Formula (7.14) implies that \begin{equation*} (v^{\textrm{top}})^\alpha+\sum_{g\ge 1}\varepsilon^{2g}w^{\alpha,[g]}\big(v^{\textrm{top}},v^{\textrm{top}}_x,\ldots\big)=(w^{\textrm{top}})^{\alpha,[g]}. \end{equation*} Therefore, |$w^\alpha (v^*_*;\varepsilon )=v^\alpha +\sum _{g\ge 1}\varepsilon ^{2g}w^{\alpha ,[g]}\in \widehat{{\mathcal{A}}}^{\textrm{rt},[0]}_{v^1,\ldots ,v^N}$|⁠. The tameness of |$w^\alpha (v^*_*;\varepsilon )$| was already explained in Section 7.1. It remains to show that |$\frac{{\partial } w^\alpha (v^*_*;\varepsilon )}{{\partial } v^1}=\delta ^{\alpha ,1}$|⁠. Let \begin{equation*} O_{\textrm{str}}:=\frac{{\partial}}{{\partial} t^1_0}-\sum_{n\ge 0}t^\gamma_{n+1}\frac{{\partial}}{{\partial} t^\gamma_n}. \end{equation*} From the string equation for the potential |$F$| it follows that |$O_{\textrm{str}}(w^{\textrm{top}})^\alpha =O_{\textrm{str}}(v^{\textrm{top}})^\alpha =\delta ^{\alpha ,1}$|⁠. Therefore, |$\frac{{\partial } w^\alpha (v^*_*;\varepsilon )}{{\partial } v^1}=\delta ^{\alpha ,1}$|⁠. The proposition is proved. 7.3.3 Final step Consider the rational Miura transformation |$v^\alpha \mapsto w^\alpha (v^*_*;\varepsilon )$| from the previous section. Since the variables |$u^\alpha $| and |${\widetilde u}^\alpha $| are related to |$w^\alpha $| by Miura transformations, we see that they are related to the variables |$v^\alpha $| by rational Miura transformations, which we denote by |$u^\alpha (v^*_*;\varepsilon )$| and |${\widetilde u}^\alpha (v^*_*;\varepsilon ),$| respectively. From equation (7.5) it follows that the operator |$K^{\textrm{DZ}}$| in the variables |$u^\alpha $| is equal to \begin{equation*} (K^{\textrm{DZ}})^{\alpha\beta}_u=\sum_{p,q\ge 0}\frac{{\partial} u^\alpha(v^*_*;\varepsilon)}{{\partial} v^\mu_p}{\partial}_x^p\circ\eta^{\mu\nu}{\partial}_x\circ(-{\partial}_x)^q\circ\frac{{\partial} u^\beta(v^*_*;\varepsilon)}{{\partial} v^\nu_q}. \end{equation*} Lemma 7.3 implies that it is sufficient to show that \begin{equation} \frac{{\partial} u^\alpha(v^*_*;\varepsilon)}{{\partial} v^1}=\delta^{\alpha,1}\quad\textrm{and}\quad u^\alpha\big(v^*_*;\varepsilon\big)^{\textrm{pol}}=v^\alpha. \end{equation} (7.17) We have \begin{equation*} {\widetilde u}^\alpha\big(v^{\textrm{top}},v^{\textrm{top}}_x,\ldots;\varepsilon\big)=\left.\eta^{\alpha\mu}\frac{{\partial}^2 F^{\textrm{red}}}{{\partial} t^\mu_0{\partial} t^1_0}\right|_{t^1_0\mapsto t^1_0+x}. \end{equation*} The string equation for |$F^{\textrm{red}}$| implies that |$O_{\textrm{str}}{\widetilde u}^\alpha (v^{\textrm{top}},v^{\textrm{top}}_x,\ldots ;\varepsilon )=\delta ^{\alpha ,1}$|⁠. Therefore, |$\frac{{\partial }{\widetilde u}^\alpha (v^*_*;\varepsilon )}{{\partial } v^1}=\delta ^{\alpha ,1}$|⁠. From the string equation for |$F^{\textrm{red}}$| and property (7.7) it follows that |$\frac{{\partial }{\widetilde u}^\alpha (u^*_*,\varepsilon )}{{\partial } u^1}=\delta ^{\alpha ,1}$|⁠. Thus, |$\frac{{\partial } u^\alpha (v^*_*;\varepsilon )}{{\partial } v^1}=\delta ^{\alpha ,1}$|⁠. Let us now prove the 2nd equation in (7.17). Let \begin{equation*} {\widetilde u}^\alpha\big(v^*_*;\varepsilon\big)=v^\alpha+\sum_{g\ge 1}\varepsilon^{2g}\sum_{k\ge -2g}\frac{P^\alpha_{g,k}(v^*_*)}{(v^1_x)^k},\quad P^\alpha_{g,k}\in{\mathcal{A}}^{[2g+k]}_{v^1,\ldots,v^N}. \end{equation*} Property (7.7) together with the string equation for |$F^{\textrm{red}}$| implies that \begin{equation*} \left.\textrm{Coef}_{\varepsilon^{2g}}{\widetilde u}^\alpha\big(v^{\textrm{top}},v^{\textrm{top}}_x,\ldots;\varepsilon\big)\right|_{x=0}\in{\mathbb{C}}\big[\big[t^*_*\big]\big]^{(2g)}. \end{equation*} Using also (7.1), we conclude that |$P^\alpha _{g,k}=0$| for |$k<0$| and \begin{equation} P^\alpha_{g,0}\big(v^*_*\big)=\sum_{n\ge 1}\sum_{d_1+\ldots+d_n=2g}\eta^{\alpha\mu}\left<\tau_0(e_\mu)\tau_0(e_1)\prod\tau_{d_i}(e_{\alpha_i})\right>^{\textrm{red}}_g \frac{\prod v^{\alpha_i}_{d_i}}{n!}. \end{equation} (7.18) Thus, \begin{equation*} {\widetilde u}^\alpha\big(u^*_*;\varepsilon\big)=\left.{\widetilde u}^\alpha\big(v^*_*;\varepsilon\big)^{\textrm{pol}}\right|_{v^\gamma_n=u^\gamma_n}. \end{equation*} The rational Miura transformation |$v^\alpha \mapsto u^\alpha (v^*_*;\varepsilon )$| is the composition of the transformations \begin{equation} v^\alpha\mapsto{\widetilde u}^\alpha\big(v^*_*;\varepsilon\big)\quad\textrm{and}\quad{\widetilde u}^\alpha\mapsto u^\alpha\big({\widetilde u}^*_*;\varepsilon\big). \end{equation} (7.19) We already know that |$\frac{{\partial }{\widetilde u}^\alpha (v^*_*;\varepsilon )}{{\partial } v^1}=\delta ^{\alpha ,1}$|⁠. Equations (7.18), (7.7), and the string and the dilaton equations for |$F^{\textrm{red}}$| imply that |$\frac{{\partial } P^\alpha _{g,0}}{{\partial } v^1_x}=0$|⁠. Therefore, |$\frac{{\partial }{\widetilde u}^\alpha (v^*_*;\varepsilon )^{\textrm{pol}}}{{\partial } v^1_x}=0$|⁠. So, the 1st transformation in (7.19) satisfies the assumptions of Lemma 7.4. Using Lemma 7.5 we see that the 2nd transformation in (7.19) also satisfies the assumptions of Lemma 7.4. We conclude that |$u^\alpha (v^*_*;\varepsilon )^{\textrm{pol}}=v^\alpha $|⁠. Theorem 7.2 is proved. 8 DR and DZ Hierarchies of Rank |$1$| In this section we focus on CohFTs of rank |$1$|⁠, that is, |$\dim V=1$|⁠, and the corresponding DR and DZ hierarchies. In Section 8.1 we recall certain definitions and the main conjecture from the work [10] about tau-symmetric deformations of the Riemann hierarchy. In Section 8.2 we show that the DR hierarchy is a standard deformation of the Riemann hierarchy in the sense of [10]. In Section 8.3 we prove an existence of a normal Miura transformation that reduces the DZ hierarchy to its unique standard form. This proves a part of the conjecture from [10] for the DZ hierarchies. In Section 8.4 we prove the strong DR/DZ equivalence conjecture at the approximation up to genus |$5$|⁠. Since |$\dim V=1$|⁠, a rank |$1$| CohFT is described by classes |$c_{g,n}=c_{g,n}(e_1^{\otimes n})\in H^{\textrm{even}}({\overline{{\mathcal{M}}}}_{g,n},{\mathbb{C}})$|⁠. Recall that, according to [28], rank |$1$| CohFTs with |$\eta _{1,1}=\alpha $| are parameterized by numbers |$s_1,s_2,\ldots $| in the following way: \begin{equation} c_{g,n}=\alpha^{1-g}e^{-\sum_{i\ge 1}\frac{(2i)!}{B_{2i}}s_i\textrm{Ch}_{2i-1}(\mathbb E)}. \end{equation} (8.1) Here |$\textrm{Ch}_{2i-1}$| denotes the |$(2i-1)$|-th component of the Chern character and we use the same rescaling of the coefficient of |$\textrm{Ch}_{2i-1}(\mathbb E)$| in the exponent, as in [10, p. 384]. Since |$\dim V=1$|⁠, we will omit Greek indices in many notations. For example, the correlators of a CohFT will be denoted by |$\left <\tau _{d_1}\ldots \tau _{d_n}\right>_g$|⁠, the Hamiltonians of the DZ hierarchy will be denoted by |$\overline h_d^{\textrm{DZ}}$|⁠,.... 8.1 Tau-symmetric deformations of the Riemann hierarchy The Riemann hierarchy is the tau-symmetric Hamiltonian hierarchy given by the Hamiltonians \begin{equation*} \overline h^{\textrm{R}}_{d}:=\int\frac{u^{d+2}}{(d+2)!}\mathrm{\,d}x,\quad d\ge 0, \end{equation*} the Hamiltonian operator |${\partial }_x$|⁠, and the tau-symmetric densities |$h^{\textrm{R}}_d:=\frac{u^{d+2}}{(d+2)!}$|⁠, |$d\ge -1$|⁠. A tau-symmetric deformation of the Riemann hierarchy is a tau-symmetric Hamiltonian hierarchy given by Hamiltonians |$\overline h_d$|⁠, |$d\ge 0$|⁠, Hamiltonian operator |$K,$| and tau-symmetric densities |$h_d$|⁠, |$d\ge -1$|⁠, such that \begin{equation*} \overline h_d|_{\varepsilon=0}=\overline h^{\textrm{R}}_d,\qquad K|_{\varepsilon=0}={\partial}_x,\qquad h_d|_{\varepsilon=0}=h^{\textrm{R}}_d,\qquad K\frac{\delta\overline h_0}{\delta u}=u_x. \end{equation*} Here the last condition means that the Hamiltonian |$\overline h_0$| generates the spatial translations. Denote by |${\mathcal{P}}_n$| the set of all partitions of |$n$|⁠. For a partition |$\lambda =(\lambda _1,\ldots ,\lambda _l)$|⁠, |$\lambda _1\ge \ldots \lambda _l\ge 1$|⁠, let |$l(\lambda ):=l$|⁠. Introduce a subset |${\mathcal{P}}^{\prime}_n\subset{\mathcal{P}}_n$| by \begin{equation*} {\mathcal{P}}^{\prime}_n:=\left\{\lambda\in{\mathcal{P}}_n\left| \begin{smallmatrix} l(\lambda)\ge 2,\\ \lambda_1=\lambda_2,\\ \lambda_i\ge 2. \end{smallmatrix}\right. \right\}. \end{equation*} For a partition |$\lambda \in{\mathcal{P}}_n$| let |$u_\lambda :=\prod _{i=1}^{l(\lambda )}u_{\lambda _i}$|⁠. A tau-symmetric deformation of the Riemann hierarchy is said to be standard if |$K={\partial }_x$| and a density |$\widetilde h_1$| for the Hamiltonian |$\overline h_1$| can be chosen in the following form: \begin{equation} \widetilde h_1=\frac{u^3}{6}-\frac{\varepsilon^2}{24}a_0 u_x^2+\sum_{g\ge 2}\varepsilon^{2g}\sum_{\lambda\in{\mathcal{P}}^{\prime}_{2g}}\alpha_\lambda u_\lambda, \end{equation} (8.2) for some complex coefficients |$a_0$| and |$\alpha _\lambda $|⁠. It is easy to show that if such a density exists, then it is unique. In [10] the authors proposed the following conjecture. Conjecture 8.1. Consider an arbitrary tau-symmetric deformation of the Riemann hierarchy. Suppose that the deformation is standard. Then for the unique density of the form (8.2) we have the following: a) If |$a_0=0$|⁠, then |$\alpha _\lambda =0$| for all |$\lambda $|⁠. b) If |$a_0\ne 0$|⁠, then all coefficients |$\alpha _\lambda $| are uniquely determined by the coefficients |$a_0$| and |$\alpha _{(2^g)}$|⁠, |$g\ge 2$|⁠. There exists a unique normal Miura transformation that transforms the hierarchy to a standard deformation. This deformation is called the standard form of the hierarchy. The authors of [10] checked the uniqueness statement in the 2nd part of the conjecture. Moreover, they verified the conjecture at the approximation up to |$\varepsilon ^{12}$|⁠. Consider a CohFT of rank |$1$| with |$\eta _{1,1}=1$|⁠. Clearly, the corresponding DR and DZ hierarchies are tau-symmetric deformations of the Riemann hierarchy. In the next section we will prove that the DR hierarchy is a standard deformation. In Section 8.3 we will prove that part 2 of Conjecture 8.1 is true for the DZ hierarchy. 8.2 DR hierarchy as a standard deformation Introduce a subset |${\mathcal{P}}^\circ _n\subset{\mathcal{P}}_n$| by \begin{equation*} {\mathcal{P}}^\circ_n:=\left\{\lambda\in{\mathcal{P}}_n\left| \begin{smallmatrix} l(\lambda)\ge 2,\\ \lambda_1=\lambda_2. \end{smallmatrix}\right. \right\}. \end{equation*} Lemma 8.2. Let |$d\ge 2$|⁠. Consider a differential polynomial |$h=\sum _{\lambda \in{\mathcal{P}}_d}h_\lambda (u)u_\lambda \in{\mathcal{A}}^{[d]}_u$|⁠, where |$h_\lambda (u)$| are formal power series in |$u$|⁠. For the local functional |$\overline h=\int h \mathrm{\,d}x$| there exists a unique density |$\widetilde h\in{\mathcal{A}}^{[d]}_u$| of the form \begin{equation} \widetilde h=\sum_{\lambda\in{\mathcal{P}}^\circ_d}\widetilde h_\lambda(u)u_\lambda, \end{equation} (8.3) where |$\widetilde h_\lambda (u)$| are formal power series in |$u$|⁠. Let |$d=2g$|⁠. Suppose that |$\frac{{\partial } h_\lambda (u)}{{\partial } u}=0$| for all |$\lambda $| and that |$h_\lambda =0$| unless |$\lambda _i\ge 2$|⁠. Then |$\frac{{\partial }\widetilde h_\lambda (u)}{{\partial } u}=0$| for all |$\lambda $| and |$\widetilde h_\lambda =0$| for |$\lambda \in{\mathcal{P}}_d\backslash{\mathcal{P}}^{\prime}_d$|⁠. Moreover, we have |$\widetilde h_{(2^g)}=h_{(2^g)}$|⁠. Proof. 1. Let us prove the existence of such density. Suppose that the set \begin{equation} \big\{\lambda\in{\mathcal{P}}_d\big\backslash{\mathcal{P}}^\circ_d\big|h_\lambda(u)\ne 0\big\} \end{equation} (8.4) is nonempty. Let |$\lambda ^{(0)}$| be the lexicographically maximal partition in the set (8.4) and |$m$| be the multiplicity of the part |$\lambda ^{(0)}_1-1$| in |$\lambda ^{(0)}$|⁠. Define a differential polynomial |$h^{(1)}$| by \begin{equation*} h^{(1)}:=h-{\partial}_x\left(\frac{u_{\lambda^{(0)}_1-1}^{m+1}}{m+1}h_{\lambda^{(0)}}(u)\prod_{i=m+2}^{l(\lambda^{(0)})}u_{\lambda^{(0)}_i}\right)=\sum_{\lambda\in{\mathcal{P}}_d}h^{(1)}_\lambda(u)u_\lambda. \end{equation*} Obviously, |$h^{(1)}$| is a density for |$\overline h$|⁠. It is also clear that the lexicographically maximal partition in the set |$\{\lambda \in{\mathcal{P}}_d\backslash{\mathcal{P}}^\circ _d|h^{(1)}_\lambda (u)\ne 0\}$| is lexicographically smaller than |$\lambda ^{(0)}$|⁠. Continuing this process, after a finite number of steps, we come to a density of |$\overline h$| of the form (8.3). The uniqueness part follows from the fact that a nonzero differential polynomial of the form (8.3) does not belong to the image of the operator |${\partial }_x$|⁠. Part 2 of the lemma is clear from the proof of part 1. Proposition 8.3. Consider an arbitrary CohFT of rank |$1$| with |$\eta _{1,1}=1$|⁠. Then we have the following: The corresponding DR hierarchy is a standard tau-symmetric deformation of the Riemann hierarchy. For the unique density |$\widetilde g_1$| for |$\overline g_1$| of the form (8.2), \begin{equation*} \widetilde g_1=\frac{u^3}{6}-\frac{\varepsilon^2}{24}a^{\textrm{DR}}_0 u_x^2+\sum_{g\ge 2}\varepsilon^{2g}\sum_{\lambda\in{\mathcal{P}}^{\prime}_{2g}}\alpha^{\textrm{DR}}_\lambda u_\lambda, \end{equation*} we have \begin{equation*} a^{\textrm{DR}}_0=1,\qquad \alpha^{\textrm{DR}}_{(2^g)}=(3g-2)\int_{{\overline{{\mathcal{M}}}}_g}\lambda_g c_{g,0}. \end{equation*} Proof. We have \begin{equation*} \overline g_1=\sum_{g\ge 0,\,n\ge 2}\frac{(-\varepsilon^2)^g}{n!}\sum_{a_1+\cdots+a_n=0}\left(\int_{\textrm{DR}_g(0,a_1,\ldots,a_n)}\lambda_g\psi_1 c_{g,n+1}\right)\prod_{i=1}^n p_{a_i}. \end{equation*} For |$g\ge 1$| and |$n\ge 2$| we have \begin{equation} \int_{\textrm{DR}_g(0,a_1,\ldots,a_n)}\lambda_g\psi_1 c_{g,n+1}=(2g-2+n)\int_{\textrm{DR}_g(a_1,\ldots,a_n)}\lambda_g c_{g,n}. \end{equation} (8.5) For |$k\le n$| denote by |$\pi _k\colon{\overline{{\mathcal{M}}}}_{g,n}\to{\overline{{\mathcal{M}}}}_{g,n-k}$| the forgetful map that forgets the last |$k$| marked points. Using (6.5), we see that if |$g=1$|⁠, then the right-hand side of (8.5) is equal to \begin{equation} n\int_{\textrm{DR}_1(a_1,\ldots,a_n)}\lambda_1 c_{1,n}= \begin{cases} 0,&\textrm{if}\ n\ge 3;\\ 2a_1^2\int_{{\overline{{\mathcal{M}}}}_{1,1}}\lambda_1 c_{1,1}\stackrel{\text{by (8.1)}}{=}\frac{a_1^2}{12},&\textrm{if}\ n=2. \end{cases} \end{equation} (8.6) Suppose |$g\ge 2$|⁠. Then \begin{equation} (2g-2+n)\int_{\textrm{DR}_g(a_1,\ldots,a_n)}\lambda_g c_{g,n}=(2g-2+n)\int_{\pi_{n*}\textrm{DR}_g(a_1,\ldots,a_n)}\lambda_g c_{g,0}. \end{equation} (8.7) Note that the right-hand side is equal to zero unless |$n\le g$|⁠. We also see that for |$n=g$| the right-hand side of (8.7) is equal to \begin{equation} (3g-2)\int_{\pi_{g*}\textrm{DR}_g(a_1,\ldots,a_g)}\lambda_g c_{g,0}=(3g-2)g!a_1^2\cdots a_g^2\int_{{\overline{{\mathcal{M}}}}_{g}}\lambda_g c_{g,0}. \end{equation} (8.8) For an arbitrary |$n\le g$| we write \begin{equation*} (2g-2+n)\int_{\pi_{n*}\textrm{DR}_g(a_1,\ldots,a_n)}\lambda_g c_{g,0}=\frac{2g-2+n}{2g-2}\int_{\pi_{n*}\textrm{DR}_g(0,a_1,\ldots,a_n)}\psi_1\lambda_g c_{g,1}. \end{equation*} The divisibility property from Section 6.4.2 implies that the integral |$\int _{\pi _{n*}\textrm{DR}_g(0,a_1,\ldots ,a_n)}\psi _1\lambda _g c_{g,1}$| can be expressed as a polynomial \begin{equation*} P(a_1,\ldots,a_n)=\sum_{d_1+\cdots+d_n=2g}P_{d_1,\ldots,d_n}a_1^{d_1}\cdots a_n^{d_n},\quad P_{d_1,\ldots,d_n}\in{\mathbb{C}}, \end{equation*} where the coefficient |$P_{d_1,\ldots ,d_n}$| is equal to zero unless |$d_i\ge 2$| for all |$1\le i\le n$|⁠. Therefore, we obtain \begin{equation*} \overline g_1=\int\left(\frac{u^3}{6}-\frac{\varepsilon^2}{24}u_x^2+\sum_{g\ge 2}\varepsilon^{2g}\sum_{\lambda\in{\mathcal{P}}_{2g}}\beta_\lambda u_\lambda\right)\mathrm{\,d}x, \end{equation*} for some constants |$\beta _\lambda \in{\mathbb{C}}$| such that |$\beta _\lambda =0$| unless |$\lambda _i\ge 2$| for all |$1\le i\le l(\lambda )$|⁠. Moreover, by (8.8), we have \begin{equation*} \beta_{(2^g)}=(3g-2)\int_{{\overline{{\mathcal{M}}}}_g}\lambda_g c_{g,0}. \end{equation*} Lemma 8.2 completes the proof of the proposition. We obtain the following formula for the constants |$\alpha ^{\textrm{DR}}_{(2^g)}$| in terms of the parameters |$s_i$| from (8.1): \begin{equation} \alpha^{\textrm{DR}}_{(2^{g})}=(3g-2)\int_{{\overline{{\mathcal{M}}}}_g}\lambda_g e^{-\sum_{i\ge 1}\frac{(2i)!}{B_{2i}}s_i\textrm{Ch}_{2i-1}(\mathbb E)}. \end{equation} (8.9) In particular, \begin{align} \alpha^{\textrm{DR}}_{(2^2)}=&-48s_1\int_{{\overline{{\mathcal{M}}}}_2}\lambda_2\lambda_1=-\frac{s_1}{120}, \end{align} (8.10) \begin{align} \alpha^{\textrm{DR}}_{(2^3)}=&\big(-4032s_1^3-840s_2\big)\int_{{\overline{{\mathcal{M}}}}_3}\lambda_3\lambda_2\lambda_1=-\frac{s_1^3}{360}-\frac{s_2}{1728}, \end{align} (8.11) \begin{align} \alpha^{\textrm{DR}}_{(2^4)}=&\big(-331776 s_1^5 - 172800 s_1^2 s_2 - 2520 s_3\big)\int_{{\overline{{\mathcal{M}}}}_4}\lambda_4\lambda_3\lambda_2=-\frac{2s_1^5}{525}-\frac{s_1^2s_2}{504}-\frac{s_3}{34560}, \end{align} (8.12) \begin{align} \alpha^{\textrm{DR}}_{(2^5)}=&s_1^7\int_{{\overline{{\mathcal{M}}}}_5}\left(\frac{207028224}{35}\lambda_5\lambda_4\lambda_3-\frac{51757056}{5}\lambda_5\lambda_4\lambda_2\lambda_1\right)\\ &+s_1^4 s_2\int_{{\overline{{\mathcal{M}}}}_5}\left(10782720\lambda_5\lambda_4\lambda_3-10782720\lambda_5\lambda_4\lambda_2\lambda_1\right)\notag\\ &+s_1^2 s_3\int_{{\overline{{\mathcal{M}}}}_5}\left(943488\lambda_5\lambda_4\lambda_3-471744\lambda_5\lambda_4\lambda_2\lambda_1\right)\notag\\ &-s_4\int_{{\overline{{\mathcal{M}}}}_5}3120\lambda_5\lambda_4\lambda_3,\notag\\ &+s_1s_2^2\int_{{\overline{{\mathcal{M}}}}_5}\left(2246400\lambda_5\lambda_4\lambda_2\lambda_1-8985600 \lambda_5\lambda_4\lambda_3\right)=\notag\\ =&-\frac{754 s_1^7}{67375}-\frac{13 s_1^4s_2}{1320}-\frac{13 s_1^2 s_3}{52800}-\frac{13 s_4}{10644480}-\frac{13 s_1 s_2^2}{22176}.\notag \end{align} (8.13) Here we use the formulas [10, 14] \begin{align*} &\int_{{\overline{{\mathcal{M}}}}_g}\lambda_g\lambda_{g-1}\lambda_{g-2}=\frac{1}{2(2g-2)!}\frac{|B_{2g-2}|}{2g-2}\frac{|B_{2g}|}{2g},\quad g\ge 2,\\[2pt] &\int_{{\overline{{\mathcal{M}}}}_5}\lambda_5\lambda_4\lambda_2\lambda_1=\frac{1}{766402560}. \end{align*} 8.3 Standard form for the DZ hierarchy of rank |$1$| Theorem 8.4. Consider a CohFT of rank |$1$| with |$\eta _{1,1}=1$|⁠. Then part 2 of Conjecture 8.1 is true for the corresponding DZ hierarchy. Proof. Consider the normal Miura transformation |$w\mapsto{\widetilde u}(w_*;\varepsilon )$| and the Miura transformation |$u\mapsto{\widetilde u}(u_*;\varepsilon )$| from Theorem 7.2. From equation (7.7) and the string equation for |$F^{\textrm{red}}$| it follows that |$\left <\tau _0^2\prod \tau _{d_i}\right>^{\textrm{red}}_g=0$|⁠, if |$\sum d_i=2g$| and |$g\ge 1$|⁠. Therefore, |${\widetilde u}(u_*;\varepsilon )=u$|⁠. By Theorem 7.2, |$K^{\textrm{DZ}}_u={\partial }_x$|⁠. Let us prove that the Hamiltonian |$\overline h_1^{\textrm{DZ}}[u]$| has a density of the form (8.2). Let \begin{equation*} u^{\textrm{red}}(x,t_*,\varepsilon):=\left.\frac{{\partial}^2 F^{\textrm{red}}}{{\partial} t_0^2}\right|_{t_0\mapsto t_0+x}. \end{equation*} Denote by |$h^{\textrm{red}}_p\in \widehat{{\mathcal{A}}}^{[0]}_u$|⁠, |$p\ge -1$|⁠, the tau-symmetric densities of the DZ hierarchy after the normal Miura transformation |$w\mapsto u(w_*;\varepsilon )$|⁠. The differential polynomial |$h^{\textrm{red}}_p$| is uniquely determined by the condition \begin{equation} h^{\textrm{red}}_p(u^{\textrm{red}},u^{\textrm{red}}_x,\ldots;\varepsilon)=\left.\frac{{\partial}^2 F^{\textrm{red}}}{{\partial} t_0{\partial} t_{p+1}}\right|_{t_0\mapsto t_0+x}. \end{equation} (8.14) The string equation for |$F^{\textrm{red}}$| implies that \begin{equation*} \frac{{\partial} h^{\textrm{red}}_p}{{\partial} u}=h^{\textrm{red}}_{p-1},\quad p\ge 0. \end{equation*} Since |$K^{\textrm{DZ}}_u={\partial }_x$| and the Hamiltonian |$\overline h^{\textrm{DZ}}_0[u]$| generates the spatial translations, we get |$u_x={\partial }_x\frac{\delta \overline h^{\textrm{DZ}}_0[u]}{\delta u}$|⁠. Therefore, \begin{equation*} \overline h^{\textrm{DZ}}_0[u]=\int\frac{u^2}{2}\mathrm{\,d}x. \end{equation*} We obtain \begin{equation*} \frac{{\partial}\overline h^{\textrm{DZ}}_1[u]}{{\partial} u}=\overline h_0^{\textrm{DZ}}[u]=\int\frac{u^2}{2}\mathrm{\,d}x. \end{equation*} Therefore, \begin{equation*} \overline h_1^{\textrm{DZ}}[u]=\int\left(\frac{u^3}{6}-\frac{\varepsilon^2}{24}a_0u_x^2\right)\mathrm{\,d}x+O(\varepsilon^4) \end{equation*} for some constant |$a_0$|⁠. Lemma 8.5. Suppose |$d\ge 4$| and |$\overline h\in \Lambda ^{[d]}_u$|⁠. Then |$\overline h$| has a density |$\widetilde h$| of the form \begin{equation} \widetilde h=\sum_{\lambda\in{\mathcal{P}}^{\prime}_d}C_\lambda u_\lambda,\quad C_\lambda\in{\mathbb{C}}, \end{equation} (8.15) if and only if \begin{equation} \frac{{\partial}\overline h}{{\partial} u}=0\quad\textrm{and}\quad\frac{{\partial}}{{\partial} u_x}\frac{\delta\overline h}{\delta u}=0. \end{equation} (8.16) Proof. Obviously, if a density of the form (8.15) exists, then equations (8.16) are satisfied. Suppose now that the conditions (8.16) are true. Consider the unique density |$\widetilde h$| for |$\overline h$| of the form (8.3). The 1st condition in (8.16) immediately implies that |$\frac{{\partial }\widetilde h_\lambda (u)}{{\partial } u}=0$|⁠. Then we compute \begin{align*} \frac{{\partial}}{{\partial} u_x}\frac{\delta\overline h}{\delta u}=&\frac{{\partial}}{{\partial} u_x}\sum_{n\ge 0}(-{\partial}_x)^n\frac{{\partial}\widetilde h}{{\partial} u_n}=-\sum_{n\ge 1}n(-{\partial}_x)^{n-1}\frac{{\partial}}{{\partial} u_n}\frac{{\partial}\widetilde h}{{\partial} u}+\sum_{n\ge 0}(-{\partial}_x)^n\frac{{\partial}}{{\partial} u_n}\frac{{\partial}\widetilde h}{{\partial} u_x}=\frac{\delta}{\delta u}\frac{{\partial}\widetilde h}{{\partial} u_x}. \end{align*} We obtain |$\frac{\delta }{\delta u}\frac{{\partial }\widetilde h}{{\partial } u_x}=0$| and, therefore, |$\frac{{\partial }\widetilde h}{{\partial } u_x}$| is |${\partial }_x$|-exact. Clearly, the differential polynomial |$\frac{{\partial }\widetilde h}{{\partial } u_x}$| has the form (8.3), so it can be |${\partial }_x$|-exact only if it is zero. Thus, |$\widetilde h$| has the form (8.15) and the lemma is proved. We see that it remains to prove that |$\frac{{\partial }}{{\partial } u_x}\frac{\delta \overline h^{\textrm{DZ}}_1[u]}{\delta u}=0$|⁠. We have (see [2, Section 3.7]) |$\frac{\delta \overline h_1^{\textrm{DZ}}[u]}{\delta u}=h^{\textrm{red}}_0$|⁠. Let us prove that \begin{equation} h^{\textrm{red}}_0=\frac{u^2}{2}+\sum_{g,n\ge 1}\frac{\varepsilon^{2g}}{n!}\sum_{d_1+\ldots+d_n=2g}\left<\tau_0\tau_1\prod\tau_{d_i}\right>^{\textrm{red}}_g\prod u_{d_i}. \end{equation} (8.17) From (7.7) and the string equation for |$F^{\textrm{red}}$| it follows that \begin{equation*} \left.u^{\textrm{red}}_d\right|_{x=0}=t_d+\delta_{d,1}+\sum_{g\ge 0}\varepsilon^{2g}R_{g,d}(t_*), \end{equation*} where |$R_{g,d}\in{\mathbb{C}}[[t_*]]^{(2g+d+1)}$|⁠. Denote the right-hand side of (8.17) by |$Q$|⁠. Using (8.14), we see that \begin{equation*} \left.\left(h_0^{\textrm{red}}\big(u^{\textrm{red}},u^{\textrm{red}}_x,\ldots;\varepsilon\big)-Q\big(u^{\textrm{red}},u^{\textrm{red}}_x,\ldots;\varepsilon\big)\right)\right|_{x=0}=\sum_{g\ge 0}\varepsilon^{2g}R_g(t_*), \end{equation*} where |$R_g\in{\mathbb{C}}[[t_*]]^{(2g+1)}$|⁠. The proof of equation (8.17) is completed by the following lemma. Lemma 8.6. Suppose for a differential polynomial |$P\in \widehat{{\mathcal{A}}}^{[0]}_u$| we have \begin{equation} \left.P\big(u^{\textrm{red}},u^{\textrm{red}}_x,\ldots;\varepsilon\big)\right|_{x=0}=\sum_{g\ge 0}\varepsilon^g T_g(t_*), \end{equation} (8.18) where |$T_g\in{\mathbb{C}}[[t_*]]^{(g+1)}$|⁠. Then |$P=0$|⁠. Proof. Suppose that \begin{equation*} P(u_*;\varepsilon)=\sum_{g\ge g_0}\varepsilon^g P_g(u_*),\quad P_g\in{\mathcal{A}}^{[g]}_u,\quad P_{g_0}\ne 0. \end{equation*} Let \begin{equation*} P_{g_0}(u_*)=\sum_{k=0}^{k_0}P_{g_0,k}(u_*)u_x^k, \end{equation*} where |$\frac{{\partial } P_{g_0,k}}{{\partial } u_x}=0$| and |$P_{g_0,k_0}\ne 0$|⁠. Clearly, we have \begin{equation} \left.P\big(u^{\textrm{red}},u^{\textrm{red}}_x,\ldots;\varepsilon\big)\right|_{x=0}=\varepsilon^{g_0}\big(P_{g_0,k_0}|_{u_d=t_d}+R(t_*)\big)+O(\varepsilon^{g_0+1}), \end{equation} (8.19) where |$R(t_*)\in{\mathbb{C}}[[t_*]]^{(g_0-k_0+1)}$|⁠. Since |$P_{g_0,k_0}\ne 0$|⁠, we see that equation (8.19) contradicts (8.18). Therefore, |$P=0$| and the lemma is proved. Equations (8.14), (7.7), and the string and the dilaton equations for |$F^{\textrm{red}}$| imply that |$\frac{{\partial } h^{\textrm{red}}_0}{{\partial } u_x}=0$|⁠. Therefore, |$\frac{{\partial }}{{\partial } u_x}\frac{\delta \overline h^{\textrm{DZ}}_1[u]}{\delta u}=0$| and the theorem is proved. 8.4 Strong DR/DZ equivalence up to genus |$5$| In Section 8.4.1 we recall a sufficient condition for the strong DR/DZ equivalence conjecture to be true. In Section 8.4.2 we consider a rank |$1$| CohFT (8.1) and show that the strong DR/DZ equivalence conjecture for general |$\alpha $| follows from the case |$\alpha =1$|⁠. Finally, in Section 8.4.3 we prove the strong conjecture at the approximation up to genus |$5$|⁠. 8.4.1 Sufficient condition for the strong DR/DZ equivalence conjecture Consider an arbitrary semisimple CohFT, |$c_{g,n}\colon V^{\otimes n}\to H^{\textrm{even}}({\overline{{\mathcal{M}}}}_{g,n},{\mathbb{C}})$|⁠, where |$\dim V=N$|⁠. Recall that by |${\widetilde u}^\alpha (u^*_*;\varepsilon )$| we denote the normal coordinates (7.8) for the DR hierarchy. Denote by |$K^{\textrm{DR}}_{\widetilde u}$| the operator |$\eta{\partial }_x$| in the coordinates |${\widetilde u}^\alpha $|⁠. In [2] we proved the following proposition. Proposition 8.7 ([2, Section 7.3]). Suppose that the Hamiltonians and the Hamiltonian operators of the DR hierarchy in the coordinates |${\widetilde u}^\alpha $| and the DZ hierarchy are related by a Miura transformation of the form \begin{equation} {\widetilde u}^\alpha\mapsto w^\alpha\big({\widetilde u}^*_*;\varepsilon\big)={\widetilde u}^\alpha+\eta^{\alpha\mu}{\partial}_x\big\{{\mathcal{Q}},\overline g_{\mu,0}[{\widetilde u}]\big\}_{K^{\textrm{DR}}_{{\widetilde u}}}, \end{equation} (8.20) where |${\mathcal{Q}}\in \widehat{{\mathcal{A}}}^{[-2]}_{{\widetilde u}^1,\ldots ,{\widetilde u}^N}$| and |$\frac{{\partial }{\mathcal{Q}}}{{\partial }{\widetilde u}^1}=\varepsilon ^2\left <\tau _0(e_1)\right>_1$|⁠. Then the strong DR/DZ equivalence conjecture is true. 8.4.2 Reduction to the case |$\alpha =1$| Consider a rank |$1$| CohFT (8.1). Then both potentials |$F$| and |$F^{\textrm{red}}$| are power series in |$t_0,t_1,\ldots $| and |$\varepsilon $| that additionally depend on the parameters |$s_1,s_2,\ldots $| and |$\alpha $|⁠. Define an operator |$O$| by |$O:=\alpha \frac{{\partial }}{{\partial }\alpha }+\frac{1}{2}\varepsilon \frac{{\partial }}{{\partial }\varepsilon }$|⁠. From Theorem 6.1 we immediately see that \begin{equation} O F^{\textrm{DR}}=F^{\textrm{DR}}. \end{equation} (8.21) Clearly, we have |$O F=F$|⁠. Since |$\eta ^{1,1}=\frac{1}{\alpha }$|⁠, we get |$O w^{\textrm{top}}=0$|⁠. Then from the construction of the reduced potential |$F^{\textrm{red}}$| in [2, Section 7.3] we can easily see that \begin{equation} OF^{\textrm{red}}=F^{\textrm{red}}. \end{equation} (8.22) Formulas (8.21) and (8.22) imply that if |$F^{\textrm{DR}}$| and |$F^{\textrm{red}}$| are equal for |$\alpha =1$|⁠, then they are equal for an arbitrary |$\alpha $|⁠. Therefore, if the strong DR/DZ equivalence conjecture is true for |$\alpha =1$|⁠, then it is true for an arbitrary |$\alpha $|⁠. 8.4.3 Proof of the equivalence up to genus |$5$| Consider a CohFT (8.1). Let us prove the strong DR/DZ equivalence conjecture at the approximation up to genus |$5$|⁠. From the previous section we know that it is enough to consider the case |$\alpha =1$|⁠. By Theorem 8.4, the normal Miura transformation \begin{equation*} w\mapsto u(w_*;\varepsilon)=w+{\partial}_x^2{\mathcal{P}}, \end{equation*} transforms the DZ hierarchy to its standard form. We have the following formula for the unique density |$\widetilde h_1$| for |$\overline h_1^{\textrm{DZ}}[u]$| of the form (8.2): \begin{equation} \begin{split} \widetilde h_{1}=&\frac{u^3}{6}-\frac{\varepsilon^2}{24}u_x^2-\frac{\varepsilon^4}{120} s_1 u_{xx}^2- \varepsilon^6\left[\left(\frac{s_1^3}{360} +\frac{s_2}{1728}\right)u_{xx}^3+\frac{s_1^2}{420}u_{xxx}^2\right]\\ &-\varepsilon^8\left[\left(\frac{2s_1^5}{525} +\frac{s_1^2 s_2}{504}+\frac{s_3}{34560}\right)u_{xx}^4+\left(\frac{11 s_1^4}{1400}+\frac{11 s_1 s_2}{6720}\right)u_{xxx}^2u_{xx}\right.\\&\qquad\quad\ \,\left. +\left(\frac{s_1^3}{1260}+\frac{s_2}{60480}\right)u_{xxxx}^2\right]\\ &-\varepsilon^{10}\left[\left(\frac{754 s_1^7}{67375}+\frac{13 s_2 s_1^4}{1320}+\frac{13 s_3 s_1^2}{52800}+\frac{13 s_2^2 s_1}{22176}+\frac{13 s_4}{10644480}\right)u_{xx}^5\right.\\ &+\left(\frac{58 s_1^6}{1375}+\frac{7s_2s_1^3}{330}+\frac{7 s_3 s_1}{26400}+\frac{s_2^2}{3168}\right)u_{xxx}^2 u_{xx}^2\\ &\left.+\left(\frac{71 s_1^5}{12600}+\frac{s_1^2s_2}{756}+\frac{s_3}{276480}\right)u_{xxxx}^2u_{xx}+\left(\frac{s_1^4}{3465}+\frac{s_2 s_1}{66528}\right)u_{xxxxx}^2\right]\\ &+O(\varepsilon^{12}). \end{split} \end{equation} (8.23) This formula is given in [10, p. 433] at the approximation up to genus |$4$|⁠, and we are grateful to the authors of [10] for providing us a software that computes the density |$\widetilde h_1$| at the approximation up to genus |$5$|⁠. We see here that |$a_0=1$| and \begin{align*} \alpha_{(2^2)}=&-\frac{s_1}{120},\\ \alpha_{(2^3)}=&-\frac{s_1^3}{360}-\frac{s_2}{1728}, \end{align*} \begin{align*} \alpha_{(2^4)}=&-\frac{2s_1^5}{525}-\frac{s_1^2 s_2}{504}-\frac{s_3}{34560},\\ \alpha_{(2^5)}=&-\frac{754 s_1^7}{67375}-\frac{13 s_2 s_1^4}{1320}-\frac{13 s_3 s_1^2}{52800}-\frac{13 s_2^2 s_1}{22176}-\frac{13 s_4}{10644480}. \end{align*} From equations 8.108.13 we see that |$\alpha _{(2^g)}=\alpha ^{\textrm{DR}}_{(2^g)}$| for |$g=2,3,4,5$|⁠. Since Conjecture 8.1 is true at the approximation up to |$\varepsilon ^{10}$|⁠, we obtain that the standard form of the DZ hierarchy coincides with the DR hierarchy up to genus |$5$|⁠. Note that |${\widetilde u}(u_*;\varepsilon )=\frac{\delta \overline g_0}{\delta u}=u$|⁠. We have \begin{equation*} \left.(F^{\textrm{red}}-F)\right|_{t_0\mapsto t_0+x}={\mathcal{P}}\big(w^{\textrm{top}},w^{\textrm{top}}_x,\ldots;\varepsilon\big). \end{equation*} From the string equations for |$F^{\textrm{red}}$| and |$F$| it follows that |$\frac{{\partial }{\mathcal{P}}}{{\partial } w^1}=-\varepsilon ^2\left <\tau _0\right>_1$|⁠. Then it is easy to see that the Miura transformation |$u\mapsto w(u_*;\varepsilon )$| has the form \begin{equation*} w(u_*;\varepsilon)=u+{\partial}_x^2{\mathcal{Q}}, \end{equation*} where |$\frac{{\partial }{\mathcal{Q}}}{{\partial } u^1}=\varepsilon ^2\left <\tau _0\right>_1$|⁠. Therefore, the sufficient condition from Proposition 8.7 is satisfied and we conclude that the strong DR/DZ equivalence conjecture is true at the approximation up to genus |$5$|⁠. Funding This work was supported by European Union’s Horizon 2020 research and innovation programme [Marie Skłodowska-Curie grant agreement No. 797635 to A.B., ERC-2012-AdG-320368-MCSK to A.B., and RFFI-16-01-00409 to A.B.]; Einstein foundation [to J.G.]. P. R. was partially supported by a Chaire CNRS/Enseignement superieur 2012–2017 grant. Acknowledgments We would like to thank Andrea Brini, Guido Carlet, Rahul Pandharipande, Sergey Shadrin and Dimitri Zvonkine for useful discussions. Communicated by Prof Igor Krichever References [1] Buryak , A. “ Double ramification cycles and integrable hierarchies .” Comm. Math. Phys. 336 , no. 3 ( 2015 ): 1085 – 1107 . [2] Buryak , A. , Dubrovin , B. , Guéré , J. , and Rossi , P. . “ Tau-structure for the Double Ramification Hierarchies .” Comm. Math. Phys. 363 , no. 1 ( 2018 ): 191 – 260 . [3] Buryak , A. and Guéré , J. . “ Towards a description of the double ramification hierarchy for Witten’s |$r$|-spin class .” J. Math. 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Leblé,, Thomas

doi: 10.1093/imrn/rnz020pmid: N/A

Abstract We prove, for any |$\beta>0$|⁠, a central limit theorem for the fluctuations of linear statistics in the |$\textrm{Sine}_{\beta }$| process, which is the infinite volume limit of the random microscopic behavior in the bulk of one-dimensional log-gases at inverse temperature |$\beta $|⁠. If |$\overline{\varphi }$| is a compactly supported test function of class |$C^4$|⁠, and |${\mathcal{C}}$| is a random point configuration distributed according to |$\textrm{Sine}_{\beta }$|⁠, the integral of |$\overline{\varphi }(\cdot / \ell )$| against the random fluctuation |$d{\mathcal{C}} - dx$| converges in law, as |$\ell $| goes to infinity, to a centered normal random variable whose standard deviation is proportional to the Sobolev |$H^{1/2}$| norm of |$\overline{\varphi }$| on the real line. The proof relies on the Dobrushin–Landford–Ruelle equations for |$\textrm{Sine}_{\beta }$| established by Dereudre–Hardy–Maïda and the author, the Laplace transform trick introduced by Johansson, and a transportation method previously used for |$\beta $|-ensembles at macroscopic scale. 1 Introduction 1.1 The Sine-beta process The |$\textrm{Sine}_{\beta }$| process is obtained as the infinite volume, or thermodynamic, limit of the microscopic behavior in the bulk of a one-dimensional log-gas. Let |$\beta> 0$| be a fixed value of the inverse temperature parameter. For |$N \geq 1$|⁠, the probability measure on |${\mathbb{R}}^N$| given by the density \begin{equation} d{\mathbb{P}}_{N, \beta}(x_1, \dots, x_N):= \frac{1}{\textrm{Z}_{N, \beta}} \exp\left( - \beta \left( \sum_{i < j} - \log |x_i - x_j| + \sum_{i=1}^N {N} \frac{x_i^2}{2} \right) \right), \end{equation} (1.1) with respect to the Lebesgue measure on |${\mathbb{R}}^N$|⁠, where |$\textrm{Z}_{N, \beta }$| is a normalization constant, is the canonical Gibbs measure of a one-dimensional log-gas at (inverse) temperature |$\beta $|⁠. It corresponds physically to a system of |$N$| particles interacting via a pairwise repulsive logarithmic potential and confined by some external field that we take here to be quadratic, for simplicity. For |$\beta = 1, 2, 4$|⁠, the density |${\mathbb{P}}_{N, \beta }$| coincides with the joint law of the |$N$| eigenvalues of certain classical models of random matrices: the Gaussian orthogonal, unitary, and symplectic ensemble, respectively, with a correct choice of the variance, due to the presence of |$\beta $| in front of |$\sum _i x_i^2$|⁠. We refer to [9] for a comprehensive survey of this connection. In fact, for every |$\beta> 0$|⁠, there exists a model of random matrices with independent entries, known as the “tridiagonal model”, discovered in [7], whose random eigenvalues behave like the particles of a log-gas at inverse temperature |$\beta $|⁠. From a statistical physics point of view, one-dimensional log-gases are interesting toy models due to the fact that interaction is singular and, most importantly, long-range; in contrast to many pair potentials studied in the literature, the logarithmic interaction does not tend rapidly to zero with the distance between the particles (in fact, not at all). Under |${\mathbb{P}}_{N, \beta }$|⁠, it is known that the particles typically arrange themselves in an interval approximately given by |$[-2, 2]$|⁠. We consider this as being the macroscopic behavior of the system. To investigate its microscopic behavior, we zoom in by a factor |$N$|⁠. We can see the random |$N$|-tuple |${\mathcal{C}}_N:= ({N}x_1, \dots , {N}x_N)$| as a random, finite point configuration in |${\mathbb{R}}$|⁠. The existence of a limit, or even of limit points, in some interesting topology, to the law of |${\mathcal{C}}_N$| is a difficult question. It was shown in [22] and [15] (for a closely related model, whose limit turns out to be the same) that when taking the thermodynamic/infinite volume limit, that is, letting |$N \to \infty $|⁠, the random, finite point configuration |${\mathcal{C}}_N$| converges in law to some random, infinite point configuration on |${\mathbb{R}}$|⁠, whose law is called the |$\textrm{Sine}_{\beta }$| process. In both cases, a description of |$\textrm{Sine}_{\beta }$| is given through a system of coupled stochastic differential equations. Finally, since the topology of convergence is local, |$\textrm{Sine}_{\beta }$| only captures the microscopic behavior “near |$0$|”. One could ask instead for the limit of |${\mathcal{C}}_N$| translated by |$cN$|⁠, where |$c$| is some parameter. It turns out that for |$c$| in |$(-2, 2)$|⁠, the law of the limit is the same, up to a scaling on the average density of points. We call this the bulk behavior. For |$c = \pm 2$|⁠, one obtains the edge behavior, whose limit is named the |$\textrm{Airy}_\beta $| process. For |$|c|> 2$|⁠, the limit point process is almost surely empty. 1.2 Main result: CLT for fluctuations of linear statistics 1.2.1 Definitions If |${\mathcal{C}}$| is a point configuration on |${\mathbb{R}}$|⁠, and |$\varphi $| a continuous, compactly supported test function, we will often use the notation |$\int \varphi (x)\ \textrm{d}{\mathcal{C}}(x)$| for \begin{equation*} \int \varphi(x) \ \textrm{d}{\mathcal{C}}(x):= \sum_{p \in{\mathcal{C}}} \varphi(p). \end{equation*} Definition 1.1. (Fluctuations of linear statistics). Let |$\varphi $| be a function of class |$C^0$|⁠, compactly supported on |${\mathbb{R}}$|⁠, and let |${\mathcal{C}}$| be a point configuration on |${\mathbb{R}}$|⁠. We define the fluctuation of the linear statistic associated to |$\varphi $| as the quantity \begin{equation} \textrm{Fluct}[\varphi]({\mathcal{C}}):= \int \varphi(x) (\textrm{d}{\mathcal{C}}(x) - \textrm{d}x). \end{equation} (1.2) Definition 1.2. (Rescaled function). Let |$\overline{\varphi }$| be a test function and |$\ell> 0$|⁠. We define the associated rescaled test function |$\varphi _{\ell }$| as \begin{equation} \varphi_{\ell}: x \mapsto \overline{\varphi} \left( \frac{x}{\ell} \right). \end{equation} (1.3) Definition 1.3. (⁠|$H^{1/2}$| norm on the real line). Whenever the following quantity is finite, we call it the |$H^{1/2}$| norm of |$\overline{\varphi }$| \begin{equation} \|\overline{\varphi}\|_{H^{\frac{1}{2}}}:= \frac{1}{2\pi} \left( \iint_{{\mathbb{R}} \times{\mathbb{R}}} \left( \frac{ \overline{\varphi}(x) - \overline{\varphi}(y)}{x-y}\right)^2 \ \textrm{d}x \ \textrm{d}y\right)^{1/2}. \end{equation} (1.4) We may observe, that for example, when |$\overline{\varphi }$| is of class |$C^1$| and compactly supported, then |$\|\overline{\varphi }\|_{H^{\frac{1}{2}}}$| is finite. Moreover, it is easy to check that the |$H^{1/2}$| norm is invariant under rescaling as in (1.3). 1.2.2 Statement of the result Theorem 1 CLT for fluctuations of linear statistics under |$\textrm{Sine}_{\beta }$|⁠. Let |$\overline{\varphi }$| be a fixed test function of class |$C^4$|⁠, compactly supported on |${\mathbb{R}}$|⁠, and for |$\ell> 0$|⁠, let |$\varphi _{\ell }$| be the rescaled function, as in Definition 1.2. Let |${\mathcal{C}}$| be a random point configuration of law |$\textrm{Sine}_{\beta }$|⁠. The following convergence holds, in law, as |$\ell \to \infty $|⁠: \begin{equation*} \textrm{Fluct}[\varphi_{\ell}]({\mathcal{C}}) \implies \text{Gaussian r.v. of mean {$0$} and variance {$\frac{2}{\beta} \|\overline{\varphi}\|_{H^{\frac{1}{2}}}^2$}}. \end{equation*} To the best of our knowledge, Theorem 1 is the 1st result concerning the fluctuations of smooth statistics for the limit process |$\textrm{Sine}_{\beta }$| at arbitrary values of |$\beta $|⁠. 1.2.3 Notation Henceforth, we let |$\overline{\varphi }$| be a fixed test function of class |$C^4$|⁠, compactly supported in |${\mathbb{R}}$|⁠, and for |$\ell> 0$| we let |$\varphi _{\ell }$| be as in Definition 1.2. For lightness of notation, we drop the subscript |$\ell $| and write |$\varphi $| instead of |$\varphi _{\ell }$|⁠. Also, for simplicity, we assume that |$\overline{\varphi }$| is supported in |$(-1,1)$|⁠, so that |$\varphi = \varphi _{\ell }$| is supported in |$(-\ell , \ell )$|⁠. We work with two parameters |$\ell , \lambda $|⁠. We will always assume that |$\ell , \lambda $| satisfy \begin{equation} 100 < \ell < \frac{\lambda}{1000}, \end{equation} (1.5) and we will use the notation |$a \preceq b$| as follows: \begin{equation*} a \preceq b \iff |a| \leq C|b|, \end{equation*} where |$C$| is some multiplicative constant independent of |$\ell , \lambda $|⁠, provided (1.5) is satisfied. We will sometimes write |$O_{\bullet }(b)$| to denote a quantity that is |$\preceq b$|⁠. Most implicit constants will depend on the test function |$\overline{\varphi }$|⁠. If |$A$| is a quantity depending on |$\ell , \lambda $|⁠, we use the notation |$A = o_{\ell , \lambda }(1)$| to denote the fact that \begin{equation*} \lim_{\ell \rightarrow \infty} \lim_{\lambda \rightarrow \infty} A = 0. \end{equation*} We let |$\Lambda $| be the interval |$(-\lambda , \lambda )$|⁠. All the expectations, denoted by |${\mathbb{E}}$|⁠, are expectations under |$\textrm{Sine}_{\beta }$|⁠, and all the probabilities, denoted by |${\mathbb{P}}$|⁠, are probabilities for |$\textrm{Sine}_{\beta }$|⁠. 1.3 Strategy of the proof and connection with other works 1.3.1 Strategy of proof The proof relies on three main ingredients: The DLR (Dobrushin–Landford–Ruelle) equations of [6]. The Laplace transform trick of [14]. The transportation method inspired by [3]. The DLR equations provide a version of the Gibbs measure (1.1) for “|$N~=~+~\infty $|” and thus give a representation |$\textrm{Sine}_{\beta }$| as an infinite-volume Gibbs measure, allowing for a “statistical physics approach”. We state these equations precisely in Section 6.2; let us think of them as describing |$\textrm{Sine}_{\beta }$|⁠, in any interval, as a mixture of Gibbs measures resembling |${\mathbb{P}}_{N, \beta }$|⁠. The CLT for fluctuations of linear statistics of log-gases has been proven by [14] in the context of Hermitian random matrices, stated as a limit in law as |$N \to \infty $| of fluctuations at macroscopic scale. A key point of the proof is the following observation: forming the Laplace transform of the fluctuations of |$\varphi $| amounts to computing the partition function of a log-gas with a perturbed external field, where |$\frac{1}{2} x_i^2$| in (1.1) is replaced by |$\frac{1}{2} x_i^2 + s \varphi (x_i)$|⁠, where |$s$| is small, and related to the parameter of the Laplace transform. More precisely, one is led to consider the ratio of the perturbed partition function and the original partition function, and the argument boils down to proving fine estimates of this ratio. One way to compare the partition functions is to use a change of variables, or transportation method, as in for example, [18], [1], and [3]. It effectively shifts the focus from the external field to the associated equilibrium measure, in the sense of logarithmic potential theory. Then the question becomes to compute the perturbed equilibrium measure, to push the original one onto the perturbed one by some change of variables (or transportation map), and to use this transport to estimate the ratio of the partition functions. This is closely related to the “loop equations” approach. Our proof is in the same spirit, with several modifications: The papers cited above treat linear statistics at macroscopic scale and consider the limit in law as |$N \to \infty $| of |$\sum _{i=1}^N \varphi (x_i)$|⁠, when |$(x_1, \dots , x_N)$| are distributed according to a Gibbs measure similar to |${\mathbb{P}}_{N, \beta }$|⁠, with a possibly more general choice of external field. The CLT is also known to hold at mesoscopic scale, when |$\varphi $| is taken as |$\overline{\varphi }(\cdot /N^{\delta })$|⁠, for |$\delta \in (0, \frac{1}{2})$|⁠, see [2].In contrast, the present work deals with the microscopic scale and with the infinite process itself. Rescaling functions as in (1.3) may be understood as considering “large microscopic scales”. Even for a compactly supported test function, the particles in the support feel the interaction of the infinite, exterior configuration, which acts on them as a random external field. This new element of the analysis is specific to working with the infinite process. When comparing the partition functions, there is usually a term (here |$\textsf{Main}_s$|⁠, see (3.14)) whose magnitude is a priori of order |$1$|⁠, and must then be studied more carefully to show that it is in fact |$o(1)$|⁠. This can be done by a technical bootstrap argument (in fact, this way, one can even obtain an all-order expansion of the partition function as in [4]), and another approach uses a trick, relying on the independent knowledge of the partition function up to order |$N$|⁠, as in [3]. Here, by recasting a priori bounds on fluctuations in terms of discrepancies, and by using good discrepancy estimates for |$\textrm{Sine}_{\beta}$|⁠, we are able to show directly that |$\textsf{Main}_s$| is |$o(1)$|⁠. 1.3.2 Connections with other works When |$\beta = 2$|⁠, the point process acquires a particularly rich determinantal structure, allowing for many explicit computations. In this case, the CLT for fluctuations of smooth enough functions was known since [20], see also [19]. Let us observe that, for |$\beta = 2$|⁠, the CLT is known to hold as soon as the test function is in |$H^{1/2}({\mathbb{R}})$|⁠, to be compared with the requirement that |$\overline{\varphi } \in C^4_c$| here. The optimal regularity condition needed in the general |$\beta $| case is an open question. Several facts concerning the number of points under |$\textrm{Sine}_{\beta }$| have been proven. A CLT follows from [16], large deviations were proven in [12, 13], and a maximal deviation result in [11]. The transportation strategy does not accommodate well to non-smooth functions like indicator functions, and we are unable to easily retrieve these results with the present techniques. The rigidity of the process in the sense of Ghosh–Peres, that is, the fact that the knowledge of the configuration outside a given compact set almost surely prescribes the number of points in that set, was proven by [5] and also obtained in [6] in a very different way. The proof of [5] follows the approach of [10] and relies on the fact that the variance of linear statistics is controlled by the |$H^{1/2}$| norm of the test function, which had been established for random matrix models and can be passed to the limit. We believe that our “statistical physics” approach could yield similar bounds and hence the rigidity result, but one needs to go over all the estimates beyond the “rescaled cases” |$\varphi = \varphi _{\ell } = \overline{\varphi }(\cdot / \ell )$| and state them in full generality, with controls depending more precisely on |$\varphi $|⁠; we do not undertake this here. 1.3.3 Plan of the paper In Section 1.6, we discuss discrepancy estimates for the |$\textrm{Sine}_{\beta }$| process and state an a priori bound on the fluctuations on linear statistics, in terms of the discrepancies. We will rely constantly on this bound in order to control the error terms in the Laplace transform expansion. In Section 2, we define the perturbation measure, which formally corresponds to the change induced on the average density of points when treating the test function |$\varphi $| as an additional external field applied to each particle. This perturbation measure is slightly singular, and we work in fact with a regularized version, the approximate perturbation measure . In Section 3, we define the perturbed measure, the transport map from the original measure (the constant density) to the perturbed one, and we expand the energy along this transport. In Section 4, we compare the interaction energy before and after transport and show that most terms are negligible. In Section 6, we combine all previous elements to give the proof of the CLT. Many parts of the argument are rather elementary but involve some lengthy computations. For legibility, we have postponed most of the computations to Section 7. 1.4 Semi-norms We will often use |$g^{(\textsf{k})}$| to denote the |$\textsf{k}$|-th derivative of |$g$|⁠. Definition 1.4. (Semi-norms and local semi-norms). Let |$g$| be a test function, compactly supported on |${\mathbb{R}}$|⁠. For |$\textsf{k} \geq 0$|⁠, if |$g$| is assumed to be of class |$C^{\textsf{k}}$|⁠, we let \begin{equation*} |g|_{\textsf{k}}:= \sup_{x} |g^{(\textsf{k})}(x)|, \end{equation*} and for |$x$| in |${\mathbb{R}}$|⁠, letting |$V_{x}$| denote the neighborhood |$V_{x}:= [x-3, x+3]$|⁠, we write \begin{equation} |g|_{\textsf{k}, V_{x}}:= \sup_{y \in V_{x}} |g^{(\textsf{k})}(y)|. \end{equation} (1.6) The following bounds will be used repeatedly: \begin{equation} |\varphi_{\ell}|_{\textsf{k}} = \frac{1}{\ell^{\textsf{k}}} |\overline{\varphi}|_{\textsf{k}}, \quad \|\varphi_{\ell}^{(\textsf{k})}\|_{L^{\textsf{p}}} = \|\overline{\varphi}\|_{L^{\textsf{p}}} \ell^{\frac{1}{\textsf{p}} - \textsf{k}}. \end{equation} (1.7) 1.5 Discrepancy and discrepancy estimates Throughout the paper, an important role is played by the discrepancy estimates, for they provide an a priori bound on the size of fluctuations that we will repeatedly use to control error terms. If |${\mathcal{C}}$| is a point configuration and |$I$| is an interval, we denote by |${\mathcal{C}}_{I}$| the restriction of |${\mathcal{C}}$| to |$I$|⁠. Definition 1.5. (Discrepancy). Let |${\mathcal{C}}$| be a point configuration on |${\mathbb{R}}$|⁠, and let |$I$| be an interval. The discrepancy of |${\mathcal{C}}$| in |$I$| is the difference between the number of points of |${\mathcal{C}}$| in |$I$| and its expected value, namely the length of |$I$|⁠. We write \begin{equation*} \textrm{Discr}_{I}:= |{\mathcal{C}}_{I}| - |I| = \int \textsf{1}_{I} (\textrm{d}{\mathcal{C}}(x) - \textrm{d}x). \end{equation*} If |$a,b$| are integers, with possibly |$a> b$|⁠, we let \begin{equation*} \textrm{Discr}_{[a,b]}:= \int_{a}^b \textsf{1}_{I} (\textrm{d}{\mathcal{C}}(x) - \textrm{d}x). \end{equation*} It is known, see for example, [17][Lemma 3.2], that if |$I$| has length at least |$1$|⁠, we have \begin{equation} {\mathbb{E}} \left[ \left(\textrm{Discr}_I\right)^2 \right] \preceq |I|, \quad{\mathbb{E}} \left[ | \textrm{Discr}_{I} | \right| \preceq \sqrt{|I|}. \end{equation} (1.8) Moreover, it was shown in [17][Remark 3.3] that, for |$\textrm{Sine}_{\beta }$|⁠, it holds \begin{equation*} \liminf_{R \to \infty} \frac{1}{2R} {\mathbb{E}} \left[ \left(\textrm{Discr}_{[-R,R]}\right)^2 \right] = 0, \end{equation*} and careful inspection of the argument yields the stronger statement, proven in [8] \begin{equation} {\mathbb{E}} \left[ \left(\textrm{Discr}_{[-R,R]}\right)^2 \right] = o(R). \end{equation} (1.9) Of course, since |$\textrm{Sine}_{\beta }$| is stationary, it implies that the variance of the number of points in any interval of length |$R$| is |$o(R)$|⁠. 1.6 A priori bound on the fluctuations We let |$\widetilde{D}_i$| be the quantity \begin{equation*} \widetilde{D}_i:= |\textrm{Discr}_{[0, i]}| + |\textrm{Discr}_{[i, i+1]}| + 1. \end{equation*} Proposition 1.6. (A priori bound on the fluctuations). Let |$g$| be a test function of class |$C^1$|⁠, compactly supported on |${\mathbb{R}}$|⁠. \begin{equation} \left| \int g(x) (\textrm{d}{\mathcal{C}} - \textrm{d}x) \right| \preceq \sum_{i=-\infty}^\infty |g|_{\textsf{1}, V_i} \widetilde{D}_i[{\mathcal{C}}]. \end{equation} (1.10) Moreover, for |$\lambda $| fixed, we may choose to replace |$\widetilde{D}_i$| by either |$\widetilde{D}^{\textrm{Left}}_i$| or |$\widetilde{D}^{\textrm{Right}}_i$|⁠, with \begin{equation*} \widetilde{D}^{\textrm{Left}}_i:= |\textrm{Discr}_{[-\lambda, i]}| + |\textrm{Discr}_{[i, i+1]}| + 1, \quad \widetilde{D}^{\textrm{Right}}_i:= |\textrm{Discr}_{[i, \lambda]}| + |\textrm{Discr}_{[i, i+1]}| + 1. \end{equation*} The proof of Proposition 1.6 is elementary; we postpone it to Section 7.1. Remark 1.7. (Bounds on |$\widetilde{D}, \widetilde{D}^{\textrm{Left}}, \widetilde{D}^{\textrm{Right}}$|⁠). In view of (1.8), for |$|i| \geq 1$|⁠, we have \begin{equation} {\mathbb{E}} \left[ \left(\widetilde{D}_i\right)^2 \right] \preceq |i|, \quad{\mathbb{E}} \left[ \widetilde{D}_i \right] \preceq \sqrt{|i|}, \end{equation} (1.11) and in fact we have, in view of (1.9), as |$|i| \to \infty $| \begin{equation} {\mathbb{E}} \left[ \left(\widetilde{D}_i\right)^2 \right] = o\left(|i|\right), \quad{\mathbb{E}} \left[ \widetilde{D}_i \right] = o \left( \sqrt{|i|} \right). \end{equation} (1.12) We obtain similar estimates for |$\widetilde{D}^{\textrm{Left}}_i$|⁠, resp. |$\widetilde{D}^{\textrm{Right}}_i$| when replacing |$|i|$| by |$|\lambda + i|$|⁠, resp. |$|\lambda - i|$|⁠. 2 The Perturbation Measure 2.1 The Cauchy principal value Definition 2.1. (Cauchy principal value). Let |$g$| be a test function of class |$C^1$|⁠, compactly supported on |${\mathbb{R}}$|⁠. For |$x$| in |${\mathbb{R}}$|⁠, we define \begin{equation} \textbf{PV} \int \frac{g(t)}{t-x} \ \textrm{d}t:= \int_{0}^{+\infty} \frac{g(x+u) - g(x-u)}{u} \ \textrm{d}u, \end{equation} (2.1) where |$\textbf{PV}$| stands for “principal value”. Definition 2.2. (The quantity |${\mathfrak{H}}_{\lambda , \varphi }$|⁠). For |$x$| in |${\mathbb{R}}$|⁠, we define |${\mathfrak{H}}_{\lambda , \varphi }(x)$| as \begin{equation} {\mathfrak{H}}_{\lambda, \varphi}(x):= \frac{1}{\pi} \textbf{PV} \int \frac{\sqrt{\lambda^2-t^2} \varphi^{\prime}(t)}{t-x} \ \textrm{d}t. \end{equation} (2.2) Remark 2.3. Since |$\varphi $| is at least, |$C^2$| and compactly supported in |$(-\ell , \ell )$|⁠, we can see |$\phi _{\Lambda }$|⁠, defined by \begin{equation} \phi_{\Lambda}: t \mapsto \sqrt{\lambda^2 - t^2} \varphi^{\prime}(t), \end{equation} (2.3) as a compactly supported function of class |$C^1$|⁠, so the “principal value” notation in (2.2) makes sense, in view of Definition 2.1. 2.2 The perturbation measure Definition 2.4. (The perturbation measure). For |$x$| in |$(-\lambda , \lambda )$|⁠, we define |${\mathfrak{m}}_{\lambda , \varphi }(x)$| as \begin{equation} {\mathfrak{m}}_{\lambda, \varphi}(x):= \frac{-1}{\pi \sqrt{\lambda^2-x^2}} {\mathfrak{H}}_{\lambda, \varphi}(x). \end{equation} (2.4) The density |${\mathfrak{m}}_{\lambda , \varphi }$| will be called the perturbation measure. Definition 2.5. (The logarithmic potential of |${\mathfrak{m}}_{\lambda , \varphi }$|⁠). For |$x$| in |${\mathbb{R}}$|⁠, we let \begin{equation} \textsf{LP}_{\lambda, \varphi}(x):= \int - \log |x-y| {\mathfrak{m}}_{\lambda, \varphi}(y) \ \textrm{d}y. \end{equation} (2.5) Lemma 2.6. (Properties of the perturbation measure). The density |${\mathfrak{m}}_{\lambda , \varphi }$| is integrable on |$\Lambda $|⁠, of total mass |$0$|⁠. The logarithmic potential generated by |${\mathfrak{m}}_{\lambda , \varphi }$| is well defined and satisfies the following equation for |$x$| in |$\Lambda $|⁠: \begin{equation} \left(\textsf{LP}_{\lambda, \varphi}\right)^{\prime}(x) = \varphi^{\prime}(x). \end{equation} (2.6) These properties are well known, and we refer to the book [21], see also Section 7.2. 2.3 Bounds on the perturbation measure Lemma 2.7. (Bounds on the perturbation measure |${\mathfrak{m}}_{\lambda , \varphi }$|⁠). We have \begin{align} {\mathfrak{m}}_{\lambda, \varphi} & \preceq \begin{cases} \frac{1}{\ell} & |x| \leq 2\ell, \\ \frac{\sqrt{\lambda} \ell}{x^2 \sqrt{\lambda-|x|}} & |x| \geq 2\ell \end{cases}, \\{\mathfrak{m}}_{\lambda, \varphi}^{(\textsf{1})} & \preceq \begin{cases} \frac{1}{\ell^2} & |x| \leq 2\ell, \\ \frac{\ell}{|x| \sqrt{\lambda} (\lambda - |x|)^{3/2}} + \frac{\sqrt{\lambda} \ell}{|x|^3 \sqrt{\lambda-|x|}} & |x| \geq 2\ell \end{cases}, \\{\mathfrak{m}}_{\lambda, \varphi}^{(\textsf{2})} & \preceq \begin{cases} \frac{1}{\ell^3} & |x| \leq 2\ell, \\ \frac{\ell}{\lambda^{3/2} (\lambda - |x|)^{5/2}} + \frac{\ell}{x^2 \lambda^{1/2} (\lambda - |x|)^{3/2}} + \frac{\sqrt{\lambda} \ell}{x^4 \sqrt{\lambda-|x|}} & |x| \geq 2\ell \end{cases}. \end{align} Lemma 2.7 follows from elementary computations, see Section 7.3. 2.4 The approximate perturbation measure The perturbation measure |${\mathfrak{m}}_{\lambda , \varphi }$| satisfies the exact relation (2.6) but is singular near |$\pm \lambda $|⁠. We will work instead with an approximate perturbation measure |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$|⁠, constructed below, which is more regular, and, in fact, vanishes near the endpoints. Of course, passing from |${\mathfrak{m}}_{\lambda , \varphi }$| to |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$| induces an error on the logarithmic potential, which we need to control. Lemma 2.8. (The approximate perturbation measure). There exists a function |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$| of class |$C^2$|⁠, compactly supported in |$(-\lambda , \lambda )$|⁠, satisfying |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$| = |${\mathfrak{m}}_{\lambda , \varphi }$| on |$[- \lambda + \ell , \lambda - \ell ]$|⁠. The masses of |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$| and |${\mathfrak{m}}_{\lambda , \varphi }$| coincide near each endpoint, that is, \begin{equation} \int_{-\lambda}^{-\lambda + \ell} \widetilde{{\mathfrak{m}}}_{\lambda, \varphi} = \int_{-\lambda}^{-\lambda + \ell} {\mathfrak{m}}_{\lambda, \varphi}, \quad \int_{\lambda - \ell}^{\lambda} \widetilde{{\mathfrak{m}}}_{\lambda, \varphi} = \int_{\lambda - \ell}^{\lambda} {\mathfrak{m}}_{\lambda, \varphi} \end{equation} (2.10) For |$x$| in |$[-\lambda , -\lambda + \ell ] \cup [\lambda - \ell , \lambda ]$| and for any |$\textsf{k} = \textsf{0}, \textsf{1}, \textsf{2}$|⁠, we have the bound \begin{equation} |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}^{(\textsf{k})}(x)| \preceq \frac{1}{\ell^{\textsf{k}}} \frac{\ell}{\lambda^{3/2} \ell^{1/2}}, \end{equation} (2.11) with implicit multiplicative constants depending on |$\textsf{k}$| and |$\overline{\varphi }$| but not on |$\ell , \lambda , x$|⁠. |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$| is identically |$0$| on |$[-\lambda , -\lambda + \ell /4]$| and on |$[\lambda - \ell /4, \lambda ]$|⁠. The construction of |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$| is given in Section 7.5. Lemma 2.9. (Additional properties of |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$|⁠). \begin{equation} \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(x) \preceq \begin{cases} \frac{1}{\ell} & |x| \leq 2\ell, \\ \frac{\ell}{x^2} & 2\ell \leq |x| \leq \lambda/2 \\ \frac{\ell}{\lambda^{3/2} \sqrt{\lambda - x}} & \lambda/2 \leq |x| \leq \lambda - \ell, \\ \frac{\ell}{\lambda^{3/2} \ell^{1/2}} & \lambda - \ell \leq |x| \leq \lambda \end{cases}, \end{equation} (2.12) \begin{align} & \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}^{(1)}(x) \preceq \begin{cases} \frac{1}{\ell^2} & |x| \leq 2 \ell \\ \frac{\ell}{|x|^3} & 2\ell \leq |x| \leq \lambda/2, \\ \frac{\ell}{\lambda^{3/2} (\lambda - |x|)^{3/2}} & \lambda/2 \leq |x| \leq \lambda - \ell, \\ \frac{\ell}{\lambda^{3/2} \ell^{3/2}} & \lambda - \ell \leq |x| \leq \lambda \end{cases}, \end{align} (2.13) \begin{equation} \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}^{(2)}(x) \preceq \begin{cases} \frac{1}{\ell^3} & |x| \leq 2 \ell \\ \frac{\ell}{x^4} & 2\ell \leq |x| \leq \lambda/2, \\ \frac{\ell}{\lambda^{3/2} (\lambda - |x|)^{5/2}} & \lambda/2 \leq |x| \leq \lambda - \ell, \\ \frac{\ell}{\lambda^{3/2} \ell^{5/2}} & \lambda - \ell \leq |x| \leq \lambda \end{cases}, \end{equation} (2.14) \begin{equation} \|\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}\|_{L^1} \preceq 1. \end{equation} (2.15) Proof. The 1st three inequalities are consequences of (2.7), (2.8), and (2.9), for |$|x| \leq 2\ell $| and |$2\ell \leq |x| \leq \lambda - \ell $|⁠, because we do not change the measure there. They follow from (2.11) for |$\lambda - \ell \leq |x| \leq \lambda $|⁠. To obtain (2.15), we split the integral into four parts: \begin{equation*} \int_{|t| \leq 2\ell} |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(t)| \ \textrm{d}t + \int_{2\ell \leq |t| \leq \lambda /2} |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(t)|\ \textrm{d}t + \int_{\lambda /2 \leq |t| \leq \lambda - \ell} |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(t)| \ \textrm{d}t + \int_{\lambda - \ell \leq |t| \leq \lambda} |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(t)| \ \textrm{d}t, \end{equation*} then (2.15) follows from (2.12) and an elementary computation. 2.5 The error on the logarithmic potential Definition 2.10. (Error on the logarithmic potential). We introduce the quantity \begin{equation} \textrm{Error}\textsf{LP}_{\lambda, \varphi}(x):= \int - \log |x-y| \left( \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y) - {\mathfrak{m}}_{\lambda, \varphi}(y) \right) \ \textrm{d}y. \end{equation} (2.16) Proposition 2.11. Error on the logarithmic potential. We have \begin{equation*} \textrm{Error}\textsf{LP}_{\lambda, \varphi} = \textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Left}} + \textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Right}}, \end{equation*} where |$\textrm{Error}\textsf{LP}_{\lambda , \varphi }^{\textrm{Left}}$|⁠, |$\textrm{Error}\textsf{LP}_{\lambda , \varphi }^{\textrm{Right}}$| satisfy \begin{equation} \textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Left}}(x) \preceq \frac{\ell^{3/2} \log(\lambda)}{\lambda^{3/2}}, \quad \textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Right}}(x) \preceq \frac{\ell^{3/2} \log(\lambda)}{\lambda^{3/2}}, \quad \textrm{ for} |x| \leq 2 \lambda \end{equation} (2.17) and \begin{equation} \begin{cases} \left(\textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Left}}\right)^{(\textsf{1})}(x) \preceq \frac{\ell^{5/2}}{\lambda^{3/2} (-\lambda - x)^2} & |x - (-\lambda)| \geq 2\ell, \\ \left(\textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Right}}\right)^{(1)}(x) \preceq \frac{\ell^{5/2}}{\lambda^{3/2} (\lambda - x)^2}, & |\lambda - x| \geq 2\ell. \end{cases} \end{equation} (2.18) The proof of Proposition 2.11 is given in Section 7.6. 2.6 The variance term Lemma 2.12. (The variance term). We have the following identity: \begin{equation} \iint - \log |x-y| \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(x) \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y) \ \textrm{d}x \ \textrm{d}y = 2 \|\overline{\varphi}\|_{H^{\frac{1}{2}}}^2 + \textrm{ErrorVar}, \end{equation} (2.19) with |$\textrm{ErrorVar}$| bounded as follows: \begin{equation} \textrm{ErrorVar} \preceq \frac{\ell^3 \log(\lambda)}{\lambda^{3}} + \frac{\ell^2}{\lambda^2}. \end{equation} (2.20) In particular, we obtain \begin{equation} s^2 \textrm{ErrorVar} = s^2 o_{\ell, \lambda}(1). \end{equation} (2.21) The proof of Lemma 2.12 is given in Section 7.7. 3 Transporting to the Perturbed Measure 3.1 The perturbed measure The |$\textrm{Sine}_{\beta }$| process has intensity |$1$|⁠. Adding a perturbative external field will formally change the average density of points from a constant density to the perturbed density |$(1 + {\mathfrak{m}}_{\lambda , \varphi }(x))dx$|⁠. Since we work with the approximate perturbation |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$|⁠, it leads to the following definition. Definition 3.1. (The perturbed measure). Let |$\textrm{s}_{\textrm{max}}$| be defined as \begin{equation} \textrm{s}_{\textrm{max}}:= \frac{1}{2} \max\left(1, |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}|_{\textsf{0}}, \|\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}\|_{L^1} \right)^{-1}. \end{equation} (3.1) For any |$s$| such that |$|s| \leq \textrm{s}_{\textrm{max}}$|⁠, we define the perturbed measure|$\mu _{s}$| as \begin{equation} \mu_{s}(x) = 1 + s \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(x). \end{equation} (3.2) Of course, |$\mu _{s}$| depends on |$\lambda , \overline{\varphi }, \ell $|⁠, but for simplicity we only keep track of the parameter |$s$|⁠. In the following, |$s$| is always assumed to satisfy |$|s| \leq \textrm{s}_{\textrm{max}}$|⁠. Lemma 3.2. (Properties of the perturbed measure). The density |$\mu _{s}$| is of class |$C^2$|⁠, is bounded above and below on |$[-\lambda , \lambda ]$| by universal positive constants, and satisfies \begin{equation*} \int_{-\lambda}^{\lambda} \mu_{s}(x)\ \textrm{d}x = \int_{-\lambda}^{\lambda} 1 \ \textrm{d}x. \end{equation*} The density |$\mu _{s}$| is equal to |$1$| on |$[-\lambda , -\lambda + \ell /4]$| and on |$[\lambda - \ell /4, \lambda ]$|⁠. Proof. This follows directly from the construction of |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$|⁠, as in Lemma 2.8, and from the choice of |$\textrm{s}_{\textrm{max}}$| as in (3.1). 3.2 Energy splitting Let |$\Lambda $| be the interval |$\Lambda = (-\lambda , \lambda )$|⁠, and let |${\mathcal{C}}$| be a point configuration in |$\Lambda $|⁠. Let |$\diamond $| be the diagonal in |$\Lambda \times \Lambda $|⁠. Lemma 3.3. (Energy splitting around |$\mu _{s}$|⁠). The following identity holds: \begin{align} & \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log |x-y| (\textrm{d}{\mathcal{C}}(x) - \textrm{d}x)(\textrm{d}{\mathcal{C}}(y) - \textrm{d}y) \nonumber\\ &= \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log |x-y| (\textrm{d}{\mathcal{C}}(x) - \textrm{d}\mu_{s}(x))(\textrm{d}{\mathcal{C}}(y) - \textrm{d}\mu_{s}(y)) \nonumber\\ &\quad+ 2 s \int_{\Lambda} \textsf{LP}_{\lambda, \varphi}(x) (\textrm{d}{\mathcal{C}} - \textrm{d}x) + 2s \int_{\Lambda} \textrm{Error}\textsf{LP}_{\lambda, \varphi}(x) (\textrm{d}{\mathcal{C}}-\textrm{d}x) \nonumber\\ &\quad - 2 s^2 \|\overline{\varphi}\|_{H^{\frac{1}{2}}}^2 - s^2 \textrm{ErrorVar}, \end{align} (3.3) where |$\textsf{LP}_{\lambda , \varphi }$| is the logarithmic potential generated by |${\mathfrak{m}}_{\lambda , \varphi }$|⁠, as in (2.5), the error term |$\textrm{Error}\textsf{LP}_{\lambda , \varphi }$| is defined in (2.16), and |$\textrm{ErrorVar}$| satisfies (2.20). We postpone the proof of Lemma 3.3 to Section 7.8; it simply consists in putting together the various definitions given above. 3.3 The transport map Definition 3.4 (Transport map). We let |$F_s$| be the cumulative distribution function of the density |$\mu _{s}$| \begin{equation*} F_s:= x \mapsto \int_{-\lambda}^x \mu_{s}(y)\ \textrm{d}y, t \end{equation*} and we define the transport map |$\Phi _s$| on |$[-\lambda , \lambda ]$| as \begin{equation} \Phi_s(x):= F_s^{-1}(x + \lambda), \end{equation} (3.4) so that |$\Phi _s$| satisfies, for |$x$| in |$[-\lambda , \lambda ]$|⁠, the identity \begin{equation*} \int_{-\lambda}^{\Phi_s(x)} \mu_{s}(y) \ \textrm{d}y = \int_{-\lambda}^x 1 \ \textrm{d}y. \end{equation*} Lemma 3.5. (Properties of the transport map). The map |$\Phi _s$| is a |$C^1$|⁠, increasing bijection from |$[-\lambda , \lambda ]$| to itself. The push-forward of the constant density |$1$| on |$[-\lambda , \lambda ]$| by |$\Phi _s$| is equal to |$\mu _{s}$|⁠, that is, for any measurable function |$f$| we have \begin{equation*} \int_{-\lambda}^{\lambda} f \circ \Phi_s(x) \ \textrm{d}x = \int_{-\lambda}^\lambda f(x) \mu_{s}(x) \ \textrm{d}x. \end{equation*} Let |$\textrm{Id}_{\lambda }$| be the identity map from |$[-\lambda , \lambda ]$| to itself. The transport map |$\Phi _s$| coincides with |$\textrm{Id}_{\lambda }$| near the endpoints, more precisely on |$[-\lambda , - \lambda + \ell /4]$| and |$[\lambda - \ell /4, \lambda ]$|⁠. We define |$\psi _s$| as \begin{equation} \psi_s:= \Phi_s - \textrm{Id}_{\lambda}. \end{equation} (3.5) The map |$\psi _s$| satisfies \begin{equation} \psi_s(x) = - s \int_{-\lambda}^{\Phi_s(x)} \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y) \ \textrm{d}y; \end{equation} (3.6) in particular, we have the rough control \begin{equation} |\psi_s|_{\textsf{0}} \leq 1, \text{ i.e.} |\Phi_s - \textrm{Id}_{\lambda}|_{\textsf{0}} \leq 1, \end{equation} (3.7) and we also obtain \begin{align} \psi_s^{(1)}(x) & \preceq s |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}|_{\textsf{0}, V_{x}}, \end{align} (3.8) \begin{align} \psi_s^{(2)}(x) & \preceq s |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}|_{\textsf{1}, V_{x}}, \end{align} (3.9) \begin{align} \psi_s^{(3)}(x) & \preceq s|\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}|_{\textsf{2}, V_{x}}. \end{align} (3.10) The proof of Lemma 3.5 is given in Section 7.9. Lemma 3.6. (Finer bound on |$\psi _s$|⁠). We have \begin{equation} \psi_s(x) \leq s \int_{-\lambda}^{x + 1} |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y)| \ \textrm{d}y, \quad \psi_s(x) \leq s \int_{x-1}^{\lambda} |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y)| \ \textrm{d}y. \end{equation} (3.11) We obtain the following bounds, which improve on (3.7): \begin{equation} \psi_s(x) \preceq s \times \begin{cases} 1 & |x| \leq 10 \ell\\ \frac{\ell}{|x|} & 10 \ell \leq |x| \leq \lambda /2, \\ \frac{\ell}{\lambda^{3/2}} \sqrt{\lambda - |x|} & |x| \geq \lambda/2. \end{cases} \end{equation} (3.12) The proof of Lemma 3.6 is given in Section 7.10 Definition 3.7. (The slope of the transport). For |$x,y$| in |$\Lambda $| we define |$\Delta _s(x,y)$| as \begin{equation} \Delta_s(x,y):= \frac{\psi_s(y) - \psi_s(x)}{y-x}, \end{equation} (3.13) with the natural convention that |$\Delta _s(x,x) = \psi_s^{\prime}(x)$|⁠. 3.4 Energy expansion along a transport We introduce the following notation: \begin{equation} \textsf{Main}_s(\eta) := \iint_{\Lambda \times \Lambda} - \log | 1 + \Delta_s(x,y) | (\textrm{d}\eta - \textrm{d}x) (\textrm{d}\eta -\textrm{d}y) \end{equation} (3.14) \begin{equation} \textsf{RE}_s := - \int \log \mu_{s}(x) \mu_{s}(x)\ \textrm{d}x \end{equation} (3.15) \begin{equation} \textrm{Flu}\textsf{RE}_s(\eta) := - \int \log \mu_{s} \circ \Phi_s(x) (\textrm{d}\eta - \textrm{d}x). \end{equation} (3.16) The term |$\textsf{Main}_s(\eta )$| will be the main term in the energy comparison below. The term |$\textsf{RE}_s$| is the relative entropy of |$\mu _{s}$|⁠, which is independent on the point configuration, and |$\textrm{Flu}\textsf{RE}_s(\eta )$| is the fluctuation of the relative entropy functional, which depends on |$\eta $|⁠. Lemma 3.8. (Energy expansion along a transport). Let |$\eta $| be a point configuration in |$\Lambda $|⁠, and let |$\eta _s$| be the push-forward of the configuration |$\eta $| by the map |$\Phi _s$|⁠. We have \begin{multline} \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log |x-y| (\textrm{d}\eta_s(x) - \mu_{s}(x) \ \textrm{d}x)(\textrm{d}\eta_s(y) - \mu_{s}(y)\ \textrm{d}y) \\ = \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log |x-y| (\textrm{d}\eta(x) - \textrm{d}x)(\textrm{d}\eta(y) - \textrm{d}y) + \textsf{Main}_s(\eta) + \textsf{RE}_s + \textrm{Flu}\textsf{RE}_s(\eta). \end{multline} (3.17) Proof of Lemma 3.8 Since, by construction, |$\Phi _s$| transports |$\eta $| onto |$\eta _s$| and the constant density |$dx$| onto |$\mu _{s}(x) dx$|⁠, we may write \begin{align*} \iint_{(\Lambda \times \Lambda) \setminus \diamond} &- \log|x-y| (\textrm{d}\eta_s(x) - \mu_{s}(x) \ \textrm{d}x)(\textrm{d}\eta_s(y) - \mu_{s}(y)\ \textrm{d}y) \\ &\quad= \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log |\Phi_s(x) - \Phi_s(y)| (\textrm{d}\eta - \textrm{d}x) (\textrm{d}\eta - \textrm{d}y) \\ &\quad= \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log|x-y| (\textrm{d}\eta - \textrm{d}x) (\textrm{d}\eta - \textrm{d}y) \\ &+ \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log | 1 + \Delta_s(x,y) | (\textrm{d}\eta - \textrm{d}x) (\textrm{d}\eta -\textrm{d}y), \end{align*} where we have used the definition |$\psi _s = \Phi _s - \textrm{Id}_{\lambda }$| and the definition of |$\Delta _s$| as in (3.13). Since |$\Delta _s$| is continuously extended by |$\psi _s^{\prime}$| on the diagonal, we may write \begin{multline*} \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log | 1 + \Delta_s(x,y) | (\textrm{d}\eta - \textrm{d}x) (\textrm{d}\eta -\textrm{d}y) \\ = \iint_{\Lambda \times \Lambda} - \log | 1 + \Delta_s(x,y) | (\textrm{d}\eta - \textrm{d}x) (\textrm{d}\eta -\textrm{d}y) + \int \log | 1 + \psi_s^{\prime}(x) | \ \textrm{d}\eta. \end{multline*} The 1st term in the right-hand side corresponds to the definition (3.14) of |$\textsf{Main}_s$|⁠. We claim that \begin{equation} \int \log | 1 + \psi_s^{\prime}(x) | (\textrm{d}\eta - \textrm{d}x) = \int \log \Phi_s^{\prime}(x) (\textrm{d}\eta - \textrm{d}x) = \textsf{RE}_s + \textrm{Flu}\textsf{RE}_s(\eta). \end{equation} (3.18) To prove (3.18), let us observe that |$1 + \psi _s^{\prime}(x) = \Phi _s^{\prime}(x)$|⁠, and, by definition of a transport, we have \begin{equation*} 1 + \psi_s^{\prime}(x) = \Phi_s^{\prime}(x) = \frac{1}{\mu_{s} \circ \Phi_s(x)}. \end{equation*} We obtain \begin{equation*} \int \log |1 + \psi_s^{\prime}(x) |\ \textrm{d}\eta = \int \log \Phi_s^{\prime}(x) = - \int \log \mu_{s} \circ \Phi_s(x) \ \textrm{d}\eta. \end{equation*} Finally, let us write \begin{equation*} -\int \log \mu_{s} \circ \Phi_s(x) \ \textrm{d}\eta = - \int \log \mu_{s} \circ \Phi_s(x) \ \textrm{d}x - \int \log \mu_{s} \circ \Phi_s(x) (\textrm{d}\eta - \textrm{d}x). \end{equation*} The 1st term in the right-hand side can be seen, using the fact that |$\Phi _s$| transports the Lebesgue density onto |$\mu _{s}$|⁠, as \begin{equation*} - \int \log \mu_{s} \circ \Phi_s(x) \ \textrm{d}x = - \int \log \mu_{s}(x) \mu_{s}(x) \ \textrm{d}x, \end{equation*} so we obtain \begin{equation} \int \log |1 + \psi_s^{\prime}(x) | \ \textrm{d}\eta(x) = \int \log \Phi_s^{\prime}(x)\ \textrm{d}\eta(x) = - \int \log \mu_{s}(x) \mu_{s}(x) \ \textrm{d}x - \int \log \mu_{s} \circ \Phi_s(x) (\ \textrm{d}\eta - \textrm{d}x). \end{equation} (3.19) Using the notation introduced above in (3.14), (3.15), and (3.16), this concludes the proof of (3.17). 4 Comparison of Energies I: The Interior–Interior Interaction 4.1 The main term in the comparison We have: Proposition 4.1 (The main term is often small). \begin{equation*} {\mathbb{E}}[ \left|\textsf{Main}_s\right| ] = s o_{\ell, \lambda}(1). \end{equation*} The proof of Proposition 4.1 is rather elementary but involves cumbersome computations. We postpone it to Section 7.11. 4.2 The relative entropy term Lemma 4.2 (The term |$\textsf{RE}_s$| is small). We have \begin{equation} \textsf{RE}_s = - \int \log \mu_{s}(x) \mu_{s}(x) \ \textrm{d}x \preceq \frac{s^2}{\ell}. \end{equation} (4.1) In particular, we obtain \begin{equation} \textsf{RE}_s = s^2 o_{\ell, \lambda}(1). \end{equation} (4.2) Proof. We write |$\mu _{s} = 1 + s \widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$|⁠, expand the |$\log $|⁠, and use the fact that |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$| has total mass |$0$|⁠. We obtain \begin{equation*} \int \log \mu_{s}(x) \mu_{s}(x) \ \textrm{d}x = \int \left(s\widetilde{{\mathfrak{m}}}_{\lambda, \varphi} + O_{\bullet}\left( s^2 \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}^2 \right)\right) (1 + s \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}) \preceq s^2 \|\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}^2\|_{L^1}. \end{equation*} Using (2.12), we see that |$\|\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }^2\|_{L^1} \preceq \frac{1}{\ell }$|⁠, which yields (4.1). 4.3 The fluctuations of the relative entropy term Lemma 4.3 (The fluctuations |$\textrm{Flu}\textsf{RE}_s(\eta )$|⁠). We have \begin{equation} \textrm{Flu}\textsf{RE}_s(\eta) = - \int \log \mu_{s} \circ \Phi_s(x) (\textrm{d}\eta - \textrm{d}x) \preceq s \sum_{i = -\lambda}^{\lambda} |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}|_{\textsf{1}, V_{x}} \widetilde{D}_i. \end{equation} (4.3) Proof of Lemma 4.3 We start by computing the derivative of |$x \mapsto \log \mu _{s} \circ \Phi _s(x)$| as \begin{equation*} \left(\log \mu_{s} \circ \Phi_s\right)^{\prime}(x) = \frac{\left[ \mu_{s}^{\prime} \circ \Phi_s(x)\right] \Phi_s^{\prime}(x)}{\mu_{s} \circ \Phi_s(x)}. \end{equation*} Using the fact that |$\Phi _s$| is bounded by |$1$|⁠, that |$\Phi _s^{\prime}$| and |$\frac{1}{\mu _{s}}$| are bounded, and that |$\mu _{s}^{\prime} = \widetilde{{\mathfrak{m}}}_{\lambda , \varphi }^{\prime}$|⁠, we obtain \begin{equation*} \left(\log \mu_{s} \circ \Phi_s\right)^{\prime}(x) \preceq s |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}|_{\textsf{1}, V_{x}}. \end{equation*} Moreover, since |$\Phi _s$| is the identity near the endpoints, and |$\mu _{s} = 1$| near the endpoints, the map |$x \mapsto \log \mu _{s} \circ \Phi _s(x)$| is compactly supported. Applying Proposition 1.6, we obtain (4.3). Corollary 4.4 (The term |$\textrm{Flu}\textsf{RE}_s(\eta )$| is often small). We have \begin{equation} {\mathbb{E}} \left[ \left| \textrm{Flu}\textsf{RE}_s(\eta) \right| \right] = s o_{\ell, \lambda}(1). \end{equation} (4.4) Proof. In view of (4.3), we use the discrepancy estimate (1.11) and the estimates (2.13) on the 1st derivative of |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$|⁠. We obtain \begin{equation*} {\mathbb{E}}\left[ \sum_{i = -\lambda}^{\lambda} |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}|_{\textsf{1}, V_{x}} \widetilde{D}_i \right] \preceq \sum_{i=0}^{2\ell} \sqrt{\ell} \frac{1}{\ell^2} + \sum_{i=2\ell}^{\lambda / 2} \sqrt{i} \frac{\ell \sqrt{\lambda}}{i^3 \sqrt{\lambda}} + \sum_{i=\lambda/2}^{\lambda - \ell} \sqrt{\lambda} \frac{\ell} {\lambda^{3/2} (\lambda -i)^{3/2}} + \sum_{i = \lambda-\ell}^{\lambda} \sqrt{\lambda} \frac{\ell}{\lambda^{3/2} \ell^{3/2}}, \end{equation*} and thus \begin{equation*} {\mathbb{E}}\left[ \sum_{i = -\lambda}^{\lambda} |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}|_{\textsf{1}, V_{x}} \widetilde{D}_i \right] \preceq \frac{1}{\sqrt{\ell}} + \frac{\ell}{\sqrt{\ell} \lambda} = o_{\ell, \lambda}(1). \end{equation*} 4.4 Conclusion Combining Proposition 4.1, Lemma 4.2, and Corollary 4.4, we obtain \begin{equation} {\mathbb{E}} \left[ \left| \textsf{Main}_s \right| + \left| \textrm{Flu}\textsf{RE}_s \right| \right] + \left|\textsf{RE}_s\right| = s o_{\ell, \lambda}(1) + s^2 o_{\ell, \lambda}(1), \end{equation} (4.5) which, in view of Lemma 3.8, says that the interior–interior interactions before and after transport are often very close. 5 Comparison of the Energies II: The Interior–Exterior Interaction 5.1 The difference field Definition 5.1 (The difference field). Let |$\eta $| be a point configuration in |$(-\lambda , \lambda )$|⁠, and let |$\eta _s$| be the push-forward of |$\eta $| by |$\Phi _s$|⁠. For |$x \notin (-\lambda , \lambda )$|⁠, we let |$\textsf{DF}_s(\eta )(x)$| be the electrostatic field created at |$x$| by the difference |$\eta _s-\eta $|⁠, that is, \begin{equation} \textsf{DF}_s(\eta)(x):= \int - \log |x-y| \left( \textrm{d}\eta_s(y) - \textrm{d}\eta(y) \right). \end{equation} (5.1) Lemma 5.2 (Decomposition of the difference field). We have \begin{equation} \textsf{DF}_s(\eta)(x) = s\textsf{LP}_{\lambda, \varphi}(x) + s\textrm{Error}\textsf{LP}_{\lambda, \varphi}(x) + \textrm{Error}\textsf{DF}_s(\eta)(x), \end{equation} (5.2) with |$\textsf{LP}_{\lambda , \varphi }$| as in (2.5), |$\textrm{Error}\textsf{LP}_{\lambda , \varphi }$| as in (2.16), and |$\textrm{Error}\textsf{DF}_s(\eta )$| defined by \begin{equation} \textrm{Error}\textsf{DF}_s(\eta)(x) = - \int \log \left( 1 - \frac{\psi_s(y)}{x-y} \right) \left(\textrm{d}\eta(y) - \textrm{d}y \right). \end{equation} (5.3) Proof. We simply write |$\eta _s - \eta = (\eta _s - \mu _{s}) + (\mu _{s} - dy) + (dy - \eta )$|⁠. We have \begin{equation*} \int - \log |x-y| (\textrm{d}\eta_s(y) - \mu_{s}(y)\ \textrm{d}y) = \int - \log |x - \Phi_s(y)| (\textrm{d}\eta(y) - \textrm{d}y), \end{equation*} and we define |$\textrm{Error}\textsf{DF}_s(\eta )(x)$| as the term such that \begin{equation} \int - \log |x - \Phi_s(y)| (\textrm{d}\eta(y) - \textrm{d}y) = \int - \log |x - y| (\textrm{d}\eta(y) - \textrm{d}y) + \textrm{Error}\textsf{DF}_s(\eta)(x), \end{equation} (5.4) which coincides with the expression given in (5.3). By definition, we obtain \begin{equation*} \textsf{DF}_s(\eta)(x) = \int - \log |x-y| (\mu_{s}(y) \textrm{d}y - \textrm{d}y) + \textrm{Error}\textsf{DF}_s(\eta)(x). \end{equation*} Since |$\mu _{s}(y) = 1 + s \widetilde{{\mathfrak{m}}}_{\lambda , \varphi }(y)$|⁠, the 1st term in the right-hand side is the logarithmic potential generated by |$s \widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$|⁠, which is given by the sum |$s \textsf{LP}_{\lambda , \varphi } + s \textrm{Error}\textsf{LP}_{\lambda , \varphi }$|⁠. Lemma 5.3. Assume |$x \geq \lambda $|⁠. We have \begin{align} \textrm{Error}\textsf{DF}_s(\eta)(x) & \preceq \sum_{i=-\lambda}^{\lambda} \left(\frac{|\psi_s(i)|}{(x-i)^2} + \frac{|\psi_s^{\prime}(i)|}{(x-i)} \right) \widetilde{D}^{\textrm{Right}}_i \end{align} (5.5) \begin{align} \left(\textrm{Error}\textsf{DF}_s(\eta)\right)^{\prime}(x) & \preceq \sum_{i = -\lambda}^{\lambda} \left(\frac{|\psi_s(i)|}{(x-i)^3} + \frac{|\psi_s^{\prime}(i)|}{(x-i)^2} \right) \widetilde{D}^{\textrm{Right}}_i. \end{align} (5.6) Proof. Let us introduce the auxiliary function \begin{equation} \textsf{H}(x,y):= - \log \left( 1 - \frac{\psi_s(y)}{x-y} \right); \end{equation} (5.7) we re-write (5.3) as \begin{equation*} \textrm{Error}\textsf{DF}_s(\eta)(x) = \int \textsf{H}(x,y) \left(\textrm{d}\eta(y) - \textrm{d}y\right). \end{equation*} In particular, for |$x \geq \lambda $|⁠, we may differentiate under the integral sign and get \begin{equation*} \left(\textrm{Error}\textsf{DF}_s(\eta)\right)^{\prime}(x) = \int \partial_x \textsf{H}(x,y) \left(\textrm{d}\eta(y) - \textrm{d}y\right). \end{equation*} Since |$\psi _s$| vanishes near the endpoints, for any |$x$| the function |$\textsf{H}(x, \cdot )$| is compactly supported with respect to the 2nd variable. Moreover, since |$\psi _s(y) = 0$| for |$y \geq \lambda - \ell /10$|⁠, and |$x \geq \lambda $|⁠, we may write \begin{equation*} x- y - \psi_s(y) \approx x - y. \end{equation*} A direct computation shows that \begin{align} \partial_y \textsf{H}(x,y) & \preceq \frac{|\psi_s^{\prime}(y)|}{|x-y|} + \frac{|\psi_s(y)|}{|x-y|^2} \end{align} (5.8) \begin{align} \partial^2_{yx} \textsf{H}(x,y) & \preceq \frac{|\psi_s^{\prime}(y)|}{|x-y|^2} + \frac{|\psi_s(y)|}{|x-y|^3}, \end{align} (5.9) as can be checked informally by treating |$\psi _s$| as a perturbation, writing \begin{equation*} \textsf{H}(x,y) \approx \frac{\psi_s(y)}{x-y}, \end{equation*} and differentiating. Using the a priori bound on fluctuations of Proposition 1.6 with discrepancy |$\widetilde{D}^{\textrm{Right}}_i$|⁠, and (5.8), resp. (5.9), we obtain (5.5), resp. (5.6). Corollary 5.4 (The contribution of |$\textrm{Error}\textsf{DF}_s$| is often small). In particular, \begin{equation} {\mathbb{E}} \left[ \left| \int_{\Lambda^c} \textrm{Error}\textsf{DF}_s({\mathcal{C}})(x) (\textrm{d}{\mathcal{C}} - \textrm{d}x) \right| \right] = s o_{\ell, \lambda}(1). \end{equation} (5.10) We give the proof of Corollary 5.4 in Section 7.12. 5.2 The logarithmic potential and its fluctuations In this section, we consider the logarithmic potential |$\textsf{LP}_{\lambda , \varphi }$| generated by |${\mathfrak{m}}_{\lambda , \varphi }$|⁠, as defined in (2.5). We state some bounds on |$\textsf{LP}_{\lambda , \varphi }$| and its 1st derivative. Lemma 5.5 (Controls on |$\textsf{LP}_{\lambda , \varphi }$|⁠). \begin{align} & \textsf{LP}_{\lambda, \varphi}(x) \preceq \frac{\ell \log^2(\lambda)}{\lambda}, \quad |x| \in [\lambda /2, 2 \lambda] \ \ \end{align} (5.11) \begin{align} & \textsf{LP}_{\lambda, \varphi}^{\prime}(x) \preceq \frac{\ell}{ \lambda^{3/2} \sqrt{x-\lambda}}, \quad \lambda \leq |x| \leq 4 \lambda \end{align} (5.12) \begin{align} & \textsf{LP}_{\lambda, \varphi}^{\prime}(x) \preceq \frac{\ell \log(\lambda)}{x^2}, \quad |x| \geq 4\lambda.\, \quad\qquad \end{align} (5.13) We give the proof of Lemma 5.5 in Section 7.13. Lemma 5.6 (Fluctuations of the logarithmic potential). We have \begin{equation} \int (\textsf{LP}_{\lambda, \varphi}(x) - \varphi(x)) (\textrm{d}{\mathcal{C}} - \textrm{d}x) \preceq \textrm{Flu}\textsf{LP}_A + \textrm{Flu}\textsf{LP}_B + \textrm{Flu}\textsf{LP}_C, \end{equation} (5.14) with \begin{align} \textrm{Flu}\textsf{LP}_A & \preceq \sum_{i = \lambda + 2 \sqrt{\lambda}}^{4 \lambda} \frac{\ell}{\lambda^{3/2} \sqrt{i - \lambda}} \widetilde{D}^{\textrm{Right}}_i + \sum_{i = 4\lambda}^{+\infty} \frac{\ell \log(\lambda)}{i^2} \widetilde{D}^{\textrm{Right}}_i + \sum_{i= \lambda + 2 \sqrt{\lambda}}^{\lambda + 4 \sqrt{\lambda}} \frac{\ell \log^2(\lambda)}{\lambda^{3/2}} \widetilde{D}^{\textrm{Right}}_i, \end{align} (5.15) \begin{align} \textrm{Flu}\textsf{LP}_B & \preceq \frac{\ell \log^2(\lambda)}{\lambda^{3/2}} \sum_{|i| = \lambda - 4 \sqrt{\lambda}}^{\lambda - 2 \sqrt{\lambda}} \widetilde{D}_i,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \end{align} (5.16) \begin{align} \textrm{Flu}\textsf{LP}_C & \preceq \frac{\ell \log^2(\lambda)}{\lambda} \left( \sqrt{\lambda} + |\textrm{Discr}_{[\lambda - 4 \sqrt{\lambda}, \lambda + 4 \sqrt{\lambda}]}| + |\textrm{Discr}_{[-\lambda - 4 \sqrt{\lambda}, -\lambda + 4 \sqrt{\lambda}]}|\right).\qquad\qquad\, \end{align} (5.17) Strictly speaking, (5.14) should be complemented with terms corresponding to the left endpoint |$-\lambda $| and involving the discrepancy |$\widetilde{D}^{\textrm{Left}}$|⁠, see below. They are bounded the same way, and for simplicity, we omit them Proof. Let |$\chi _1$| be a smooth non-negative function such that \begin{equation} \chi_1 = 1 \textrm{ on } [\lambda + 4\sqrt{\lambda}, +\infty) \quad \chi_1 = 0 \textrm{ on } (-\infty, \lambda + 2 \sqrt{\lambda}], \end{equation} (5.18) with |$|\chi _1|_{\textsf{0}} \leq 1$| and |$|\chi _1|_{\textsf{1}} \preceq \frac{1}{\sqrt{\lambda }}$|⁠. Let |$\chi _2$| be a smooth non-negative function such that \begin{equation} \chi_2 = 1 \textrm{ on } [-\infty, - \lambda - 4\sqrt{\lambda}) \quad \chi_2= 0 \textrm{ on } [-\lambda - 2 \sqrt{\lambda}, + \infty), \end{equation} (5.19) with |$|\chi _2|_{\textsf{0}} \leq 1$| and |$|\chi _2|_{\textsf{1}} \preceq \frac{1}{\sqrt{\lambda }}$|⁠. Finally, let |$\chi _3$| be a smooth non-negative function such that \begin{equation} \chi_3 = 1 \textrm{ on } [-\lambda + 4 \sqrt{\lambda}, \lambda - 4 \sqrt{\lambda}] \quad \chi_3 = 0 \textrm{ outside } [- \lambda + 2 \sqrt{\lambda}, \lambda - 2 \sqrt{\lambda}], \end{equation} (5.20) with |$|\chi _3|_{\textsf{0}} \leq 1$| and |$|\chi _3|_{\textsf{1}} \preceq \frac{1}{\sqrt{\lambda }}$|⁠. We write trivially |$\textsf{LP}_{\lambda , \varphi }(x)$| as the sum \begin{equation*} \textsf{LP}_{\lambda, \varphi}(x) \chi_1(x) + \textsf{LP}_{\lambda, \varphi}(x) \chi_2(x) + \textsf{LP}_{\lambda, \varphi}(x) \chi_3(x) + \textsf{LP}_{\lambda, \varphi}(x) (1 - \chi_1(x) - \chi_2(x) - \chi_3(x)), \end{equation*} and we integrate these terms against |$d{\mathcal{C}} - dx$|⁠. The |$\chi _1, \chi _2$| terms. We have \begin{equation*} \left(\textsf{LP}_{\lambda, \varphi} \chi_1\right)^{\prime}(x) = \textsf{LP}_{\lambda, \varphi}^{\prime}(x) \chi_1(x) + \textsf{LP}_{\lambda, \varphi}(x) \chi_1^{\prime}(x), \end{equation*} where |$\chi _1^{\prime}$| is supported on |$[\lambda + 2 \sqrt{\lambda }, \lambda + 4 \sqrt{\lambda }]$| and bounded by |$O_{\bullet }\left ( \frac{1}{\sqrt{\lambda }} \right )$|⁠. Applying Proposition 1.6, we obtain \begin{equation*} \int \left(\textsf{LP}_{\lambda, \varphi} \chi_1\right) (\textrm{d}{\mathcal{C}} - \textrm{d}x) \preceq \sum_{i = \lambda + 2 \sqrt{\lambda}}^{+ \infty} |\textsf{LP}_{\lambda, \varphi}|_{\textsf{1}, V_i} \widetilde{D}^{\textrm{Right}}_i \\ + \sum_{i= \lambda + 2 \sqrt{\lambda}}^{\lambda + 4 \sqrt{\lambda}} |\textsf{LP}_{\lambda, \varphi}|_{\textsf{0}, V_i} \widetilde{D}^{\textrm{Right}}_i \frac{1}{\sqrt{\lambda}}, \end{equation*} and using (5.11), (5.12), and (5.13), we may write \begin{align*} \int \left(\textsf{LP}_{\lambda, \varphi} \chi_1\right) (\textrm{d}{\mathcal{C}} - \textrm{d}x) \preceq& \sum_{i = \lambda + 2 \sqrt{\lambda}}^{4 \lambda} \frac{\ell}{\lambda^{3/2} \sqrt{i - \lambda}} \widetilde{D}^{\textrm{Right}}_i \\ &+ \sum_{i = 4\lambda}^{+\infty} \frac{\ell \log(\lambda)}{i^2} \widetilde{D}^{\textrm{Right}}_i + \sum_{i= \lambda + 2 \sqrt{\lambda}}^{\lambda + 4 \sqrt{\lambda}} \frac{\ell \log^2(\lambda)}{\lambda^{3/2}} \widetilde{D}^{\textrm{Right}}_i. \end{align*} Of course, |$\textsf{LP}_{\lambda , \varphi } \chi _2$| satisfies the same inequality, with |$\widetilde{D}^{\textrm{Left}}_i$| instead of |$\widetilde{D}^{\textrm{Right}}_i$|⁠, and this yields (5.15) (we only keep track of the “right-hand” term; the estimates on “left-hand” term are the same). The |$\chi _3$| term. We have \begin{equation*} \left((\textsf{LP}_{\lambda, \varphi} - \varphi) \chi_3\right)^{\prime}(x) = (\textsf{LP}_{\lambda, \varphi} - \varphi)^{\prime}(x) \chi_3(x) + (\textsf{LP}_{\lambda, \varphi}(x) - \varphi) \chi_3^{\prime}(x), \end{equation*} but we know by (2.6) that |$(\textsf{LP}_{\lambda , \varphi } - \varphi )^{\prime} = 0$| on the support of |$\chi _3$|⁠, and moreover, |$\varphi $| is supported outside the support of |$\chi _3^{\prime}$|⁠, so we have in fact \begin{equation*} \left((\textsf{LP}_{\lambda, \varphi} - \varphi) \chi_3\right)^{\prime}(x) = \textsf{LP}_{\lambda, \varphi}(x) \chi_3^{\prime}(x), \end{equation*} which is supported on |$[-\lambda + 2 \sqrt{\lambda }, - \lambda + 4 \sqrt{\lambda }] \cup [\lambda - 4 \sqrt{\lambda }, \lambda - 2 \sqrt{\lambda }]$|⁠. We use Proposition 1.6 and (5.11) and the fact that |$|\chi _3|_{\textsf{1}} \preceq \frac{1}{\sqrt{\lambda }}$| to get \begin{equation*} \int \left(\textsf{LP}_{\lambda, \varphi} - \varphi\right) \chi_3 (\textrm{d}{\mathcal{C}} - \textrm{d}x) \preceq \sum_{|i| = \lambda - 4 \sqrt{\lambda}}^{\lambda - 2 \sqrt{\lambda}} \widetilde{D}_i \frac{\ell \log^2(\lambda)}{\lambda} \frac{1}{\sqrt{\lambda}}, \end{equation*} which yields (5.16). The |$1-\chi _1-\chi _2 - \chi _3$| term. The function |$1-\chi _1-\chi _2 - \chi _3$| is supported near the endpoints, on |$[\lambda - 4 \sqrt{\lambda }, \lambda + 4 \sqrt{\lambda }]$| and on the symmetric interval. We use (5.11) to get \begin{multline*} \int \textsf{LP}_{\lambda, \varphi} (1-\chi_1-\chi_2 - \chi_3) (\textrm{d}{\mathcal{C}} - \textrm{d}x) \\ \preceq \frac{\ell \log^2(\lambda)}{\lambda} \left( \sqrt{\lambda} + |\textrm{Discr}_{[\lambda - 4 \sqrt{\lambda}, \lambda + 4 \sqrt{\lambda}]}| + |\textrm{Discr}_{[-\lambda - 4 \sqrt{\lambda}, -\lambda + 4 \sqrt{\lambda}]}|\right), \end{multline*} where the last parenthesis is, up to a multiplicative constant, a bound on the the number of the points in the intervals and on the length of the intervals. This yields (5.17). Corollary 5.7 (The contribution of |$\textsf{LP}_{\lambda , \varphi } - \varphi $| is often small). We have \begin{equation} {\mathbb{E}} \left[ \left| \int (\textsf{LP}_{\lambda, \varphi}(x)- \varphi(x)) (\textrm{d}{\mathcal{C}} - \textrm{d}x) \right| \right] \preceq \frac{\ell \log^2(\lambda)}{\sqrt{\lambda}}. \end{equation} (5.21) In particular, \begin{equation} {\mathbb{E}} \left[ \left| \int (\textsf{LP}_{\lambda, \varphi}(x)- \varphi(x)) (\textrm{d}{\mathcal{C}} - \textrm{d}x) \right| \right] = o_{\ell, \lambda}(1). \end{equation} (5.22) Proof. For |$\textrm{Flu}\textsf{LP}_A$|⁠, we use the estimate (1.12) in the form \begin{equation*} {\mathbb{E}}[\widetilde{D}^{\textrm{Right}}_i] \preceq \sqrt{|i- \lambda|}, \end{equation*} and we get \begin{equation*} {\mathbb{E}}[\textrm{Flu}\textsf{LP}_A] \preceq \sum_{i = \lambda + 2 \sqrt{\lambda}}^{4 \lambda} \frac{\ell}{\lambda^{3/2}} \frac{ \sqrt{i - \lambda}}{\sqrt{i - \lambda}} + \sum_{i = 4 \lambda}^{+ \infty} \ell \log(\lambda) \frac{\sqrt{i}}{i^2} + \sum_{i = \lambda + 2 \sqrt{\lambda}}^{\lambda + 4 \sqrt{\lambda}} \frac{\ell \log^2(\lambda)}{\lambda^{3/2}} \sqrt{i - \lambda}, \end{equation*} which, after computation, gives \begin{equation*} {\mathbb{E}}[\textrm{Flu}\textsf{LP}_A] \preceq \frac{\ell \log(\lambda)}{\sqrt{\lambda}}. \end{equation*} For |$\textrm{Flu}\textsf{LP}_B$|⁠, we use the estimate (1.12), in the form \begin{equation*} {\mathbb{E}}[\widetilde{D}_i] \preceq \sqrt{|i|}, \end{equation*} and we get \begin{equation*} {\mathbb{E}}[\textrm{Flu}\textsf{LP}_B] \preceq \frac{\ell \log^2(\lambda)}{\lambda^{3/2}} \lambda = \frac{\ell \log^2(\lambda)}{\sqrt{\lambda}}. \end{equation*} Finally, using the discrepancy estimate (1.8), we have \begin{equation*} {\mathbb{E}}[\textrm{Flu}\textsf{LP}_C] \preceq \frac{\ell \log^2(\lambda)}{\sqrt{\lambda}}. \end{equation*} The dominant error term is thus |$\frac{\ell \log ^2(\lambda )}{\sqrt{\lambda }}$|⁠, which proves the result. 5.3 Fluctuations of the error on the logarithmic potential Lemma 5.8 (Fluctuations of |$\textrm{Error}\textsf{LP}_{\lambda , \varphi }$|⁠). We have \begin{equation} \int \textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Left}} (\textrm{d}{\mathcal{C}} - \textrm{d}x) \preceq \textsf{A} + \textsf{B} + \textsf{C}, \end{equation} (5.23) with \begin{align*} \textsf{A} & \preceq \left( \sum_{i = -\lambda + 3\ell}^{+ \infty} + \sum_{i = -\infty}^{- \lambda - 3 \ell} \right) \frac{\ell^{3/2} \ell}{\lambda^{3/2} (- \lambda -i)^2} \widetilde{D}^{\textrm{Left}}_i \\ \textsf{B} & \preceq \sum_{i=-\lambda - 4 \ell}^{-\lambda + 4 \ell} \frac{\ell \log(\lambda)}{\lambda^{3/2} \sqrt{\ell}} \widetilde{D}^{\textrm{Left}}_i \\ \textsf{C} & \preceq \left( \ell + |\textrm{Discr}_{[-\lambda - 4\ell, - \lambda + 4 \ell]}| \right) \frac{\ell \sqrt{\ell} \log(\lambda)}{\lambda^{3/2}}, \end{align*} and similarly for |$\textrm{Error}\textsf{LP}_{\lambda , \varphi }^{\textrm{Right}}$|⁠, replacing |$\widetilde{D}^{\textrm{Left}}_i$| by |$\widetilde{D}^{\textrm{Right}}_i$|⁠. Proof of Lemma 5.8 Let |$\chi $| be a smooth non-negative function such that \begin{equation} \chi = 1 \textrm{ on } [-\lambda - 3\ell, -\lambda + 3\ell], \quad \chi = 0 \textrm{ outside } [-\lambda - 4\ell, -\lambda + 4\ell], \end{equation} (5.24) with |$|\chi |_{\textsf{0}} \leq 1$| and |$|\chi |_{\textsf{1}} \preceq \frac{1}{\ell }$|⁠. We write trivially \begin{equation*} \int \textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Left}} (\textrm{d}{\mathcal{C}} - \textrm{d}x) = \int \textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Left}}(x) \chi(x) (\textrm{d}{\mathcal{C}} - \textrm{d}x) + \int \textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Left}}(x) \left(1-\chi(x)\right) (\textrm{d}{\mathcal{C}} - \textrm{d}x). \end{equation*} We have \begin{equation} \left( \textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Left}} (1-\chi) \right)^{^{\prime}}(x) = \left(\textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Left}}\right)^{^{\prime}}(x) (1-\chi(x)) + \textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Left}}(x) (1-\chi)^{\prime}(x). \end{equation} (5.25) Let us observe that |$1-\chi $| is supported outside |$[-\lambda - 3 \ell , \lambda + 3\ell ]$| and bounded by |$1$|⁠, while |$(1-\chi )^{\prime}$| is supported on |$[-\lambda - 4\ell , -\lambda - 3\ell ] \cup [-\lambda + 3\ell , -\lambda + 4\ell ]$| and is bounded by |$O_{\bullet }\left (\frac{1}{\ell } \right )$|⁠. Using Proposition 1.6, we obtain \begin{multline} \int \textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Left}}(x) (1-\chi(x)) (\textrm{d}{\mathcal{C}} - \textrm{d}x) \leq \left( \sum_{i=-\lambda+3\ell}^{+ \infty} + \sum_{i = - \infty}^{-\lambda - 3\ell} \right) |\textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Left}}|_{\textsf{1}, \textsf{V}(i)} \widetilde{D}^{\textrm{Left}}_i, \\ + \sum_{i=-\lambda - 4\ell}^{-\lambda + 4\ell} |\textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Left}}|_{\textsf{0}} \frac{1}{\ell} \widetilde{D}^{\textrm{Left}}_i. \end{multline} (5.26) On the other hand, since |$\chi $| is supported on |$[-\lambda -4\ell , -\lambda + 4\ell ]$|⁠, we have a trivial bound \begin{equation} \int \textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Left}}(x) \chi(x) ( \textrm{d}{\mathcal{C}} - \textrm{d}x) \preceq \left(\ell + |\textrm{Discr}_{[-\lambda-4\ell, -\lambda + 4\ell]}| \right) |\textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Left}}|_{\textsf{0}}, \end{equation} (5.27) where |$\ell + |\textrm{Discr}_{[-\lambda -4\ell , -\lambda + 4\ell ]}|$| is (up to a multiplicative constant) a bound on the number of the points in the interval and on the length of the interval. We let |$\textsf{A}$| be the 1st line of (5.26), |$\textsf{B}$| be the 2nd line of (5.26), and |$\textsf{C}$| be the right-hand side of (5.27), and we use the bounds of Proposition 2.11 to obtain (5.23). Corollary 5.9 (The contribution of |$\textrm{Error}\textsf{LP}_{\lambda , \varphi }$| is often small). We have \begin{equation} {\mathbb{E}} \left[ \int \textrm{Error}\textsf{LP}_{\lambda, \varphi} (\textrm{d}{\mathcal{C}} - \textrm{d}x) \right] \preceq \frac{\ell^{5/2} \log(\lambda)}{\lambda^{3/2}}. \end{equation} (5.28) In particular, \begin{equation} {\mathbb{E}} \left[ \int \textrm{Error}\textsf{LP}_{\lambda, \varphi} (\textrm{d}{\mathcal{C}} - \textrm{d}x) \right] = o_{\ell, \lambda}(1). \end{equation} (5.29) Proof. It is of course enough to prove (5.28) for |$\textrm{Error}\textsf{LP}_{\lambda , \varphi }^{\textrm{Left}}$|⁠. We use the discrepancy estimate (1.11) in the form |${\mathbb{E}}[\widetilde{D}^{\textrm{Left}}_i] \preceq \sqrt{|i + \lambda |}$| and write \begin{equation*} \textsf{A} \preceq \frac{\ell^{3/2} \ell}{\lambda^{3/2}} \sum_{i=3 \ell}^{+ \infty} \frac{\sqrt{i}}{i^2} = \frac{\ell \ell}{\lambda^{3/2}}. \end{equation*} Using again |${\mathbb{E}}[\widetilde{D}^{\textrm{Left}}_i] \preceq \sqrt{|i + \lambda |}$|⁠, we have \begin{equation*} \textsf{B} \preceq \ell \frac{\ell \log(\lambda)}{\lambda^{3/2} \ell^{1/2}} \ell^{1/2} = \frac{\ell \ell \log(\lambda)}{\lambda^{3/2}}. \end{equation*} Finally, we use the discrepancy estimate (1.8) to get \begin{equation*} \textsf{C} \preceq \frac{\ell \ell^{3/2} \log(\lambda)}{\lambda^{3/2}}, \end{equation*} and this is the dominant term. 6 Proof of the Central Limit Theorem 6.1 A good event Lemma 6.1 (Defining a good event). For any point configuration |${\mathcal{C}}$|⁠, and |$\Lambda = (-\lambda , \lambda )$| fixed, let us decompose |${\mathcal{C}}$| as |${\mathcal{C}} = \nu \cup \gamma _{\Lambda ^c}$|⁠, where \begin{equation*} \nu = {\mathcal{C}} \cap \Lambda, \quad \gamma_{\Lambda^c} = {\mathcal{C}} \cap \Lambda^c. \end{equation*} We let |$\nu _s$| be the push-forward of |$\nu $| by |$\Phi _s$|⁠. We will consider |${\mathcal{C}}$| and |${\mathcal{C}}_s$|⁠, where (in fact, since |$\Phi _s$| is the identity outside |$\Lambda $|⁠, |${\mathcal{C}}_s$| itself is also the push-forward of |${\mathcal{C}}$| by |$\Phi _s$|⁠): \begin{equation*} {\mathcal{C}}_s:= \nu_s \cup \gamma_{\Lambda^c}. \end{equation*} There exists an |$\textsf{Event}_{\lambda , \ell }$| satisfying \begin{equation} {\mathbb{P}}\left({\mathcal{C}} \in \textsf{Event}_{\lambda, \ell}\right) = 1 - o_{\ell, \lambda}(1), \quad{\mathbb{P}}\left({\mathcal{C}}_s \in \textsf{Event}_{\lambda, \ell}\right) = 1 - o_{\ell, \lambda}(1), \end{equation} (6.1) such that, if |${\mathcal{C}} \in \textsf{Event}_{\lambda , \ell }$|⁠, \begin{align*} \textsf{Main}_s(\nu) & = s o_{\ell, \lambda}(1),\\ \textrm{Flu}\textsf{RE}_s(\nu) & = s o_{\ell, \lambda}(1), \\ \int_{\Lambda^c} \textrm{Error}\textsf{DF}_s(\nu)(x) (\textrm{d}\gamma_{\Lambda^c} - \textrm{d}x) & = s o_{\ell, \lambda}(1), \\ s \int \left(\textsf{LP}_{\lambda, \varphi}(x) - \varphi\right) (\textrm{d} \left[\nu_s \cup \gamma_{\Lambda^c}\right] - \textrm{d}x) & = s o_{\ell, \lambda}(1), \\ s \int \textrm{Error}\textsf{LP}_{\lambda, \varphi}(x) (\textrm{d}\left[\nu_s \cup \gamma_{\Lambda^c}\right] -\textrm{d}x) & = s o_{\ell, \lambda}(1), \end{align*} and moreover, \begin{equation} |{\mathcal{C}} \cap (-\ell, \ell)| \preceq \ell^2. \end{equation} (6.2) Proof. The control (6.2) is needed for technical reasons, in order to ensure that the number of poins in |$(-\ell , \ell )$| is bounded. Since the mean number of points is |$2\ell $|⁠, the event (6.2) is of course very likely. Using Proposition 4.1, Corollary 4.4, Corollary 5.4, Corollary 5.7, and Corollary 5.9, and applying Markov’s inequality, we see that there exists an event |$E$| of probability |$1 - o_{\ell , \lambda }(1)$| on which the three 1st bounds hold, and moreover, \begin{align*} s \int \left(\textsf{LP}_{\lambda, \varphi}(x) - \varphi\right) (\textrm{d} \left[\nu \cup \gamma_{\Lambda^c}\right] - \textrm{d}x) & = s o_{\ell, \lambda}(1), \\ s \int \textrm{Error}\textsf{LP}_{\lambda, \varphi}(x) (\textrm{d}\left[\nu \cup \gamma_{\Lambda^c}\right] -\textrm{d}x) & = s o_{\ell, \lambda}(1). \end{align*} Moreover, we argue that \begin{equation*} {\mathbb{P}}\left( \left( \nu_s \cup \gamma_{\Lambda^c} \right) \in E \right) = 1 - o_{\ell, \lambda}(1). \end{equation*} Indeed, we know, by construction, that the transport map |$\Phi _s$| is close to the identity map, with |$\Phi _s - \textrm{Id}_{\lambda }$| bounded by |$1$|⁠, see (3.7). So if |${\mathcal{C}}_s = \nu _s \cup \gamma _{\Lambda ^c}$| is the push-forward of |${\mathcal{C}} = \nu \cup \gamma _{\Lambda ^c}$| by |$\Phi _s$|⁠, we have for any |$x,y \in{\mathbb{R}}$| \begin{equation*} |\textrm{Discr}_{[x, y]}|[{\mathcal{C}}_s] \preceq |\textrm{Discr}_{[x-1,y+1]}|({\mathcal{C}}) + 1. \end{equation*} Any estimate involving the discrepancies of |${\mathcal{C}}$| can thus be converted into the estimate on |${\mathcal{C}}_s$|⁠. We then take |$\textsf{Event}_{\lambda , \ell }$| to be the intersection \begin{equation*} E \cap \left\lbrace{\mathcal{C}}_s \in E\right\rbrace, \end{equation*} for which the last two bounds hold as stated. 6.2 The DLR equations The DLR formalism for |$\textrm{Sine}_{\beta }$| is a statistical physics representation of the point process as an infinite volume Gibbs measure. Before stating the result of [6] in a convenient fashion for the present paper, we need to introduce some notation. Definition 6.2 (Infinite volume Gibbs kernel). Let |$\lambda> 0$|⁠, and let |$\Lambda : = (-\lambda , \lambda )$|⁠. Let |$\gamma $| be a point configuration in |${\mathbb{R}}$| and |$\eta $| be a point configuration in |$\Lambda $|⁠. We aim at defining the energy of the point configuration |$\eta \cup \gamma _{\Lambda ^c}$| formed by |$\eta $| in |$\Lambda $| and |$\gamma $| in |$\Lambda ^c:= {\mathbb{R}} \backslash \Lambda $|⁠. In fact, we only want to compare these energies for a fixed |$\gamma $| and a variable |$\eta $|⁠, so we may work up to (possibly infinite) additive constants, which formally disappear in the comparison. The interaction energy of |$\eta $| with itself is denoted by |$\textsf{H}_{\Lambda }(\eta )$|⁠. \begin{equation} \widetilde{\textsf{H}}_{\Lambda}(\eta):= \frac{1}{2} \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log |x-y| (\textrm{d}\eta(x) - \textrm{d}x) (\textrm{d}\eta(y) - \textrm{d}y). \end{equation} (6.3) The following quantity encodes the interaction energy of the configuration |$\eta $| in |$\Lambda $| with the configuration |$\gamma $| outside |$\Lambda $|⁠. In fact, we compute the interaction of |$\eta - \gamma $| in |$\Lambda $| with |$\gamma - dx$| outside |$\Lambda $|⁠. The 1st modification only plays the role of a (possibly infinite) additive constant (for fixed |$\gamma $|⁠), and the 2nd modification is technical. \begin{equation} \widetilde{{\mathcal{M}}}_{\Lambda}(\eta, \gamma):= \lim_{p \to \infty} \int_{x \in ([-p,p] \backslash \Lambda)} \int_{y \in \Lambda} - \log |x-y| (\textrm{d}\eta(y) - \textrm{d}\gamma_{\Lambda}(y)) (\textrm{d}\gamma(x) - \textrm{d}x). \end{equation} (6.4) We denote Bernoulli point processes by |$\textbf{B}$|⁠. In particular, |$\textbf{B}_{|\gamma _\Lambda |, \Lambda }$| is the law of the Bernoulli point process with |$|\gamma _{\Lambda }|$| points in |$\Lambda $|⁠, that is, the law of a random point configuration made of |$|\gamma _{\Lambda }|$| points drawn uniformly and independently in |$\Lambda $|⁠. We may now form the Boltzmann factor associated to the sum of these energies, given by \begin{equation*} \exp \left( - \beta \left(\widetilde{\textsf{H}}_{\Lambda}(\eta) + \widetilde{{\mathcal{M}}}_{\Lambda}(\gamma, \eta) \right) \right) \end{equation*} and the associated partition function \begin{equation} Z_{\Lambda, \beta}(\gamma):= \int \exp \left(- \beta\left( \widetilde{\textsf{H}}_{\Lambda}(\eta) + \widetilde{{\mathcal{M}}}_{\Lambda}(\eta, \gamma) \right) \right) \ \textrm{d}\textbf{B}_{|\gamma_\Lambda|, \Lambda}(\eta). \end{equation} (6.5) Finally, for |$\gamma $| fixed, we define |$\textsf{Gibbs}_{\Lambda , \beta }(\eta ; \gamma )$| as a probability measure on random point configurations |$\eta $| in |$\Lambda $| given by \begin{equation} d\textsf{Gibbs}_{\Lambda, \beta}(\eta ; \gamma):= \frac{1}{ Z_{\Lambda, \beta}( \gamma)} \exp \left(- \beta\left( \widetilde{\textsf{H}}_{\Lambda}(\eta) + \widetilde{{\mathcal{M}}}_{\Lambda}(\eta, \gamma) \right) \right) d\textbf{B}_{|\gamma_\Lambda|, \Lambda}(\eta). \end{equation} (6.6) The following is a re-writing of the main result in [6]. Proposition 6.3 DLR equations for |$\textrm{Sine}_{\beta }$|⁠. Let |$f$| be a bounded, measurable function on the space of point configurations, and |$\lambda> 0$|⁠, we have \begin{equation} {\mathbb{E}}[f] = \int \ \textrm{d}\textrm{Sine}_{\beta}(\gamma) \int f(\eta \cup \gamma_{\Lambda^c}) \ \textrm{d}\textsf{Gibbs}_{\Lambda, \beta}(\eta; \gamma). \end{equation} (6.7) Proof. The only difference with [6] is that we chose here to include the background in the definition of the energy. The result of [6] is stated with |$\textsf{H}_{\Lambda }$| and |${\mathcal{M}}_{\Lambda }$| instead of |$\widetilde{\textsf{H}}_{\Lambda }$| and |$\widetilde{{\mathcal{M}}}_{\Lambda }$|⁠, respectively, where \begin{equation*} \textsf{H}_{\Lambda}(\eta):= \frac{1}{2} \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log |x-y| \ \textrm{d}\eta(x) \ \textrm{d}\eta(y), \end{equation*} \begin{equation*} {\mathcal{M}}_{\Lambda}(\gamma, \eta):= \lim_{p \to \infty} \int_{x \in ([-p,p] \backslash \Lambda)} \int_{y \in \Lambda} - \log |x-y| (\textrm{d}\eta(y) - \textrm{d}\gamma_{\Lambda}(y)) \ \textrm{d}\gamma(x). \end{equation*} It is easy to check that the difference between these two formulations is an additive constant (for fixed |$\Lambda , \gamma $|⁠), which is absorbed by the partition function, plus the term \begin{equation*} \lim_{p \to \infty} \int_{y \in [-p, p]} \int_{x \in \Lambda} - \log |x-y| (\textrm{d}\eta(x) - \textrm{d}\gamma(x)) \ \textrm{d}y, \end{equation*} which is almost surely zero. 6.3 The Laplace transform of the fluctuations We introduce the function |${\mathcal{L}}_P$| \begin{equation*} t \mapsto{\mathcal{L}}_P(t):= {\mathbb{E}} \left[ \exp\left(t \textrm{Fluct}[\varphi]({\mathcal{C}}) \textsf{1}_{\textsf{Event}_{\lambda, \ell}}({\mathcal{C}})\right) \right], \end{equation*} which is the Laplace transform of the fluctuations of |$\varphi $|⁠, up to the indicator function |$\textsf{1}_{\textsf{Event}_{\lambda , \ell }}$|⁠, and, by construction, |$\textsf{Event}_{\lambda , \ell }$| is very likely. Using |${\mathbb{P}}(\textsf{Event}_{\lambda , \ell }) = 1 - o_{\ell , \lambda }(1)$| as stated in (6.1), we may of course re-write |${\mathcal{L}}_P$| as \begin{equation*} {\mathcal{L}}_P(t) = {\mathbb{E}} \left[ \exp\left(t \textrm{Fluct}[\varphi]({\mathcal{C}}) \right) \textsf{1}_{\textsf{Event}_{\lambda, \ell}}({\mathcal{C}}) \right] + o_{\ell, \lambda}(1), \end{equation*} and we now focus on the 1st term in the right-hand side, which we denote by \begin{equation} \widetilde{\mathcal{L}}_{{\varphi}, \ {\ell}, \ {\lambda}}(t):= {\mathbb{E}} \left[ \exp\left(t \textrm{Fluct}[\varphi]({\mathcal{C}}) \right) \textsf{1}_{\textsf{Event}_{\lambda, \ell}}({\mathcal{C}}) \right]. \end{equation} (6.8) The map |${\mathcal{C}} \mapsto \exp \left (t \textrm{Fluct}[\varphi ]({\mathcal{C}}) \right ) \textsf{1}_{\textsf{Event}_{\lambda , \ell }}({\mathcal{C}})$| is bounded, because by construction, on |$\textsf{Event}_{\lambda , \ell }$|⁠, the number of points of |${\mathcal{C}}$| in the support of |$\varphi $| is bounded, see (6.2). Using DLR equations (6.7), we write \begin{align} \widetilde{\mathcal{L}}_{{\varphi}, \ {\ell}, \ {\lambda}}(t) &= \int \ \textrm{d}\textrm{Sine}_{\beta}(\gamma) \frac{1}{Z_{\Lambda, \beta}(\gamma)} \nonumber\\\nonumber &\quad\times \int \exp\left(t \textrm{Fluct}[\varphi](\eta) \right) \textsf{1}_{\textsf{Event}_{\lambda, \ell}}\left(\eta \cup \gamma_{\Lambda^c}\right) \\ &\qquad\qquad\exp \left(- \beta\left( \widetilde{\textsf{H}}_{\Lambda}(\eta) + \widetilde{{\mathcal{M}}}_{\Lambda}(\eta, \gamma) \right) \right) \ \textrm{d}\textbf{B}_{|\gamma_\Lambda|, \Lambda}(\eta), \end{align} (6.9) where we have used the fact that, since |$\varphi $| is supported inside |$\Lambda $|⁠, we may write \begin{equation*} \textrm{Fluct}[\varphi](\eta \cup \gamma_{\Lambda^c}) = \textrm{Fluct}[\varphi](\eta). \end{equation*} Combining both exponential terms and using the definition (6.3) of |$\widetilde{\textsf{H}}_{\Lambda }$|⁠, we obtain, in the exponent, \begin{equation*} -\beta \left( \frac{1}{2} \left( \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log |x-y| (\textrm{d}\eta(x) - \textrm{d}x) (\textrm{d}\eta(y) - \textrm{d}y) - \frac{2t}{\beta} \textrm{Fluct}[\varphi](\eta) \right) + \widetilde{{\mathcal{M}}}_{\Lambda}(\eta, \gamma) \right), \end{equation*} and we let \begin{equation} s:= \frac{t}{\beta}. \end{equation} (6.10) 6.4 Laplace transform I. Energy splitting In view of the “energy splitting” identity stated in Lemma 3.3, we may write \begin{multline*} \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log |x-y| (\textrm{d}\eta(x) - \textrm{d}x)(\textrm{d}\eta(y) - \textrm{d}y) - 2s \textrm{Fluct}[\varphi](\eta) \\ = \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log |x-y| (\textrm{d}\eta(x) - \textrm{d}\mu_{s}(x))(\textrm{d}\eta(y) - \textrm{d}\mu_{s}(y)) \\ + 2 s \int_{\Lambda} \left(\textsf{LP}_{\lambda, \varphi}(x) - \varphi\right) (\textrm{d}\eta - \textrm{d}x) + 2s \int_{\Lambda} \textrm{Error}\textsf{LP}_{\lambda, \varphi}(x) (\textrm{d}\eta -\textrm{d}x) \\ - 2 s^2 \|\overline{\varphi}\|_{H^{\frac{1}{2}}}^2 - s^2 \textrm{ErrorVar}. \end{multline*} The term |$\textrm{ErrorVar}$| is bounded as in (2.20), hence we have \begin{align} &\exp\left(t \textrm{Fluct}[\varphi](\eta) \right) \exp \left(- \beta\left( \widetilde{\textsf{H}}_{\Lambda}(\eta) + \widetilde{{\mathcal{M}}}_{\Lambda}(\eta, \gamma) \right) \right) \textsf{1}_{\textsf{Event}_{\lambda, \ell}}(\eta \cup \gamma_{\Lambda^c}) \nonumber\\\nonumber &\quad= \exp \left( - \beta \left( \frac{1}{2} \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log |x-y| (\textrm{d}\eta(x) - \textrm{d}\mu_{s}(x))(\textrm{d}\eta(y) - \textrm{d}\mu_{s}(y)) \right) \right)\\ &\qquad\times \exp\left( - \beta \left( s \int_{\Lambda} \left(\textsf{LP}_{\lambda, \varphi}(x) - \varphi\right) (\textrm{d}\eta - \textrm{d}x) + s \int_{\Lambda} \textrm{Error}\textsf{LP}_{\lambda, \varphi}(x) (\textrm{d}\eta -\textrm{d}x) + \widetilde{{\mathcal{M}}}_{\Lambda}(\eta, \gamma) \right)\! \right) \nonumber\\ &\qquad\times \exp\left( s^2 \beta \|\overline{\varphi}\|_{H^{\frac{1}{2}}}^2 \right) \times \exp\left( s o_{\ell, \lambda}(1) + s^2 o_{\ell, \lambda}(1) \right) \textsf{1}_{\textsf{Event}_{\lambda, \ell}}\left(\eta \cup \gamma_{\Lambda^c}\right). \end{align} (6.11) Inserting this expansion into (6.9), we obtain \begin{align} &\widetilde{\mathcal{L}}_{{\varphi}, \ {\ell}, \ {\lambda}}(t) = \exp\left( \frac{s^2}{2} \left(2\beta \|\overline{\varphi}\|_{H^{\frac{1}{2}}}^2\right) + s o_{\ell, \lambda}(1) + s^2 o_{\ell, \lambda}(1) \right) \int \textrm{d}\textrm{Sine}_{\beta}(\gamma) \frac{1}{Z_{\Lambda, \beta}(\gamma)} \nonumber\\ &\quad\times \int \exp \left( - \beta \left( \frac{1}{2} \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log |x-y| (\textrm{d}\eta(x) - \textrm{d}\mu_{s}(x))(\textrm{d}\eta(y) - \textrm{d}\mu_{s}(y)) \right) \right)\nonumber\\ &\quad\times \exp\left( - \beta \left( s \int_{\Lambda} \left(\textsf{LP}_{\lambda, \varphi}(x) - \varphi\right) (\textrm{d}\eta - \textrm{d}x) + s \int_{\Lambda} \textrm{Error}\textsf{LP}_{\lambda, \varphi}(x) (\textrm{d}\eta -\textrm{d}x) + \widetilde{{\mathcal{M}}}_{\Lambda}(\eta, \gamma) \right) \right) \nonumber\\ &\quad\times \textsf{1}_{\textsf{Event}_{\lambda, \ell}}\left(\eta \cup \gamma_{\Lambda^c}\right) \ \textrm{d}\textbf{B}_{|\gamma_\Lambda|, \Lambda}(\eta). \end{align} (6.12) 6.5 Laplace transform II. Change of variables We now perform a change of variables on |$\eta $|⁠. For |$N$| fixed, we may consider the map |$\Phi_{s} : \Lambda ^{N} \to \Lambda ^{N} $| given by \begin{equation*} \Phi_{s}(x_1, \dots, x_N):= (\Phi_s(x_1), \dots, \Phi_s(x_N)), \end{equation*} where |$\Phi _s$| is the transport map from the constant density to |$\mu _{s}$|⁠. Since |$\Phi _s$| is a bijection, so is |$\Phi_{s} $|⁠. We let |$\nu = \Phi_{s}^{-1}(\eta )$|⁠, so that |$\eta = \Phi_{s} (\nu )$| is the push-forward of |$\nu $| by |$\Phi _s$|⁠, which we will now denote by |$\nu _s$|⁠. The innermost integral in (6.12) becomes \begin{align} &\int \exp \left( - \beta \left( \frac{1}{2} \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log |x-y| (\textrm{d}\nu_s(x) - \textrm{d}\mu_{s}(x))(\textrm{d}\nu_s(y) - \textrm{d}\mu_{s}(y)) \right) \right)\nonumber\\\nonumber &\quad\times \exp\left( - \beta \left( s \int_{\Lambda} \left(\textsf{LP}_{\lambda, \varphi}(x) - \varphi\right) (\textrm{d}\nu_s - \textrm{d}x) + s \int_{\Lambda} \textrm{Error}\textsf{LP}_{\lambda, \varphi}(x) (\textrm{d}\nu_s -\textrm{d}x) + \widetilde{{\mathcal{M}}}_{\Lambda}(\nu_s, \gamma) \right) \right) \\ &\quad\times \exp\left(\int \log \Phi_s^{\prime}(x) \ \textrm{d}\nu(x) \right) \textsf{1}_{\textsf{Event}_{\lambda, \ell}}\left(\nu_s \cup \gamma_{\Lambda^c}\right) \ \textrm{d}\textbf{B}_{|\gamma_\Lambda|, \Lambda}(\nu), \end{align} (6.13) where the term |$\exp \left (\int \log \Phi _s^{\prime}(x) \ \textrm{d}\nu (x)\right )$| is the Jacobian of the transformation. In view of (3.18), we have \begin{equation*} \int \log \Phi_s^{\prime}(x) \ \textrm{d}\nu(x) = \textsf{RE}_s + \textrm{Flu}\textsf{RE}_s(\nu); \end{equation*} we know from (4.2) that |$\textsf{RE}_s = s^2 o_{\ell , \lambda }(1)$|⁠, and we know from Lemma 6.1 that \begin{equation*} \nu_s \cup \gamma_{\Lambda^c} \in \textsf{Event}_{\lambda, \ell} \implies \textrm{Flu}\textsf{RE}_s(\nu) = s o_{\ell, \lambda}(1), \end{equation*} hence the Jacobian only contributes to an error term |$\exp (so_{\ell , \lambda }(1) + s^2 o_{\ell , \lambda }(1))$|⁠. 6.6 Laplace transform III. The interior–interior energy Using Lemma 3.8, we have \begin{multline*} \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log |x-y| (\textrm{d}\nu_s(x) - \textrm{d}\mu_{s}(x))(\textrm{d}\nu_s(y) - \textrm{d}\mu_{s}(y)) \\ = \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log |x-y| (\textrm{d}\nu(x) - \textrm{d}x)(\textrm{d}\nu(y) - \textrm{d}y) + \textsf{Main}_s(\nu) + \textsf{RE}_s + \textrm{Flu}\textsf{RE}_s(\nu). \end{multline*} We know from (4.2) that |$\textsf{RE}_s = s^2 o_{\ell , \lambda }(1)$|⁠, and we know from Lemma 6.1 that \begin{equation*} \nu_s \cup \gamma_{\Lambda^c} \in \textsf{Event}_{\lambda, \ell} \implies \textrm{Flu}\textsf{RE}_s(\nu) = s o_{\ell, \lambda}(1), \ \textsf{Main}_s(\nu) = s o_{\ell, \lambda}(1). \end{equation*} We may thus write (6.13) as \begin{multline} \int \exp \left( - \beta \left( \frac{1}{2} \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log |x-y| (\textrm{d}\nu(x) - \textrm{d}x)(\textrm{d}\nu(y) - \textrm{d}y) \right) \right)\\ \times \exp\left( - \beta \left( s \int_{\Lambda} \left(\textsf{LP}_{\lambda, \varphi}(x) - \varphi\right) (\textrm{d}\nu_s - \textrm{d}x) + s \int_{\Lambda} \textrm{Error}\textsf{LP}_{\lambda, \varphi}(x) (\textrm{d}\nu_s -\textrm{d}x) + \widetilde{{\mathcal{M}}}_{\Lambda}(\nu_s, \gamma) \right) \right) \\ \times \exp\left(s o_{\ell, \lambda}(1) + s^2 o_{\ell, \lambda}(1) \right) \textsf{1}_{\textsf{Event}_{\lambda, \ell}}\left(\nu_s \cup \gamma_{\Lambda^c}\right) \ \textrm{d}\textbf{B}_{|\gamma_\Lambda|, \Lambda}(\nu). \end{multline} (6.14) 6.7 Laplace transform IV. The interior–exterior energy Let us recall that |$\widetilde{{\mathcal{M}}}_{\Lambda }$| is defined in (6.4) by \begin{equation*} \widetilde{{\mathcal{M}}}_{\Lambda}(\eta, \gamma):= \lim_{p \to \infty} \int_{x \in ([-p,p] \backslash \Lambda)} \int_{y \in \Lambda} - \log |x-y| (\textrm{d}\eta(y) - \textrm{d}\gamma_{\Lambda}(y)) (\textrm{d}\gamma(x) - \textrm{d}x). \end{equation*} A direct computation shows that \begin{equation} \widetilde{{\mathcal{M}}}_{\Lambda}(\nu_s, \gamma) = \widetilde{{\mathcal{M}}}_{\Lambda}(\nu, \gamma) + \int_{\Lambda^c} \textsf{DF}_s(\nu)(x) (\textrm{d}\gamma - \textrm{d}x), \end{equation} (6.15) where |$\textsf{DF}_s(\nu )$| is the difference field generated by |$\nu _s - \nu $| as in (5.1). Using the decomposition \begin{equation*} \textsf{DF}_s(\nu) = s\textsf{LP}_{\lambda, \varphi}(x) + s\textrm{Error}\textsf{LP}_{\lambda, \varphi}(x) + \textrm{Error}\textsf{DF}_s(\nu)(x), \end{equation*} as in (5.2), we may write \begin{multline} \widetilde{{\mathcal{M}}}_{\Lambda}(\nu_s, \gamma) = \widetilde{{\mathcal{M}}}_{\Lambda}(\nu, \gamma) + s \int_{\Lambda^c} \textsf{LP}_{\lambda, \varphi}(x) (\textrm{d}\gamma - \textrm{d}x) \\ + s \int_{\Lambda^c} \textrm{Error}\textsf{LP}_{\lambda, \varphi}(x) (\textrm{d}\gamma - \textrm{d}x) + \int_{\Lambda^c} \textrm{Error}\textsf{DF}_s(\nu)(x) (\textrm{d}\gamma - \textrm{d}x), \end{multline} (6.16) so in particular, the middle line in (6.14) reads as \begin{align}& s \int_{\Lambda} \left(\textsf{LP}_{\lambda, \varphi}(x) - \varphi\right) (\textrm{d}\nu_s - \textrm{d}x) + s \int_{\Lambda} \textrm{Error}\textsf{LP}_{\lambda, \varphi}(x) (\textrm{d}\eta -\textrm{d}x) + \widetilde{{\mathcal{M}}}_{\Lambda}(\nu_s, \gamma) \nonumber\\ &\quad= \widetilde{{\mathcal{M}}}_{\Lambda}(\nu, \gamma) + s \int \left(\textsf{LP}_{\lambda, \varphi}(x) - \varphi\right) (\textrm{d} \left[\nu_s \cup \gamma_{\Lambda^c}\right] \!-\! \textrm{d}x) + s \int \textrm{Error}\textsf{LP}_{\lambda, \varphi}(x) (\textrm{d}\left[\nu_s \cup \gamma_{\Lambda^c}\right]\! -\!\textrm{d}x) \nonumber\\&\qquad+ \int_{\Lambda^c} \textrm{Error}\textsf{DF}_s(\nu)(x) (\textrm{d}\gamma - \textrm{d}x). \end{align} (6.17) We know from Lemma 6.1 that \begin{equation*} \nu_s \cup \gamma_{\Lambda^c} \in \textsf{Event}_{\lambda, \ell} \implies \left\lbrace \begin{array}{@{}lll} \displaystyle{\int_{\Lambda^c} \textrm{Error}\textsf{DF}_s(\nu)(x) (\textrm{d}\gamma - \textrm{d}x)} & = & s o_{\ell, \lambda}(1) \\ \displaystyle{s \int \left(\textsf{LP}_{\lambda, \varphi}(x) - \varphi\right) (\textrm{d} \left[\nu_s \cup \gamma_{\Lambda^c}\right] - \textrm{d}x)} & = & s o_{\ell, \lambda}(1) \\ \displaystyle{s \int \textrm{Error}\textsf{LP}_{\lambda, \varphi}(x) (\textrm{d}\left[\nu_s \cup \gamma_{\Lambda^c}\right] -\textrm{d}x)} & = & s o_{\ell, \lambda}(1) \end{array}\right., \end{equation*} so we may re-write (6.14) as \begin{multline} \int \exp \left( - \beta \left( \frac{1}{2} \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log |x-y| (\textrm{d}\nu(x) - \textrm{d}x)(\textrm{d}\nu(y) - \textrm{d}y) \right) \right) \times \exp\left( - \beta \left( \widetilde{{\mathcal{M}}}_{\Lambda}(\nu, \gamma) \right) \right) \\ \times \exp\left(s o_{\ell, \lambda}(1) + s^2 o_{\ell, \lambda}(1) \right) \textsf{1}_{\textsf{Event}_{\lambda, \ell}}\left(\nu_s \cup \gamma_{\Lambda^c}\right) \ \textrm{d}\textbf{B}_{|\gamma_\Lambda|, \Lambda}(\nu). \end{multline} (6.18) 6.8 Conclusion Using (6.18) recognizing |$\widetilde{\textsf{H}}_{\Lambda }(\nu )$| in the 1st exponent (as defined in (6.3)) and coming back to the expression (6.12) of |$\widetilde{\mathcal{L}}_{{\varphi}, \ {\ell}, \ {\lambda}} (t)$|⁠, we get \begin{multline} \widetilde{\mathcal{L}}_{{\varphi}, \ {\ell}, \ {\lambda}}(t) = \exp\left( \frac{s^2}{2} \left(2\beta \|\overline{\varphi}\|_{H^{\frac{1}{2}}}^2\right) + s o_{t}(1) + s^2 o_{\ell, \lambda}(1) \right) \int \ \textrm{d}\textrm{Sine}_{\beta}(\gamma) \frac{1}{Z_{\Lambda, \beta}(\gamma)} \\ \times \int \exp \left( - \beta \left(\widetilde{\textsf{H}}_{\Lambda}(\nu) + \widetilde{{\mathcal{M}}}_{\Lambda}(\nu, \gamma) \right) \right) \times \textsf{1}_{\textsf{Event}_{\lambda, \ell}}\left(\nu_s \cup \gamma_{\Lambda^c}\right) \ \textrm{d}\textbf{B}_{|\gamma_\Lambda|, \Lambda}(\nu). \end{multline} (6.19) By the DLR equations (6.7), we may write \begin{align*} &\int \ \textrm{d}\textrm{Sine}_{\beta}(\gamma) \frac{1}{Z_{\Lambda, \beta}(\gamma)} \times \int \exp \left( - \beta \left(\widetilde{\textsf{H}}_{\Lambda}(\nu) + \widetilde{{\mathcal{M}}}_{\Lambda}(\nu, \gamma) \right) \right) \times \textsf{1}_{\textsf{Event}_{\lambda, \ell}}\left(\nu_s \cup \gamma_{\Lambda^c}\right) \ \textrm{d}\textbf{B}_{|\gamma_\Lambda|, \Lambda}(\nu)\\ &\quad= {\mathbb{E}} \left[ \textsf{1}_{\nu_s \cup \gamma_{\Lambda^c} \in \textsf{Event}_{\lambda, \ell}} \right] = {\mathbb{P}} \left[ \left\lbrace\nu_s \cup \gamma_{\Lambda^c} \in \textsf{Event}_{\lambda, \ell} \right\rbrace \right], \end{align*} and by Lemma 6.1, this quantity is |$1 - o_{\ell , \lambda }(1)$|⁠. Doing a final replacement of |$s$| by |$\frac{t}{\beta }$|⁠, we obtain \begin{equation} \widetilde{\mathcal{L}}_{{\varphi}, \ {\ell}, \ {\lambda}}(t) = \exp\left( \frac{t^2}{2} \times \frac{2}{\beta} \|\overline{\varphi}\|_{H^{\frac{1}{2}}}^2 + t o_{\ell, \lambda}(1) + t^2 o_{\ell, \lambda}(1) \right) ( 1 - o_{\ell, \lambda}(1)). \end{equation} (6.20) In particular, for |$t$| such that |$\frac{|t|}{\beta } \leq \textrm{s}_{\textrm{max}}$| as in (3.1), we get, uniformly in |$t$|⁠, \begin{equation} \lim_{\ell \to \infty} \lim_{\lambda \to \infty} {\mathcal{L}}_P(t) = \exp \left( \frac{t^2}{2} \times \frac{2}{\beta} \|\overline{\varphi}\|_{H^{\frac{1}{2}}}^2 \right). \end{equation} (6.21) We have thus obtained that, sending |$\lambda \to \infty $| then |$\ell \to \infty $|⁠, the Laplace transform of the random variable \begin{equation*} \textrm{Fluct}[\varphi]({\mathcal{C}}) \textsf{1}_{\textsf{Event}_{\lambda, \ell}}({\mathcal{C}}) \end{equation*} converges to \begin{equation*} t \mapsto \exp\left( \frac{t^2}{2} \times \frac{2}{\beta}\|\overline{\varphi}\|_{H^{\frac{1}{2}}}^2 \right), \end{equation*} which is the Laplace transform of a centered Gaussian variable with variance |$ \frac{2}{\beta } \|\overline{\varphi }\|_{H^{\frac{1}{2}}}^2$|⁠. The convergence is uniform for values of the parameter in some open interval around |$0$|⁠. It is well known that this convergence implies convergence in law. Moreover, since we know by (6.1) that |${\mathbb{P}}(\textsf{Event}_{\lambda , \ell }) = 1 - o_{\ell , \lambda }(1)$|⁠, the convergence in law of |$\textrm{Fluct}[\varphi ]({\mathcal{C}}) \textsf{1}_{\textsf{Event}_{\lambda , \ell }}({\mathcal{C}})$| implies the convergence in law of the fluctuations themselves. This concludes the proof of the central limit theorem. Remark 6.4 (Lack of moderate deviations bounds). Since |$|\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }|_{\textsf{0}} \preceq \frac{1}{\ell }$|⁠, taking |$s$| of order as large as |$\ell $| still guarantees that |$\mu _{s}$| will be a positive density. The transport map |$\Phi _s$| may now move points at a distance |$O_{\bullet }(\ell )$|⁠, but, in fact, this is harmless because a careful inspection reveals that our estimates are insensitive to a displacement of the points of order |$\ell $|⁠. Taking |$s$| large is tempting because it yields a control on the Laplace transform of the fluctuations for large values of the parameter, which in turn implies strong concentration bounds with exponential (in |$\ell $|⁠) tails. However, our argument relies on the discrepancy estimate (1.9), which is not quantitative, and raises an obstacle for obtaining such moderate deviations bounds on the fluctuations. 7 Auxiliary Proofs 7.1 Proof of Proposition 1.6 Proof of Proposition 1.6 We write \begin{equation*} \int g(x) (\textrm{d}{\mathcal{C}} - \textrm{d}x) = \sum_{k=-\infty}^{\infty} \int_{k}^{k+1} g(x) (\textrm{d}{\mathcal{C}} - \textrm{d}x). \end{equation*} Since |$g$| is assumed to be compactly supported, all the sums are finite. On |$[k, k+1]$| we may write, using the mean value theorem, |$g(x) = g(k) + O_{\bullet }\left (|g|_{\textsf{1},V_{k}}\right )$|⁠, and we obtain \begin{equation*} \int_{k}^{k+1} g(x) (\textrm{d}{\mathcal{C}} - \textrm{d}x) = g(k) \textrm{Discr}_{[k, k+1]} + O_{\bullet}\left(|g|_{\textsf{1},V_{k}}\right)\left(1 + |\textrm{Discr}_{[k, k+1]}|\right). \end{equation*} We have, of course, \begin{equation} \textrm{Discr}_{[k, k+1]} = \textrm{Discr}_{[0, k+1]} - \textrm{Discr}_{[0, k]}, \end{equation} (7.1) so a summation by parts yields \begin{equation*} \sum_{k=-\infty}^{\infty} g(k) \textrm{Discr}_{[k, k+1]} = \sum_{k = -\infty}^{\infty} \left(g(k-1) - g(k) \right) \textrm{Discr}_{[0, k]}. \end{equation*} Using the mean value theorem again, we get \begin{align*} \left|\sum_{k = -\infty}^{\infty} \left(g(k-1) - g(k) \right) \textrm{Discr}_{[0, k]}\right| \leq \sum_{k = -\infty}^{\infty} |g|_{\textsf{1}, V_{k}} |\textrm{Discr}_{[0, k]}|. \end{align*} We have thus obtained \begin{equation*} \int g(x) (\textrm{d}{\mathcal{C}} - \textrm{d}x) \preceq \sum_{k=-\infty}^{\infty} |g|_{\textsf{1}, V_{k}} \left(|\textrm{Discr}_{[0, k]}| + |\textrm{Discr}_{[k, k+1]}| +1 \right), \end{equation*} which yields the result. Finally, if |$\lambda $| is fixed, we could choose to write, instead of (7.1), \begin{equation*} \textrm{Discr}_{[k, k+1]} = \textrm{Discr}_{[-\lambda, k+1]} - \textrm{Discr}_{[-\lambda, k]}, \quad \textrm{Discr}_{[k, k+1]} = \textrm{Discr}_{[k+1, \lambda]} - \textrm{Discr}_{[k, \lambda]} \end{equation*} so we can replace |$\widetilde{D}$| by |$\widetilde{D}^{\textrm{Left}}$| or |$\widetilde{D}^{\textrm{Right}}$| as claimed. 7.2 Proof of Lemma 2.6 Proof of Lemma 2.6 It is easy to check that |${\mathfrak{H}}_{\lambda , \varphi }$| is bounded, and |$x \mapsto \frac{1}{\sqrt{\lambda ^2-x^2}}$| is integrable, thus so is |${\mathfrak{m}}_{\lambda , \varphi }$|⁠. Moreover, for any |$x$| in |$(-\lambda , \lambda )$|⁠, the map \begin{equation*} y \mapsto \log |x-y| \frac{1}{\sqrt{\lambda^2-x^2}} \end{equation*} is also integrable, hence the logarithmic potential is well defined. The fact that |${\mathfrak{m}}_{\lambda , \varphi }$| has total mass |$0$| follows from the well-known identity \begin{equation*} \textbf{PV} \int \frac{1}{\sqrt{\lambda^2-x^2}} \frac{1}{t-x} \ \textrm{d}x = 0 \text{ for {$t$} in {$(-\lambda, \lambda)$},} \end{equation*} which can be proven by elementary means, see for example, [21, Sec. 4.3, eq. (7)]. The fact that the logarithmic potential satisfies (2.6) is a also a well-known result, which can be obtained by integrating the identity \begin{equation} \textbf{PV} \int \frac{{\mathfrak{m}}_{\lambda, \varphi}(t)}{t-x} \ \textrm{d}t = \varphi^{\prime}(x), \end{equation} (7.2) valid for any |$x$| in |$(-\lambda , \lambda )$|⁠. This is known as the airfoil equation, and we refer again to [21, Sec. 4.3, eq. (12)]. 7.3 Proof of Lemma 2.7 We start by the following bounds concerning |${\mathfrak{H}}_{\lambda , \varphi }$|⁠. Lemma 7.1 (Bounds on |${\mathfrak{H}}_{\lambda , \varphi }$| and its derivatives). We have \begin{align} & {\mathfrak{H}}_{\lambda, \varphi}(x) \preceq \begin{cases} \frac{\lambda}{\ell} & |x| \leq 2\ell, \\ \frac{\lambda \ell}{x^2} & |x| \geq 2\ell. \end{cases} \end{align} (7.3) \begin{align} & {\mathfrak{H}}_{\lambda, \varphi}^{(\textsf{1})}(x) \preceq \begin{cases} \frac{\lambda}{\ell^2} & |x| \leq 2\ell, \\ \frac{\lambda \ell}{x^3} & |x| \geq 2\ell. \end{cases} \end{align} (7.4) \begin{align} & {\mathfrak{H}}_{\lambda, \varphi}^{(\textsf{2})}(x) \preceq \begin{cases} \frac{\lambda}{\ell^3} & |x| \leq 2\ell, \\ \frac{\lambda \ell}{x^4} & |x| \geq 2\ell. \end{cases} \end{align} (7.5) with implicit constants depending on |$\overline{\varphi }$|⁠. Proof of Lemma 7.1 We start with the following claim. Claim 7.2. Let |$g$| be a test function of class |$C^1$|⁠, supported on |$(-\ell , \ell )$|⁠. Then for any |$x$| such that |$|x| \leq 2\ell $| we have \begin{equation} \textbf{PV} \int \frac{g(t)}{t-x} \ \textrm{d}t \preceq \ell^{1/2} \|g^{\prime}\|_{L^{2}}. \end{equation} (7.6) Proof of the claim. Let |$x$| be such that |$|x| \leq 2\ell $|⁠. Let us use the definition (2.1) of the Cauchy principal value and write \begin{equation*} \textbf{PV} \int \frac{g(t)}{t-x} \ \textrm{d}t = \int_{0}^{+ \infty} \frac{g(x+u) -g(x-u)}{u} \ \textrm{d}u = \int_{u \in I_x} \frac{g(x+u) -g(x-u)}{u} \ \textrm{d}u, \end{equation*} where |$I_x$| is the set of positive real numbers |$u$| such that |$g(x+u)$| or |$g(x-u)$| is not zero. This set depends on |$x$|⁠, but since |$g$| is supported on |$(-\ell , \ell )$|⁠, the set |$I_x$| is included in a union of intervals whose total length is bounded by |$4 \ell $|⁠. Using the elementary identity \begin{equation*} g(x+u) - g(x-u) = \int_{x-u}^{x+u} g^{\prime}(v) \ \textrm{d}v, \end{equation*} and applying Fubini’s theorem, we get (⁠|$x-u \leq v \leq x+u$| is equivalent to |$u \geq |x-v|$|⁠) \begin{align*} \int_{u \in I_x} \frac{g(x+u) -g(x-u)}{u} \ \textrm{d}u = \int_{u \in I_x} \frac{\textrm{d}u}{u} \int_{x-u}^{x+u} g^{\prime}(v) \ \textrm{d}v = \int_{-\ell}^{\ell} g^{\prime}(v) \ \textrm{d}v \int_{u \geq |x-v|, u \in I_{x}} \frac{1}{u} \ \textrm{d}u. \end{align*} Since |$u \mapsto \frac{1}{u}$| is decreasing, and |$I_x$| has its length bounded by |$4 \ell $|⁠, the innermost integral satisfies \begin{equation*} \int_{u \geq |x-v|, u \in I_{x}} \frac{1}{u} \ \textrm{d}u \leq \int_{|x-v|}^{|x-v| + 4 \ell} \frac{1}{u} \ \textrm{d}u = \log \left(\frac{4 \ell + |x-v|}{|x-v|}\right), \end{equation*} so we have \begin{equation*} \int_{-\ell}^{\ell} g^{\prime}(v) \ \textrm{d}v \int_{u \geq |x-v|, u \in I_{x}} \frac{1}{u} \ \textrm{d}u \leq \int_{-\ell}^{\ell} |g^{\prime}(v)|\log \left(\frac{4 \ell + |x-v|}{|x-v|}\right) \ \textrm{d}v. \end{equation*} Applying Cauchy–Schwarz’s inequality, we get \begin{equation*} \textbf{PV} \int \frac{g(t)}{t-x} \ \textrm{d}t \leq \|g^{\prime}\|_{L^{\textsf{2}}} \left( \int_{-\ell}^\ell \left|\ln \left(\frac{4 \ell + |x-v|}{|x-v|} \right) \right|{}^{2} \ \textrm{d}v \right)^{1/2}. \end{equation*} A linear change of variables |$w = \frac{x-v}{\ell }$| shows that, for |$|x| \leq 2\ell $|⁠, we have \begin{equation*} \left( \int_{-\ell}^{\ell} \left|\ln \left(\frac{4 \ell + |x-v|}{|x-v|} \right) \right|{}^{2} \ \textrm{d}v \right)^{1/2} \preceq \ell^{1/2}, \end{equation*} which proves (7.6). We recall that, by definition, \begin{equation} {\mathfrak{H}}_{\lambda, \varphi}(x) = \frac{1}{\pi} \textbf{PV} \int \frac{\phi_{\Lambda}(t)}{t-x} \ \textrm{d}t, \end{equation} (7.7) with |$\phi _{\Lambda }$| defined as \begin{equation*} \phi_{\Lambda}: t \mapsto \sqrt{\lambda^2-t^2} \varphi^{\prime}(t), \end{equation*} and we compute the 1st derivatives of |$\phi _{\Lambda }$| as \begin{equation} \phi_{\Lambda}^{(1)}(t) = \frac{-t}{\sqrt{\lambda^2-t^2}} \varphi^{(1)}(t) + \sqrt{\lambda^2 - t^2} \varphi^{(2)}(t) \end{equation} (7.8) \begin{equation} \phi_{\Lambda}^{(2)}(t) = \left( \frac{-1}{\sqrt{\lambda^2 -t^2}} - \frac{t^2}{(\lambda^2-t^2)^{3/2}} \right) \varphi^{(1)}(t) + \left( \frac{-2t}{\sqrt{\lambda^2 - t^2}} \right) \varphi^{(2)}(t) + \sqrt{\lambda^2-t^2} \varphi^{(3)}(t), \end{equation} (7.9) and (this is the only moment where we need the |$C^4$| regularity of |$\varphi $|⁠) \begin{align} \phi_{\Lambda}^{(\textsf{3})}(t) &= \left( - \frac{3t}{(\lambda^2-t^2)^{3/2}} - \frac{3t^3}{(\lambda^2 - t^2)^{5/2}} \right) \varphi^{(\textsf{1})}(t) \nonumber\\\nonumber &\quad+ \left( \frac{-3}{\sqrt{\lambda^2 -t^2}} - \frac{3t^2}{(\lambda^2-t^2)^{3/2}} \right) \varphi^{(\textsf{2})}(t) \\ &\quad+ \left( \frac{-3t}{\sqrt{\lambda^2-t^2}} \right) \varphi^{(\textsf{3})}(t) + \sqrt{\lambda^2-t^2} \varphi^{(\textsf{4})}(t). \end{align} (7.10) Let us observe that, for |$\textsf{k} \geq 1$|⁠, if |$g$| is a test function of class |$C^{\textsf{k} +1}$|⁠, we have \begin{equation} \left( \textbf{PV} \int \frac{g(t)}{t - \cdot} \ \textrm{d}t \right)^{(\textsf{k})}(x) = \int_0^{+\infty} \frac{g^{(\textsf{k})}(x+u) - g^{(\textsf{k})}(x-u)}{u} \ \textrm{d}u = \textbf{PV} \int \frac{g^{(\textsf{k})}(t)}{t-x} \ \textrm{d}t. \end{equation} (7.11) In view of (7.6), (7.7), and (7.11), we get that, for |$|x| \leq 2\ell $|⁠, \begin{equation*} {\mathfrak{H}}_{\lambda, \varphi}(x) \preceq \ell^{1/2} \| \phi_{\Lambda}^{(1)} \|_{L^2}, \quad \big({\mathfrak{H}}_{\lambda, \varphi}\big)^{(1)}(x) \preceq \ell^{1/2} \| \phi_{\Lambda}^{(2)} \|_{L^2}, \quad \big({\mathfrak{H}}_{\lambda, \varphi}\big)^{(2)}(x) \preceq \ell^{1/2} \| \phi_{\Lambda}^{(3)} \|_{L^2}. \end{equation*} Since |$\varphi $| is supported in |$(-\ell , \ell )$| and |$\ell $| satisfies |$0 < \ell < \frac{1}{10} \lambda $|⁠, it is easy to check, from (7.8), (7.9), that \begin{align*} & \big|\phi_{\Lambda}^{(\textsf{1})}(t)\big| \leq \frac{\ell}{\lambda} \big|\varphi^{(\textsf{1})}(t)\big| + \lambda \big|\varphi^{(\textsf{2})}(t)\big|, \\ & \big|\phi_{\Lambda}^{(\textsf{2})}(t)\big| \leq \frac{1}{\lambda} \big|\varphi^{(\textsf{1})}(t)\big| + \frac{\ell}{\lambda} \big|\varphi^{(\textsf{2})}(t)\big| + \lambda \big|\varphi^{(\textsf{3})}(t)\big| \\ & \big|\phi_{\Lambda}^{(\textsf{3})}(t)\big| \leq \frac{\ell}{\lambda^3} \big|\varphi^{(\textsf{1})}(t)\big| + \frac{1}{\lambda} \big|\varphi^{(\textsf{2})}(t)\big| + \frac{\ell}{\lambda} \big|\varphi^{(\textsf{3})}(t)\big| + \lambda \big|\varphi^{(\textsf{4})}(t)\big|. \end{align*} Using finally the homogeneity bounds (1.7), we see that the dominant term is the last one in each line, and we obtain the controls for |$|x| \leq 2\ell $| as in (7.3), (7.4), and (7.5). We now turn to the case |$|x| \geq 2\ell $|⁠. Bound on |$\big ({\mathfrak{H}}_{\lambda , \varphi }\big )^{(\textsf{1})}$|⁠. Since |$\varphi $| is supported on |$(-\ell , \ell )$|⁠, so is |$\phi _{\Lambda }$|⁠, and for |$|x| \geq 2\ell $| the integral defining (2.2) (or its derivatives) can be understood in the standard sense as a Riemann integral. In particular, we have \begin{align*} {\mathfrak{H}}_{\lambda, \varphi}(x) = \frac{1}{\pi} \int \frac{\sqrt{\lambda^2- t^2} \varphi^{\prime}(t)}{t-x} \ \textrm{d}t = \frac{1}{\pi} \int \frac{\lambda \left(1 + O_{\bullet}\left(\frac{\ell^2}{\lambda^2}\right) \right) \varphi^{\prime}(t)}{x\left(1+ O_{\bullet}\left(\frac{\ell}{x}\right) \right)} \ \textrm{d}t \end{align*} The 1st-order term vanishes because |$\int \varphi ^{\prime}(t) = 0$|⁠. We are left with \begin{equation*} {\mathfrak{H}}_{\lambda, \varphi}(x) \preceq \frac{\lambda}{x} \left(\frac{\ell^2}{\lambda^2} + \frac{\ell}{x} \right) \|\varphi^{\prime}\|_{L^1} \preceq \frac{\lambda \ell}{x^2} \|\varphi^{\prime}\|_{L^1}, \end{equation*} which yields the control for |$|x| \geq 2\ell $| as in (7.3). Bound on the 1st derivative. To treat |$\big ({\mathfrak{H}}_{\lambda , \varphi }\big )^{(\textsf{1})}(x)$|⁠, we write it as \begin{equation*} \left({\mathfrak{H}}_{\lambda, \varphi}\right)^{(\textsf{1})}(x) = \frac{1}{\pi} \int \frac{\phi_{\Lambda}^{\prime}(t)}{t-x} \ \textrm{d}t = \frac{1}{\pi} \frac{1}{x} \int \phi_{\Lambda}^{(1)}(t) \left( 1 - \frac{t}{x} + O_{\bullet}\left( \frac{\ell^2}{x^2} \right) \right) \ \textrm{d}t. \end{equation*} The 1st-order term vanishes because |$\int \phi _{\Lambda }^{(1)} (t) = 0$|⁠. Using (7.8), we may thus write \begin{equation*} \big({\mathfrak{H}}_{\lambda, \varphi}\big)^{(\textsf{1})}(x) = \frac{1}{\pi} \frac{1}{x} \int \left( \frac{-t}{\sqrt{\lambda^2-t^2}} \varphi^{(1)}(t) + \sqrt{\lambda^2 - t^2} \varphi^{(\textsf{2})}(t) \right) \left( - \frac{t}{x} + O_{\bullet}\left( \frac{\ell^2}{x^2} \right) \right) \ \textrm{d}t. \end{equation*} First, we compute \begin{equation} \frac{1}{x} \int \frac{-t}{\sqrt{\lambda^2-t^2}} \varphi^{(1)}(t)\left( \frac{t}{x} +\! O_{\bullet}\left( \frac{\ell^2}{x^2} \right) \right) \, \textrm{d}t \!=\! \frac{1}{x} \int \frac{-t}{\sqrt{\lambda^2-t^2}} \varphi^{(1)}(t) O_{\bullet}\!\left( \frac{\ell}{x} \right) \, \textrm{d}t \\ \preceq \frac{\ell^2}{\lambda x^2} \| \varphi \|_{L^1} \!\preceq \frac{\ell^2}{\lambda x^2}. \end{equation} (7.12) Next, we write \begin{equation} \frac{1}{x} \int\! \sqrt{\lambda^2 - t^2} \varphi^{(\textsf{2})}(t)\! \left( - \frac{t}{x} + O_{\bullet}\!\left( \frac{\ell^2}{x^2} \right) \right) \, \textrm{d}t \\ \!=\! \frac{1}{x} \int \lambda \left(1 \!+\! O_{\bullet} \left(\frac{\ell^2}{\lambda^2}\right) \right) \varphi^{(\textsf{2})}(t) \left( - \frac{t}{x} + O_{\bullet}\left( \frac{\ell^2}{x^2} \right) \right) \, \textrm{d}t. \end{equation} (7.13) The 1st-order term vanishes because |$\int t \varphi ^{(2)}(t) = 0$|⁠. We are left with \begin{equation*} \frac{\lambda}{x} \int \varphi^{(2)}(t) \left( \frac{\ell^3}{\lambda^2 x} + \frac{\ell^2}{x^2} \right) \ \textrm{d}t \preceq \frac{\lambda \ell^2}{x^3} \| \varphi^{(2)} \|_{L^1} \preceq \frac{\lambda \ell} {x^3}. \end{equation*} Combining (7.12) and (7.13), the dominant term is |$\frac{\lambda \ell } {x^3}$|⁠, and we obtain the control on |$\big ({\mathfrak{H}}_{\lambda , \varphi }\big )^{(\textsf{1})}(x)$| for |$|x| \geq 2\ell $|⁠, as in (7.4). Bound on |$\big ({\mathfrak{H}}_{\lambda , \varphi }\big )^{(\textsf{2})}$|⁠. The proof is similar to the one for |$\big ({\mathfrak{H}}_{\lambda , \varphi }\big )^{(\textsf{1})}$|⁠, except that we push the expansions to the next order and use the fact that |$\int \phi _{\lambda }^{(2)}(t) \!=\! 0$| and |$\int t^2 \varphi ^{(3)}(t)\! =\! 0$|⁠. We may now give the proof of Lemma 2.7. Proof of Lemma 2.7 We compute \begin{align*} & {\mathfrak{m}}_{\lambda, \varphi}(x) = \frac{-1}{\pi} \left[ \frac{1}{\sqrt{\lambda^2 - x^2}} {\mathfrak{H}}_{\lambda, \varphi}(x) \right], \\ & {\mathfrak{m}}_{\lambda, \varphi}^{(\textsf{1})}(x) = \frac{-1}{\pi} \left[ \frac{-x}{(\lambda^2-x^2)^{3/2}} {\mathfrak{H}}_{\lambda, \varphi}(x) + \frac{1}{\sqrt{\lambda^2-x^2}} {\mathfrak{H}}_{\lambda, \varphi}^{(\textsf{1})}(x) \right], \end{align*} and the 2nd derivative is given by \begin{align*} {\mathfrak{m}}_{\lambda, \varphi}^{(\textsf{2})}(x) = \frac{-1}{\pi} \left[ \left(\frac{-1}{(\lambda^2-x^2)^{3/2}} - \frac{3x^2}{(\lambda^2-x^2)^{5/2}} \right) {\mathfrak{H}}_{\lambda, \varphi}(x) \right. \\ \left. - \frac{3x}{(\lambda^2-x^2)^{3/2}} {\mathfrak{H}}_{\lambda, \varphi}^{(\textsf{1})}(x) + \frac{1}{\sqrt{\lambda^2-x^2}} {\mathfrak{H}}_{\lambda, \varphi}^{(\textsf{2})}(x)\right], \end{align*} and we use (7.3), (7.4), and (7.5) together with the simple observation that \begin{equation*} \frac{1}{\sqrt{\lambda^2-x^2}} \preceq \frac{1}{\sqrt{\lambda}\sqrt{\lambda - |x|}}, \end{equation*} which allows for a slight simplification in the formulas. We obtain \begin{align*} &{\mathfrak{m}}_{\lambda, \varphi}(x) \preceq \frac{1}{\lambda} \frac{\lambda}{\ell}, \quad |x| \leq 2\ell, \\ &{\mathfrak{m}}_{\lambda, \varphi}(x) \preceq \frac{1}{\sqrt{\lambda} \sqrt{\lambda - |x|}} \frac{\lambda \ell}{x^2}, \quad |x| \geq 2\ell, \\ & {\mathfrak{m}}_{\lambda, \varphi}^{(\textsf{1})}(x) \preceq \frac{\ell}{\lambda^3} \frac{\lambda}{\ell} + \frac{1}{\lambda} \frac{\lambda}{\ell^2}, \quad |x| \leq 2\ell, \\ & {\mathfrak{m}}_{\lambda, \varphi}^{(\textsf{1})}(x) \preceq \frac{|x|}{\lambda^{3/2} (\lambda - |x|)^{3/2}} \frac{\lambda \ell}{x^2} + \frac{1}{\sqrt{\lambda} \sqrt{\lambda - |x|}} \frac{\lambda \ell}{x^3}, \quad |x| \geq 2\ell, \\ & {\mathfrak{m}}_{\lambda, \varphi}^{(\textsf{2})}(x) \preceq \left( \frac{1}{\lambda^3} + \frac{\ell^2}{\lambda^5} \right) \frac{\lambda}{\ell} + \frac{\ell}{\lambda^3} \frac{\lambda}{\ell^2} + \frac{1}{\lambda}\frac{\lambda}{\ell^3}, \quad |x| \leq 2\ell, \end{align*} and, for |$|x| \geq 2\ell $|⁠, \begin{equation*} {\mathfrak{m}}_{\lambda, \varphi}^{(\textsf{2})}(x) \preceq \left( \frac{1}{\lambda^{3/2} (\lambda - |x|)^{3/2}} + \frac{x^2}{\lambda^{5/2} (\lambda - |x|)^{5/2}}\right) \frac{\lambda \ell}{x^2} + \frac{|x|}{\lambda^{3/2} (\lambda - |x|)^{3/2}} \frac{\lambda \ell}{x^3} + \frac{1}{\sqrt{\lambda} \sqrt{\lambda - |x|}} \frac{\lambda \ell}{x^4}. \end{equation*} We obtain (2.7), (2.8), and (2.9). 7.4 Two intermediate results Lemma 7.3 (A decomposition of |$\phi _{\Lambda }$|⁠). Let |$\phi _{\Lambda }: t \mapsto \sqrt{\lambda ^2-t^2} \varphi ^{\prime}(t)$|⁠. We have \begin{equation} \phi_{\Lambda}(t) = \lambda \varphi^{\prime}(t) + \textrm{Er}(t), \end{equation} (7.14) where |$\textrm{Er}$| is a |$C^1$| function, supported in |$[-\ell , \ell ]$|⁠, satisfying \begin{equation} |\textrm{Er}|_{\textsf{0}} \preceq \frac{\ell}{\lambda}, \quad |\textrm{Er}|_{\textsf{1}} \preceq \frac{1}{\lambda}. \end{equation} (7.15) Proof of Lemma 7.3 To see that (7.14) holds with (7.15), we simply expand \begin{equation*} \sqrt{\lambda^2-t^2}\varphi^{\prime}(t) = \lambda \varphi^{\prime}(t) + O_{\bullet}\left( \frac{\ell^2}{\lambda} \right) \varphi^{\prime}(t), \end{equation*} and since |$|\varphi |_{\textsf{1}} \preceq \frac{1}{\ell }$|⁠, we obtain the 1st bound in (7.15). We may then compute \begin{equation*} \phi_{\Lambda}^{\prime}(t) = \frac{-t}{\sqrt{\lambda^2-t^2}} \varphi^{\prime}(t) + \sqrt{\lambda^2-t^2} \varphi^{\prime\prime}(t) = O_{\bullet}\left( \frac{\ell}{\lambda} \right) \varphi^{\prime}(t) + \lambda \varphi^{\prime\prime}(t) + O_{\bullet}\left( \frac{\ell^2}{\lambda} \right) \varphi^{\prime\prime}(t), \end{equation*} which yields the 2nd bound in (7.15). Lemma 7.4 (The integral of |${\mathfrak{H}}_{\lambda , \varphi }$| on large intervals.). Let |$a$| be in |$[10 \ell , \lambda /2]$|⁠. We have \begin{align} & \int_{-a}^{a} {\mathfrak{H}}_{\lambda, \varphi}(y) \ \textrm{d}y \preceq \frac{\ell \lambda}{a}, \end{align} (7.16) \begin{align} & \int_{-a}^a y^2 |{\mathfrak{H}}_{\lambda, \varphi}(y)| \ \textrm{d}y \preceq \ell \lambda a. \end{align} (7.17) Proof of Lemma 7.4 Preliminary. We use the definition (2.1) of the Cauchy principal value and write \begin{equation*} {\mathfrak{H}}_{\lambda, \varphi}(y) = \frac{1}{\pi} \textbf{PV} \int \frac{\varphi^{\prime}(t) \sqrt{\lambda^2-t^2}}{y-t} \ \textrm{d}t = \int_{0}^{+\infty} \frac{ \phi_{\Lambda}(y+u) - \phi_{\Lambda}(y-u)}{u} \ \textrm{d}u, \end{equation*} where |$\phi _{\Lambda }: t \mapsto \sqrt{\lambda ^2-t^2} \varphi ^{\prime}(t)$|⁠. We use Lemma 7.3 and decompose |$\phi _{\Lambda }$| as |$\phi _{\Lambda } = \lambda \varphi ^{\prime}(t) + \textrm{Er}(t)$|⁠. We may thus write \begin{align} {\mathfrak{H}}_{\lambda, \varphi}(y) = \frac{\lambda}{\pi} \int_{u=0}^{+ \infty} \frac{\varphi^{\prime}(y+u) - \varphi^{\prime}(y-u)}{u} \ \textrm{d}u + \int_{u=0}^{+ \infty} \frac{\textrm{Er}(y+u) - \textrm{Er}(y-u)}{u} \ \textrm{d}u. \end{align} (7.18) The 2nd term in the right-hand side of (7.18) can be bounded using (7.6). We obtain \begin{equation*} \int_{u=0}^{+ \infty} \frac{\textrm{Er}(y+u) - \textrm{Er}(y-u)}{u} \ \textrm{d}u = \textbf{PV} \int \frac{\textrm{Er}(t)}{y-t} \ \textrm{d}t \preceq \ell^{1/2} \|\textrm{Er}^{\prime}\|_{L^2}, \end{equation*} and in view of the 2nd inequality in (7.15), we get \begin{equation} \textbf{PV} \int \frac{\textrm{Er}(t)}{y-t} \ \textrm{d}t \preceq \frac{\ell}{\lambda}. \end{equation} (7.19) Proof of (7.16). We use (7.18) and write \begin{equation} \int_{-a}^a {\mathfrak{H}}_{\lambda, \varphi}(y) \ \textrm{d}y = \frac{\lambda}{\pi} \int_{-a}^a \int_{u=0}^{+ \infty} \frac{\varphi^{\prime}(y+u) - \varphi^{\prime}(y-u)}{u} \ \textrm{d}u + \frac{1}{\pi} \int_{-a}^a \left(\textbf{PV} \int \frac{\textrm{Er}(t)}{y-t} \ \textrm{d}t\right) \ \textrm{d}{y}. \end{equation} (7.20) We may bound the 2nd term in the right-hand side of (7.20), using (7.19), as \begin{equation} \int_{-a}^a \left(\textbf{PV} \int \frac{\textrm{Er}(t)}{y-t} \ \textrm{d}t\right) \ \textrm{d}y \preceq \frac{a \ell}{\lambda}. \end{equation} (7.21) We now turn to the 1st term in the right-hand side of (7.20). It can be expressed, using Fubini’s theorem, as \begin{align*} \lambda \int_{-a}^{a} \int_{u=0}^{+ \infty}& \frac{\varphi^{\prime}(y+u) - \varphi^{\prime}(y-u)}{u} \ \textrm{d}u \\ &= \lambda \int_{0}^{+\infty} \frac{\varphi(u + a ) - \varphi(u-a) - \varphi(a-u) + \varphi(-a -u)}{u} \ \textrm{d}u \\ &= \lambda \left( \textbf{PV} \int \frac{\varphi(t)}{a-t} \ \textrm{d}t - \textbf{PV} \int \frac{\varphi(t)}{-a-t} \ \textrm{d}t\right). \end{align*} Since |$\varphi $| is supported on |$(-\ell , \ell )$|⁠, and since |$a \geq 10 \ell $|⁠, these are standard Riemann integrals, and we have \begin{equation*} \textbf{PV} \int \frac{\varphi(t)}{a-t} \ \textrm{d}t \preceq \frac{1}{a} \|\varphi\|_{L^1} \preceq \frac{\ell}{a} \end{equation*} and similarly for the other term. We get \begin{equation} \lambda \int_{-a}^{a} \int_{u=0}^{+ \infty} \frac{\varphi^{\prime}(y+u) - \varphi^{\prime}(y-u)}{u} \ \textrm{d}u \preceq \frac{\ell \lambda}{a}. \end{equation} (7.22) Combining (7.22) with (7.21) (which is at most of the same order) yields (7.16). Proof of (7.17). We simply use the bounds of (7.3). 7.5 Proof of Lemma 2.8 Proof of Lemma 2.8 First, we define |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$| as |${\mathfrak{m}}_{\lambda , \varphi }$| on |$[- \lambda + \ell , \lambda - \ell ]$|⁠. On |$[-\lambda + \ell /2, - \lambda + \ell ]$|⁠. Let |$S_a, S_b, S_c$| be three smooth, non-negative functions defined on |$[-1,0]$| such that \begin{align*} & \forall \textsf{k} \geq 0, S_a^{(\textsf{k})}(-1) = 0, \quad S_a(0) = 1, \quad \forall \textsf{k} \geq 1, S_a^{(\textsf{k})}(0) = 0 \\ & \forall \textsf{k} \geq 0, S_b^{(\textsf{k})}(-1) = 0, \quad S_b(0) = 1, \quad S_b^{(\textsf{1})}(0) = 1, \quad \forall \textsf{k} \geq 2, S_b^{(\textsf{k})}(0) = 0 \\ & \forall \textsf{k} \geq 0, S_c^{(\textsf{k})}(-1) = 0, \quad S_c(0) = 1, \quad S_c^{(\textsf{1})}(0) = 0, \quad S_c^{(\textsf{2})}(0) = 1, \quad \forall \textsf{k} \geq 2, S_b^{(\textsf{k})}(0) = 0. \end{align*} We let \begin{equation*} \textsf{P} = - \lambda + \ell, \quad \textsf{size} = \frac{\ell}{2}, \end{equation*} and we define \begin{equation*} \textsf{D}_0 = {\mathfrak{m}}_{\lambda, \varphi}\left(\textsf{P} \right), \quad \textsf{D}_1 = {\mathfrak{m}}_{\lambda, \varphi}^{(\textsf{1})}(\textsf{P}), \quad \textsf{D}_2 = {\mathfrak{m}}_{\lambda, \varphi}^{(\textsf{2})}(\textsf{P}). \end{equation*} We have, in view of (2.7), (2.8), and (2.9), \begin{equation*} \textsf{D}_0 \preceq \frac{\ell}{\lambda^{3/2} \ell^{1/2}}, \quad \textsf{D}_1 \preceq \frac{\ell}{\lambda^{3/2} \ell^{3/2}}, \quad \textsf{D}_2 \preceq \frac{\ell}{\lambda^{3/2} \ell^{5/2}}. \end{equation*} Finally, wet let |$\textsf{R}$| be the function \begin{equation*} \textsf{R}(x):= \textsf{D}_0 S_a\left( (x- \textsf{P}) \frac{1}{\textsf{size}} \right) S_b\left( (x- \textsf{P}) \frac{\textsf{D}_1}{\textsf{D}_0} \right) S_c\left( (x- \textsf{P}) \sqrt{\frac{\textsf{D}_2}{\textsf{D}_0}} \right). \end{equation*} The function |$\textsf{R}$| is defined on |$[\textsf{P} - \textsf{size}, \textsf{P}] = [-\lambda + \ell /2, - \lambda + \ell ]$|⁠, and on this interval, we let |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }(x) = \textsf{R}(x)$|⁠. By construction, the derivatives of order |$\textsf{0}, \textsf{1}, \textsf{2}$| of |$\textsf{R}$| and |${\mathfrak{m}}_{\lambda , \varphi }$| coincide at |$\textsf{P}$|⁠, so the piece-wise definition is |$C^2$| at |$\textsf{P}$|⁠. Moreover, it can be checked that for |$\textsf{k} = \textsf{0}, \textsf{1}, \textsf{2}$|⁠, we have \begin{equation} \textsf{R}^{(\textsf{k})}(x) \preceq \frac{\ell}{\ell^{\textsf{k}} \lambda^{3/2} \ell^{1/2}}. \end{equation} (7.23) This is easy to see for |$\textsf{R}^{(\textsf{0})}$|⁠, since |$\textsf{D}_0 \preceq \frac{\ell }{\lambda ^{3/2} \ell ^{1/2}}$|⁠. For the 1st and 2nd derivatives, we use the fact that |$\frac{1}{\textsf{size}}, \frac{\textsf{D}_1}{\textsf{D}_0}$| and |$\sqrt{\frac{\textsf{D}_2}{\textsf{D}_0}}$| have the same order |$\frac{1}{\ell }$|⁠. On |$[-\lambda + \ell /4, - \lambda + \ell /2]$|⁠. Let |$S_d$| be a smooth, non-negative function defined on |$[-1, 0]$| such that \begin{equation*} \forall \textsf{k} \geq 0, S_d^{(\textsf{k})}(-1) = S_d^{(\textsf{k})}(0) = 0, \quad \int S_d(x) \ \textrm{d}x = 1. \end{equation*} We overwrite the definition above and let now \begin{equation*} \textsf{P}:= - \lambda + \ell/2, \quad \textsf{size}:= \frac{\ell}{4}, \end{equation*} and we introduce \begin{equation*} \textsf{D}_{-1}:= \int_{-\lambda}^{-\lambda + \ell} {\mathfrak{m}}_{\lambda, \varphi}(x) \ \textrm{d}x - \int_{-\lambda + \ell/2}^{-\lambda + \ell} \textsf{R}(x) \ \textrm{d}x. \end{equation*} Finally, we let |$\textsf{T}$| be the function \begin{equation*} \textsf{T}(x) = \frac{\textsf{D}_{-1}}{\textsf{size}} S_d\left( (x - \textsf{P}) \frac{1}{\textsf{size}} \right). \end{equation*} The function |$\textsf{T}$| is defined on |$[-\lambda + \ell /4, - \lambda + \ell /2]$|⁠, and on this interval, we let |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }(x) = \textsf{T}(x)$|⁠. By construction, all the derivatives of |$\textsf{T}$| and |$\textsf{R}$| are equal (to |$0$|⁠) at the point |$\textsf{P}$|⁠, so we still get a |$C^2$| function. Finally, we let |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }(x) = 0$| on |$[-\lambda , - \lambda + \ell /4]$|⁠, and this connects with the previous definition in a |$C^2$| way because all the derivatives of |$\textsf{T}$| vanish at |$-\lambda + \ell /4$|⁠. We define |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$| similarly near the other endpoint. Checking the statements. By construction, |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$| and |${\mathfrak{m}}_{\lambda , \varphi }$| coincide on a large interior part, and |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$| vanishes near the endpoints, so the 1st and 4th statements of the lemma are satisfied. Also by construction, we have \begin{equation*} \int \textsf{T}(x) \ \textrm{d}x + \int \textsf{R}(x) = \int_{-\lambda}^{-\lambda + \ell} {\mathfrak{m}}_{\lambda, \varphi}(x) \ \textrm{d}x, \end{equation*} so the total masses of |${\mathfrak{m}}_{\lambda , \varphi }$| and |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$| are equal near the endpoints, and (2.10) holds. We have already checked (2.11) for the 1st part of the construction, see (7.23). On |$[-\lambda + \ell /4, - \lambda + \lambda /2]$| we have \begin{equation} |\textsf{T}^{(\textsf{k})}(x)| \preceq \frac{1}{\textsf{size}^{\textsf{k}+1}} \textsf{D}_{-1}, \end{equation} (7.24) and we observe that |$\textsf{D}_{-1}$| is of order |$\textsf{size} \times \textsf{D}_0$|⁠, with |$\textsf{D}_0 \preceq \frac{\ell }{\lambda ^{3/2}\ell ^{1/2}}$|⁠, which yields the result. 7.6 Proof of Proposition 2.11 Proof of Proposition 2.11 We recall that, by definition, we have \begin{equation*} \textrm{Error}\textsf{LP}_{\lambda, \varphi}(x): = \int - \log |x-y| (\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y)- {\mathfrak{m}}_{\lambda, \varphi}(y))\ \textrm{d}y. \end{equation*} We may split |$\textrm{Error}\textsf{LP}_{\lambda , \varphi }$| as the sum |$\textrm{Error}\textsf{LP}_{\lambda , \varphi }(x) = \textrm{Error}\textsf{LP}_{\lambda , \varphi }^{\textrm{Left}}(x) + \textrm{Error}\textsf{LP}_{\lambda , \varphi }^{\textrm{Right}}(x)$|⁠, where |$\textrm{Error}\textsf{LP}_{\lambda , \varphi }^{\textrm{Left}}$| (resp. |$\textrm{Error}\textsf{LP}_{\lambda , \varphi }^{\textrm{Right}}$|⁠) is the contribution coming from the left (resp. right) endpoint, that is, \begin{align} & \textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Left}}(x): = \int_{-\lambda}^{-\lambda + \ell} \log |x-y| ({\mathfrak{m}}_{\lambda, \varphi}(y) - \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y))\ \textrm{d}y, \end{align} (7.25) \begin{align} & \textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Right}}(x): = \int_{\lambda - \ell}^{\lambda} \log |x-y| ({\mathfrak{m}}_{\lambda, \varphi}(y) - \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y))\ \textrm{d}y.\quad\ \ \end{align} (7.26) The sup norm, for |$x$| close to the endpoints. Let us start with a rough bound: \begin{equation} \textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Left}}(x) \preceq \int_{-\lambda}^{-\lambda + \ell} \left|\log |x-y| \right| |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y)| \ \textrm{d}y + \int_{-\lambda}^{-\lambda + \ell} |\log|x-y|| |{\mathfrak{m}}_{\lambda, \varphi}(y)| \ \textrm{d}y. \end{equation} (7.27) Of course, if |$x$| is far from the endpoints, this is sub-optimal because we do not use the fact that |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi } -{\mathfrak{m}}_{\lambda , \varphi }$| has mass zero; in fact we will use this inequality only for |$x$| at distance |$O_{\bullet }(\ell )$| of an endpoint. Using (2.12), we see that \begin{equation} \int_{-\lambda}^{-\lambda + \ell} \left|\log |x-y| \right| |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y)| \ \textrm{d}y \preceq \frac{\ell \sqrt{\ell} \log \lambda}{\lambda^{3/2}}. \end{equation} (7.28) It remains to bound the 2nd integral in (7.27). We use (2.7) to write \begin{equation*} \int_{-\lambda}^{-\lambda + \ell} |\log|x-y|| {\mathfrak{m}}_{\lambda, \varphi}(y) \ \textrm{d}y \preceq \int_{-\lambda}^{-\lambda + \ell} \frac{\ell \sqrt{\lambda} |\log|x-y||}{\lambda^2 \sqrt{\lambda - |y|}} \ \textrm{d}y \preceq \frac{\ell}{\lambda^{3/2}} \int_{-\lambda}^{-\lambda + \ell} \frac{|\log|x-y||}{\sqrt{\lambda - |y|}} \ \textrm{d}y. \end{equation*} We use Hölder’s inequality and write \begin{equation*} \int_{-\lambda}^{-\lambda + \ell} \frac{|\log|x-y||}{\sqrt{\lambda - |y|}} \ \textrm{d}y \leq \left(\int_{-\lambda}^{-\lambda + \ell} |\log|x-y||^{3} \right)^{1/3} \left(\int_{-\lambda}^{-\lambda + \ell} \frac{1}{(\lambda - |y|)^{3/4}}\right)^{2/3} \ \textrm{d}y. \end{equation*} An elementary computation shows that for |$x$| in |$(-2\lambda , 2\lambda )$|⁠, |$\left (\int _{-\lambda }^{-\lambda + \ell } |\log |x-y||^{3} \right )^{1/3} \preceq \ell ^{1/3} \log \lambda $| and that, on the other hand, |$\left (\int _{-\lambda }^{-\lambda + \ell } \frac{1}{(\lambda - |y|)^{3/4}}\right )^{2/3} \preceq \ell ^{1/6}$|⁠, hence we obtain \begin{equation} \int_{-\lambda}^{-\lambda + \ell} |\log|x-y|| {\mathfrak{m}}_{\lambda, \varphi}(y) \ \textrm{d}y \preceq \frac{\ell \sqrt{\ell} \log \lambda}{\lambda^{3/2}}. \end{equation} (7.29) Combining (7.27), (7.28), and (7.29), we obtain (2.17). The derivative, for |$x$| far from the endpoints. We now turn to proving (2.18). Let |$x$| such that |$|x-\lambda | \geq 2\ell $|⁠. We have, by definition, \begin{equation*} \textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Right}}(x) = \int_{\lambda-\ell}^{\lambda} - \log |x-y| (\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y) - {\mathfrak{m}}_{\lambda, \varphi}(y)) \ \textrm{d}y. \end{equation*} We may differentiate under the integral sign and write \begin{equation*} \left(\textrm{Error}\textsf{LP}_{\lambda, \varphi}^{\textrm{Right}} \right)^{\prime}(x) = \int_{\lambda - \ell}^{\lambda} \frac{-1}{x-y} (\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y) - {\mathfrak{m}}_{\lambda, \varphi}(y)) \ \textrm{d}y. \end{equation*} A Taylor’s expansion yields \begin{multline} \int_{\lambda - \ell}^{\lambda} \frac{-1}{x-y} (\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y) - {\mathfrak{m}}_{\lambda, \varphi}(y)) \ \textrm{d}y = \frac{1}{x-\lambda} \int_{\lambda-\ell}^{\lambda} (\widetilde{{\mathfrak{m}}}_{\lambda, \varphi} - {\mathfrak{m}}_{\lambda, \varphi})(y) \ \textrm{d}y \\ + O_{\bullet}\left( \frac{\ell}{(\lambda-x)^2} \right) \int_{\lambda-\ell}^{\lambda} (|\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y)| + |{\mathfrak{m}}_{\lambda, \varphi}(y)|)\ \textrm{d}y. \end{multline} (7.30) The 1st term in the right-hand side of (7.30) vanishes because, by construction as in (2.10), |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$| and |${\mathfrak{m}}_{\lambda , \varphi }$| have the same mass on |$[\lambda - \ell , \lambda ]$|⁠. We can estimate the integral in the 2nd term directly, and we obtain \begin{equation*} \int_{\lambda - \ell}^{\lambda} \frac{-1}{x-y} (\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y) - {\mathfrak{m}}_{\lambda, \varphi}(y)) \ \textrm{d}y \preceq \frac{\ell^{3/2} \ell}{\lambda^{3/2} (\lambda - |x|)^2}. \end{equation*} The same argument holds near the other endpoint, which proves (2.18). 7.7 Proof of Lemma 2.11 Proof of Lemma 2.11 Let us introduce |$\textsf{V}_{\lambda , \varphi }$| as \begin{equation} \textsf{V}_{\lambda, \varphi}:= \frac{-1}{\pi^2} \int \frac{\varphi(x)}{\sqrt{\lambda^2-x^2}} \int\frac{\sqrt{\lambda^2-t^2} \varphi^{\prime}(t)}{t-x} \ \textrm{d}t \ \textrm{d}x, \end{equation} (7.31) which is equal to \begin{equation*} \iint - \log |x-y| {\mathfrak{m}}_{\lambda, \varphi}(x) {\mathfrak{m}}_{\lambda, \varphi}(y) \ \textrm{d}x \ \textrm{d}y. \end{equation*} The error due to |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$|⁠. The 1st step in the proof is to show that \begin{align} \iint - \log |x-y| \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(x) \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y) \ \textrm{d}x \ \textrm{d}y = \textsf{V}_{\lambda, \varphi} + O_{\bullet}\left( \frac{\ell^2 \ell \log(\lambda)}{\lambda^{3}} \right). \end{align} (7.32) We decompose the left-hand side of (2.19) as \begin{align} \!\!\!\!\!\!\iint - \log |x-y| \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(x) \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y) \ \textrm{d}x \ \textrm{d}y =& \iint - \log |x-y| {\mathfrak{m}}_{\lambda, \varphi}(x){\mathfrak{m}}_{\lambda, \varphi}(y) \ \textrm{d}x \ \textrm{d}y \nonumber\\ &+ \iint - \log |x-y| (\widetilde{{\mathfrak{m}}}_{\lambda, \varphi} - {\mathfrak{m}}_{\lambda, \varphi})(x) (\widetilde{{\mathfrak{m}}}_{\lambda, \varphi} - {\mathfrak{m}}_{\lambda, \varphi})(y)\nonumber \\ &+ 2\iint - \log |x-y| {\mathfrak{m}}_{\lambda, \varphi}(y) (\widetilde{{\mathfrak{m}}}_{\lambda, \varphi} - {\mathfrak{m}}_{\lambda, \varphi})(x). \end{align} (7.33) Using the fact that |${\mathfrak{m}}_{\lambda , \varphi }$| satisfies (2.6) and has total mass |$0$|⁠, we may write \begin{align} \iint - \log |x-y| {\mathfrak{m}}_{\lambda, \varphi}(x){\mathfrak{m}}_{\lambda, \varphi}(y) \ \textrm{d}x \ \textrm{d}y &= \int \varphi(x) {\mathfrak{m}}_{\lambda, \varphi}(x) \ \textrm{d}x \nonumber \\ &= \frac{-1}{\pi^2} \int \frac{\varphi(x)}{\sqrt{\lambda^2-x^2}} \int\frac{\sqrt{\lambda^2-t^2} \varphi^{\prime}(t)}{t-x} \ \textrm{d}t \ \textrm{d}x. \end{align} (7.34) which is equal to |$\textsf{V}_{\lambda , \varphi }$| as defined in (7.31). Using again (2.6), and the fact that, by construction, |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi } - {\mathfrak{m}}_{\lambda , \varphi }$| has total mass |$0$|⁠, we write that \begin{equation*} \iint - \log |x-y| {\mathfrak{m}}_{\lambda, \varphi}(y) (\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(x) - {\mathfrak{m}}_{\lambda, \varphi}(x)) \ \textrm{d}y \ \textrm{d}x = \int \varphi(x) (\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(x) - {\mathfrak{m}}_{\lambda, \varphi}(x)) \ \textrm{d}x, \end{equation*} but by construction, |$\varphi $| vanishes on the support of |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }- {\mathfrak{m}}_{\lambda , \varphi }$|⁠, hence this is equal to |$0$|⁠. Finally, we write \begin{multline*} \iint - \log |x-y| (\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(x) - {\mathfrak{m}}_{\lambda, \varphi}(x)) (\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y) - {\mathfrak{m}}_{\lambda, \varphi}(y)) \ \textrm{d}x \ \textrm{d}y \\ = \int \textrm{Error}\textsf{LP}_{\lambda, \varphi}(x) (\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(x) - {\mathfrak{m}}_{\lambda, \varphi}(x)) \ \textrm{d}x, \end{multline*} where |$\textrm{Error}\textsf{LP}_{\lambda , \varphi } = \textrm{Error}\textsf{LP}_{\lambda , \varphi }^{\textrm{Left}} + \textrm{Error}\textsf{LP}_{\lambda , \varphi }^{\textrm{Right}}$| as in Proposition 2.11. We can use (2.17) and the fact that |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }(x) - {\mathfrak{m}}_{\lambda , \varphi }(x)$| is supported near the endpoints of |$(-\lambda , \lambda )$| to write \begin{equation*} \textrm{ErrorVar} \preceq \frac{\ell \sqrt{\ell} \log(\lambda)}{\lambda^{3/2}} \int_{-\lambda}^{-\lambda + \ell} (|\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(x)| + |{\mathfrak{m}}_{\lambda, \varphi}(x)|) \ \textrm{d}x \preceq \frac{\ell \sqrt{\ell} \log(\lambda)}{\lambda^{3/2}} \frac{\ell \sqrt{\ell}}{\lambda^{3/2}}, \end{equation*} which yields (7.32). The error due to |$\lambda $| finite. Now, we compare |$\textsf{V}_{\lambda , \varphi }$| with the norm |$\|\overline{\varphi }\|_{H^{\frac{1}{2}}}$|⁠; we claim that \begin{equation} \textsf{V}_{\lambda, \varphi} = 2 \|\overline{\varphi}\|_{H^{\frac{1}{2}}}^2 + O_{\bullet}\left(\frac{\ell^2}{\lambda^2}\right). \end{equation} (7.35) Indeed, we may write, since |$\varphi $| is supported in |$(-\ell , \ell )$|⁠, \begin{equation*} \int \frac{\varphi(x)}{\sqrt{\lambda^2-x^2}} \textbf{PV} \int \frac{\sqrt{\lambda^2-t^2} \varphi^{\prime}(t)}{t-x} \ \textrm{d}t \ \textrm{d}x = \int \varphi(x) \frac{1}{\lambda} \left( 1 + O_{\bullet}\left( \frac{\ell^2}{\lambda^2} \right) \right) {\mathfrak{H}}_{\lambda, \varphi}(x) \ \textrm{d}x, \end{equation*} and we can use (7.3) to write this as \begin{equation*} \frac{1}{\lambda} \int \varphi(x) {\mathfrak{H}}_{\lambda, \varphi}(x) \ \textrm{d}x + O_{\bullet}\left( \frac{\ell^2}{\lambda^2} \right). \end{equation*} Now, we have, by definition, \begin{equation*} {\mathfrak{H}}_{\lambda, \varphi}(x) = \frac{1}{\pi} \textbf{PV} \int \frac{\phi_{\Lambda}(t)}{x-t} \ \textrm{d}t, \end{equation*} and |$\phi _{\Lambda }$| admits the decomposition as in (7.14) and (7.15). It implies that \begin{equation*} \frac{1}{\lambda} \int \varphi(x) {\mathfrak{H}}_{\lambda, \varphi}(x) \ \textrm{d}x = \frac{1}{\pi \lambda} \int \varphi(x) \textbf{PV} \frac{\lambda \varphi^{\prime}(t)}{x-t} \ \textrm{d}t \ \textrm{d}x + O_{\bullet}\left( \frac{\ell^2}{\lambda^2} \right). \end{equation*} We may thus write |$\textsf{V}_{\lambda , \varphi }$| as \begin{equation*} \textsf{V}_{\lambda, \varphi} = \frac{-1}{\pi^2} \int \varphi(x) \textbf{PV} \int \frac{\varphi^{\prime}(t)}{x-t} \ \textrm{d}t \ \textrm{d}x + O_{\bullet}\left( \frac{\ell^2}{\lambda^2} \right), \end{equation*} and the result follows from the identity \begin{equation*} \frac{-1}{\pi^2} \int \varphi(x) \textbf{PV} \int \frac{\varphi^{\prime}(t)}{x-t} \ \textrm{d}t \ \textrm{d}x = 2 \|\overline{\varphi}\|_{H^{\frac{1}{2}}}^2, \end{equation*} with |$\|\overline{\varphi }\|_{H^{\frac{1}{2}}}$| as in (1.4), which can be checked by elementary means. 7.8 Proof of Lemma 3.3 Proof of Lemma 3.3 For simplicity, we will use the notation |$\times $| as follows: \begin{align*} A \times B = \iint_{(\Lambda \times \Lambda) \setminus \diamond} - \log |x-y| A(x) B(y). \end{align*} We have \begin{align*} \left(d{\mathcal{C}} - \mu_{s}\right) \times \left(d{\mathcal{C}} - \mu_{s} \right) &= \left(d{\mathcal{C}} - 1 - s \widetilde{{\mathfrak{m}}}_{\lambda, \varphi} \right) \times \left(d{\mathcal{C}} - 1 - s \widetilde{{\mathfrak{m}}}_{\lambda, \varphi} \right) \\ &= \left(d{\mathcal{C}} - 1\right) \times (d{\mathcal{C}} - 1) + s^2 \cdot \widetilde{{\mathfrak{m}}}_{\lambda, \varphi} \times \widetilde{{\mathfrak{m}}}_{\lambda, \varphi} - 2 s \cdot \widetilde{{\mathfrak{m}}}_{\lambda, \varphi} \times (d{\mathcal{C}} - 1). \end{align*} By Lemma 2.12, we have \begin{equation*} \widetilde{{\mathfrak{m}}}_{\lambda, \varphi} \times \widetilde{{\mathfrak{m}}}_{\lambda, \varphi} = 2 \|\overline{\varphi}\|_{H^{\frac{1}{2}}}^2 + \textrm{ErrorVar}. \end{equation*} Next, we write \begin{equation*} \widetilde{{\mathfrak{m}}}_{\lambda, \varphi} \times (d{\mathcal{C}} - 1) = {\mathfrak{m}}_{\lambda, \varphi} \times (d{\mathcal{C}} - 1) + (\widetilde{{\mathfrak{m}}}_{\lambda, \varphi} - {\mathfrak{m}}_{\lambda, \varphi}) \times (d{\mathcal{C}} -1). \end{equation*} We recall that |$\textsf{LP}_{\lambda , \varphi }$| is the logarithmic potential generated by |${\mathfrak{m}}_{\lambda , \varphi }$| and that |$\textrm{Error}\textsf{LP}_{\lambda , \varphi }$| is the logarithmic potential generated by the difference |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }- {\mathfrak{m}}_{\lambda , \varphi }$|⁠. So \begin{equation*} {\mathfrak{m}}_{\lambda, \varphi} \times (d{\mathcal{C}} - 1) = \iint_{(\Lambda \times \Lambda) \setminus \diamond} -\log |x-y| {\mathfrak{m}}_{\lambda, \varphi}(y) (\textrm{d}{\mathcal{C}}(x) - \textrm{d}x) = \int_{\Lambda} \textsf{LP}_{\lambda, \varphi}(x) (\textrm{d}{\mathcal{C}}-\textrm{d}x), \end{equation*} and similarly, \begin{align*} (\widetilde{{\mathfrak{m}}}_{\lambda, \varphi} - {\mathfrak{m}}_{\lambda, \varphi}) \times (d{\mathcal{C}} -1) &= \iint_{(\Lambda \times \Lambda) \setminus \diamond} -\log |x-y| (\widetilde{{\mathfrak{m}}}_{\lambda, \varphi} - {\mathfrak{m}}_{\lambda, \varphi})(y) (\textrm{d}{\mathcal{C}}(x) - \textrm{d}x) \\ &= \int_{\Lambda} \textrm{Error}\textsf{LP}_{\lambda, \varphi}(x) (\textrm{d}{\mathcal{C}}-\textrm{d}x). \end{align*} 7.9 Proof of Lemma 3.5 Proof of Lemma 3.5 Since |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$| is continuous and bounded as in Lemma 2.8, and |$\textrm{s}_{\textrm{max}}$| is chosen as in (3.1), we see that |$1 + s \widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$| is a continuous, positive function on |$\Lambda $|⁠. Consequently, |$F_s$| is |$C^1$| and increasing, thus it is a |$C^1$| bijection, and so is |$\Phi _s$|⁠. The fact that |$\Phi _s$| transports the constant density onto |$\mu _{s}$| results from the definition; in fact, |$\Phi _s$| is the “monotone rearrangement” of the constant density onto |$\mu _{s}$|⁠. By construction, |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$| has total mass |$0$| and vanishes near the endpoints; therefore, |$F_s(x) = x+\lambda $| near the endpoints, which implies that |$\Phi _s$| coincides with the identity map near the endpoints. We now turn to proving estimates on |$\psi _s$|⁠. We may write, by definition, that for any |$x$| in |$[-\lambda , \lambda ]$| we have \begin{equation*} \int_{-\lambda}^{\Phi_s(x)} \left(1 + s\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y)\right) \ \textrm{d}y = x + \lambda, \end{equation*} and we thus obtain, as claimed in (3.6), \begin{equation} \psi_s(x) = \Phi_s(x) - x = - s \int_{-\lambda}^{\Phi_s(x)} \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y) \ \textrm{d}y. \end{equation} (7.36) Bound on |$\psi _s$|⁠. We easily deduce |$|\psi _s|_{\textsf{0}} \leq s \|\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }\|_{L^1}$|⁠, and since |$|s| \leq \textrm{s}_{\textrm{max}}$| as in (3.1), we have \begin{equation} |\psi_s|_{\textsf{0}} \leq 1. \end{equation} (7.37) Finer bounds on |$\psi _s$| are the goal of another lemma. Bound on |$\psi _s^{(\textsf{1})}$|⁠. Let us differentiate (7.36) with respect to |$x$|⁠: \begin{equation*} \Phi_s^{\prime}(x) - 1 = -s \widetilde{{\mathfrak{m}}}_{\lambda, \varphi} \circ \Phi_s(x) \cdot \Phi_s^{\prime}(x), \end{equation*} and we obtain \begin{equation} \Phi_s^{\prime}(x) = \frac{1}{1 + s \widetilde{{\mathfrak{m}}}_{\lambda, \varphi} \circ \Phi_s(x)}. \end{equation} (7.38) The denominator is bounded below by a positive constant, and a Taylor’s expansion yields \begin{equation*} \left|\Phi_s^{\prime}(x) - 1\right| \preceq s |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi} \circ \Phi_s(x)|, \end{equation*} hence, since by definition |$\psi _s^{\prime} = \Phi _s^{\prime} - 1$|⁠, we get \begin{equation*} \psi_s^{\prime}(x) \preceq s |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi} \circ \Phi_s(x)|. \end{equation*} By definition, |$|\Phi _s(x) -x| = |\psi _s(x)|$|⁠, and (7.37) holds; we may thus write \begin{equation*} |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi} \circ \Phi_s(x)| \leq \sup_{y \in [x-1, x+1]} |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(x)| \leq |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}|_{\textsf{0}, V_{x}}, \end{equation*} with the notation of (1.6). This yields (3.8). In particular, |$|\psi _s|_{\textsf{1}} \preceq s|\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }|_{\textsf{0}}$| and is thus bounded, and so is |$\Phi _s^{\prime}$|⁠. Bound on |$\psi _s^{(2)}$|⁠. We differentiate (7.38) again and write \begin{equation} \Phi_s^{(\textsf{2})}(x) = \frac{-s \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}^{\prime} \circ \Phi_s(x) \Phi_s^{\prime}(x)}{(1 + s\widetilde{{\mathfrak{m}}}_{\lambda, \varphi} \circ \Phi_s(x))^2}. \end{equation} (7.39) We have previously established that for |$|s| \leq \textrm{s}_{\textrm{max}}$|⁠, we have |$\Phi _s^{\prime} \preceq 1$|⁠, and the quantity |$(1 + s\widetilde{{\mathfrak{m}}}_{\lambda , \varphi } \circ \Phi _s(x))$| is bounded above and below by a positive constant. We obtain \begin{equation} \psi_s^{(\textsf{2})}(x) = \Phi_s^{(\textsf{2})}(x) \preceq s \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}^{\prime} \circ \Phi_s(x) \preceq s |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}|_{\textsf{1}, V_{x}}, \end{equation} (7.40) which yields (3.9). Bound on |$\psi _s^{(\textsf{3})}$|⁠. Finally, differentiating (7.39) again, we get \begin{equation} \Phi_s^{(\textsf{3})}(x) = \frac{-s \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}^{(\textsf{2})} \circ \Phi_s(x) \left(\Phi_s^{\prime}(x)\right)^2 - s \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}^{\prime} \circ \Phi_s(x) \Phi_s^{(\textsf{2})}(x)}{(1+ s\widetilde{{\mathfrak{m}}}_{\lambda, \varphi} \circ \Phi_s(x))^2} + \frac{2s^2 \left(\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}^{\prime} \circ \Phi_s(x)\right)^2 \left(\Phi_s^{\prime}(x)\right)^2}{(1 + s\widetilde{{\mathfrak{m}}}_{\lambda, \varphi} \circ \Phi_s(x))^3}. \end{equation} (7.41) Using the fact that |$\Phi _s^{\prime}$| is bounded, that |$\Phi _s^{(\textsf{2})}(x)$| is of order |$s |\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }|_{\textsf{1}, V_{x}}$| (see (7.40)), and that the quantity |$1 + s\widetilde{{\mathfrak{m}}}_{\lambda , \varphi } \circ \Phi _s(x)$| is bounded below by a positive constant, we obtain \begin{equation*} \psi_s^{(\textsf{3})}(x) = \Phi_s^{(\textsf{3})}(x) \preceq s |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}|_{\textsf{2}, V_{x}} + s^2 |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}|^2_{\textsf{1}, V_{x}}, \end{equation*} and one can check from (2.13) and (2.14) that the dominant term in the right-hand side is the 1st one, which yields (3.10). 7.10 Proof of Lemma 3.6 Proof of Lemma 3.6 The 1st inequality in (3.11) follows from (3.6) combined with (3.7), and the 2nd one is obtained similarly, using the fact that |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$| has total mass |$0$|⁠. We now turn to proving the inequalities of (3.12). The case |$|x| \leq 10 \ell $|⁠. Since |$\|\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }\|_{L^1} \preceq 1$|⁠, as observed in (2.15), we have |$|\psi _s|_{\textsf{0}} \preceq s$|⁠, which in particular yields the bound for |$|x| \leq 10\ell $| as stated in (3.12). The case |$|x| \geq \lambda /2$|⁠. For |$|x| \geq \lambda /2$|⁠, we may combine (3.11) with the estimates on |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$| as in (2.12), and we obtain \begin{equation*} |\psi_s(x)| \preceq s \frac{\ell}{\lambda^{3/2}} \sqrt{\lambda + 1 - |x|}, \end{equation*} as stated in (3.12). The case |$10 \ell \leq |x| \leq \lambda /2$|⁠. Finally, let us assume that |$x$| is in |$[10 \ell , \lambda /2]$| (the case |$x \in [-\lambda /2, -10\ell ]$| being, of course, similar). We may write \begin{equation*} \int_{-\lambda}^{\Phi_s(x)} \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(t) \ \textrm{d}t = \int_{-\lambda}^{-\Phi_s(x)} \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(t) \ \textrm{d}t + \int_{-\Phi_s(x)}^{\Phi_s(x)} \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(t) \ \textrm{d}t. \end{equation*} Using (2.12), we can write \begin{equation} \int_{-\lambda}^{-\Phi_s(x)} \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(t) \ \textrm{d}t \preceq \frac{\ell^{3/2}}{\lambda^{3/2}} + \frac{\ell}{\lambda} + \frac{\ell}{\Phi_s(x)}, \end{equation} (7.42) and the dominant term is the last one. Next, we write, for |$|t|$| in |$[10\ell , \lambda /2]$|⁠, \begin{equation*} \widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(t) = \frac{1}{\lambda} {\mathfrak{H}}_{\lambda, \varphi}(t) + O_{\bullet}\left( \frac{t^2}{\lambda^3} \right) |{\mathfrak{H}}_{\lambda, \varphi}(t)|, \end{equation*} and we use Lemma 7.4. First, we apply (7.16) with |$a = -\Phi _s(x)$| and |$b = \Phi _s(x)$|⁠; we obtain \begin{equation} \frac{1}{\lambda} \int_{-\Phi_s(x)}^{\Phi_s(x)} {\mathfrak{H}}_{\lambda, \varphi}(t) \ \textrm{d}t \preceq \frac{\ell}{\Phi_s(x)}. \end{equation} (7.43) Secondly, we use (7.17) to get \begin{equation*} \frac{1}{\lambda^3} \int_{-\Phi_s(x)}^{\Phi_s(x)} t^2|{\mathfrak{H}}_{\lambda, \varphi}(t)| \ \textrm{d}st \preceq \frac{\ell \Phi_s(x)}{\lambda^2}. \end{equation*} The term in (7.43) is the dominant one. Combining it with a similar one in (7.42), and since we know that |$|\Phi _s(x) - x| \leq 1$|⁠, it yields, as desired \begin{equation*} |\psi(x)| \preceq s \frac{\ell}{|x|}. \end{equation*} 7.11 Proof of Proposition 4.1 We extend the notation of (1.6) as follows: if |$g$| is a function of two variables, we let \begin{equation*} |g|_{V(x,y)}:= \sup_{a \in V_x, b \in V_y} |g(x,y)|. \end{equation*} We introduce the auxiliary function \begin{equation} \textsf{F}(x,y):= - \log \left( 1 + \Delta_s(x,y) \right), \end{equation} (7.44) so that, in view of definition (3.14), we have \begin{equation*} \textsf{Main}_s(\eta) = \iint_{\Lambda \times \Lambda} \textsf{F}(x,y) (\textrm{d}\eta(x)-\textrm{d}x) (\textrm{d}\eta(y)-\textrm{d}y). \end{equation*} Lemma 7.5 (Energy comparison—the main term). We have \begin{equation} \textsf{Main}_s(\eta) \preceq \textsf{Main}_s^{\circ}(\eta) + \textsf{Main}_s^{A}(\eta) + \textsf{Main}_s^{B}(\eta) + \textsf{Main}_s^{C}(\eta) + \textsf{Main}_s^{D}(\eta), \end{equation} (7.45) where the terms in the right-hand side are defined as \begin{align*} \textsf{Main}_s^{\circ}(\eta) & = \sum_{i = - \lambda}^{\lambda} \sum_{j = -\lambda}^{\lambda} \widetilde{D}_i \widetilde{D}_j |\partial^2_{xy} F|_{V(i,j)}, \\ \textsf{Main}_s^{A}(\eta) & = \sum_{i=-\lambda}^{\lambda} \sum_{|j| = \lambda - \ell}^{\lambda - \ell/10} \widetilde{D}_i \widetilde{D}_j \frac{|\partial_x \textsf{F}|_{V(i,j)}}{\ell}, \\ \textsf{Main}_s^{B}(\eta) & = \sum_{|i| = \lambda - \ell}^{\lambda - \ell/10} \sum_{|j| = \lambda - \ell}^{\lambda - \ell/10} \widetilde{D}_i \widetilde{D}_j \frac{|\textsf{F}|_{V(i,j)}}{\ell^2}, \\ \textsf{Main}_s^{C}(\eta) & = \left( \ell + \left|\textrm{Discr}_{|x| \in [\lambda - \ell/8, \lambda]}\right| \right) \sum_{j=-\lambda}^\lambda \sup_{|x| \in [\lambda - \ell/8, \lambda]} |\partial_y \textsf{F}|_{V(x,j)} \widetilde{D}_j, \\ \textsf{Main}_s^{D}(\eta) & = \left( \ell + \left|\textrm{Discr}_{|x| \in [\lambda - \ell/8, \lambda]}\right| \right) \sum_{|j| = \lambda - \ell}^{\lambda - \ell/10} \frac{\sup_{|x| \in [\lambda - \ell/8, \lambda]} |\textsf{F}|_{V(x,j)}}{\ell} \widetilde{D}_j. \end{align*} Proof. Let |$\chi $| be a cut-off function equal to |$1$| on |$[-\lambda + \ell /4, \lambda - \ell /4]$|⁠, vanishing outside |$[-\lambda + \ell /8, \lambda - \ell /8]$|⁠, bounded by |$1$| and whose derivative is bounded by |$O_{\bullet }\left (\frac{1}{\ell }\right )$|⁠. We may write \begin{align} \textsf{Main}_s(\eta) =& \iint_{\Lambda \times \Lambda} \chi(x) \textsf{F}(x,y) \chi(y) (\textrm{d}\eta - \textrm{d}x) (\textrm{d}\eta -\textrm{d}y) \nonumber\\ &+ 2 \iint_{\Lambda \times \Lambda} \left(1-\chi(x)\right) \textsf{F}(x,y) \chi(y) (\textrm{d}\eta - \textrm{d}x) (\textrm{d}\eta -\textrm{d}y)\nonumber\\ &+ \iint_{\Lambda \times \Lambda} \left(1 - \chi(x)\right) \textsf{F}(x,y) \left(1- \chi(y) \right) (\textrm{d}\eta - \textrm{d}x) (\textrm{d}\eta -\textrm{d}y). \end{align} (7.46) The last term in the right-hand side vanishes, because |$\psi _s$| vanishes on the support of |$1-\chi $|⁠, and so |$\Delta _s(x,y) = 0$|⁠, and thus |$\textsf{F}(x,y) = 0$|⁠, when both |$x$| and |$y$| belong to the support of |$1 - \chi $|⁠. We now study the two 1st terms in the right-hand side of (7.46) separately. Claim 7.6 (The “|$\chi $|⁠,|$\chi $|” term). We claim that \begin{multline} \iint_{\Lambda \times \Lambda} \chi(x) \textsf{F}(x,y) \chi(y) (\textrm{d}\eta - \textrm{d}x) (\textrm{d}\eta -\textrm{d}y) \\ \preceq \sum_{i = - \lambda}^{\lambda} \sum_{j = -\lambda}^{\lambda} \widetilde{D}_i \widetilde{D}_j |\partial^2_{xy} \textsf{F}|_{V(i,j)} + \sum_{i=-\lambda}^{\lambda} \sum_{|j| = \lambda - \ell}^{\lambda - \ell/10} \widetilde{D}_i \widetilde{D}_j \frac{|\partial_x \textsf{F}|_{V(i,j)}}{\ell} + \sum_{|i| = \lambda - \ell}^{\lambda - \ell/10} \sum_{|j| = \lambda - \ell}^{\lambda - \ell/10} \widetilde{D}_i \widetilde{D}_j \frac{|\textsf{F}|_{V(i,j)}}{\ell^2}. \end{multline} (7.47) Proof of Claim 7.6 For a fixed configuration |$\eta $|⁠, and |$x$| in |$(-\lambda , \lambda )$|⁠, let us define \begin{equation} \textsf{G}_{\eta}(x):= \int \chi(y) \textsf{F}(x,y) (\textrm{d}\eta(y) - \textrm{d}y). \end{equation} (7.48) We have \begin{equation} \iint_{\Lambda \times \Lambda} \chi(x) \textsf{F}(x,y) \chi(y) (\textrm{d}\eta - \textrm{d}x) (\textrm{d}\eta -\textrm{d}y) = \int \textsf{G}_{\eta}(x) \chi(x) (\textrm{d}\eta - \textrm{d}x). \end{equation} (7.49) By construction, the map |$x \mapsto \textsf{G}_{\eta }(x) \chi (x)$| is compactly supported. Using the a priori bounds of Proposition 1.6, we obtain \begin{equation*} \iint_{\Lambda \times \Lambda} \chi(x) \textsf{F}(x,y) \chi(y) (\textrm{d}\eta - \textrm{d}x) (\textrm{d}\eta -\textrm{d}y) \preceq \sum_{i = -\lambda}^{\lambda} \left| \textsf{G}_{\eta} \chi \right|{}_{\textsf{1}, V_i} \widetilde{D}_i. \end{equation*} We have, of course, differentiating a product, \begin{equation*} \big| \textsf{G}_{\eta} \chi \big|{}_{\textsf{1}, V_i} \preceq \big| \textsf{G}_{\eta} \big|{}_{\textsf{1}, V_i} \big| \chi \big|{}_{\textsf{0}, V_i} + \big| \textsf{G}_{\eta} \big|{}_{\textsf{0}, V_i} \big| \chi \big|{}_{\textsf{1}, V_i}, \end{equation*} and we use the fact that |$\chi $| is bounded by |$1$|⁠, and that |$\chi ^{\prime}(x)$| is bounded by |$\ell ^{-1}$| and supported on |$\{|x| \in [\lambda - \ell /4, \lambda - \ell /8]\}$|⁠. We obtain \begin{equation} \int_{\Lambda} \textsf{G}_{\eta}(x) \chi(x) (\textrm{d}\eta - \textrm{d}x) \preceq \sum_{i = -\lambda}^{\lambda} |\textsf{G}_{\eta}|_{\textsf{1}, V_i} \widetilde{D}_i + \sum_{|i| = \lambda - \ell}^{\lambda - \ell/2}\frac{|\textsf{G}_{\eta}|_{\textsf{0}, V_i}}{\ell} \widetilde{D}_i. \end{equation} (7.50) Let us now study |$\textsf{G}_{\eta }$| itself. We have \begin{align*} \textsf{G}_{\eta}(x) = \int \chi(y) \textsf{F}(x,y) (\textrm{d}\eta(y) - \textrm{d}y), \quad \textsf{G}_{\eta}^{\prime}(x) = \int \chi(y) \partial_x \textsf{F}(x,y) (\textrm{d}\eta(y) - \textrm{d}y). \end{align*} We have, of course, differentiating with respect to |$y$| for |$x$| fixed, \begin{equation*} |\chi \textsf{F}(x,\cdot)|_{1, V_{j}} \preceq |\partial_y \textsf{F}|_{V(x,j)} |\chi|_{0, V_{j}} + |\textsf{F}|_{V(x,j)} |\chi|_{1, V_{j}}, \end{equation*} and similarly, \begin{equation*} |\chi \partial_x \textsf{F}(x,\cdot)|_{1, V_{j}} \preceq |\partial^2_{yx} \textsf{F}|_{V(x,j)} |\chi|_{0, V_{j}} + |\partial_x \textsf{F}|_{V(x,j)} |\chi|_{1, V_{j}}. \end{equation*} We use the a priori bounds of Proposition 1.6 again and use again the fact that |$\chi $| is bounded by |$1$|⁠, that |$\chi ^{\prime}(y)$| is zero outside |$\{|y| \in [\lambda - \ell /4, \lambda - \ell /8]\}$| and bounded by |$\ell ^{-1}$|⁠. We obtain \begin{align} \textsf{G}_{\eta}(x) & \preceq \sum_{j=-\lambda}^\lambda |\partial_y \textsf{F}|_{V(x,j)} \widetilde{D}_j + \sum_{|j| = \lambda - \ell}^{\lambda - \ell/10} \frac{|\textsf{F}|_{V(x,j)}}{\ell} \widetilde{D}_j, \end{align} (7.51) \begin{align}\quad \ \textsf{G}_{\eta}^{\prime}(x) & \preceq \sum_{j=-\lambda}^\lambda |\partial^2_{xy} \textsf{F}|_{V(x,j)} \widetilde{D}_j + \sum_{|j| = \lambda - \ell}^{\lambda - \ell/10} \frac{|\partial_x \textsf{F}|_{V(x,j)}}{\ell} \widetilde{D}_j. \end{align} (7.52) Combining (7.49), (7.50) and (7.51), and (7.52), we obtain the expression (7.47). Claim 7.7 (The “|$\chi $|⁠,|$(1-\chi )$|” term). We claim that \begin{align} \iint_{\Lambda \times \Lambda} \left(1-\chi(x)\right) &\textsf{F}(x,y) \chi(y) (\textrm{d}\eta - \textrm{d}x) (\textrm{d}\eta -\textrm{d}y)\nonumber \\ &\preceq \left( \ell + \left|\textrm{Discr}_{|x| \in [\lambda - \ell/8, \lambda]}\right| \right) \sum_{j=-\lambda}^\lambda \sup_{|x| \in [\lambda - \ell/8, \lambda]} |\partial_y \textsf{F}|_{V(x,j)} \widetilde{D}_j \nonumber\\ &+ \left( \ell + \left|\textrm{Discr}_{|x| \in [\lambda - \ell/8, \lambda]}\right| \right) \sum_{|j| = \lambda - \ell}^{\lambda - \ell/10} \frac{\sup_{|x| \in [\lambda - \ell/8, \lambda]} |\textsf{F}|_{V(x,j)}}{\ell} \widetilde{D}_j. \end{align} (7.53) Proof of Claim 7.7 With the notation |$\textsf{G}_{\eta }$| of (7.48), we write \begin{equation} \iint_{\Lambda \times \Lambda} \left(1-\chi(x)\right) \textsf{F}(x,y) \chi(y) (\textrm{d}\eta - \textrm{d}x) (\textrm{d}\eta -\textrm{d}y) = \int_{\Lambda} \left(1-\chi(x)\right) \textsf{G}_{\eta}(x) (\textrm{d}\eta - \textrm{d}x). \end{equation} (7.54) By construction, |$1- \chi (x)$| is supported on |$\{|x| \in [\lambda - \ell /8, \lambda ] \}$|⁠, so we have, using a rough bound on |$\textsf{G}_{\eta }$| and the mass of |$d\eta - dx$| in |$\{|x| \in [\lambda - \ell /8, \lambda ]\}$|⁠, \begin{align} \int_{\Lambda} \left(1-\chi(x)\right) \textsf{G}_{\eta}(x) (\textrm{d}\eta - \textrm{d}x) \preceq \left( \ell + \left|\textrm{Discr}_{|x| \in [\lambda - \ell/8, \lambda]}\right| \right) \sup_{|x| \in [\lambda - \ell/8, \lambda]} \left| \textsf{G}_{\eta}(x)\right|. \end{align} (7.55) Using (7.51) in (7.55), we obtain \begin{multline*} \int_{\Lambda} \left(1-\chi(x)\right) \textsf{G}_{\eta}(x) (\textrm{d}\eta - \textrm{d}x) \preceq \left( \ell + \left|\textrm{Discr}_{|x| \in [\lambda - \ell/8, \lambda]}\right| \right) \sum_{j=-\lambda}^\lambda \sup_{|x| \in [\lambda - \ell/8, \lambda]} |\partial_y \textsf{F}|_{V(x,j)} \widetilde{D}_j \\ + \left( \ell + \left|\textrm{Discr}_{|x| \in [\lambda - \ell/8, \lambda]}\right| \right) \sum_{|j| = \lambda - \ell}^{\lambda - \ell/10} \frac{\sup_{|x| \in [\lambda - \ell/8, \lambda]} |\textsf{F}|_{V(x,j)}}{\ell} \widetilde{D}_j, \end{multline*} which yields (7.53). The estimate (7.45) is simply the combination of (7.46) and the two claims above. Proof of Proposition 4.1 We recall that \begin{equation*} \textsf{F}(x,y) = - \log \left(1 + \frac{\psi_s(y)-\psi_s(x)}{y-x} \right). \end{equation*} Claim 7.8 (The magnitude of |$\textsf{F}$| and its derivatives). We have \begin{align} \textsf{F}(x,y) & \preceq \frac{|\psi_s(x)| + |\psi_s(y)|}{|x-y|}\qquad\qquad\qquad\qquad\qquad \end{align} (7.56) \begin{align} \textsf{F}(x,y) & \preceq \sup_{t \in [x,y]} |\psi_s^{(\textsf{1})}(t)|\ \ \quad\qquad\qquad\qquad\qquad\qquad\end{align} (7.57) \begin{align} \partial_x \textsf{F}(x,y) & \preceq \frac{|\psi_s^{\prime}(x)|}{|y-x|} + \frac{|\psi_s(x)| + |\psi_s(y)|}{(y-x)^2} \qquad\qquad\qquad\ \ \end{align} (7.58) \begin{align} \partial_x \textsf{F}(x,y) & \preceq \sup_{t \in [x,y]} |\psi_s^{(\textsf{2})}(t)|.\ \qquad\qquad\qquad\qquad\qquad\qquad \end{align} (7.59) \begin{align} \partial^2_{xy} \textsf{F}(x,y) & \preceq \frac{|\psi_s^{\prime}(x)|}{(x-y)^2} + \frac{|\psi_s^{\prime}(y)|}{(x-y)^2} + \frac{|\psi_s(x)|}{|x-y|^3} + \frac{|\psi_s(y)|}{|x-y|^3}, \end{align} (7.60) \begin{align} \partial^2_{xy} \textsf{F}(x,y) & \preceq \sup_{t \in [x,y]} |\psi_s^{(3)}(t)| + \sup_{t \in [x,y]} |\psi_s^{(2)}(t)|^2.\quad\qquad\qquad \end{align} (7.61) Proof of Claim 7.8 The bounds (7.56) and (7.57) are straightforward. We then perform the following simple computation: \begin{align} \partial_x \textsf{F} & = \frac{\partial_x \Delta_s}{1 + \Delta_s},\qquad\qquad\qquad\qquad\qquad\qquad\quad \end{align} (7.62) \begin{align} \partial^2_{xy} \textsf{F} & = \frac{ - \left(\partial^2_{xy} \Delta_s\right) (1 + \Delta_s) + \left(\partial_x \Delta_s\right) \left(\partial_y \Delta_s\right)}{(1 + \Delta_s)^2}. \end{align} (7.63) Moreover, we have \begin{equation} \partial_x \Delta_s(x,y) = \frac{- \psi_s^{\prime}(x)}{y-x} + \frac{\psi_s(y) - \psi_s(x)}{(y-x)^2}, \quad \partial_y \Delta_s(x,y) = \frac{ \psi_s^{\prime}(y)}{y-x} - \frac{\psi_s(y) - \psi_s(x)}{(y-x)^2}. \end{equation} (7.64) \begin{equation*} \partial^2_{xy} \Delta_s(x,y) = \frac{\psi_s^{\prime}(x) + \psi_s^{\prime}(y)}{(y-x)^2} - 2 \frac{\psi_s(y) - \psi_s(x)}{(y-x)^3}. \end{equation*} From (7.62) and the fact that |$1 + \Delta _s$| is bounded below by a positive constant (because |$|\Delta _s|_{\textsf{0}}$| is bounded by |$|\psi _s^{\prime}|_{\textsf{0}}$|⁠, itself bounded by |$s|\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }|_{\textsf{0}}$|⁠, and |$\textrm{s}_{\textrm{max}}$| is chosen as in (3.1)) we see that \begin{equation} \partial_x \textsf{F} \preceq \partial_x \Delta_s, \end{equation} (7.65) and using (7.64) we obtain (7.58). Using again the fact that |$1 + \Delta _s$| is bounded below by a positive constant, we get \begin{equation} \partial^2_{xy} \textsf{F} \preceq - \left(\partial^2_{xy} \Delta_s\right) (1 + \Delta_s) + \left(\partial_x \Delta_s\right) \left(\partial_y \Delta_s\right), \end{equation} (7.66) and after some algebra, we obtain \begin{equation} \partial^2_{xy} \textsf{F}(x,y) \preceq \frac{\psi_s^{\prime}(x)}{(x-y)^2} + \frac{\psi_s^{\prime}(y)}{(x-y)^2} - \frac{\psi_s^{\prime}(x) \psi_s^{\prime}(y)}{(x-y)^2} + \frac{\Delta_s^2(x,y)}{(x-y)^2} - \frac{2\Delta_s(x,y)}{(x-y)^2}. \end{equation} (7.67) Since |$\psi _s^{\prime}$| is bounded, and so is |$\Delta _s(x,y) = \frac{\psi _s(x) - \psi _s(y)}{x-y}$|⁠, we may certainly write \begin{equation*} \partial^2_{xy} \textsf{F}(x,y) \preceq \frac{|\psi_s^{\prime}(x)|}{(x-y)^2} + \frac{|\psi_s^{\prime}(y)|}{(x-y)^2} + \frac{|\psi_s(x)|}{|x-y|^3} + \frac{|\psi_s(y)|}{|x-y|^3}, \end{equation*} which is (7.60). It remains to prove (7.59) and (7.61). Using the identity \begin{equation*} \Delta_s(x,y) = \frac{1}{y-x} \int_{x}^y \psi_s^{\prime}(s) \ \textrm{d}s, \end{equation*} an elementary computation yields \begin{equation} \partial_x \Delta_s(x,y) \preceq \sup_{t \in [x,y]} \left|\psi_s^{(\textsf{2})}(t)\right|, \quad \partial_x \Delta_s(x,y) \preceq \sup_{t \in [x,y]} \left|\psi_s^{(\textsf{2})}(t)\right|, \partial^2_{xy} \Delta_s(x,y)| \preceq \sup_{t \in [x,y]} \left|\psi_s^{(\textsf{3})}(t)\right|. \end{equation} (7.68) We may then derive (7.59) from (7.65) and (7.68) and (7.61) from (7.66) and (7.68). General strategy and convention for the proof. We estimate the expectations of the all terms in Proposition 4.1. They involve (double) sums with coefficients of the type \begin{equation*} \widetilde{D}_i \widetilde{D}_j A(i,j), \end{equation*} where |$A(i,j)$| is a non-random quantity related to |$\textsf{F}$| or one of its derivatives. We will use the estimates of Claim 7.8 to control the terms |$A(i,j)$|⁠. Typically, the estimates (7.56), (7.58), and (7.60) will be used when |$i$| and |$j$| are far away, and the estimates (7.57), (7.59), and (7.61) will be used for |$i$| and |$j$| close to each other. The expectation of |$\widetilde{D}_i \widetilde{D}_j$| can be controlled using the discrepancy estimates (1.11) and (1.12). Using Cauchy–Schwarz’s inequality, we see that (it is easy to check that the fact that, strictly speaking, the inequality is not true for |$i= 0$| or |$j=0$| is irrelevant) \begin{equation} {\mathbb{E}}\left[ \widetilde{D}_i \widetilde{D}_j \right] \preceq \sqrt{|i|} \sqrt{|j|}, \end{equation} (7.69) and we will replace all occurrences of |$\widetilde{D}_i$|⁠, resp. |$\widetilde{D}_j$| by |$\sqrt{|i|}$|⁠, resp. |$\sqrt{|j|}$|⁠. For most estimates, this is enough, and we obtain terms that are |$o_{\ell , \lambda }(1)$|⁠. A couple of terms are seen this way to be only bounded but perhaps not vanishing, as |$\lambda \to \infty , \ell \to \infty $|⁠, which we denote by |$O(1)$|⁠. For these terms, we use (1.12) instead of (1.11) and write that \begin{equation} {\mathbb{E}}\left[ \widetilde{D}_i \widetilde{D}_j \right] \preceq o_{|i| \to \infty} \left(\sqrt{|i|}\right) o_{|j| \to \infty} \left(\sqrt{|j|}\right), \end{equation} (7.70) which allows us to improve the bound to |$o_{\ell , \lambda }(1)$|⁠. The term |$\textsf{Main}_s^{\circ }$|⁠. We recall that \begin{equation*} \textsf{Main}_s^{\circ}(\eta) = \sum_{i = - \lambda}^{\lambda} \sum_{j = -\lambda}^{\lambda} \widetilde{D}_i \widetilde{D}_j |\partial^2_{xy} F|_{V(i,j)}. \end{equation*} Using symmetries, it is enough to study \begin{equation*} \sum_{i=0}^{\lambda} \sum_{i \leq |j| \leq \lambda} \widetilde{D}_i \widetilde{D}_j |\partial^2_{xy} F|_{V(i,j)}. \end{equation*} Let us start with the region |$0 < x < 2\ell $| and |$x < y < 4\ell $|⁠. We want to prove that \begin{equation*} {\mathbb{E}} \left[ \sum_{i=0}^{2\ell} \sum_{j = i}^{4\ell} \widetilde{D}_i \widetilde{D}_j |\partial^2_{xy} F|_{V(i,j)} \right] = s o_{\ell, \lambda}(1). \end{equation*} In this case, since |$i,j$| are close, we use (7.61) to control |$|\partial ^2_{xy} F|_{V(i,j)}$|⁠. By (3.9), (3.10), we know that |$\psi _s^{(\textsf{2})}$| is controlled by |$s \widetilde{{\mathfrak{m}}}_{\lambda , \varphi }^{(\textsf{1})}$| and that |$\psi _s^{(\textsf{3})}$| is controlled by |$s \widetilde{{\mathfrak{m}}}_{\lambda , \varphi }^{(\textsf{2})}$|⁠, and we refer to the bounds (2.13) and (2.14) to see that \begin{equation*} \sup_{|t| \leq 4\ell} |\psi_s^{(\textsf{2})}(t)|^2 \preceq \frac{s^2}{\ell^4} \quad \sup_{|t| \leq 4\ell} |\psi_s^{(\textsf{3})}(t)| \preceq \frac{s}{\ell^3}; \end{equation*} the dominant term is obviously the 2nd one, so we may simply study \begin{equation*} \sum_{i=0}^{2\ell} \sum_{j = i}^{4\ell} \widetilde{D}_i \widetilde{D}_j \frac{s}{\ell^3}. \end{equation*} Taking the expectation and using (7.69), we are left with \begin{equation*} \sum_{i=0}^{2\ell} \sum_{j = i}^{4\ell} \sqrt{i} \sqrt{j} \frac{s}{\ell^3} \preceq s \ell^2 \times \ell \times \frac{s}{\ell^3} = s O(1). \end{equation*} This is an example where the bound (7.69) is not sufficient, and we replace it by (7.70). By well-known results on divergent series, we have \begin{equation*} \sum_{i=0}^{2\ell} \sum_{j = i}^{4\ell} o_{i}\left(\sqrt{i}\right) o_{j}\left(\sqrt{j}\right) = o_{l \to \infty} (\ell^3), \end{equation*} and thus we have, as desired, \begin{equation*} {\mathbb{E}} \left[ \sum_{i=0}^{2\ell} \sum_{j = i}^{4\ell} \widetilde{D}_i \widetilde{D}_j \frac{s}{\ell^3} \right] = s o_{\ell, \lambda}(1). \end{equation*} \begin{equation*} \star \star \star \end{equation*} For |$2\ell < x < \frac{\lambda }{2}$|⁠, |$x < y < \frac{4}{3} x$|⁠. We study the expectation of \begin{align*} \sum_{i=2\ell}^{\lambda/2} \sum_{j = i}^{\frac{4}{3} i} \widetilde{D}_i \widetilde{D}_j |\partial^2_{xy} F|_{V(i,j)}. \end{align*} We use (7.61) to control |$|\partial ^2_{xy} F|_{V(i,j)}$|⁠. We control again |$\psi _s^{(\textsf{2})}$| by |$s\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }^{(\textsf{1})}$| (and read (2.13)), and |$\psi _s^{(\textsf{3})}$| by |$s\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }^{(\textsf{2})}$| (and read (2.14)), we get \begin{equation*} \sup_{t \in [x,y]} \left|\psi_s^{(\textsf{2})}(t) \right|{}^2 \preceq s^2 \frac{\ell^2}{x^6}, \sup_{t \in [x,y]} \left| \psi_s^{(\textsf{3})}(t) \right| \preceq s\frac{\ell}{x^4}. \end{equation*} Since |$x> \ell $|⁠, the dominant term is the 2nd one. Finally, we take the expectation, use the discrepancy estimates and replace |$\widetilde{D}_i \widetilde{D}_j$| by |$\sqrt{i} \sqrt{j}$|⁠. Comparing the sum with an integral, we are left to study \begin{equation*} s \int_{2\ell}^{\lambda/2} \int_{x}^{\frac{4}{3}x} \sqrt{x} \sqrt{y} \frac{\ell}{x^4} \ \textrm{d}x \ \textrm{d}y. \end{equation*} Replacing |$\sqrt{y}$| by |$\sqrt{x}$| (since |$x < y < \frac{4}{3} x$|⁠), it yields \begin{equation*} s\int_{2\ell}^{\lambda/2} \int_{x}^{\frac{4}{3}x} \sqrt{x} \sqrt{x} \frac{\ell}{x^4} \ \textrm{d}y \ \textrm{d}x = s\int_{2\ell}^{\lambda/2} \frac{\ell}{x^2} = sO(1). \end{equation*} We are again in a case where (7.69) is not enough and must be replaced by the discrepancy estimates (7.70), which improves the bound from |$s O(1)$| to |$s o_{\ell , \lambda }(1)$|⁠. \begin{equation*} \star \star \star \end{equation*} For |$2\ell < x < \frac{\lambda }{2}$|⁠, |$\frac{4}{3} x < y$|⁠. We study the expectation of \begin{equation*} \sum_{i=2\ell}^{\lambda/2} \sum_{j = \frac{4}{3} i} ^{\lambda} \widetilde{D}_i \widetilde{D}_j |\partial^2_{xy} F|_{V(i,j)}. \end{equation*} Since |$i$| and |$j$| are far from each other, we use (7.60) to control |$|\partial ^2_{xy} F|_{V(i,j)}$|⁠. Taking the expectations, using (7.69) and comparing the sum with an integral, we are left to study \begin{equation*} \int_{2\ell}^{\lambda/2} \sqrt{x} \int_{\frac{4}{3}x}^{\lambda} \sqrt{y} \left( \frac{|\psi_s^{\prime}(x)|}{(x-y)^2} + \frac{|\psi_s^{\prime}(y)|}{(x-y)^2} + \frac{|\psi_s(x)|}{(x-y)^3} + \frac{|\psi_s(y)|}{(x-y)^3} \right) \ \textrm{d}y. \end{equation*} Since |$\frac{4}{3} x < y$| we can replace |$x-y$| by |$y$|⁠, and we split the integrand in four parts. (a) Using (3.8) to control |$\psi _s^{\prime}$| by |$s\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$|⁠, and (2.12) to control |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$|⁠, we get \begin{equation*} \int_{2\ell}^{\lambda/2} \sqrt{x} |\psi_s^{\prime}(x)| \int_{\frac{4}{3}x}^{\lambda} \frac{\sqrt{y}}{y^2} \ \textrm{d}y \preceq s \int_{2\ell}^{\lambda /2} |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(x)| \textrm{d}x = s O(1). \end{equation*} Using again (7.70) instead of (7.69), we may replace |$\sqrt{x}, \sqrt{y}$| by |$o_x(\sqrt{x}), o_y(\sqrt{y})$|⁠, and we obtain in fact |$so_{\ell , \lambda }(1)$|⁠. (b) Using (3.8) to control |$\psi _s^{\prime }$| by |$s \widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$|⁠, and (2.12) to control |$\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }$|⁠, and splitting the domain of integration in two parts, we see that \begin{align*}& \int_{2\ell}^{\lambda/2} \sqrt{x} \ \textrm{d}x \int_{\frac{4}{3}x}^{\lambda} \frac{\sqrt{y} |\psi_s^{\prime}(y)|}{y^2} \ \textrm{d}y\\ &\qquad\preceq s \int_{2\ell}^{\lambda /2} \sqrt{x} \ \textrm{d}x \left[\int_{\frac{4}{3}x}^{\lambda/2} \frac{\sqrt{y} \ell}{y^{2} y^2} \ \textrm{d}y + \int_{\lambda/2}^{\lambda} \frac{\ell \sqrt{y}}{\lambda^{3/2} y^2 \sqrt{\lambda - y}} \ \textrm{d}y \right] \end{align*} The 1st contribution is \begin{align*} s\int_{2\ell}^{\lambda/2} \sqrt{x} \int_{\frac{4}{3}x}^{\lambda/2} \frac{\sqrt{y} \ell}{y^{2} y^2} \ \textrm{d}y \ \textrm{d}x &= s\int_{2\ell}^{\lambda/2} \sqrt{x} \int_{\frac{4}{3}x}^{\lambda/2} \frac{\ell}{y^{7/2}} \ \textrm{d}y \ \textrm{d}x\\ &\preceq s \int_{2\ell}^{\lambda/2} \sqrt{x} \frac{\ell}{x^{5/2}} \ \textrm{d}x = sO(1), \end{align*} and for the 2nd one, since |$y \in [\lambda /2,\lambda ]$|⁠, we may replace |$y$| by |$\lambda $| and compute \begin{equation*} s \int_{2\ell}^{\lambda/2} \sqrt{x} \int_{\lambda/2}^{\lambda} \frac{\ell}{\lambda^{3} \sqrt{\lambda-y}} \ \textrm{d}y \ \textrm{d}x = s O(1). \end{equation*} Again, this can be improved to |$s o_{\ell , \lambda }(1)$|⁠. (c) Using (3.12) to control |$\psi _s(x)$|⁠, we have \begin{equation*} \int_{2\ell}^{\lambda/2} \sqrt{x} |\psi_s(x)| \int_{\frac{4}{3}x}^{\lambda} \frac{\sqrt{y}}{y^{3}} \ \textrm{d}y \preceq s \int_{2\ell}^{\lambda/2} \sqrt{x} \frac{\ell}{|x|} \frac{1}{x^{3/2}} \ \textrm{d}x = s O(1), \end{equation*} which can be improved to |$o_{\ell , \lambda }(1)$|⁠. (d) Using (3.12) to control |$\psi _s(y)$| and splitting the domain of integration (on |$y$|⁠) in two parts, we have \begin{align*}& \int_{2\ell}^{\lambda/2} \sqrt{x} \ \textrm{d}x \int_{\frac{4}{3}x}^{\lambda} \frac{\sqrt{y} |\psi_s(y)|}{y^{3}} \ \textrm{d}y\\ &\quad\preceq \int_{2\ell}^{\lambda/2} \sqrt{x} \ \textrm{d}x \left[ \int_{\frac{4}{3}x}^{\lambda/2} \frac{\ell}{y^{7/2}} \ \textrm{d}y + \int_{\lambda/2}^{\lambda} \frac{\ell}{\lambda^{3/2}} \frac{\sqrt{\lambda - y}}{\lambda^{7/2}} \ \textrm{d}y \right] \\ &\quad\preceq s \int_{2\ell}^{\lambda/2} \sqrt{x} \left[ \frac{\ell}{x^{5/2}} + \frac{\ell}{\lambda^{5/2}} \right] \ \textrm{d}x = s O(1). \end{align*} Similarly, this can be improved to |$s o_{\ell , \lambda }(1)$|⁠. \begin{equation*} \star \star \star \end{equation*} For |$\frac{\lambda }{2} < x < \lambda - \ell $| and |$y-x < \frac{1}{2} (\lambda - x)$|⁠. We study the expectation of \begin{equation*} \sum_{i=\lambda/2}^{\lambda - \ell} \sum_{0 <j-i < \frac{1}{2} (\lambda - i)} \widetilde{D}_i \widetilde{D}_j |\partial^2_{xy} F|_{V(i,j)}. \end{equation*} Since |$i,j$| are close we use (7.61) to control |$|\partial ^2_{xy} F|_{V(i,j)}$|⁠. We see (the now usual way) that \begin{equation*} \sup_{t \in [x,y]} \left| \psi_s^{(2)}(t) \right|{}^2 \preceq s^2 \frac{\ell^2}{\lambda^{3} (\lambda - x)^{3}}, \quad \sup_{t \in [x,y]} \left| \psi_s^{(3)}(t) \right| \preceq s \frac{\ell}{\lambda^{3/2} (\lambda - x)^{5/2}}, \end{equation*} and the dominant term is the 2nd one. We take the expectation, we use the discrepancy estimates, we compare the sum to an integral, we replace |$\sqrt{x}, \sqrt{y}$| by |$\sqrt{\lambda }$|⁠, and we are left to compute \begin{align*}& s \int_{\lambda/2}^{\lambda - \ell} \lambda \ \textrm{d}x \int_{0 < y-x < \frac{1}{2} (\lambda - x)} \frac{\ell}{\lambda^{3/2} (\lambda - x)^{5/2}} \ \textrm{d}y \\&\qquad\quad\preceq s \int_{\lambda/2}^{\lambda - \ell} \frac{\ell}{\sqrt{\lambda}} \frac{1}{(\lambda -x)^{3/2}} \preceq s \frac{\ell}{\sqrt{\lambda}\sqrt{\ell}} = s o_{\ell, \lambda}(1). \end{align*} \begin{equation*} \star \star \star \end{equation*} For |$\frac{\lambda }{2} < x < \lambda - \ell $| and |$y-x> \frac{1}{2} (\lambda - x)$|⁠. We use (7.60) to control |$|\partial ^2_{xy} F|_{V(i,j)}$|⁠. We take the expectation and compare the sum to a series; we are left to study \begin{align*} \int_{\lambda/2}^{\lambda} \ \textrm{d}x \sqrt{x} \int_{y-x> \frac{\lambda -x}{2}} \sqrt{y} \left( \frac{|\psi_s^{\prime}(x)|}{(x-y)^2} + \frac{|\psi_s^{\prime}(y)|}{(x-y)^2} + \frac{|\psi_s(x)|}{(x-y)^3} + \frac{|\psi_s(y)|}{(x-y)^3} \right) \ \textrm{d}y. \end{align*} We replace |$\sqrt{x}, \sqrt{y}$| by |$\sqrt{\lambda }$| and split the integrand in four parts. (a) Using (3.8) to control |$\psi _s^{\prime}(x)$| by |$s\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }(x)$|⁠, and (2.12), we have \begin{equation*} |\psi_s^{\prime}(x)| \preceq s\frac{\ell}{\lambda^{3/2} \sqrt{\lambda-x}}, \end{equation*} and thus consider \begin{align*}& \int_{\lambda/2}^{\lambda-\ell} \ \textrm{d}x \sqrt{\lambda} \int_{y-x> \frac{\lambda -x}{2}} \sqrt{\lambda} \frac{|\psi_s^{\prime}(x)|}{(y-x)^2}\\ &\qquad\preceq s \int_{\lambda/2}^{\lambda} \ \textrm{d}x \lambda \frac{\ell}{\lambda^{3/2} \sqrt{\lambda-x}} \int_{y-x > \frac{\lambda -x}{2}} \frac{1}{(y-x)^2} \ \textrm{d}y \\ &\qquad\preceq s \int_{\lambda/2}^{\lambda-\ell} \ \textrm{d}x \lambda \frac{\ell}{\lambda^{3/2} \sqrt{\lambda - x}} \frac{1}{\lambda-x} \preceq s \frac{\ell}{\sqrt{\lambda} \sqrt{\ell}} = s o_{\ell, \lambda}(1). \end{align*} (b) Using (3.8) to control |$\psi _s^{\prime}(y)$| by |$s\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }(y)$|⁠, and (2.12), we have \begin{align*} |\psi_s^{\prime}(y)| \preceq s\frac{\ell}{\lambda^{3/2} \sqrt{\lambda-y}}, \end{align*} and thus consider \begin{align*}& \int_{\lambda/2}^{\lambda-\ell} \ \textrm{d}x \sqrt{\lambda} \int_{y-x> \frac{\lambda -x}{2}} \sqrt{\lambda} \frac{|\psi_s^{\prime}(y)|}{(y-x)^2} \ \textrm{d}y\\&\qquad \preceq s \int_{\lambda/2}^{\lambda-\ell} \ \textrm{d}x \lambda \int_{y-x > \frac{\lambda -x}{2}} \frac{\ell}{\lambda^{3/2} \sqrt{\lambda - y} (y-x)^2} \ \textrm{d}y. \end{align*} Since |$y-x> \frac{\lambda -x}{2}$|⁠, we may replace |$\frac{1}{(y-x)^2}$| by |$\frac{1}{(\lambda - x)^2}$|⁠, and we now study \begin{align*}& s \int_{\lambda/2}^{\lambda-\ell} \ \textrm{d}x \frac{\lambda}{(\lambda-x)^2} \int_{x + \frac{\lambda-x}{2}}^{\lambda} \frac{\ell}{\lambda^{3/2} \sqrt{\lambda-y}} \ \textrm{d}y\\ &\qquad\preceq s \int_{\lambda/2}^{\lambda-\ell} dx \frac{\lambda}{(\lambda-x)^2} \frac{\ell}{\lambda^{3/2}} \sqrt{\lambda-x} \ \textrm{d}x\\&\qquad \preceq s \frac{\ell}{\sqrt{\lambda} \sqrt{\ell}} = s o_{\ell, \lambda}(1). \end{align*} (c) Using (3.12) to control |$\psi _s(x)$| by |$s \frac{\ell \sqrt{\lambda -x}}{\lambda ^{3/2}}$|⁠, we write \begin{align*}& \int_{\lambda/2}^{\lambda-\ell} \ \textrm{d}x \sqrt{\lambda} \int_{y-x> \frac{\lambda -x}{2}} \sqrt{\lambda} \frac{|\psi_s(x)|}{(y-x)^3} \ \textrm{d}y\\&\qquad \preceq s \int_{\lambda/2}^{\lambda-\ell} \ \textrm{d}x \lambda \frac{\ell \sqrt{\lambda -x}}{\lambda^{3/2}} \int_{|y-x| > \frac{\lambda - x}{2}} \frac{1}{(y-x)^3} \ \textrm{d}y \\ &\qquad\preceq s \int_{\lambda/2}^{\lambda-\ell} \lambda \frac{\ell \sqrt{\lambda -x}}{\lambda^{3/2}} \frac{1}{(\lambda-x)^2} \ \textrm{d}x \preceq s \frac{\ell}{\sqrt{\ell} \sqrt{\lambda}} = s o_{\ell, \lambda}(1). \end{align*} (d) Using (3.12) to control |$\psi _s(y)$| by |$s \frac{\ell \sqrt{\lambda -y}}{\lambda ^{3/2}}$|⁠, we write \begin{align*}& \int_{\lambda/2}^{\lambda-\ell} \ \textrm{d}x \sqrt{\lambda} \int_{y-x> \frac{\lambda -x}{2}} \sqrt{\lambda} \frac{|\psi_s(y)|}{(y-x)^3} \ \textrm{d}y\\&\qquad \preceq s \int_{\lambda/2}^{\lambda} \ \textrm{d}x \frac{\lambda}{(\lambda-x)^3} \int_{x + \frac{\lambda-x}{2}}^{\lambda-\ell} \frac{\ell}{\lambda^{3/2}} \sqrt{\lambda -y} \ \textrm{d}y \\&\qquad \preceq s \int_{\lambda/2}^{\lambda-\ell} \ \textrm{d}x \frac{\lambda}{(\lambda-x)^3} \frac{\ell}{\lambda^{3/2}} (\lambda - x)^{3/2} \preceq s \frac{\ell}{\sqrt{\lambda} \sqrt{\ell}} = so_{\ell, \lambda}(1). \end{align*} \begin{equation*} \star \star \star \end{equation*} For |$\lambda - \ell < x < y < \lambda $|⁠. We use (7.64), the computation is similar to the case |$\frac{\lambda }{2} < x < \lambda - \ell $| and |$y-x < \frac{1}{2} (\lambda - x)$| above, and we obtain again an error as |$s \frac{\ell }{\sqrt{\ell } \sqrt{\lambda }}$|⁠, which is |$s o_{\ell , \lambda }(1)$|⁠. \begin{equation*} \star \star \star \end{equation*} For |$x,y$| in |$(-4\ell , 4\ell )$| the proof is as in the very 1st case. \begin{equation*} \star \star \star \end{equation*} For |$0 < x < 2 \ell $|⁠, and |$4\ell < |y| < \lambda $|⁠, we use (7.64), we write |$\frac{1}{|x-y|} \preceq \frac{1}{|y|}$|⁠, and we are left with \begin{align*} \int_{0}^{2\ell} \sqrt{x} \ \textrm{d}x \int_{4\ell}^{\lambda} \sqrt{y} \left( \frac{|\psi_s^{\prime}(x)|}{y^2} + \frac{|\psi_s^{\prime}(y)|}{y^2} + \frac{|\psi_s(x)|}{y^3} + \frac{|\psi_s(y)|}{y^3} \right) \ \textrm{d}y. \end{align*} We have |$\psi _s(x) \preceq s$|⁠, |$\psi _s^{\prime}(x) \preceq \frac{s}{\ell }$|⁠, and the corresponding terms give \begin{align*} \int_{0}^{2\ell} \sqrt{\ell} \int_{4\ell}^{\lambda} \sqrt{y} \left( \frac{1}{\ell y^2} + \frac{1}{y^3} \right) \ \textrm{d}y = s O(1). \end{align*} For the two other terms, we obtain \begin{equation*} \int_{0}^{2\ell} \sqrt{x} \ \textrm{d}x \int_{4\ell}^{\lambda} \sqrt{y} \frac{|\psi_s^{\prime}(y)|}{y^2} \ \textrm{d}y \preceq s \ell^{3/2} \int_{4\ell}^{\lambda} \frac{|\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y)|}{y^{3/2}} \ \textrm{d}y, \end{equation*} and using the bounds (2.12) we see that this is |$s O(1)$|⁠. All these terms are in fact improved to |$s o_{\ell , \lambda }(1)$| as above. \begin{equation*} \star \star \star \end{equation*} For |$2\ell < x < \frac{\lambda }{2}$| and |$x < - y$|⁠, the computation is similar to the case |$2\ell < x < \frac{\lambda }{2}$| and |$\frac{4}{3} x < y$|⁠, since we can write |$\frac{1}{|x-y|} \leq \frac{1}{|y|}$|⁠. \begin{equation*} \star \star \star \end{equation*} Finally, for |$\frac{\lambda }{2} < x < \lambda $| and |$-\lambda < y < - \frac{\lambda }{2}$|⁠, we use (7.61), and we are left to bound, after replacing |$\sqrt{x}, \sqrt{y}$| by |$\sqrt{\lambda }$|⁠, and |$|y-x|$| by |$\lambda $|⁠; the quantity \begin{equation*} \int_{\lambda/2}^{\lambda} \sqrt{\lambda} \ \textrm{d}x \int_{\lambda/2}^{\lambda} \sqrt{\lambda} \left( \frac{|\psi_s^{\prime}(y)|}{\lambda^2} + \frac{|\psi_s(y)|}{\lambda^3} \right) \ \textrm{d}y, \end{equation*} where we use the symmetry in |$x,y$| to forget about the |$\psi _s^{\prime}(x), \psi _s(x)$| terms. We have \begin{equation*} \lambda^2 \int_{\lambda/2}^{\lambda} \frac{|\psi_s^{\prime}(y)|}{\lambda^2} \ \textrm{d}y \preceq \int_{\lambda/2}^{\lambda} |\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(y)| \ \textrm{d}y = s O(1), \end{equation*} and using (3.12), \begin{equation*} \lambda^2 \int_{\lambda/2}^{\lambda} \frac{|\psi_s(y)|}{\lambda^3} \ \textrm{d}y \preceq \frac{1}{\lambda} \frac{s \ell}{\lambda^{3/2}} \int_{\lambda/2}^{\lambda} \sqrt{\lambda - y} \ \textrm{d}y \preceq \frac{s \ell}{\lambda} = s o_{\ell, \lambda}(1). \end{equation*} This concludes the study of |$\textsf{Main}_s^{\circ }$|⁠. \begin{equation*} \star \star \star \star \star \end{equation*} The term |$\textsf{Main}_s^{A}$|⁠. We recall that \begin{align*} \textsf{Main}_s^{A}(\eta) = \sum_{i=-\lambda}^{\lambda} \sum_{|j| = \lambda - \ell}^{\lambda - \ell/10} \widetilde{D}_i \widetilde{D}_j \frac{|\partial_x \textsf{F}|_{V(i,j)}}{\ell}. \end{align*} Using symmetries, it is enough to study \begin{equation*} \sum_{i = - \lambda}^{\lambda} \sum_{j = \lambda-\ell}^{\lambda} \widetilde{D}_i \widetilde{D}_j \frac{|\partial_x \textsf{F}|_{V(i,j)}}{\ell}. \end{equation*} When taking the expectation, we use the discrepancy estimates and replace |$\widetilde{D}_i \widetilde{D}_j$| by |$\sqrt{i} \sqrt{j}$|⁠, keeping in mind that any |$O(1)$| can be improved to |$o_{\ell , \lambda }(1)$| by using (7.70) instead of (7.69). We split the 1st sum into |$i \leq \lambda - 3 \ell $| and |$i \geq \lambda - 3\ell $|⁠. For the 1st sum, we use (7.58) and study \begin{equation*} \sum_{i = - \lambda}^{\lambda-3\ell} \sum_{j = \lambda-\ell}^{\lambda} \frac{\sqrt{i} \sqrt{j}}{\ell} \left( \frac{|\psi_s^{\prime}(i)}{|j-i|} + \frac{|\psi_s(i)| + |\psi_s(j)|}{(j-i)^2} \right). \end{equation*} Replacing |$\sqrt{j}$| by |$\sqrt{\lambda }$|⁠, |$|\psi _s(j)|$| by |$\frac{\ell \sqrt{\ell }}{\lambda ^{3/2}}$| (in view of (3.12)), and |$j-i$| by |$\lambda - i$|⁠, we are left with \begin{equation} \sum_{i = - \lambda}^{\lambda-3\ell} \frac{\ell \sqrt{\lambda}}{\ell} \sqrt{i} \left( \frac{|\psi_s^{\prime}(i)|}{\lambda - i} + \frac{|\psi_s(i)|}{(\lambda-i)^2} + \frac{\sqrt{\ell} \ell}{\lambda^{3/2} (\lambda -i)^2} \right). \end{equation} (7.71) We decompose the sum further. (a) For |$- \lambda \leq i \leq 2\ell $|⁠, we use the fact that |$\psi _s^{\prime }(i) \preceq s \widetilde{{\mathfrak{m}}}_{\lambda , \varphi }(i)$| and |$|\widetilde{{\mathfrak{m}}}_{\lambda , \varphi }|_{L^1} \preceq 1$|⁠, that |$\psi _s(i) \preceq s$|⁠, we replace |$\lambda -i$| by |$\lambda $|⁠, and we bound |$\sqrt{|i|}$| by |$\sqrt{\lambda }$|⁠. We obtain \begin{equation*} s \sum_{i = - \lambda}^{2\ell} \frac{\ell \sqrt{\lambda}}{\ell} \sqrt{\lambda} \left( \frac{|\widetilde{{\mathfrak{m}}}_{\lambda, \varphi}(i)|}{\lambda} + \frac{1}{\lambda^2} + \frac{\sqrt{\ell} \ell}{\lambda^{3/2} \lambda^2} \right) = s O(1). \end{equation*} (b) For |$2\ell \leq i \leq \lambda /2$|⁠, we use the fact that |$\psi _s^{\prime}(i) \preceq s \widetilde{{\mathfrak{m}}}_{\lambda , \varphi }(i) \preceq s \frac{\ell }{i^2}$|⁠, that |$\psi _s(i) \preceq s \frac{\ell }{i}$|⁠, and we replace |$\lambda -i$| by |$\lambda $|⁠. We obtain \begin{equation*} s \sum_{i = 2\ell}^{\lambda/2} \frac{\ell \sqrt{\lambda}}{\ell} \sqrt{i} \left( \frac{\ell}{i^2 \lambda} + \frac{\ell}{i \lambda^2} + \frac{\sqrt{\ell} \ell}{\lambda^{3/2} \lambda^2} \right) = s o_{\ell, \lambda}(1). \end{equation*} (c) For |$\lambda /2 \leq i \leq \lambda - 3\ell $|⁠, we use the fact that |$\psi _s^{\prime}(i) \preceq s \widetilde{{\mathfrak{m}}}_{\lambda , \varphi }(i) \preceq s\frac{\ell }{\lambda ^{3/2} \sqrt{\lambda -i}}$|⁠, that |$\psi _s(i) \preceq s \frac{\ell \sqrt{\lambda -i}}{\lambda ^{3/2}}$|⁠, and we replace |$\sqrt{i}$| by |$\sqrt{\lambda }$|⁠. We obtain \begin{equation*} s\sum_{i = \lambda/2}^{\lambda-3\ell} \frac{\ell \sqrt{\lambda}}{\ell} \sqrt{\lambda} \left( \frac{\ell}{\lambda^{3/2} (\lambda - i)^{3/2}} + \frac{\ell \sqrt{\lambda -i}}{\lambda^{3/2} (\lambda-i)^2} + \frac{\sqrt{\ell} \ell}{\lambda^{3/2} (\lambda -i)^2} \right) = so_{\ell, \lambda}(1). \end{equation*} For the 2nd sum, we use (7.59), observe that near |$\lambda $| we have, in view of (3.9) and (2.13), \begin{equation*} \sup_{t \in [\lambda - 4 \ell, \lambda]} |\psi_s^{(\textsf{2})}(t)| \preceq s \frac{\ell}{\lambda^{3/2} \ell^{3/2}}, \end{equation*} and study \begin{equation*} s \sum_{i = \lambda - 3\ell}^{\lambda} \sum_{j = \lambda-\ell}^{\lambda} \frac{\sqrt{i}\sqrt{j}}{\ell} \frac{\ell}{\lambda^{3/2} \ell^{3/2}}. \end{equation*} We replace |$\sqrt{i} \sqrt{j}$| by |$\lambda $| and get |$s o_{\ell , \lambda }(1)$| by direct computation. This concludes the study of |$\textsf{Main}_s^{A}$|⁠. \begin{equation*} \star \star \star \star \star \end{equation*} The term |$\textsf{Main}_s^{B}$|⁠. We recall that \begin{equation*} \textsf{Main}_s^{B}(\eta) = \sum_{|i| = \lambda - \ell}^{\lambda - \ell/10} \sum_{|j| = \lambda - \ell}^{\lambda - \ell/10} \widetilde{D}_i \widetilde{D}_j \frac{|\textsf{F}|_{V(i,j)}}{\ell^2}. \end{equation*} Using symmetries, it is enough to study \begin{equation*} \sum_{i = \lambda - \ell}^{\lambda - \ell/10} \sum_{j = \lambda - \ell}^{\lambda - \ell/10} \widetilde{D}_i \widetilde{D}_j \frac{|\textsf{F}|_{V(i,j)}}{\ell^2} + \sum_{i = -\lambda} ^{-\lambda + \ell/10} \sum_{j = \lambda - \ell}^{\lambda - \ell/10} \widetilde{D}_i \widetilde{D}_j \frac{|\textsf{F}|_{V(i,j)}}{\ell^2}. \end{equation*} When taking the expectation, we use the discrepancy estimates and replace |$\widetilde{D}_i \widetilde{D}_j$| by |$\sqrt{i} \sqrt{j}$|⁠. Here, we replace further |$\sqrt{i} \sqrt{j}$| by |$\lambda $|⁠. For the 1st sum, we use (7.57) and observe that, near |$\lambda $|⁠, we have (in view of (3.8) and (2.12)) \begin{equation*} \sup_{t \in \lambda - 2\ell, \lambda} |\psi_s^{(\textsf{1})}(t)| \preceq s \frac{\ell}{\lambda^{3/2} \sqrt{\ell}}, \end{equation*} hence we obtain \begin{equation*} s \sum_{i = \lambda - \ell}^{\lambda - \ell/10} \sum_{j = \lambda - \ell}^{\lambda - \ell/10} \lambda \frac{|\textsf{F}|_{V(i,j)}}{\ell^2} \preceq \ell^2 \lambda \frac{\ell}{\ell^2 \lambda^{3/2} \sqrt{\ell}} = s o_{\ell, \lambda}(1). \end{equation*} For the 2nd sum, we use (7.56) and observe that, in view of (3.12), we have, for |$i$| near |$-\lambda $| and |$j$| near |$\lambda $|⁠, \begin{equation*} \frac{|\psi_s(i)| + |\psi_s(j)|}{(j-i)} \preceq s \frac{\ell \sqrt{\ell}}{\lambda^{3/2} \lambda}, \end{equation*} and we thus obtain \begin{equation*} s \sum_{i = -\lambda} ^{-\lambda + \ell/10} \sum_{j = \lambda - \ell}^{\lambda - \ell/10} \lambda \frac{\ell \sqrt{\ell}}{\lambda^{3/2} \ell^2 \lambda} = s o_{\ell, \lambda}(1). \end{equation*} This concludes the study of |$\textsf{Main}_s^{B}$|⁠. \begin{equation*} \star \star \star \star \star \end{equation*} The term |$\textsf{Main}_s^{C}$|⁠. We recall that \begin{equation*} \textsf{Main}_s^{C}(\eta) = \left( \ell + \left|\textrm{Discr}_{|x| \in [\lambda - \ell/8, \lambda]}\right| \right) \sum_{j=-\lambda}^\lambda \sup_{|x| \in [\lambda - \ell/8, \lambda]} |\partial_y \textsf{F}|_{V(x,j)} \widetilde{D}_j. \end{equation*} Taking the expectation, we use the discrepancy estimates and get \begin{equation*} {\mathbb{E}} \left[ \left( \ell + \left|\textrm{Discr}_{|x| \in [\lambda - \ell/8, \lambda]}\right| \right) \widetilde{D}_j \right] \preceq \ell \sqrt{j}, \end{equation*} so we study \begin{equation*} \sum_{j=-\lambda}^\lambda \ell \sqrt{j} \sup_{|x| \in [\lambda - \ell/8, \lambda]} |\partial_y \textsf{F}|_{V(x,j)}. \end{equation*} We split the sum into |$j \leq \lambda - 3\ell $| and |$j \geq \lambda - 3\ell $|⁠. For |$j \leq \lambda - 3\ell $|⁠, we use (7.58) (switching the roles of |$x$| and |$y$|⁠) and write, for |$x$| in |$[\lambda - \ell /8, \lambda ]$|⁠, \begin{equation*} |\partial_y \textsf{F}|_{V(x,j)} \preceq \frac{|\psi_s^{\prime}(j)|}{|j-x|} + \frac{|\psi_s(x)|}{(j-x)^2} + \frac{|\psi_s(j)|}{(j-x)^2}. \end{equation*} We replace |$j-x$| by |$\lambda -j$| and (in view of (3.12)) |$|\psi _s(x)|$| by |$s \frac{\ell \sqrt{\ell }}{\lambda ^{3/2}}$|⁠, and we obtain \begin{align*} \sup_{|x| \in [\lambda - \ell/8, \lambda]} |\partial_y \textsf{F}|_{V(x,j)} \preceq \frac{|\psi_s^{\prime}(j)|}{\lambda - j} + \frac{s \ell \sqrt{\ell}}{\lambda^{3/2} (\lambda-j)^2} + \frac{|\psi_s(j)|}{(\lambda-j)^2}; \end{align*} we are thus left to study \begin{equation*} \sum_{j=-\lambda}^{\lambda - 3\ell} \ell \sqrt{j} \left( \frac{|\psi_s^{\prime}(j)|}{\lambda - j} + \frac{s \ell \sqrt{\ell}}{\lambda^{3/2} (\lambda-j)^2} + \frac{|\psi_s(j)|}{(\lambda-j)^2} \right). \end{equation*} This is actually much smaller than (7.71), which was already treated. For |$j \geq \lambda - 3\ell $|⁠, we use (7.59) and write, for |$x$| in |$[\lambda - \ell /8, \lambda ]$|⁠, \begin{equation*} |\partial_y \textsf{F}|_{V(x,j)} \preceq \sup_{t \in [\lambda - 4\ell, \lambda]} |\psi_s^{(\textsf{2})}(t)| \preceq s \frac{\ell}{\lambda^{3/2} \ell^{3/2}}, \end{equation*} and a direct computation gives \begin{equation*} s \sum_{j=\lambda-3\ell}^\lambda \ell \sqrt{j} \sup_{|x| \in [\lambda - \ell/8, \lambda]} |\partial_y \textsf{F}|_{V(x,j)} \preceq s \ell^2 \sqrt{\lambda} \frac{\ell}{\lambda^{3/2} \ell^{3/2}} = s o_{\ell, \lambda}(1). \end{equation*} This concludes the study of |$\textsf{Main}_s^{C}$|⁠. \begin{equation*} \star \star \star \star \star \end{equation*} The term |$\textsf{Main}_s^{D}$|. We recall that \begin{equation*} \textsf{Main}_s^{D}(\eta) = \left( \ell + \left|\textrm{Discr}_{|x| \in [\lambda - \ell/8, \lambda]}\right| \right) \sum_{|j| = \lambda - \ell}^{\lambda - \ell/10} \frac{\sup_{|x| \in [\lambda - \ell/8, \lambda]} |\textsf{F}|_{V(x,j)}}{\ell} \widetilde{D}_j. \end{equation*} For the same reasons as above, we are led to study \begin{equation*} \sum_{|j| = \lambda - \ell}^{\lambda - \ell/10} \frac{\sup_{|x| \in [\lambda - \ell/8, \lambda]} |\textsf{F}|_{V(x,j)}}{\ell} \ell \sqrt{\lambda}, \end{equation*} and we split the sum in two parts: |$j$| near |$-\lambda $| and |$j$| near |$\lambda $|⁠. For the 1st part, we use (7.56), and for the 2nd part we use (7.57) to control |$|\textsf{F}|_{V(x,j)}$|⁠. After some computation, we obtain |$s o_{\ell , \lambda }(1)$|⁠. This concludes the study of |$\textsf{Main}_s^{D}$| and the proof of the proposition. 7.12 Proof of Corollary 5.4 Proof of Corollary 5.4 We can split |$\Lambda ^c$| into |$\{ x \geq \lambda \}$| and |$\{ x \leq - \lambda \}$|⁠, both parts yield an equivalent contribution, so we only consider the 1st one. We need an adaptation of the a priori bound (1.10) to a slightly different context. Claim 7.9 (A priori bound—“hard edge” and decay assumption). Let |$g$| be a |$C^1$| function such that \begin{equation} \limsup_{x \to \infty} |x g(x)| < + \infty, \quad \limsup_{x \to \infty} x^2 |g^{\prime}(x)| < + \infty, \end{equation} (7.72) then, |$\textrm{Sine}_{\beta }$|-a.s. both sides of the following inequality are finite, and the inequality holds \begin{equation*} \int_{\lambda}^{+\infty} g(x) (\textrm{d}{\mathcal{C}} - \textrm{d}x) \preceq \sum_{j=\lambda}^{+\infty} |g|_{\textsf{1}, V_{j}} \widetilde{D}^{\textrm{Right}}_j + g(\lambda) |\textrm{Discr}_{[\lambda, \lambda+1]}|. \end{equation*} Proof of Claim 7.9 We follow the same lines as for the proof of Proposition 1.6. We split the domain of integration into unit intervals and use the mean value theorem, in order to get, for |$M> \lambda $| fixed, \begin{align*} \int_{\lambda}^M g(x) (\textrm{d}{\mathcal{C}} - \textrm{d}x) &= \sum_{k=\lambda}^{M-1} \int_{k}^{k+1} g(x) (\textrm{d}{\mathcal{C}} - \textrm{d}x)\\& \preceq \sum_{k = \lambda}^{M-1} g(k) \textrm{Discr}_{[k, k+1]} + O_{\bullet}\left(|g|_{\textsf{1},V_{k}}\right)\left(1 + |\textrm{Discr}_{[k, k+1]}|\right). \end{align*} We write, for any |$k$|⁠, |$\textrm{Discr}_{[k, k+1]} = \textrm{Discr}_{[\lambda , k+1]} - \textrm{Discr} _{[\lambda , k]}$| and perform a summation by parts to get \begin{equation*} \sum_{k = \lambda}^{M-1} g(k) \textrm{Discr}_{[k, k+1]} = \sum_{k = \lambda+1}^{M-1} \left( g(k-1) - g(k) \right) \textrm{Discr}_{[\lambda, k]} + g(M-1) \textrm{Discr}_{[\lambda, M]} + g(\lambda) \textrm{Discr}_{[\lambda, \lambda+1]}. \end{equation*} In view of (7.72), the boundary term |$g(M-1) \textrm{Discr}_{[\lambda , M]}$| tends almost surely to |$0$| as |$M \to \infty $| because |$\frac{1}{M} \textrm{Discr}_{[\lambda , M]}$| tends almost surely to |$0$|⁠. On the other hand, the series \begin{equation*} \sum_{k = \lambda}^{+\infty} |g|_{\textsf{1},V_{k}} \left(1 + |\textrm{Discr}_{[k, k+1]}|\right) \end{equation*} is almost surely convergent because we have, in view of (1.11) and (7.72), \begin{equation*} \limsup_{k \to \infty} {\mathbb{E}} \left[ \left( k^{2} |g|_{\textsf{1},V_{k}} \left(1 + |\textrm{Discr}_{[k, k+1]}| \right)\right)^2 \right] < + \infty. \end{equation*} Sending |$M \to \infty $| yields the result. We can easily check that |$\textrm{Error}\textsf{DF}_s$| satisfies the decay assumption (7.72). Using Claim 7.9, we get \begin{equation*} \int_{\lambda}^{+\infty} \textrm{Error}\textsf{DF}_s({\mathcal{C}})(x) (\textrm{d}{\mathcal{C}} - \textrm{d}x) \preceq s \sum_{j = \lambda}^{+\infty} |\textrm{Error}\textsf{DF}_s|_{\textsf{1}, V_{j}} \widetilde{D}^{\textrm{Right}}_j + |\textrm{Error}\textsf{DF}_s(\lambda)| \textrm{Discr}_{[\lambda, \lambda+1]}. \end{equation*} The boundary term. We claim that \begin{equation} {\mathbb{E}} \left[ \left|\textrm{Error}\textsf{DF}_s(\lambda) \textrm{Discr}_{[\lambda, \lambda+1]}\right| \right] \preceq s \frac{\ell \log(\lambda)}{\lambda^{3/2}} = s o_{\ell, \lambda}(1). \end{equation} (7.73) Indeed, using (5.5) and the discrepancy estimates (1.11) for |$\widetilde{D}^{\textrm{Right}}_i$|⁠, we obtain \begin{multline*} {\mathbb{E}} \left[ |\textrm{Error}\textsf{DF}_s(\lambda)| \textrm{Discr}_{[\lambda, \lambda+1]} \right] \preceq \sum_{i = - \lambda}^{\lambda - \ell} \left(\frac{|\psi_s(i)|}{(x-i)^2} + \frac{|\psi_s^{\prime}(i)|}{(x-i)} \right) {\mathbb{E}}\left[\widetilde{D}^{\textrm{Right}}_i \textrm{Discr}_{[\lambda, \lambda+1]} \right] \\ \preceq \sum_{i = - \lambda}^{\lambda - \ell} \left(\frac{|\psi_s(i)|}{(x-i)^2} + \frac{|\psi_s^{\prime}(i)|}{(x-i)} \right) \sqrt{\lambda - i}. \end{multline*} We use (3.8) and (2.12) to control the contribution of the |$\psi _s^{\prime}(i)$| terms and (3.12) to control the contribution of the |$\psi _s(i)$| terms. For example, we have \begin{equation*} \sum_{i = \lambda/2}^{\lambda - \ell} \frac{|\psi_s^{\prime}(i)|}{(x-i)} \sqrt{\lambda - i} \preceq s \sum_{i = \lambda/2}^{\lambda - \ell} \frac{ \ell}{\lambda^{3/2}} \frac{\sqrt{\lambda - i}} {(\lambda - i)^2} \sqrt{\lambda - i} \preceq s \frac{\ell \log(\lambda)}{\lambda^{3/2}}. \end{equation*} The main contribution. We now claim that \begin{equation} {\mathbb{E}} \left[ \sum_{j = \lambda}^{+\infty} |\textrm{Error}\textsf{DF}_s|_{\textsf{1}, V_{j}} \widetilde{D}^{\textrm{Right}}_j \right] = s o_{\ell, \lambda}(1). \end{equation} (7.74) To prove (7.74), we use (5.6) and write \begin{align*} \sum_{j = \lambda}^{+\infty} |\textrm{Error}\textsf{DF}_s|_{\textsf{1}, V_{j}} \widetilde{D}^{\textrm{Right}}_j \preceq \sum_{j = \lambda}^{+\infty} \sum_{i = - \lambda}^{\lambda - \ell} \left(\frac{|\psi_s(i)|}{(j-i)^3} + \frac{|\psi_s^{\prime}(i)|}{(j-i)^2} \right) \widetilde{D}^{\textrm{Right}}_i \widetilde{D}^{\textrm{Right}}_j. \end{align*} Taking the expectation and using Cauchy–Schwarz’s inequality, we get \begin{multline} {\mathbb{E}}\left[ \sum_{j = \lambda}^{+\infty} |\textrm{Error}\textsf{DF}_s|_{\textsf{1}, V_{j}} \widetilde{D}^{\textrm{Right}}_j \right] \\ \preceq \sum_{j = \lambda}^{+\infty} \sum_{i = - \lambda}^{\lambda - \ell} \left(\frac{|\psi_s(i)|}{(j-i)^3} + \frac{|\psi_s^{\prime}(i)|}{(j-i)^2} \right) {\mathbb{E}}\left[\left(\widetilde{D}^{\textrm{Right}}_i\right)^2\right]^{1/2} {\mathbb{E}}\left[\left(\widetilde{D}^{\textrm{Right}}_j\right)^2\right]^{1/2} \end{multline} (7.75) Using the discrepancy estimate (1.11) we obtain \begin{equation*} {\mathbb{E}}\left[ \sum_{j = \lambda}^{+\infty} |\textrm{Error}\textsf{DF}_s|_{\textsf{1}, V_{j}} \widetilde{D}^{\textrm{Right}}_j \right] \preceq \sum_{j = \lambda}^{+\infty} \sum_{i = - \lambda}^{\lambda-\ell} \left(\frac{|\psi_s(i)|}{(j-i)^3} + \frac{|\psi_s^{\prime}(i)|}{(j-i)^2} \right) \sqrt{|\lambda - i|}\sqrt{|\lambda - j|}. \end{equation*} Let us keep in mind that, as in the proof of Proposition 4.1, we may use the sharper discrepancy estimates (1.12) instead of (1.11) and take advantage of the fact that \begin{equation*} {\mathbb{E}}\left[\left(\widetilde{D}^{\textrm{Right}}_j\right)^2\right] = o_{|j - \lambda| \to \infty}\left(j-\lambda\right). \end{equation*} The terms |$i$| far from |$\lambda $| We first treat the case |$-\lambda \leq i \leq \lambda /2$|⁠. For any |$j \geq \lambda $|⁠, using the estimates (3.12) on |$\psi _s$|⁠, we may write \begin{equation*} \sum_{i = - \lambda}^{\lambda /2} \frac{|\psi_s(i)|}{(j-i)^3} \sqrt{\lambda - i} \preceq s \frac{\sqrt{\lambda}}{j^3} \sum_{i = - \lambda}^{\lambda /2} |\psi_s(i)| \preceq s \frac{ \sqrt{\lambda} \ell \log(\lambda)}{j^3}. \end{equation*} We thus get \begin{equation} \sum_{j = \lambda}^{+\infty} \sum_{i = - \lambda}^{\lambda /2} \frac{|\psi_s(i)|}{(j-i)^3} \sqrt{\lambda - i} \sqrt{j - \lambda} \preceq s \sum_{j = \lambda}^{+\infty} \frac{ \sqrt{\lambda} \ell \log(\lambda)}{j^{5/2}} \preceq s \frac{ \ell \log(\lambda)}{ \lambda} = s o_{\ell, \lambda}(1). \end{equation} (7.76) For any |$j \geq \lambda $|⁠, using (3.8) and (2.12), we write \begin{align*} \sum_{i = - \lambda}^{\lambda /2} \frac{|\psi_s^{\prime}(i)|}{(j-i)^2} \sqrt{\lambda - i} \preceq \frac{\sqrt{\lambda}}{j^2} \sum_{i = - \lambda}^{\lambda /2} |\psi_s^{\prime}(i)| \preceq s \frac{\sqrt{\lambda}}{j^2}. \end{align*} We thus get \begin{equation*} \sum_{j = \lambda}^{+\infty} \sum_{i = - \lambda}^{\lambda /2} \frac{|\psi_s^{\prime}(i)|}{(j-i)^2} \sqrt{\lambda - i} \sqrt{j - \lambda} \preceq s \sum_{j = \lambda}^{+\infty} \frac{ \sqrt{\lambda} \sqrt{j - \lambda}}{j^2}. \end{equation*} A rough bound would only yield a |$O(1)$| contribution here. Instead, we split the sum into \begin{equation*} \sum_{j = \lambda}^{\lambda + \log(\lambda)} \frac{ \sqrt{\lambda} \sqrt{j - \lambda}}{j^2} \preceq \frac{1}{\lambda^{3/2}} \sum_{j = 0}^{\log(\lambda)} \sqrt{k} = o_{\ell, \lambda}(1) \end{equation*} and the remainder where |$j- \lambda \geq \log (\lambda )$|⁠, in which we use (1.12) instead of (1.11), which allows us to replace |$\sqrt{j - \lambda }$| by |$o_{\lambda }(\sqrt{j - \lambda })$|⁠, and \begin{equation*} \sum_{j = \lambda + \log(\lambda)}^{+\infty} \frac{ \sqrt{\lambda} \times o_{\lambda} \left(\sqrt{j - \lambda}\right)}{j^2} = o_{\ell, \lambda}(1). \end{equation*} Hence \begin{equation} \sum_{j = \lambda}^{+\infty} \sum_{i = - \lambda}^{\lambda /2} \frac{|\psi_s^{\prime}(i)|}{(j-i)^2} \sqrt{\lambda - i} \times o\left( \sqrt{j - \lambda} \right) = s o_{\ell, \lambda}(1). \end{equation} (7.77) Combining (7.76) and (7.77), we see that the contribution in (7.75) coming from the terms “|$i$| far from |$\lambda $|”, that is, here |$-\lambda \leq i \leq \lambda /2$| is |$s o_{\ell , \lambda }(1)$|⁠. The terms |$i$| close to |$\lambda $| We now consider |$\lambda /2 \leq i \leq \lambda - \ell $|⁠. For any |$j \geq \lambda $|⁠, using the estimates (3.12) on |$\psi _s$|⁠, we may write \begin{equation*} \sum_{i = \lambda/2}^{\lambda - \ell} \frac{|\psi_s(i)|}{(j-i)^3} \sqrt{\lambda - i} \preceq s \sum_{i = \lambda/2}^{\lambda - \ell} \frac{\ell}{\lambda^{3/2}} \frac{\lambda -i}{(j-\lambda + \lambda-i)^3}. \end{equation*} We distinguish the cases |$\lambda - i \leq j - \lambda $| and |$\lambda -i \geq j - \lambda $|⁠. (a) We have (the sum being non-empty only if |$j - \lambda \geq \ell $|⁠) \begin{align*}& \sum_{i \in [\lambda/2, \lambda- \ell] | \lambda -i \leq j - \lambda} \frac{\ell}{\lambda^{3/2}} \frac{\lambda -i}{(j-\lambda + \lambda-i)^3}\\&\qquad\qquad\qquad \preceq \frac{\ell}{\lambda^{3/2}} \frac{1}{(j-\lambda)^2} \sum_{i \in [\lambda/2, \lambda- \ell] | \lambda -i \leq j - \lambda} 1 \preceq \frac{\ell}{\lambda^{1/2}} \frac{1}{(j-\lambda)^2}. \end{align*} (b) On the other hand (the sum being non-empty only if |$j - \lambda \leq \lambda /2$|⁠), \begin{align*}& \sum_{i \in [\lambda/2, \lambda - \ell] | \lambda -i> j - \lambda} \frac{\ell}{\lambda^{3/2}} \frac{\lambda -i}{(j-\lambda + \lambda-i)^3} \\&\qquad\qquad\qquad\preceq \frac{\ell}{\lambda^{3/2}} \sum_{i \in [\lambda/2, \lambda - \ell] | \lambda -i > j - \lambda} \frac{1}{(\lambda-i)^2} \preceq \frac{\ell}{\lambda^{3/2}} \frac{1}{j-\lambda}. \end{align*} We may thus write \begin{align*} \sum_{j = \lambda}^{+ \infty} \sum_{i = \lambda/2}^{\lambda - \ell} \frac{|\psi_s(i)|}{(j-i)^3} \sqrt{\lambda - i} \sqrt{j - \lambda} \preceq&\ s \sum_{j = \lambda + \ell}^{+ \infty} \frac{\ell}{\lambda^{1/2}} \frac{1}{(j-\lambda)^{3/2}} \\&+ s \sum_{j= \lambda}^{3 \lambda /2} \frac{\ell}{\lambda^{3/2}} \frac{1}{\sqrt{j-\lambda}} \preceq s \frac{\ell}{\lambda^{1/2} \ell^{1/2}} + s \frac{\ell}{\lambda}. \end{align*} We obtain \begin{equation} \sum_{j = \lambda}^{+\infty} \sum_{i = \lambda/2}^{\lambda - \ell} \frac{|\psi_s(i)|}{(j-i)^3} \sqrt{\lambda - i} \sqrt{j - \lambda} = s o_{\ell, \lambda}(1). \end{equation} (7.78) 2 Concerning the terms |$\psi _s^{\prime}(i)$|⁠, we use (3.7) and (2.12) and write, for any |$j \geq \lambda $|⁠, \begin{equation*} \sum_{i = \lambda/2}^{\lambda - \ell} \frac{|\psi_s^{\prime}(i)|}{(j-i)^2} \sqrt{\lambda - i} \preceq s \sum_{i = \lambda/2}^{\lambda - \ell} \frac{\ell}{\lambda^{3/2} \sqrt{\lambda - i} (j-i)^2} \sqrt{\lambda - i} \preceq s \frac{\ell}{\lambda^{1/2}} \frac{1}{(j-\lambda + \ell)^2}. \end{equation*} We thus have \begin{equation*} \sum_{\lambda}^{+ \infty} \sum_{i = \lambda/2}^{\lambda - \ell} \frac{|\psi_s^{\prime}(i)|}{(j-i)^2} \sqrt{\lambda - i} \sqrt{j - \lambda} \preceq s\sum_{\lambda}^{+ \infty}\frac{\ell}{\lambda^{1/2}} \frac{\sqrt{j - \lambda}}{(j-\lambda + \ell)^2} \preceq s\frac{\ell}{\lambda^{1/2} \ell^{1/2}}, \end{equation*} and we obtain \begin{equation} \sum_{j = \lambda}^{+\infty} \sum_{i = \lambda/2}^{\lambda - \ell} \frac{|\psi_s^{\prime}(i)|}{(j-i)^2} \sqrt{\lambda - i} \sqrt{j - \lambda} = s o_{\ell, \lambda}(1). \end{equation} (7.79) Combining (7.78) and (7.79), we see that the contribution in (7.75) coming from the terms “|$i$| close to |$\lambda $|”, that is, here |$\lambda /2 \leq i \leq \lambda - \ell $|⁠, is |$o_{\ell , \lambda }(1)$|⁠. This concludes the proof of (7.75), which, combined with (7.73), yields (5.10). 7.13 Proof of Lemma 5.5 Proof of Lemma 5.5 For |$3\lambda /4 \leq |x| \leq 4 \lambda $|⁠. For simplicity, we consider |$\textsf{LP}_{\lambda , \varphi }(\lambda )$|⁠; the proof extends readily to |$\textsf{LP}_{\lambda , \varphi }(x)$| for |$3\lambda /4 \leq |x| \leq 4 \lambda $|⁠. We have, by definition, \begin{equation*} \textsf{LP}_{\lambda, \varphi}(\lambda) = \int - \log(\lambda -y) {\mathfrak{m}}_{\lambda, \varphi}(y) \ \textrm{d}y. \end{equation*} The |$|y| \leq \frac{1}{2} \lambda $| part. We want to show \begin{equation} \int_{-\lambda/2}^{\lambda/2} \log(\lambda - y) {\mathfrak{m}}_{\lambda, \varphi}(y) \ \textrm{d}y \preceq \frac{ \ell \log^2(\lambda)}{\lambda}. \end{equation} (7.80) We recall that, by definition, \begin{equation*} {\mathfrak{m}}_{\lambda, \varphi}(y) = \frac{-1}{\pi} \frac{1}{\sqrt{\lambda^2-y^2}} {\mathfrak{H}}_{\lambda, \varphi}(y) = \frac{-1}{\pi} \frac{1}{\sqrt{\lambda^2-y^2}} \textbf{PV} \int \frac{\varphi^{\prime}(t) \sqrt{\lambda^2-t^2}}{y-t} \ \textrm{d}t. \end{equation*} For |$|y| \leq \frac{1}{2} \lambda $|⁠, we write \begin{align*} \log |\lambda - y| \frac{1}{\sqrt{\lambda^2-y^2}} &= \left( \log(\lambda) + O_{\bullet}\left( \frac{|y|}{\lambda} \right) \right) \left( \frac{1}{\lambda} + O_{\bullet}\left(\frac{|y|^2}{\lambda^3} \right) \right) \\ &= \frac{\log(\lambda)}{\lambda} + O_{\bullet} \left( \frac{|y|}{\lambda^2} + \frac{|y|^2 \log(\lambda)}{\lambda^3} \right) = \frac{\log(\lambda)}{\lambda} + O_{\bullet} \left( \frac{\log(\lambda) |y|}{\lambda^2}\right), \end{align*} and thus \begin{equation} \log |\lambda - y| \frac{1}{\sqrt{\lambda^2-y^2}} {\mathfrak{H}}_{\lambda, \varphi}(y) = \frac{\log(\lambda)}{\lambda} {\mathfrak{H}}_{\lambda, \varphi}(y) + O_{\bullet} \left( \frac{\log (\lambda) |y|}{\lambda^2}\right) |{\mathfrak{H}}_{\lambda, \varphi}(y)|. \end{equation} (7.81) Using (7.18), we get \begin{equation} \int_{-\lambda/2}^{\lambda/2} \frac{\log(\lambda)}{\lambda} {\mathfrak{H}}_{\lambda, \varphi}(y) \preceq \frac{\ell \log (\lambda)}{\lambda}. \end{equation} (7.82) Using the bounds (2.7), we can check that \begin{align*} \int_{-\lambda/2}^{\lambda/2} |y| |{\mathfrak{H}}_{\lambda, \varphi}(y)| \ \textrm{d}y \preceq \ell \lambda \log(\lambda), \end{align*} and thus \begin{equation} \int_{-\lambda/2}^{\lambda/2} \frac{\log (\lambda)}{\lambda^2} |y| |{\mathfrak{H}}_{\lambda, \varphi}(y)| \ \textrm{d}y \preceq \frac{\ell \log^2(\lambda)}{\lambda}. \end{equation} (7.83) We obtain (7.80). The |$|y| \geq \lambda / 2$| part. We want to show \begin{equation} \int_{|y| \geq \lambda / 2} \log |\lambda -y| {\mathfrak{m}}_{\lambda, \varphi}(y) \ \textrm{d}y \preceq \frac{\ell \log(\lambda)}{\lambda}. \end{equation} (7.84) We use (2.7) and an elementary computation. The mass of |${\mathfrak{m}}_{\lambda , \varphi }$| outside |$[-\lambda /2, \lambda /2]$| is indeed |$O_{\bullet }\left (\frac{\ell }{\lambda }\right )$|⁠. Combining (7.80) and (7.84), we obtain (5.11). For |$\lambda \leq |x| \leq 4 \lambda $|⁠. We can, for example, assume that |$\lambda \leq x \leq 4\lambda $|⁠. Write |$\textsf{LP}_{\lambda , \varphi }^{\prime}(x)$| as \begin{equation*} \textsf{LP}_{\lambda, \varphi}^{\prime}(x) = \int \frac{1}{x-t} {\mathfrak{m}}_{\lambda, \varphi}(t) \ \textrm{d}t, \end{equation*} and we use (2.7). We have \begin{equation*} \int_{-\lambda}^{\lambda^2} \frac{1}{x-t} |{\mathfrak{m}}_{\lambda, \varphi}(t)| \leq \frac{1}{\lambda}, \end{equation*} and we focus on the remaining part |$t \in [\lambda /2, \lambda ]$|⁠. We write \begin{equation*} \int_{\lambda^2}{\lambda} \frac{1}{x-t} |{\mathfrak{m}}_{\lambda, \varphi}(t)| \ \textrm{d}t \preceq \int_{\lambda^2}^{\lambda} \frac{1}{x-\lambda + \lambda-t} \frac{\ell}{\lambda^{3/2} \sqrt{\lambda - t}} \ \textrm{d}t = \frac{\ell}{\lambda^{3/2}} \int_{0}^{\lambda/2} \frac{1}{(x-\lambda + v) \sqrt{v}} \ \textrm{d}v. \end{equation*} An elementary computation shows that \begin{equation*} \int_{0}^{\lambda/2} \frac{1}{(x-\lambda + v) \sqrt{v}} \preceq \frac{1}{\sqrt{x-\lambda}}, \end{equation*} which yields (5.12). For |$|x| \geq 4 \lambda $|⁠. We write |$\textsf{LP}_{\lambda , \varphi }^{\prime}(x)$| as \begin{equation*} \textsf{LP}_{\lambda, \varphi}^{\prime}(x) = \int \frac{1}{x-t} {\mathfrak{m}}_{\lambda, \varphi}(t) \ \textrm{d}t, \end{equation*} and, since |${\mathfrak{m}}_{\lambda , \varphi }$| has total mass |$0$|⁠, a 1st-order expansion yields \begin{equation*} \textsf{LP}_{\lambda, \varphi}^{\prime}(x) \preceq \frac{1}{x^2} \int |t| |{\mathfrak{m}}_{\lambda, \varphi}(t)|. \end{equation*} We can use (2.7) to compute |$\int |t| |{\mathfrak{m}}_{\lambda , \varphi }(t)| \textrm{d}t \preceq \ell \log (\lambda ),$| which yields (5.13). Communicated by Prof. Kurt Johansson References [1] Bekerman , F. , A. Figalli , and A. Guionnet . “ Transport maps for |$\beta $|-matrix models and universality .” Comm. Math. Phys. 338 , no. 2 ( 2015 ): 589 – 619 . Google Scholar Crossref Search ADS [2] Bekerman , F. and A. Lodhia . “ Mesoscopic central limit theorem for general |$\beta $|-ensembles .” Ann. Inst. Henri Poincaré Probab. 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Ganguly,, Shirshendu;Srivastava,, Nikhil

doi: 10.1093/imrn/rnz008pmid: N/A

Abstract We prove improved bounds on how localized an eigenvector of a high girth regular graph can be, and present examples showing that these bounds are close to sharp. This study was initiated by Brooks and Lindenstrauss [6] who relied on the observation that certain suitably normalized averaging operators o nhigh girth graphs are hyper-contractive and can be used to approximate projectors onto the eigenspaces of such graphs. Informally, their delocalization result in the contrapositive states that for any |$\varepsilon \in (0,1)$| and positive integer |$k,$| if a |$(d+1)-$|regular graph has an eigenvector that supports |$\varepsilon $| fraction of the |$\ell _2^2$| mass on a subset of |$k$| vertices, then the graph must have a cycle of size |$\log _{d}(k)/\varepsilon ^2)$|⁠, up to multiplicative universal constants and additive logarithmic terms in |$1/\varepsilon $|⁠. In this paper, we improve the upper bound to |$\log _{d}(k)/\varepsilon $| up to similar logarithmic correction terms; and present a construction showing a lower bound of |$\log _d(k)/\varepsilon $| up to multiplicative constants. Our construction is probabilistic and involves gluing together a pair of trees while maintaining high girth as well as control on the eigenvectors and could be of independent interest. 1 Introduction Spectral graph theory studies graphs via associated linear operators such as the Laplacian and the adjacency matrix. While the extreme eigenvectors of these operators are relatively well understood and correspond to sparse cuts and colorings, much less is known about the combinatorial meaning of the interior eigenvectors. Most of the literature about them falls into two categories: 1. Analysis of eigenvectors of random graphs. For example, Dekel et al. [7] prove that any eigenvector of a dense random graph has a bounded number of nodal domains, that is, connected components where the eigenvector does not change sign. Following a sequence of results by various authors, in a recent breakthrough work Bauerschmidt et al. [4], among various other things, show that with high probability, any “bulk” eigenvector |$v$| of a random regular graph with |$n$| vertices and a large enough but fixed constant degree, is |$\ell _{\infty }$| delocalized in the following sense: \begin{equation*} ||v||_\infty\le \frac{\log^C(n)}{\sqrt{n}}||v||_2, \end{equation*} where |$||\cdot ||_2$|⁠, and |$||\cdot ||_\infty $| denote the usual |$\ell _2$| and |$\ell _{\infty }$| norms, respectively, and |$C$| is a constant. For a more precise statement see Theorem 1.2 in [4]. In another line of work, Backhausz and Szegedy [3] establish Gaussian behavior of the entry distribution of eigenvectors of random regular graphs by studying factors of i.i.d. processes on the regular infinite tree. In all of these works the randomness of the model is used heavily, and weaker notions of delocalization are also considered (see, e.g., [9]). We refer the reader to [15] for a survey of recent developments on delocalization of eigenvectors of random matrices. 2. A parallel story based on asymptotic analysis of sequences of deterministic graphs. The driving force for this is the so called Quantum Unique Ergodicity (QUE) conjecture by Rudnick and Sarnak [16]. The QUE conjecture states that on any compact negatively curved manifold all high energy eigenfunctions of the Laplacian equi-distribute. The conjecture is still widely open having been verified in only a few cases; perhaps most notably for the Hecke orthonormal basis on an arithmetic surface by Lindenstrauss [12]. Brooks–Lindenstrauss [6] initiated the study of graph-theoretic analogues of this conjecture. The analogue of negatively curved manifolds are high girth regular graphs—the girth is defined as the length of the shortest cycle in a graph. Subsequently, Anantharaman and Le–Masson [2] proved an asymptotic version of quantum ergodicity for regular expanders, which converge (in the Benjamini–Schramm local topology) to the infinite |$d-$|regular tree. The starting point of this paper is the beautiful result of [6]. Since the statement is a bit technical and could be hard to parse at first read we first explain the content informally in words. The theorem roughly says that if a graph does not have many short cycles, then eigenvectors cannot localize on small sets: for any eigenvector, any subset of the vertices representing a fraction of the |$\ell _2^2$| mass must have size |$n^\delta $| for some |$\delta $| depending on the fraction. The condition of not having many cycles is articulated as hyper-contractivity (i.e., control of |$\|\cdot \|_{p\rightarrow q}$| norms for some |$p<q$|⁠, a more formal definition is added at the beginning of Section 2) of certain spherical mean operators on the graph, where |$\|\cdot \|_{p\rightarrow q}$|denotes the norm of the naturally associated operator from |$\ell _p$| to |$\ell _{q}.$| Theorem 1.1 ([6]). Suppose |$G=(V,E)$| is a |$(d+1)-$|regular graph with adjacency matrix |$A$|⁠. Let \begin{equation*} S_n(\,f)(x):=\frac1{d^{n/2}}\sum_{{\textsf{dist}}(x,y)=n} f(y) \end{equation*} and suppose \begin{equation}\|S_n\|_{p\rightarrow q}\le Cd^{-\alpha n}\end{equation} (1) for all |$n\le N$|⁠, for some |$1\le p\le q\le \infty $| and |$\alpha \in [0,1]$|⁠. Then for any normalized |$\ell _2$| eigenvector |$v=(v_x)_{x\in V},$| of |$A$| and |$S\subset V$| with |$\|v_S\|_2^2:=\sum _{x\in S}v^2_x\ge \varepsilon ,$| \begin{equation*} |S|\ge \Omega_{d,C,\alpha}\Big(\varepsilon^{\frac{2p}{2-p}}d^{\delta N}\Big), \end{equation*} where |$\delta =2^{-7}\frac{\alpha p}{2-p}\varepsilon ^2$| and the constant in the |$\Omega $| notation depends on the parameters |$d,C,\alpha $|⁠. In particular, the condition (1) is satisfied with |$p=1,q=\infty ,\alpha =1/2,C=d$| and |$N=\lceil g/2\rceil -1$| for a graph of girth |$g$|⁠. Viewed in the contrapositive, the theorem therefore says that the existence of an eigenvector of |$A$| with |$\varepsilon $| fraction of its mass on |$k=|S|$| coordinates implies that the graph must contain a cycle of length |$O(\log _d(k/\varepsilon )/\varepsilon ^2)$|⁠. In fact, a close examination of the proof reveals that it gives an upper bound, which varies between |$O(\log _d(k/\varepsilon )/\varepsilon )$| and |$O(\log _d(k/\varepsilon )/\varepsilon ^2)$| depending on the Diophantine properties of the eigenvalue being considered. In this paper, we contribute to the understanding of this phenomenon in two ways. First, we improve the above bound to |$O(\log _d(k/\varepsilon )/\varepsilon )$| for all eigenvalues of |$d+1$|-regular(we work with |$(d+1)$|-regular rather than |$d-$|regular graphs to avoid repeatedly writing |$d-1$|⁠) graphs, irrespective of the number theoretic properties of the eigenvalue. The proof involves replacing the approximation-theoretic component of their proof by a simpler and more efficient method. Specifically, we prove the following theorem in Section 2. Theorem 1.2. Suppose |$G$| is a |$(d+1)$|-regular graph of girth |$g$| and |$v$| is a normalized eigenvector of the adjacency matrix of |$G$|⁠. Then any subset |$S$| with |$\|v_S\|_2^2\ge \varepsilon $| must have \begin{equation*} |S|\ge \frac{d^{\varepsilon g/4}\varepsilon}{2d^2}. \end{equation*} The contrapositive of the above theorem implies that if there exists |$\varepsilon $| and |$k$| and |$S$| such that |$|S|=k$| and |$\|v_S\|_2^2=\varepsilon ,$| then \begin{equation*} g\le \frac{4\log_d(k/\varepsilon)+O(1)}{\varepsilon}. \end{equation*} Before proceeding further some remarks are in order. Remark 1.3 (Choice of hypercontractive norms). The paper [6] works with general |$p\rightarrow q$| norms, but in this paper we will work solely with the |$1\rightarrow \infty $| since it is reveals all of the ideas and is easier to interpret combinatorially. Our proof of Theorem 1.2 can easily be modified to work with |$p\rightarrow q$| norms, if desired. Remark 1.4 (Entropy bounds from delocalization). As already observed in [5, Corollary 1], it is quite straightforward to obtain a lower bound on the entropy of an eigenvector |$v$| from a delocalization result such as Theorem 1.2, where the entropy of |$v$| is |$-\sum _{x\in V}v_x^2\\\log _{d}v_x^2.$| Remark 1.5 (Tempered and untempered eigenvalues). Eigenvalues of |$A$| in the interval |$[-2\sqrt{d},2\sqrt{d}]$| are referred to as tempered (indicating wave-like behavior) and those outside are called untempered (indicating exponential growth) in the QUE literature. It is known that for all eigenvalues that are strictly untempered, that is, bounded away from |$\{-2\sqrt{d},2\sqrt{d}\}$|⁠, a much stronger delocalization result, with dependence roughly |$g=O(\log _d(k/\varepsilon ))$|⁠, can be proven using elementary arguments—see, for example, [5, page 59] or the arguments of [11]. Note that any sequence of graphs with girth going to infinity must have a vanishingly small fraction of untempered eigenvalues. We will present bounds for arbitrary eigenvalues in this paper, without focusing on the distinction between tempered and untempered. A slightly more elaborate discussion is presented in Remark 2.3. Moreover, for every |$d\ge 2$|⁠, sufficiently large |$k$|⁠, and |$\varepsilon \in (0,1)$|⁠, we exhibit a |$(d+1)-$|regular graph with a localized eigenvector that has girth at least |$\Omega (\log _d(k)/\varepsilon )$|⁠, showing that our improved bound is sharp up to an additive |$\log (1/\varepsilon )$| factor in the numerator, which is negligible whenever |$k=\Omega (1/\varepsilon ^c)$| for any |$c$|⁠. We are able to construct such eigenvectors for a dense subset of eigenvalues in |$(-2\sqrt{d},2\sqrt{d})$|⁠. The proof is probabilistic, and involves gluing together two trees without introducing any short cycles and while controlling their eigenvectors, which may be of independent interest. Theorem 1.6. For every |$d\ge 2$|⁠, sufficiently large |$k$| and all |$\varepsilon>0$|⁠, there is a finite |$(d+1)-$|regular graph |$G$| with the following properties: |$A_G$| has a normalized eigenvector |$v$| with eigenvalue |$\lambda \in (-2\sqrt{d},2\sqrt{d})$| and \begin{equation*} \|v_S\|_2^2=\Omega_\lambda(\varepsilon) \end{equation*} for a set |$S$| of size |$k$|⁠, where the implicit constant |$C_{\lambda }$| depends on |$\lambda $| and is bounded away from zero on any subinterval of |$(-2\sqrt{d},2\sqrt{d})$|⁠, that is, \begin{equation*} \inf_{\lambda \in [-2\sqrt{d}+\delta,2\sqrt{d}-\delta]}C_{\lambda}> 0, \end{equation*} for any small enough positive |$\delta .$| |$G$| has girth at least \begin{equation*} \Omega\left(\frac{\log_{d}(k)}{\varepsilon}\right). \end{equation*} For every fixed |$\varepsilon $| (or for every fixed, sufficiently large |$k$|⁠), the set of eigenvalues attained by the above graphs is dense in |$(-2\sqrt{d},2\sqrt{d})$|⁠. The proof of Theorem 1.6 appears in Section 3. Notice that the above theorem does not provide any bound as the eigenvalue |$\lambda $| approaches one of the edges |$\pm 2\sqrt{d}$|⁠, which is consistent with Remark 1.5. Remark 1.7 (Connections to other notions of delocalization). Various other notions of delocalization for eigenvectors have been studied—|$\ell _\infty $| delocalization as mentioned above, lower bounds on the |$\ell _1$| norm, and “no-gaps” delocalization [8, 18, 19] (see the surveys [15, 17] for details). Note that taking |$\varepsilon =\frac{C\log _d\log _d(n)}{\log _{d}(n)}$| for a suitably large constant |$C$| in Theorem 1.2 and |$k=1$| implies an |$\ell _\infty $| bound of |$O\Big(\sqrt{\frac{\log _d\log _d(n)}{\log _d(n)}}\Big)$| for any eigenvector of any |$d+1-$|regular graph with girth |$\Omega (\log _d(n)).$| Moreover, the examples we construct in Theorem 1.6 show that one cannot expect to do much better. This is a much weaker result than the known bounds for random |$d+1-$|regular graphs where the corresponding bound is |$\tilde O\left ({\frac{1}{\sqrt n}}\right )$| suppressing logarithmic terms (see [4]). This establishes that the delocalization properties of high girth graphs are weaker than those of random graphs. Remark 1.8. In [2], the authors prove a quantum ergodicity result (“most” eigenfunctions are delocalized in some quantitative sense) for large regular graphs that are expanders (i.e., the spectral gap of the associated random walk operator is bounded away from 0 uniformly in the size of the graph) and have a few short cycles. A canonical model satisfying the above two properties are uniformly chosen random regular graphs. A discussion about the results of this paper in the context of the results in [2] is presented in Remark 3.6. 1.1 Connection between localization and low girth: |$\varepsilon =1$| case. Before proceeding to the proofs of these theorems, we give a quick proof of the upper bound in the extreme case |$\varepsilon =1$|⁠, that is, when the entire mass is supported on a small set, to give some intuition about why a localized eigenvector implies a short cycle. Assume |$G$| is a |$(d+1)$|-regular graph with adjacency matrix |$A$| and |$Av=\lambda v$| for a vector |$v$| with exactly |$k$| nonzero entries. Let |$H$| be the induced subgraph of |$G$| supported on the nonzero vertices. Observe that for the eigenvector equation to hold for any vertex |$s\notin H$|⁠, we must have \begin{equation*} \sum_{t\in V}v(t)A(s,t)=\sum_{t\in H}v(t)A(s,t)=0, \end{equation*} so in particular any such |$s$| must have at least two neighbors (of opposite signs for the value of |$v$|⁠) inside |$H$|⁠. Thus, for every edge |$ts$| with |$t\in H$| leaving |$H$|⁠, there must be some |$t^{\prime}\in H$| such that |$tst^{\prime}$| is a path of length |$2$| in |$G$|⁠. Replace all such paths by new edges |$tt^{\prime}$| to obtain a graph |$H^{\prime}$| on the vertices of |$H$| (possibly creating multi-edges). Note that every vertex in this graph has degree at least |$(d+1)$| ( the degree can in fact exceed |$d+1$| if one edge |$ts$| is a part of many paths |$tst^{\prime}$|⁠). Now, if |$H^{\prime}$| has girth |$g$|⁠, then any ball of radius |$g/2-1$| does not contain cycles. Growing a ball from any vertex, we find that \begin{equation*} d^{g/2-1}\le |H^{\prime}|\le k, \end{equation*} which implies that |$g\le 2\log _d(k)+2.$| Since every edge in |$H^{\prime}$| corresponds to a path of length at most |$2$| in |$G$|⁠, |$G$| must contain a cycle of length at most |$4\log _d(k)+4$|⁠. Theorem 1.2 shows that this continues to happen even when |$\varepsilon =o(1)$|⁠. Note that since the girth of a |$(d+1)$|-regular graph on |$n$| vertices is at most |$O(\log _d(n))$| by a similar argument, the only interesting regime is when |$\varepsilon =\Omega (1/\log _d(n))$|⁠. 2 Improved Upper Bound In this section we prove Theorem 1.2, at a high level following the approach of [6]. The main ingredient is the following hyper-contractivity estimate. Just for completeness we include the following definition of hyper-contractivity: for any |$1\le p < q \le \infty ,$| an operator |$S$| from |$\ell _p$| to |$\ell _q$| is said to be hyper-contractive if its operator norm |$\|S\|_{p\to q}$| is bounded by |$1$|⁠. Let |$T_m$| be the Chebyshev polynomials of the 1st kind, that is, |$T_m(\cos \theta )=\cos (m\theta ).$| Lemma 2.1 (Hypercontractivity of Chebyshev polynomials, [6]). If |$A$| is a |$d+1$|-regular graph with girth |$g$|⁠, then for all even |$m<g/2$|⁠, \begin{equation*} \left\|T_m\big(A/(2\sqrt{d})\big)\right\|_{1\rightarrow\infty}= \frac{d-1}{2d^{m/2}}. \end{equation*} The proof appearing in [6] is based on a spectral decomposition in terms of spherical functions on trees. For completeness we give a quick proof of the above using connections to non-backtracking walks instead. Proof. Let |$U_m(\cdot )$| defined by |$U_m(\cos \theta )=\frac{\sin ((m+1)\theta )}{\sin \theta }$| be the Chebyshev polynomials of the second kind. It is well known that for any |$m,$| (see for, e.g., Section 2 in [1]) \begin{equation*} B^{(m)}=d^{m/2}\left(U_m\big(A/(2\sqrt d)\big)-\frac{1}{d}{U_{m-2}\big(A/(2\sqrt d)\big)}\right), \end{equation*} where for any pair of vertices |$u,v \in V$|⁠, the entry |$B^{(m)}(u,v),$| is the number of non-backtracking walks of length |$m$| between |$u$| and |$v$|⁠. At this point we use the following well- known relation between the Chebyshev Polynomials of the 1st and 2nd kind: |$T_m=\frac{1}{2} (U_m-U_{m-2} ).$| Putting the above together we get \begin{equation} T_{m}(A/(2\sqrt d))=\frac{1}{d}\left(T_{m-2}(A/(2\sqrt d))\right)+\frac{1}{2}\left(\frac{B^{(m)}}{d^{m/2}}-\frac{B^{(m-2)}}{d^{m/2-1}}\right).\end{equation} (2) Now note that |$ \|T_m (A/(2\sqrt{d}) ) \|_{1\rightarrow \infty }$| is nothing but the maximum entry of the corresponding matrix. Since |$m<g/2$| by hypothesis, for all |$j\le m$| and for all |$u,v \in V$| we have |$B^{(\,j)}(u,v)=\delta _{j,{\textsf{dist}}(u,v),}$| where |${\textsf{dist}}(u,v)$| is the graph distance between |$u$| and |$v.$| Summing (2) over |$2, 4,\ldots , m$| after multiplying both sides of the equation corresponding to |$m-2j$| by |$\frac{1}{d^j}$| and using the last observation completes the proof. Using the above lemma, the next approximation result is at the heart of the proof of Theorem 1.2. As will be clear soon, given any eigenvalue |$\lambda _0$| of |$A/(2\sqrt d)$|⁠, the proof of Theorem 1.2 demands the existence of a polynomial |$f$|⁠, with the following two properties: |$f(A/(2 \sqrt d))$| is hyper-contractive. |$f(\lambda _0)$| is large, and |$f(\lambda )$| is not too negative for any other eigenvalue |$\lambda .$| The key insight then is that |$f(A/(2\sqrt d))$| in some approximate sense acts as a projector onto the |$\lambda _0$|-eigenspace of |$A/2\sqrt{d}$|⁠, and at the same time is hyper-contractive. By analyzing the action of the operator |$f(A/(2\sqrt d))$| on the corresponding eigenvector one can then show that the latter cannot be localized. The following lemma states that such a polynomial exists. It is in the proof of this lemma that we achieve the required estimates needed to improve the bounds in [6]. Lemma 2.2 (Hypercontractive polynomial approximation). If |$A$| has girth |$g$|⁠, then for all positive integers |$r,m$| such that |$r$| is even, |$mr<g/2$| and all |$\lambda \in{{\mathbb{R}}}$| there exists a polynomial |$f$| such that |$f(\lambda )\ge \frac{m}{4}-1$|⁠. |$f(x)\ge -1$| on |${{\mathbb{R}}}$|⁠. |$\|\,f(A/2\sqrt{d})\|_{1\rightarrow \infty }\le \frac{2(d-1)}{d^{r/2}}$|⁠. Fig. 1. View largeDownload slide An example of the polynomial |$f$| in Lemma 2.2, with parameters |$m=8,r=2,\phi =\pi /3.$| Fig. 1. View largeDownload slide An example of the polynomial |$f$| in Lemma 2.2, with parameters |$m=8,r=2,\phi =\pi /3.$| Proof. Assume first that |$\lambda \in [-1,1]$|⁠. We will use the Fejer kernel of order |$m,$| \begin{equation*} F_m(\theta):=\sum_{j=-m}^m (1-|\,j|/m)e^{ij\theta}=1+2\sum_{j=1}^m(1-j/m)\cos(\,j\theta). \end{equation*} Recall that |$F_m(\theta )\ge 0$| and |$F_m(0)=m$|⁠. Let |$\lambda =\cos \phi $| for |$\phi \in [0,\pi ]$| and define \begin{equation*} K_\phi(\theta):=\frac12\left(F_m(r(\theta-\phi))+F_m(r(\theta+\phi))\right)-1, \end{equation*} and notice that |$K_\phi (\theta )\ge -1$| and, \begin{equation}K_\phi(\phi)\ge \frac{1}{2} (F_m(0)+0)-1=m/2-1,\end{equation} (3) and \begin{align*} K_\phi(\theta) &= \sum_{j=1}^m(1-j/m)\cos(\,jr(\theta-\phi))+\cos(\,jr(\theta+\phi))\\ &=\sum_{j=1}^m 2(1-j/m)\cos(\,jr\phi)\cos(\,jr\theta)\\ &=2\sum_{j=1}^m (1-j/m)\cos(\,jr\phi)T_{jr}(\cos(\theta)). \end{align*} Let \begin{equation*} f(x):=\sum_{j=1}^m (1-j/m)(\cos(\,jr\phi)+1)T_{jr}(x), \end{equation*} so that \begin{equation*} f(\cos(\theta))=\frac{K_\phi(\theta)+K_0(\theta)}{2}. \end{equation*} The 1st property is implied by (3) and |$K_0(\theta )\ge -1$|⁠, \begin{equation*} f(\lambda)=K_\phi(\phi)/2+K_0(\phi)/2\ge \frac{m}4-1. \end{equation*} The 2nd property holds for |$x=\cos (\theta )\in [-1,1]$| since |$K_\phi (\theta ),K_0(\theta )\ge -1$|⁠. For |$x\notin [-1,1]$| we observe that |$f$| is a nonnegative linear combination of Chebyshev polynomials of even degree, which are nonnegative outside |$[-1,1]$|⁠. For the 3rd property, we observe that \begin{align*}\big\|\,f(A/2\sqrt{d})\big\|_{1\rightarrow\infty} &\le \sum_{j=1}^m |(1-j/m)(\cos(\,jr\phi)+1)|\big\|T_{jr}(A/2\sqrt{d})\big\|_{1\rightarrow\infty} \\&\le 2(d-1)\left(\frac{1}{2d^{r/2}}+\frac1{2d^{2r/2}}+\ldots\right) \\&\le\frac{2(d-1)}{d^{r/2}},\end{align*} by Lemma 2.1, as desired. If |$\lambda \notin [-1,1]$|⁠, then we simply use the polynomial |$f$| corresponding to |$\lambda =1$| (which by symmetry is the same as the one for |$\lambda =-1$|⁠). Properties (2) and (3) continue to hold, and property (1) holds because |$f$| is a nonnegative linear combination of even degree Chebyshev polynomials, which are increasing on |$[1,\infty )$| and decreasing on |$(-\infty ,1]$|⁠. Thus, for such an |$f$| and |$\lambda $| it follows that |$f(\lambda )\ge f(1)=f(-1)\ge \frac{m}{4}-1.$| We now finish the proof of Theorem 1.2. Proof of Theorem 1.2 Let |$\lambda $| be an eigenvalue of |$A/2\sqrt{d}$| with normalized eigenvector |$v$|⁠. Let |$f$| be the polynomial from Lemma 2.2 applied to |$\lambda $|⁠, |$m=\lceil 4/\varepsilon \rceil +4$|⁠, and |$r=\lceil g/2m\rceil -1$| or |$\lceil g/2m\rceil -2$|⁠, whichever is even. Taking |$K=f(A/2\sqrt{d})$|⁠, we then have \begin{align} \langle v1_S,K v1_S\rangle \le \|K\|_{1\rightarrow\infty} \|v1_S\|_1^2\le 2d\cdot d^{-\varepsilon g/4+1}\cdot\|v1_S\|_1^2 \nonumber\\[8pt] \le 2d\cdot d^{-\varepsilon g/4+1}\cdot |S|\|v1_S\|_2^2 = 2d^{2-\varepsilon g/4} |S|\varepsilon, \end{align} (4) since |$r\ge \varepsilon g/8-2$|⁠, by Property (3) of Lemma 2.2 and |$\|v1_S\|_1^2\le |S|\|v1_S\|_2^2$| (Cauchy–Schwarz inequality). On the other hand, decompose |$v1_S$| as |$av+bw,$| where |$w$| is a unit vector orthogonal to |$v$| and |$a,b$| are scalars. Observe that \begin{equation*} a=\langle v1_S,v\rangle=\|v_S\|^2=\varepsilon, \end{equation*} and \begin{equation*} b^2 = \|v1_S\|^2-a^2=\varepsilon(1-\varepsilon). \end{equation*} Since |$\langle v, Kw\rangle =0$|⁠, we have \begin{align*} \langle v1_S, K v1_S\rangle &= a^2\langle v, K v\rangle + b^2\langle w, K w\rangle \\& \ge a^2(1/\varepsilon) - b^2 \quad\textrm{by (1) and (2) of Lemma 2.2} \\&= \varepsilon-\varepsilon(1-\varepsilon)=\varepsilon^2. \end{align*} Combining this with (4), we obtain \begin{equation*} |S|\ge \frac{d^{\varepsilon g/4}\varepsilon}{2d^2}, \end{equation*} as desired. Remark 2.3 (Improvement in the untempered case). The proof of Lemma 2.2 is clearly wasteful with regards to eigenvalues of |$A$| outside |$[-2\sqrt{d},2\sqrt{d}]$|⁠; in particular, noting that Chebyshev polynomials blow up exponentially outside |$[-1,1]$|⁠, one can considerably improve the approximation bound for untempered |$\lambda $| that are bounded away from the edge values |$\{-2\sqrt{d},2\sqrt{d}\},$| (see Remark 1.5), and obtain a significantly stronger delocalization result in this case. Remark 2.4. Note that in the previous arguments, the high girth assumption was only used in Lemma 2.1, to prove a bound on |$\left \|T_m\left (A/(2\sqrt{d})\right )\right \|_{1\rightarrow \infty }$|⁠. However, the proof of the lemma shows that there is a lot of room to relax our assumptions. For example, let |$G$| be a random regular graph of size |$n$| and degree |$d+1,$| a model also considered in [2], for which it is known (for, e.g., see [13, Lemma 2.1]) that with high probability, there exists at most one cycle in the |$\rho =\frac{1}{5}\log _{d-1} n$| neighborhood of any vertex. This implies that for any |$j\le \frac{\rho }{2}$| and vertices |$u,v$|⁠, the number of non-backtracking walks from |$u$| to |$v$| of length |$j$| (⁠|$B^{(j)}(u,v)$| according to the notation in Lemma 2.1) is at most |$2$|⁠. One can check that this implies Lemma 2.1 holds with the bound |$ \frac{d-1}{2d^{m/2}}$| replaced by |$\frac{O(m)}{d^{m/2}}.$| This yields a similar lower bound on |$|S|$| as in the conclusion of Theorem 1.2, up to certain polynomial factors in |$\varepsilon .$| 3 Lower Bound In this section, we prove the Theorem 1.6, which shows that the logarithmic dependence on |$k$| and polynomial dependence on |$\varepsilon $| in Theorem 1.2 are sharp up to a |$\log (1/\varepsilon )$| term. The starting point is to observe that certain eigenvectors of finite trees already have good localization properties. For the remainder of the section, we will refer to a complete tree of finite depth |$D$| (i.e., |$D+1$| levels of vertices including the root that corresponds to level |$0$|⁠) in which every non-leaf vertex has degree |$d+1$| as a |$d$|-ary tree. We will pay special attention to eigenvectors of the adjacency matrices of |$d$|-ary trees which are radial, which means that they assign the same value to vertices in a given level. We begin by recording some facts about eigenvalues and eigenvectors of rooted |$d-$|ary trees. Recall that the eigenvalues of a |$d$|-ary tree are contained in the interval |$(-2\sqrt{d},2\sqrt{d})$| [10, Section 5]. For our purposes we will only consider radial eigenvectors, and we will refer to the corresponding eigenvalues as radial eigenvalues. Lemma 3.1 (Radial eigenvalues). For any positive integer |$D\ge 2,$||$A_k$| the adjacency matrix of |$T_D,$| a |$d$|-ary tree of depth |$D,$| has exactly |$D+1$| radial eigenvalues counting multiplicities. Proof. Let the vector space of all the radial functions on the vertices of |$T_D$| be called |$\mathscr{S}_{\textrm{sym}}.$| Clearly |$\mathscr{S}_{\textrm{sym}}$| has dimension |$D+1.$| Let |$n$| be the total number of vertices in |$T_D.$| Thus, |$\mathscr{S}_{\textrm{sym}}\subset{{\mathbb{C}}}^n.$| Now notice that |$A_D$| keeps |$\mathscr{S}_{\textrm{sym}}$| invariant and because it is a self-adjoint operator the same is true for its orthogonal complement |$\mathscr{S}_{\textrm{sym}}^{\perp }$|⁠. The proof now follows by conjugating |$A_D$| by an orthogonal matrix |$U$| whose 1st |$D+1$| rows are formed by an orthonormal basis of |$\mathscr{S}_{\textrm{sym}}$| and the remaining rows are formed by an orthonormal basis of |$\mathscr{S}^{\perp }_{\textrm{sym}}.$| This transforms |$A_D$| into a block diagonal self adjoint matrix with two blocks of size |$D+1$| and |$n-(D+1)$| respectively. The proof is now complete by standard spectral theory of self-adjoint operators. Lemma 3.2 (Eigenvalues of |$d$|-ary Trees). The set of all radial eigenvalues of any infinite sequence of distinct finite |$d$|-ary trees is dense in the interval |$(-2\sqrt{d},2\sqrt{d})$|⁠. Proof. Let |$T_1,T_2,\ldots ,T_m,\ldots $| be an infinite sequence of |$d$|-ary trees. Let |$T$| be the infinite |$(d+1)-$|regular tree with root |$r$| and observe that there are sets |$S_1\subset S_2,\ldots $| such that |$T_m$| is the induced subgraph of |$T$| on |$S_m$|⁠. Let |$A_m$| be the adjacency matrix of |$T_m$| and let |$A$| be the adjacency matrix of |$T$|⁠. Assume for contradiction that there is a closed interval \begin{equation*} I=[\lambda-\eta,\lambda+\eta]\subset (-2\sqrt{d},2\sqrt{d}) \end{equation*} such that every |$A_m$| has no radial eigenvalues in |$I$|⁠. We derive a contradiction with the fact [10,Theorem 5.2] that for |$\lambda \in \mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}(A)=[-2\sqrt{d},2\sqrt{d}]$|⁠, there is no vector |$v\in \ell _2$| such that |$(\lambda I - A_T)v=e_r$|⁠, where |$e_r$| is the indicator of |$r$|⁠. Our assumption along with the invariance of |$\mathscr{S}_{\textrm{sym}},$| implies that the operator |$A_{\textrm{sym}}(\lambda )$| that is the restriction of the operator |$(\lambda I - A_{m})$| on the subspace |$\mathscr{S}_{\textrm{sym}}$| is invertible, and |$\|A_{\textrm{sym}}(\lambda )^{-1}\|\le \eta ^{-1},$| for all |$m.$| Now since |$P_{m}e_r$| is a radial vector where |$P_m:\ell _2\rightarrow{{\mathbb{C}}}^{S_m}$| is the restriction onto |${{\mathbb{C}}}^{S_m}$|⁠, there must be a sequence of finite-dimensional radial vectors |$\{v_m\}$| with |$v_m\in{{\mathbb{C}}}^{S_m}$| such that \begin{equation*} A_{\textrm{sym}}(\lambda)v_m=P_mA_{\textrm{sym}}(\lambda)P_m^Tv_m=P_me_r. \end{equation*} Moreover, we have the uniform bound \begin{equation*} \big\|P_m^Tv_m\big\|=\|v_m\|\le \eta^{-1} \end{equation*} for all |$m$|⁠. The Banach-Alaoglu Theorem implies that |$\{P_m^Tv_m\}$| must have a weakly convergent subsequence; let |$v\in \ell _2$| be the weak limit of this subsequence, and note that |$v$| must be radial as well. Moreover, |$v$| must satisfy: \begin{align*} e_j^T(\lambda I - A)v &= \lim_{m\rightarrow\infty} e_j^TP_m^TP_m(\lambda I - A)P_m^Tv_m\\ &= e_j^TP_m^TA_{\textrm{sym}}(\lambda)v_m \\ &= \lim_{m\rightarrow\infty} e_j^TP_m^TP_me_r=e_j^Te_r, \end{align*} for every vertex |$j\in T$|⁠, so in fact we must have |$(\lambda I - A)v=e_r$|⁠, which is impossible. The next lemma lower bounds the relative mass of an eigenvector on different levels of a tree. The key reason behind such a result is the propagation of mass across levels via the eigenvalue equation. Lemma 3.3 (Eigenvectors of |$d$|-ary Trees). Assume |$d\ge 2$| and let |$T$| be a |$d$|-ary tree of depth |$D$| with root |$r$|⁠. Let |$S_0=\{r\},S_1,\ldots ,S_D\subset T$| be the vertices at levels |$0,1,\ldots , D$| of the tree and let |$v$| be a radial eigenvector of its adjacency matrix with eigenvalue |$\lambda =2\sqrt{d}\cos \theta \in (-2\sqrt{d},2\sqrt{d})$|⁠. Then every pair of adjacent levels has approximately the same total |$\ell _2^2$| mass as the root \begin{equation*} \Omega(\sin^2\theta) = \frac{\|v_{S_i}\|_2^2+\|v_{S_{i+1}}\|_2^2}{\|v(r)\|_2^2}=O(1/\sin^{2}\theta). \end{equation*} Proof. Suppose |$v$| has value |$x_i$| for all vertices in |$S_i$|⁠, and for convenience assume that the root has value |$x_0=1$| (although this makes |$v$| un-normalized). The eigenvector equation at the non-leaf vertices yields the following quadratic recurrence: \begin{align*} \lambda x_0 &= (d+1)x_1,\\ \lambda x_i &= x_{i-1}+dx_{i+1}\quad 1\le i\le D-1, \end{align*} which must be satisfied by any eigenvector (ignoring the boundary condition at the leaves). Since we are interested in the total |$\ell _2^2$| mass at each level, it will be more convenient to work with the quantities \begin{equation*} m_0=x_0=1\qquad\textrm{and}\qquad m_i = \sqrt{|S_i|}x_i = \sqrt{(d+1)d^{i-1}}x_i\quad 1\le i\le D, \end{equation*} which satisfy |$m_i^2 = \|v_{S_i}\|_2^2$|⁠. Rewriting the recurrence in terms of the |$m_i$|⁠, we obtain \begin{align*} m_1 &= \frac{\lambda}{\sqrt{d+1}} m_0,\\ m_{i+1} &= \frac{\lambda m_i}{d}\cdot\sqrt{\frac{|S_{i+1}|}{|S_i|}} - \frac{m_{i-1}}{d}\cdot\sqrt{\frac{|S_{i+1}|}{|S_{i-1}|}} = \frac{\lambda}{\sqrt{d}}m_i - m_{i-1},\quad 1\le i\le D+1. \end{align*} Letting |$\lambda =2\sqrt{d}\cos \theta $| and writing the above in matrix form, we have \begin{equation*} w_{i+1}:=\begin{bmatrix} m_{i+1} \\ m_i\end{bmatrix} = \begin{bmatrix} 2\cos\theta & -1 \\ 1 & 0\end{bmatrix}\begin{bmatrix} m_i\\m_{i-1}\end{bmatrix} =: PDP^{-1}w_i, \end{equation*} since the matrix above is diagonalizable for |$\theta \neq 0,\pi $|⁠, with \begin{equation*} D:= \begin{bmatrix} e^{i\theta} & 0\\ 0 & e^{-i\theta}\end{bmatrix},\quad P = \begin{bmatrix} 1 & 1\\ e^{-i\theta} & e^{i\theta}\end{bmatrix}. \end{equation*} Since |$D$| is unitary we have |$\|P^{-1}w_{i}\|=\|P^{-1}w_0\|$| for all |$i$|⁠. Observe that \begin{equation*} \|P\|\le 2,\quad \|P^{-1}\|\le \frac{1}{\sin\theta}. \end{equation*} Whence \begin{equation*} \frac{\|w_i\|}{\|w_0\|}\in \left[\frac{\sin\theta}{2},\frac{2}{\sin\theta}\right]. \end{equation*} Noting that |$|m_1|\le 2|m_0|$| and squaring yields the claim. Let |$T$| be a |$d$|-ary tree of depth |$D$|⁠. Choosing |$S$| to be the top |$\lfloor \varepsilon D\cdot (d/d+1)\rfloor $| levels of |$T$| and applying the above lemma to any eigenvector with eigenvalue bounded away from |$\pm 2\sqrt{d}$|⁠, we find that |$\|v_S\|_2^2=\Theta (\varepsilon )$| and |$|S|=O((d+1)^{\varepsilon D})=O(n^\varepsilon )$|⁠, where |$n$| is the number of vertices in the tree. This is exactly the kind of localization we want for our lower bound. Unfortunately, finite |$d$|-ary trees are not regular because they have leaves. The rest of this section is devoted to showing that we can nonetheless embed these trees in |$(d+1)$|-regular graphs without disturbing their eigenvectors or creating any short cycles, thereby establishing Theorem 1.6. The main device in doing this is the following lemma that shows that it is possible to identify the leaves of two trees in a manner that does not introduce short cycles. Lemma 3.4 (Pairing of trees). Suppose |$T_1$| and |$T_2$| are two |$d$|-ary trees of depth |$D$|⁠, each with |$n=(d+1)d^{D-1}$| leaf vertices |$V_1$| and |$V_2$|⁠. Then for sufficiently large |$n$| there is a bijection |$\pi :V_1\rightarrow V_2$| such that the graph obtained by identifying |$v$| with |$\pi (v)$| (i.e., the new graph is obtained by first adding an edge between a vertex |$v$| and |$\pi (v)$| for |$v\in V_1$| and then contracting the added edges) has girth at least |$\log _d(n)/4$|⁠. We defer the proof of this lemma. However, such a graph obtained is still not regular. The 2nd ingredient is the following. Lemma 3.5 (Degree-fixing gadget). For every degree |$d\ge 3$| and sufficiently large |$n$|⁠, there is a graph |$H$| on |$n$| vertices with the following properties: |$H$| has one distinguished vertex of degree |$d-2$| and the remaining vertices have degree |$d$|⁠. |$H$| has girth at least |$\log _{d-1}(n)/3$|⁠. Proof. According to Corollary 2 of [14], the number of |$d$|-regular graphs with girth at least |$g$| is asymptotic to \begin{equation*} \frac{(dn)!}{(dn/2)!2^{dn/2}(d!)^n}\cdot\exp\left(-\sum_{r=1}^g\frac{(d-1)^r}{2r}+o(1)\right) \ge \exp\left(dn/2-g(d-1)^r+o(1)\right), \end{equation*} whenever |$(d-1)^{2g-1}=o(n).$| Taking |$g=\log _{d-1}(n)/3+1$| we find that this condition is satisfied and the right-hand side is positive for large enough |$n$|⁠. Let |$G$| be a |$d$|-regular graph on |$n$| vertices with girth at least |$g$|⁠. Let |$v$| be any vertex of |$G$| and let |$u_1,u_2$| be two of its neighbors. Let |$H$| be the graph obtained by deleting the edges |$vu_1$| and |$vu_2$| and adding the edge |$u_1u_2$|⁠. Observe that |$v$| has degree |$d-2$| and every other vertex has degree |$d$| in |$H$|⁠. Moreover, since we replaced a path of length |$2$| by an edge, the length of every cycle decreases by at most |$1$|⁠, so |$H$| must have girth at least |$\log _{d-1}(n)/3$|⁠, as desired. Equipped with the above lemmas we now complete our construction and hence prove Theorem 1.6. Proof of Theorem 1.6 Let |$d\ge 2$|⁠, and let |$k$| be any integer larger than the |$n$| required for Lemmas 3.4 and 3.5 to apply. Suppose |$\varepsilon \in (0,1)$| is given. Let |$t$| be the largest integer such that |$(d+1)d^{t-1}\le k$|⁠. Choose |$D-1=\lceil t/\varepsilon \rceil $| and let |$T_1$| and |$T_2$| be two disjoint |$d$|-ary trees with |$D-1$| levels. Let |$S_1$| and |$S_2$| be the sets of vertices consisting of the top |$t$| levels of |$T_1$| and |$T_2$|⁠, respectively. Let |$\lambda $| be an eigenvalue of |$A_{T_1}$| and let |$f$| be the corresponding normalized radial eigenvector, (⁠|$\,f$| has the same value on every vertex within a level) as in Lemma 3.3. By Lemma 3.3, we know \begin{equation*} \big\|\,f_{S_1}\big\|_2^2=\big\|\,f_{S_2}\big\|_2^2=\Omega_\lambda(\varepsilon), \end{equation*} where the implicit constant is |$\Omega (\sin ^4\theta )$| for |$\lambda =2\cos \theta $|⁠. Construct |$T_1^{\prime}$| and |$T_2^{\prime}$| by attaching |$d$| new marked vertices to each leaf of |$T_1$| and |$T_2$|⁠, respectively, so that they are |$d$|-ary trees of depth |$D$|⁠, with |$n=(d+1)d^{D-1}$| leaves each, corresponding to the marked vertices. Apply Lemma 3.4 to pair these marked leaves; call the resulting graph |$H$|⁠. Notice that |$H$| is |$d+1$|-regular except for the marked vertices, which have degree two. Applying Lemma 3.5 with degree parameter |$d+1$| and size |$n$|⁠, we obtain a disjoint collection of graphs |$W_v$| (one for every marked vertex |$v$|⁠) each with a single distinguished vertex of degree |$d-1$|⁠, all remaining vertices of degree |$d+1$|⁠, and girth |$\log _d(n)/3.$| Finally, let |$G$| be the graph obtained by identifying each marked vertex |$v$| with the distinguished vertex of |$W_v$|⁠. Observe that |$G$| has girth at least |$\log _d(n)/4=\Omega (D)=\Omega (\log _d(k)/\varepsilon )$|⁠, since |$H$| has girth at least this much by Lemma 3.4 and attaching disjoint copies of |$W$| at single vertices does not create any new cycles. Using the symmetry in the above construction we now prescribe an eigenvector of |$G.$| Let |$\nu $| be the function equal to |$f$| on vertices of |$T_1$|⁠, |$-f$| on vertices of |$T_2$|⁠, and zero elsewhere. We claim that |$\nu $| is an eigenvector of |$G$| with eigenvalue |$\lambda $|⁠. To see this one has to verify the eigenvector equation at vertices of three kinds: At every vertex of |$T_1$| and |$T_2$| because all new neighbors of those vertices are assigned a value of |$0$| in |$\nu $|⁠. It is also satisfied at the marked vertices, because every such vertex is adjacent to exactly one leaf in |$T_1$| and one leaf in |$T_2$|⁠, which have the same values with opposite signs, The remaining vertices in copies of |$W$| have value zero, so the eigenvector equation is trivially satisfied. Observing that |$\|\nu _{S_1\cup S_2}\|_2^2=\Omega _\lambda (\varepsilon )$| with |$|S_1\cup S_2|=2k$| finishes the construction. Since this construction is valid for infinitely many |$n$|⁠, Lemma 3.2 implies that the set of radial eigenvalues is dense in |$(-2\sqrt{d},2\sqrt{d})$|⁠, as desired. Remark 3.6. Comparing the results in this paper to the ones in [2], we wish to make the following two remarks: The results in [2] imply that in a certain quantitative way, all but a vanishing fraction of eigenfunctions are close to a constant vector in a weak sense. In contrast, in this article we show that the radial eigenvectors of finite trees lead to localized eigenvectors for certain |$d+1$| regular graphs with high girth. However, as discussed in Lemma 3.1, the number of such eigenvectors is rather small, and constitutes a vanishing fraction. There could be other non-radial localized eigenvectors as well, but as the results in [2] show, if the graph |$G$| is an expander and has few short cycles, such eigenvectors must form a vanishing fraction of all eigenvectors. Note that in the above proof, even though the graph |$H$| obtained from gluing |$T_1$| and |$T_2$| has nice expansion properties, the final graph |$G$| obtained from |$H$| by adding the degree correcting gadgets are not expanders since the gadgets only have an edge boundary of size |$2$|⁠. We outline a more complicated alternate approach that could conceivably produce high girth expanders in Section 3.2, after the proof of Lemma 3.4, but do not pursue this in the present paper. 3.1 Proof of Lemma 3.4 Probabilistic model. Our construction is probabilistic, and inspired by the switching argument of [14], but with much cruder estimates since we are not interested in precise asymptotic enumeration, but only in showing that a certain probability is not zero. Let |$T_1$| and |$T_2$| be two |$d$|-ary trees of depth |$D$|⁠, each with exactly |$n=(d+1)d^{D-1}$| leaf vertices, henceforth denoted |$V_1$| and |$V_2$|⁠. Consider the random graph |$G$| obtained by taking the union of |$T_1$| and |$T_2$| and a random perfect matching between |$V_1$| and |$V_2$| (the edges induced by the matching will henceforth be called matching edges). Fig. 2. View largeDownload slide This describes the construction of gluing two 3-regular trees by a random matching of the leaves. A cycle in the glued graph is illustrated where the blue paths denote the excursions into the trees and the red edges denote the jump from one tree to the other. However, in the picture note that all the interior vertices in the trees have degree three but the roots only have degree two (since the 3rd edge is not significant for the purposes of the illustration, it was omitted just to avoid cluttering in the figure). Fig. 2. View largeDownload slide This describes the construction of gluing two 3-regular trees by a random matching of the leaves. A cycle in the glued graph is illustrated where the blue paths denote the excursions into the trees and the red edges denote the jump from one tree to the other. However, in the picture note that all the interior vertices in the trees have degree three but the roots only have degree two (since the 3rd edge is not significant for the purposes of the illustration, it was omitted just to avoid cluttering in the figure). Graph-theoretic terminology. A cycle in a graph is an oriented closed walk with no repeated edges. We will consider cyclic shifts and reversals of a cycle to be the same cycle. Note that because no edge is repeated, a cycle in |$G$| has a natural decomposition into a alternating sequence of paths and matching edges (will be called matching edge traversals to incorporate the orientation as well), where the paths are entirely a subset of |$T_1$| or |$T_2$| alternately. Naturally, we will call such paths as tree excursions. We will formally denote this as a sequence of alternating matching edge traversals |$e_i$| and tree excursions|$\gamma _i$| (see Figure 2) \begin{equation*} e_1,\gamma_1,e_2,\gamma_2,\ldots,e_{k},\gamma_k \end{equation*} Fig. 3. View largeDownload slide This illustrates the use of the degree correcting gadget. The two red vertices in the 1st figure denote two leaf vertices in |$T_1$| and |$T_2$|⁠, respectively, connected via the identified marked vertex (see Figure 2). The marked vertex (yellow) is then identified with a degree |$d-1$| vertex in the gadget. Fig. 3. View largeDownload slide This illustrates the use of the degree correcting gadget. The two red vertices in the 1st figure denote two leaf vertices in |$T_1$| and |$T_2$|⁠, respectively, connected via the identified marked vertex (see Figure 2). The marked vertex (yellow) is then identified with a degree |$d-1$| vertex in the gadget. (or equivalently |$\gamma _1,e_2,\gamma _2,\ldots ,e_{k},\gamma _k, e_1$|⁠), where the |$\gamma _i$| are simple paths in either |$T_1$| or |$T_2$| with endpoints at leaves, and |$\gamma _k$| ends where |$e_1$| begins. We will follow the convention that |$\gamma _1,\gamma _3,\ldots $| are excursions in |$T_1$| and |$\gamma _2,\gamma _4,\ldots $| are excursions in |$T_2$|⁠. The total number of edges in a cycle will be called its length. We begin by establishing some preliminary facts about short cycles in |$G$|⁠. Lemma 3.7 (Number and overlaps of short cycles). Let |$c<1/2$| be a constant. Then for sufficiently large |$n$|⁠, with positive probability (independent of |$n$|⁠) we have both of the following properties simultaneously: |$G$| does not contain two cycles of length at most |$L,$| which share a matching edge, |$G$| contains at most |$B=O(n^{c(1+o(1))})$| cycles of length at most |$L$|⁠, where |$L:=2c\log _d(n)$|⁠. Proof. Let |$v\in V_1$| be a leaf vertex. We will first show that \begin{equation}{{\mathbb{P}}}[v\ \textrm{occurs in}\ \ge 1 \textrm{cycle of length} \le L]=O\big(n^{-1+c(1+o(1))}\big).\end{equation} (5) Call a cycle that occurs in |$G$| with nonzero probability a potential cycle. Every potential cycle consists of |$k$| matching traversals and |$k$| tree excursions for some even |$k$|⁠. Observe that every excursion of length |$h$| has even length and consists of |$h/2$| upward steps towards the root of the tree and |$h/2$| downward steps back down to the leaves. Given a starting vertex for the excursion, the upward steps are uniquely determined, and there are at most |$d$| choices for each of the downward steps (since backtracking is not allowed, and the root has degree |$d+1$|⁠). Since there are at most |$n$| choices for each matching traversal given one of its endpoints, the total number of potential cycles containing |$v$| with exactly |$k$| matching traversals and excursions of lengths |$h_1,\ldots ,h_k$| is at most \begin{equation}d^{h_1/2}\cdot n\cdot d^{h_2/2}\cdot n\ldots d^{h_k/2} = d^{(h_1+\ldots+h_k)/2}\cdot n^{k-1}\le n^{k-1+c},\end{equation} (6) since the last matching traversal is determined by the starting vertex |$v$|⁠. Every such potential cycle fixes |$k$| matching edges, so the probability that it occurs in a random matching is at most |$(n-k)!/n!$|⁠. Taking a union bound over all |$k\le L$|⁠, ordered partitions |$h_1+\ldots +h_k\le L-k$|⁠, and potential cycles with those parameters, we have \begin{equation*} {{\mathbb{P}}}[v\ \textrm{occurs in a cycle of length} \le L]\le \sum_{k=1}^L e^{O(\sqrt{L})}k!\cdot\frac{n^{k-1+c}(n-k)!}{n!}=O(n^{-1+c(1+o(1))}). \end{equation*} Next, we use a similar argument to estimate the probability that any leaf vertex |$v$| occurs in more than one cycle. Fix |$v\in V_1$| and observe that a pair of potential cycles of length at most |$L$| both containing |$v$| can be specified by the following choices: Lengths |$s,s^{\prime}\le L$|⁠, matching traversal counts |$k,k^{\prime}\le L$|⁠, and tuples of excursion lengths |$h_1,\ldots ,h_k$| and |$h_1^{\prime},\ldots ,h_{k^{\prime}}^{\prime}$| for both cycles. The common matching edge |$e$| incident to |$v$| contained in both cycles. The excursions made in both cycles. The remaining |$k-2$| and |$k^{\prime}-2$| matching edges in both cycles (noting that the final edge is not required once all excursions are specified). Since any particular pair fixes |$k+k^{\prime}-1$| matching edges, the probability that it occurs in |$G$| is at most |$(n-(k+k^{\prime}-1))!/n!$|⁠. Bounding excursions as in (6) and taking a union bound, we have \begin{align*}&{{\mathbb{P}}}[v\ \textrm{occurs in } \ge 2 \textrm{cycles of length} \le L] \\&\le L^4e^{O(\sqrt{L})}\cdot n\cdot n^{2c}\cdot \sup _{k,k^{\prime}}\left (k!(k^{\prime})! n^{k+k^{\prime}-4}\cdot\frac{(n-(k+k^{\prime}-1))!}{n!}\right) \\&=O(n^{2c(1+o(1))-2}). \end{align*} Taking a union bound over all leaf vertices |$v$|⁠, and observing that cycles in |$G$| share a matching edge if and only if they pass through a common leaf vertex we conclude that \begin{align*} {{\mathbb{P}}}[&G\ \textrm{contains two cycles sharing a matching edge, of length} \le L]\\&=O\big(n^{2c(1+o(1))-1}\big)=o(1). \end{align*} For the 2nd claim we sum (5) over all |$v\in{V_1}$| and apply Markov’s inequality to obtain \begin{equation*} {{\mathbb{P}}}[|V_{C}|>O(n^{c(1+o(1))})]<1/2, \end{equation*} where |$|V_{C}|$| denotes the set of vertices contained in at least one cycle of length |$\le L$|⁠. Taking a union bound with our previous conclusion, we have that with probability |$1/3$| all cycles in |$G$| are matching edge disjoint and |$|V_C|=O(n^{c(1+o(1))})$|⁠. Since every cycle contains at least one vertex, this gives the 2nd claim. Proof of Lemma 3.4 Let |$c=1/4$| and |$L:=2c\log _d n$|⁠, and choose |$n$| sufficiently large so that Lemma 3.7 applies with |$B=O(n^{c(1+o(1))})\le n^{1/3}$|⁠. Let |$\Gamma $| be the set of graphs in the support of the random variable |$G$| (i.e., the set of all the graphs which |$G$| is equal to with nonzero probability) such that both conditions of Lemma 3.7 are satisfied. Clearly the support size is equal to the number of matchings which is |$n!$|⁠. Now, since by Lemma 3.7|${{\mathbb{P}}}(G\in \Gamma )>0$| (independent of |$n$|⁠), it follows that, \begin{equation} |\Gamma|=\Omega(n!).\end{equation} (7) For integers |$z_2,z_4,\ldots ,z_L\le B$| let \begin{equation*} \Gamma(z_2,\ldots,z_L) \end{equation*} denote the subset of |$\Gamma $| containing graphs with exactly |$z_j$| cycles of length |$j$|⁠, noting that in our model there are never any odd cycles. Our goal is to show that |$\Gamma (0,\ldots ,0)$| is not empty. Following [14], our strategy will be to establish the following two claims: Let |$[B]=\{0,1,\ldots ,B\}.$| Claim 3.8. There exists a |$z^*\in [B]^L$| such that \begin{equation*} |\Gamma(z^*_2,\ldots,z^*_L)|\ge \exp(\Omega(n\log n)). \end{equation*} Claim 3.9. For every |$z\in [B]^L$| such that |$z_k>0$|⁠: \begin{equation*} \frac{|\Gamma(z_2,\ldots,z_{k}-1,\ldots,z_L)|}{|\Gamma(z_2,\ldots,z_k,\ldots,z_L)|}= \Omega(n^{-3c}). \end{equation*} Iterating the above claims yields \begin{equation*} |\Gamma(0,\ldots,0)|\ge \exp\left(Cn\log n-O(\log(n))\sum_{i=2,\ldots L} z_i\right)=\exp(Cn\log n-\Omega(n^{1/3}\log^2 n))>1, \end{equation*} so that with nonzero probability |$G$| has no cycles of length at most |$L$|⁠. Contracting all matching edges shrinks the length of every cycle by at most a factor of |$2$|⁠, yielding the desired pairing. To establish Claim 3.8, we observe that the tuple |$z^*\in [B]^L$| that maximizes |$|\Gamma (z^*)|$| must have cardinality at least \begin{equation*} \frac{\Omega(n!)}{B^L}\ge\Omega\left(\exp\left(n\log n-O(n)-O(\log^2 n)\right)\right). \end{equation*} For Claim 3.9 we use a switching argument, that is, we will switch some edges in a graph in |$\Gamma (z_2,\ldots , z_k,\ldots , z_L)$| to construct a graph in |$\Gamma (z_2,\ldots , z_k-1,\ldots , z_L)$|⁠. Given a graph |$H\in \Gamma (z_2,\ldots , z_k,\ldots , z_L)$| with |$z_k>0$|⁠, a forward switching is defined as the following operation: Choose the lexicographically (fix two arbitrary orderings of edges and cycles) 1st matching edge |$e=st$| in the lexicographically 1st cycle |$C$| of length |$k$| in |$H$|⁠. Choose any matching edge |$f=uv$| in |$H$| at distance at least |$2L$| from |$e$| that is not contained in any cycle of length at most |$L$|⁠. Remove |$st$| and |$uv$| from the matching and add |$sv$| and |$ut$|⁠. Observe that since every matching edge is contained in at most one cycle of length |$\le L$| and |$f$| does not belong to any cycle of length |$\le L$|⁠, removing |$e$| and |$f$| destroys only the cycle |$C$| among all the cycles of length |$\le L$|⁠. Since the endpoints of |$e$| and |$f$| are at distance |$2L$| in |$H$|⁠, adding |$sv$| and |$ut$| does not create any cycles of length at most |$L$|⁠. Thus, the outcome of a forward switching is a graph |$H^{\prime}\in \Gamma (z_2,\ldots , z_k-1,\ldots ,z_L)$|⁠, which has exactly the same cycle counts except with one less cycle of length |$k$|⁠. Let |${{\mathcal{F}}}(H)$| denote the set of foward switchings of a graph |$H\in \Gamma (z_2,\ldots ,z_L)$|⁠. Observe that the only choice in the switching is the choice of the 2nd matching edge |$f$|⁠. The number of matching edges contained in cycles of length at most |$L$| is bounded by |$BL=o(n)$| and the number of edges within distance |$2L$| of |$e$| is at most |$(d+1)^{2L}=o(n)$|⁠. Therefore, for every |$H\in \Gamma (z_2,\ldots ,z_L)$|⁠, we have \begin{equation*} |{{\mathcal{F}}}(H)|=\Omega(n). \end{equation*} We now investigate how many forward switchings can map to a given graph |$H^{\prime}$| in |$\Gamma (z_2,\ldots ,z_k-1,\ldots ,z_L)$|⁠. Given such an |$H^{\prime}$|⁠, a backward switching is defined as the following operation: Choose two vertices |$u$| and |$v$| at distance exactly |$k+1$| in |$H^{\prime}$|⁠, such that (a) the extreme edges |$ut$| and |$vs$| of the |$uv-$|path |$p$| are matching edges. (b) The distance between |$u$| and |$v$| in |$H^{\prime}$| along any path other than |$p$| is at least |$L$|⁠. Delete the edges |$ut$| and |$vs$| from the matching and add edges |$st$| and |$uv$|⁠. Observe that a backward switching always yields a graph |$H\in \Gamma (z_2,\ldots ,z_L)$|⁠, and that all graphs |$H$| with a forward switching equal to |$H^{\prime}$| may be achieved in this manner. The number of backward switchings of any graph |$H^{\prime}$| is upper bounded by \begin{equation*} |{{\mathcal{B}}}(H^{\prime})|\le n\cdot (d+1)^{L+1} = O(n^{1+3c}), \end{equation*} where we have overcounted by ignoring the conditions (a) and (b) in the definition of a backward switching. A double counting argument now yields \begin{equation*} \frac{|\Gamma(z_2,\ldots,z_{k}-1,\ldots,z_L)|}{|\Gamma(z_2,\ldots,z_k,\ldots,z_L)|}\ge \frac{\min_H|{{\mathcal{F}}}(H)|}{\max_{H^{\prime}}|{{\mathcal{B}}}(H^{\prime})|}=\Omega(n^{-3c}), \end{equation*} as desired. Fig. 4. View largeDownload slide This illustrates the switching argument: forward switching takes a cycle and an edge and produces a path while the backward switching does the reverse. Fig. 4. View largeDownload slide This illustrates the switching argument: forward switching takes a cycle and an edge and produces a path while the backward switching does the reverse. 3.2 Constructing expanders: a possible approach We will follow the notations in the proof of Theorem 1.6 and assume for simplicity that |$d+1=2k$| for some positive integer |$k.$| Consider |$T_1, T_1^{\prime}$| and |$T_2,T_2^{\prime}$| as in the proof of Theorem 1.6. For our purposes we will require |$k$| copies of |$T_1^{\prime}$| as well as |$T_2^{\prime}.$| Let us call them |$T^{\prime}_{1,1}, T^{\prime}_{1,2},\ldots T^{\prime}_{1,k}$| and similarly |$T^{\prime}_{2,1}, T^{\prime}_{2,2},\ldots T^{\prime}_{2,k}.$| Now the number of leaves in |$T_1^{\prime}$| is equal to |$dm,$| where |$m$| is the number of leaves in |$T_1.$| It would be useful to think of the the leaves of |$T^{\prime}_1$| as half edges hanging from the leaves of |$T_1$| and similarly for |$T_2$| and |$T_2^{\prime}.$| Also let us denote the natural copy of |$T_1$| sitting inside |$T^{\prime}_{1,i}$| by |$T_{1,i}$| and similarly we define the notation |$T_{2,i}.$| Moreover, consider |$(2k-1)m$| new vertices |$W=\{w_1,w_2,\ldots , w_{(2k-1)m}\}$| with |$k$| half edges on each of them. Similarly consider |$(2k-1)m$| new vertices |$W^{\prime}=\{w^{\prime}_1,w^{\prime}_2,\ldots , w^{\prime}_{(2k-1)m}\}$| with |$k$| half edges on each of them as well. Now as before we glue the half edges on the leaves of |$\{T^{\prime}_{1,i}:i=1,2,\ldots k\}$| with the half edges on the vertices in |$W$| using a random matching. Note that the number of half edges incident on |$\{T^{\prime}_{1,i}:i=1,2,\ldots k\}$| is |$kdm$| that is precisely the number of half edges incident on |$W.$| Similarly we glue the half edges on the leaves of |$\{T^{\prime}_{2,i}:i=1,2,\ldots k\}$| with the half edges on the vertices in |$W^{\prime}$| using an independent random matching. Finally, to obtain a high girth graph |$H,$| identify every vertex |$w_i$| with the vertex |$w_i^{\prime}$| and call it say, |$u_i.$| Thus, the degree of |$u_i$| is the sum of the degrees of |$w_i$| and |$w_i^{\prime}$| that is |$d+1.$| Thus, the resulting graph |$H$| is a |$d+1$| regular graph on the vertex set formed by the union of the vertices in |$\{T_{1,i}, T_{2,i}: i=1,2,\ldots ,k\}$| as well as |$U=\{u_1,u_2,\ldots , u_{(2k-1)m}\}.$| A localized eigenvector can now be constructed as follows. Pick a radial eigenvector |$f$| corresponding to some radial eigenvalue |$\lambda $| of |$A_{T_1}.$| Now consider the function |$\nu $| that is equal to |$f$| on each copy |$T_{1,i}$| and is equal to |$-f$| on each copy |$T_{2,i}$| and is |$0$| on each |$u_i.$| It can be easily checked that the |$\nu $| is an eigenvector corresponding to eigenvalue |$\lambda $| for the large graph |$H.$| Thus, the above construction avoids the gadgets. However, that the matchings indeed lead to a high girth graph with nonzero probability as in Lemma 3.4 still remains to be checked and will not be pursued here. The above discussion suggests that the general strategy of gluing trees could lead to various interesting constructions and investigating them and other related phenomena is left for future work. Funding This work was supported by a Miller Research Fellowship (to S. Ganguly), NSF Grant (CCF-1553751 to N. Srivastava), and a Sloan Research Fellowship. Acknowledgments We would like to thank Assaf Naor and Mark Rudelson for helpful conversations, and MSRI and the Simons Institute for the Theory of Computing, where this work was partially carried out. We are also grateful to two anonymous referees whose comments greatly improved the paper. Communicated by Prof. Assaf Naor References [1] Alon , N. , I. Benjamini , E. Lubetzky , and S. Sodin . “ Non-backtracking random walks mix faster .” Commun. Contemp. 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Anal. 26 , no. 6 ( 2016 ): 1716 – 1776 . Google Scholar Crossref Search ADS © The Author(s) 2019. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

Rivière,, Tristan

doi: 10.1093/imrn/rnz012pmid: N/A

Abstract The goal of the present work is two-fold. First we prove the existence of a Hilbert manifold structure on the space of immersed oriented closed surfaces with three derivatives in |$L^2$| in an arbitrary compact submanifold |$M^m$| of an Euclidian space |${{\mathbb{R}}}^Q$|⁠. Second, using this Hilbert manifold structure, we prove a lower semi-continuity property of the index for sequences of conformal immersions, critical points to the viscous approximation of the area satisfying a Struwe entropy estimate and a bubble tree strongly converging in |$W^{1,2}$| to a limiting minimal surface as the viscous parameter is going to zero. 1 Introduction Let |$M^m$| be a smooth |$m$|-dimensional submanifold of a Euclidian space |${{\mathbb{R}}}^Q$| and denote by |$\pi _{M^m}$| the orthogonal projection onto |$M^m$| defined in a neighborhood of |$M^m$|⁠. Let |$\Sigma $| be an arbitrary closed oriented 2D manifold. We define the Sobolev space of maps between |$\Sigma $| and |$M^m$| with three derivatives in |$L^2$| as follows: \begin{equation*} W^{3,2}(\Sigma, M^m):=\left\{ u\in W^{3,2}(\Sigma,{{\mathbb{R}}}^Q)\quad;\quad u(x)\in M^m\quad \forall\, x\in \Sigma \right\}, \end{equation*} where the Sobolev space of |$W^{3,2}$| functions on |$\Sigma $| is defined with respect to any arbitrary reference metric (they are all equivalent due to the compactness of |$\Sigma $|⁠). Since |$ 3\times 2=6>2=\mbox{dim}(\Sigma )$| the space |$W^{3,2}(\Sigma , M^m)$| inherits a natural Hilbert manifold structure (see [11] lecture 2). Within this manifold we are considering the open subset of |$W^{3,2}$|-immersions \begin{equation*} \mbox{Imm}^{3,2}(\Sigma, M^m):=\left\{ \vec{\Phi}\in W^{3,2}(\Sigma, M^m)\quad;\quad \mathrm{d}\vec{\Phi}\wedge \mathrm{d}\vec{\Phi}\ne 0\quad \mbox{ in} \Sigma \right\}. \end{equation*} Since there is no ambiguity on the regularity we are choosing we shall simply omit the superscripts |$3,2$| and denote |$\mbox{Imm}(\Sigma , M^m)$| for |$\mbox{Imm}^{3,2}(\Sigma , M^m)$|⁠. For a given oriented closed surface |$\Sigma $| we denote by |$b(\Sigma )$| the sum of the 1st three Betti Numbers of |$\Sigma $| \begin{equation*} b(\Sigma):=b_0(\Sigma)+b_1(\Sigma)+b_2(\Sigma). \end{equation*} There are obviously finitely many classes modulo diffeomorphisms of surfaces |$\Sigma $| such that |$b(\Sigma )\le b$|⁠. We will work from now on with one fixed representative in each of these classes. For any |$b\in{{\mathbb{N}}}^\ast $| we denote by |$\mbox{Imm}_b(M^m)$| the Hilbert manifold obtained by taking the disjoined union of the Hilbert manifold of |$W^{3,2}$|-immersions of the finitely many surfaces such that |$b(\Sigma )\le b$|⁠. Starting from |$\mbox{Imm}_b(M^m)$| we are constructing a Hilbert manifold of |$W^{3,2}$|-immersed surfaces in the following way. We are first marking each surface |$\Sigma $| by fixing respectively 3, 1, or 0 distinct points on each component of |$\Sigma $| of genus respectively 0, 1, and |$>1$|⁠. We are then considering the quotients of |$\mbox{Imm}(\Sigma ,M^m)$| by |$\mbox{Diff}^{\,\ast }_+(\Sigma )$|⁠, the positive |$W^{3,2}$|-diffeomorphisms of |$\Sigma $| preserving the points we have fixed and isotopic to the identity. Then we denote by \begin{equation*} {\mathfrak M}_b(M^m):=\bigsqcup_{ b(\Sigma)\le b}\!\mbox{Imm}(\Sigma,\!M^m)/\mbox{Diff}^{\,\ast}_+(\Sigma)\quad \mbox{and} \quad{\mathfrak M}(M^m):=\bigsqcup_{ b(\Sigma)<+\infty}\!\mbox{Imm}(\Sigma,\!M^m)\!/\mbox{Diff}^{\,\ast}_+(\Sigma). \end{equation*} Our 1st main result is the following. Theorem 1.1. For any |$b\in{{\mathbb{N}}}\cup \{\infty \}$| there exists a Hilbert manifold Structure on |${\mathfrak M}_b(M^m)$| such that the canonical projection \begin{equation*} \Pi \: \ \mbox{Imm}_{b}(M^{m}) \ \longrightarrow \ \mathfrak{M}_{b}(M^{m}) \end{equation*} is a smooth map. |$\Box $| Since |$\mbox{Diff}^{\, \ast }_+(\Sigma )$| misses to be a Banach–Lie group but is only a topological group (On the space of |$W^{3,2}$|-diffeomorphisms the right multiplication is smooth but the left multiplication is not differentiable, the inverse mapping is not |$C^1$|⁠, there is no canonical chart in the neighborhood of the identity, the exponential map is continuous but not |$C^1$|⁠, it is not locally surjective in a neighborhood of the identity, the Bracket operation in the Tangent space to the identity is not continuous..., etc. See a description of all these “pathological behavior” for instance in [4] or [8].) with a Hilbert manifold structure the existence of a differentiable Hilbert structure on the quotient |$\mbox{Imm}(\Sigma ,M^m)/\mbox{Diff}^{\, \ast }_+(\Sigma )$| is not the result of classical consideration and deserves to be studied with care (Progresses in this direction are given in [2] for |$W^{3,2}$|-embeddings but we are not going to follow this approach and the one we choose is more specific to surfaces but more precise too). We shall in fact make the previous theorem more precise and to that aim we introduce some notations. Let |$\Sigma $| be a closed oriented 2D manifold, and |$\vec{\Phi }\in W_{imm}^{3,2}(\Sigma , M^m)$| and let |$g_{\vec{\Phi }}:=\vec{\Phi }^{\,\ast } g_{M^m}$|⁠. Denote by |$\wedge ^{1,0}\Sigma $| the canonical bundle of 1-0 forms over the Riemann surface issued from |$(\Sigma ,g_{\vec{\Phi }})$| and denote by |$P_{\vec{\Phi }}$| the |$L^2$| projection orthogonal projection from |$(\wedge ^{(1,0)}\Sigma )^{\otimes ^2}$| onto the space of holomorphic quadratic forms|$\mbox{Hol}_Q(\Sigma ,g_{\vec{\Phi }})$| on |$(\Sigma ,g_{\vec{\Phi }})$| and by |$P_{\vec{\Phi }}^\perp :=\mbox{Id}-P_{\vec{\Phi }}$|⁠. Define the linear map \begin{equation*} \begin{array}{rcl} {D}^\ast_{\vec{\Phi}}\ : \ T_{\vec{\Phi}}\mbox{Imm}(\Sigma,M^m) &\longrightarrow &W^{2,2}\left( (\wedge^{(1,0)}\Sigma)^{\otimes^2} \right)\\[5mm] \vec{w} &\longrightarrow & P_{\vec{\Phi}}^\perp\left({\partial}\vec{w}\ \dot{\otimes}\ {\partial}\vec{\Phi}\right)\quad , \end{array} \end{equation*} where in local complex coordinates for |$\vec{\Phi }$| we denote \begin{equation*} {\partial}\vec{w}\ \dot{\otimes}\ {\partial}\vec{\Phi}:={\partial}_{{z}}\vec{w}\, \cdot\, {\partial}_{{z}}\vec{\Phi}\ \ \mathrm{d}{z}\otimes \mathrm{d}{z}\quad. \end{equation*} We are now going to prove the following theorem Theorem 1.2. Let |$\vec{\Phi }\in \mbox{Imm}(\Sigma ,M^m)$|⁠, then there exists an open neighborhood |${\mathcal O}_{\vec{\Phi }}$| of |$\vec{\Phi }$| in |$\mbox{Imm}(\Sigma ,M^m)$| invariant under the action of |$\mbox{Diff}^{\, \ast }_+(\Sigma )$| and two smooth maps on |${\mathcal O}_{\vec{\Phi }}$|⁠, equivariant under the action of |$\mbox{Diff}^{\, \ast }_+(\Sigma )$|⁠, \begin{equation*} \left\{ \begin{array}{rcl} \displaystyle{\vec{w}}_{\vec{\Phi}}\ : \ {\mathcal O}_{\vec{\Phi}} &\longrightarrow & \mbox{Ker}( {D}^\ast_{\vec{\Phi}})\subset T_{\vec{\Phi}}\mbox{Imm}(\Sigma,M^m) \\[5mm] \displaystyle{\Psi}_{\vec{\Phi}}\ : \ {\mathcal O}_{\vec{\Phi}} & \longrightarrow & \mbox{Diff}^{\, \ast}_+(\Sigma) \end{array} \right. \end{equation*} satisfying \begin{equation*} \forall\ \vec{\Xi}\in{\mathcal O}_{\vec{\Phi}}\quad\quad\vec{\Xi}\circ\Psi_{\vec{\Phi}}(\vec{\Xi})=\pi_{M^m}\left(\vec{\Phi}+\vec{w}_{\vec{\Phi}}(\vec{\Xi})\right), \end{equation*} where |$\pi _{M^m}$| is the orthogonal projection onto |$M^m$| and |${\mathcal T}_{\vec{\Phi }}:=({\vec{w}}_{\vec{\Phi }},{\Psi }_{\vec{\Phi }})$| realizes a diffeomorphism from |${\mathcal O}_{\vec{\Phi }}$| onto |$U_{\vec{\Phi }}\times \mbox{Diff}^{\, \ast }_+(\Sigma )$|⁠, where |$U_{\vec{\Phi }}$| is a neighborhood of |$0$| in |$\mbox{Ker}( {D}^\ast _{\vec{\Phi }})$|⁠. Moreover, the map |${\mathcal T}_{\vec{\Phi }}$| satisfies the following equivariance property: |$\forall \, \vec{\Xi }\in{\mathcal O}_{\vec{\Phi }}$| and for all |$\Psi _0\in \mbox{Diff}^{\, \ast }_+(\Sigma )$| one has \begin{equation*} \Psi_{\vec{\Phi}}(\vec{\Xi}\circ{\Psi}_0)=\Psi_0^{-1}\circ\Psi_{\vec{\Phi}}(\vec{\Xi})\quad\mbox{ and} \quad\vec{w}_{\vec{\Phi}}(\vec{\Xi}\circ\Psi_0)=\vec{w}_{\vec{\Phi}}(\vec{\Xi})\quad. \end{equation*} The space |${\mathfrak M}_{\Sigma }(M^m):=\mbox{Imm}(\Sigma ,M^m)/\mbox{Diff}^{\, \ast }_+(\Sigma )$| is Hausdorff and defines a Hilbert manifold such that the projection map \begin{equation*} \mbox{Imm}(\Sigma,M^m)\longrightarrow{\mathfrak M}_{\Sigma}(M^m) \end{equation*} defines a |$\mbox{Diff}^{\, \ast }_+(\Sigma )$|-bundle for which |$({\mathcal T}_{\vec{\Phi }})_{\vec{\Phi }}$| represents a local trivialization. |$\Box $| Remark 1.1. The condition \begin{equation*} D^\ast_{\vec{\Phi}}\vec{w}:=P_{\vec{\Phi}}^\perp\left({\partial}\vec{w}\ \dot{\otimes}\ {\partial}\vec{\Phi}\right)=0 \end{equation*} corresponds (Observe that |${\partial }\vec{w}\ \dot{\otimes }\ {\partial }\vec{\Phi }=0$| is the condition which, starting from a conformal immersion |$\vec{\Phi }$|⁠, preserves the conformality of |$\vec{\Phi }+t\vec{w}$| at the 1st order for the same Riemman structure on |$\Sigma $|⁠. Similarly, |$d^\ast _Aa=0$| is the 1st-order condition, which, starting from a Coulomb gauge |$d_A^\ast A=0$|⁠, preserves the Coulomb condition for |$A+ta$| for the same covariant co-differentiation |$d^\ast _A$|⁠. Moreover, it is well known that the conformality of an immersion |$\vec{\Phi }$| can be interpreted as a Coulomb condition (see for instance [12]) with respect to the action of the “gauge group” |$\mbox{Diff}^{\, \ast }_+(\Sigma )$|⁠.) to the Coulomb slice condition \begin{equation*} d^\ast_Aa=0 \end{equation*} in the gauge theory while studying the Hilbert bundle structure of the quotient of |$H^s$| connections by the Gauge group for |$s>n/2$| and away from reducible connections (see [5]). |$\Box $| Once this Hilbert bundle structure will be established we shall be considering the following application to the viscosity method for the area functional introduced by the author in [13]. For any immersion |$\vec{\Phi }\in \mbox{Imm}(\Sigma ,M^m)$| we denote \begin{equation*} F(\vec{\Phi}):=\int_{\Sigma}\left[1+|{\vec{\mathbb I}}_{\vec{\Phi}}|^2\right]^2\ \mathrm{d}vol_{g_{\vec{\Phi}}}, \end{equation*} where |${\vec{\mathbb I}}_{\vec{\Phi }}=\pi _{\vec{n}}(\nabla ^M\mathrm{d}\vec{\Phi })$| is the 2nd fundamental form of the immersion |$\vec{\Phi }$| in |$M^m$|⁠. Observe that \begin{equation} \forall\ b\in{{\mathbb{N}}} \quad\quad\exists\ C_{b}>0\quad\quad F(\vec{\Phi})<C_{b}\quad\Longrightarrow\quad\mbox{b}\,(\Sigma)\le b. \end{equation} (1.1) This is a direct consequence of Gauss–Bonnet theorem and Cauchy–Schwartz inequality. It is clear that |$F(\vec{\Phi })$| only depends on the equivalence class |$[\vec{\Phi }]$| of |$\vec{\Phi }$| in |${\mathfrak M}_\Sigma (M^m)$|⁠. Since |$F$| is a smooth functional on |$\mbox{Imm}(\Sigma ,M^m)$| (see [13]) it descends to a smooth functional on |${\mathfrak M}_\Sigma (M^m)$|⁠. We shall prove the following theorem. Proposition 1.1. Let |$[\vec{\Phi }]$| be a critical point of |$F$| in |${\mathfrak M}(M^m)$|⁠. Then the 2nd derivative of |$F$| at |$[\vec{\Phi }]$| defines a Fredholm and elliptic operator. |$\Box $| The viscosity method consists in studying the variations of the area Lagrangian \begin{equation*} \mbox{Area}(\vec{\Phi})=\int_{\Sigma}\, \mathrm{d}vol_{g_{\vec{\Phi}}} \end{equation*} by considering relaxations of the form \begin{equation*} A^\sigma(\vec{\Phi}):=\mbox{Area}(\vec{\Phi})+\sigma^2\, F(\vec{\Phi})=\mbox{Area}(\vec{\Phi})+\sigma^2\, \int_{\Sigma}\left[1+|{\vec{\mathbb I}}_{\vec{\Phi}}|^2\right]^2\ \mathrm{d}vol_{g_{\vec{\Phi}}}, \end{equation*} where |$\sigma>0$|⁠. The work [13] has been devoted to the asymptotic analysis of sequences of critical points of |$A^{\sigma _k}$|⁠, with uniformly bounded |$A^{\sigma _k}$| energy and satisfying Struwe’s entropy condition \begin{equation} \sigma_k^2 F(\vec{\Phi}_{k})= o\left(\frac{1}{\log\sigma_k^{-1}}\right) \quad\mbox{ as {$\sigma_k$} goes to zero.} \end{equation} (1.2) It is proved in these two works that, modulo extraction of a subsequence, the immersions |$\vec{\Phi }_{k}$|varifold converges toward a 2D integer rectifiable stationary varifold|$\mathbf v_\infty $| of |$M^m$| that is parametrized. In [14] and [10] the regularity of the parametrized integer rectifiable stationary varifold is established. Hence, we have the following theorem. Theorem 1.3. [[13], [14], [10]] Let |$\vec{\Phi }_k$| be a sequence of immersions of a closed surface |$\Sigma $|⁠, critical points of |$A^{\sigma _k}$| and such that \begin{equation*} \limsup_{k\rightarrow+\infty}A^{\sigma_k}(\vec{\Phi}_k)<+\infty\quad\mbox{and} \quad\sigma_k^2\int_{\Sigma^g}(1+|\vec{\mathbb I}_{\vec{\Phi}_k}|^2)^2\ \mathrm{d}vol_{g_{\vec{\Phi}_k}}=o\left( \frac{1}{\log\sigma_k^{-1}} \right). \end{equation*} Then there exists a subsequence |$\vec{\Phi }_{k^{\prime}}$| such that the corresponding associated varifold (The associated varifold |$\mathbf v$| associated to an immersion |$\vec{\Phi }$| of |$\Sigma ^g$| is given by \begin{equation*} \forall \ \phi\in C^0(G_2TM^m)\quad{\mathbf v}(\phi):=\int_{\Sigma^g} \phi(\vec{\Phi}_\ast T_x \Sigma^g)\ \mathrm{d}vol_{\vec{\Phi}^{\,\ast} g_{M^m}}) \end{equation*}|${\mathbf v}_k$| converges toward the varifold |${\mathbf v}_\infty $| associated to a smooth, possibly branched, conformal minimal immersion |$\vec{\Psi }_\infty $| of a constant Gauss curvature nodal surface |$(S_\infty ,h)$| equipped with a locally constant odd multiplicity |$N_\infty \in C^\infty (S_\infty ,2\,{{\mathbb{N}}}+1)$| and such that \begin{equation*} \mbox{genus}\,(S_\infty)\le g\quad\mbox{ and} \quad\lim_{k\rightarrow+\infty}\mbox{Area}(\vec{\Phi}_k)=\frac{1}{2}\int_{S_\infty}N_\infty\ |\mathrm{d}\vec{\Psi}_\infty|_h^2\ \mathrm{d}vol_h. \end{equation*}|$\Box $| The question of comparing the Morse index of the limiting surface for the area with the Morse index of the sequence |$\vec{\Phi }_k$| for the relaxed functionals |$A^{\sigma _k}$| was left open in these works. The 2nd main result of the present work is the following lower semi-continuity of the index. Theorem 1.5. Let |$\vec{\Phi }_k$| be a sequence of immersions of a closed surface |$\Sigma $|⁠, critical points of |$A^{\sigma _k}$| and such that \begin{equation*} \limsup_{k\rightarrow+\infty}A^{\sigma_k}(\vec{\Phi}_k)<+\infty\quad\mbox{and} \quad\sigma_k^2\int_{\Sigma^g}(1+|\vec{\mathbb I}_{\vec{\Phi}_k}|^2)^2\ \mathrm{d}vol_{g_{\vec{\Phi}_k}}=o\left( \frac{1}{\log\sigma_k^{-1}} \right). \end{equation*} Then there exists a subsequence |$\vec{\Phi }_{k^{\prime}}$| such that the corresponding immersed surface converges in varifolds toward a parametrized integer rectifiable stationary varifold |${\mathbf v}_\infty :=(S_\infty ,\vec{\Psi }_\infty ,N_\infty )$|⁠. If |$N_\infty \equiv 1$| then we have \begin{equation} \mbox{Ind} (\vec{\Psi}_\infty)\le \liminf_{k\rightarrow \infty} \mbox{Ind}^{\,\sigma_{k^{\prime}}}(\vec{\Phi}_{k^{\prime}}), \end{equation} (1.3) where |$\mbox{Ind} (\vec{\Psi }_\infty )$| is the maximal dimension of a subspace of |$T_{[\vec{\Psi }_\infty ]}{\mathfrak M}$| on which |$D^2\mbox{Area}(\vec{\Psi }_\infty )$| is strictly negative and |$\mbox{Ind}^{\,\sigma _k}(\vec{\Phi }_k)$| is the maximal dimension of a subspace of |$T_{[\vec{\Phi }_k]}{\mathfrak M}$| on which |$D^2 A^{\sigma _k}(\vec{\Phi }_k)$| is strictly negative. |$\Box $| Remark 1.2. After the present work has been completed, the author in collaboration with Alessandro Pigati proved that the condition |$N_\infty \equiv 1$| always holds for the varifold limit of sequences of critical points of |$F_{\sigma _k}$| satisfying the entropy condition (2) (see [9]). Hence, the lower semi-continuity of the index always holds. Combining this result with the main theorem of [7] the authors in [9] establish that the Morse index of any minimal surface realizing the minmax of a k-dimensional homological (or cohomological) family obtained by the viscosity method is bounded by |$k$|⁠. |$\Box $| The paper is organized as follows. In the next section we are proving Theorem 1.2 from which we deduce Theorem 1.1. In a short intermediate section we establish Proposition 1.3. Then, in Section 4, we are proving the lower semi-continuity of the index in the viscosity method (i.e., Theorem 1.5). 2 A Proof of Theorem 1.2. Let |$\Sigma ^g$| be a closed connected oriented surface of genus |$g$|⁠. Let |$\mbox{Diff}_+(\Sigma ^g)$| be the topological group of positive |$W^{3,2}$|-diffeomorphisms of |$\Sigma $|⁠, isotopic to the identity. This can be seen as an open subspace of |$W^{3,2}(\Sigma ,\Sigma )$| that itself defines a Hilbert manifold (see [11]). For |$g=0$| we are marking three distinct points, which we denote |$a_1,a_2,a_3$|⁠, for |$g=1$| we are marking one point that we denote |$a$|⁠, and for |$g>1$| no point is marked. We denote by |$\mbox{Diff}^\ast _+(\Sigma ^g)$| the subgroup of |$\mbox{Diff}_+(\Sigma ^g)$|⁠, which is fixing the marked points. In particular for |$g>1$| we have |$\mbox{Diff}^\ast _+(\Sigma ^g)=\mbox{Diff}_+(\Sigma ^g)$|⁠. We have the following lemma. Lemma 2.1. The action of |$\mbox{Diff}^\ast _+(\Sigma ^g)$| on |$\mbox{Imm}(\Sigma ^g,M^m)$| is free. |$\Box $|Proof of Lemma 2.1. We first claim that every element in |$\mbox{Diff}^\ast _+(\Sigma ^g)$| possess at least one fixed point. This is included in the definition for |$g=0,1$|⁠. For |$g>1$| we have that for any diffeomorphism |$\Psi $| isotopic to the identity the Lefschetz number|$L(\Psi )$| is given by definition by \begin{equation*} L(\Psi)=\mbox{Tr}(\Psi| H_0(\Sigma^g))-\mbox{Tr}(\Psi| H_1(\Sigma^g))+\mbox{Tr}(\Psi| H_2(\Sigma^g))=2-2g. \end{equation*} Hence, for |$g>1$| we have |$L(\Psi )\ne 0$| and then |$\Psi $| must have at least one fixed point. Due to Lemma 1.3 in [3] we deduce that for any |$g\in{{\mathbb{N}}}$| the action of |$\mbox{Diff}^{\, \ast }_+(\Sigma ^g)$| on |$\mbox{Imm}(\Sigma ^g,M^m)$| is free. |$\Box $| Proof of Theorem 1.2. Let |$\vec{\Phi }\in \mbox{Imm}(\Sigma , M^m)$|⁠. A basis of neighborhoods of |$\vec{\Phi }$| is given by \begin{equation*} {\mathcal V}^\varepsilon_{\vec{\Phi}}:=\left\{\vec{\Xi}=\pi_{M^m}\left(\vec{\Phi}+\vec{v}\right)\quad;\quad \vec{v}\in \Gamma^{3,2}(\vec{\Phi}^\ast TM^m)\ \mbox{ and} \ \|\vec{v}\|_{W^{3,2}}<\varepsilon\right\}, \end{equation*} for |$\varepsilon>0$| small enough and where |$\Gamma ^{3,2}(\vec{\Phi }^\ast TM^m)$| denotes the |$W^{3,2}$|-sections of the pullback bundle |$\vec{\Phi }^\ast TM^m$|⁠, which is the subvector space of |$\vec{v}\in W^{3,2}(\Sigma ,{{\mathbb{R}}}^Q)$| such that |$\vec{v}(x)\in T_{\vec{\Phi }(x)}\Sigma ^g$| for any |$x\in \Sigma ^g$|⁠. For any |$\vec{v}\in \Gamma ^{3,2}(\vec{\Phi }^\ast TM^m)$| we consider the tensor in |$\Gamma ^{2,2}((T^\ast \Sigma ^g)^{(0,1)}\otimes (T\Sigma ^g)^{(1,0)})$| given by \begin{equation*} \overline{D}^\ast_{\vec{\Phi}}\vec{v}\ |\!\_\!\_ \, g^{-1}_{\vec{\Phi}} \quad\mbox{ where} \quad g_{\vec{\Phi}}^{-1}=e^{-2\lambda}\ [\partial_z\otimes\partial_{\overline{z}}+\partial_{\overline{z}}\otimes \partial_z], \end{equation*} where \begin{equation*} \overline{D}^\ast_{\vec{\Phi}}\vec{v}=\overline{P}_{\vec{\Phi}}^\perp\left(\overline{\partial}\vec{v}\ \dot{\otimes}\ \overline{\partial}\vec{\Phi}\right). \end{equation*}|$\overline{P}_{\vec{\Phi }}$| is the |$L^2$| projection orthogonal projection from |$(\wedge ^{(0,1)}\Sigma )^{\otimes ^2}$| onto the space of anti-holomorphic quadratic forms|$\mbox{AHol}_Q(\Sigma ,g_{\vec{\Phi }})$| on |$(\Sigma ,g_{\vec{\Phi }})$| and by |$\overline{P}_{\vec{\Phi }}^\perp :=\mbox{Id}-\overline{P}_{\vec{\Phi }}$|⁠, and where |$|\!\_\!\_ $| is the contraction operator between covariant and contravariant tensors. In particular we have in local complex coordinates \begin{equation*} \left(b\ \mathrm{d}{\overline{ z}}\otimes \mathrm{d}{\overline{ z}}\right)|\!\_\!\_ g^{-1}_{\vec{\Phi}} =e^{-2\lambda}\ b\ \mathrm{d}{\overline{ z}}\otimes \partial_z. \end{equation*} We denote \begin{equation} {\mathcal I}:=\left\{ \overline{D}^\ast_{\vec{\Phi}}\vec{v}\ |\!\_\!\_\, g^{-1}_{\vec{\Phi}} \ ;\quad \vec{v}\in \Gamma^{3,2}(\vec{\Phi}^\ast TM^m)\right\}. \end{equation} (2.4) Recall the definition of the |$\overline{\partial }$| operator on |$\wedge ^{(1,0)}\Sigma ^g$| given in local coordinates by \begin{equation*} \overline{\partial}\left(a\,\partial_z\right)=\partial_{\overline{z}}a\ \mathrm{d}\overline{z}\otimes\partial_z. \end{equation*} Denote |$\mbox{Hol}_1(\Sigma ^g)$| the finite-dimensional subspace of |$\Gamma ^{3,2}(\wedge ^{(1,0)}\Sigma ^g)$| made of holomorphic sections (Due to the Riemann–Hurwitz theorem, the holomorphic tangent bundle |$T^{(1,0)}\Sigma ^g$|⁠, which is the inverse of the canonical bundle of the Riemann surface defined by |$(\Sigma ,g_{\vec{\Phi }})$|⁠, has a degree given by \begin{equation*} \mbox{deg}\left(T^{(1,0)}\Sigma^g\right)=\int_{\Sigma^g}c_1\left(T^{(1,0)}\Sigma^g\right)=2-2g; \end{equation*} therefore, |$\mbox{Hol}_1(\Sigma ^g)\ne 0$||$\Rightarrow $||$g<2$|⁠.) of |$T^{(1,0)}\Sigma ^g$|⁠. We shall now prove the following lemma. Lemma 2.2. Under the previous notations we have that \begin{equation} \forall\ \vec{v}\in W^{3,2}(\Sigma,{{\mathbb{R}}}^Q)\quad \exists\, !\, f\in (\mbox{Hol}_1(\Sigma^g))^\perp\cap \Gamma^{3,2}(\wedge^{(1,0)}\Sigma^g)\quad\quad\mbox{s. t.} \quad\overline{\partial}f=\overline{D}^\ast_{\vec{\Phi}}\vec{v}\ |\!\_\!\_\, g^{-1}_{\vec{\Phi}}. \end{equation} (2.5) Moreover, \begin{equation} \|f\|_{W^{3,2}}\le C_{\vec{\Phi}}\ \|\vec{v}\|_{W^{3,2}}. \end{equation} (2.6)|$\Box $|Proof of Lemma 2.2. We have for any |$\alpha =a\, \partial _z\in \Gamma ^{3,2}((T\Sigma ^g)^{1,0})$| and |$\beta =b\ d\overline{z}\otimes \partial _z\in \Gamma ^{2,2}((T^\ast \Sigma ^g)^{(0,1)}\otimes (T\Sigma ^g)^{(1,0)})$| \begin{equation*} \begin{array}{l} \displaystyle\int_{\Sigma^g}\left<\overline{\partial}\left(a\,\partial_z\right), b\ \mathrm{d}\overline{z}\otimes \partial_z\right>_{g_{\vec{\Phi}}}\ \mathrm{d}\mbox{vol}_{g_{\vec{\Phi}}}=\Re\left[\frac{i}{2}\int_{\Sigma^g} {\partial}\overline{a}\ b\ e^{2\lambda}\, \mathrm{d}z\wedge \mathrm{d}\overline{z}\right]\\[5mm] \displaystyle\quad=\Re\left[\frac{i}{2}\int_{\Sigma^g} \mathrm{d}[\overline{a}\ b\ e^{2\lambda}]\wedge \mathrm{d}\overline{z}\right]-\Re\left[\frac{i}{2}\int_{\Sigma^g} \overline{a}\ {\partial}_z[b\ e^{2\lambda}]\ \mathrm{d}z\wedge \mathrm{d}\overline{z}\right]\\[5mm] \displaystyle\quad=\int_{\Sigma^g} \left<\alpha,\left(\partial\left(\beta|\!\_\!\_ g_{\vec{\Phi}}\right)|\!\_\!\__2\, g_{\vec{\Phi}}^{-1}\right)|\!\_\!\_ g_{\vec{\Phi}}^{-1}\right>_{g_{\vec{\Phi}}}\ \mathrm{d}\mbox{vol}_{g_{\vec{\Phi}}}, \end{array} \end{equation*} where \begin{align*} g_{\vec{\Phi}}^{-1}=e^{-2\lambda}\ [\partial_z\otimes\partial_{\overline{z}}+\partial_{\overline{z}}\otimes \partial_z] \quad\mbox{,} \quad (b \ \mathrm{d}z\otimes \mathrm{d}\overline{z}\otimes \mathrm{d}\overline{z})|\!\_\!\__2 g_{\vec{\Phi}}^{-1}=e^{-2\lambda}\, b\ \mathrm{d}\overline{z},\quad\mbox{and} \quad\\ (e^{-2\lambda}\, b\ \mathrm{d}\overline{z})|\!\_\!\_ g_{\vec{\Phi}}^{-1}=e^{-4\lambda}\, b\ \partial_z, \end{align*} where |$|\!\_\!\_ $| and |$|\!\_\!\_ _2$| are, respectively, again the simple and double contractions between covariant and contravariant tensors. Hence, we have proved that the adjoint of |$\overline{\partial }$| on |$\Gamma ((T^\ast \Sigma ^g)^{(0,1)}\otimes (T\Sigma ^g)^{(1,0)})$| is given by \begin{equation*} \overline{\partial}^\ast\ : \ \beta\in \Gamma((T^\ast\Sigma^g)^{(0,1)}\otimes(T\Sigma^g)^{(1,0)})\ \longrightarrow\ \overline{\partial}^\ast\beta=\left(\partial\left(\beta|\!\_\!\_ g_{\vec{\Phi}}\right)|\!\_\!\__2\, g_{\vec{\Phi}}^{-1}\right)|\!\_\!\_ g_{\vec{\Phi}}^{-1}\in \Gamma(\wedge^{(1,0)}\Sigma^g). \end{equation*} We have |$\mbox{Im}\,\overline{\partial }=(\mbox{Ker}\,\overline{\partial }^\ast )^\perp $|⁠. We have that \begin{equation} \mbox{Ker}\,\overline{\partial}^\ast=\left\{\beta\in \Gamma((T^\ast\Sigma^g)^{(0,1)}\otimes(T\Sigma^g)^{(1,0)})\quad;\quad\beta|\!\_\!\_ g_{\vec{\Phi}}\in \mbox{AHol}_Q(\Sigma^g,g_{\vec{\Phi}}))\right\}. \end{equation} (2.7) Observe that \begin{equation*} \forall\,\gamma\in \Gamma\left(\left((T^\ast\Sigma^g)^{(0,1)}\right)^{\otimes^2}\right)\,\ \forall\,\beta\in \Gamma((T^\ast\Sigma^g)^{(0,1)}\otimes(T\Sigma^g)^{(1,0)})\quad \left<\gamma|\!\_\!\_ g^{-1}_{\vec{\Phi}},\beta\right>_{g_{\vec{\Phi}}}=\left<\gamma,\beta|\!\_\!\_ g_{\vec{\Phi}}\right>_{g_{\vec{\Phi}}}. \end{equation*} This implies \begin{equation} \gamma|\!\_\!\_ g^{-1}_{\vec{\Phi}}\in \mbox{Im}\,\overline{\partial}\quad\Longleftrightarrow\quad \gamma\in \left(\mbox{AHol}_Q(\Sigma^g,g_{\vec{\Phi}})\right)^\perp. \end{equation} (2.8) We deduce (2.5) from (2.8) and (2.6) follows by classical estimates for elliptic complexes in Sobolev spaces. |$\Box $| Continuation of the proof of Theorem 1.2. To |$f\in (\mbox{Hol}_1(\Sigma ^g))^\perp \cap \Gamma ^{3,2}(\wedge ^{(1,0)}\Sigma ^g)$| solving (2.5) we assign \begin{equation*} X:=2\,\Re(f)=2\,\Re((f_1+i\,f_2)\ \partial_z)=(f_1\,\partial_{x_1}+f_2\,\partial_{x_2})=X_1\, \partial_{x_1}+X_2\,\partial_{x_2}. \end{equation*} Observe that, if we denote |$\vec{X}:=\mathrm{d}\vec{\Phi }\cdot X$|⁠, we have \begin{equation*} \overline{\partial}\left(\vec{X}\cdot \overline{\partial}\vec{\Phi}|\!\_\!\_ g_{\vec{\Phi}}^{-1}\right)=\overline{\partial}\left(e^{2\lambda} (X_1+i\,X_2) \ d\overline{z}|\!\_\!\_ g_{\vec{\Phi}}^{-1}\right)=\overline{\partial} f. \end{equation*} Observe also that, since |$\vec{X}$| is tangent to the immersion |$\vec{X}\cdot \overline{\partial }\left ( \overline{\partial }\vec{\Phi }|\!\_\!\_ g_{\vec{\Phi }}^{-1}\right )=0$|⁠. Indeed in local conformal coordinates we have \begin{equation*} \overline{\partial}\left( \overline{\partial}\vec{\Phi}|\!\_\!\_ g_{\vec{\Phi}}^{-1}\right)=\partial_{\overline{z}}(e^{-2\lambda}\,{\partial_{\overline{z}}}\vec{\Phi})\ \mathrm{d}\overline{z}\otimes\partial_{z}, \end{equation*} and |$\vec{h}^0:=\partial _{\overline{z}}(e^{-2\lambda }\,\partial _{\overline{z}}\vec{\Phi })\ \mathrm{d}\overline{z}\otimes \mathrm{d}\overline{z}=e^{-2\lambda }\, \pi _{\perp }({\partial }^2_{\overline{z}^2}\vec{\Phi })\ \mathrm{d}\overline{z}\otimes \mathrm{d}\overline{z}$| is nothing but the trace-free part of the 2nd fundamental form (We denote by |${\pi _\perp }$| the orthogonal projection onto |$(\vec{\Phi }_\ast T\Sigma ^g)^\perp $| in |$T{{\mathbb{R}}}^Q$|⁠.) of the immersion viewed as an immersion into |${{\mathbb{R}}}^Q$| and by which is then orthogonal to the immersion. Hence, \begin{equation*} \overline{\partial}f=(\overline{\partial}\vec{X}\cdot\overline{\partial}\vec{\Phi})|\!\_\!\_ g_{\vec{\Phi}}^{-1}. \end{equation*} Using |$\mbox{Im}\,\overline{\partial }=(\mbox{Ker}\,\overline{\partial }^\ast )^\perp $| and the characterization of |$\mbox{Ker}\,\overline{\partial }^\ast $| given by (4), we have |$\overline{\partial }\vec{X}\cdot \overline{\partial }\vec{\Phi }=P^\perp _{\vec{\Phi }} (\overline{\partial }\vec{X}\cdot \overline{\partial }\vec{\Phi })$| and hence \begin{equation} \overline{\partial}f=\overline{D}_{\vec{\Phi}}^\ast\vec{X} |\!\_\!\_ g_{\vec{\Phi}}^{-1}. \end{equation} (2.9) We denote \begin{equation*} \left\{ \begin{array}{l} \displaystyle{\mathcal X}^{3,2}(S^2):=\left\{{X}\in \Gamma^{3,2}(TS^2)\ ;\ X(a_i)=0\quad i=1,2,3\right\},\\[3mm] \displaystyle{\mathcal X}^{3,2}(T^2):=\left\{{X}\in \Gamma^{3,2}(T T^2)\ ;\ X(a)=0\right\},\\[3mm] \displaystyle{\mathcal X}^{3,2}(\Sigma^g)=\Gamma^{3,2}(\Sigma^g)\quad\mbox{ for} g>1. \end{array} \right. \end{equation*} The space of the holomorphic vector field on |$T^{(1,0)}S^2$| is a 3D complex vector space given in |${{\mathbb{C}}}$|⁠, after the stereographic projection, by \begin{equation*} h(z)= (\alpha+\beta\, z+\gamma\, z^2)\ \partial_z\quad\mbox{ where} (\alpha,\beta,\gamma)\in{{\mathbb{C}}}^3. \end{equation*} Whereas the space of the holomorphic vector field on |$T^{(1,0)}T^2$| is a one-dimensional complex vector space given in |$\mathbb{C}$| by \begin{equation*} h(z)=\alpha\ \partial_z\quad\mbox{ where }\alpha\in{{\mathbb{C}}}, \end{equation*} while for |$g>1$| we have |$\mbox{Hol}_1(\Sigma ^g)=\{0\}$|⁠. Hence, for any |$g\in{{\mathbb{N}}}$| and any \begin{equation} f\in(\mbox{Hol}_1(\Sigma^g))^\perp\cap \Gamma^{3,2}(\wedge^{(1,0)}\Sigma^g)\quad\exists\ !\ h_f\in \mbox{Hol}_1(\Sigma^g)\quad\mbox{ s. t.} \quad\Re(f+h_f)\in{\mathcal X}^{3,2}(\Sigma^g); \end{equation} (2.10) moreover, the map |$f\rightarrow h_f$| from |$(\mbox{Hol}_1(\Sigma ^g))^\perp \cap \Gamma ^{3,2}(\wedge ^{(1,0)}\Sigma ^g)$| into |$\mbox{Hol}_1(\Sigma ^g)$| is linear and smooth. Hence, we can summarize what we have proved so far in the following lemma. Lemma 2.3. Let |$\vec{\Phi }$| be a |$W^{3,2}$|-immersion. Then the following holds: \begin{equation*} \begin{array}{l} \displaystyle\forall\, \vec{v}\in \Gamma^{3,2}(\vec{\Phi}^\ast TM^m)\quad\exists \ !\ X\in{\mathcal X}^{3,2}(\Sigma^g) \quad\mbox{ s. t.} \quad\\[3mm] \displaystyle\overline{\partial}\left(X-\,i\,X^\perp\right)=\overline{D}_{\vec{\Phi}}^\ast\vec{X}|\!\_\!\_ g_{\vec{\Phi}}^{-1}=\overline{D}_{\vec{\Phi}}^\ast\vec{v}|\!\_\!\_ g_{\vec{\Phi}}^{-1}, \end{array} \end{equation*} where |$\vec{X}=d\vec{\Phi }\cdot X$| and such that \begin{equation*} \|{X}\|_{W^{3,2}}\le C_{\vec{\Phi}}\ \|\vec{v}\|_{W^{3,2}}. \end{equation*}|$\Box $|End of the proof of Theorem 1.2. Let |$g_0$| be a smooth reference metric on |$\Sigma ^g$| and denote by |$\exp ^{g_0}$| the smooth exponential map from |$T\Sigma $| into |$\Sigma $| associated to |$g_0$|⁠. Let |$\varepsilon>0$| be small and denote \begin{equation*} {\mathcal X}^{3,2}_\varepsilon(\Sigma^g):=\left\{{X}\in{\mathcal X}^{3,2}(\Sigma^g)\ ;\ \|X\|_{W^{3,2}}<\varepsilon\right\} \end{equation*} and \begin{equation*} {\mathcal D}_\varepsilon:=\left\{\Psi\in \mbox{Diff}_+^\ast(\Sigma)\ ;\quad \exists\, X\in{\mathcal X}^{3,2}_\varepsilon(\Sigma^g)\quad \mbox{s.t}\quad \Psi(x)= \exp^{g_0}_x(X(x)) \right\}. \end{equation*} We define \begin{equation*} \begin{array}{rcl} \Lambda_{\vec{\Phi}}\ : \ {\mathcal V}_{\vec{\Phi}}^\varepsilon\times{\mathcal D}_\varepsilon &\longrightarrow& \Gamma^{2,2}((T^\ast\Sigma )^{(0,1)}\otimes (T^\ast\Sigma )^{(0,1)})\\[3mm] (\vec{\Xi},\Psi) &\longrightarrow& \overline{D}_{\vec{\Phi}}^\ast\left(\vec{\Xi}\circ\Psi\right)|\!\_\!\_ g^{-1}_{\vec{\Phi}}. \end{array} \end{equation*} The map is clearly |$C^1$| and Lemma 2.3 gives that \begin{equation*} \left.\partial_{\Psi}\Lambda_{\vec{\Phi}}\right|{}_{(\vec{\Phi},0)}\cdot X=\overline{D}_{\vec{\Phi}}^\ast\left(d\vec{\Phi}\cdot X\right)|\!\_\!\_ g^{-1}_{\vec{\Phi}} \end{equation*} realizes an isomorphism between |${\mathcal X}^{3,2}$| and |${\mathcal I}$| (defined in (1)). The implicit function theorem gives then the existence of a |$C^1$| map |$\Psi _{\vec{\Phi }}(\vec{\Xi })$| defined in a neighborhood of |$\vec{\Phi }$| such that \begin{equation*} \overline{D}_{\vec{\Phi}}^\ast\left(\vec{\Xi}\circ\Psi_{\vec{\Phi}}(\vec{\Xi})\right)|\!\_\!\_ g^{-1}_{\vec{\Phi}}=0, \end{equation*} and we denote |$\vec{w}_{\vec{\Phi }}(\vec{\Xi }):=\vec{\Xi }\circ \Psi _{\vec{\Phi }}(\vec{\Xi })-\vec{\Phi }$|⁠. For any element |$\Psi _0\in \mbox{Diff}_+^\ast (\Sigma )$| close to the identity and |$\vec{\Xi }$| close enough to |$\vec{\Phi }$| one has trivially \begin{equation*} \overline{D}_{\vec{\Phi}}^\ast\left(\vec{\Xi}\circ\Psi_0\circ\Psi_0^{-1}\circ\Psi_{\vec{\Phi}}(\vec{\Xi})\right)|\!\_\!\_ g^{-1}_{\vec{\Phi}}=0. \end{equation*} Because of the local uniqueness of |$\Psi _{\vec{\Phi }}(\vec{\Xi })$| given by the implicit function theorem, we deduce the equivariance property \begin{equation*} \Psi_{\vec{\Phi}}(\vec{\Xi}\circ{\Psi}_0)=\Psi_0^{-1}\circ\Psi_{\vec{\Phi}}(\vec{\Xi})\quad\mbox{ and} \quad\vec{w}_{\vec{\Phi}}(\vec{\Xi}\circ\Psi_0):=\vec{\Xi}\circ\Psi_{\vec{\Phi}}(\vec{\Xi})-\vec{\Phi}=\vec{w}_{\vec{\Phi}}(\vec{\Xi}). \end{equation*} We then naturally extend, by equivariance, the map |${\mathcal T}_{\vec{\Phi }}:=(\vec{w}_{\vec{\Phi }},\Psi _{\vec{\Phi }})$| on a neighborhood |${\mathcal O}_{\vec{\Phi }}$| of |$\vec{\Phi }$| invariant under the action of |$\mbox{Diff}_+^\ast (\Sigma )$|⁠. We are now proving the Hausdorff property for |${\mathfrak M}_{g}(\Sigma ^g,M^m):=\mbox{Imm}(\Sigma ^g,M^m)/ \mbox{Diff}^{\, \ast }_+(\Sigma ^g)$|⁠. Following classical considerations (see the arguments in [15, proof of Lemma 2.9.9]) it suffices to prove that \begin{equation*} \Gamma:=\left\{ (\vec{\Phi},\vec{\Phi}\circ\Psi)\quad;\quad\vec{\Phi}\in \mbox{Imm}(\Sigma^g,M^m)\ \mbox{ and }\ \Psi\in \mbox{Diff}^{\, \ast}_+(\Sigma^g)\right\} \end{equation*} is closed in |$(\mbox{Imm}(\Sigma ^g,M^m))^2$|⁠. This follows from the 1st part of the proof of the theorem. Let |$(\vec{\Phi }_k,\vec{\Xi }_k:=\vec{\Phi }_k\circ \Psi _k)\rightarrow (\vec{\Phi }_\infty ,\vec{\Xi }_\infty )$| in |$W^{3,2}$|⁠. For |$k$| large enough both |$\vec{\Phi }_k$| and |$\vec{\Xi }_k$| are included in |${\mathcal O}_{\vec{\Phi }_\infty }$|⁠. Because of the continuity of the map |$\vec{w}_{\vec{\Phi }_\infty }$| we have, respectively, \begin{equation*} \vec{w}_{\vec{\Phi}_\infty}(\vec{\Phi}_k)\rightarrow \vec{w}_{\vec{\Phi}_\infty}(\vec{\Phi}_\infty)=0\quad\mbox{ and }\quad\vec{w}_{\vec{\Phi}_\infty}(\vec{\Xi}_k)\rightarrow \vec{w}_{\vec{\Phi}_\infty}(\vec{\Xi}_\infty). \end{equation*} The equivariance of |$\vec{w}_{\vec{\Phi }_\infty }$| gives |$\vec{w}_{\vec{\Phi }_\infty }(\vec{\Xi }_k)=\vec{w}_{\vec{\Phi }_\infty }(\vec{\Phi }_k)$|⁠, hence |$\vec{w}_{\vec{\Phi }_\infty }(\vec{\Xi }_\infty )=0$|⁠. Thus, |$\vec{\Xi }_\infty \circ \Psi _{\vec{\Phi }_\infty }=\vec{\Phi }_\infty $| and this shows that |$\Gamma $| is closed and then |${\mathfrak M}(\Sigma ^g,M^m)$| defines a Hausdorff Hilbert manifold and Theorem 1.2 is proved. |$\Box $| 3 A Proof of Proposition 1.1 From [6] (see also an alternative approach in [1]) we know that under the assumptions that |$\vec{\Phi }$| is a critical point of |$F$|⁠, it defines a smooth immersion in conformal coordinates. We shall be working in the chart in the neighborhood of |$[\vec{\Phi }]$| in |${\mathfrak M}(\Sigma ^g,M^m)$| given by |$\vec{w}_{\vec{\Phi }}$| from Theorem 1.2. In other words we identify \begin{equation} T_{[\vec{\Phi}]}{\mathfrak M}\simeq\left\{\vec{w}\in \Gamma^{3,2}(\vec{\Phi}^\ast T M^m)\quad;\quad P^\perp_{\vec{\Phi}}\left(\overline{\partial}\vec{w}\,\dot{\otimes}\,\overline{\partial}\vec{\Phi}\right)=0\right\}. \end{equation} (3.11) For such a |$\vec{w}$| we denote by |$q_{\vec{w}}$| the holomorphic quadratic form given by \begin{equation*} \overline{\partial}\vec{w}\,\dot{\otimes}\,\overline{\partial}\vec{\Phi}=q_{\vec{w}}. \end{equation*} After contracting with the tensor |$g_{\vec{\Phi }}^{-1}$|⁠, this equation becomes \begin{equation*} \overline{\partial}\left(\vec{w}\cdot\overline{\partial}\vec{\Phi}|\!\_\!\_ g_{\vec{\Phi}}^{-1}\right)=-\pi_{\perp}(\vec{w})\cdot \vec{h}^0+q_{\vec{w}}|\!\_\!\_ g_{\vec{\Phi}}^{-1}, \end{equation*} where we recall that |$\pi _\perp $| is the orthogonal projection onto the normal space to |$\vec{\Phi }_\ast T\Sigma $| in |$T_{\Phi }{{\mathbb{R}}}^Q$| and |$\vec{h}^0$| is the trace-free part of the 2nd fundamental form of the immersion in |${{\mathbb{R}}}^Q$|⁠, which is orthogonal to the tangent plane of the immersion and given in local coordinates by \begin{equation*} \vec{h}^0_{\vec{\Phi}}=\partial_{\overline{z}}\left(e^{-2\lambda}\partial_{\overline{z}}\vec{\Phi}\right)\ \mathrm{d}\overline{z}\otimes \partial_z. \end{equation*} Observe that since |$\vec{w}$| is tangent to |$T_{\vec{\Phi }}M^m$| we have |$\pi _\perp (\vec{w})=\pi _{\vec{n}}(\vec{w})$| where |$\pi _{\vec{n}}$| is the orthogonal projection onto the normal space to |$\vec{\Phi }_\ast T\Sigma $| in |$T_{\vec{\Phi }}M^m$|⁠. Using the characterization of |$\mbox{Im}\,\overline{\partial }=(\mbox{Ker}\,\overline{\partial }^\ast )^\perp $| given by (5) we deduce \begin{equation*} \overline{\partial}\left(\vec{w}\cdot\overline{\partial}\vec{\Phi}|\!\_\!\_ g_{\vec{\Phi}}^{-1}\right)=-{\mathfrak P}_{\vec{\Phi}}\left(\pi_{\vec{n}}(\vec{w})\cdot \vec{h}^0_{\vec{\Phi}}\right), \end{equation*} where |${\mathfrak P}_{\vec{\Phi }}$| is the orthogonal projection onto |$(\mbox{Hol}_Q(\Sigma ^g,g_{\vec{\Phi }})|\!\_\!\_ g_{\vec{\Phi }}^{-1})^\perp $|⁠. Denote |$\vec{X}_{\vec{w}}$| the projection of |$\vec{w}$| onto the tangent plane (i.e., |$\vec{X}_{\vec{w}}=\vec{w}-\pi _{\vec{n}}(\vec{w})$|⁠) and let |$X_{\vec{w}}$| be the vector field on |$\Sigma $| such that |$\mathrm{d}\vec{\Phi }\cdot X_{\vec{w}}=\vec{X}_{\vec{w}}$|⁠. Following the computations from the previous subsection we deduce \begin{equation} \overline{\partial}\left(X_{\vec{w}}-\,i\,X_{\vec{w}}^\perp\right)=-{\mathfrak P}_{\vec{\Phi}}\left(\pi_{\vec{n}}(\vec{w})\cdot \vec{h}^0_{\vec{\Phi}}\right). \end{equation} (3.12) Denote \begin{equation*} \begin{array}{rcl} \displaystyle\pi_T\ : \ \Gamma^{3,2}(\vec{\Phi}^\ast TM^m) &\longrightarrow & \Gamma^{3,2}((T\Sigma)^{(1,0)})\\[3mm] \displaystyle\vec{w} &\longrightarrow & X_{\vec{w}}-\,i\,X_{\vec{w}}^\perp. \end{array} \end{equation*} In view of the expression of the 2nd derivative |$D^2F$| given by (20) we have that, modulo compact operators (remembering that |$\vec{\Phi }$| is smooth), we are reduced (The sum of a Fredholm operator with a compact operator is Fredholm.) to study the Fredholm nature of the operator generated by the following quadratic form: \begin{equation*} \begin{array}{l} Q_{\vec{\Phi}}(\vec{w})= \,\, \int_{\Sigma}(1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}}) \left|\pi_{\vec{n}}\left(D^{g_{\vec{\Phi}}}\mathrm{d}\vec{w}\right)\right|{}_{g_{\vec{\Phi}}}^2\ \mathrm{d}vol_{g_{\vec{\Phi}}} +2\, \int_{\Sigma}\, \left|\left<\vec{\mathbb I}_{\vec{\Phi}},D^{g_{\vec{\Phi}}}\mathrm{d}\vec{w}\right>_{g_{\vec{\Phi}}}\right|{}^2\ \mathrm{d}vol_{g_{\vec{\Phi}}} \end{array} \end{equation*} combined with (2). Hence, the symbols of the generated operator, in local conformal coordinates, is given by \begin{equation*} \left\{ \begin{array}{l} 2 e^{-2\lambda} \pi_{\vec{n}}\circ\left[(1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}}) |\xi|^4+2 e^{-4\lambda}\sum_{i,j,k,l} \vec{\mathbb I}_{kl}\otimes\vec{\mathbb I}_{kl} \ \xi_i\,\xi_j\,\xi_k\,\xi_l\right]\circ\pi_{\vec{n}}\\[5mm] (\xi_1+i\,\xi_2)\circ\pi_T. \end{array} \right. \end{equation*} This is clearly the symbol defining an elliptic operator on |$\Gamma ^{3,2}(\vec{\Phi }^\ast TM^m)$| and |$D^2F$| is Fredholm on |$T_{[\vec{\Phi }]}{\mathfrak M}$|⁠. This concludes the proof of Proposition 1.3. 4 A Proof of Theorem 1.4, the Lower Semi-continuity of the Index We shall assume that |$\Sigma ^g$| is connected. We shall present the computations for |$M^m=S^m$|⁠. The general constraint generates lower-order terms whose abundance could mask the true reason why the theorem is true whereas the same terms in the |$M^m=S^m$| case are easier to present. The 1st part of the theorem is the main results of [13]. It remains to prove the inequality (3) under the assumption of Theorem 1.5. The 1st derivative of the area of an immersion (possibly branched) of a closed surface |$\Sigma ^g$| into |${{\mathbb{R}}}^Q$| is given by (see [13]) \begin{equation*} D\mbox{Area}(\vec{\Phi})\cdot\vec{w}=\int_{\Sigma^g} \left<\mathrm{d}\vec{\Phi}\,;\,\mathrm{d}\vec{w}\right>_{g_{\vec{\Phi}}}\ \mathrm{d}\mbox{vol}_{g_{\vec{\Phi}}} \end{equation*} and the 2nd derivative (A reader familiar to the rich literature in geometry on minimal surface theory in three dimensions might not immediately recognize the most commonly used expression of the 2nd derivative of the area by the mean of the Jacobi field. This classical presentation of |$D^2\mbox{Area}$| has the advantage to “reduce” this operator to an operator on function by introducing the decomposition |$\vec{w}=w\, \vec{n}$|⁠. This decomposition however is not “analytically” favorable since |$\vec{n}$| has a priori one degree of regularity less than |$\vec{w}$|⁠. This observation is at the base of the analysis of the Willmore functional as it has been developed by the author in the recent years.) \begin{equation*} D^2 \mbox{Area}(\vec{\Phi})\cdot(\!\vec{w},\vec{w}\!)=\int_\Sigma \left[\!\left<\mathrm{d}\vec{w}\,;\,\mathrm{d}\vec{w}\right>_{g_{\vec{\Phi}}}+\left|\left<\mathrm{d}\vec{\Phi}\,;\,\mathrm{d}\vec{w}\right>_{g_{\vec{\Phi}}}\right|{}^2 - 2^{-1}\!\left|\mathrm{d}\vec{\Phi}\dot{\otimes}\, \!\mathrm{d}\vec{w}+\mathrm{d}\vec{w}\dot{\otimes}\, \mathrm{d}\vec{\Phi}\right|{}^2\!\right]\ \mathrm{d}\mbox{vol}_{g_{\vec{\Phi}}}, \end{equation*} where we recall that in coordinates |$\mathrm{d}\vec{\Phi }\,\dot{\otimes } \,\mathrm{d}\vec{w}:=\sum _{i,j}\partial _{x_i}\vec{\Phi }\cdot \partial _{x_j}\vec{w}\ \mathrm{d}x_i\otimes \mathrm{d}x_j$|⁠. Since we are assuming |$N_\infty \equiv 1$| a.e. on |$S_\infty $| and, following the proof of the main theorem of [13], we can extract a subsequence that we keep denoting |$\vec{\Phi }_{k}$| such that we have a bubble tree strong |$W^{1,2}$| convergence of |$\vec{\Phi }_{k}$| toward a minimal (possibly branched) immersion |$\vec{\Psi }_\infty $| of a surface |$S_\infty $|⁠, which is the union of nodal surfaces and spheres. More precisely, if one denotes |$\{S^j_\infty \}_{j\in J}$| to be the connected components of |$S_\infty $|⁠, for every |$j\in J$| there exists |$N^j$| points |$\{a^{j,l}\}_{l=1\cdots N^j}$| (containing in particular the possible branched points of |$\vec{\Psi }_\infty $| and a converging sequence of constant scalar curvature metrics |$h^j_{k}$| of volume one and for any |$\delta>0$| a sequence of conformal embeddings |$\phi ^j_{k}$| from |$(S^j_\infty \setminus \cup _{l=1}^{N^j} B_\delta (a^{j,l}), h^j_{k})$| into |$(\Sigma ^g,g_{\vec{\Phi }_{k}})$| such that \begin{equation} \vec{\Psi}^j_{k}:=\vec{\Phi}_{k}\circ \phi^j_{k}\longrightarrow{\vec{\Psi}_\infty}\quad\quad\mbox{ strongly in} W^{1,2}_{loc}(S^j_\infty\setminus \cup_{l=1}^{N^j} B_\delta(a^{j,l})). \end{equation} (4.13) For |$\delta $| small enough and |$k^{\prime}$| large enough the subdomains |$\Omega _{k}^j(\delta ):=\phi ^j_{k}(S^j_\infty \setminus \cup _{l=1}^{N^j} B_\delta (a^{j,l}))$| are disjoint and \begin{equation*} \lim_{\delta\rightarrow 0}\lim_{k\rightarrow +\infty}\mbox{Area}\left(\vec{\Phi}_{k}\left(\Sigma^g\setminus \bigcup_{j\in J}\Omega_{k}^j(\delta)\right)\right)=0. \end{equation*} Let |$\vec{w}_1\cdots \vec{w}_N$| a family of |$N$| independent smooth vectors in |$W^{3,2}(\vec{\Psi }_\infty ^\ast T{M}^m)$| representing |$N$| independent directions in |$T_{[\vec{\Psi }_\infty ]}{\mathfrak M}$| on the span of which |$D^2\mbox{Area}$| is strictly negative. We can assume without loss of generality that the |$\vec{w}_i$| are |$C^\infty $|⁠. One modifies each of these vectors in the following way. For each |$i\in \{1\cdots Q\}$| for each |$j\in J$| and each |$l\in \{1\cdots N^j\}$| we introduce (after identifying for each |$j$| and |$l$| the tangent planes to |$M^m$| around |$\vec{\Phi }_\infty (a^{j,l})$| with the one at exactly |$\vec{\Phi }_\infty (a^{j,l})$|⁠) \begin{equation*} \vec{w}^{\,\delta}_i(x)=\left\{\begin{array}{l} \vec{w}_i(x)\quad\quad\quad\quad \mbox{for} |a^{j,l}-x|\ge \sqrt{\delta}\\[3mm] \vec{w}_i(x)\ \chi^{\,\delta}(|x-a^{j,l}|)\quad \quad\mbox{ for }\delta\le |a^{j,l}-x|\le \sqrt{\delta}\\[3mm] 0\quad\quad\quad\quad \mbox{ for } |a^{j,l}-x|\le \delta, \end{array} \right. \end{equation*} where we take |$\chi ^{\,\delta }(s)$| to be a slight smoothing of |$ \log ({s}/{\delta })/\log ({1}/{\sqrt{\delta }}) $|⁠. A short computation gives that \begin{equation*} \vec{w}_i^{\,\delta}\longrightarrow\vec{w}_i\quad\mbox{ strongly in }\quad W^{1,2}(S_\infty,{{\mathbb{R}}}^Q). \end{equation*} Therefore, in view of the explicit expression of |$D^2 \mbox{Area}(\vec{\Psi }_\infty )\cdot (\vec{w},\vec{w})$|⁠, there exists |$\delta $| small enough such that |$\vec{w}^{\,\delta }_1\cdots \vec{w}^{\,\delta }_N$| realizes a family of |$N$| independent smooth vectors in |$W^{3,2}(\vec{\Psi }_\infty ^\ast TM^m)$| on the span of which |$D^2\mbox{Area}$| is strictly negative. We fix such a |$\delta $|⁠. Let |$\rho>0$| small enough such that for any |$z\in M^m$| the map |$\vec{\Psi }_\infty $| is injective on each components of |$\vec{\Psi }^{-1}_\infty (\overline{B^Q_\rho (p)})\subset S^j_\infty $|⁠. Let |$\{\chi _s(p)\}_{s\in \{1\cdots N\}}$| be a finite smooth partition of unity of |$M^m\subset{{\mathbb{R}}}^Q$| such that the support of every |$\chi _s$| is included in an |$m$|-ball of radius |$\rho $|⁠. We denote the connected components of |$\vec{\Psi }_\infty ^{-1}(\mbox{Supp}(\chi _s))$| in |$S_\infty $| by |$\Omega _{s}^t$| for |$t=1\cdots n_{s}$| and |$\omega _s^t$| are the corresponding characteristic functions. We have that |$\chi _s(\vec{\Psi }_\infty (x))\ \omega _s^t(x)$| is smooth for any |$s\in \{1\cdots N\}$| and any |$t\in \{1\cdots n_s\}$| and moreover \begin{equation*} \mathrm{d}(\chi_s(\vec{\Psi}_\infty(x))\ \omega_s^t(x))=\mathrm{d}(\chi_s(\vec{\Psi}_\infty(x)))\ \omega_s^t(x). \end{equation*} We can then write each |$\vec{w}_i^{\,\delta }$| in the form \begin{equation*} \vec{w}_i^{\,\delta}(x)=\sum_{s=1}^N\chi_s(\vec{\Psi}_\infty(x))\sum_{t=1}^{n_{s}}\vec{v}_{i,s}^{\,t}(\vec{\Psi}_\infty(x))\ \omega_s^t, \end{equation*} where |$\vec{v}^{\,t}_{i,s}$| are smooth functions (This is due to the fact that |$\vec{\Psi }_\infty $| is smooth embedding on each open set |$\Omega _{s}^t$|⁠). For any |$s=\in \{1\cdots N\}$| since the components |$\overline{\Omega _s^t}$| are disjoint to each other for |$t\in \{1\cdots n_s\}$| we can include them in strictly larger disjoint open sets |$\overline{\Omega _s^t}\subset \tilde{\Omega }_s^t$|⁠; moreover, because of the strong |$W^{1,2}$|-convergence of |$\vec{\Phi }_k$| toward |$\vec{\Phi }_\infty $| in |$S^j_\infty \setminus \cup _{l=1}^{N^j} B_\delta (a^{j,l})$| \begin{equation} \|\vec{\Psi}_k-\vec{\Psi}_\infty\|_{L^\infty(\partial \tilde{\Omega_s^t})}\longrightarrow 0. \end{equation} (2) We denote |$\tilde{\omega }_s^t$| the characteristic functions of |$\tilde{\Omega }_s^t$|⁠: |$\tilde{\omega }_s^t:={\mathbf 1}_{\tilde{\Omega }_s^t}$|⁠. We still have of course \begin{equation*} \vec{w}_i^{\,\delta}(x)=\sum_{s=1}^N\chi_s(\vec{\Psi}_\infty(x))\sum_{t=1}^{n_{s}}\vec{v}_{i,s}^{\,t}(\vec{\Psi}_\infty(x))\ \tilde{\omega}_s^t. \end{equation*} Because of (2), we have that \begin{equation*} \mbox{dist}\left(\vec{\Psi}_k(\partial \tilde{\Omega}_s^t),\vec{\Psi}_\infty(\partial \tilde{\Omega}_s^t\right)\longrightarrow 0. \end{equation*} Since |$\chi _s$| is zero in an open neighborhood of each |$\vec{\Psi }_\infty (\partial \tilde{\Omega }_s^t)$| and since |$\vec{\Psi }_k$| is smooth, we have for every |$s$| and |$t$| and for |$k$| large enough \begin{equation} \chi_s\circ\vec{\Psi}_k\equiv 0 \quad \mbox{ in an open neighborhood} U_{k,s}^t\mbox{ of } \partial \tilde{\Omega}_s^t. \end{equation} (4.15) Hence, in particular we have for every |$s$| and |$t$| and |$k$| large enough \begin{equation*} \mathrm{d}(\chi_s(\vec{\Psi}_k(x))\ \tilde{\omega}_s^t(x))=\mathrm{d}(\chi_s(\vec{\Psi}_k(x)))\ \tilde{\omega}_s^t(x). \end{equation*} It is then clear that \begin{equation} \vec{w}_{i,k}^{\,\delta}(x):=\sum_{s=1}^N\chi_s(\vec{\Psi}_k(x))\sum_{t=1}^{n_{s}}\vec{v}_{t,s}(\vec{\Psi}_k(x))\ \tilde{\omega}_s^t\ \longrightarrow\ \vec{w}_i^{\,\delta}(x)\mbox{ strongly in} W^{1,2}_{loc}(S^j_\infty\setminus \cup_{l=1}^{N^j} B_\delta(a^{j,l})). \end{equation} (4) Using the compositions with the maps |$(\phi ^{j,k})^{-1}$| we extend the |$\vec{w}^{\,\delta }_{i,k}$|⁠, which we still denote |$\vec{w}_{i,k}^{\,\delta }$| to the whole of |$\Sigma ^g$| by taking |$\vec{w}_{i,k}^{\,\delta }=0$| on |$\Sigma ^g\setminus \bigcup _{j\in J}\Omega _{k^{\prime}}^j(\delta )$|⁠. We see |$\vec{w}^{\,\delta }_{i,k}$| as vectors in |${{\mathbb{R}}}^Q$| and we denote by |$\pi ^j_{k^{\prime}}$| the map from |$S^j_\infty \setminus \cup _{l=1}^{N^j} B_\delta (a^{j,l})$| into the space of projection matrices that to |$x\in S^j_\infty \setminus \cup _{l=1}^{N^j} B_\delta (a^{j,l})$| assigns the orthogonal projection from |$T_{\vec{\Psi }^j_{k^{\prime}}(x)}{{\mathbb{R}}}^Q$| into |$T_{\vec{\Psi }^j_{k^{\prime}}(x)}M^m$|⁠. In other words, let |$P_z$| be the |$C^1$| map from |$M^m$| into the space of |$Q\times Q$| matrices that assigns the orthogonal projection onto |$T_zM^m$|⁠, we have |$\pi ^j_{k^{\prime}}(x):=P_{\vec{\Psi }^j_{k^{\prime}}(x)}$| and we have \begin{equation} \pi^j_{k^{\prime}}\longrightarrow P_{\vec{\Psi}_\infty}\quad\quad\mbox{ strongly in} \ \ W^{1,2}_{loc}(S^j_\infty\setminus \cup_{l=1}^{N^j} B_\delta(a^{j,l})). \end{equation} (5) On |$S^j_\infty \setminus \cup _{l=1}^{N^j} B_\delta (a^{j,l})$| we denote |$\vec{u}_{i,k^{\prime}}^{\,\delta }(x):=\pi ^j_{k}(x)(\vec{w}_{i,k}^{\,\delta })$|⁠. Because of (5) we have \begin{equation} \vec{u}_{i,k}^{\,\delta}\longrightarrow \vec{w}_i^{\,\delta}\quad\quad\mbox{ strongly in}\ \ W^{1,2}(S^j_\infty). \end{equation} (6) Consider now the symmetric matrix \begin{equation*} \begin{array}{l} D^2 \mbox{Area}(\vec{\Phi}_{k})(\vec{u}_{i,k}^{\,\delta},\vec{u}_{i^{\prime},k}^{\,\delta})=\\[3mm] \displaystyle\quad\quad\sum_{j=1}^{\mbox{card}(J)}\int_{S^j_\infty} \left[\left<\mathrm{d}\vec{u}_{i,k}^{\,\delta}\,;\,\mathrm{d}\vec{u}_{i^{\prime},k}^{\,\delta}\right>_{g_{\vec{\Psi}^j_{k}}}+\left<\mathrm{d}\vec{\Psi}^j_{k}\,;\,\mathrm{d}\vec{u}^{\,\delta}_{i,k}\right>_{g_{\vec{\Psi}_{k}^j}}\left<\mathrm{d}\vec{\Psi}_{k}^j\,;\,\mathrm{d}\vec{u}_{i^{\prime},k}^{\,\delta}\right>_{g_{\vec{\Psi}_{k}^j}} \right]\ \mathrm{d}\mbox{vol}_{g_{\vec{\Psi}_{k}^j}}\\[3mm] \displaystyle- \ 2^{-1} \sum_{j=1}^{\mbox{card}(J)}\int_{S^j_\infty} \left<\mathrm{d}\vec{\Psi}_{k}^j\dot{\otimes}\, \mathrm{d}\vec{u}_{i,k}^{\,\delta}+\mathrm{d}\vec{u}_{i,k}^{\,\delta}\dot{\otimes}\, \mathrm{d}\vec{\Psi}_{k}^j, \mathrm{d}\vec{\Psi}_{k}^j\dot{\otimes}\, \mathrm{d}\vec{u}_{i^{\prime},k}^{\,\delta}+\mathrm{d}\vec{u}^{\,\delta}_{i^{\prime},k}\dot{\otimes}\, \mathrm{d}\vec{\Psi}_{k}^j\right>\ \mathrm{d}\mbox{vol}_{g_{\vec{\Psi}_{k}^j}}. \end{array} \end{equation*} Let |$f$| and |$g$| be two smooth functions supported on |$M^m\setminus \cup _{l=1}^{N^j} \Psi _\infty (B_\delta (a^{j,l}) )$| then one has \begin{equation*} \int_{S^j_\infty}<\mathrm{d}(f(\vec{\Psi}_k)), \mathrm{d}(g(\vec{\Psi}_k))>_{g_{\vec{\Psi}_{k}}}\ \mathrm{d}\mbox{vol}_{g_{\vec{\Psi}_{k}}}=\int_{S^j_\infty}<\mathrm{d}(f(\vec{\Psi}_k)), \mathrm{d}(g(\vec{\Psi}_k))>_{h^j_k}\ \mathrm{d}\mbox{vol}_{h^j_k}, \end{equation*} and since |$h_{k}^j$| converges in any norms toward |$h^j_\infty $|⁠, because of the strong |$W^{1,2}$| convergence of |$\vec{\Psi }_k$| on |$S_\infty ^j\setminus \cup _{l=1}^{N^j} B_\delta (a^{j,l})$| one has \begin{equation} \int_{S^j_\infty}<\mathrm{d}(f(\vec{\Psi}_k)), \mathrm{d}(g(\vec{\Psi}_k))>_{g_{\vec{\Psi}_{k}}}\ \mathrm{d}\mbox{vol}_{g_{\vec{\Psi}_{k}}}\longrightarrow \int_{S^j_\infty}<\mathrm{d}(f(\vec{\Psi}_\infty)), \mathrm{d}(g(\vec{\Psi}_\infty))>_{g_{\vec{\Psi}_{\infty}}}\ \mathrm{d}\mbox{vol}_{g_{\vec{\Psi}_{\infty}}}. \end{equation} (7) In a conformal chart for |$h_{k}^j$| we denote |$e^{\lambda ^j_{k^{\prime}}}:=|\partial _{x_1}\vec{\Psi }_{k^{\prime}}^j|=|\partial _{x_2}\vec{\Psi }_{k^{\prime}}^j|$|⁠. Because of the strong |$W^{1,2}$| convergence (1) we have \begin{equation*} e^{\lambda^j_{k^{\prime}}}\longrightarrow e^{\lambda^j_\infty}=|\partial_{x_1}\vec{\Psi}_\infty|=|\partial_{x_2}\vec{\Psi}_\infty|\quad\mbox{ a. e. in} \quad S^j_\infty. \end{equation*} Since |$e^{\lambda ^j_\infty }>0$| almost everywhere on |$S^j_\infty $| we have |$e^{-\lambda ^j_{k^{\prime}}}\longrightarrow e^{-\lambda ^j_\infty }$| almost everywhere and then for |$i=1,2$| \begin{equation*} \partial_{x_i}\vec{\Psi}^j_{k}/e^{\lambda^j_{k}}\longrightarrow \partial_{x_i}\vec{\Psi}^j_{\infty}/e^{\lambda^j_{\infty}}\quad\mbox{ almost everywhere.} \end{equation*} Let |$f$|⁠,|$g$|⁠, |$\phi $|⁠, and |$\psi $| be four arbitrary smooth functions on |$M^m$|⁠. Assume that both |$f$| and |$g$| are supported on |$M^m\setminus \cup _{l=1}^{N^j}\vec{\Psi} _\infty (B_\delta (a^{j,l}) )$|⁠. One has in local conformal coordinates \begin{equation*} \begin{array}{l} \displaystyle<\mathrm{d}(f(\vec{\Psi}^j_k))\otimes \mathrm{d}(\phi(\vec{\Psi}^j_k)), \mathrm{d}(g(\vec{\Psi}^j_k))\otimes \mathrm{d}(\psi(\vec{\Psi}^j_k))>_{g_{\vec{\Psi}^j_{k}}}\ \mathrm{d}\mbox{vol}_{g_{\vec{\Psi}^j_{k}}}=\\[3mm] \displaystyle\sum_{\mu,\nu=1,2} e^{-2\lambda^j_k} \partial_{x_\mu}f(\vec{\Psi}^j_k)\ \partial_{x_\nu}\phi(\vec{\Psi}^j_k)\ \partial_{x_\mu}g(\vec{\Psi}^j_k)\ \partial_{x_\nu}\psi(\vec{\Psi}^j_k)\ \mathrm{d}x_1\wedge \mathrm{d}x_2. \end{array} \end{equation*} Because of the above \begin{equation*} e^{-2\lambda^j_k} \partial_{x_\mu}f(\vec{\Psi}^j_k)\ \partial_{x_\nu}\phi(\vec{\Psi}^j_k)\ \partial_{x_\mu}g(\vec{\Psi}^j_k)\ \partial_{x_\nu}\psi(\vec{\Psi}^j_k)\!\longrightarrow\! e^{-2\lambda^j_\infty} \partial_{x_\mu}f(\vec{\Psi}^j_\infty)\ \partial_{x_\nu}\phi(\vec{\Psi}^j_\infty)\ \partial_{x_\mu}g(\vec{\Psi}^j_\infty)\ \partial_{x_\nu}\psi(\vec{\Psi}^j_\infty) \end{equation*} almost everywhere and we have moreover \begin{equation*} |e^{-2\lambda^j_k} \partial_{x_\mu}f(\vec{\Psi}^j_k)\ \partial_{x_\nu}\phi(\vec{\Psi}^j_k)\ \partial_{x_\mu}g(\vec{\Psi}^j_k)\ \partial_{x_\nu}\psi(\vec{\Psi}^j_k)|\le C\ |\nabla\vec{\Psi}^j_k|^2\rightarrow |\nabla\vec{\Psi}^j_\infty|^2\quad\mbox{strongly in} \ L^1. \end{equation*} Hence, the generalized dominated convergence theorem implies \begin{equation*} \begin{array}{l} \displaystyle\int_{S^j_\infty}<\mathrm{d}(f(\vec{\Psi}^j_k))\otimes \mathrm{d}(\phi(\vec{\Psi}^j_k)), \mathrm{d}(g(\vec{\Psi}^j_k))\otimes \mathrm{d}(\psi(\vec{\Psi}^j_k))>_{g_{\vec{\Psi}^j_{k}}}\ \mathrm{d}\mbox{vol}_{g_{\vec{\Psi}^j_{k}}}\\[3mm] \displaystyle\longrightarrow\quad\int_{S^j_\infty}<\mathrm{d}(f(\vec{\Psi}^j_\infty))\otimes \mathrm{d}(\phi(\vec{\Psi}^j_\infty)), \mathrm{d}(g(\vec{\Psi}^j_\infty))\otimes \mathrm{d}(\psi(\vec{\Psi}^j_\infty))>_{g_{\vec{\Psi}^j_{\infty}}}\ \mathrm{d}\mbox{vol}_{g_{\vec{\Psi}^j_{\infty}}}. \end{array} \end{equation*} Similarly we also have \begin{equation*} \begin{array}{l} \displaystyle\int_{S^j_\infty}\left<\mathrm{d}(f(\vec{\Psi}^j_k)), \mathrm{d}(g(\vec{\Psi}^j_k))\right>_{g_{\vec{\Psi}_k}}\ \left<\mathrm{d}(\phi(\vec{\Psi}^j_k)), \mathrm{d}(\psi(\vec{\Psi}^j_k))\right>_{g_{\vec{\Psi}_k}} \ \mathrm{d}\mbox{vol}_{g_{\vec{\Psi}^j_{k}}}\\[3mm] \displaystyle\longrightarrow\quad\int_{S^j_\infty}\left<\mathrm{d}(f(\vec{\Psi}^j_\infty)), \mathrm{d}(g(\vec{\Psi}^j_\infty))\right>_{g_{\vec{\Psi}_\infty}}\ \left<\mathrm{d}(\phi(\vec{\Psi}^j_\infty)), \mathrm{d}(\psi(\vec{\Psi}^j_\infty))\right>_{g_{\vec{\Psi}_\infty}} \ \mathrm{d}\mbox{vol}_{g_{\vec{\Psi}^j_{\infty}}}. \end{array} \end{equation*} We have \begin{equation*} \vec{u}_{i,k}:=\sum_{s=1}^N\chi_s(\vec{\Psi}_k(x))\sum_{t=1}^{n_{s}}P_{\vec{\Psi}_k}\left(\vec{v}_{i,s}^{\,t}(\vec{\Psi}_k(x))\right)\ \omega_s^t. \end{equation*} Because of (3) we have obviously from (20) that for any choice of |$s,t,s^{\prime},t^{\prime}$| \begin{equation*} \begin{array}{l} D^2 F(\vec{\Phi})\left(\chi_s(\vec{\Psi}_k(x))P_{\vec{\Psi}_k}\left(\vec{v}_{i,s}^{\,t}(\vec{\Psi}_k(x))\right)\ \tilde{\omega}_s^t, \chi_{s^{\prime}}(\vec{\Psi}_k(x))P_{\vec{\Psi}_k}\left(\vec{v}_{i,s^{\prime}}^{\,t^{\prime}}(\vec{\Psi}_k(x))\right)\ \tilde{\omega}_{s^{\prime}}^{t^{\prime}}\right)\\[5mm] \displaystyle\quad\le \int_{\tilde{\Omega}_s^t\cap \tilde{\Omega}_{s^{\prime}}^{t^{\prime}}}(1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}})^2\ \mathrm{d}vol_{g_{\vec{\Phi}}}. \end{array} \end{equation*} Combining all the above gives \begin{equation} D^2 \mbox{Area}(\vec{\Phi}_{k})(\vec{u}_{i,k}^{\,\delta},\vec{u}_{i^{\prime},k}^{\,\delta})\quad\longrightarrow\quad D^2 \mbox{Area}(\vec{\Phi}_{\infty})(\vec{w}_{i}^{\,\delta},\vec{w}_{i^{\prime}}^{\,\delta}). \end{equation} (4.20) Hence, for |$k$| large enough |$(D^2 A(\vec{\Phi }_{k})(\vec{u}_{i,k}^{\,\delta },\vec{u}_{i^{\prime},k}^{\,\delta }))_{i,i^{\prime}=1\cdots N}$| defines a strictly negative quadratic form. Using now Lemma A.1 below we deduce that for any |$i,i^{\prime}\in \{1\cdots N\}$| \begin{equation} \sigma_k^2 \left|D^2 F(\vec{\Phi}_{k})(\vec{u}_{i,k}^{\,\delta},\vec{u}_{i^{\prime},k}^{\,\delta})\right|\le C\sigma_k^2 \left[ F(\vec{\Phi}_{k})+\mbox{Area}(\vec{\Phi}_{k})^{1/4}\ F(\vec{\Phi}_{k})^{3/4}\right]=o(1). \end{equation} (4.21) Combining (8) and (9) we obtain that for |$k$| large enough |$(D^2 A^{\sigma _k}(\vec{\Phi }_{k})(\vec{u}_{i,k}^{\,\delta },\vec{u}_{i^{\prime},k}^{\,\delta }))_{i,i^{\prime}=1\cdots N}$| defines a strictly negative quadratic form. This implies inequality (1.3) and Theorem 1.4 is proved. |$\Box $| Lemma A.1. Let |$M^m$| be a closed submanifold of the euclidian space |${{\mathbb{R}}}^Q$|⁠. For any |$W^{2,4}$|-immersion |$\vec{\Phi }$| of an oriented closed surface |$\Sigma $| we denote \begin{equation*} F(\vec{\Phi}):=\int_\Sigma\left(1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}}\right)^2\ \mathrm{d}vol_{g_{\vec{\Phi}}}, \end{equation*} where |$\vec{\mathbb I}_{\vec{\Phi }}$| is the 2nd fundamental form of the immersion into |$M^m$|⁠. The Lagrangian |$F$| is |$C^2$| and there exists a constant |$C$| depending only on |$M^m$| such that for any perturbation |$\vec{w}$| of the form |$\vec{v}\circ \vec{\Phi }$| one has \begin{equation} |DF(\vec{\Phi})(\vec{v}(\vec{\Phi}))|\le C\,\int_{\Sigma}\left(1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}}\right)\ \left[ (1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}})\ |\partial \vec{v}|(\vec{\Phi})+ |\vec{\mathbb I}_{\vec{\Phi}}|_{g_{\vec{\Phi}}}\ |\partial^2 \vec{v}|(\vec{\Phi})\right]\ \mathrm{d}vol_{g_{\vec{\Phi}}}, \end{equation} (A.1) and \begin{equation} |D^2F(\vec{\Phi})(\vec{v}(\vec{\Phi}),\vec{v}(\vec{\Phi}))|\le C\,\int_{\Sigma}\left(1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}}\right)\ \left[ (1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}})\ |\partial \vec{v}|^2(\vec{\Phi})+\ |\partial^2 \vec{v}|^2(\vec{\Phi})\right]\ \mathrm{d}vol_{g_{\vec{\Phi}}}. \end{equation} (A.2)|$\Box $|Proof of Lemma A.1. We give the proof of inequalities (1) and (2) in the case of immersions into |${{\mathbb{R}}}^Q$|⁠. The terms coming from the fact we restrict to immersions into |$M^m\subset{{\mathbb{R}}}^Q$| are of lower order and do not contribute in clarifying the argument and the successive estimates. In local coordinates we denote the 2nd fundamental form \begin{equation*} \vec{\mathbb I}_{\vec{\Phi}}=\pi_{\vec{n}}\left(\mathrm{d}^2\vec{\Phi} \right)=\pi_{\vec{n}}\left(\partial^2_{x_ix_j}\vec{\Phi}\right)\ \mathrm{d}x_i\otimes \mathrm{d}x_j \end{equation*} we have \begin{equation} |\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}}:=\left|\pi_{\vec{n}}\left(d^2\vec{\Phi} \right)\right|{}^2_{g_{\vec{\Phi}}}=\sum_{i,j,k,l}g^{ik}g^{jl}\pi_{\vec{n}}\partial^2_{x_ix_j}\vec{\Phi}\cdot\pi_{\vec{n}}\partial^2_{x_kx_l}\vec{\Phi}. \end{equation} (A.3) Denote |$\pi _T$| the projection onto the tangent plane of the immersion. We have in local coordinates \begin{equation} \pi_T(\vec{X})=\sum_{i,j=1}^2 g^{ij}\ \partial_{x_i}\vec{\Phi}\cdot\vec{X}\ \partial_{x_j}\vec{\Phi}. \end{equation} (A.4) Hence, \begin{equation} \left.\pi_{\vec{n}}\frac{\mathrm{d}\pi_{\vec{n}}}{\mathrm{d}t}\right|{}_{t=0}(\vec{X})=-\sum_{i,j=1}^2 g^{ij}\ \partial_{x_i}\vec{\Phi}\cdot\vec{X}\ \pi_{\vec{n}}\left(\partial_{x_j}\vec{w}\right). \end{equation} (A.5) We have clearly \begin{equation*} \frac{\mathrm{d} g_{ij}}{\mathrm{d}t}=\partial_{x_i}\vec{\Phi}\cdot\partial_{x_j}\vec{w}+\partial_{x_i}\vec{w}\cdot\partial_{x_j}\vec{\Phi}. \end{equation*} Hence, \begin{equation} \frac{\mathrm{d} g^{ij}}{\mathrm{d}t}=-\,g^{ik} g^{jl}\,\left[\partial_{x_k}\vec{\Phi}\cdot\partial_{x_l}\vec{w}+\partial_{x_k}\vec{w}\cdot\partial_{x_l}\vec{\Phi}\right]:= -2\,(\mathrm{d}\vec{\Phi}\dot{\otimes}_S\mathrm{d}\vec{w})^{ij}. \end{equation} (A.6) We have then \begin{equation} \left.\frac{\mathrm{d}|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}}}{\mathrm{d}t}\right|{}_{t=0}=2\, \left<\pi_{\vec{n}}\left(\mathrm{d}^2\vec{\Phi} \right),\pi_{\vec{n}}\left(D^{g_{\vec{\Phi}}}\mathrm{d}\vec{w}\right)\right>_{g_{\vec{\Phi}}}-\,4\left( g\otimes(\mathrm{d}\vec{\Phi}\dot{\otimes}_S\mathrm{d}\vec{w})|\!\_\!\_ \vec{{\mathbb I}}_{\vec{\Phi}}\dot{\otimes}\vec{{\mathbb I}}_{\vec{\Phi}}\right), \end{equation} (A.7) where |$|\!\_\!\_ $| is the contraction operator between |$4$|-contravariant and |$4$|-covariant tensors and \begin{equation} D^{g_{\vec{\Phi}}}\mathrm{d}\vec{w}:=\left[\partial^2_{x_ix_j}\vec{w} -\sum_{rs=1}^2g^{rs}\partial_{x_r}\vec{\Phi}\cdot \partial^2_{x_ix_j}\vec{\Phi}\ \partial_{x_s}\vec{w}\right]\, \mathrm{d}x_i\otimes \mathrm{d}x_j. \end{equation} (A.8) This gives in particular that \begin{equation} \begin{array}{l} \displaystyle\left.\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Sigma}(1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}})^2\ \mathrm{d}vol_{g_{\vec{\Phi}}}\right|{}_{t=0}=DF(\vec{\Phi})(\vec{w})\\[5mm] \displaystyle\quad=4\, \int_{\Sigma}(1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}}) \left[\left<\vec{\mathbb I}_{\vec{\Phi}},D^{g_{\vec{\Phi}}}\mathrm{d}\vec{w}\right>_{g_{\vec{\Phi}}}-\,2\left( g\otimes(\mathrm{d}\vec{\Phi}\dot{\otimes}_S\mathrm{d}\vec{w})\right)|\!\_\!\_ \left(\vec{{\mathbb I}}_{\vec{\Phi}}\dot{\otimes}\vec{{\mathbb I}}_{\vec{\Phi}}\right)\right]\ \mathrm{d}vol_{g_{\vec{\Phi}}}\\[5mm] \displaystyle\quad+\int_{\Sigma}(1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}})^2 \left<\mathrm{d}\vec{\Phi};\mathrm{d}\vec{w}\right>_{g_{\vec{\Phi}}}\ \mathrm{d}vol_{g_{\vec{\Phi}}}. \end{array} \end{equation} (A.9) For |$\vec{w}:=\vec{v}\left (\vec{\Phi }\right )$| we have \begin{equation*} \begin{array}{l} \displaystyle\partial^2_{x_ix_j}\vec{w} -\sum_{rs=1}^2g^{rs}\partial_{x_r}\vec{\Phi}\cdot \partial^2_{x_ix_j}\vec{\Phi}\ \partial_{x_s}\vec{w}=\sum_{\alpha,\beta=1}^Q\partial^2_{z_\alpha z_\beta}\vec{v}(\vec{\Phi})\, \partial_{x_i}\vec{\Phi}^{\,\alpha}\,\partial_{x_j}\vec{\Phi}^{\,\beta}\\[5mm] \displaystyle+\sum_{\alpha=1}^Q\partial_{z_\alpha} \vec{v}(\vec{\Phi})\, \left[\partial^2_{x_ix_j}\vec{\Phi}^{\,\alpha} -\displaystyle\sum_{rs=1}^2g^{rs}\partial_{x_r}\vec{\Phi}\cdot \partial^2_{x_ix_j}\vec{\Phi}\ \partial_{x_s}\vec{\Phi}^{\,\alpha}\right]. \end{array} \end{equation*} We have \begin{equation} \pi_{T}\left(\partial^2_{x_ix_j}\vec{\Phi} \right)=\sum_{rs=1}^2g^{rs}\,\partial^2_{x_ix_j}\vec{\Phi}\cdot\partial_{x_r}\vec{\Phi}\ \partial_{x_s}\vec{\Phi}. \end{equation} (A.10) Hence, \begin{equation*} \partial^2_{x_ix_j}\vec{\Phi}-\sum_{rs=1}^2g^{rs}\partial_{x_r}\vec{\Phi}\cdot \partial^2_{x_ix_j}\vec{\Phi}\ \partial_{x_s}\vec{\Phi}=\pi_{\vec{n}}(\partial^2_{x_ix_j}\vec{\Phi})=\vec{\mathbb I}_{ij}. \end{equation*} This implies that \begin{equation} D^{g_{\vec{\Phi}}}\mathrm{d}\vec{w}=\sum_{\alpha,\beta=1}^Q\partial^2_{z_\alpha z_\beta}\vec{v}(\vec{\Phi})\, \mathrm{d}\vec{\Phi}^{\,\alpha}\otimes \mathrm{d} \vec{\Phi}^{\,\beta}+\sum_{\alpha=1}^Q\partial_{z_\alpha} \vec{v}(\vec{\Phi})\, \vec{\mathbb I}_{ij}^{\, \alpha}. \end{equation} (A.11) We deduce \begin{equation} |DF(\vec{\Phi})(\vec{v}(\vec{\Phi}))|\le C\,\int_{\Sigma}\left(1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}}\right)\ \left[ (1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}})\ |\partial \vec{v}|(\vec{\Phi})+ |\vec{\mathbb I}_{\vec{\Phi}}|_{g_{\vec{\Phi}}}\ |\partial^2 \vec{v}|(\vec{\Phi})\right]\ \mathrm{d}vol_{g_{\vec{\Phi}}}. \end{equation} (A.12) We now compute the 2nd derivative \begin{equation} \begin{array}{l} \displaystyle\left.\frac{\mathrm{d}}{\mathrm{d}t}DF(\vec{\Phi}_t)(\vec{w})\right|{}_{t=0}\\[5mm] \displaystyle=4\, \int_{\Sigma}(1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}}) \left[\left<\vec{\mathbb I}_{\vec{\Phi}},D^{g_{\vec{\Phi}}}\mathrm{d}\vec{w}\right>_{g_{\vec{\Phi}}}-\,2\left( g\otimes(\mathrm{d}\vec{\Phi}\dot{\otimes}_S\mathrm{d}\vec{w})\right)|\!\_\!\_ \left(\vec{{\mathbb I}}_{\vec{\Phi}}\dot{\otimes}\vec{{\mathbb I}}_{\vec{\Phi}}\right)\right]\ \left<\mathrm{d}\vec{\Phi};\mathrm{d}\vec{w}\right>_{g_{\vec{\Phi}}}\ \mathrm{d}vol_{g_{\vec{\Phi}}}\\[5mm] \displaystyle+\int_{\Sigma}\!|1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}}|^2 \left|\left<\mathrm{d}\vec{\Phi};\mathrm{d}\vec{w}\right>_{g_{\vec{\Phi}}}\right|{}^2+8\! \left|\left<\vec{\mathbb I}_{\vec{\Phi}},D^{g_{\vec{\Phi}}}\mathrm{d}\vec{w}\right>_{g_{\vec{\Phi}}}-\,2\!\left(\! g\otimes(\mathrm{d}\vec{\Phi}\dot{\otimes}_S\mathrm{d}\vec{w})\!\right)|\!\_\!\_ \left(\vec{{\mathbb I}}_{\vec{\Phi}}\dot{\otimes}\vec{{\mathbb I}}_{\vec{\Phi}}\right)\right|\!{}^2\ \mathrm{d}vol_{g_{\vec{\Phi}}}\\[5mm] \displaystyle+\,4\, \int_{\Sigma}(1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}}) \frac{\mathrm{d}}{\mathrm{d}t}\left[\left<\vec{\mathbb I}_{\vec{\Phi}_t},D^{g_{\vec{\Phi}_t}}\mathrm{d}\vec{w}\right>_{g_{\vec{\Phi}_t}}-\,2\left( g_{\vec{\Phi}_t}\otimes(\mathrm{d}\vec{\Phi}_t\dot{\otimes}_S\mathrm{d}\vec{w})\right)|\!\_\!\_ \left(\vec{{\mathbb I}}_{\vec{\Phi}_t}\dot{\otimes}\vec{{\mathbb I}}_{\vec{\Phi}_t}\right)\right]\ \mathrm{d}vol_{g_{\vec{\Phi}}}\\[5mm] \displaystyle+\int_{\Sigma}(1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}})^2 \left<\mathrm{d}\vec{w};\mathrm{d}\vec{w}\right>_{g_{\vec{\Phi}}}\ \mathrm{d}vol_{g_{\vec{\Phi}}}-2\,\int_{\Sigma} (1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}})^2 \ \left(\mathrm{d}\vec{\Phi}\dot{\otimes}_S \mathrm{d}\vec{w}\right)|\!\_\!\_ \left(\mathrm{d}\vec{\Phi}\otimes \mathrm{d}\vec{w}\right)\ \mathrm{d}vol_{g_{\vec{\Phi}}}. \end{array} \end{equation} (A.13) We have on one hand \begin{equation} \pi_{\vec{n}}\frac{\mathrm{d}}{\mathrm{d}t}\vec{\mathbb I}_{\vec{\Phi}_t}=\pi_{\vec{n}}\left(D^{g_{\vec{\Phi}}}\mathrm{d}\vec{w}\right), \end{equation} (A.14) on the other hand \begin{equation} \begin{array}{l} \displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left( \sum_{r=1}^2g^{rs}_{\vec{\Phi}_t}\ \partial_{x_r}\vec{\Phi}_t\cdot \partial^2_{x_ix_j}\vec{\Phi}_t \right)=\sum_{r=1}^2g^{rs}\ \partial_{x_r}\vec{\Phi}\cdot \partial^2_{x_ix_j}\vec{w}+g^{rs}\ \partial_{x_r}\vec{w}\cdot \partial^2_{x_ix_j}\vec{\Phi}+\frac{\mathrm{d} g^{rs}}{\mathrm{d}t} \partial_{x_r}\vec{\Phi}\cdot \partial^2_{x_ix_j}\vec{\Phi}\\[5mm] \displaystyle\quad=\sum_{r=1}^2g^{rs}\ \partial_{x_r}\vec{\Phi}\cdot \partial^2_{x_ix_j}\vec{w}+g^{rs}\ \partial_{x_r}\vec{w}\cdot \pi_{\vec{n}}(\partial^2_{x_ix_j}\vec{\Phi})+\frac{\mathrm{d} g^{rs}}{\mathrm{d}t} \partial_{x_r}\vec{\Phi}\cdot \partial^2_{x_ix_j}\vec{\Phi}\\[5mm] \displaystyle\quad+\sum_{r,k,l=1}^2g^{rs}\ g^{kl}\ \partial_{x_r}\vec{w}\cdot \partial_{x_k}\vec{\Phi}\ \partial_{x_l}\vec{\Phi}\cdot\partial^2_{x_ix_j}\vec{\Phi}; \end{array} \end{equation} (A.15) we have \begin{equation} \begin{array}{l} \displaystyle\sum_{r=1}^2\frac{\mathrm{d} g^{rs}}{\mathrm{d}t} \partial_{x_r}\vec{\Phi}\cdot \partial^2_{x_ix_j}\vec{\Phi}=- \sum_{r,k,l=1}^2 g^{rk}\,g^{sl}\,\partial_{x_r}\vec{\Phi}\cdot \partial^2_{x_ix_j}\vec{\Phi}\, \left[\partial_{x_k}\vec{w}\cdot\partial_{x_l}\vec{\Phi}+ \partial_{x_l}\vec{w}\cdot\partial_{x_k}\vec{\Phi} \right]\\[5mm] \displaystyle=- \sum_{r,k,l=1}^2 g^{lk}\,g^{sr}\,\partial_{x_l}\vec{\Phi}\cdot \partial^2_{x_ix_j}\vec{\Phi}\ \partial_{x_k}\vec{w}\cdot\partial_{x_r}\vec{\Phi} + g^{lk}\,g^{sr}\,\partial_{x_l}\vec{\Phi}\cdot \partial^2_{x_ix_j}\vec{\Phi}\ \partial_{x_r}\vec{w}\cdot\partial_{x_k}\vec{\Phi}. \end{array} \end{equation} (A.16) Combining (A.15) and (A.16) we obtain \begin{equation} \displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left( \sum_{r=1}^2g^{rs}_{\vec{\Phi}_t}\ \partial_{x_r}\vec{\Phi}_t\cdot \partial^2_{x_ix_j}\vec{\Phi}_t \right)=\sum_{r=1}^2g^{rs}\ \partial_{x_r}\vec{\Phi}\cdot (D^{g_{\vec{\Phi}}}\mathrm{d}\vec{w})_{ij}+\sum_{r=1}^2 g^{rs}\, \partial_{x_r}\vec{w}\cdot\vec{\mathbb I}_{ij}. \end{equation} (A.17) Thus, \begin{equation} \displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(D^{g_{\vec{\Phi}_t}}d\vec{w}\right)=-\sum_{i,j=1}^2\left[ \sum_{r=1}^2g^{rs}\ \partial_{x_r}\vec{\Phi}\cdot (D^{g_{\vec{\Phi}}}\mathrm{d}\vec{w})_{ij}\ \partial_{x_s}\vec{w}+ g^{rs}\, \partial_{x_r}\vec{w}\cdot\vec{\mathbb I}_{ij}\ \partial_{x_s}\vec{w}\right] \ \mathrm{d}x_i\otimes \mathrm{d}x_j. \end{equation} (A.18) Combining (A.6), (A.14), and (A.18) we obtain \begin{equation} \begin{array}{l} \frac{\mathrm{d}}{\mathrm{d}t}\left[\left<\vec{\mathbb I}_{\vec{\Phi}_t},D^{g_{\vec{\Phi}_t}}\mathrm{d}\vec{w}\right>_{g_{\vec{\Phi}_t}}-\,2\left( g_{\vec{\Phi}_t}\otimes(\mathrm{d}\vec{\Phi}_t\dot{\otimes}_Sd\vec{w})\right)|\!\_\!\_ \left(\vec{{\mathbb I}}_{\vec{\Phi}_t}\dot{\otimes}\vec{{\mathbb I}}_{\vec{\Phi}_t}\right)\right]\\[5mm] \displaystyle=\left|\pi_{\vec{n}}\left(D^g\mathrm{d}\vec{w}\right)\right|{}_{g_{\vec{\Phi}}}^2-\left<\vec{\mathbb I}\ ;\sum_{i,j=1}^2g^{ij}\partial_{x_i}\vec{\Phi}\cdot D^g \mathrm{d}\vec{w}\ \partial_{x_j}\vec{w}\right>_{g_{\vec{\Phi}}}\!+ 4\! \left[(\mathrm{d}\vec{\Phi}\otimes_S \mathrm{d}\vec{w})\otimes(\mathrm{d}\vec{\Phi}\otimes_S \mathrm{d}\vec{w})\right]|\!\_\!\_ (\vec{\mathbb I}\dot{\otimes}\vec{\mathbb I})\\[5mm] \displaystyle-4\, \left(g\otimes (\mathrm{d}\vec{\Phi}\otimes_S \mathrm{d}\vec{w})\right)|\!\_\!\_ \left(\vec{\mathbb I}\otimes \pi_{\vec{n}}(D^g \mathrm{d}\vec{w})+\pi_{\vec{n}}(D^g \mathrm{d}\vec{w})\otimes\vec{\mathbb I}\right)-2\ \left(g\otimes (\mathrm{d}\vec{w}\otimes_S \mathrm{d}\vec{w})\right)|\!\_\!\_ (\vec{\mathbb I}\dot{\otimes}\vec{\mathbb I}). \end{array} \end{equation} (A.19) Combining (A.13) and (A.19) gives \begin{equation} \begin{array}{l} D^2F(\vec{\Phi})(\vec{w},\vec{w})=\\[5mm] \displaystyle4\, \int_{\Sigma}(1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}}) \left[\left<\vec{\mathbb I}_{\vec{\Phi}},D^{g_{\vec{\Phi}}}\mathrm{d}\vec{w}\right>_{g_{\vec{\Phi}}}-\,2\left( g_{\vec{\Phi}}\otimes(\mathrm{d}\vec{\Phi}\dot{\otimes}_S\mathrm{d}\vec{w})\right)|\!\_\!\_ \left(\vec{{\mathbb I}}_{\vec{\Phi}}\dot{\otimes}\vec{{\mathbb I}}_{\vec{\Phi}}\right)\right]\ \left<\mathrm{d}\vec{\Phi};\mathrm{d}\vec{w}\right>_{g_{\vec{\Phi}}}\ \mathrm{d}vol_{g_{\vec{\Phi}}}\\[5mm] \displaystyle+\int_{\Sigma}|1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}}|^2 \left|\left<\mathrm{d}\vec{\Phi};\mathrm{d}\vec{w}\right>_{g_{\vec{\Phi}}}\right|{}^2+\!8\! \left|\left<\vec{\mathbb I}_{\vec{\Phi}},D^{g_{\vec{\Phi}}}\mathrm{d}\vec{w}\right>_{g_{\vec{\Phi}}}\!-\!2\!\left( g\otimes(\mathrm{d}\vec{\Phi}\dot{\otimes}_S\mathrm{d}\vec{w})\right)|\!\_\!\_ \left(\vec{{\mathbb I}}_{\vec{\Phi}}\dot{\otimes}\vec{{\mathbb I}}_{\vec{\Phi}}\right)\right|{}^2\ \mathrm{d}vol_{g_{\vec{\Phi}}}\\[5mm] \displaystyle+\,4\, \int_{\Sigma}(1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}}) \left[\left|\pi_{\vec{n}}\left(D^{g_{\vec{\Phi}}}\mathrm{d}\vec{w}\right)\right|{}_{g_{\vec{\Phi}}}^2-\left<\vec{\mathbb I}_{\vec{\Phi}}\ ;\sum_{i,j=1}^2g^{ij}\partial_{x_i}\vec{\Phi}\cdot D^g_{g_{\vec{\Phi}}} \mathrm{d}\vec{w}\ \partial_{x_j}\vec{w}\right>_{g_{\vec{\Phi}}}\right] \ \mathrm{d}vol_{g_{\vec{\Phi}}}\\[5mm] \displaystyle+\,16\, \int_{\Sigma}(1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}}) \left[(\mathrm{d}\vec{\Phi}\otimes_S \mathrm{d}\vec{w})\otimes(\mathrm{d}\vec{\Phi}\otimes_S \mathrm{d}\vec{w})\right]|\!\_\!\_ (\vec{\mathbb I}_{\vec{\Phi}}\dot{\otimes}\vec{\mathbb I}_{\vec{\Phi}}) \ \mathrm{d}vol_{g_{\vec{\Phi}}}\\[5mm] \displaystyle-\,16\, \int_{\Sigma}(1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}}) \ \left(g_{\vec{\Phi}}\otimes (\mathrm{d}\vec{\Phi}\otimes_S \,\mathrm{d}\vec{w})\right)|\!\_\!\_ \left(\vec{\mathbb I}_{\vec{\Phi}}\otimes \pi_{\vec{n}}(D^{g_{\vec{\Phi}}} \mathrm{d}\vec{w}) +\pi_{\vec{n}}(D^{g_{\vec{\Phi}}} \mathrm{d}\vec{w})\otimes\vec{\mathbb I}_{\vec{\Phi}}\right) \ \mathrm{d}vol_{g_{\vec{\Phi}}}\\[5mm] \displaystyle-\, 8\, \int_{\Sigma}(1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}}) \left(g_{\vec{\Phi}}\otimes (\mathrm{d}\vec{w}\otimes_S \mathrm{d}\vec{w})\right)|\!\_\!\_ (\vec{\mathbb I}_{\vec{\Phi}}\dot{\otimes}\vec{\mathbb I}_{\vec{\Phi}}) \ \mathrm{d}vol_{g_{\vec{\Phi}}}\\[5mm] \displaystyle+\int_{\Sigma}(1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}})^2 \left<\mathrm{d}\vec{w};\mathrm{d}\vec{w}\right>_{g_{\vec{\Phi}}}\ \mathrm{d}vol_{g_{\vec{\Phi}}}-2\,\int_{\Sigma} (1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}})^2 \ \left(\mathrm{d}\vec{\Phi}\dot{\otimes}_S \mathrm{d}\vec{w}\right)|\!\_\!\_ \left(\mathrm{d}\vec{\Phi}\otimes \mathrm{d}\vec{w}\right)\ \mathrm{d}vol_{g_{\vec{\Phi}}}. \end{array} \end{equation} (A.20) For |$\vec{w}:=\vec{v}(\vec{\Phi })$|⁠, using (11), we deduce \begin{equation} |D^2F(\vec{\Phi})(\vec{v}(\vec{\Phi}),\vec{v}(\vec{\Phi}))|\le C\,\int_{\Sigma}\left(1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}}\right)\ \left[ (1+|\vec{\mathbb I}_{\vec{\Phi}}|^2_{g_{\vec{\Phi}}})\ |\partial \vec{v}|^2(\vec{\Phi})+\ |\partial^2 \vec{v}|^2(\vec{\Phi})\right]\ \mathrm{d}vol_{g_{\vec{\Phi}}}. \end{equation} (A.21) This concludes the proof of Lemma .1. |$\Box $| References [1] Bernard , Y. and T. 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[11] Rivière , T. “ Minmax Methods in the Calculus of Variations of Curves and Surfaces .” Mini-course Columbia University , 2016 . https://people.math.ethz.ch/∼riviere/minimax. [12] Rivière , T. “ Weak Immersions of Surfaces with |${L}^2$|-bounded Second Fundamental Form .” In Geometric Analysis , 303 – 384 . IAS/Park City Mathematics Series 22 . Providence, RI : American Mathematical Society , 2016 . [13] Rivière , T. “ A viscosity method in the min-max theory of minimal surfaces .” Publ. Math. Inst. Hautes Études Sci. 126 ( 2017 ): 177 – 246 . Google Scholar Crossref Search ADS [14] Rivière , T. “ The regularity of conformal target harmonic maps .” Calc. Var. Partial Differ. Equ. 56 , no. 4 , Art. 117 ( 2017 ): 15 . [15] Varadarajan , V. S. Lie Groups, Lie Algebras, and Their Representations . ( Reprint of the 1974 edition ). Graduate Texts in Mathematics 102 . New York : Springer , 1984 . © The Author(s) 2019. Published by Oxford University Press. All rights reserved. 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Jordan,, David;Vazirani,, Monica

doi: 10.1093/imrn/rnz030pmid: N/A

Abstract Given a module |$M$| for the algebra |${\mathcal{D}}_{\mathtt{q}}(G)$| of quantum differential operators on |$G$|⁠, and a positive integer |$n$|⁠, we may equip the space |$F_n^G(M)$| of invariant tensors in |$V^{\otimes n}\otimes M$|⁠, with an action of the double affine Hecke algebra of type |$A_{n-1}$|⁠. Here |$G= SL_N$| or |$GL_N$|⁠, and |$V$| is the |$N$|-dimensional defining representation of |$G$|⁠. In this paper, we take |$M$| to be the basic |${\mathcal{D}}_{\mathtt{q}}(G)$|-module, that is, the quantized coordinate algebra |$M= {\mathcal{O}}_{\mathtt{q}}(G)$|⁠. We describe a weight basis for |$F_n^G({\mathcal{O}}_{\mathtt{q}}(G))$| combinatorially in terms of walks in the type |$A$| weight lattice, and standard periodic tableaux, and subsequently identify |$F_n^G({\mathcal{O}}_{\mathtt{q}}(G))$| with the irreducible “rectangular representation” of height |$N$| of the double affine Hecke algebra. 1 Introduction This article furthers the study of “elliptic Schur–Weyl duality”—a functorial relationship between modules for the algebra |${\mathcal{D}}_{\mathtt{q}}(G)$| of quantum differential operators on |$G=GL_N$| or |$SL_N$| and modules for type |$A$| double affine Hecke algebras. This functor has proved very useful because, while the representation theory of double affine Hecke algebras is rather well understood in terms of type |$A$| algebraic combinatorics, the representation theory of |${\mathcal{D}}_{\mathtt{q}}(G)$| is much less well understood. Let us now recall the basic construction in more detail before stating our main results. On one side of the duality, an important role is played by the ad-equivariant quantized algebra |${\mathcal{O}}_{\mathtt{q}}(G)$|⁠, a deformation of |${\mathcal{O}}(G)$| along the so-called Semenov-Tian-Shansky Poisson bracket. Likewise, |${\mathcal{D}}_{\mathtt{q}}(G)$| is a simultaneous |$\mathtt{q}$|-deformation of the algebra |$D(G)$| of differential operators on |$G$|⁠, and of functions on |$G\times G$|⁠, with respect to the Heisenberg double Poisson bracket. The quantized coordinate algebra |${\mathcal{O}}_{\mathtt{q}}(G)$| is naturally a module for |${\mathcal{D}}_{\mathtt{q}}(G)$|⁠, which we call the basic |${\mathcal{D}}_{\mathtt{q}}(G)$|-module. On the other side of the duality lie Cherednik’s double affine Hecke algebras of type |$A_{n-1}$|⁠, |${\mathbb{H}}_{q,t}$|⁠, which we will also refer to as the DAHA (because the parameters |$\mathtt{q}$| in |$U_{\mathtt{q}}({\mathfrak{g}})$|⁠, |$q$| in |${\mathbb{H}}_{q,t}(GL_n)$| and |$\boldsymbol{q}$| in |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$| do not coincide, yet each notation is well established, we distinguish them by using different typeface). There is a variant for |$G=GL_n$|⁠, and for |$G=SL_n$|⁠; we will consider both. Each is a universal deformation of the semi-direct product, |${\mathbb{C}}[{{{\mathfrak{S}}}}_n]\rtimes{\mathcal{D}}_{\mathtt{q}}(H)$|⁠, of the group algebra of the symmetric group |${{{\mathfrak{S}}}}_n$| and the algebra of difference operators on the Cartan subgroup |$H \subset G$|⁠. The |$t$| parameter deforms the reflection action of |${\mathbb{C}}[{{{\mathfrak{S}}}}_n]$| to an action of the finite Hecke algebra, while the parameter |$q$| corresponds to the step for the difference action on |$H$|⁠. In [15], the 1st author constructed functors \begin{equation} F_n^{SL}: {\mathcal{D}}_{\mathtt{q}}(SL_N)\textrm{-mod} \to{\mathbb{H}}_{\boldsymbol{q},t}(SL_n)\textrm{-mod} \end{equation} (1.1) from the category of |${\mathcal{D}}_{\mathtt{q}}(SL_N)$|-modules to the category of modules for |${\mathbb{H}}_{\boldsymbol{q},t}(SL_{n})$|⁠, specialized at |$t=\mathtt{q}$|⁠, |$\boldsymbol{q}=\mathtt{q}^{1/N}$|⁠. The underlying vector space on which |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$| acts is \begin{equation} F^{SL}_n(M) = (V^{\otimes n} \otimes M)^{inv} \end{equation} (1.2) of |$U_{\mathtt{q}}(\mathfrak{sl}_N)$|-invariants in the tensor product, where we make |$M$| a module over |$U_{\mathtt{q}}(\mathfrak{sl}_N)$| via the quantum moment map, and we take invariants with respect to that action. The action of |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$| is given by certain explicit operators, which we recall in Section 7.1. Here |$n$| and |$N$| are independent integer parameters, though in this paper we restrict to the case where |$n=kN$|⁠, for some positive integer |$k$|⁠. The functor |$F_n^{SL}$| has a natural modification to a functor \begin{equation*}F^{GL}_n: {\mathcal{D}}_{\mathtt{q}}(GL_N)\textrm{-mod} \to{\mathbb{H}}_{q,t}(GL_n)\textrm{-mod},\end{equation*} where the parameters are specialized at |$t=\mathtt{q}$|⁠, |$q=\mathtt{q}^{-2k}$|⁠. While the |$SL$| version of the functor is simpler to define, the output is simpler to analyze in the |$GL$| case. The main results are essentially the same in the two cases, but with subtle minor modifications. For this reason, in this paper we construct and analyze the functor in both settings in parallel, paying special care to what happens when we pass between the two settings. Our main results all concern the case when |$M$| is the basic |${\mathcal{D}}_{\mathtt{q}}(G)$|-module |$M={\mathcal{O}}_{\mathtt{q}}(G)$|⁠; we compute the outputs |$F^{GL}_n({\mathcal{O}}_{\mathtt{q}}(GL_N))$| and |$F^{SL}_n({\mathcal{O}}_{\mathtt{q}}(SL_N))$| of the functor in this case. This is easily seen to lie in Category |${\mathcal{O}}$| for |${\mathbb{H}}_{q,t}$|⁠, a particularly nice subcategory of |${\mathbb{H}}_{q,t}$|-modules, upon which a commutative subalgebra of |${\mathbb{H}}_{q,t}$| acts locally finitely (see Section 5). Our main result (given in more detail in Theorem 7.11) is Theorem. We have isomorphisms \begin{equation*}{F_{n}^{GL}}({\mathcal{O}}_{\mathtt{q}}(GL_{N}))\cong L(k^{N}),\quad{F_{n}^{SL}}({\mathcal{O}}_{\mathtt{q}}(SL_N))\cong \overline{L}(k^{N}),\end{equation*} where |$L(k^N)$| and |$\overline{L}(k^N)$| denote the so-called rectangular representations of |${\mathbb{H}}_{q,t}(GL_n)$| and |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$|⁠, respectively. We note that if |$k= \frac nN$| is not a positive integer, then |$F^G_n({\mathcal{O}}_{\mathtt{q}}(G)) = 0$|⁠, see Remark 7.5. First, let us discuss the |$G=GL$| case. The rectangular representations are irreducible modules in Category |${\mathcal{O}}$| that are moreover |${\mathcal{Y}}$|-semisimple, where |${\mathcal{Y}}$| is the commutative subalgebra appearing in the definition of Category |${\mathcal{O}}$|⁠, see Section 6. The proof of the main theorem relies on identifying the weights of each module and then appealing to the structure of |${\mathcal{Y}}$|-semisimple modules, to yield the isomorphism. In particular, these modules have simple spectrum and so come with distinguished basis up to scaling. This lets us focus on a combinatorial analysis of the |${\mathcal{Y}}$|-weights to fully understand the module. The computation of the |${\mathcal{Y}}$|-weights involves three well-known elements of type |$A$| combinatorics: walks on the weight lattice, tableaux on skew diagrams, and periodic tableaux on an |$N\times \infty$| strip. Schur–Weyl duality considerations, together with the Peter–Weyl decomposition, lead us naturally to a |${\mathcal{Y}}$|-weight basis of |$F^{GL}_n({\mathcal{O}}_{\mathtt{q}}(GL_N))$| indexed by so-called looped walks in the dominant chamber of the weight lattice. In fact, each Peter–Weyl component is naturally a simple submodule upon restriction to the affine Hecke algebra. We compute the |${\mathcal{Y}}$|-weights, following a technique of Orellana and Ram [19], involving well-known formulas for the action of the ribbon element of the quantum group. The most natural combinatorial expression for the |${\mathcal{Y}}$|-weights comes after we identify the set of looped walks with the set of standard tableaux on size |$n$| skew diagrams with |$k$| boxes in each of its |$N$| rows. Via this bijection, the |${\mathcal{Y}}$|-weights can be read off as the values of certain “diagonal labels” on the tableaux (a variant of what is sometimes called “content” in the literature). Finally, we introduce a procedure called “periodization”, which turns a tableau on such a skew diagram |$D$| into a periodic tableau on the |$N\times \infty$| strip. Roughly, |$D$| gets identified with a fundamental domain in the |$N\times \infty$| strip, under horizontal shifting, and then the periodization map sends a standard tableau on |$D$| to a standard periodic tableau on the |$N\times \infty$| strip, by extending the entries according to a periodicity rule with respect to the shifting action. We appeal to Cherednik’s classification [9] of irreducible |${\mathcal{Y}}$|-semisimple |${\mathbb{H}}_{q,t}(GL_n)$|-modules via periodic Young diagrams, and in particular to the enumeration of the |${\mathcal{Y}}$|-weights via periodic tableaux on such diagrams, described combinatorially in the follow up paper, [24]. In this framework, the set of standard periodic tableaux on the |$N\times \infty$| strip index a weight basis for a unique irreducible representation |$L(k^N)$|⁠, hence allowing us to prove the theorem. For |$G=SL$|⁠, the above story holds with minor modifications. The role of the commutative algebra |${\mathcal{Y}}$| is played instead by an algebra |${\mathcal{Z}}$|⁠, which we may regard as a quotient of |${\mathcal{Y}}$| by a determinant-equals-one relation. We consider periodic tableaux on the |$N\times \infty$| strip modulo a natural equivalence relation of horizontal shifting by |$k$| steps. The diagonal labeling of these must be modified so as to be compatible with this periodicity. Once this is done, analogous bijections hold as in the |$GL$| case, and the |${\mathcal{Z}}$|-weights can once again be read off from the (modified) diagonal labeling. Cherednik’s classification no longer holds on the nose, as the construction does not distinguish irreducible |${\mathcal{Z}}$|-semisimple modules from those obtained by scaling the action of the shifting generator |$\pi$| by a root of unity. Once we account for this, the |$SL$| statement in the theorem follows. 1.1 Rational degeneration of |$F^{SL}_n$| The functor |$F^{SL}_n$| is a |$\mathtt{q}$|-deformation of a similar functor \begin{equation*}\textrm{F}_n: D(\mathfrak{sl}_N)\textrm{-mod} \to \textrm{RCA}_n(c)\textrm{-mod},\end{equation*} introduced by Calaque, Enriquez, and Etingof in [7]. Here |$\textrm{RCA}_n(c)$| denotes the rational Cherednik algebra of type |$A_{n-1}$| with parameter |$c=\frac{N}{n}$|⁠. Their functor in turn builds on a similar construction of Arakawa and Suzuki [1] for the degenerate affine Hecke algebra. In [7, Theorem 8.8], the authors compute the image of their functor on the basic |${\mathcal{D}}$|-module |$M={\mathcal{O}}(\mathfrak{sl}_N)$| and identify it as a unique simple quotient of an induced rectangular module. The techniques we use here in the non-degenerate case are, however, completely different than in the degenerate setting. One can connect these theorems more directly, following [15, Section 6] and [6, Section 6]. One can define a suitable degeneration |${\mathcal{D}}_{\mathtt{q}}(SL_N)\leadsto D(\mathfrak{sl}_N)$|⁠, and a degeneration |${\mathbb{H}}_{q,t}\leadsto \textrm{RCA}_n(c)$|⁠, compatible in such a way that we obtain a degeneration of the functors |$F^{SL}_n\leadsto \textrm{F}_n$|⁠. Tracing through these degenerations, one may obtain from Theorem 7.11 an independent proof of Calaque, Enriquez, and Etingof’s description of |$\textrm{F}_n({\mathcal{O}}(\mathfrak{sl}_N))$|⁠. Category |${\mathcal{O}}$| for |$\textrm{RCA}_n(c)$| is a highest weight category and its simples can be obtained as unique simple quotients of induced modules. While the construction of the DAHA-module |$L(k^N)$| (resp. |$\overline{L}(k^N)$|⁠) combinatorially via its weight basis was a key ingredient in the proof of the main theorems, we also give an alternate construction as a quotient of an induced module, motivated by the parallel construction for the rational Cherednik algebra. Another motivation for this alternate construction is that |${\mathcal{O}}_{\mathtt{q}}(G)$| is most naturally constructed as an induced |${\mathcal{D}}_{\mathtt{q}}(G)$|-module. More precisely, we construct |$L(k^N)$| as the unique simple quotient of the module |$M(k^N)$|⁠, induced from the |$N\times k$| rectangular representation for the affine Hecke algebra in Corollary 6.11. In Theorem 6.14, we modify the construction of |$\overline{L}(k^N)$| as a quotient of the module |$\overline{M}(k^N)$|⁠, induced from the |$N\times k$| rectangular representation for the appropriate quotient |$H({\mathcal{Z}})$| of the affine Hecke algebra. It is no longer the unique simple quotient, as |$\overline{M}(k^N)$| also has quotients that are obtained from |$\overline{L}(k^N)$| by twisting by an automorphism of |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$|⁠, see also Remark 6.15. Let us also mention here an interesting parallel approach to elliptic Schur–Weyl duality, initiated in [2], and continued in [23], [26], [18], in which the role of (classical or quantum) differential operators on |$G$| is played rather by the (classical, quantum) affine Lie algebra, but double affine Hecke algebras appear (at present, only in their rational and trigonometric degenerations). While we are unaware of a precise connection between those constructions and the present framework, the basic structure of the functor—extending the ordinary Schur–Weyl duality on |$V^{\otimes n}$| by taking instead group covariants from |$V^{\otimes n}$| into a module for some auxiliary algebra—is the same, and it seems to be a very interesting question how to relate the two approaches, in particular, how to relate quantum differential operators to affine Lie algebras. 1.2 Relation to factorization homology The action of |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$| on |$F^{SL}_n(M)$| was initially defined via generators and relations. Following [3, 4], it may be re-cast as a consequence of a much more general construction of the algebra |${\mathcal{D}}_{\mathtt{q}}(G)$|⁠, via the factorization homology of surfaces valued in braided tensor categories. Since such a description does not yet appear explicitly in the literature, we outline it here. This context is not technically necessary for any of the proofs in this paper, so we will keep the discussion informal. Recall from [3], we have the following equivalence of categories: \begin{equation*}\int_{T^2\backslash D^2} \operatorname{Rep}_{\mathtt{q}}(SL_N) \simeq{\mathcal{D}}_{\mathtt{q}}(SL_N)\textrm{-mod}_{\operatorname{Rep}_{\mathtt{q}} SL_N},\end{equation*} where the left-hand side denotes the factorization homology of a once-punctured torus with coefficients in the braided tensor category |$\operatorname{Rep}_{\mathtt{q}}(SL_N)$| of integrable |$U_{\mathtt{q}}(\mathfrak{sl}_N)$|-modules, and the right-hand side denotes the category of |${\mathcal{D}}_{\mathtt{q}}(SL_N)$|-modules equipped with an equivariance structure. In analogy with the theory of |$D$|-modules, these are called “weakly equivariant” |${\mathcal{D}}_{\mathtt{q}}(SL_N)$|-modules. Recall from [4], we have a further equivalence of categories \begin{equation*}\int_{T^2} \operatorname{Rep}_{\mathtt{q}}(SL_N) \simeq{\mathcal{D}}_{\mathtt{q}}\left(\frac{SL_N}{SL_N}\right)\textrm{-mod},\end{equation*} where now the right-hand side denotes the category of |${\mathcal{D}}_{\mathtt{q}}(SL_N)$|-modules such that the quantum adjoint action, obtained via the “quantum moment map” \begin{equation*}\mu_{\mathtt{q}}:U_{\mathtt{q}}(\mathfrak{sl}_N)^{lf}\to{\mathcal{D}}_{\mathtt{q}}(SL_N)\end{equation*} makes |${\mathcal{D}}_{\mathtt{q}}(SL_N)$| into an integrable |$U_{\mathtt{q}}(\mathfrak{sl}_N)$|-module. Here |$U_{\mathtt{q}}(\mathfrak{sl}_N)^{lf}$| denotes the sub-algebra consisting of elements that are locally finite for the quantum adjoint representation, see Section 3 for more details. In analogy with the theory of |$D$|-modules, these are called “strongly equivariant” |${\mathcal{D}}_{\mathtt{q}}(SL_N)$|-modules; a weakly equivariant |${\mathcal{D}}_{\mathtt{q}}(G)$|-module is strong if its weak equivariance structure coincides with that coming from the quantum moment map. Alternatively, a |${\mathcal{D}}_{\mathtt{q}}(G)$|-module without a specified equivariance structure is a strong |${\mathcal{D}}_{\mathtt{q}}(G)$|-module if the |$U_{\mathtt{q}}({\mathfrak{g}})$|-action given by the quantum moment map is integrable. It is an immediate consequence of formula (1.2) that |$F_n^{SL}$| factors through the functor |$M\to M^{lf}$|⁠, of taking locally finite vectors for the quantum adjoint action, hence we may just as well regard it as a functor from |${\mathcal{D}}_{\mathtt{q}}(\frac{SL_N}{SL_N})\textrm{-mod}$|⁠. Indeed, it is more natural to expect the factorization homology of the closed torus to appear, since we regard the double affine Hecke algebra as a quotient of the braid group of the closed torus (also known as the elliptic braid group) by the quadratic Hecke relations. The |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$| action in the definition of |$F^{SL}_n$| most naturally arises as an action of the braid group of the closed torus, which descends to its quotient |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$| in important special cases. Let us now explain the origins of the elliptic braid group action in factorization homology terms. Given an embedding |$\iota ^n: (D^2)^{\sqcup n}\to T^2$|⁠, that is, a collection of |$n$| disjoint discs in the torus, factorization homology produces a functor, \begin{equation*}\iota^n_*:\operatorname{Rep}_{\mathtt{q}}(SL_N)^{\boxtimes n}\to \int_{T^2}\operatorname{Rep}_{\mathtt{q}}(SL_N)\simeq{\mathcal{D}}_{\mathtt{q}}\left(\frac{SL_N}{SL_N}\right)\textrm{-mod}.\end{equation*} This functor moreover carries an action, by natural isomorphisms, of the elliptic braid group |$B_n^{Ell}$|⁠. The functors |$\iota ^n_*$| have been identified in [3] and [4] with free module functors to their respective categories of |${\mathcal{D}}_{\mathtt{q}}(SL_N)$|-modules. It follows from these descriptions that, for any strongly equivariant |${\mathcal{D}}_{\mathtt{q}}(SL_N)$|-module |$M$|⁠, we may identify \begin{equation*}F^{SL}_n(M) \cong \operatorname{Hom}\!\left(\iota^n_*\left((^*V)^{\boxtimes n}\right),M\right),\end{equation*} where |$V\in \operatorname{Rep}_{\mathtt{q}}(SL_N)$| denotes the defining representation and |$(^*V)$| denotes its left dual. The action of the torus braid group |$B_n^{Ell}$|—and hence of |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$|—on |$F^{SL}_n(M)$| is that induced by the action of |$B_n^{Ell}$| on the functor |$\iota ^n_*$| itself. From this perspective, the |${\mathcal{D}}_{\mathtt{q}}(G)$|-module |${\mathcal{O}}_{\mathtt{q}}(G)$| can be understood to arise from the solid torus 3-manifold |$D^2\times S^1$|⁠, with boundary the two-dimensional torus. For this reason, |${\mathcal{O}}_{\mathtt{q}}(G)$| can be expected to play an important role in the associated four-dimensional topological field theory, more precisely in computing the invariants associated to 3-manifolds obtained by surgery. In the |$GL$| case, we modify the construction above by instead fixing an inclusion |$\iota ^{n+1}$|⁠, of |$n+1$| discs into the torus. Instead of applying |$\iota ^n_*$| to |$(^*V)^{\boxtimes n}$|⁠, we apply |$\iota ^{n+1}_*$| to |$\operatorname{det}_{\mathtt{q}}(V)^{\otimes k} \boxtimes{}^*V\boxtimes \cdots \boxtimes{}^*V$|⁠, where |$\operatorname{det}_{\mathtt{q}}(V):= \bigwedge \nolimits _{\mathtt{q}}^N(V)$|⁠. Instead of |$B^{Ell}_{n}$|⁠, the relevant braid group becomes |$B^{Ell}_{n,1}$|⁠. See Section 4.2. In future work, we plan to extend the results of the present paper to understand the structure of |${\mathcal{D}}_{\mathtt{q}}(\frac{G}{G})$|-mod more generally, through the functor |$F_n^G$|⁠. As a starting point, we will introduce the so-called quantum Springer sheaf, of which |${\mathcal{O}}_{\mathtt{q}}(G)$| is a distinguished simple quotient, and in fact is a simple summand occurring with multiplicity one. 1.3 Outline In Section 2, we collect some preliminaries about the weight lattices for type |$A$|⁠, their |$GL$| and |$SL$| variants, in particular, the set of walks in the dominant chamber and a bijection to an appropriate set of standard skew tableaux. In Section 3, we recall the quantum algebras we will study: |$U_{\mathtt{q}}({\mathfrak{g}})$|⁠, |${\mathcal{O}}_{\mathtt{q}}(G)$|⁠, and |${\mathcal{D}}_{\mathtt{q}}(G)$|⁠, and we recall some facts from the representation theory of |$U_{\mathtt{q}}({\mathfrak{g}})$|⁠. In Section 4, we recall the elliptic braid group, the double affine Hecke algebras of |$GL$| and |$SL$| type, and how these all relate to one another. In Section 5, we recall some structural results about Category |${\mathcal{O}}$| for the double affine Hecke algebra and in particular the |${\mathcal{Y}}$|-semisimple modules. In Section 6, we recall the classification of irreducible |${\mathcal{Y}}$|-semisimple modules in Category |${\mathcal{O}}$| in terms of periodic skew diagrams and their bases, indexed by standard tableaux; for expedience of exposition, we only focus on the case we will need, of the so-called “rectangular representations”. We also discuss the construction of the rectangular representation as a quotient of an induced module. In Section 7, we state and prove our main results: we give the GL modification of the functor |$F^G_n$|⁠, we identify a weight basis for |$F^G_n({\mathcal{O}}_{\mathtt{q}}(G))$|⁠, for |$G=GL_N$| and |$G=SL_N$|⁠, with the set of periodic standard tableaux on an infinite strip, and we subsequently identify |$F^G_n({\mathcal{O}}_{\mathtt{q}}(G))$| as an |${\mathbb{H}}_{q,t}$|-module with the rectangular representation. 2 Combinatorics in Type |$A$|⁠: Lattice Walks and Skew Tableaux In this section, we review a number of well-known combinatorial constructions related to the weight lattice in type |$A$|⁠. 2.1 The |$GL_N$| root system Let |$\mathfrak{g}= \mathfrak{gl}_N =\mathfrak{gl}(N,\mathbb{C})$| be the Lie algebra of the general linear algebraic group |$G=GL_N =GL(N,\mathbb{C})$|⁠. Let |${\mathbb{E}}_{\mathfrak{gl}_N} =\mathbb{R}^N$|⁠, with standard basis, \begin{equation*}\mathcal{E} = \mathcal{E}_{\mathfrak{gl}_N} = \{\epsilon_{i} \quad | \quad i=1,\ldots, N\}\end{equation*} and symmetric form |$\langle \,, \, \rangle$| with respect to which |$\mathcal{E}$| is an orthonormal basis (technically, we should be using the pairing |${\mathbb{E}}^* \times{\mathbb{E}} \to{\mathbb{R}}$|⁠, particularly if we were to consider non-simply laced type. But for convenience we identify |${\mathbb{E}}^*$| with |${\mathbb{E}}$| and express our formulas with respect to the symmetric form we thus chose to denote |$\langle \;, \; \rangle$| over the more standard |$( \; \mid \; ))$|⁠. The weight lattice of |$\mathfrak{gl}_N$| is \begin{equation*}\Lambda_{\mathfrak{gl}_N} = \bigoplus_{i=1}^N {\mathbb{Z}} \epsilon_{i} = {\mathbb{Z}}^N.\end{equation*} Elements of |$\Lambda _{\mathfrak{gl}_N}$| are called integral weights. The dominant integral weights are \begin{equation*}\Lambda^+_{\mathfrak{gl}_N} = \{ m_1\epsilon_{1} + \cdots + m_N\epsilon_{N}\,\, |\,\, m_i\in\mathbb{Z}, m_1\geq \cdots \geq m_N\}.\end{equation*} Let us denote |$\textbf{d}:= \epsilon _{1} + \epsilon _{2} + \cdots + \epsilon _{N}$|⁠, which is the weight of the trace representation in |$\mathfrak{gl}_N$|⁠, corresponding to the character of the determinant representation of |$GL_N$|⁠. Definition 2.1. Given a dominant integral weight |$\lambda = \sum _i m_i \epsilon _{i} \in \Lambda ^+_{\mathfrak{gl}_N}$|⁠, we denote by |$\operatorname{YD}(\lambda )$| the diagram (or integer partition) with fewer than |$N$| parts, \begin{equation*}\operatorname{YD}(\lambda) = (m_1-m_N, m_2-m_N, \ldots, m_{N-1}-m_N, 0).\end{equation*} The construction of |$\operatorname{YD}(\lambda )$| highlights a representative of |$\lambda \bmod \textbf{d}$| with |$m_N=0$|⁠. We can recover |$\lambda$| by remembering an additional datum, that of a diagonal labeling on |$\operatorname{YD}(\lambda )$|⁠. The diagonals of a Young diagram run down and to the right through the NW and SE corners of its boxes, and a diagonal labeling is an assignment of an integer to each diagonal. The diagonal through the upper left box of |$\operatorname{YD}(\lambda )$| is called the principal diagonal, and we decree that this diagonal is labeled with |$m_N$|⁠. The other diagonals are labeled consecutively, so that the next diagonal to the right is labeled |$m_N+1$|⁠, etc. Equivalently, we can say that the upper left box is in row |$1$| and column |$m_N+1$|⁠, and then the diagonal is the column number minus the row number. These labels will be discussed further in Section 2.4. Note the diagram |$\operatorname{YD}(\lambda +r \textbf{d})$| is that of |$\operatorname{YD}(\lambda )$| shifted |$r$| units right, and so its diagonal labels are incremented |$+r$|⁠. Hence, although we draw the same diagram for |$\lambda$| as well as |$\lambda + r\textbf{d}$|⁠, they are distinguished by their diagonal labels. Note that if we think of the “main” diagonal of an integer partition |$\gamma = (m_1, \ldots , m_N)$| with |$m_N \ge 0$| as the diagonal through the box in its upper leftmost corner, that is the one in its 1st row and 1st column, then we traditionally assign its label or “content” to be |$0$|⁠. This is consistent with our labeling as the |$(m_N+1)$|st column of |$\gamma$| agrees with the leftmost, that is, |$(m_N+1)$|st, column of |$\operatorname{YD}(\sum _i m_i \epsilon _{i})$|⁠. Or in other words, the 1st column of |$\operatorname{YD}(\sum _i m_i \epsilon _{i})$| is also the 1st column of |$\gamma$|⁠. Further, our conventions let us make sense of partitions with negative parts. See Figure 14. Given |$\lambda = \sum _i m_i \epsilon _{i}$|⁠, its dual weight is |$\lambda ^*:= \sum _i -m_i \epsilon _{N+1-i}$|⁠, in terms of coordinate vectors |$(m_1,\ldots ,m_N)^* =(-m_N, \ldots , -m_1)$|⁠. Observe therefore that if one takes |$\operatorname{YD}(\lambda ^*)$| and rotates it 180 degrees, then it is the complement to |$\operatorname{YD}(\lambda )$| in a |$N\times (m_1 - m_N)$| rectangle. See Figure 1. Let us describe the diagonal labels in terms of the inner product on |$\Lambda ^+_{\mathfrak{gl}_N}$|⁠. Consider |$\lambda$| as compared to |$\lambda + \epsilon _{i}$|⁠. The diagram has one extra box and we claim the diagonal of that box is labeled \begin{equation} \langle \lambda, \epsilon_{i} \rangle +1-i = \langle \lambda+\epsilon_{i}, \epsilon_{i} \rangle -i. \end{equation} (2.1) The new box in the |$i$|th row. Note that |$m_i = \langle \lambda , \epsilon _{i} \rangle$|⁠. The |$i$|th row of |$\lambda$| has “length” |$m_i -m_N$|⁠, which is to say it ends |$m_i-m_N$| units to the right of the leftmost column, so the new box is in column |$m_i+1 = (1+m_N) + (m_i-m_N)$|⁠, yielding that it is on the |$m_i+1-i$| diagonal. We introduce a special weight |$\rho$| given by \begin{equation*} \rho = \frac 12 ((N-1)\epsilon_{1} + (N-3)\epsilon_{2} + (N-5)\epsilon_{3} + \cdots + (1-N) \epsilon_{N}).\end{equation*} Observe |$2\rho \in \Lambda ^+_{\mathfrak{gl}_N}$|⁠, although |$\rho$| might not be depending on the parity of |$N$|⁠. We also note \begin{equation} \langle 2\rho,\epsilon_{i} \rangle = N +1-2i. \end{equation} (2.2) Hence, another way to describe the diagonal label of the box above is by \begin{equation} m_i+1-i= \langle \lambda, \epsilon_{i} \rangle + \langle \rho,\epsilon_{i} \rangle - \langle \rho,\epsilon_{1} \rangle. \end{equation} (2.3) Remark 2.2. We remark that |$\mathcal{E}$| are the weights of the |$N$|-dimensional defining representation |$V = V_{\epsilon _{1}}$| of |$G$|⁠. 2.2 The |$SL_N$| root system Let |$\mathfrak{g}= \mathfrak{sl}_N =\mathfrak{sl}(N,\mathbb{C})$| be the Lie algebra of the special linear algebraic group |$G=SL_N =SL(N,\mathbb{C})$|⁠. Here \begin{equation*}{\mathbb{E}}_{\mathfrak{sl}_N} = {\mathbb{E}}_{\mathfrak{gl}_N}\Big/ {\mathbb{R}} \cdot (\epsilon_{1} + \cdots + \epsilon_{N})\end{equation*} and the weight lattice is \begin{equation*}\Lambda_{\mathfrak{sl}_N} = \sum_{i=1}^N {\mathbb{Z}} \epsilon_{i} \Big/ {\mathbb{Z}} \cdot (\epsilon_{1} + \cdots + \epsilon_{N})\end{equation*} that is a free |${\mathbb{Z}}$|-module of rank |$N-1$|⁠. Let \begin{equation*}\mathcal{E} = \mathcal{E}_{\mathfrak{sl}_N} = \{\varepsilon_{i} \quad | \quad 1 \le i \le N\},\end{equation*} where |$\varepsilon _{i} = \epsilon _{i} + {\mathbb{Z}} \cdot (\epsilon _{1} + \cdots + \epsilon _{N})$|⁠. As before these are the weights of |$V = V_{\varepsilon _{1}}$|⁠. Note |$\varepsilon _{1} + \cdots + \varepsilon _{N}=0$|⁠. |${\mathbb{E}}_{\mathfrak{sl}_N}$| has three bases we consider of which the 1st two are also |${\mathbb{Z}}$|-bases of |$\Lambda _{\mathfrak{sl}_N}$|⁠. The 1st basis is |$\{\varepsilon _{i} \quad | \quad 1 \le i < N\} \subseteq \mathcal{E}$|⁠. The 2nd basis is |$\{ \omega _i \mid 1\le i<N \}$| consisting of the fundamental weights |$\omega _i = \varepsilon _{1} + \cdots + \varepsilon _{i}$|⁠. The weight lattice can also be expressed as \begin{equation*}\Lambda_{\mathfrak{sl}_N} = \bigoplus_{i=1}^{N-1} {\mathbb{Z}} \omega_i = \bigoplus_{i=1}^{N-1} {\mathbb{Z}} \varepsilon_{i}.\end{equation*} The set of dominant integral weights is \begin{gather*} \Lambda^+_{\mathfrak{sl}_N} = \sum_{i=1}^{N-1} {\mathbb{Z}}_{\ge 0} \omega_i = \{ m_1\varepsilon_{1}+ \cdots + m_{N-1}\varepsilon_{N-1}\,\, \mid \,\, m_i\in\mathbb{Z},\, m_1\geq \cdots \geq m_{N-1} \ge 0\}. \end{gather*} Our 3rd basis is the set of simple roots \begin{equation*}\Pi = \Pi^{A_{N-1}}=\{\alpha_{i} \mid 1 \le i < N \},\end{equation*} where |$\alpha _i = \varepsilon _{i} - \varepsilon _{i+1}$|⁠. The root lattice is |$Q =Q^{A_{N-1}} = \bigoplus _{i=1}^{N-1} {\mathbb{Z}} \alpha _i$| and it is a sublattice of index |$N = [\Lambda _{\mathfrak{sl}_N}: Q^{A_{N-1}} ]$|⁠. The set of roots, and the set of positive roots, are \begin{equation*}\Delta = \Delta^{A_{N-1}} = \{ \varepsilon_{i}-\varepsilon_{j} \mid 1 \le i, j \le N, i\neq j \},\qquad \Delta_+ = \Delta^{A_{N-1}}_+ = \{ \varepsilon_{i}-\varepsilon_{j} \mid 1 \le i < j \le N \}.\end{equation*} This is the root system of type |$A_{N-1}$|⁠. We have a symmetric form |$\langle \,, \, \rangle$| on |${\mathbb{E}}_{\mathfrak{sl}_N}$| for which |$[ \langle \alpha _i, \alpha _j \rangle ]_{i,j =1}^{N-1}$| yields the Cartan matrix of type |$A_{N-1}$|⁠. It is useful to note \begin{equation} \langle \varepsilon_{i}, \varepsilon_{j} \rangle = \delta_{ij} - \frac{1}{N}. \end{equation} (2.4) We have |$\rho \in \Lambda ^+_{\mathfrak{sl}_N}$| given by \begin{equation*} \rho = \frac 12 \sum_{\beta \in \Delta_+} \beta = \frac 12 \left( (N-1)\varepsilon_{1} + (N-3)\varepsilon_{2} + (N-5)\varepsilon_{3} + \cdots + (1-N) \varepsilon_{N} \right) = \sum_{i=1}^{N-1} \omega_i.\end{equation*} We note \begin{equation} \langle \varepsilon_{i}, 2\rho \rangle = N - (2i-1). \end{equation} (2.5) The Young diagram |$\operatorname{YD}(\lambda )$| is defined the same way as for |$GL$|⁠, and the only difference is in the diagonal labeling. We say the size of |$\lambda \in \Lambda ^+_{\mathfrak{sl}_N}$| is \begin{equation*}|\lambda| = \sum_{i=1}^{N} (m_i-m_N)\end{equation*} that corresponds to the number of boxes in |$\operatorname{YD}(\lambda )$|⁠. We label the principal diagonal |$-\frac{|\lambda |}{N}$|⁠. Hence, the diagonal labels will lie in |$-\frac pN + {\mathbb{Z}}$| if |$\lambda$| is in the coset |$\omega _p + Q$|⁠. Note that while |$|\lambda |$| is well defined, that is, |$|\lambda |=|\lambda +\textbf{d}|$|⁠, the sum |$\sum _i m_i = |\lambda | + N m_N$| is not. Another natural choice of representative for |$\lambda$| would be to choose the (possibly fractional) representative with |$\sum _i m_i = 0,$| as follows: \begin{equation*}\lambda = \left(m_1 - \frac{\sum_i m_i}{N}, \ldots, m_{N-1} - \frac{\sum_i m_i}{N}, m_N - \frac{\sum_i m_i}{N}\right).\end{equation*} One may consider its upper left corner to lie in row 1, but column |$1 + \big(m_N - \frac{\sum _i m_i}{N}\big) = 1-\frac{|\lambda |}{N}$|⁠. This explains in part the appearance of fractional diagonal labels. Another useful calculation is \begin{equation} \langle \varepsilon_{j}, \lambda \rangle = m_j - \frac{\sum_i m_i}{N}= m_j -m_N - \frac 1N |\lambda|. \end{equation} (2.6) 2.3 Walks on the weight lattice For the following definition, we may take |$G =GL_N, SL_N$|⁠. Correspondingly, we let |$\Lambda ^+=\Lambda ^+_{\mathfrak{gl}_N}$| or |$\Lambda ^+_{\mathfrak{sl}_N}$|⁠. Since |$\epsilon _{i}$| is a representative of |$\varepsilon _{i}$|⁠, we will use |$\epsilon _{i}$| below to denote either one. Definition 2.3. A walk in |$\Lambda ^+$| of length |$n$|, from weight |$\lambda$| to weight |$\mu$| is a finite sequence, \begin{equation*}\underline{u}=(\lambda = u_0, u_1,\ldots, u_n=\mu),\end{equation*} where each |$u_i\in \Lambda ^+$| and each difference |$u_{i}-u_{i-1}$| lies in |$\mathcal{E}$|⁠. We denote by |$\delta _i(\underline{u})$| the index of |$u_i-u_{i-1}\in \mathcal{E}$|⁠, so that |$u_i-u_{i-1} = \epsilon _{\delta _i(\underline{u})}$|⁠. Remark 2.4. The condition that each |$u_i\in \Lambda ^+$| means that we could call these “dominant walks” or “walks in the dominant chamber”. Since these are the only kinds of walks we will consider, we drop mention of the adjective dominant henceforth. Definition 2.5. A walk in |$\Lambda ^+_{\mathfrak{sl}_N}$| that begins and ends at the same |$\lambda$| is called a looped walk at |$\lambda$|⁠. A walk in |$\Lambda ^+_{\mathfrak{gl}_N}$| that begins at |$\lambda$| and ends at |$\lambda +k \textbf{d}$| is also called a looped walk at |$\lambda$|⁠. We denote by |${\mathcal{W}}^{N,k}_{\lambda }$|⁠, the set of all looped walks at |$\lambda$| of length |$n=Nk$|⁠. See Figures 3, 4, and 5 for examples of looped walks. Remark 2.6. Since |$\textbf{d} = 0 \in \Lambda ^+_{\mathfrak{sl}_N}$|⁠, we can say in either case that a looped walk begins at |$\lambda$| and ends at |$\lambda +k\textbf{d}$|⁠. Remark 2.7. Note that the multiset |$\{ \epsilon _{\delta _i(\underline{u})} \mid 1 \le i \le n \}$| of steps taken on any looped walk |$\underline{u}$| consists of |${\mathcal{E}}$| with integer multiplicity |$k=n/N$|⁠. Fig. 1. View largeDownload slide The skew diagram |$\textrm{D}^{7,2}_{\lambda }$| in the case |$N=7$|⁠, |$k=2$|⁠, |$n=14$|⁠. Fig. 1. View largeDownload slide The skew diagram |$\textrm{D}^{7,2}_{\lambda }$| in the case |$N=7$|⁠, |$k=2$|⁠, |$n=14$|⁠. 2.4 Skew tableaux from walks on the weight lattice For the rest of the paper, unless otherwise noted, we let |$k \in{\mathbb{Z}}_{>0}$| and |$n=Nk$|⁠. We shall now recall an alternative combinatorial description of the set |${\mathcal{W}}^{N,k}_{\lambda }$| of looped walks at |$\lambda$| in terms of skew tableaux. To begin, we associate to a weight |$\lambda \in \Lambda ^+$| a skew diagram |$\textrm{D}^{N,k}_{\lambda }$| constructed in either of the following clearly equivalent ways: (Figure 1): as the skew diagram |$\textrm{D}^{N,k}_{\lambda } = (\operatorname{YD}(\lambda ) +(k^N))/\operatorname{YD}(\lambda )$| as the skew diagram |$\textrm{D}^{N,k}_{\lambda }$| obtained by removing |$\operatorname{YD}(\lambda )$| from the upper left, and |$\operatorname{YD}(\lambda ^*)$|⁠, rotated 180 degrees, from the lower right, of the |$N\times (k+m_1-m_N)$| rectangular diagram. The skew diagram |$\textrm{D}^{N,k}_{\lambda }$| inherits diagonal labels from |$\operatorname{YD}(\lambda )$| as well as choice of principal diagonal. Fig. 2. View largeDownload slide |$G=SL_3$|⁠, |$k=1$|⁠. We list |${\mathcal{T}} \in \mathcal{SK}^{N,k}_{\lambda }$| with principal diagonal labeled |$-\frac{|\lambda |}{3}$|⁠. Once periodized via |$\overline{{\mathcal{P}}\!er}({\mathcal{T}})$|⁠, these tableaux form a |$\pi$| orbit, see Figure 12. Fig. 2. View largeDownload slide |$G=SL_3$|⁠, |$k=1$|⁠. We list |${\mathcal{T}} \in \mathcal{SK}^{N,k}_{\lambda }$| with principal diagonal labeled |$-\frac{|\lambda |}{3}$|⁠. Once periodized via |$\overline{{\mathcal{P}}\!er}({\mathcal{T}})$|⁠, these tableaux form a |$\pi$| orbit, see Figure 12. Fig. 3. View largeDownload slide A looped walk |$\underline{u}$| at |$\lambda = \epsilon _{1}\in \Lambda ^+_{\mathfrak{gl}_N}$| with four steps, its projection in |$\Lambda ^+_{\mathfrak{sl}_N}$| for |$N=2$|⁠, and the skew tableau |${\mathcal{T}}\!ab(\underline{u})$| with the principal diagonal indicated by a dashed line. For |$\Lambda ^+_{\mathfrak{gl}_N}$| the principal diagonal is labeled |$0$|⁠, while for |$\Lambda ^+_{\mathfrak{sl}_N}$| it is labeled |$-\frac{1}{2}$|⁠. Fig. 3. View largeDownload slide A looped walk |$\underline{u}$| at |$\lambda = \epsilon _{1}\in \Lambda ^+_{\mathfrak{gl}_N}$| with four steps, its projection in |$\Lambda ^+_{\mathfrak{sl}_N}$| for |$N=2$|⁠, and the skew tableau |${\mathcal{T}}\!ab(\underline{u})$| with the principal diagonal indicated by a dashed line. For |$\Lambda ^+_{\mathfrak{gl}_N}$| the principal diagonal is labeled |$0$|⁠, while for |$\Lambda ^+_{\mathfrak{sl}_N}$| it is labeled |$-\frac{1}{2}$|⁠. Fig. 4. View largeDownload slide The case |$G=SL_2$|⁠, |$n=4$|⁠. The 1st column lists the four-step walks in |${\mathcal{W}}^{2,2}_{m}$|⁠, of which there are six when |$m$| is large. The 2nd column lists the standard skew tableaux |$\mathcal{SK}^{2,2}_{m}$|⁠. The final column lists fundamental domains of the six corresponding standard periodic tableaux in |$\textrm{P}_{2}\textrm{SYT}(2^2)\big / \pi ^4$| (see Section 6.4), in the case |$m=3$|⁠. For purposes of illustration, we also take |$m=3$| across the entire 1st row; otherwise |$m$| is free. The placement of rows indicates the bijections |$\overline{{\mathcal{T}}\!ab}$| of Definition 2.9 and |$\overline{{\mathcal{P}}\!er}$| of Definition 7.3. Fig. 4. View largeDownload slide The case |$G=SL_2$|⁠, |$n=4$|⁠. The 1st column lists the four-step walks in |${\mathcal{W}}^{2,2}_{m}$|⁠, of which there are six when |$m$| is large. The 2nd column lists the standard skew tableaux |$\mathcal{SK}^{2,2}_{m}$|⁠. The final column lists fundamental domains of the six corresponding standard periodic tableaux in |$\textrm{P}_{2}\textrm{SYT}(2^2)\big / \pi ^4$| (see Section 6.4), in the case |$m=3$|⁠. For purposes of illustration, we also take |$m=3$| across the entire 1st row; otherwise |$m$| is free. The placement of rows indicates the bijections |$\overline{{\mathcal{T}}\!ab}$| of Definition 2.9 and |$\overline{{\mathcal{P}}\!er}$| of Definition 7.3. Fig. 5. View largeDownload slide The set |${\mathcal{W}}^{3,1}_{\omega _{1}}$| of three-step looped walks in |$\Lambda ^+_{\mathfrak{sl}_N}$| at |$\omega _1$|⁠. The allowed steps |$\varepsilon _{1}$|⁠, |$\varepsilon _{2}$|⁠, and |$\varepsilon _{3}$| are shown to the right of the 1st column. As in Figure 4, the 2nd and 3rd columns list the corresponding tableaux in |$\mathcal{SK}^{3,1}_{\omega _1}$| and |$\textrm{P}_{3}\textrm{SYT}(3^1) \big / \pi ^3$| under |$\overline{{\mathcal{T}}\!ab}$| from Definition 2.9 and |$\overline{{\mathcal{P}}\!er}$| from Definition 7.3. Fig. 5. View largeDownload slide The set |${\mathcal{W}}^{3,1}_{\omega _{1}}$| of three-step looped walks in |$\Lambda ^+_{\mathfrak{sl}_N}$| at |$\omega _1$|⁠. The allowed steps |$\varepsilon _{1}$|⁠, |$\varepsilon _{2}$|⁠, and |$\varepsilon _{3}$| are shown to the right of the 1st column. As in Figure 4, the 2nd and 3rd columns list the corresponding tableaux in |$\mathcal{SK}^{3,1}_{\omega _1}$| and |$\textrm{P}_{3}\textrm{SYT}(3^1) \big / \pi ^3$| under |$\overline{{\mathcal{T}}\!ab}$| from Definition 2.9 and |$\overline{{\mathcal{P}}\!er}$| from Definition 7.3. Fig. 6. View largeDownload slide The above is not a walk as it exits the dominant chamber when |$m=1$|⁠; likewise, the corresponding skew tableau assigned by |$\overline{{\mathcal{T}}\!ab}$| is not standard. Fig. 6. View largeDownload slide The above is not a walk as it exits the dominant chamber when |$m=1$|⁠; likewise, the corresponding skew tableau assigned by |$\overline{{\mathcal{T}}\!ab}$| is not standard. Recall for a (skew) diagram with |$n$| boxes that a standard tableau is a filling of its boxes with |$\{1, 2, \ldots , n\}$| such that entries increase across rows and down columns. Definition 2.8. Given a weight |$\lambda \in \Lambda ^+$|⁠, we denote by |$\mathcal{SK}^{N,k}_{\lambda }$| the set of all standard tableaux of diagonal-labeled skew shape |$\textrm{D}^{N,k}_{\lambda }$|⁠. Definition 2.9. Define the map |${\mathcal{T}}\!ab:{\mathcal{W}}^{N,k}_{\lambda } \to \mathcal{SK}^{N,k}_{\lambda },$| from length |$n =Nk$| looped walks at |$\lambda \in \Lambda ^+$| to standard skew tableaux of shape |$\textrm{D}^{N,k}_{\lambda }$| as follows: for each |$i=1,\ldots , n$| fill the leftmost vacant box in the |$\delta _i(\underline{u})$|-th row of |$\textrm{D}^{N,k}_{\lambda }$| with the symbol |$i$|⁠. We note that the map |${\mathcal{T}}\!ab$| doesn’t reference the extra data of the diagonal labeling. When it is important to differentiate the |$SL$| setting, we will use |$\overline{{\mathcal{T}}\!ab}(\lambda )$| for |$\lambda \in \Lambda ^+_{\mathfrak{sl}_N}$|⁠. The following is essentially proved in [19]: Proposition 2.10. The map |${\mathcal{T}}\!ab:{\mathcal{W}}^{N,k}_{\lambda } \xrightarrow{\sim } \mathcal{SK}^{N,k}_{\lambda }$| is a bijection. Example 2.11. For the first walk in Figure 4, \begin{equation*}\underline{u} = (m, m+{\color{red} \varepsilon_{1}}, m+\varepsilon_{1} + {\color{red} \varepsilon_{1}}, m+\varepsilon_{1} + \varepsilon_{1} + {\color{green!80!black}\varepsilon_{2}}, m+\varepsilon_{1} + \varepsilon_{1} + \varepsilon_{2} +{\color{green!80!black} \varepsilon_{2}} = m),\end{equation*} and so the sequence \begin{equation*}(\delta_1(\underline{u}), \delta_2(\underline{u}), \delta_3(\underline{u}), \delta_4(\underline{u})) = ({\color{red}1},{\color{red} 1}, {\color{green!80!black} 2}, {\color{green!80!black} 2}).\end{equation*} Compare this to the 1st skew tableau |${\mathcal{T}} = {\mathcal{T}}\!ab(\underline{u})$| in Figure 4 that places 1 and 2 in the 1st row, 3 and 4 in the 2nd row. See Figure 6 to see how leaving the dominant chamber results in the tableau becoming non-standard. In particular, if |$m>1$|⁠, then the walk would not leave the chamber and the 3 would not be directly above the 2. 3 Quantum Algebras: |$U_{\mathtt{q}}({\mathfrak{g}})$|⁠, |${\mathcal{O}}_{\mathtt{q}}(G),$| and |${\mathcal{D}}_{\mathtt{q}}(G)$| 3.1 The quantum groups |$U_{\mathtt{q}}(\mathfrak{gl}_N)$|⁠, |$U_{\mathtt{q}}(\mathfrak{sl}_N)$| We will consider the braided tensor categories |$\operatorname{Rep}_{\mathtt{q}}(GL_N)$| and |$\operatorname{Rep}_{\mathtt{q}}(SL_N)$| of integrable |$U_{\mathtt{q}}(\mathfrak{gl}_N)$|-modules (resp. |$U_{\mathtt{q}}(\mathfrak{sl}_N)$|-modules). We refer to [17] for detailed definitions, in particular the Serre presentation of the quantum groups |$U_{\mathtt{q}}(\mathfrak{gl}_N)$| and |$U_{\mathtt{q}}(\mathfrak{sl}_N)$|⁠, the formulas for |$R$|-matrices, and the Peter–Weyl theorem. Recall that a |$U_{\mathtt{q}}(\mathfrak{g})$|-module is called integrable if the Cartan generators |$K_i$| act diagonalizably with eigenvalues in |$\mathtt{q}^{\mathbb{Z}}$|⁠, and each vector lies in a finite-dimensional submodule. For each |$\lambda \in \Lambda ^+_{\mathfrak{gl}_N}$| or |$\Lambda ^+_{\mathfrak{sl}_N}$|⁠, we denote by |$V_\lambda$| the unique simple module of highest weight |$\lambda$|⁠. Note that we have an isomorphism |$(V_\lambda )^* \cong V_{\lambda ^*}$|⁠. 3.2 The vector representation The representation |$V_{\epsilon _{1}}\cong{\mathbb{C}}^N$| for either |$U_{\mathtt{q}}(\mathfrak{gl}_N)$| and |$U_{\mathtt{q}}(\mathfrak{sl}_N)$| will be simply denoted |$V$|⁠. We fix |$e_{1},\ldots , e_N$| to be the standard basis for |$V$|⁠, and we denote by |$\rho _V$| the associated homomorphism to |$\operatorname{End}(V)$|⁠. Recall that the |$GL_N$||$R$|-matrix for the vector representation can be expressed explicitly \begin{equation} R:=(\rho_V\otimes\rho_V)({\mathcal{R}}) = \left(\mathtt{q}\sum_{i}E_i^i\otimes E_i^i +\sum_{i\neq j}E_i^i\otimes E_j^j+ (\mathtt{q}-\mathtt{q}^{-1})\sum_{i>j}E_i^j\otimes E_j^i\right). \end{equation} (3.1) We define |$R^{ik}_{jl},(R^{-1})^{ik}_{jl}\in{\mathbb{C}}$|⁠, for |$i,j,k,l=1,\ldots ,N$| by \begin{eqnarray*} R(e_{i}\otimes e_{j}) =\sum_{k,l}R_{ij}^{kl}(e_{k}\otimes e_{l}),\quad R^{-1}(e_{i}\otimes e_{j}) =\sum_{k,l}(R^{-1})_{ij}^{kl}(e_{k}\otimes e_{l}). \end{eqnarray*} We can write the coefficients explicitly as follows: \begin{equation} R_{ij}^{kl}=\left\{\begin{array}{ccc}\mathtt{q}, & & i=j=k=l; \\1, & & i=k\neq j=l; \\\mathtt{q}-\mathtt{q}^{-1}, & & i=l<j=k; \\0, & &\textrm{otherwise};\end{array}\right.\quad (R^{-1})_{ij}^{kl}=\left\{\begin{array}{ccc}\mathtt{q}^{-1}, & & i=j=k=l; \\1, & & i=k\neq j=l; \\\mathtt{q}^{-1}-\mathtt{q}, & & i=l<j=k; \\0, & &\textrm{otherwise}. \end{array}\right. \end{equation} (3.2) Let |$\tau :V\otimes V$| denote the tensor flip, |$\tau (v\otimes w)=w\otimes v$|⁠. The braiding, |$\sigma _{V,V}=\tau \circ R$|⁠, for |$U_{\mathtt{q}}(\mathfrak{gl}_N)$| satisfies a Hecke relation \begin{equation*}(\sigma_{V,V}-\mathtt{q})(\sigma_{V,V}+\mathtt{q}^{-1})=0.\end{equation*} The |$SL_N$||$R$|-matrix on the vector representation is equal to the |$GL_N$||$R$|-matrix, multiplied by a factor of |$\mathtt{q}^{-\frac{1}{N}}$|⁠. To avoid confusion, we will reserve the notation |$R$| for the |$GL_N$||$R$|-matrix and write |$\mathtt{q}^{-\frac{1}{N}}R$| to reference the |$SL_N$||$R$|-matrix. Hence, the resulting braiding |$\sigma _{V,V}=\tau \circ (\mathtt{q}^{-\frac{1}{N}} R)$| for |$U_{\mathtt{q}}(\mathfrak{sl}_N)$| satisfies the shifted Hecke relation \begin{equation*}\big(\sigma_{V,V}-\mathtt{q}^{-\frac{1}{N}}\mathtt{q}\big)\big(\sigma_{V,V}+\mathtt{q}^{-\frac{1}{N}}\mathtt{q}^{-1}\big)=0.\end{equation*} 3.3 The determinant representation We will denote by |$\operatorname{det}_{\mathtt{q}}(V)=\bigwedge \nolimits _q^N(V)$| the one-dimensional determinant representation of |$U_{\mathtt{q}}(\mathfrak{gl}_N)$|⁠, and we will abbreviate \begin{equation*}\operatorname{det}_{\mathtt{q}}^k(V) = \left\{\begin{array}{ll}(\operatorname{det}_{\mathtt{q}}(V))^{\otimes k},& k\geq 0,\\ (\operatorname{det}_{\mathtt{q}}(V)^*)^{\otimes k},& k<0.\end{array}\right.\end{equation*} 3.4 The ribbon element Recall the ribbon element (note that axioms for the ribbon element and its inverse are swapped between [15] and [19]; we will follow the conventions of [19]) |$\nu \in U_{\mathtt{q}}(\mathfrak g)$|⁠. The ribbon element is central and satisfies the identity \begin{equation}\Delta(\nu)=R_{21}R_{12}(\nu\otimes \nu). \end{equation} (3.3) It acts on any irreducible representation |$V_\lambda$|⁠, and hence on any isotypic component |$X[\lambda ]$|⁠, by the scalar \begin{equation} \nu|_{X[\lambda]} =\mathtt{q}^{\langle\lambda+2\rho,\lambda\rangle}. \end{equation} (3.4) 3.5 The quantum coordinate algebra Definition 3.1. The reflection equation algebra of type |$GL_N$|⁠, denoted |${\mathcal{O}}_{\mathtt{q}}(Mat_N)$|⁠, is the algebra generated by symbols |$a^i_j$|⁠, for |$i,j = 1,\ldots N$|⁠, subject to the relations, \begin{equation*}R_{21}A_1R_{12}A_2 = A_2R_{21}A_1R_{12},\end{equation*} where |$A:= \sum _{i,j} a^i_j E^j_i$| is a matrix with entries the generators |$a^i_j$|⁠, and for a matrix |$X$|⁠, we write |$X_1=X\otimes \operatorname{Id}_V$|⁠, |$X_2=\operatorname{Id}_V\otimes X$|⁠, so that the matrix equation above is equivalent to the list of relations, for |$i,j,n,r\in \{1,\ldots ,N\}$| \begin{equation}\sum_{k,l,m,p}R^{ij}_{kl}a^l_mR^{mk}_{np}a^p_r = \sum_{s,t,u,v} a^i_sR^{sj}_{tu}a^u_vR^{vs}_{nr}\end{equation} (3.5) Remark 3.2 We note that, since |$R$| appears quadratically on both sides of the defining relation, the relations are unchanged by replacing |$R \leadsto \mathtt{q}^{-\frac{1}{N}}R$|⁠. Proposition 3.3 ([16]). The element \begin{equation*}\operatorname{det}_{\mathtt{q}}(A):= \sum_{\sigma\in{{{\mathfrak{S}}}}_N} (-\mathtt{q})^{\ell(\sigma)}\cdot \mathtt{q}^{e(\sigma)} a^1_{\sigma(1)}\cdots a^N_{\sigma(N)}\end{equation*} is central in |${\mathcal{O}}_{\mathtt{q}}(Mat_N)$|⁠. Here |$\ell (\sigma )$| denotes the length, that is, the number of pairs |$i<j$| such that |$\sigma (i)>\sigma (j)$|⁠, and |$e(\sigma )$| denotes the excedence, that is, the number of elements |$i$| such that |$\sigma (i)>i$|⁠. Definition 3.4. The quantum coordinate algebras |${\mathcal{O}}_{\mathtt{q}}(GL_N)$| and |${\mathcal{O}}_{\mathtt{q}}(SL_N)$| are the algebras obtained from |${\mathcal{O}}_{\mathtt{q}}(Mat_N)$|⁠, by inverting, respectively specializing to one, the central element |$\operatorname{det}_{\mathtt{q}}$|⁠. That is, \begin{equation*}{\mathcal{O}}_{\mathtt{q}}(GL_N) = {\mathcal{O}}_{\mathtt{q}}(Mat_N)[\operatorname{det}_{\mathtt{q}}(A)^{-1}],\qquad{\mathcal{O}}_{\mathtt{q}}(SL_N) = {\mathcal{O}}_{\mathtt{q}}(Mat_N)/\langle \operatorname{det}_{\mathtt{q}}(A)-1 \rangle.\end{equation*} Theorem 3.5 (Peter–Weyl decomposition). As a module for |$U_{\mathtt{q}}(\mathfrak{gl}_N)$| and |$U_{\mathtt{q}}(\mathfrak{sl}_N)$|⁠, respectively, we have isomorphisms \begin{equation*}{\mathcal{O}}_{\mathtt{q}}(GL_N) \cong \bigoplus_{\lambda\in \Lambda^+_{\mathfrak{gl}_N}} V_\lambda^* \otimes V_\lambda, \qquad{\mathcal{O}}_{\mathtt{q}}(SL_N) \cong \bigoplus_{\lambda\in \Lambda^+_{\mathfrak{sl}_N}} V_\lambda^* \otimes V_\lambda.\end{equation*} 3.6 Quantum differential operators The algebra of quantum differential operators on |$G$|⁠, which we denote by |${\mathcal{D}}_{\mathtt{q}}(G)$|⁠, was studied in many different settings. The presentation below as a twisted tensor product is adapted from the paper [25] (see also [6]), and hence matches the conventions of [15] (see footnote 3 there, however). Definition 3.6. For |$G=GL_N$|⁠, or |$SL_N$|⁠, the algebra |${\mathcal{D}}_{\mathtt{q}}(G)$| is the twisted tensor product, \begin{gather} {\mathcal{D}}_{\mathtt{q}}(G) = {\mathcal{O}}_{\mathtt{q}}(G)\widetilde{\otimes} {\mathcal{O}}_{\mathtt{q}}(G), \end{gather} (3.6) with cross relations, \begin{align*}D_2R_{21}A_1 = R_{21}A_1R_{12}D_2R_{21}, &\qquad \textrm{if }G=GL_N\\ D_2R_{21}A_1 = R_{21}A_1R_{12}D_2R_{21}\mathtt{q}^{-2/N}, &\qquad \textrm{if }G=SL_N, \end{align*} where |$A= \sum _{i,j}a^i_jE^j_i$| and |$D=\sum _{i,j}\partial ^i_j E^j_i$| denote matrices of generators of each tensor factor, so that the matrix equation above is equivalent to the list of cross relations, for |$i,k,l,n\in \{1,\ldots ,N\}$| \begin{equation*}\sum_{j,m} \partial^i_jR^{jk}_{lm}a^m_n = \sum_{p,r,s,t,u,v}R^{ik}_{pr}a^r_sR^{sp}_{tu}\partial^u_vR^{vt}_{ln}.\end{equation*} We denote by |$\ell$| and |$\partial _{\lhd }$| the inclusions into the 1st and 2nd tensor factor of (3.6), so that |$\ell \otimes \partial _\lhd :{\mathcal{O}}_{\mathtt{q}}(G)\otimes{\mathcal{O}}_{\mathtt{q}}(G)\to{\mathcal{D}}_{\mathtt{q}}(G)$| is the tautological isomorphism of |$U_{\mathtt{q}}({\mathfrak{g}})$|-modules (however, it is not an algebra homomorphism). This is a |$\mathtt{q}$|-deformation of the tensor decomposition |${\mathcal{D}}(G) \cong{\mathcal{O}}(G)\otimes U({\mathfrak{g}})$| into functions on |$G$|⁠, and the vector fields of left translation, hence the notation. 4 Double Affine Hecke Algebras of Type |$A$| Let |${\mathcal{K}}$| denote a field of characteristic zero, and let |$q,t\in{\mathcal{K}}^\times$|⁠, and assume neither |$q$| nor |$t$| is a root of unity. Typical instances are |${\mathcal{K}}={\mathbb{C}}$|⁠, |${\mathbb{C}}(t)$|⁠, or |${\mathbb{C}}(q,t)$|⁠. 4.1 The extended affine symmetric group Definition 4.1. The extended affine symmetric group is (we drop the first relation when |$n=2$|⁠) \begin{equation*}\widehat{\mathfrak{S}}_n = \left\langle \pi, s_i, \,\, i\in{\mathbb{Z}}/ n{\mathbb{Z}} \left|\quad \begin{array}{ll} s_is_{i+1}s_i = s_{i+1}s_is_{i+1} & \textrm{for }i\in{\mathbb{Z}}/n{\mathbb{Z}},\\ s_is_j=s_js_i &\textrm{for }j \not\equiv i \pm 1 \bmod n,\\ \pi s_i = s_{i+1}\pi &\textrm{for }i\in{\mathbb{Z}}/n{\mathbb{Z}}, \\ s_i^2 = 1 & \textrm{for }i\in{\mathbb{Z}}/n{\mathbb{Z}} \end{array}\right.\right\rangle .\end{equation*} We can associate to |$SL_n$| the quotient |$\overline{{\mathfrak{S}}}_n = \widehat{{\mathfrak{S}}}_n/\langle \pi ^n \rangle$|⁠. Then we can think of the image, |$\overline \pi$|⁠, of |$\pi$| as the Dynkin diagram automorphism or the generator of the cyclic group |$\Lambda _{\mathfrak{sl}_N}/Q$|⁠. Note both |$\widehat{{\mathfrak{S}}}_n$| and |$\overline{{\mathfrak{S}}}_n$| have as a subgroup the affine symmetric group |$\langle s_i \mid i \in{\mathbb{Z}}/n{\mathbb{Z}} \rangle$|⁠. We recall that |$\widehat{{\mathfrak{S}}}_n$| acts on |${\mathbb{Z}}$| by |$n$|-periodic permutations, that is, bijections |$\sigma :{\mathbb{Z}}\to{\mathbb{Z}}$| such that |$\sigma (i+n) = \sigma (i)+n$|⁠. It also acts on the set |$({\mathcal{K}}^\times )^n$| via \begin{align} s_i\cdot(a_1,\ldots,a_i,a_{i+1},\ldots a_n) &= (a_1, \ldots, a_{i+1}, a_i, \ldots, a_n)\nonumber \\ s_0\cdot(a_1,a_2,\ldots, a_{n-1},a_n) &= (q a_n, a_2, \ldots, a_{n-1}, q^{-1} a_1) \\ \pi \cdot (a_1, \ldots, a_n) &= (q a_n, a_1, a_2, \ldots, a_{n-1}).\nonumber \end{align} (4.1) This action is relevant to the |$GL_n$| double affine Hecke algebra (see Section 4.3). We modify the action of |$\widehat{{\mathfrak{S}}}_n$| on |$({\mathcal{K}}^\times )^n$| for |$SL_n$| as follows, so that it will factor through the quotient |$\overline{{\mathfrak{S}}}_n$|⁠. Let |$\boldsymbol{q} \in{\mathcal{K}}^\times$| be another constant that we assume is not a root of unity. \begin{align} s_i\cdot(z_1,\ldots,z_i,z_{i+1},\ldots z_n) &= (z_1,\ldots,z_{i+1},z_{i},\ldots z_n),\nonumber\\ s_0\cdot(z_1,z_2,\ldots, z_{n-1},z_n) &= (\boldsymbol{q}^{-2n}z_n,z_2,\ldots, z_{n-1},\boldsymbol{q}^{2n}z_1),\\ \pi \cdot(z_1,\ldots, z_n) &= (\boldsymbol{q}^{-2n+2}z_n, \boldsymbol{q}^2z_1,\boldsymbol{q}^2z_2,\ldots, \boldsymbol{q}^2z_{n-1}).\nonumber \end{align} (4.2) It is easy to check that in this case, the action of the generators satisfy the defining relations of |$\widehat{{\mathfrak{S}}}_n$|⁠, as well as the additional relation |$\pi ^n=\operatorname{Id}$|⁠. See Section 4.5 for a discussion of the relationship between |$\boldsymbol{q}^{-2n}$| and |$q$| that unifies the action of |$s_0$| in (4.1) with that in (4.2). 4.2 The elliptic braid group Definition 4.2. The elliptic braid group|$B_n^{Ell}$| is the fundamental group of the configuration space of |$n$| points on the torus |$T^2$|⁠. The marked elliptic braid group|$B_{n,1}^{Ell}$| is the fundamental group of the configuration space of |$n$| points on the punctured torus |$T^2\backslash D^2$|⁠. Proposition 4.3 ([5, 22]). The elliptic braid group |$B_n^{Ell}$| is presented by Pairwise commuting generators |$\textsf{X}_1,\ldots , \textsf{X}_n$|⁠. Pairwise commuting generators |$\textsf{Y}_1,\ldots , \textsf{Y}_n$|⁠. The braid group of the plane \begin{equation*}B_n = \left\langle \quad \textsf{T}_1, \ldots, \textsf{T}_{n-1} \quad \left| \begin{array}{cl}\textsf{T}_i\textsf{T}_{i+1}\textsf{T}_i = \textsf{T}_{i+1}\textsf{T}_i\textsf{T}_{i+1}, &i=1,\ldots n-2,\\ \textsf{T}_i\textsf{T}_j=\textsf{T}_j\textsf{T}_i, &|i-j|\geq 2\end{array}\right.\right\rangle,\end{equation*} with the cross relations \begin{gather*} \textsf{T}_i\textsf{X}_i\textsf{T}_i = \textsf{X}_{i+1},\,\, \textsf{T}_i\textsf{Y}_i\textsf{T}_i=\textsf{Y}_{i+1}, \quad i=1,\ldots,n-1,\\ \textsf{X}_1\textsf{Y}_2=\textsf{Y}_2\textsf{X}_1\textsf{T}_1^2,\qquad \left(\prod_i\textsf{X}_i\right)\textsf{Y}_j = \textsf{Y}_j\left(\prod_i\textsf{X}_i\right), \quad j=1,\ldots, n,\\ \textsf{X}_i\textsf{T}_j=\textsf{T}_j\textsf{X}_i,\,\, \textsf{Y}_i\textsf{T}_j=\textsf{T}_j\textsf{Y}_i, \quad \textrm{for }|i-j|>1.\end{gather*} It is well known that |$B_{n,1}^{Ell}$| embeds into |$B_{n+1}^{Ell}$|⁠, as the subgroup generated by |$\textsf{X}_1,\ldots ,\textsf{X}_n,$||$\textsf{Y}_1,\ldots ,\textsf{Y}_n,$||$\textsf{T}_1,\ldots , \textsf{T}_{n-1}$| and by |$\textsf{T}_n^2$|⁠. 4.3 Double affine Hecke algebra for |$GL_n$| Definition 4.4. The |$GL_n$| double affine Hecke algebra |${\mathbb{H}}_{q,t}(GL_n)$| is the |${\mathcal{K}}$|-algebra presented by generators \begin{equation*}T_0,T_1,\ldots T_{n-1}, \pi^{\pm 1}, Y_1^{\pm 1},\ldots, Y_n^{\pm 1},\end{equation*} subject to relations (as with |$\widehat{{\mathfrak{S}}}_n$|⁠, we drop the relations on the second line when |$n=2$|⁠) \begin{align} &(T_i-t)(T_i+t^{-1})=0 \quad (i=0,\ldots, n-1),& && \end{align} (4.3) \begin{align} &T_iT_jT_i = T_jT_iT_j\quad (\,j\equiv i\pm 1 \bmod n),& &T_iT_j = T_jT_i \quad (\textrm{otherwise}),&\\ &\pi T_i\pi^{-1} = T_{i+1} \quad (i=0,\ldots, n-2),& &\pi T_{n-1}\pi^{-1}=T_0,&\nonumber\\ &T_iY_iT_i=Y_{i+1} \quad (i=1,\ldots, n-1),& &T_0Y_nT_0 = q^{-1}Y_1&\nonumber\\ &T_i Y_j = Y_j T_i \quad (\,j \not\equiv i, i+1 \bmod n),& &&\nonumber\\ &\pi Y_i\pi^{-1} = Y_{i+1} \quad (i=1,\ldots, n-1),& &\pi Y_{n}\pi^{-1}= q^{-1}Y_1.&\nonumber \end{align} (4.4) Any |$\sigma \in \widehat{{\mathfrak{S}}}_n$| has a canonical lift |$T_\sigma \in{\mathbb{H}}_{q,t}(GL_n)$|⁠, defined as follows: if |$\sigma$| is written as a reduced word |$\sigma =\pi ^r s_{i_1}\cdots s_{i_k}$| of the generators, then we set |$T_\sigma = \pi ^r T_{i_1}\cdots T_{i_k}$|⁠. This expression for |$T_\sigma$| is well defined because the |$T_i$| satisfy the same braid relations as |$s_i \in \widehat{{\mathfrak{S}}}_n$|⁠. We abuse notation and abbreviate |$\pi = T_\pi$|⁠. For |$\beta = (b_1, b_2, \ldots , b_n) \in{\mathbb{Z}}^n$|⁠, we denote by |$Y^\beta = Y_1^{b_1} \cdots Y_n^{b_n}$| the corresponding monomial. We note that |${\mathbb{H}}_{q,t}(GL_n)$| has basis |$\{ T_{\sigma } Y^\beta \mid \sigma \in \widehat{{\mathfrak{S}}}_n, \beta \in{\mathbb{Z}}^n \}$|⁠. Given the combinatorial viewpoint of this paper, the presentation above involving |$\pi$| is the most convenient for us. However, it is sometimes desirable to define \begin{gather*} X_1 = \pi T_{n-1}^{-1} \cdots T_{2}^{-1} T_{1}^{-1},\qquad X_{i+1} = T_i X_i T_i \quad (i=1, \ldots, n-1). \end{gather*} Then it is not hard to show |$X_i X_j =X_j X_i$| and that these elements generate a Laurent polynomial subalgebra |${\mathcal{K}}[X_1^{\pm 1},\ldots , X_n^{\pm 1}] \subseteq{\mathbb{H}}_{q,t}(GL_n)$|⁠. It is also easy to show that |$X_1 X_2 \cdots X_n = \pi ^n$|⁠, and that this element |$q$|-commutes with each |$Y_i$|⁠. We thus have two commutative subalgebras, \begin{equation*}{\mathcal{X}} = {\mathcal{K}}\big[X_1^{\pm 1},\ldots, X_n^{\pm 1}\big] \qquad \textrm{and} \qquad{\mathcal{Y}} = {\mathcal{K}}\big[Y_1^{\pm 1},\ldots, Y_n^{\pm 1}\big],\end{equation*} of |${\mathbb{H}}_{q,t}(GL_n)$|⁠, each isomorphic to a Laurent polynomial ring. Further, the DAHA |${\mathbb{H}}_{q,t}(GL_n)$| has two distinguished subalgebras, \begin{equation*}H({\mathcal{Y}}) = \big\langle T_1, \ldots, T_{n-1}, Y_1^{\pm 1},\ldots, Y_n^{\pm 1} \big\rangle,\qquad H({\mathcal{X}}) = \big\langle T_0, T_1, \ldots, T_{n-1}, \pi^{\pm 1} \big\rangle,\end{equation*} each of which is isomorphic to the extended affine Hecke algebra of type |$A$|⁠. Finally, we will denote by |$H_n$| the finite Hecke algebra, generated by |$T_i$| and subject to the relations (4.3) and (4.4). The finite Hecke algebra sits as a common subalgebra of |$H({\mathcal{X}})$| and |$H({\mathcal{Y}})$|⁠, but it is also naturally realized as a quotient of |$H({\mathcal{Y}})$| via the homomorphism |$H({\mathcal{Y}}) \to H_n$| determined by \begin{gather} T_i \mapsto T_i,\qquad Y_1 \mapsto 1. \end{gather} (4.5) In this way, we can inflate any |$H_n$|-module to be a |$H({\mathcal{Y}})$|-module. We note that the center |$Z(H({\mathcal{Y}})) = {\mathcal{K}}[Y_1^{\pm 1}, \ldots , Y_n^{\pm 1}]^{{{{\mathfrak{S}}}}_n}$| consists of the symmetric Laurent polynomials. In particular, the product |$Y_1 Y_2 \cdots Y_n$| commutes with all |$T_i$|⁠. (It even commutes with |$T_0$| that is not in |$H({\mathcal{Y}})$|⁠, but does not commute with |$\pi$|⁠.) Proposition 4.5. There exists a unique isomorphism, \begin{equation*}\phi:{\mathcal{K}}\left[B_{n,1}^{Ell}\right]\Big/\left\langle \textsf{T}^2_n=q,\qquad (\textsf{T}_i-t)(\textsf{T}_i+t^{-1})=0\quad (i=1,\ldots, n-1) \right\rangle \to{\mathbb{H}}_{q,t}(GL_n),\end{equation*} such that \begin{equation*}\phi(\textsf{T}_i)=T_i, \quad (i=1,\ldots n-1),\qquad \phi(\textsf{X}_1) = \pi T_{n-1}^{-1} \cdots T_{2}^{-1} T_{1}^{-1}, \qquad \phi(\textsf{Y}_1) = Y_1.\end{equation*} Proof. We shall mostly leave this to the reader, but let us explain the relation |$\textsf{T}^2_n=q$| here. In |$B_{n+1}^{Ell}$|⁠, we have |$\textsf{X}_{n+1}\textsf{Y}_{n}=\textsf{T}_n^2\textsf{Y}_{n}\textsf{X}_{n+1}$|⁠. Recall too that the |$\textsf{X}_i$| commute with each other and further |$\textsf{T}_n$| commutes with |$\textsf{X}_n \textsf{X}_{n+1}$|⁠. Hence, \begin{eqnarray*} \textsf{X}_{n+1} \textsf{Y}_n (\textsf{X}_1 \cdots \textsf{X}_n) &=& \textsf{T}_n^2\textsf{Y}_{n}\textsf{X}_{n+1} (\textsf{X}_1 \cdots \textsf{X}_n) \\ &=& \textsf{T}_n^2\textsf{X}_1 \cdots \textsf{X}_n\textsf{X}_{n+1} \textsf{Y}_{n}\\ &=& \textsf{X}_1 \cdots \textsf{X}_n\textsf{X}_{n+1} \textsf{T}_n^2 \textsf{Y}_{n}\\ &&\text{multiplying both sides on the left by }\textsf{X}_{n+1}^{-1}\text{ gives}\\ \textsf{Y}_n (\textsf{X}_1 \cdots \textsf{X}_n) &=& (\textsf{X}_1 \cdots \textsf{X}_n) \textsf{T}_n^2\textsf{Y}_{n}. \end{eqnarray*} It is easy to check |$\phi (\textsf{X}_1 \cdots \textsf{X}_n) = \pi ^n$|⁠. Hence, setting |$\phi (\textsf{T}_n^2)=q$| is consistent with our |$\pi ^n q Y_n \pi ^{-n} = Y_n$| relation. We leave it to the reader to check the other relations. 4.4 Double affine Hecke algebra for |$SL_n$| Let us fix further constants |$\boldsymbol{r}, \boldsymbol{q} \in{\mathcal{K}}^\times$|⁠, which we assume are not roots of unity. However, for convenience, we will assume |${\mathcal{K}}^\times$| contains primitive |$n$|th roots of unity. Definition 4.6. The |$SL_n$| double affine Hecke algebra |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$| is presented by generators \begin{equation*}T_0,T_1,\ldots T_{n-1}, \pi^{\pm 1}, Z_1^{\pm 1},\ldots, Z_n^{\pm 1},\end{equation*} subject to relations \begin{align*} &(T_i-t)(T_i+t^{-1})=0 \quad (i=0,\ldots, n-1),& &&\\ &T_iT_jT_i = T_jT_iT_j\quad (\,j\equiv i\pm 1 \bmod n),& &T_iT_j = T_jT_i \quad (\textrm{otherwise}),&\\ &\pi T_i\pi^{-1} = T_{i+1} \quad (i=0,\ldots, n-2),& &\pi T_{n-1}\pi^{-1}=T_0,&\\ &T_iZ_iT_i=Z_{i+1} \quad (i=1,\ldots, n-1),& &T_i Z_j = Z_j T_i \quad (\,j \not\equiv i, i+1 \bmod n),& \\ &T_0Z_nT_0 = \boldsymbol{q}^{2n}Z_1,& &Z_1 Z_2 \cdots Z_n = \boldsymbol{q}^{n(n-1)}\boldsymbol{r}^n, & & \pi^n=1,&\\ &\pi Z_i\pi^{-1} = \boldsymbol{q}^{-2} Z_{i+1} \quad (i=1,\ldots, n-1),& &\pi Z_{n}\pi^{-1}= \boldsymbol{q}^{2n-2}Z_1.& \end{align*} Similar to the |$GL$| case, we have two commutative subalgebras, \begin{equation*}{\mathcal{X}} = \big\langle X_1^{\pm 1},\ldots, X_n^{\pm 1}\big\rangle \qquad \textrm{and} \qquad{\mathcal{Z}} = \big\langle Z_1^{\pm 1},\ldots, Z_n^{\pm 1}\big\rangle\end{equation*} of |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$|⁠, each isomorphic to the quotient of a Laurent polynomial ring by a single relation on the product of generators, as in Proposition 4.9 below. We let |$H({\mathcal{Z}})$| denote the subalgebra generated by |${\mathcal{Z}}$| and |$H_n$|⁠. We can realize |$H({\mathcal{Z}})$| as a quotient of the extended affine Hecke algebra by the relation |$Y_1\cdots Y_n= \boldsymbol{q}^{n(n-1)}\boldsymbol{r}^n$| \begin{align*}H({\mathcal{Y}})&\to H({\mathcal{Z}}),\\ T_i&\mapsto T_i &\qquad (i=1, \ldots, n-1) \\ Y_i&\mapsto Z_i&\qquad (i=1, \ldots, n). \end{align*} Definition 4.7. Let |$a \in{\mathcal{K}}^\times$| satisfy |$a^n = 1$|⁠. Given an |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$|-module |$M$|⁠, denote by |$M^a$| the twist of |$M$| by the automorphism of |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$| that sends |$T_i \mapsto T_i$|⁠, |$Z_i \mapsto Z_i$|⁠, |$\pi \mapsto a \pi$|⁠. It may happen for some modules |$M$| that |$M \cong M^a$|⁠. See Section 6.4. Remark 4.8. For any |$a,b \in{\mathcal{K}}^\times$|⁠, simultaneously rescaling all |$X_i\mapsto a X_i$| and all |$Y_i\mapsto b Y_i$| defines an automorphism of |${\mathbb{H}}_{q,t}(GL_n)$|⁠, and |${\mathcal{K}}[B_n^{Ell}]$|⁠, compatible with |$\phi$|⁠. For |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$|⁠, however, rescaling |$\pi \leadsto a\pi$| and |$Z_i\leadsto bZ_i$|’s changes the relations by |$(\pi ^n=1)\leadsto (\pi ^n=a^n)$| and |$\boldsymbol{r}\leadsto b\boldsymbol{r}$|⁠. Hence, in order to define an isomorphism |$a$| must be an |$n$|th root of unity as in Definition 4.7 above. On the other hand, this shows that the parameter |$\boldsymbol{r}$| is inessential and can be chosen at will by re-scaling the |$Z_i$|’s. In Section 6, we will take |$n$| to be a multiple of a fixed integer |$N$|⁠, and will fix |$\boldsymbol{r}=\boldsymbol{q}^{1-N^2}=\mathtt{q}^{1/N-N}$|⁠, the inverse value of the ribbon element on the defining representation of |$SL_N$|⁠. This is a matter of combinatorial convenience. One could also introduce such an inessential parameter for the value of |$\pi ^n$|⁠, but |$\pi ^n=1$| is the most natural choice combinatorially, so we fix this convention now. This forces the scalar factor in front of |$\phi (\textsf{X}_1)$| below. Proposition 4.9. There exists a unique isomorphism \begin{equation*}\phi: {\mathcal{K}}\left[B_n^{Ell}\right]\Big/\left\langle \begin{array}{c} \left(\textsf{T}_i-\boldsymbol{q}^{-1}t\right)\left(\textsf{T}_i+\boldsymbol{q}^{-1}t^{-1}\right)=0 \quad(i=1,\ldots, n-1),\\\textsf{X}_1\cdots \textsf{X}_n=\boldsymbol{r}^n,\quad \textsf{Y}_1\cdots \textsf{Y}_n=\boldsymbol{r}^n\end{array} \right\rangle \to{\mathbb{H}}_{\boldsymbol{q},t}(SL_n),\end{equation*} such that \begin{equation*}\phi(\textsf{T}_i)=\boldsymbol{q}^{-1}T_i, \quad (i=1,\ldots n-1),\qquad \phi(\textsf{X}_1) = \boldsymbol{r}\cdot \boldsymbol{q}^{n-1}\cdot \pi T_{n-1}^{-1} \cdots T_{2}^{-1} T_{1}^{-1}, \qquad \phi(\textsf{Y}_1) = Z_1.\end{equation*} We note that this implies |$\phi (\textsf{Y}_i) = \boldsymbol{q}^{2(1-i)} Z_i$|⁠. Remark 4.10. In [15], a variant of Proposition 4.9 is taken as the definition of |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$|⁠, because of the central role played in the paper by the elliptic braid group. 4.5 Relationship between DAHAs for |$GL_n$| and |$SL_n$| The relationship between the |$GL_n$| and |$SL_n$| DAHA is not entirely straightforward; the |$SL_n$| DAHA may be realized as a sub-quotient of a degree |$n$|-extension of |$GL_n$| DAHA. This can be found in [10], but we review it here in the notation of this paper. First, we note that in |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$|⁠, we have the relation |$\pi ^n=1$|⁠, while in |${\mathbb{H}}_{q,t}(GL_n)$|⁠, we cannot simply set |$\pi ^n=1$|⁠, because of the relation |$\pi ^n Y_i \pi ^{-n} = q^{-1} Y_i.$| However, we note that |$\pi ^n$| commutes with the |$T_i$|’s and with the Laurent polynomials in the |$Y_i$|’s of total degree zero. We enlarge the degree zero part of |${\mathcal{Y}}$|⁠, by adjoining an element, \begin{equation*}\widetilde{Y}\quad ``=^{\prime\prime}\quad (Y_1 \cdots Y_n)^{-1/n}.\end{equation*} More precisely, this means we define a new algebra |$\widetilde{{\mathbb{H}}_{q,t}}(GL_n)$|⁠, by adding to |${\mathcal{K}}$|⁠, if necessary, a |$2n$|th root of |$q$|⁠, and adjoining to |${\mathbb{H}}_{q,t}(GL_n)$| an element |$\widetilde{Y}$| in degree |$-1$|⁠, such that |$\widetilde{Y}$| commutes with all |$T_i, \quad (i=0\ldots n-1),$| and all |$Y_i, \quad (i=1,\ldots , n)$|⁠, and subject to the further relations \begin{equation*}\widetilde{Y}^n=(Y_1 \cdots Y_n)^{-1},\qquad \pi \widetilde{Y} \pi^{-1} = q^{1/n} \widetilde{Y}.\end{equation*} We let |$\widetilde{{\mathcal{Y}}}$| denote the resulting subalgebra of degree |$0$| polynomials, that is, the subalgebra generated by |$(Y_i \widetilde{Y})^{\pm 1}$|⁠. The algebra generated by |$\pi , T_1, \widetilde{{\mathcal{Y}}}$| now has |$\pi ^n-1$| in its center, so we may quotient by the ideal it generates (and below mark elements of the quotient with a bar). We can identify |${\mathbb{H}}_{q^{-1/2n},t}(SL_n)$| with this quotient via \begin{align*} T_i \mapsto \overline{T_i},\qquad Z_i \mapsto q^{\frac{1-n}{2n}}\boldsymbol{r} \overline{Y_i \widetilde{Y}},\qquad \pi \mapsto \overline \pi. \end{align*} Hence, we can identify the parameter |$\boldsymbol{q}$| of |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$| with |$q^{-1/2n}$| of |${\mathbb{H}}_{q,t}(GL_n)$|⁠. To avoid fractional exponents, we chose to make |$\boldsymbol{q}$| a separate parameter. 5 Category |${\mathcal{O}}$| and |${\mathcal{Y}}$|-semisimple Representations In his landmark paper [9], Cherednik gave a complete classification of irreducible |${\mathcal{Y}}$|-semisimple representations, that is, those |${\mathbb{H}}_{q,t}$|-modules for which the |${\mathcal{Y}}$|-action can be diagonalized. His classification builds on the parallel story for the affine Hecke algebra [11–13], [20, 21]. Subsequently, the paper [24] built on Cherednik’s classification via periodic skew diagrams combinatorially, connecting standard tableaux on the diagrams to |${\mathcal{Y}}$|-weights. We recall these constructions below. We will give basic definitions in parallel for both |$GL$| and |$SL$|⁠. Theorems 5.3 and 5.4 were proved in [11–13], [20] for affine Hecke algebras and in [8, 9], [24] for |${\mathbb{H}}_{q,t}(GL_n)$|⁠. We will also state without proof the analogous results for |$SL_N$|⁠, which follow by straightforward modifications of the proofs, keeping track of the additional relation |$\pi ^n=1$|⁠. We will write |${\mathbb{H}}_{q,t}$| in this section when we do not need to distinguish between the |$GL$| or |$SL$| case. Definition 5.1. Category |${\mathcal{O}}$| for |${\mathbb{H}}_{q,t}(GL_n)$| (resp. |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$|⁠) is the full subcategory of finitely generated |${\mathbb{H}}_{q,t}$|-modules |$M$|⁠, such that for each vector |$m\in M$|⁠, its orbit |${\mathcal{Y}}\cdot m$| (resp. |${\mathcal{Z}}\cdot m$|⁠) is finite dimensional. A tuple |$\underline{z} = (z_1,\ldots , z_n)\in ({\mathcal{K}}^\times )^n$| will be called a weight for |${\mathcal{Y}}$|⁠. A tuple |$\underline{z}$| satisfying further that |$\prod _iz_i=\boldsymbol{q}^{n(n-1)}\boldsymbol{r}^n$| will be called a |${\mathcal{Z}}$|-weight. We will make use of the actions (4.1) and (4.2) of the extended affine symmetric group on each type of weights. A |${\mathcal{Y}}$|-weight |$\underline{z}$| defines a homomorphism |${\mathcal{Y}}\to{\mathcal{K}}$|⁠, sending |$Y_i\mapsto z_i$|⁠, and hence a one-dimensional representation of |${\mathcal{Y}}$|⁠. Similarly, a |${\mathcal{Z}}$|-weight |$\underline{z}$| defines a homomorphism |${\mathcal{Z}}\to{\mathcal{K}}$|⁠, sending |$Z_i\mapsto z_i$|⁠, and hence a one-dimensional representation of |${\mathcal{Z}}$|⁠. Let |$M$| be an |${\mathbb{H}}_{q,t}$|-module. We define its |$\underline{z}$|-weight space to be \begin{equation*}M[\underline{z}] = \{ v\in M \,\,|\,\, hv=\underline{z}(h)v,\,\, \forall h\in{\mathcal{Y}}\ \textrm{(resp.} {\mathcal{Z}})\}.\end{equation*} A non-zero |$v\in M[\underline{z}]$| is called a weight vector or |$\underline{z}$|-weight vector. Its generalized weight space is \begin{equation*}M^{gen}[\underline{z}] = \{v\in M \,\,|\,\, (h-\underline{z}(h))^m v=0, \textrm{for all}\ h\in{\mathcal{Y}}\ \textrm{(resp.} {\mathcal{Z}}), m\gg 0 \}.\end{equation*} Assume for any |$M \in{\mathcal{O}}$| that |$M$| splits over |${\mathcal{K}}$|⁠. Then we have an isomorphism, \begin{equation*}M \cong \bigoplus_{\underline{z}}M^{gen}[\underline{z}],\end{equation*} as vector spaces and in fact as |${\mathcal{Y}}$|- or |${\mathcal{Z}}$|-modules. We will call \begin{equation*}\operatorname{supp}(M) = \{\underline{z} \mid M^{gen}[\underline{z}]\neq 0\} = \{\underline{z} \mid M[\underline{z}]\neq 0\}\end{equation*} the support of |$M$|⁠. Definition 5.2. We say that |$M$| is |${\mathcal{Y}}$|-semisimple (resp. |${\mathcal{Z}}$|-semisimple) if we have an isomorphism \begin{equation*}M\cong \bigoplus_{\underline{z}}M[\underline{z}],\end{equation*} as |${\mathcal{Y}}$|- (resp. |${\mathcal{Z}}$|-) modules, that is, if |$\operatorname{Res}^{{\mathbb{H}}_{q,t}}_{{\mathcal{Y}}}(M)$| is semisimple as a |${\mathcal{Y}}$|- (resp. |${\mathcal{Z}}$|-) module. We will write “|${\mathcal{Y}}\text{-}{\mathcal{Z}}$|-semisimple” to mean either |${\mathcal{Y}}$|- or |${\mathcal{Z}}$|-semisimple, depending on |$G$|⁠. Such |$M$| are called calibrated in [20]. Note that if |$M$| is |${\mathcal{Y}}\text{-}{\mathcal{Z}}$|-semisimple, it has a weight basis: a basis consisting of weight vectors. The notion of |${\mathcal{Y}}\text{-}{\mathcal{Z}}$|-semisimplicity makes sense whether |$M$| is a representation of the affine Hecke algebra or of the double affine Hecke algebra. The structure of |${\mathcal{Y}}\text{-}{\mathcal{Z}}$|-semisimple modules in both cases is extremely rigid. Given a |${\mathcal{Y}}$|-semisimple module, one can read its composition factors directly from its support. Further if |$M$| is both simple and |${\mathcal{Y}}$|-semisimple, one need only determine a single |$\underline{z}\in \operatorname{supp}(M)$| in order to determine all of |$\operatorname{supp}(M)$|⁠, and hence the isomorphism type of |$M$|⁠. For simple and |${\mathcal{Z}}$|-semisimple |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$|-modules, similar statements hold, once we allow twisting by an automorphism that rescales |$\pi$| by a root of unity. Theorem 5.3 ([8, 11–13], [20], [24]). Let |$M$| be a simple and |${\mathcal{Y}}\text{-}{\mathcal{Z}}$|-semisimple |${\mathbb{H}}_{q,t}$|-module. Then for all |$\underline{z}\in \operatorname{supp}(M)$|⁠, we have |$\dim M[\underline{z}]=1$|⁠. Hence, a simple and |${\mathcal{Y}}\text{-}{\mathcal{Z}}$|-semisimple |${\mathbb{H}}_{q,t}$|-module has a weight basis that is unique up to rescaling each vector individually, that is, the underlying vector space has a canonical decomposition as a direct sum of lines. For |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$| one must modify the proof of Theorem 5.3 for |${\mathbb{H}}_{q,t}(GL_n)$| to include the case that the stabilizer of |$\underline{z}$| via the action (4.2) contains an element outside of |$\langle s_0, s_1, \ldots , s_{n-1} \rangle$|⁠. In that case, it contains a subgroup conjugate to |$\langle \pi ^r \rangle$| for some |$r \mid n$|⁠. One may then appeal to minimal idempotents for the cyclic group |$\langle \pi ^r \rangle \big / \langle \pi ^n \rangle$| to get the required multiplicity one result. The supports of simple |${\mathcal{Y}}\text{-}{\mathcal{Z}}$|-semisimple modules have a very nice combinatorial structure. It is easy to show that if |$M$| is simple, then all its support is contained in a single |$\widehat{{\mathfrak{S}}}_n$|-orbit. If additionally |$M$| is |${\mathcal{Y}}\text{-}{\mathcal{Z}}$|-semisimple, we can say exactly what subset of the |$\widehat{{\mathfrak{S}}}_n$|-orbit we get, that is, we can completely determine |$\operatorname{supp}(M)$|⁠. More precisely, given |$\underline{z}\in \operatorname{supp}(M)$|⁠, one can determine the set |$S\subset \widehat{{\mathfrak{S}}}_n$| such that \begin{equation*}\operatorname{supp}(M) = \{w\cdot\underline{z} \,\, | \,\, w\in S\}.\end{equation*} The following theorem uniquely characterizes the set |$S$|⁠, which depends on choice of |$\underline{z}$|⁠: Theorem 5.4 ([9, 11–13], [20]). Let |$M$| be a simple and |${\mathcal{Y}}\text{-}{\mathcal{Z}}$|-semisimple |${\mathbb{H}}_{q,t}$|-module. Let |$\underline{z}\in \operatorname{supp}(M)$|⁠. We have For |$1\leq i < n,$| we have |$M[s_i\cdot \underline{z}]=0$| if, and only if, |$z_i/z_{i+1}\in \{t^2,t^{-2}\}$|⁠. Further, we have \begin{equation*}T_iM[\underline{z}] \subset M[\underline{z}] \oplus M[s_i\cdot\underline{z}].\end{equation*} We have |$M[s_0\cdot \underline{z}]=0$| if, and only if, |$q z_n/z_1\in \{t^2,t^{-2}\}$| for |${\mathbb{H}}_{q,t}(GL_n)$|⁠, |$z_n/\boldsymbol{q}^{2n}z_1\in \{t^2,t^{-2}\}$| for |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$|⁠. Further, we have \begin{equation*}T_0 M[\underline{z}] \subset M[\underline{z}]\oplus M[s_0\cdot \underline{z}].\end{equation*} We have |$M[\pi \cdot \underline{z}]\neq 0$| and |$\pi M[\underline{z}] = M[\pi \cdot \underline{z}]$|⁠. Remark 5.5. Conditions (1) and (2) of Theorem 5.4 may be written more uniformly as |$\frac{ (s_i \cdot \underline{z})_{i+1}} {z_{i+1}} \in \{t^2, t^{-2}\}$|⁠, but not much clarity is gained from this reformulation. Note that Theorem 5.4 allows us to precisely describe the action of the |${\mathbb{H}}_{q,t}(GL_n)$|-generators on a weight basis, once we have chosen a sensible normalization/scaling. The proof of this theorem uses the theory of “intertwiners” [9], for which the reader may also consult [24]. For |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$|⁠, this is true once we pin down the action of |$\pi$| (see Corollary 5.6). As a consequence of this rigid structure, we also have the following: Corollary 5.6. Let |$M,N$| be simple |${\mathbb{H}}_{q,t}(GL_n)$|-modules, which are both |${\mathcal{Y}}$|-semisimple. Then either |$M\cong N$| or |$\operatorname{supp}(M)\cap \operatorname{supp}(N)=\emptyset$|⁠. Corollary 5.7. Let |$M,N$| be simple |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$|-modules, which are both |${\mathcal{Z}}$|-semisimple. Then either |$M\cong N^a$| for some |$a \in{\mathcal{K}}^\times$| with |$a^n=1$| or |$\operatorname{supp}(M)\cap \operatorname{supp}(N)=\emptyset$|⁠. We will use these corollaries to identify the module |$F^G_n({\mathcal{O}}_{\mathtt{q}}(G))$| in Section 7. 6 The Rectangular Representations In this section we will detail a very special case of Cherednik’s construction, when the Young diagram indexing the irreducible module is an |$N\times k$| rectangle \begin{equation*}\mu= (k^N) = (\underbrace{k,\ldots, k}_{N})\end{equation*} and the periodicity is purely horizontal, so that the shape is not actually skew. We begin, however, by recalling the simpler setting of the affine Hecke algebra. 6.1 The rectangle and the affine Hecke algebra One can associate to the partition |$\mu =(k^N)$| a finite dimensional irreducible representation of the finite Hecke algebra |$H_n$|⁠, with |$n=kN$|⁠. A basis for this representation is indexed by the set of standard Young tableaux of shape |$(k^N)$|⁠. We denote by |$\operatorname{Rect}(N,k)$| the |$H({\mathcal{Y}})$|-module obtained by inflating this module via the homomorphism (4.5). It is well known for generic |$t$| (i.e., away from small roots of unity) that |$\operatorname{Rect}(N,k)$| is |${\mathcal{Y}}$|-semisimple. This is stated precisely in Proposition 6.4 below. Identifying the |$N \times k$| rectangular diagram |$(k^N)$| with |$\textrm{D}^{N,k}_{0}$|⁠, that is, |$\lambda = 0$|⁠, we assign diagonal labels as in Section 2. Hence, its principal diagonal is labeled |$0$|⁠, and a box in the |$j$|th row and |$m$|th column is on the |$m-j$| diagonal. This agrees with the traditional notion of content. Definition 6.1. Given a standard tableau |$R$| of shape |$(k^N)$|⁠, we define a map |$\textrm{diag}_R: \{1, \ldots , n\} \to{\mathbb{Z}}$| such that |$\textrm{diag}_R(i)$| is the label of the diagonal on which lies. See Figure 7. Fig. 7. View largeDownload slide The diagonal labels of a rectangular tableau. The dashed line traverses the principal diagonal. Fig. 7. View largeDownload slide The diagonal labels of a rectangular tableau. The dashed line traverses the principal diagonal. Definition 6.2. The weight|$\textrm{wt}(R) \in ({{\mathcal{K}}}^{\times })^n$| of |$R\in \textrm{SYT}(k^N)$|⁠, is the tuple, \begin{equation*}\textrm{wt}(R) = \big(t^{2 \textrm{diag}_R(1)},t^{2 \textrm{diag}_R(2)},\ldots,t^{2 \textrm{diag}_R(n)} \big) = t^{\left(2 \textrm{diag}_R(1), 2 \textrm{diag}_R(2),\ldots,2 \textrm{diag}_R(n)\right)}.\end{equation*} Example 6.3. Let |$N=3, k=2$|⁠. There are five standard tableau of shape |$(2^3)$|⁠. We list them below with their corresponding weights. Note that for all |$R \in \textrm{SYT}(k^N)$|⁠, box is always in the upper left corner, so |$\textrm{diag}_R(1)=0$| and is always in the lower right corner, so |$\textrm{diag}_R(n)=k-N$|⁠. Proposition 6.4. We have: The irreducible |$H({\mathcal{Y}})$|-module |$\operatorname{Rect}(N,k)$| has a |${\mathcal{Y}}$|-weight basis, \begin{equation*}\{ v_R \mid R \in \textrm{SYT}(k^N)\}\end{equation*} when |$t$| is generic, such that each |$v_R$| has weight |$\textrm{wt}(R)$|⁠, that is, \begin{equation*}Y_i v_R = t^{2 \textrm{diag}_R(i)}v_R.\end{equation*} In particular, the central element |$\prod _{i=1}^n Y_i$| acts as the scalar |$t^{n(k-N)}$|⁠. Set |$\boldsymbol{r} = \boldsymbol{q}^{1-n} t^{k-N}$|⁠. Then |$\operatorname{Rect}(N,k)$| descends to a representation of |$H({\mathcal{Z}})$|⁠, such that \begin{equation*}Z_i v_R = t^{2 \textrm{diag}_R(i)}v_R.\end{equation*} In Section 7.1, we will consider |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$|⁠, with the specialization |$t=\boldsymbol{q}^N=\mathtt{q}$|⁠. In light of Proposition 6.4, we will therefore further specialize |$\boldsymbol{r}=\boldsymbol{q}^{1-n}t^{(k-N)}=\boldsymbol{q}^{1-N^2}=\mathtt{q}^{1/N-N}$| in the definition of |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$|⁠. 6.2 Induction of the rectangular representation to the DAHA for |$GL$| In this section, we prove that when we induce |$\operatorname{Rect}(N,k)$| from the affine Hecke algebra to |${\mathbb{H}}_{q,t}(GL_n)$| under the specialization |$q=t^{-2k}$| it has unique simple quotient. Let \begin{equation*}M(k^N) = \operatorname{Ind}_{H({\mathcal{Y}})}^{{\mathbb{H}}_{q,t}(GL_n)} \operatorname{Rect}(N,k)\end{equation*} denote the induction of |$\operatorname{Rect}(N,k)$| from |$H({\mathcal{Y}})$| to |${\mathbb{H}}_{q,t}(GL_n)$|⁠. This module has basis \begin{equation*}\{ T_\sigma \otimes v_R \mid \sigma \in \widehat{{\mathfrak{S}}}_n/{{{\mathfrak{S}}}}_n, R \in \textrm{SYT}(k^N) \}\end{equation*} that can be ordered (refining weak Bruhat order) so that with respect to this basis |${\mathcal{Y}}$| acts triangularly. |$M(k^N)$| thus has support \begin{equation} \operatorname{supp}\big(M(k^N)\big) = \big\{ \sigma \cdot \textrm{wt}(R) \mid \sigma \in \widehat{{\mathfrak{S}}}_n/{{{\mathfrak{S}}}}_n, R \in \textrm{SYT}(k^N)\big\}. \end{equation} (6.1) Theorem 6.5. Let |$q = t^{-2k}$|⁠. Then |$M(k^N)$| has unique simple quotient. In the next section, we will explicitly construct this unique simple quotient and thereby show it is |${\mathcal{Y}}$|-semisimple. Proof. We will show below that, for any fixed |$R \in \textrm{SYT}(k^N)$|⁠, the |$\textrm{wt}(R)$|-weight space of |$M(k^N)$| is one-dimensional; hence, any weight vector of weight |$\textrm{wt}(R)$| generates the induced module |$M(k^N)$| and likewise any proper submodule must avoid it. This shows it has a unique maximal submodule and hence unique simple quotient. Alternatively, the following argument is rather standard for induced modules. Suppose that |$L$| is simple and we have a non-zero map \begin{equation*}M(k^N) = \operatorname{Ind}_{H({\mathcal{Y}})}^{{\mathbb{H}}_{q,t}(GL_n)} \operatorname{Rect}(N,k) \to L.\end{equation*} By adjointness of induction and restriction (Frobenius reciprocity) |$L$| contains a weight vector of weight |$\textrm{wt}(R)$|⁠. If there were another simple quotient |$L^{\prime}$|⁠, then |$L \oplus L^{\prime}$| would also be a quotient but with two-dimensional |$\textrm{wt}(R)$|-weight space, a contradiction to the |$\textrm{wt}(R)$|-weight space of |$M(k^N)$| being one-dimensional. So let us now compute the |$\textrm{wt}(R)$|-weight space of |$M(k^N)$|⁠. Because |$\operatorname{Rect}(N,k)$| is |${\mathcal{Y}}$|-semisimple, it suffices to determine that |$\sigma \in \widehat{{\mathfrak{S}}}_n / {{{\mathfrak{S}}}}_n$| stabilize |$\textrm{wt}(R)$|⁠; the size of the stabilizer is the dimension of the weight space. On the one hand, the weights of |$M(k^N)$| are given in (6.1). On the other hand, each double coset |${{{\mathfrak{S}}}}_n \backslash \widehat{{\mathfrak{S}}}_n / {{{\mathfrak{S}}}}_n$| contains “translation by a dominant weight”—that is to say, any weight in the support of |$M(k^N)$| has the form \begin{equation}\tau\cdot \big(q^{\gamma_1} t^{2a_1}, \ldots, q^{\gamma_n} t^{2a_n}\big) = \tau\cdot \big( t^{2a_1- 2k \gamma_1}, \ldots, t^{2a_n- 2k \gamma_n}\big), \end{equation} (6.2) where |$\tau \in{{{\mathfrak{S}}}}_n$|⁠, |$\gamma _i \in{\mathbb{Z}}$|⁠, |$\gamma _1 \le \gamma _2 \le \cdots \le \gamma _n$|⁠, and |$t^{(2a_1, \ldots , 2a_n)} = \textrm{wt}(R)$| for some |$R \in \textrm{SYT}(k^N)$|⁠. All such |$R$| satisfy \begin{equation} a_1 = 0, \qquad 1-N \le a_i \le k-1\quad (i=2,\ldots n), \qquad a_n = k-N \end{equation} (6.3) see Figure 8. We now claim that the only possible |$\gamma$| that can appear in equation (6.2) is the trivial one, \begin{equation*}\gamma_1 = \gamma_2 = \cdots = \gamma_n = 0.\end{equation*} Since |$\gamma$| is dominant, it suffices to show that |$\gamma _1\geq 0$| and |$\gamma _n\leq 0$|⁠. Indeed, suppose for the sake of contradiction that |$\gamma _1 < 0$|⁠, then we have |$a_1-k \gamma _1 \ge k> k-1$|⁠, hence |$a_1-k \gamma _1 \neq a_i$| for any |$i$|⁠, by the restrictions (6.3). Likewise if we suppose |$\gamma _n>0$|⁠, we have |$a_n-k \gamma _n \le -N < 1-N$| so |$a_n-k \gamma _n \neq a_i$| for any |$i$|⁠, again by (6.3). Given that |$\sigma \in \widehat{{\mathfrak{S}}}_n/{{{\mathfrak{S}}}}_n$| is taken to be of minimal length, this implies |$\sigma = \operatorname{Id}$|⁠. In other words, for any |$R \in \textrm{SYT}(k^N)$| the |$\textrm{wt}(R)$| weight space of |$M(k^N)$| is one-dimensional. Fig. 8. View largeDownload slide The contents of the corners of the rectangle imply the restrictions on |$a_i=\textrm{diag}_R(i)$| in equation (6.3). Fig. 8. View largeDownload slide The contents of the corners of the rectangle imply the restrictions on |$a_i=\textrm{diag}_R(i)$| in equation (6.3). Let us remark that the technique used in the proof of Theorem 6.5 is more general: for an irreducible representation |$A$| of |$H({\mathcal{Y}})$|⁠, if one shows the (generalized) |${\mathcal{Y}}$|-weight space of |$A$| of weight |$\underline{z}$| has the same dimension as that of |$\operatorname{Ind}_{H({\mathcal{Y}})}^{{\mathbb{H}}_{q,t}(GL_n)} A$|⁠, then by the same Frobenius reciprocity argument as above, the latter must have unique simple quotient. To illustrate, below we give a non-example, that is, an example of an induced representation that has two distinct simple quotients. Example 6.6. In contrast to the situation in Theorem 6.5 consider the following example of a representation of |${\mathbb{H}}_{q,t}(GL_n)$| at |$n=N=2$|⁠, |$k=1$|⁠, |$q=t^{-2}$|⁠. It is an induced representation with two distinct simple quotients, and furthermore one of those quotients is not |${\mathcal{Y}}$|-semisimple. Let |$\lambda = 2\epsilon _{1} = (2,0)$|⁠, and as in Section 2.4 consider the skew shape |$\textrm{D}^{N,k}_{\lambda } = \textrm{D}^{2,1}_{(2,0)}$| and the two skew tableaux |$\mathcal{SK}^{2,1}_{(2,0)}$| of that shape that correspond to the |${\mathcal{Y}}$|-weights |$(t^2, t^{-4})$| and |$(t^{-4}, t^2)$|⁠. This data is attached to a two-dimensional irreducible representation |$A$| of |$H({\mathcal{Y}})$|⁠. In particular, |$A \simeq \operatorname{Ind}_{{\mathcal{Y}}}^{H({\mathcal{Y}})} t^2 \boxtimes t^{-4}$|⁠. Let \begin{equation*}M:= \operatorname{Ind}_{H({\mathcal{Y}})}^{{\mathbb{H}}_{q,t}(GL_{2})} A \simeq \operatorname{Ind}_{{\mathcal{Y}}}^{{\mathbb{H}}_{q,t}(GL_{2})} t^2 \boxtimes t^{-4}.\end{equation*} We claim |$M$| has two distinct simple quotients. In fact |$M$| is isomorphic to the direct sum of two irreducible |${\mathbb{H}}_{q,t}(GL_{2})$|-modules. We leave the details of most computations to the reader, but summarize them below. The interested reader can verify the following assertions. While the |$(t^2, t^{-4})$|-weight space of |$A$| is one-dimensional, the corresponding weight space of |$M$| is two-dimensional. It is easy to verify this by considering the |$\widehat{\mathfrak{S}}_2$|-orbit of |$(t^2, t^{-4})$| when |$q = t^{-2}$|⁠. Hence, our example does not exhibit the properties or hypotheses used in the proof of Theorem 6.5. One simple quotient is our |$L(1^2) \simeq M(1^2)$|⁠, whose |${\mathcal{Y}}$|-weights are computed in Theorem 6.10 below. (In fact, comparing with equation (6.1) this computation also implies at |$k=1$| that |$M(1^2)$| is simple and coincides with |$L(1^2)$|⁠.) Its |$(t^2, t^{-4})$|-weight space is one-dimensional, yielding that it occurs as a quotient of |$M$| by Frobenius reciprocity. Another simple quotient is |$K:= \operatorname{Ind}_{H({\mathcal{Y}})}^{{\mathbb{H}}_{q,t}(GL_{2})} B$|⁠, where |$B$| is the one-dimensional “trivial” module on which |$Y_1- t^{-2}$|⁠, |$Y_2 - t^0$|⁠, |$T_1 - t$| each vanish. An appropriate name for this |$K$| would be |$L(2) = M(2)$|⁠, except its spectrum is slightly shifted from what one might expect, as |$B$| is inflated from the trivial |$H_2$| module along a different homomorphism than (4.5). (Nonetheless, by abuse of notation we will refer to |$K$| as |$L(2)$| here and in Example 6.7 below.) Note |$L(2)$| is not|${\mathcal{Y}}$|-semisimple. For instance, its |$(t^{-2}, t^0)$|-weight space is two-dimensional. Again the computation of its |$(t^2, t^{-4})$|-weight space gives it as a quotient of |$M$|⁠. Further, on closer examination of |${\mathcal{Y}}$|-weight spaces of all the above representations, it in fact follows |$M \simeq L(1^2) \bigoplus L(2)$|⁠. Example 6.7. As another example, in a forthcoming paper we construct an isomorphism, \begin{equation}F_2^{GL}(\mathtt{Spr}) \simeq \operatorname{Ind}_{{\mathcal{Y}}}^{{\mathbb{H}}_{q,t}(GL_{2})} t^0 \boxtimes t^{-2} \simeq L(1^2) \oplus L(2),\end{equation} (6.4) where |$\mathtt{Spr}$| is the so-called quantum Springer–Hotta–Kashiwara sheaf, a |${\mathcal{D}}_{\mathtt{q}}(G)$|-module of considerable interest, which |$\mathtt{q}$|-deforms the Hotta–Kashiwara presentation [14] of the classical Springer sheaf. Unlike in Example 6.6 above, the two-dimensional |$H({\mathcal{Y}})$|-representation |$\operatorname{Ind}_{{\mathcal{Y}}}^{H({\mathcal{Y}})} t^{0} \boxtimes t^{-2}$| is reducible; we have a non-split short exact sequence \begin{equation} 0 \longrightarrow B \longrightarrow \operatorname{Ind}_{{\mathcal{Y}}}^{H({\mathcal{Y}})}t^{0} \boxtimes t^{-2} \longrightarrow \operatorname{Rect}(2,1) \longrightarrow 0,\end{equation} (6.5) where |$B$| is as in Example 6.6 above. It is very interesting therefore that upon further induction by the functor |$\operatorname{Ind}_{H({\mathcal{Y}})}^{{\mathbb{H}}_{q,t}(GL_{2})}$|⁠, we obtain the splitting (6.4). Thus the |${\mathbb{H}}_{q,t}(GL_{2})$|-endomorphism algebra of |$F_2^{GL}(\mathtt{Spr})$| is isomorphic to the finite Hecke algebra |$H_2$|⁠. More generally, we obtain analogous results for quantum Springer–Hotta–Kashiwara sheaves associated to |$GL_N$| for any |$N$|⁠. 6.3 The periodic rectangle and the double affine Hecke algebra The construction of irreducible |${\mathcal{Y}}$|-semisimple |${\mathbb{H}}_{q,t}(GL_n)$|-modules starts with the notion of a periodic skew diagram [9], and the subsequent notion of a periodic skew tableau. Let us recall the construction in [24] here, modified to conform to the conventions of this paper. Given a skew Young diagram |$\mu /\lambda$|⁠, we make it periodic as follows. We embed the |$n$| cells of a skew diagram |$\mu /\lambda$| into |${\mathbb{Z}}^2$| using matrix style coordinates |$(\textrm{row},\textrm{column})$| (this is a standard choice for Young diagrams in English notation, in place of Cartesian coordinates). For each |$r$|⁠, we have the |$r$|-shifted diagram \begin{equation*}(\mu/\lambda)[r] = \mu/\lambda + r(-\ell,k),\end{equation*} where |$k = \mu _1=$| the number of columns of |$\mu$|⁠, and |$\ell$| is determined by |$t=q^{-2(k + \ell )}$|⁠. The condition that |$\mu /\lambda [0] \cup \mu /\lambda [1]$| again forms a skew diagram forces \begin{equation} l(\mu) - \textrm{mult}(\mu_1) \le \ell,\qquad l(\lambda) \le \ell, \end{equation} (6.6) where |$l$| denotes the number of rows of the diagram. The case we consider in this paper is very special; we have |$\lambda = \emptyset$| and |$\mu = (k^N)$|⁠, so the 1st condition in (6.6) is vacuous, as |$l(\mu ) - \textrm{mult}(\mu _1) = N - N = 0$|⁠. Further, for |${\mathbb{H}}_{q,t}(GL_n)$| we specialize |$q=t^{-2k}$| and so |$\ell =0$|⁠. In other words \begin{equation*}(k^N) [r] = (k^N) + r(0, k)\end{equation*} so the boxes of the periodic (skew) diagram form an |$N \times \infty$| strip, see Figure 9. Fig. 9. View largeDownload slide The diagram |$\mu = (2^3)$| is made periodic by shifting horizontally. Fig. 9. View largeDownload slide The diagram |$\mu = (2^3)$| is made periodic by shifting horizontally. In the |$GL_n$| case, we always consider the fundamental domain |$\mu [0]$| to be anchored on the |$0$|-diagonal. In other words, the |$(1,1)$| cell is always the upper left corner of |$\mu [0]$|⁠. Definition 6.8. Let |$n=kN$|⁠. An |$n$|-periodic standard tableaux of shape |$\mu =(k^N)$| is a bijection |$R:{\mathbb{Z}} \to \{$|boxes of |$N \times \infty$| strip|$\}$| such that fillings increase across rows and down columns the fillings of |$\mu [0]$| are distinct |$\bmod$||$n$| the fillings of |$\mu [r]$| are those of |$\mu [0]$||$+ nr$|⁠. We will denote the set of all such tableaux |$\textrm{P}_n\textrm{SYT}(k^N)$|⁠. An |$R\in \textrm{P}_n\textrm{SYT}(k^N)$| is completely determined by the fillings of |$\mu [0]$|⁠, see Figure 10. However, it may happen that the filling of |$\mu [0]$| is row- and column-increasing, but its periodization is not standard, see Figure 11. Fig. 10. View largeDownload slide The filling of |$\mu [0]$| completely determines the filling of |$R \in \textrm{P}_{6}\textrm{SYT}(2^3)$|⁠, via the periodicity constraint. Fig. 10. View largeDownload slide The filling of |$\mu [0]$| completely determines the filling of |$R \in \textrm{P}_{6}\textrm{SYT}(2^3)$|⁠, via the periodicity constraint. Fig. 11. View largeDownload slide The filling of |$\mu [0]$| is row- and column-increasing, but its periodization is not standard. Fig. 11. View largeDownload slide The filling of |$\mu [0]$| is row- and column-increasing, but its periodization is not standard. The diagonal and weight functions are defined for periodic skew tableaux |$R \in \textrm{P}_n\textrm{SYT}(k^N)$| similarly to those for standard Young tableaux, in Definitions 6.1 and 6.2. The diagonal function is |$\textrm{diag}_R(i) = m-j$|⁠, when lies in row |$j$| and column |$m$|⁠. Note this is defined for all |$i \in{\mathbb{Z}}$| and satisfies |$\textrm{diag}_R(i+n) = \textrm{diag}_R(i) + k$|⁠. The weight, |$wt(R)$|⁠, of |$R$| is the tuple |$(t^{2 \textrm{diag}_R(1)}, t^{2 \textrm{diag}_R(2)}, \ldots , t^{2 \textrm{diag}_R(n)})$|⁠. The |$\widehat{{\mathfrak{S}}}_n$| action on |${\mathbb{Z}}$| descends to an action on periodic tableaux as follows. We set |$\sigma \cdot R$| to be the tableau where is replaced with . The function |$\textrm{diag}_R$| is compatible with this action \begin{equation*}\textrm{diag}_{\sigma \cdot R}( \sigma(i) ) = \textrm{diag}_R( i)\end{equation*} for any |$\sigma \in \widehat{{\mathfrak{S}}}_n$|⁠. Furthermore, the action intertwines the action of |$\widehat{{\mathfrak{S}}}_n$| on |$({\mathcal{K}}^\times )^n$| described in (4.1), we have |$\textrm{wt}(\sigma \cdot R) = \sigma \cdot \textrm{wt}(R).$| We note that |$\sigma \cdot R$| need not be standard, even if |$R$| is. Remark 6.9. We note that any domain for the |$n$|-periodicity in Definition 6.8 is also a domain for the |$\pi ^n$|-action. Note that |$\pi ^n$| shifts the |$N\times \infty$| strip |$k$| steps horizontally. Theorem 6.10. ([9], [24]). Let |$L(k^N)$| denote the linear span over |${\mathcal{K}}$| of \begin{equation*}\{v_R \mid R\in \textrm{P}_n\textrm{SYT}(k^N)\}.\end{equation*} When |$q=t^{-2k}$| there exists a unique irreducible representation of |${\mathbb{H}}_{q,t}(GL_n)$| on |$L(k^N)$|⁠, such that each |$v_R$| is a |${\mathcal{Y}}$|-weight vector of weight |$\textrm{wt}(R)$|⁠, that is, \begin{equation*}Y_iv_R = t^{2\textrm{diag}_R(i)}v_R, \quad(i=1,\ldots, n).\end{equation*} Corollary 6.11. When |$q=t^{-2k}$|⁠, the unique simple quotient of |$M(k^N)$| is |$L(k^N)$|⁠; in particular, it is |${\mathcal{Y}}$|-semisimple and its support is given in Theorem 6.10. Proof. It follows by Frobenius reciprocity that there exists a non-zero map from |$M(k^N)$| to |$L(k^N)$|⁠, hence |$L(k^N)$| is indeed the unique simple quotient guaranteed by Theorem 6.5. Both Theorem 6.10 and Corollary 6.11 justify associating the irreducible |${\mathbb{H}}_{q,t}(GL_n)$|-module with the |$N \times k$| rectangle. 6.4 |$SL$| modifications When considering |$SL$| versus |$GL$|⁠, we need to make the following modifications both to Theorems 6.5 and 6.10 and the underlying combinatorics. Further, we specialize |$t=\boldsymbol{q}^N$| and |$\boldsymbol{r}=\boldsymbol{q}^{1-N^2}$|⁠. They are consistent with the modified |$\widehat{{\mathfrak{S}}}_n$| action on |$({\mathcal{K}}^\times )^n$| described in (4.2). The role of periodic skew diagram |$\textrm{P}_n\textrm{SYT}(k^N)$| in the |$GL$| case is now played by the set |$\textrm{P}_n\textrm{SYT}(k^N)\big /\pi ^n$| of equivalence classes: we impose the equivalence relation |$R \sim \pi ^n \cdot R$| and denote equivalence classes as |${\overline{R}}$|⁠. For |${\overline{R}} \in \textrm{P}_n\textrm{SYT}(k^N)\big /\pi ^n$| we modify the function |$\textrm{diag}_R$| to |$\textrm{diag}_{{\overline{R}}}$| as follows. When a fundamental domain such as |$\mu [0]$| is filled with any |$R_0 \in \textrm{SYT}(k^N)$|⁠, it has “filling sum”, \begin{equation*}{{\mathfrak{s}}} = \sum_{i=1}^n i.\end{equation*} For |$R \in \textrm{P}_n\textrm{SYT}(k^N)$|⁠, any of its fundamental domains (under |$\pi ^n$|-shifts, see Remark 9) has filling sum |${{\mathfrak{s}}} +np$| for some |$p \in{\mathbb{Z}}$| (even if the domain is not of the form |$\mu [r]$|⁠). For any |$N \times k$| rectangle of |${\overline{R}}$| we label with |$\frac pN$| the diagonal through its northwest corner, we will call this its NW diagonal. To see that this gives a well-defined diagonal labeling to all of |$R$|⁠, we note that if a domain has filling sum |${{\mathfrak{s}}}+np$|⁠, then the domain that is shifted one unit right has filling sum |${{\mathfrak{s}}} + np + nN$| and |$\frac{p+N}{N} = \frac pN +1$|⁠. More precisely, if sits in the domain |$\mu [r]$| in the |$j$|th row and |$(m + rk)$|th column, \begin{gather} \textrm{diag}_{{\overline{R}}}(i) = m-j + \frac pN \end{gather} (6.7) where |$p$| is determined as above with respect to |$\mu [r]$|⁠. In particular |$\textrm{diag}_{{\overline{R}}}(i)$| is independent of |$r$|⁠, and |$\textrm{diag}_{{\overline{R}}}(i) = \textrm{diag}_{\pi ^n \cdot{\overline{R}}}(i)$| since now sits in domain |$\mu [r+1]$| of |$\pi ^n \cdot{\overline{R}}$| but the local information of |$p$| stays the same. The function |$\textrm{diag}_{{\overline{R}}}$| is defined for all |$i \in{\mathbb{Z}}$| and satisfies |$\textrm{diag}_{{\overline{R}}}(i+n) = \textrm{diag}_{{\overline{R}}}(i) + k$|⁠. The weight of |${\overline{R}} \in \textrm{P}_n\textrm{SYT}(k^N)\big /\pi ^n$| is the tuple |$(t^{2 \textrm{diag}_{\overline{R}}(1)}, t^{2 \textrm{diag}_{\overline{R}}(2)}, \ldots , t^{2 \textrm{diag}_{\overline{R}}(n)})$|⁠. Example 6.12. See Figure 12 for an example of how the diagonal labels change within a |$\pi$| orbit. One should also compare this to Figure 2, where only the locations of filling by {1,2,3} are marked. We let |$N=3, k=1$|⁠, so |${{\mathfrak{s}}} = 6$|⁠. In the first periodic tableau |$R$|⁠, |$1+5+6 = {{\mathfrak{s}}} + 3(2)$| so the chosen fundamental domain has NW diagonal labeled |$\frac 23$|⁠. In the second, |$2+6+7 = {{\mathfrak{s}}} + 3(3)$| and so on. In the fourth periodic tableau, the chosen domain has diagonal labeled |$\frac 53$| and passes through , but the diagonal one step left has label |$\frac 23$| and passes through and indeed |$R \sim \pi ^3 \cdot R$|⁠. In other words, |$\textrm{diag}_{{\overline{R}}}(1) = \frac 23 = \textrm{diag}_{\overline{\pi ^3 \cdot R}}(1).$| We also list |$\textrm{wt}({\overline{R}})$| in Figure 12. Note that \begin{equation*}\pi \cdot (t^{4/3}, t^{-8/3}, t^{-14/3}) = \big(\boldsymbol{q}^{-2(3)+2} t^{-14/3}, \boldsymbol{q}^2 t^{4/3}, \boldsymbol{q}^2 t^{-8/3}\big) = \big(t^{-6}, t^{2}, t^{-2}\big),\end{equation*} as |$t=\boldsymbol{q}^3$|⁠. Fig. 12. View largeDownload slide The label of the NW diagonal of a fundamental rectangle depends on its filling sum in type |$SL$|⁠. Fig. 12. View largeDownload slide The label of the NW diagonal of a fundamental rectangle depends on its filling sum in type |$SL$|⁠. Just as in the |$GL_n$| case, |$\widehat{{\mathfrak{S}}}_n$| acts on |$\textrm{P}_n\textrm{SYT}(k^N)\big /\pi ^n$|⁠. It is no longer true that |$\textrm{diag}_{\overline{\sigma \cdot R}}(\sigma (i))$| agrees with |$\textrm{diag}_{{\overline{R}}}(i)$| for |$\sigma \in \widehat{{\mathfrak{S}}}_n$| (in particular for |$\sigma =\pi$|⁠). However, we do still have the intertwining property \begin{equation*}\textrm{wt}(\sigma \cdot{\overline{R}}) = \sigma \cdot \textrm{wt}({\overline{R}})\end{equation*} using the |$SL_n$| modified action of |$\widehat{{\mathfrak{S}}}_n$| on |$({\mathcal{K}}^\times )^n$| described in (4.2). Let |$\overline{L}(k^N)$| denote the linear span over |${\mathcal{K}}$| of \begin{equation*}\{v_{{\overline{R}}} \mid{\overline{R}}\in \textrm{P}_n\textrm{SYT}(k^N)\big/\pi^n\}.\end{equation*} Theorem 6.13. When |$\boldsymbol{q}^N=t$| and |$\boldsymbol{r}=\boldsymbol{q}^{1-N^2}$| there exists a unique irreducible representation of |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$| on |$\overline{L}(k^N)$| such that each |$v_{{\overline{R}}}$| is a |${\mathcal{Z}}$|-weight vector of weight |$\textrm{wt}({\overline{R}})$|⁠, that is, \begin{equation*}Z_iv_{{\overline{R}}} = t^{2\textrm{diag}_{{{\overline{R}}}}({i})}v_{{\overline{R}}}, \quad(i=1,\ldots, n),\end{equation*} and such that \begin{equation*}\pi^N v_{{\overline{R}}_0} = v_{{\overline{R}}_0},\end{equation*} where |${\overline{R}}_0$| is as in Figure 13 below. Fig. 13. View largeDownload slide A periodic skew tableau |${\overline{R}}_0$| stabilized by |$\pi ^N$|⁠. Fig. 13. View largeDownload slide A periodic skew tableau |${\overline{R}}_0$| stabilized by |$\pi ^N$|⁠. The existence of the module |$\overline{L}(k^N)$| has been established above, constructing it combinatorially via equivalence classes of standard periodic tableaux. Unlike the |$GL$| case, |$\overline{L}(k^N)$| is not the unique simple quotient of the induced module |$\overline{M}(k^N)$|⁠. Rather we have Theorem 6.14. Let |$\boldsymbol{q}^N=t$| and |$\boldsymbol{r}=\boldsymbol{q}^{1-N^2}$|⁠. Let |$a \in{\mathcal{K}}^\times$| be a primitive |$n$|th root of unity. The induced module |$\overline{M}(k^N) = \operatorname{Ind}_{H({\mathcal{Z}})}^{{\mathbb{H}}_{\boldsymbol{q},t}(SL_n)} \operatorname{Rect}(N,k)$| has maximal semisimple quotient \begin{equation*}\overline{L}(k^N) \oplus \left(\overline{L}(k^N)\right)^{a} \oplus \cdots \oplus \left(\overline{L}(k^N)\right)^{a^{k-1}}.\end{equation*} We have |$\overline{L}(k^N)^a \cong \overline{L}(k^N)^b$| if and only if |$a^N = b^N$|⁠. These |$k$| twists are all the representations of |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$| that have the same support as |$\overline{L}(k^N)$|⁠. Their isomorphism type is distinguished by the extra data of |$\pi ^N v_{{\overline{R}}_0} = a^N v_{{\overline{R}}_0}.$| Proof. Along the lines of Theorem 6.5, one sees that for |${\overline{R}}_0$| depicted in Figure 13, |$\textrm{wt}({\overline{R}}_0)$| has stabilizer |$\langle \pi ^N \rangle$| via the action (4.2), and hence, the |$\textrm{wt}({\overline{R}}_0)$|-weight space of |$\overline{M}(k^N)$| has dimension |$k=n/N$|⁠. Let us denote by |$v_{a}$|⁠, a basis vector of the one-dimensional |$\textrm{wt}({\overline{R}}_0)$| weight space of |$(\overline{L}(k^N))^a$|⁠. Note that |$\pi ^N$| preserves the weight space, hence it scales |$v_{a}$| by some scalar. Since any homomorphism |$\varphi : (\overline{L}(k^N))^a \to (\overline{L}(k^N))^b$| commutes with the action of |$Z_i$|⁠, it sends |$v_{a}$| to some multiple of |$v_{b}$|⁠, say |$\varphi (v_{a}) = c v_{b}$|⁠. Then \begin{equation*}a^N c v_{a} = \varphi(a^N v_{a}) = \varphi(\pi^N v_{a}) = \pi^N \varphi(v_{a}) = c b^N v_{b}.\end{equation*} If |$\varphi \neq 0$| then |$a^N = b^N$|⁠. This proves the “only if” part of (2). As in the |$GL$| case, Frobenius reciprocity gives a map, \begin{equation*}\overline{M}(k^N) = \operatorname{Ind}_{H({\mathcal{Z}})}^{{\mathbb{H}}_{\boldsymbol{q},t}(SL_n)} \operatorname{Rect}(N,k) \to \left(\overline{L}(k^N)\right)^{a},\end{equation*} as the restriction of |$\left (\overline{L}(k^N)\right )^{a}$| to |$H({\mathcal{Z}})$| contains |$\operatorname{Rect}(N,k)$|⁠. As each of these quotients has a one-dimensional |$\textrm{wt}({\overline{R}}_0)$|-weight space, and |$\overline{M}(k^N)$| has |$k$|-dimensional |$\textrm{wt}({\overline{R}}_0)$|-weight space, |$\overline{M}(k^N)$| can have at most |$k$| simple quotients. Now using the pigeon-hole principle, we complete the proof of (2), as well as (1). In particular, the discussion on |$\varphi$| shows (3) as well. Remark 6.15. Alternatively, we could have replaced |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$| by its subalgebra generated by |$\langle T_0, T_1, \ldots , T_{n-1}, Z_1^{\pm 1} \rangle$|⁠, so that the weight |$\textrm{wt}({\overline{R}}_0)$| (and indeed the weight of any |${\overline{R}}\in \textrm{SYT}(k^N)$|⁠) would once again have trivial stabilizer. Hence, defining |$\overline{M}(k^N)$| by induction in the same way would yield a module with unique simple quotient |$\overline{L}(k^N)$|⁠. We note that |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$| is a free module of rank |$n$| over this subalgebra with basis |$\{1, \pi , \ldots , \pi ^{n-1} \}$|⁠. This subalgebra is in some ways more similar to the RCA of type |$SL$|⁠. 7 The Functor and the Isomorphism 7.1 The functor |$F_n^G$| Let |$G=SL_N$| or |$GL_N$|⁠, and let |$M$| be a |${\mathcal{D}}_{\mathtt{q}}(G)$|-module. Let |$n\in \mathbb{N}$|⁠, and consider the space, \begin{align*}F^{SL}_n(M) &= \Big(\underset{n}{V}\otimes \cdots \otimes \underset{1}{V}\otimes M\Big)^{U_{\mathtt{q}}(\mathfrak{sl}_N)},\quad\textrm{and}\\ F^{GL}_n(M) &= \Big(\operatorname{det}_{\mathtt{q}}^{-k}(V)\otimes\underset{n}{V}\otimes \cdots \otimes \underset{1}{V}\otimes M\Big)^{U_{\mathtt{q}}(\mathfrak{gl}_N)}, \end{align*} of invariants, respectively |$\operatorname{det}_{\mathtt{q}}^{k}$|-variants, in the tensor product of |$M$| with the |$n$|-fold tensor power of the defining |$N$|-dimensional representation, |$V = V_{\epsilon _{1}}$| of |$U_{\mathtt{q}}({\mathfrak{g}})$|⁠. In order to match various conventions (while breaking others), we index the tensor factors from right to left, as indicated in the subscripts above. We note that |$-k$| is the unique possible tensor exponent of |$\operatorname{det}_{\mathtt{q}}(V)$| such that |$F^{GL}_n({\mathcal{O}}_{\mathtt{q}}(GL_N))$| is non-zero and that similarly |$n$| must be an integer multiple of |$N$| in the |$SL$| case, see Remark 7.5. In [15], an action of the double affine Hecke algebra was constructed on the space |$F^{SL}_n$|⁠. Let us recall the construction here and formulate its |$GL$|-modification. Theorem 7.1 ([15]). Let |$G=SL_N$|⁠, or |$GL_N$|⁠, respectively. Let |$M$| be a module for |${\mathcal{D}}_{\mathtt{q}}(G)$|⁠, let |$k$| be a positive integer, and let |$n=kN$|⁠. There is a unique representation of |$B_n^{Ell}$| (resp. |$B_{n,1}^{Ell}$|⁠) on |$F^G_n(M)$| such that Each generator |$\textsf{T}_i$||$(i=1, \ldots , n-1)$| acts by the braiding |$\sigma _{V,V}$| on the |$\underset{i+1}{V}\otimes \underset{i}{V}$| factors. For |$GL_N$|⁠, |$\textsf{T}_n^2$| acts by the double braiding on |$\operatorname{det}_{\mathtt{q}}^{-k}(V) \otimes \underset{n}{V}$|⁠. The operator |$\textsf{Y}_1$| acts only in the rightmost two tensor factors, |$\underset{1}{V}\otimes M$|⁠, via \begin{equation*}\textsf{Y}_1=\sigma_{M_\lhd,V}\circ\sigma_{V,M_\lhd}, \textrm{(the double-braiding of }V\text{ and }M,\text{ using }\partial_\lhd).\end{equation*} The operator |$\textsf{X}_1$| acts only in the rightmost two tensor factors |$\underset{1}{V}\otimes M$|⁠, via \begin{equation*}\textsf{X}_1 = V\otimes M \xrightarrow{\Delta_{V}\otimes \operatorname{Id}_{M}} V\otimes{\mathcal{O}}_{\mathtt{q}}(G)\otimes M \xrightarrow{\operatorname{Id}_{V}\otimes \operatorname{act}_{M}} V\otimes M,\end{equation*} where in the second arrow, |${\mathcal{O}}_{\mathtt{q}}(G)$| acts on |$M$| via the homomorphism |$\ell :{\mathcal{O}}_{\mathtt{q}}(G)\to{\mathcal{D}}_{\mathtt{q}}(G)$|⁠. Remark 7.2. Since |$B_n^{Ell}$| and |$B_{n,1}^{Ell}$| are generated by the operators listed above, their action is uniquely determined. That these operators satisfy the defining relations listed in Proposition 4.3 is established in [15, Theorem 22 and Corollary 24]. Corollary 7.3. The |$B_n^{Ell}$|-action on |$F_n^{SL}(M)$| is natural in |$M$| and |$V$|⁠, satisfies the additional relations \begin{equation*}\big(\textsf{T}_i-\mathtt{q}^{-1/N}\mathtt{q}\big)\big(\textsf{T}_i+\mathtt{q}^{-1/N}\mathtt{q}^{-1}\big)=0, \qquad \textsf{Y}_1\cdots \textsf{Y}_n=\mathtt{q}^{n(1/N-N)}.\end{equation*} Hence by Proposition 4.9 we have an exact functor, \begin{equation*}F^{SL}_n: {\mathcal{D}}_{\mathtt{q}}(SL_N)\textrm{-mod}\to{\mathbb{H}}_{\boldsymbol{q},t}(SL_n)\textrm{-mod},\end{equation*} for |$\boldsymbol{q}=\mathtt{q}^{1/N}$|⁠, |$t=\mathtt{q}$|⁠, and |$\boldsymbol{r} = \mathtt{q}^{1/N - N}$|⁠. Proof. In [15], Proposition 30, it is proved that |$\textsf{Y}_1\cdots \textsf{Y}_n$| acts as |$(\nu ^{-1}|_V)^n$|⁠. We compute, using (3.4), that \begin{equation*}\nu^{-1}|_V = \mathtt{q}^{-\langle\varepsilon_{1} +2\rho,\varepsilon_{1}\rangle} = \mathtt{q}^{1/N-N},\end{equation*} as desired. Corollary 7.4. The |$B_{n,1}^{Ell}$|-action on |$F_n^{GL}(M)$| is natural in |$M$| and |$V$|⁠, satisfies the additional relations \begin{equation}(\textsf{T}_i-\mathtt{q})(\textsf{T}_i+\mathtt{q}^{-1})=0, \qquad \textsf{T}_n^2=\mathtt{q}^{-2k}.\end{equation} (7.1) Hence by Proposition 4.5 we have an exact functor, \begin{equation*}F^{GL}_n: {\mathcal{D}}_{\mathtt{q}}(GL_N)\textrm{-mod}\to{\mathbb{H}}_{q,t}(GL_n)\textrm{-mod},\end{equation*} for |$q=\mathtt{q}^{-2k}$| and |$t=\mathtt{q}$|⁠. Proof. We outline how to modify the proof from [15], to obtain relations (7.1) when |$SL_N$| is replaced by |$GL_N$|⁠. The differing form of the Hecke relation for |$\textsf{T}_1,\ldots , \textsf{T}_{n-1}$| is clear from the quantum |$R$|-matrix for |$GL_N$| compared to |$SL_N$|⁠. The quantum determinant representation |$\operatorname{det}_{\mathtt{q}}(V)$| and all its tensor powers are invertible, which means that |$\operatorname{det}_{\mathtt{q}}^{-k}(V)\otimes V$| is irreducible, so that |$\textsf{T}_n^2$| must act as a scalar; this scalar can be computed using the ribbon element to be |$\mathtt{q}^{-2k}$| as in the proof of Theorem 7.4 above. 7.2 The case |$M={\mathcal{O}}_{\mathtt{q}}(G)$| In the special case |$M={\mathcal{O}}_{\mathtt{q}}(G)$|⁠, the action is compatible with Peter–Weyl decomposition in the following sense. Let \begin{equation*}W^n_\lambda = \left\{\begin{array}{@{}ll}(V^{\otimes n} \otimes V_\lambda^*\otimes V_\lambda)^{U_{\mathtt{q}}({\mathfrak{g}})},& G=SL_N\\ (\operatorname{det}_{\mathtt{q}}^{-k}(V)\otimes V^{\otimes n} \otimes V_\lambda^*\otimes V_\lambda)^{U_{\mathtt{q}}({\mathfrak{g}})},& G=GL_N\end{array}\right.,\end{equation*} and consider the vector space decomposition, \begin{equation*}F_n^G({\mathcal{O}}_{\mathtt{q}}(G)) \cong \bigoplus_{\lambda\in \Lambda^+} W^n_\lambda,\end{equation*} induced by the Peter–Weyl decomposition, Theorem 3.4. Remark 7.5. Recall that we have restricted to the case |$n=kN$| for a positive integer |$k$|⁠. It is elementary to see that otherwise all spaces |$W^n_\lambda$|⁠, and hence |$F_n^G({\mathcal{O}}_{\mathtt{q}}(G))$|⁠, are zero. It follows from the definition of |$Y_1$| and |$T_i$| as braiding operators that they preserve the Peter–Weyl decomposition and the passage to invariants. Hence, each subspace |$W^n_\lambda \subset F_n^G({\mathcal{O}}_{\mathtt{q}}(G))$| is a finite-dimensional submodule for the action of the affine Hecke algebra |${\mathcal{H}}({\mathcal{Y}})$|⁠. 7.3 Walks, skew diagrams, and a weight basis of the invariants In this section, we identify the |${\mathcal{H}}({\mathcal{Y}})$|-modules |$W^n_\lambda$| with those constructed in [19]. Let |$\lambda \in \Lambda ^+_{\mathfrak{sl}_N}$| or |$\Lambda ^+_{\mathfrak{gl}_N}$|⁠. To each walk |$\underline{u}\in{\mathcal{W}}^{N,k}_{\lambda }$|⁠, we associate a unique line |$L_{\underline{u}}$| in |$W^n_\lambda$| as follows. First, in the |$SL$| case we have natural isomorphisms, \begin{align*}W^n_\lambda &= \operatorname{Hom}_{U_{\mathtt{q}}(\mathfrak{sl}_N)}(\textbf{1},V^{\otimes n} \otimes V_\lambda^*\otimes V_\lambda)\\ &\cong \operatorname{Hom}_{U_{\mathtt{q}}(\mathfrak{sl}_N)}(\textbf{1},V^{\otimes n} \otimes V_\lambda\otimes V_\lambda^*)\\ &\cong \operatorname{Hom}_{U_{\mathtt{q}}(\mathfrak{sl}_N)}\left(V_\lambda,V^{\otimes n} \otimes V_\lambda \right),\end{align*} where we have first applied the braiding |$\sigma _{V^*, V}$| and then the canonical isomorphism |$Hom(\textbf{1},X\otimes Y^*)\cong Hom(Y,X)$|⁠. In the |$GL$| case, a similar series of isomorphisms \begin{equation*}W^n_\lambda \cong \operatorname{Hom}_{U_{\mathtt{q}}(\mathfrak{gl}_N)}(\operatorname{det}_{\mathtt{q}}^{k}(V)\otimes V_\lambda, V^{\otimes n} \otimes V_\lambda).\end{equation*} The Pieri rule gives a multiplicity-free decomposition, \begin{equation*}V\otimes V_\alpha \cong \bigoplus_{\beta} V_\beta,\end{equation*} where each |$\beta$| is a dominant weight, which differs from |$\alpha$| by an |$\epsilon _i$|⁠. Hence |$(V \otimes V_\alpha ) [\beta ] \cong V_\beta$|⁠, where the notation |$X[\lambda ]$| denotes the |$\lambda$|-isotypic component of a representation |$X$|⁠. In particular, |$\dim \operatorname{Hom} (V_\beta , V \otimes V_\alpha ) = 1$|⁠. Hence, given a looped walk |${\underline u}$| of length |$n$| from |$\lambda$| to |$\lambda +k\textbf{d}$|⁠, the space \begin{equation*} \operatorname{Hom}\left(V_{\lambda+k\textbf{d}}, \bigcap_{i=0}^n V^{\otimes(n-i)}\otimes \left(\big(V^{\otimes i}\otimes V_\lambda\big)[u_i]\right)\right) \subset \operatorname{Hom}\big(V_{\lambda+k\textbf{d}}, V^{\otimes n}\otimes V_{\lambda}\big),\end{equation*} is also one-dimensional. We define |$L_{\underline u}$| to be this line. In other words, |$L_{\underline u}$| is the subspace of |$\operatorname{Hom}(V_{\lambda +k\textbf{d}},V^{\otimes n}\otimes V_{\lambda })$|⁠, consisting of vectors whose component in each tensor subfactor |$V^{\otimes k}\otimes V_{\lambda }$| has isomorphism type |$u_k$|⁠. By construction, we have an isomorphism of vector spaces, \begin{equation*}W^n_\lambda \cong \bigoplus_{\underline{u}\in{\mathcal{W}}^{N,k}_{\lambda}} L_{\underline{u}}.\end{equation*} Theorem 7.6. Let |$G=GL_N$| or |$SL_N$|⁠. For any |$v\in L_{\underline{u}}$| we have \begin{equation}\textsf{Y}_iv = \mathtt{q}^{ \langle u_i + 2 \rho, u_i \rangle - \langle u_{i-1} + 2 \rho, u_{i-1} \rangle - \langle \epsilon_{1}+ 2 \rho, \epsilon_{1} \rangle} v.\end{equation} (7.2) Proof. This is essentially the content of [19, Proposition 3.6]. We recall the proof for the reader here, in our conventions. By its construction, |$\textsf{Y}_k=\textsf{T}_{k-1}\cdots \textsf{T}_1 \textsf{Y}_1 \textsf{T}_1 \cdots \textsf{T}_{k-1}$| is the double-braiding of |$V$| around |$V^{\otimes k-1}\otimes M$|⁠. Hence, applying equation (3.3) we have \begin{equation*}\textsf{Y}_k = \operatorname{Id}_{V^{\otimes n-k}} \otimes \left(\Delta^{(k+1)}(\nu)\cdot\big(\nu^{-1} \otimes \Delta^{(k)}(\nu^{-1})\big)\right)\!\Big|_{V^{\otimes k}\otimes M}.\end{equation*} By definition of the line |$L_{\underline{u}}$|⁠, and equation (3.4), we have \begin{align*}\left(\operatorname{Id}_{V^{\otimes n-k}} \otimes\Delta^{(k+1)}(\nu)\right)\Big|_{L_{\underline{u}}} &= \mathtt{q}^{ \langle u_i + 2 \rho, u_i \rangle},\\ \left(\operatorname{Id}_{V^{\otimes n-k}} \otimes\left(\nu^{-1} \otimes \Delta^{(k)}(\nu^{-1})\right)\right)\Big|_{L_{\underline{u}}} &= \mathtt{q}^{- \langle \epsilon_{1}+ 2 \rho, \epsilon_{1} \rangle- \langle u_{i-1} + 2 \rho, u_{i-1} \rangle}.\end{align*} Hence, we compute that |$\textsf{Y}_i$| scales the line |$L_{\underline{u}}$| as claimed. Remark 7.7. Recall that the bijection |${\mathcal{T}}\!ab$| from Section 2.4 identifies looped walks at |$\lambda$| with skew tableaux in the diagram |$\textrm{D}^{N,k}_{\lambda }$|⁠. The diagram |$\textrm{D}^{N,k}_{\lambda }$| is natural in light of Theorem 7.6, tensoring |$L_{\underline{u}}$| with |$V_{\lambda ^*}$| corresponds to filling in the lower right corner of the diagram, thus producing an invariant (in the |$SL$| case), or a multiple of the determinant (in the GL case), within |$V^{\otimes n}\otimes V_{\lambda ^*}\otimes V_{\lambda }$|⁠. 7.4 From skew tableaux to periodic tableaux While the natural basis of lines coming from Lie theory is indexed by looped walks of length |$n$|⁠, and hence skew tableaux of size |$n$|⁠, the weight basis for simple and |${\mathcal{Y}}$|-semisimple modules for |${\mathbb{H}}_{q,t}(GL_n)$|-modules is indexed by periodic tableaux on infinite skew diagrams. For simple and |${\mathcal{Z}}$|-semisimple |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$|-modules, the weight basis is indexed by shift-equivalence classes of periodic tableaux. In this section, we construct a bijection between the two bases. Recall from Section 2.4 the definition of |$\textrm{D}^{N,k}_{\lambda }$| and |$\mathcal{SK}^{N,k}_{\lambda }$|⁠. We first observe that |$\textrm{D}^{N,k}_{\lambda }$| is also a fundamental domain of the periodization of |$\mu =(k^N)$|⁠, that is, of the |$N\times \infty$| strip (see Remark 6.9). Similarly, “periodizing” a standard skew tableau in |${\mathcal{T}} \in \mathcal{SK}^{N,k}_{\lambda }$| (i.e., filling in the rest of the entries according to the periodicity constraint) yields a well-defined standard periodic tableau in |$\textrm{P}_n\textrm{SYT}(k^N)$|⁠, as soon as we specify the compatibility with the diagonal labeling. See Figure 14. We formalize this as follows: Definition 7.8. The periodization maps, \begin{gather} {\mathcal{P}}\!er: \bigsqcup_{\lambda \in \Lambda^+_{\mathfrak{gl}_N}} \mathcal{SK}^{N,k}_{\lambda} \to \textrm{P}_n\textrm{SYT}(k^N), \qquad \qquad \overline{{\mathcal{P}}\!er}: \bigsqcup_{\lambda \in \Lambda^+_{\mathfrak{sl}_N}} \mathcal{SK}^{N,k}_{\lambda} \to \textrm{P}_n\textrm{SYT}(k^N)\big/\pi^n, \end{gather} (7.3) send |${\mathcal{T}}$| to the unique periodic tableau in |$\textrm{P}_n\textrm{SYT}(k^N)$| (resp. |$\textrm{P}_n\textrm{SYT}(k^N)\big /\pi ^n$|⁠) agreeing with |${\mathcal{T}}$| in the fundamental domain of shape |$\textrm{D}^{N,k}_{\lambda }$| located along the |$N\times \infty$| strip so that diagonal labels coincide. Note that as the filling of |${\mathcal{T}}$| is |$\{1, \ldots , n\}$| it is easy to see its periodization is standard. While it’s clear then that |${\mathcal{P}}\!er$| is well defined, we need the following for |$\overline{{\mathcal{P}}\!er}$|⁠. Proposition 7.9. The map |$\overline{{\mathcal{P}}\!er}$| is well defined. Proof. Let |$\lambda =\sum _i m_i \varepsilon _{i} \in \Lambda _{\mathfrak{sl}_N}$| and consider |${\mathcal{T}} \in \mathcal{SK}^{N,k}_{\lambda }$|⁠. Recall its principal diagonal is labeled |$-\frac{|\lambda |}{N}$|⁠. We need to check this agrees with the diagonal labels assigned by (6.7) once we extend |${\mathcal{T}}$| periodically to the |$N\times \infty$| strip, see Remark 6.9. In other words, we need to check the |$N \times k$| rectangle whose NW diagonal agrees with the principal diagonal of |${\mathcal{T}}$| has filling sum |${{\mathfrak{s}}} -|\lambda |n$|⁠. Consider the |$k$| entries in the 1st row of |${\mathcal{T}}$|⁠. They lie in |$\{1,\ldots ,n\}$| and hence contribute to the sum |${{\mathfrak{s}}}$|⁠; they also sit on diagonals labeled |$m_1-m_N - \frac{|\lambda |}{N}, \ldots , m_1-m_N+k-1 - \frac{|\lambda |}{N}$|⁠. Likewise after periodizing, the |$k$| entries on diagonals labeled \begin{equation*}-kr + m_1-m_N - \frac{|\lambda|}{N}, \ldots, -kr + m_1-m_N+k-1 - \frac{|\lambda|}{N}\end{equation*} lie in the set |$\{-nr+1, \ldots , -nr+n\}$| and so contribute toward sum |${{\mathfrak{s}}} - krn$|⁠. However, the above makes sense not only for |$r \in{\mathbb{Z}}$| but in the case |$kr \in{\mathbb{Z}}$|⁠. The entries in the 1st row that start on principal diagonal labeled |$-\frac{|\lambda |}{N}$| correspond to left shift by |$r = \frac{m_1-m_N}{k}$|⁠. Considering now all |$N$| rows, the filling sum of this rectangle is |${{\mathfrak{s}}} - \sum _i k(\frac{m_i-m_N}{k})n = {{\mathfrak{s}}} - |\lambda |n$|⁠, and so our |$p = -|\lambda |$| as in (6.7), as desired. Proposition 7.10. The maps |${\mathcal{P}}\!er$| and |$\overline{{\mathcal{P}}\!er}$| are bijections. Proof. The inverse map |${\mathcal{P}}\!er^{-1}$| can be described as follows. Given |$R\in \textrm{P}_n\textrm{SYT}(k^N)$| the boxes filled with |$\{1,2,\ldots , n\}$| form a skew diagram |${\mathcal{T}}$|⁠. Since the box |$k$| steps to the right of is , |${\mathcal{T}}$| has shape |$(\gamma + (k^N))/\gamma$| for some partition with |$\gamma _N=0$|⁠. The principal diagonal of |$\gamma$| has some diagonal label |$r$|⁠. Then set |$\lambda = (\sum _i \gamma _i \epsilon _{i}) + r \textbf{d}$|⁠. We may consider this diagonal-labeled skew diagram |${\mathcal{T}}$| to be in |$\mathcal{SK}^{N,k}_{\lambda }$|⁠. We do similarly for |$\overline{{\mathcal{P}}\!er}^{-1}$|⁠, setting |$\lambda = \sum _{i=1}^{N-1} \gamma _i \varepsilon _{i}$|⁠. Having already checked the diagonal labels for |$\overline{{\mathcal{P}}\!er}$| are well defined, the principal diagonal will already have inherited label |$-\frac{|\lambda |} {N}$|⁠. For an illustration of |${\mathcal{P}}\!er$| in the |$GL$| case see Figure 14. In that example, note the skew tableaux are only differentiated by their diagonal labels and likewise for the periodic tableaux. For an example of |$\overline{{\mathcal{P}}\!er}$| in the |$SL$| case, compare Figure 2 to Figure 12. Fig. 14. View largeDownload slide Here |$G=GL_2, k=2$|⁠. The fundamental rectangle of |${\mathcal{P}}\!er({\mathcal{T}})$| is chosen so that the |$0$|th diagonal matches that of |${\mathcal{T}}\in \mathcal{SK}^{N,k}_{\lambda }$|⁠. Fig. 14. View largeDownload slide Here |$G=GL_2, k=2$|⁠. The fundamental rectangle of |${\mathcal{P}}\!er({\mathcal{T}})$| is chosen so that the |$0$|th diagonal matches that of |${\mathcal{T}}\in \mathcal{SK}^{N,k}_{\lambda }$|⁠. 7.5 The isomorphism type of |$F_n^G({\mathcal{O}}_{\mathtt{q}}(G))$| We are finally ready to prove our main theorem. Theorem 7.11. Let |$\underline{u}$| be a looped walk in |$\Lambda ^+_{\mathfrak{gl}_N}$|⁠, respectively |$\Lambda ^+_{\mathfrak{sl}_N}$|⁠. Each subspace |$L_{\underline{u}}$| is moreover a simultaneous |${\mathcal{Y}}\text{-}{\mathcal{Z}}$|-weight vector, for any |$v\in L_{\underline{u}}$|⁠, we have: \begin{equation*}Y_i v = t^{2\textrm{diag}_{{{\mathcal{T}}\!ab(\underline{u})}}({i})} v, \qquad Z_i v = t^{2\textrm{diag}_{{\overline{{\mathcal{T}}\!ab}(\underline{u})}}({i})} v.\end{equation*} We have isomorphisms of |${\mathbb{H}}_{q,t}(GL_n)$|-modules, and |${\mathbb{H}}_{\boldsymbol{q},t}(SL_n)$|-modules, respectively: \begin{equation*}F_n^{GL}({\mathcal{O}}_q(GL_N))\cong L(k^N), \qquad F_n^{SL}({\mathcal{O}}_q(SL_N))\cong \overline{L}(k^N).\end{equation*} Proof. For |$GL_N$|⁠, all that remains is to simplify the exponent of |$\mathtt{q}$| appearing in Theorem 7.6. We compute first for |$GL_N$| Hence, we may re-write the scalar (7.2) by which |$Y_i$||$= \phi (\textsf{Y}_i)$| acts as \begin{equation*}\mathtt{q}^{2\textrm{diag}_{{{\mathcal{T}}\!ab(\underline{u})}}({i})} = t^{2\textrm{diag}_{{{\mathcal{T}}\!ab(\underline{u})}}({i})}.\end{equation*} Having matched their support, the isomorphism |$F_n^{GL}({\mathcal{O}}_{\mathtt{q}}(GL_N))\cong L(k^N)$| then follows from Theorem 6.10 and Corollary 5.6. In the case |$G=SL_N$|⁠, a similar computation gives: The 2nd equality follows from (2.6), where we write |$u_{i-1} = \sum _{j=1}^{N} m_j \varepsilon _{j}$|⁠. Since |$m_j-m_N$| is the length of the |$j$|th row of |$u_{i-1}$|⁠, |$\overline{{\mathcal{T}}\!ab}(\underline{u})$| places in column |$m_{\delta _i(\underline{u})} -m_N+ 1$| and row |$\delta _i(\underline{u})$|⁠. Recall from Section 2 that the principal diagonal of |$\overline{{\mathcal{T}}\!ab}(\underline{u})$| is labeled |$- \frac{|u_{0}|}{N}$|⁠. Hence, we may re-write the scalar (7.2) by which |$Z_i$||$=\phi (\boldsymbol{q}^{2(i-1)}\textsf{Y}_i)$| acts as \begin{equation*}\boldsymbol{q}^{2(i-1)}\mathtt{q}^{2 \frac{1-i}{N}}\mathtt{q}^{2\textrm{diag}_{{\overline{{\mathcal{P}}\!er}\overline{{\mathcal{T}}\!ab}(u)}}({i})} =(\mathtt{q}^{\frac 1N})^{2(i-1)}\mathtt{q}^{2 \frac{1-i}{N}}\mathtt{q}^{2\textrm{diag}_{{\overline{{\mathcal{P}}\!er}\overline{{\mathcal{T}}\!ab}(u)}}({i})} = t^{2\textrm{diag}_{{\overline{{\mathcal{P}}\!er}\overline{{\mathcal{T}}\!ab}(u)}}({i})}\end{equation*} since we have specialized |$t=\mathtt{q}, \boldsymbol{q}=\mathtt{q}^{1/N}$|⁠. This completely determines the support of |$F_n^{SL}({\mathcal{O}}_q(SL_N))$|⁠. Comparing with Theorems 6.13 and 6.14 we see that \begin{equation*} F_n^{SL}({\mathcal{O}}_q(SL_N)) \cong \overline{L}(k^N)^a,\end{equation*} for some |$a \in{\mathcal{K}}^\times$| with |$a^n =1$|⁠. In order to determine the twist parameter |$a$|⁠, it suffices to consider a weight vector whose weight |$\textrm{wt}({\overline{R}}_0)$| is stabilized by |$\pi ^N$|⁠, the constant by which |$\pi ^N$| scales such a vector is |$a^N$|⁠, and this determines the isomorphism type of the twist as in Theorem 6.14. Hence, in order to conclude that |$a=1$|⁠, it suffices to check that |$\pi ^N v_{{\overline{R}}_0} = v_{{\overline{R}}_0}$| with |${\overline{R}}_0$| as in Figure 13. We note that |${\overline{R}}_0 = \overline{{\mathcal{P}}\!er}(\overline{{\mathcal{T}}\!ab}(\underline{u}_0))$| where |$\underline{u}_0$| is the looped walk at |$\lambda =0$| whose steps satisfy |$\delta _i(\underline{u}_0) = i \bmod N$|⁠. In other words, |$\underline{u}_0$| is formed from the unique |$N$|-step looped walk |$\underline{u}$| at |$\lambda =0$|⁠, concatenated with itself |$k$| times. We have \begin{equation*}v_{{\overline{R}}_0} \in L_{\underline{u}_0} = \underbrace{W_0^N \otimes \cdots \otimes W_0^N \otimes W_0^N}_{k} \subseteq W_0^{kN} = W_0^n,\end{equation*} where we recall that |$W_0^N$| denotes the space |$\operatorname{Hom}(\textbf{1},V^{\otimes N})$| of invariants in |$V$|⁠. It was shown in [15] that |$\textsf{X}_1 \cdots \textsf{X}_N|_{W_0^N} = (\nu ^{-1}|_{V})^N \mathtt{q}^{N(\frac 1N -N)},$| hence \begin{equation*}\textsf{X}_1 \cdots \textsf{X}_N v_{{\overline{R}}_0} = \mathtt{q}^{N(\frac 1N -N)} v_{{\overline{R}}_0}.\end{equation*} We observe that |$\phi (\boldsymbol{r}^{-N} \textsf{X}_1 \cdots \textsf{X}_N \textsf{T}_w) = \pi ^N,$| where |$w$| is the length |$N^2(k-1)$| permutation obtained by raising an |$n$|-cycle to the |$N$|th power. This sends \begin{gather*} \underset{k}{W_0^N} \otimes \underset{k-1}{W_0^N} \otimes \cdots \otimes \underset{1}{W_0^N} \qquad \textrm{ to} \qquad \underset{1}{W_0^N} \otimes \underset{k}{W_0^N} \otimes \cdots \otimes \underset{2}{W_0^N}, \end{gather*} and in fact |$\textsf{T}_w$| acts trivially on |$v_{{\overline{R}}_0}$|⁠. This computation is also consistent with |$\mathtt{q}^{N(k-1)} \mathtt{q}^{(-\frac 1N) \ell (w)} =1$| in the |$GL$| setting where the |$(k-1)$| crossings of |$\det$| contribute the 1st term. Hence, as we are taking |$\boldsymbol{r} = \boldsymbol{q}^{1-N^2} = \mathtt{q}^{1/N-N}$|⁠, we have \begin{equation*}\pi^N v_{{\overline{R}}_0} = \boldsymbol{r}^{-N} \mathtt{q}^{N(\frac 1N -N)} v_{{\overline{R}}_0} = \mathtt{q}^{-N(1/N-N)}\mathtt{q}^{N(1/N-N)} v_{{\overline{R}}_0} = v_{{\overline{R}}_0}.\end{equation*} As a consequence of Theorem 7.11, we can give a combinatorial description of the action of |$\pi$| on the weight basis of |$F_n^G({\mathcal{O}}_{\mathtt{q}}(G))$|⁠, as follows. Definition 7.12. Given a looped walk |$\underline{u}$| of length |$n$|⁠, we may construct a new looped walk |$\varpi (\underline{u})$|⁠, by setting \begin{align*} \delta_i(\varpi(\underline{u})) = \delta_{i-1}(\underline{u}) \quad (2\le i \le n),\qquad &\delta_1(\varpi(\underline{u})) = \delta_{n}(\underline{u}),\\ \varpi(\underline{u})_i = \underline{u}_{i-1} \quad (1\le i \le n),\qquad &\varpi(\underline{u})_0 = \varpi(\underline{u})_{1} - \epsilon_{\delta_n(\underline{u})}. \end{align*} In other words, |$\varpi (\underline{u})$| is the looped walk ending at |$\underline{u}_{n-1}$| instead of |$\underline{u}_n$|⁠. We note that |$\varpi ^n(\underline{u}) = \underline{u}$| when |$\underline{u}$| is a looped walk of length |$n$| in |$\Lambda ^+_{\mathfrak{sl}_N}$|⁠, while in |$\Lambda ^+_{\mathfrak{gl}_N}$| we have |$\varpi ^n(\underline{u})_i =\underline{u}_i - k \textbf{d}$|⁠. Corollary 7.13. We have |$\pi (W^n_\lambda ) \subset \bigoplus _i W^n_{\lambda +\epsilon _i}$|⁠, and moreover \begin{equation*}\pi L_{\underline{u}} = L_{\varpi(\underline{u})}.\end{equation*} While the corollary follows immediately from |${\mathcal{Y}}\text{-}{\mathcal{Z}}$|-semisimplicity of |$F_n^G({\mathcal{O}}_{\mathtt{q}}(G))$| and the results of Section 5, it would not be straightforward to prove in purely Lie theoretic terms. One can also directly deduce the action of the |$X_i$| on the module combinatorially, in a similar fashion. Funding This collaboration began at the Mathematical Sciences Research Institute (MSRI) program, “Geometric Representation Theory”, in 2014, funded by National Science Foundation (NSF) [grant number DMS-1440140]. This work was supported by the European Research Council under the European Union’s Horizon 2020 research and innovation programme [grant agreement number 637618] and by the Simons Foundation. Acknowledgments The authors wish to thank Pavel Etingof and Peter Samuelson for helpful discussions about the |$GL$|-modification of the functor |$F_n^{SL}$|⁠, Siddhartha Sahi for suggesting we extend results from |$SL$| to |$GL$| in the first place, and Stephen Griffeth for pointing out that Theorem 6.5 might not hold for |$SL$|⁠. References [1] Arakawa , T. and Suzuki , T. . “ Duality between |${\mathfrak{sl}}_n\left (\textbf{C}\right )$| and the degenerate affine Hecke algebra ”. J. Algebra 209 , no. 1 ( 1969 ): 288 – 304 . [2] Arakawa , T. , Suzuki , T. , and Tsuchiya , A. . “ Degenerate Double Affine Hecke Algebra and Conformal field theory .” In Topological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996), vol. 160 of Progr. Math ,– 34 . Boston, MA : Birkhäuser Boston , 1998 . [3] Ben-Zvi , D. , Brochier , A. , and Jordan , D. . “ Integrating quantum groups over surfaces .” J. Topol. 11 , no. 4 ( 2018a ): 873 – 916 . 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Vasserot . “ Cyclotomic double affine Hecke algebras and affine parabolic category |$\mathcal{O}$| .” Adv. Math. 225, no. 3 ( 2010a ): 1523 – 1588 . © The Author(s) 2019. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

Alonso-Gutiérrez,, David;Hernández Cifre, María, A;Roysdon,, Michael;Yepes Nicolás,, Jesús;Zvavitch,, Artem

doi: 10.1093/imrn/rnz010pmid: N/A

Abstract In this paper we prove a series of Rogers–Shephard type inequalities for convex bodies when dealing with measures on the Euclidean space with either radially decreasing densities or quasi-concave densities attaining their maximum at the origin. Functional versions of classical Rogers–Shephard inequalities are also derived as consequences of our approach. 1 Introduction and Main Results We denote the length of a vector |$x \in{\mathbb{R}}^n$| by |$|x|$|⁠. We represent by |$B_n=\left \{x\in{\mathbb{R}}^n:|x|\leq 1\right \}$| the |$n$|-dimensional Euclidean unit ball, by |${\mathbb{S}}^{n-1}$| its boundary, and |$\sigma $| will denote the standard surface area measure on |${\mathbb{S}}^{n-1}$|⁠. The |$n$|-dimensional volume of a measurable set |$M\subset{\mathbb{R}}^n$|⁠, that is, its |$n$|-dimensional Lebesgue measure, is denoted by |$\textrm{vol}(M)$| or |$\textrm{vol}_n(M)$| if the distinction of the dimension is useful (when integrating, as usual, |$\textrm{d} x$| will stand for |$\textrm{d} \textrm{vol}(x)$|⁠). With |$\operatorname{int} M$|⁠, |$\textrm{bd} M$|⁠, and |$\textrm{conv} M$| we denote the interior, boundary, and convex hull of |$M$|⁠, respectively, and we set |$[x,y]$| for |$\textrm{conv} \{x,y\}$|⁠, |$x,y\in{\mathbb{R}}^n$|⁠. The set of all |$i$|-dimensional linear subspaces of |${\mathbb{R}}^n$| is denoted by |$\textrm{G}(n,i)$|⁠, and for |$H\in \textrm{G}(n,i)$|⁠, the orthogonal projection of |$M$| onto |$H$| is denoted by |$P_HM$|⁠. Moreover, |$H^{\bot }\in \textrm{G}(n,n-i)$| represents the orthogonal complement of |$H$|⁠. Finally, let |${\mathcal{K}}^n$| be the set of all |$n$|-dimensional convex bodies, that is, compact convex sets with nonempty interior, in |${\mathbb{R}}^n$|⁠. We will frequently refer to [3], [17], and [36] for general references for convex bodies and their properties. The Minkowski sum of two nonempty sets |$A,B\subset{\mathbb{R}}^n$| denotes the classical vector addition of them, |$A+B=\{a+b:\, a\in A, \, b\in B\}$|⁠, and we write |$A-B$| for |$A+(-B)$|⁠. One of the most famous relations involving the volume and the Minkowski addition is the Brunn–Minkowski inequality (we refer to [16] for an extensive survey of this inequality). One form of it states that if |$K,L\in{\mathcal{K}}^n$|⁠, then \begin{equation} \textrm{vol}(K+L)^{1/n}\geq \textrm{vol}(K)^{1/n}+\textrm{vol}(L)^{1/n}, \end{equation} (1) and equality holds if and only if |$K$| and |$L$| are homothetic. The Brunn–Minkowski inequality was generalized to different types of measures, including the case of log-concave measures [24, 31], a very powerful generalization to the case of Gaussian measures [9, 10, 13, 14, 37], to |$p$|-concave measures and many other extensions (see, e.g., [8] and [11]). It is interesting to note that it was proved by Borell [7, 8] that most of such generalizations would require a |$p$|-concavity assumption on the underlined measure and its density (see (6) below for the precise definition). Following those works, recently, many classical results in convex geometry were generalized to the case of log-concave (and in some cases |$p$|-concave) functions. We mention, among others, the Blaschke–Santaló inequality [4, 5, 15], the Bourgain–Milman, and the reverse Brunn–Minkowski inequality [21], the general works on duality and volume [5, 6], as well as the Grünbaum inequality [28, 29] and others [18, 26, 27, 30, 32]. In the particular case when |$L=-K$|⁠, (1) gives \begin{equation*} \textrm{vol}(K-K)\geq 2^n\textrm{vol}(K), \end{equation*} with equality if and only if |$K$| is centrally symmetric, that is, there exists a point |$x \in{\mathbb{R}}^n$| such that |$K-x=-(K-x)$|⁠. An upper bound for the volume of |$K-K$| is given by the Rogers–Shephard inequality, originally proven in [34, Theorem 1]. For more details about this inequality, we also refer the reader to [36, Section 10.1] or [3]. Theorem A (The Rogers-Shephard inequality) Let |$K\in{\mathcal{K}}^n$|⁠. Then \begin{equation} \textrm{vol}(K-K)\leq \binom{2n}{n}\textrm{vol}(K), \end{equation} (2) with equality if and only if |$K$| is a simplex. Similarly to the Brunn–Minkowski inequality (1), it is natural to wonder about the possibility of extending (2) for measures associated with certain densities. The most natural candidates would be the classes of |$p$|-concave measures. Nevertheless, it was noticed recently that a number of results in convex geometry and geometric tomography can be generalized to a class of measures whose densities have no concavity assumption. This includes the solution of the Busemann–Petty problem for general measures [38], the Koldobsky slicing inequality [19, 20, 22, 23], as well as Shephard’s problem for general measures [25]. First, we observe that one cannot expect to obtain \begin{equation} \mu(K-K)\leq \binom{2n}{n}\mu(K) \end{equation} (3) without having certain control on the “position” of the body |$K$|⁠. Indeed, it is enough to consider the standard |$n$|-dimensional Gaussian measure |$\gamma _n$| given by \begin{equation*} \textrm{d}\gamma_n(x)=\frac{1}{(2\pi)^{n/2}}e^{\frac{-|x|^2}{2}}\textrm{d} x, \end{equation*} and |$K=x+B_n$| for |$|x|$| large enough. In this case it is clear that |$\gamma _n(K-K)=\gamma _n(2B_n)>0$|⁠, whereas |$\gamma _n(K)$| can be arbitrarily small. One option to get control on the right-hand side of (3) might be to exchange |$\mu (K)$| with a mean of the measures of all the translated copies of |$K$| with respect to |$-K$|⁠. To this end, given a measure |$\mu $| on |${\mathbb{R}}^n$|⁠, we define its translated-average|$\overline{\mu }$| as \begin{equation*} \overline{\mu}(K)=\dfrac{1}{\textrm{vol}(K)}\int_{K}\mu(-y+K)\,\textrm{d} y, \end{equation*} for any |$K\in{\mathcal{K}}^n$|⁠. With this notion, our 1st main result reads as follows. Theorem 1.1. Let |$K\in{\mathcal{K}}^n$|⁠. Let |$\mu $| be a measure on |${\mathbb{R}}^n$| given by |$\textrm{d}\mu (x)=\phi (x)\,\textrm{d} x$|⁠, where |$\phi :{\mathbb{R}}^n\longrightarrow [0,\infty )$| is radially decreasing. Then \begin{equation} \mu(K-K)\leq \binom{2n}{n}\min\left\{\overline{\mu}(K),\overline{\mu}(-K)\right\}. \end{equation} (4) Moreover, if |$\phi $| is continuous at the origin then equality holds in (4) if and only if |$\mu $| is a constant multiple of the Lebesgue measure on |$K-K$| and |$K$| is a simplex. A function |$\phi :{\mathbb{R}}^n\longrightarrow [0,\infty )$| is said to be radially decreasing if |$\phi (tx)\geq \phi (x)$| for any |$t\in [0,1]$| and any point |$x\in{\mathbb{R}}^n$|⁠. A lower bound for |$\mu (K-K)$| when the density function of |$\mu $| is even and |$p$|-concave (see the definition below), |$p\geq -1/n$|⁠, can be directly obtained from the results by Borell and Brascamp–Lieb [8, 11]: \begin{equation} \mu(K-K)\geq\mu(2K). \end{equation} (5) Here we extend (5) to the case of measures with even and quasi-concave densities (Theorem 2.6). We recall that a function |$\phi :{\mathbb{R}}^n\longrightarrow [0,\infty )$| is |$p$|-concave, for |$p\in{\mathbb{R}}\cup \{\pm \infty \}$|⁠, if \begin{equation} \phi\left((1-\lambda)x+\lambda y\right)\geq M_p\left(\phi(x),\phi(y),\lambda\right) \end{equation} (6) for all |$x,y\in{\mathbb{R}}^n$| and any |$\lambda \in (0,1)$|⁠. Here |$M_p$| denotes the |$p$|-mean of two nonnegative numbers: \begin{equation*} M_p(a,b,\lambda)=\left\{\!\! \begin{array}{ll} \left((1-\lambda)a^p+\lambda b^p\right)^{1/p} & \textrm{if}\ p\neq 0,\pm\infty,\\[1mm] a^{1-\lambda}b^\lambda & \textrm{if}\ p=0,\\[1mm] \max\{a,b\} & \textrm{if}\ p=\infty,\\[1mm] \min\{a,b\} & \textrm{if}\ p=-\infty, \end{array}\right. \end{equation*} for |$ab>0$|⁠; |$M_p(a,b,\lambda )=0$|⁠, when |$ab=0$| and |$p\in{\mathbb{R}}\cup \{\pm \infty \}$|⁠. A |$0$|-concave function is usually called log-concave whereas a |$(-\infty )$|-concave function is called quasi-concave. Quasi-concavity is equivalent to the fact that the superlevel sets \begin{equation} {\mathcal{C}}_t(\phi)=\left\{x\in\textrm{supp}\ \phi:\phi(x)\geq t\|\phi\|_{\infty}\right\} \end{equation} (7) are convex for |$t\in [0,1]$|⁠. Here |$\textrm{supp} \phi $| denotes the support of |$\phi $|⁠, that is, the closure of the set |$\left \{x\in{\mathbb{R}}^n:\phi (x)>0\right \}$|⁠, and with |$\|\cdot \|_{\infty }$| we mean \begin{equation*} \|\phi\|_{\infty}=\textrm{ess sup}_{x\in{\mathbb{R}}^n}\phi(x)=\inf\Bigl\{t\in{\mathbb{R}}:\textrm{vol}\left(\{x\in{\mathbb{R}}^n:\phi(x)>t\}\right)=0\Bigr\}. \end{equation*} We notice that if |$\phi $| is |$p$|-concave, then |$\textrm{supp}\ \phi $| is a closed convex set. Furthermore, if a function |$\phi $| is quasi-concave and such that |$\max _{x\in{\mathbb{R}}^n}\phi (x)=\phi (0)$| then it is radially decreasing. Although the Rogers–Shephard inequality (2) has been recently extended to the functional setting (see, e.g., [1], [2], and [12] and the references therein), there seems to be no direct way to derive inequality (4) from the above-mentioned functional versions just by considering the function |$\chi _{_K}\,\phi $|⁠, where |$\phi $| is the density of the given measure and |$\chi _{_K}$| is the characteristic function of a convex body |$K$| (Remark 2.10). More precisely, in [12, Theorems 4.3 and 4.5], Colesanti extended (2) to the more general functional inequality \begin{equation} \int_{{\mathbb{R}}^n}\sup_{x=x_1+x_2}\left(f(x_1)^p+f(-x_2)^p\right)^{1/p}\,\textrm{d} x\leq \binom{2n}{n}\int_{{\mathbb{R}}^n} f(x)\,\textrm{d} x, \end{equation} (8) for any |$p$|-concave integrable function, with |$p\in [-\infty ,0)$|⁠. Here, the case |$p=-\infty $| has to be understood as |$\min \left \{f(x_1),f(-x_2)\right \}$|⁠. In Section 2 we will also generalize (8) to general measures (Theorem 2.9). In [35], in addition to |$K-K$|⁠, Rogers and Shephard considered two other centrally symmetric convex bodies associated with |$K$|⁠. The 1st one is \begin{equation*} CK=\left\{(x,\theta)\in{\mathbb{R}}^{n+1}:\, x\in(1-\theta)K+\theta(-K),\,\theta\in[0,1]\right\}, \end{equation*} whose volume is given by \begin{equation*} \textrm{vol}_{n+1}(CK)=\int_0^1\textrm{vol}\left((1-\theta)K+\theta(-K)\right)\,\textrm{d}\theta. \end{equation*} The 2nd one is just |$\textrm{conv}\ \left (K\cup (-K)\right )$|⁠. The relation of the volumes of |$CK$| and |$\textrm{conv}\ \left (K\cup (-K)\right )$| to the volume of |$K$| was proved in [35]: Theorem B Let |$K\in{\mathcal{K}}^n$| be a convex body containing the origin. Then \begin{equation} \int_0^1\textrm{vol}\bigl((1-\theta)K+\theta(-K)\bigr)\,\textrm{d}\theta \leq\frac{2^n}{n+1}\,\textrm{vol}(K), \end{equation} (9) with equality if and only if |$K$| is a simplex. Moreover, \begin{equation} \textrm{vol}\Bigl(\textrm{conv} \bigl(K\cup(-K)\bigr)\Bigr)\leq2^n\,\textrm{vol}(K), \end{equation} (10) with equality if and only if |$K$| is a simplex with the origin as a vertex. Here we will show an analog of the above result in the setting of measures with radially decreasing density: Theorem 1.2. Let |$K\in{\mathcal{K}}^n$| be a convex body containing the origin and let |$\mu $| be a measure on |${\mathbb{R}}^n$| given by |$\textrm{d}\mu (x)=\phi (x)\,\textrm{d} x$|⁠, where |$\phi :{\mathbb{R}}^n\longrightarrow [0,\infty )$| is radially decreasing. Then \begin{equation} \int_0^1\mu\bigl((1-\theta)K+\theta(-K)\bigr)\,\textrm{d}\theta \leq\frac{2^n}{n+1}\,\sup_{\substack{y\in K\\\theta\in(0,1]}}\frac{\mu\bigl((1-\theta)y-\theta K\bigr)}{\theta^n} \end{equation} (11) and moreover, if |$\phi$| is quasi-concave, \begin{equation} \mu\Bigl(\textrm{conv}\bigl(K\cup(-K)\bigr)\Bigr)\leq 2^n\,\sup_{\substack{y\in K\\\theta\in(0,1]}}\frac{\mu\bigl((1-\theta)y-\theta K\bigr)}{\theta^n}. \end{equation} (12) Moreover, if |$\phi $| is continuous at the origin then equality holds in (11) if and only if |$\mu $| is a constant multiple of the Lebesgue measure on |$\textrm{conv} \bigl (K\cup (-K)\bigr )$| and |$K$| is a simplex, and equality holds in (12) if and only if |$\mu $| is a constant multiple of the Lebesgue measure on |$\textrm{conv} \bigl (K\cup (-K)\bigr )$| and |$K$| is a simplex with the origin as a vertex. We note that the upper bounds in Theorem 1.2 are bounded and can be restated using |$\|\phi \|_{\infty }\textrm{vol}(K)$|⁠; indeed, |$\mu \bigl ((1-\theta )y-\theta K\bigr )/\theta ^n$| is bounded from above by |$\|\phi \|_{\infty }\textrm{vol}(K)$|⁠. In [35, Theorem 1], Rogers and Shephard also gave the following lower bound for the volume of |$K$| in terms of the volumes of a projection and a maximal section of |$K$|⁠: Theorem C Let |$k\in \{1,\dots ,n-1\}$|⁠, |$H\in \textrm{G}(n,n-k)$| and |$K\in{\mathcal{K}}^n$|⁠. Then \begin{equation} \textrm{vol}_{n-k}\bigl(P_HK\bigr)\max_{x_0\in H} \textrm{vol}_k\bigl(K\cap\bigl(x_0+H^{\bot}\bigr)\bigr)\leq\binom{n}{k}\textrm{vol}(K). \end{equation} (13) In this paper we will show that the above result remains true for products of measures associated with quasi-concave densities, provided that |$P_HK\subset K$|⁠, that is, |$P_HK=K\cap H$|⁠. The assumption on the projection is necessary, as pointed out in Example 4.2. In particular, this hypothesis does not allow one to prove Theorem 1.2 by directly following the proof of Theorem B, see [35, Theorems 2 and 3]: there, the authors constructed a suitable higher dimensional set to which (13) was applied. This will be not possible here. Before stating the result, we fix the following notation: given a convex body |$K$| and |$x\in P_HK$|⁠, we write |$K(x)=(K-x)\cap H^{\bot }$|⁠. We will use the definition of superlevel set |${\mathcal{C}}_t(\phi )$| given by (7). Theorem 1.3. Let |$k\in \{1,\dots ,n-1\}$| and |$H\in \textrm{G}(n,n-k)$|⁠. Given a continuous at the origin and quasi-concave function |$\phi _k:{\mathbb{R}}^k\longrightarrow [0,\infty )$| with |$\|\phi _k\|_{\infty }=\phi _k(0)$| and a radially decreasing function |$\phi _{n-k}:{\mathbb{R}}^{n-k}\longrightarrow [0,\infty )$|⁠, let |$\mu _n=\mu _{n-k}\times \mu _{k}$| be the product measure on |${\mathbb{R}}^n$| given by |$\textrm{d}\mu _{n-k}(x)=\phi _{n-k}(x)\,\textrm{d} x$| and |$\textrm{d}\mu _{k}(y)=\phi _k(y)\,\textrm{d} y$|⁠. Let |$K\in{\mathcal{K}}^n$| with |$P_HK\subset K$| and so that |$\textrm{vol}_k\bigl ({\mathcal{C}}_t(\phi _k)\cap K(x)\bigr )$| attains its maximum at |$x=0$| for every |$t\in (0,1)$|⁠. Then \begin{equation} \mu_{n-k}\bigl(P_HK\bigr)\mu_k\bigl(K\cap H^{\bot}\bigr)\leq\binom{n}{k}\mu_n(K). \end{equation} (14) The above assumption on the maximal section |$K(0)$| of |$K$| can be omitted when the density of the product measure is also quasi-concave, as shown in Theorem 4.1, which is a straightforward consequence of the following functional version of (13). Theorem 1.4. Let |$k\in \{1,\dots ,n-1\}$| and |$H\in \textrm{G}(n,n-k)$|⁠. Let |$f:{\mathbb{R}}^n\longrightarrow [0,\infty )$| be a bounded quasi-concave function such that |$\textrm{vol}_k\bigl ({\mathcal{C}}_t(f)\cap (x+H^{\bot })\bigr )$|⁠, |$x\in H$|⁠, attains its maximum at |$x=0$| for every |$t\in (0,1)$|⁠, and let |$g:H\longrightarrow [0,\infty )$| be a radially decreasing function. Then, \begin{equation*} \int_{H}g(x)P_Hf(x)\,\textrm{d} x\int_{H^{\bot}}f(y)\,\textrm{d} y \leq\binom{n}{k}\|f\|_{\infty}\int_{{\mathbb{R}}^n}g(P_Hx)f(x)\,\textrm{d} x. \end{equation*} Here, the projection function |$P_Hf:H\longrightarrow [0,\infty )$| of |$f$| is defined by |$P_Hf(x)=\sup _{y\in H^{\bot }}f(x+y)$|⁠. In the particular case of a log-concave integrable function |$f$|⁠, this result has been recently obtained in [1, Theorem 1.1]. The paper is organized as follows. Section 2 is mainly devoted to the proofs of Theorems 1.1 and 1.2 as well as the functional analogs of these results. We start Section 3 by deriving a general result for functions with certain concavity conditions, which will play a relevant role along the manuscript. As a consequence of this result we prove, in particular, Theorem 1.4. Next, in Section 4, we study Rogers–Shephard type inequalities for measures with quasi-concave densities and prove Theorem 1.3. Finally, in Section 5, we present another Rogers–Shephard type inequality when assuming a further concavity for the density of the involved measure. 2 Rogers–Shephard Type Inequalities for Measures with Radially Decreasing Densities 2.1 The case of convex sets As pointed out in the previous section, one cannot expect to obtain (3) without having control on the translations of the set |$K$|⁠. Moreover, certain requirements on the density of the measure |$\mu $| must be made (see also the comments after Remark 2.4 and Example 2.5). To this regard, in Section 4, we will show that one may consider quasi-concave densities with maximum at the origin. In this setting, we will also obtain other Rogers–Shephard type inequalities. Let us now follow a different approach. First, we will prove an extension of (2) for the more general case of radially decreasing densities, collected in Theorem 1.1. Before showing it, we need the following auxiliary result. Lemma 2.1. Let |$\phi :[0,\infty )\longrightarrow [0,\infty )$| be a decreasing function and let |$n,m\in{\mathbb{N}}$|⁠. Then, for every |$x\in (0,\infty )$|⁠, \begin{equation*} \int_0^x \left(1-\frac{t}{x} \right)^n t^{m-1} \phi(t)\,\textrm{d} t \geq \binom{n+m}{n}^{-1} \int_0^x t^{m-1}\phi(t) \,\textrm{d} t, \end{equation*} with equality if and only if |$\phi $| is constant on |$(0,x)$|⁠. Proof. Considering the function |$F:(0,\infty )\longrightarrow [0,\infty )$| given by \begin{equation*} F(x)=\binom{n+m}{n}^{-1}\int_0^x t^{m-1}\phi(t) \,\textrm{d} t -\int_0^x \left(1-\frac{t}{x} \right)^n t^{m-1} \phi(t)\,\textrm{d} t, \end{equation*} we need to show that it is nonpositive. Expanding the binomial |$\left (1 - t/x \right )^n$| we may assert on one hand that |$F(x)\to 0$| as |$x\to 0^+$|⁠. On the other hand, and jointly with Lebesgue’s differentiation theorem, we get that the derivative of |$F$| exists for almost every |$x\in (0,\infty )$| and further \begin{equation*} F^{\prime}(x)=\binom{n+m}{n}^{-1}\,x^{m-1}\phi(x)-n\int_0^x\left(1-\frac{t}{x}\right)^{n-1}\frac{t^m}{x^2}\,\phi(t) \,\textrm{d} t. \end{equation*} Now, applying the change of variable |$u = t/x$|⁠, we get \begin{equation*} n\int_0^x\left(1-\frac{t}{x}\right)^{n-1}t^m\,\textrm{d} t=\dfrac{n\,\Gamma(n)\Gamma(m+1)}{\Gamma(n+m+1)}x^{m+1}=\binom{n+m}{n}^{-1}x^{m+1}, \end{equation*} where |$\Gamma $| represents the gamma function. This together with the fact that |$\phi $| is decreasing implies that |$F^{\prime}(x)\leq 0$|⁠, with equality if and only if |$\phi $| is constant on |$(0,x)$|⁠. Since |$F$| is absolutely continuous on every interval |$[a,b]\subset (0,\infty )$|⁠, because it arises as a finite sum of products of absolutely continuous functions, \begin{equation*} F(x)=F(a)+\int_a^xF^{\prime}(s)\,\textrm{d} s\leq F(a) \end{equation*} for all |$x>0$| and any |$0<a\leq x$|⁠. Taking into account that |$\lim _{a\to 0^+}F(a)=0$| we then have \begin{equation*} F(x)= \int_0^x F^{\prime}(s)\,\textrm{d} s\leq 0, \end{equation*} with equality if and only if |$F^{\prime}\equiv 0$| almost everywhere or, equivalently, when |$\phi $| is constant on |$(0,x)$|⁠. Next we prove Theorem 1.1. We follow the idea of the original proof of the Rogers–Shephard inequality [34], with the main difference of the application of Lemma 2.1 in (18). Proof of Theorem 1.1. Let |$f:{\mathbb{R}}^n\longrightarrow [0,\infty )$| be the function given by \begin{equation*} f(x)=\textrm{vol}\bigl(K\cap(x+K)\bigr). \end{equation*} Observe that |$\textrm{supp} f=K-K$| and |$f$| vanishes on |$\textrm{bd}(K-K)$|⁠. Furthermore, using the Brunn–Minkowski inequality (1) together with the inclusion \begin{equation} K\cap\bigl[(1-\lambda)x+\lambda y+K\bigr] \supset(1-\lambda)\bigl[K\cap(x+K)\bigr]+\lambda\bigl[K\cap(y+K)\bigr], \end{equation} (15) which holds for all |$\lambda \in [0,1]$| and |$x,y\in K-K$|⁠, we get that |$f$| is |$(1/n)$|-concave. On the one hand, by Fubini’s theorem, we have \begin{equation} \int_{K-K}f(x) \,\textrm{d}\mu(x)= \int_{{\mathbb{R}}^n}\int_{{\mathbb{R}}^n}\chi_{_K}(y)\chi_{_{y-K}}(x)\,\phi(x)\,\textrm{d} y\,\textrm{d} x=\int_{K}\mu(y-K)\,\textrm{d} y=\textrm{vol}(K)\,\overline{\mu}(-K). \end{equation} (16) On the other hand, we define the function |$g:K-K\longrightarrow [0,\infty )$| given by \begin{equation*} g(x)=f(0)\left[1-\frac{|x|}{\rho_{_{\!K-K}}\bigl(x/|x|\bigr)} \right]^n, \quad \textrm{for every}\ x\neq0, \end{equation*} and |$g(0)=f(0)$|⁠, where \begin{equation*} \rho_{_{\!L}}(u)=\max\{\rho\geq 0:\rho u\in L\},\quad u\in{\mathbb{S}}^{n-1}, \end{equation*} stands for the radial function of |$L\in{\mathcal{K}}^n$|⁠. Notice that |$g^{1/n}$| is affine on |$\bigl [0,\rho _{_{\!K-K}}(u)u\bigr ]$|⁠, for all |$u\in{\mathbb{S}}^{n-1}$|⁠, and so |$g(0)^{1/n}=f(0)^{1/n}$| and \begin{equation*} g\bigl(\rho_{_{\!K-K}}(u)u\bigr)^{1/n}=0=f\bigl(\rho_{_{\!K-K}}(u)u\bigr)^{1/n}. \end{equation*} Hence, since |$f^{1/n}$| is concave, it follows that |$f^{1/n}\geq g^{1/n}$| on |$\bigl [0,\rho _{_{\!K-K}}(u)u\bigr ]$|⁠. Therefore, using polar coordinates, we have \begin{equation} \begin{split} \int_{K-K}f(x)\,\textrm{d}\mu(x) & =\int_{{\mathbb{S}}^{n-1}}\int_0^{\rho_{_{\!K-K}}(u)} r^{n-1} f(r u) \phi(r u)\,\textrm{d} r\,\textrm{d}\sigma(u)\\ & \geq f(0)\int_{{\mathbb{S}}^{n-1}}\int_0^{\rho_{_{\!K-K}}(u)} \left(1 - \frac{r}{\rho_{_{\!K-K}}(u)}\right)^n r^{n-1} \phi(r u) \,\textrm{d} r \,\textrm{d}\sigma(u). \end{split} \end{equation} (17) Now, from (17) and Lemma 2.1 we obtain \begin{equation} \int_{K-K} f(x) \,\textrm{d}\mu(x) \geq \frac{1}{\binom{2n}{n}}f(0)\int_{{\mathbb{S}}^{n-1}}\int_0^{\rho_{_{\!K-K}}(u)}r^{n-1} \phi(r u)\,\textrm{d} r\,\textrm{d}\sigma(u)=\frac{1}{\binom{2n}{n}}\textrm{vol}(K)\mu(K-K), \end{equation} (18) which, together with (16), yields \begin{equation*} \mu(K-K)\leq\binom{2n}{n}\overline{\mu}(-K). \end{equation*} By replacing |$K$| with |$-K$|⁠, we obtain the desired inequality. Finally, we notice that equality holds in (4) only if there is equality in (18). This implies, by Lemma 2.1, that |$\phi (ru)$| is constant on |$\bigl (0,\rho _{_{\!K-K}}(u)\bigr )$| for |$\sigma $|-almost every |$u\in{\mathbb{S}}^{n-1}$|⁠. Since |$\phi $| is continuous at the origin, |$\mu $| is a constant multiple of the Lebesgue measure on |$K-K$| and, by Theorem A, |$K$| is a simplex. The converse immediately follows from Theorem A. Remark 2.2. From the proof of the equality case in the above result (and the corresponding one of Lemma 2.1), we notice that the assumption of continuity at the origin for |$\phi $| is necessary in order to “recover” the Lebesgue measure (up to a constant). Indeed, one could consider a simplex |$K$| and a function |$\phi $| that is constant on |$\bigl (0,\rho _{_{\!K-K}}(u)\bigr )$| for every |$u\in{\mathbb{S}}^{n-1}$|⁠, but not necessarily constant on |$K-K$|⁠, and thus (4) would hold with equality. The next theorem is obtained just by repeating the same argument given in the proof of Theorem 1.1 but replacing |$-K$| with |$L$|⁠. Theorem 2.3. Let |$K, L\in{\mathcal{K}}^n$| be such that |$K+L$| contains the origin, and let |$\mu $| be a measure on |${\mathbb{R}}^n$| given by |$\textrm{d}\mu (x)=\phi (x)\,\textrm{d} x$|⁠, where |$\phi :{\mathbb{R}}^n\longrightarrow [0,\infty )$| is radially decreasing. Then \begin{equation*} \mu(K+L)\textrm{vol}\bigl(K\cap(-L)\bigr)\leq \binom{2n}{n}\int_K\mu(x+L)\textrm{d} x. \end{equation*} Remark 2.4. As a straightforward consequence of Theorem 1.1, we get the following statement. Let |$K\in{\mathcal{K}}^n$| and let |$\mu $| be a measure on |${\mathbb{R}}^n$| given by |$\textrm{d}\mu (x)=\phi (x)\,\textrm{d} x$|⁠, where |$\phi :{\mathbb{R}}^n\longrightarrow [0,\infty )$| is radially decreasing. Then \begin{equation} \mu(K-K)\leq \binom{2n}{n} \min\left\{\sup_{x\in{\mathbb{R}}^n}\mu(x+K),\sup_{x\in{\mathbb{R}}^n}\mu(x-K)\right\}. \end{equation} (19) The above fact trivially holds in dimension |$n=1$| for an arbitrary measure. Indeed, given |$K=[a,b]$|⁠, then \begin{equation*} \mu(K-K)=\mu\bigl([a-b,b-a]\bigr)=\mu\bigl([a,b]-a\bigr)+\mu\bigl([a,b]-b\bigr)\!\leq 2\min\!\left\{\sup_{x\in{\mathbb{R}}}\mu(x+K),\,\sup_{x\in{\mathbb{R}}}\mu(x-K)\right\}\!. \end{equation*} However, in dimension |$n\geq 2$| the radial decay assumption cannot be omitted, as the following example shows. Fig. 1. View largeDownload slide Constructing a measure for which (19) does not hold. Fig. 1. View largeDownload slide Constructing a measure for which (19) does not hold. Example 2.5. Fix |$0<\varepsilon <\delta <2$|⁠. Consider the measure |$\mu $| on |${\mathbb{R}}^2$| with density \begin{equation*} \phi(x)=\left\{\begin{array}{ll} 1 & \textrm{if}\ x\in\delta B_2\cup\bigl(2B_2\setminus(2-\varepsilon)B_2\bigr)\\ 0 & \textrm{ otherwise} \end{array}\right. \end{equation*} (Figure 1). Then \begin{equation} \mu(B_2-B_2)>6\sup_{x\in{\mathbb{R}}^2}\mu(x+B_2). \end{equation} (20) Note that (20) contradicts (19). Indeed, on the one hand, \begin{equation*} \mu(B_2-B_2)=\mu(2B_2)=\pi\delta^2+\bigl(4-(2-\varepsilon)^2\bigr)\pi=4\pi\varepsilon+\pi(\delta^2-\varepsilon^2). \end{equation*} On the other hand, we note that we need at least six copies of the unit disc in order to cover |$\textrm{bd} (2B_2)$|⁠, which can be seen by considering a regular hexagon inscribed in |$2B_2$| (Figure 1). Moreover, if we would cover |$\textrm{bd} (2B_2)$| with exactly six translated copies of |$B_2$|⁠, then the covering discs would stay away from the origin. Thus, for |$\varepsilon>0$| small enough, \begin{equation*} \sup_{x\in{\mathbb{R}}^2}\textrm{vol}\Bigl((x+B_2)\cap\bigl(2B_2\setminus(2-\varepsilon)B_2\bigr)\Bigr)=\frac{1}{6}4\pi\varepsilon+o(\varepsilon). \end{equation*} Taking, for example, |$\delta =\sqrt{\varepsilon }/100$| we get, for |$\varepsilon $| small enough, that |$\delta>\varepsilon $| and also that |$4\pi \varepsilon /6>\pi \delta ^2$| and |$o(\varepsilon )<\delta ^2$|⁠. Thus, \begin{equation*} 6\sup_{x\in{\mathbb{R}}^2}\mu(x+B_2)=6\sup_{x\in{\mathbb{R}}^2}\textrm{vol}\Bigl((x+B_2)\cap\bigl(2B_2\setminus(2-\varepsilon)B_2\bigr)\Bigr) =4\pi\varepsilon+o(\varepsilon)<4\pi\varepsilon+\pi(\delta^2-\varepsilon^2). \end{equation*} Moreover, since |$\sup _{x\in{\mathbb{R}}^2}\mu (x+B_2)>\overline{\mu }(B_2)$|⁠, this example shows that the radial decay assumption is also needed in Theorem 1.1. Regarding a reverse inequality for Theorem 1.1 (or (19)), we have the following result, which extends (5). Theorem 2.6. Let |$K\in{\mathcal{K}}^n$|⁠. Let |$\mu $| be a measure on |${\mathbb{R}}^n$| given by |$\textrm{d}\mu (x)=\phi (x)\,\textrm{d} x$|⁠, where |$\phi :{\mathbb{R}}^n\longrightarrow [0,\infty )$| is an even quasi-concave function. Then \begin{equation} \mu(K-K)\geq\mu(2K). \end{equation} (21) Equality holds in (21) only if |$K\cap (\textrm{supp}\ \phi )/2$| is centrally symmetric. Moreover, if |$K$| is centrally symmetric with respect to the origin, then equality holds in (21). Proof. We write |$\overline{K}_t=(2K)\cap{\mathcal{C}}_t(\phi )$| for every |$t\in [0,1]$|⁠. On the one hand, by Fubini’s theorem, we have \begin{align} \mu(2K) & = \int_{2K} \phi(x) \,\textrm{d} x =\|\phi\|_{\infty}\int_{2K}\int_0^{\frac{\phi(x)}{\|\phi\|_{\infty}}}\,\textrm{d} t \,\textrm{d} x =\|\phi\|_{\infty} \int_0^1 \int_{2K} \chi_{_{{\mathcal{C}}_t(\phi)}}(x) \,\textrm{d} x \,\textrm{d} t\nonumber \\&=\|\phi\|_{\infty} \int_0^1 \textrm{vol}\bigl(\overline{K}_t\bigr) \,\textrm{d} t \leq\|\phi\|_{\infty} \,2^{-n} \int_0^1 \textrm{vol}\bigl(\overline{K}_t-\overline{K}_t\bigr) \,\textrm{d} t, \end{align} (22) where in the last inequality we have used the Brunn–Minkowski inequality (cf. (1)). On the other hand, since |$\phi $| is quasi-concave and even, then |${\mathcal{C}}_t(\phi )$| is convex and centrally symmetric (with respect to the origin), and hence |$\overline{K}_t-\overline{K}_t\subset (2K-2K)\cap 2{\mathcal{C}}_t(\phi )=2\bigl ((K-K)\cap{\mathcal{C}}_t(\phi )\bigr )$|⁠. Thus, we get \begin{equation*} \begin{split} \mu(2K) &\leq \|\phi\|_{\infty} \,2^{-n} \int_0^1 \textrm{vol}\bigl(\overline{K}_t-\overline{K}_t\bigr) \,\textrm{d} t \leq \|\phi\|_{\infty} \int_0^1 \textrm{vol} \bigl( (K-K) \cap{\mathcal{C}}_t(\phi)\bigr) \,\textrm{d} t\\ &= \|\phi\|_{\infty} \int_0^1 \int_{{\mathbb{R}}^n} \chi_{_{(K-K) \cap{\mathcal{C}}_t(\phi)}}(x) \,\textrm{d} x \,\textrm{d} t = \mu(K-K). \end{split} \end{equation*} For the equality case, we note that the identity |$\mu (2K)=\mu (K-K)$| implies that (22) holds with equality, and thus |$\textrm{vol}\bigl (\overline{K}_t\bigr )=2^{-n}\textrm{vol}\bigl (\overline{K}_t-\overline{K}_t\bigr )$| for almost every |$t\in [0,1]$|⁠. Then, there exists a decreasing sequence |$(t_m)_m\subset [0,1]$| with |$t_m\to 0$| and such that |$\textrm{vol}\bigl (\overline{K}_{t_m}\bigr )=2^{-n}\textrm{vol}\bigl (\overline{K}_{t_m}-\overline{K}_{t_m}\bigr )$| for all |$m\in{\mathbb{N}}$|⁠. Therefore, since the boundary of a convex set has null (Lebesgue) measure, we get \begin{equation} \begin{split} \textrm{vol}\bigl((2K)\cap\textrm{supp}\ \phi\bigr) & =\textrm{vol}\left(\bigcup_{m=1}^\infty\overline{K}_{t_m}\right) =\lim_m \textrm{vol}\bigl(\overline{K}_{t_m}\bigr) =\lim_m 2^{-n}\textrm{vol}\bigl(\overline{K}_{t_m}-\overline{K}_{t_m}\bigr)\\ & =2^{-n}\textrm{vol}\left(\bigcup_{m=1}^\infty\bigl(\overline{K}_{t_m}-\overline{K}_{t_m}\bigr)\right)\\& =2^{-n}\textrm{vol}\Bigl(\bigl((2K)\cap \textrm{supp}\ \phi\bigr)-\bigl((2K)\cap \textrm{supp}\ \phi\bigr)\Bigr). \end{split} \end{equation} (23) Since |$\textrm{supp}\ \phi $| is an |$n$|-dimensional convex set containing the origin then |$\mu (2K)=\mu (K-K)>0$|⁠, and so |$\textrm{vol}\bigl ((2K)\cap \textrm{supp}\ \phi \bigr )>0$|⁠. Therefore, (23) implies that |$(2K)\cap \textrm{supp}\ \phi $| is centrally symmetric. The sufficient condition is evident. If we apply (21) to the set |$K+x/2$| then |$\mu (K-K)\geq \sup _{x\in{\mathbb{R}}^n}\mu (x+2K)$| also holds. We observe, however, that we cannot expect a general reverse inequality for (19) in the non-even case, as the following example shows. Example 2.7. Let |$\theta>0$| and consider |$W_{\theta }=\bigl \{r(\cos t,\sin t):0\leq t\leq \theta ,\,r\geq 0\bigr \}\subset{\mathbb{R}}^2$|⁠. Let |$\mu _{\theta }$| be the measure on |${\mathbb{R}}^2$| with density |$\phi _{\theta }(x)=\chi _{_{W_\theta }}(x)$| (Figure 2). By letting |$\theta \to 0$|⁠, we can move a set |$K$| far enough, but keeping the measure of the shifts of |$K$| constant, while the measure of |$K-K$| will be arbitrarily small. So the left-hand side of (19) tends to zero whereas the right-hand side is fixed. Fig. 2. View largeDownload slide A construction for which |$\mu (K-K)\to 0$|⁠. Fig. 2. View largeDownload slide A construction for which |$\mu (K-K)\to 0$|⁠. A way to strengthen inequality (19) would be to replace |$\mu (K-K)$| by |$\sup _{\omega \in{\mathbb{R}}^n} \mu (K-K +\omega )$|⁠. Question: Given a measure |$\mu $| on |${\mathbb{R}}^n$|⁠, is it true that for every |$K\in{\mathcal{K}}^n$| \begin{equation*} \sup_{\omega\in{\mathbb{R}}^n} \mu(K-K +\omega)\leq\binom{2n}{n}\min\left\{\sup_{x\in{\mathbb{R}}^n}\mu(x+K),\sup_{x\in{\mathbb{R}}^n}\mu(x-K)\right\}? \end{equation*} The following result partially solves this question, in the setting of quasi-concave densities, by exploiting the approach carried out in the proof of Theorem 1.1. The idea relies on the possibility of finding a point, for each translated copy of |$K-K$|⁠, from which the density is radially decreasing over the given translation of |$K-K$|⁠. The negative counterpart is the apparent necessity of including a factor jointly with the measure of the shift of |$K-K$|⁠. Nevertheless, we observe that the supremum on the right-hand side can be taken over |$K$|⁠. In Section 4, we will provide a different solution to this issue (Theorem 4.5). Proposition 2.8. Let |$K\in{\mathcal{K}}^n$| and let |$\mu $| be a measure on |${\mathbb{R}}^n$| given by |$\textrm{d}\mu (x)=\phi (x)\,\textrm{d} x$|⁠, where |$\phi :{\mathbb{R}}^n\longrightarrow [0,\infty )$| is a quasi-concave function whose restriction to its support is continuous. Then, for every |$\omega \in{\mathbb{R}}^n$|⁠, \begin{equation} c(\omega)\mu(K-K+\omega) \leq\binom{2n}{n}\sup_{y\in K}\mu(y+\omega-K), \end{equation} (24) where |$c(\omega )=\textrm{vol}\bigl (K\cap (\omega ^{\prime}-\omega +K)\bigr )\textrm{vol}(K)^{-1}$|⁠, and |$\omega ^{\prime}\in K-K+\omega $| is such that |$\phi (\omega ^{\prime})=\max _{x\in K-K+\omega }\phi (x)$|⁠. Moreover, equality holds for some |$\omega _0\in{\mathbb{R}}^n$| if and only if |$\mu $| is a constant multiple of the Lebesgue measure on |$K-K+\omega _0$|⁠, |$c(\omega _0)=1$| and |$K$| is a simplex. Proof. Let |$f:{\mathbb{R}}^n\longrightarrow [0,\infty )$| be defined as |$f(x)=\textrm{vol}\bigl (K\cap (x-\omega +K)\bigr )$|⁠. As before, we get that |$\textrm{supp}\ f=K-K+\omega $| and |$f$| is |$(1/n)$|-concave (see (1) and (15)). On the one hand, by Fubini’s theorem, we have \begin{equation} \int_{K-K+\omega}f(x) \,\textrm{d}\mu(x)= \int_{{\mathbb{R}}^n}\int_{{\mathbb{R}}^n}\chi_{_K}(y)\chi_{_{y+\omega-K}}(x)\,\phi(x)\,\textrm{d} y\,\textrm{d} x=\int_{K}\mu(y+\omega-K)\,\textrm{d} y. \end{equation} (25) On the other hand, from the continuity of |$\phi $| on |$\textrm{supp}\ \phi $|⁠, we know that there exists a point |$\omega ^{\prime}\in (K-K+\omega )\cap \textrm{supp}\ \phi $|⁠, which is a compact set, such that |$\phi (\omega ^{\prime}) = \max _{x\in K-K+\omega }\phi (x)$|⁠. This, together with the quasi-concavity of |$\phi $|⁠, implies that it radially decays from |$\omega ^{\prime}$| on |$K-K+\omega $|⁠, that is, |$\phi \bigl (\omega ^{\prime}+t(x-\omega ^{\prime})\bigr )\geq \phi (x)$| for any |$t\in [0,1]$| and all |$x\in K-K+\omega $|⁠. Now we define the function |$g:K-K+\omega \longrightarrow [0,\infty )$| given by \begin{equation*} g(x)=f(\omega^{\prime})\left[1-\frac{|x-\omega^{\prime}|}{\rho_{_{\!K-K+\omega-\omega^{\prime}}}\bigl((x-\omega^{\prime})/|x-\omega^{\prime}|\bigr)} \right]^n, \quad \textrm{ for every}\ x\neq\omega^{\prime}, \end{equation*} and |$g(\omega ^{\prime})=f(\omega ^{\prime})$|⁠. Since |$f^{1/n}$| is concave, it follows that |$f^{1/n}\geq g^{1/n}$| on |$\bigl [\omega ^{\prime},\omega ^{\prime}+\rho _{_{\!K-K+\omega -\omega ^{\prime}}}(u)u\bigr ]$|⁠, and so, via the polar coordinates |$z=x-\omega ^{\prime}=ru$|⁠, we get \begin{equation*} \begin{split} \int_{K-K+\omega} f(x) \,\textrm{d}\mu(x) & =\int_{K-K+\omega-\omega^{\prime}} f(\omega^{\prime}+z)\phi(\omega^{\prime}+z)\,\textrm{d} z\\ & =\int_{{\mathbb{S}}^{n-1}}\int_0^{\rho_{_{\!K-K+\omega-\omega^{\prime}}}(u)}r^{n-1}f(\omega^{\prime}+ru)\phi(\omega^{\prime}+ru)\,\textrm{d} r\,\textrm{d}\sigma(u)\\ & \geq f(\omega^{\prime})\int_{{\mathbb{S}}^{n-1}}\int_0^{\rho_{_{\!K-K+\omega-\omega^{\prime}}}(u)} \left[1-\frac{r}{\rho_{_{\!K-K+\omega-\omega^{\prime}}}(u)}\right]^nr^{n-1}\phi(\omega^{\prime}+ru)\,\textrm{d} r \,\textrm{d}\sigma(u). \end{split} \end{equation*} Then Lemma 2.1 yields \begin{equation} \begin{split} \int_{K-K+\omega} f(x) \,\textrm{d}\mu(x) & \geq\frac{f(\omega^{\prime})}{\binom{2n}{n}}\int_{{\mathbb{S}}^{n-1}}\int_0^{\rho_{_{\!K-K+\omega-\omega^{\prime}}}(u)} r^{n-1} \phi(\omega^{\prime}+r u)\,\textrm{d} r\,\textrm{d}\sigma(u)\\ & =\frac{1}{\binom{2n}{n}}\textrm{vol}\bigl(K\cap(\omega^{\prime}-\omega+K)\bigr)\mu(K-K+\omega), \end{split} \end{equation} (26) which, together with (25), gives \begin{equation*} \mu(K-K+\omega)\textrm{vol}\bigl(K\cap(\omega^{\prime}-\omega+K)\bigr) \leq\binom{2n}{n}\int_{K}\mu(y+\omega-K)\,\textrm{d} y \leq\binom{2n}{n}\textrm{vol}(K)\sup_{y\in K}\mu(y+\omega-K). \end{equation*} Finally, we notice that equality holds in (24) for some |$\omega _0\in{\mathbb{R}}^n$| only if there is equality in (26). This implies, by Lemma 2.1, that |$\phi (\omega ^{\prime}+r u)$| is constant on |$\bigl (0,\rho _{_{\!K-K+\omega _0-\omega ^{\prime}}}(u)\bigr )$| for |$\sigma $|-almost every |$u\in{\mathbb{S}}^{n-1}$|⁠. Since |$\phi $| is continuous at |$\omega ^{\prime}\in \textrm{supp}\ \phi $|⁠, |$\mu $| is a constant multiple of the Lebesgue measure on |$K-K+\omega _0$| and, by Theorem A, |$K$| is a simplex (in particular, |$c(\omega _0)=1$|⁠). The converse immediately follows from Theorem A. 2.2 The functional case In this subsection we draw a consequence of Theorem 1.1 regarding integrals of quasi-concave functions, which extends two results of Colesanti [12, Theorems 4.3 and 4.5] and is collected in Theorem 2.9. To this end, given a quasi-concave function |$f:{\mathbb{R}}^n\longrightarrow [0,\infty )$|⁠, we define the (⁠|$-\infty $|⁠)-difference of |$f$|⁠, which remains quasi-concave (cf. [12, Proposition 4.2]), by \begin{equation*} \Delta_{-\infty}\,f(z)=\sup_{z=x-y}\min\bigl\{f(x),f(y)\bigr\}. \end{equation*} Besides |$\Delta _{-\infty } f$|⁠, we also consider the (difference) functions |$\Delta _{-\infty ,\theta } f$| (for some |$\theta \in [0,1]$|⁠) and |$\widetilde{\Delta }_{-\infty } f$| given by \begin{equation*} \begin{split} \Delta_{-\infty,\theta} f(z) & =\sup_{z=(1-\theta)x-\theta y}\min\bigl\{f(x),f(y)\bigr\},\\ \widetilde{\Delta}_{-\infty} f(z) & =\sup_{\substack{z=(1-\theta)x-\theta y\\\theta\in[0,1]}}\min\bigl\{f(x),f(y)\bigr\}. \end{split} \end{equation*} These functions can be regarded as the (quasi-concave) functional counterparts of |$K-K$|⁠, |$(1-\theta )K-\theta K$| and |$\textrm{conv} \bigl (K\cup (-K)\bigr )$|⁠, respectively, as it is shown via their (strict) superlevel sets. For the sake of brevity we will write, for a function |$f:{\mathbb{R}}^n\longrightarrow [0,\infty )$| and |$t\in [0,\infty )$|⁠, \begin{equation*} S_{>t}(f)=\bigl\{x\in{\mathbb{R}}^n:f(x)>t\bigr\}; \end{equation*} analogously, |$S_{\geq t}(f)=\bigl \{x\in{\mathbb{R}}^n:f(x)\geq t\bigr \}$|⁠. We observe that if |$f:{\mathbb{R}}^n\longrightarrow [0,\infty )$| is a quasi-concave function, then \begin{equation} \begin{split} (i) & \qquad S_{>t}\bigl(\Delta_{-\infty}\,f\bigr)=S_{>t}(f)-S_{>t}(f),\\ (ii) & \qquad S_{>t}\bigl(\Delta_{-\infty,\theta}f\bigr)=(1-\theta)S_{>t}(f)-\theta S_{>t}(f),\\ (iii) & \qquad S_{>t}\left(\widetilde{\Delta}_{-\infty}\,f\right)=\textrm{conv} \Bigl(S_{>t}(f)\cup\bigl(-S_{>t}(f)\bigr)\Bigr). \end{split} \end{equation} (27) Indeed, (i), (ii), and (iii) are completely analogous. To see (i), let |$z\in S_{>t}\bigl (\Delta _{-\infty }f\bigr )$|⁠. Then there exist |$x$|⁠,|$y$| such that |$z=x-y$| and |$\min \bigl \{f(x),f(y)\bigr \}>t$|⁠, which shows the inclusion \begin{equation*} S_{>t}\bigl(\Delta_{-\infty}\,f\bigr)\subset S_{>t}(f)-S_{>t}(f). \end{equation*} For the reverse inclusion, if |$z\in S_{>t}(f)-S_{>t}(f)$| then there exist |$x,y\in{\mathbb{R}}^n$|⁠, with |$z=x-y$|⁠, such that |$f(x)>t$| and |$f(y)>t$|⁠. Since |$\min \bigl \{f(x),f(y)\bigr \}>t$| and |$z=x-y$|⁠, we get that |$\Delta _{-\infty }f(z)>t$|⁠, as desired. Now we collect the above-mentioned consequence of (4), which may be seen as its functional version. Theorem 2.9. Let |$f:{\mathbb{R}}^n\longrightarrow [0,\infty )$| be an integrable quasi-concave function. Let |$\mu $| be a measure on |${\mathbb{R}}^n$| given by |$\textrm{d}\mu (x)=\phi (x)\,\textrm{d} x$|⁠, where |$\phi :{\mathbb{R}}^n\longrightarrow [0,\infty )$| is radially decreasing. Then \begin{equation} \int_{{\mathbb{R}}^n}\Delta_{-\infty}\,f(x)\,\textrm{d}\mu(x)\leq\binom{2n}{n}\int_0^{\infty}\min\Bigl\{\overline{\mu}\bigl(S_{\geq t}(f)\bigr),\overline{\mu}\bigl(-S_{\geq t}(f)\bigr)\Bigr\}\,\textrm{d} t. \end{equation} (28) In particular, by choosing |$\textrm{d}\mu (x)=\textrm{d} x$|⁠, the Lebesgue measure, we get \begin{equation*} \int_{{\mathbb{R}}^n}\Delta_{-\infty}\,f(x)\,\textrm{d} x \leq \binom{2n}{n}\int_{{\mathbb{R}}^n}f(x)\,\textrm{d} x. \end{equation*} Proof. The proof follows the general ideas of those of [12, Theorems 4.3 and 4.5]. Using Fubini’s theorem, together with (i) in (27), we may write \begin{equation*} \Delta_{-\infty}\,f(x)=\int_0^{\infty}\chi_{_{S_{>t}(f)-S_{>t}(f)}}(x) \,\textrm{d} t \end{equation*} and, consequently, \begin{equation} \int_{{\mathbb{R}}^n}\Delta_{-\infty}\,f(x)\,\textrm{d}\mu(x) =\int_{{\mathbb{R}}^n}\int_0^{\infty}\chi_{_{S_{>t}(f)-S_{>t}(f)}}(x)\,\textrm{d} t\,\textrm{d}\mu(x)\leq\int_0^{\infty}\mu\bigl(S_{\geq t}(f)-S_{\geq t}(f)\bigr)\,\textrm{d} t. \end{equation} (29) Since |$f$| is quasi-concave and integrable, the closure of the superlevel sets |$S_{\geq t}(f)$| are convex bodies for all |$0<t<\|f\|_{\infty }$|⁠. Thus, we may apply (4) to |$S_{\geq t}(f)$| (since the boundary of a convex set has null measure) which, together with (29), allows us to obtain (28). Now we note that, if |$\textrm{d}\mu (x)=\,\textrm{d} x$|⁠, then we have \begin{equation*} \min\Bigl\{\overline{\textrm{vol}}\bigl(S_{\geq t}(f)\bigr),\overline{\textrm{vol}}\bigl(-S_{\geq t}(f)\bigr)\Bigr\}=\textrm{vol}\bigl(S_{\geq t}(f)\bigr), \end{equation*} which completes the proof. Given a |$p$|-concave function |$f:{\mathbb{R}}^n\longrightarrow [0,\infty )$|⁠, for |$p\in [-\infty ,0)$|⁠, one can define the |$p$|-difference of |$f$|⁠, which remains |$p$|-concave (cf. [12, Proposition 4.2]), by \begin{equation*} \Delta_pf(z)=\sup_{z=x+y}\bigl(f(x)^p+f(-y)^p\bigr)^{1/p} =\sup_{z=x-y}\bigl(f(x)^p+f(y)^p\bigr)^{1/p}, \end{equation*} where the case |$p=-\infty $| is understood as the minimum between both values. Theorem 2.9 can be established for any |$p\in (-\infty ,0)$|⁠. It suffices to note that if |$f$| is |$p$|-concave then it is also quasi-concave, and then, we may apply inequality (28) for |$p=-\infty $| together with the fact that |$(a^p+b^p)^{1/p}\leq \min \{a,b\}$| for each |$a,b\geq 0$|⁠. Hence |$\Delta _{p}f\leq \Delta _{-\infty }f$|⁠. Remark 2.10. As mentioned before, Theorem 2.9 is an application of Theorem 1.1. It is a natural and interesting question whether (4) could be directly derived from previous functional versions as (8). Just considering |$\chi _{_K}\,\phi $| this is not possible because of item (i) in (27): the integral of |$\Delta _{-\infty }f$| does not provide (in general) the measure of |$K-K$| with respect to the density |$\phi $|⁠. 2.3 Rogers–Shephard type inequalities for |$CK$| and |$\textrm{conv} \bigl (K\cup (-K)\bigr )$| and their functional versions Now we prove the corresponding Rogers–Shephard type inequalities for |$CK$| and |$\textrm{conv} \bigl (K\cup (-K)\bigr )$|⁠, as well as their equality cases. Proof of Theorem 1.2. Let |$f:{\mathbb{R}}^n\times [0,1]\longrightarrow [0,\infty )$| be the function given by \begin{equation*} f(x,\theta)=\textrm{vol}\Bigl(\bigl((1-\theta)K\bigr)\cap(x+\theta K)\Bigr). \end{equation*} Note that |$f$| is |$(1/n)$|-concave by (1), and |$\textrm{supp}\ f=CK$|⁠. On the one hand, taking the measure |$\mu _{n+1}$| on |${\mathbb{R}}^{n+1}$| given by |$\textrm{d}\mu _{n+1}(x,\theta )=\phi (x)\,\textrm{d} x \,\textrm{d} \theta $|⁠, Fubini’s theorem and the change of variable |$z=(1-\theta )y$| yield \begin{equation} \begin{split} \int_{CK}f(x,\theta)\,\textrm{d}\mu_{n+1}(x,\theta) & =\int_0^1\int_{{\mathbb{R}}^n}\textrm{vol}\Bigl(\bigl((1-\theta)K\bigr)\cap(x+\theta K)\Bigr)\phi(x)\,\textrm{d} x\,\textrm{d}\theta\\ & =\int_0^1\int_{{\mathbb{R}}^n}\int_{{\mathbb{R}}^n}\chi_{_{(1-\theta)K}}(z)\chi_{_{x+\theta K}}(z)\,\phi(x)\,\textrm{d} z\,\textrm{d} x\,\textrm{d}\theta\\ & =\int_0^1\int_{(1-\theta)K}\int_{{\mathbb{R}}^n}\chi_{_{z-\theta K}}(x)\,\phi(x)\,\textrm{d} x\,\textrm{d} z\,\textrm{d}\theta\\ & =\int_0^1(1-\theta)^n\int_{K}\mu\bigl((1-\theta)y-\theta K\bigr)\textrm{d} y\,\textrm{d}\theta\\ & \leq \textrm{vol}(K)\int_0^1(1-\theta)^n\theta^n\,\textrm{d}\theta \sup_{\substack{y\in K\\\theta\in(0,1]}}\frac{\mu\bigl((1-\theta)y-\theta K\bigr)}{\theta^n}\\ & =\frac{1}{\binom{2n+1}{n}}\frac{\textrm{vol}(K)}{n+1} \sup_{\substack{y\in K\\\theta\in(0,1]}}\frac{\mu\bigl((1-\theta)y-\theta K\bigr)}{\theta^n}. \end{split} \end{equation} (30) Now we define the function |$g:CK\longrightarrow [0,\infty )$| given by \begin{equation*} g(x,\theta)=f\left(0,\frac{1}{2}\right)\left[1-\frac{\left|(x,\theta)-\bigl(0,\frac{1}{2}\bigr)\right|} {\rho_{_{\!CK-(0,\frac{1}{2})}}\Bigl(\bigl((x,\theta)-(0,\frac{1}{2})\bigr)/\bigl|(x,\theta)-(0,\frac{1}{2})\bigr|\Bigr)} \right]^n, \end{equation*} for every |$(x,\theta )\neq (0,1/2)$| and |$g(0,1/2)=f(0,1/2)=\textrm{vol}(K)/2^n$|⁠. Since |$f^{1/n}$| is concave, then |$f^{1/n}\geq g^{1/n}$| on |$\left [(0,1/2),(0,1/2)+\rho _{_{\!CK-(0,\frac{1}{2})}}(u)u\right ]$|⁠, and so, via the polar coordinates |$(x,\theta ^{\prime})=(x,\theta )-(0,1/2)=ru$|⁠, we get \begin{equation*} \begin{split} \int_{CK}f(x,\theta)\,\textrm{d}\mu_{n+1}(x,\theta) & =\int_{CK-(0,\frac{1}{2})} f\left(x,\theta^{\prime}+\frac{1}{2}\right) \phi(x)\,\textrm{d} x\,\textrm{d}\theta^{\prime}\\ & =\int_{{\mathbb{S}}^{n}}\int_0^{\rho_{_{\!CK-(0,\frac{1}{2})}}(u)} r^{n}f\left(\Bigl(0,\frac{1}{2}\Bigr)+ru\right)\phi\bigl(r P_Hu\bigr)\,\textrm{d} r\,\textrm{d}\sigma(u)\\ & \geq f\left(0,\frac{1}{2}\right)\int_{{\mathbb{S}}^{n}}\int_0^{\rho_{_{\!CK-(0,\frac{1}{2})}}(u)} \left(1-\frac{r}{\rho_{_{\!CK-(0,\frac{1}{2})}}(u)}\right)^n r^{n}\phi\bigl(r P_Hu\bigr)\,\textrm{d} r \,\textrm{d}\sigma(u), \end{split} \end{equation*} where |$H=\bigl \{(x,\theta )\in{\mathbb{R}}^{n+1}:\theta =0\bigr \}$|⁠. Then, Lemma 2.1 yields \begin{equation} \begin{split} \int_{CK}f(x,\theta)\,\textrm{d}\mu_{n+1}(x,\theta) & \geq\frac{f\left(0,\frac{1}{2}\right)}{\binom{2n+1}{n}}\int_{{\mathbb{S}}^n}\int_0^{\rho_{_{\!CK-(0,\frac{1}{2})}}(u)}r^n \phi\bigl(r P_Hu\bigr)\,\textrm{d} r\,\textrm{d}\sigma(u)\\ & =\frac{1}{\binom{2n+1}{n}}\frac{\textrm{vol}(K)}{2^n}\mu_{n+1}(CK), \end{split} \end{equation} (31) which, together with (30), gives (11). Finally, we notice that equality holds in (11) only if there is equality in (31). This implies, by Lemma 2.1, that |$\phi \bigl (r P_Hu\bigr )$| is constant on |$\left (0,\rho _{_{\!CK-(0,\frac{1}{2})}}(u)\right )$| for |$\sigma $|-almost every |$u\in{\mathbb{S}}^{n}$|⁠. Since |$\phi $| is continuous at the origin, |$\mu _{n+1}$| is a constant multiple of the Lebesgue measure on |$CK$| and hence |$\mu $| is so on |$P_H(CK)=\textrm{conv} \bigl (K\cup (-K)\bigr )$| because |$\mu _{n+1}$| is a product measure. Since |$(1-\theta )y-\theta K\subset CK$| for all |$y\in K$| and any |$\theta \in [0,1]$|⁠, there is equality in (9) and therefore, by Theorem B, |$K$| is a simplex. The converse is a direct consequence of Theorem B. Next we prove (12). We notice that \begin{equation*} P_H\Bigl(CK\cap\bigl({\mathcal{C}}_t(\phi)\times[0,1]\bigr)\Bigr)=\textrm{conv}\bigl(K\cup(-K)\bigr)\cap{\mathcal{C}}_t(\phi) \end{equation*} and, since |$0\in K$|⁠, then \begin{equation*} CK\cap\bigl({\mathcal{C}}_t(\phi)\times[0,1]\bigr)\cap H^{\bot}=[0,1]. \end{equation*} Hence, Theorem C yields \begin{equation*} \textrm{vol}_{n+1}\Bigl(CK\cap\bigl({\mathcal{C}}_t(\phi)\times[0,1]\bigr)\Bigr)\geq \frac{1}{n+1}\textrm{vol}\Bigl(\textrm{conv}\bigl(K\cup(-K)\bigr)\cap{\mathcal{C}}_t(\phi)\Bigr), \end{equation*} which, together with Fubini’s theorem, gives \begin{equation*} \begin{split} \mu_{n+1}(CK) & =\|\phi\|_{\infty}\int_{CK}\int_0^1\chi_{_{{\mathcal{C}}_t(\phi)}}(x)\,\textrm{d} t\,\textrm{d} x\,\textrm{d}\theta =\|\phi\|_{\infty}\int_0^1\int_{CK}\chi_{_{{\mathcal{C}}_t(\phi)\times[0,1]}}(x,\theta)\,\textrm{d} x \,\textrm{d}\theta\,\textrm{d} t\\ & =\|\phi\|_{\infty}\int_0^1\textrm{vol}_{n+1}\Bigl(CK\cap\bigl({\mathcal{C}}_t(\phi)\times[0,1]\bigr)\Bigr)\,\textrm{d} t \\&\geq\|\phi\|_{\infty}\frac{1}{n+1}\int_0^1\!\!\textrm{vol}\Bigl(\textrm{conv}\bigl(K\cup(-K)\bigr)\cap{\mathcal{C}}_t(\phi)\Bigr)\,\textrm{d} t\\ & =\|\phi\|_{\infty}\frac{1}{n+1}\int_0^1\int_{\textrm{conv}(K\cup(-K))}\chi_{_{{\mathcal{C}}_t(\phi)}}(x)\,\textrm{d} x \, \textrm{d} t \\&=\|\phi\|_{\infty}\frac{1}{n+1}\int_{\textrm{conv}\ (K\cup(-K))}\int_0^{\frac{\phi(x)}{\|\phi\|_{\infty}}}\textrm{d} t\,\textrm{d} x\\ & =\frac{1}{n+1}\int_{\textrm{conv}\ (K\cup(-K))}\phi(x)\,\textrm{d} x=\frac{\mu\Bigl(\textrm{conv}\ \bigl(K\cup(-K)\bigr)\Bigr)}{n+1}. \end{split} \end{equation*} This, together with (11), shows (12). Equality in (12) implies, in particular, equality in (11) and thus |$\mu $| is a constant multiple of the Lebesgue measure on |$\textrm{conv} \bigl (K\cup (-K)\bigr )$|⁠. The proof is now concluded from the equality case of (10). Remark 2.11. Taking the function |$f(x,\theta )=\textrm{vol}\Bigl (\bigl ((1-\theta )K\bigr )\cap \bigl (x+\theta (-L)\bigr )\Bigr )$|⁠, and arguing as in the proof of Theorem 1.2, an analogous result can be obtained for two arbitrary convex bodies instead of |$K$| and |$-K$|⁠. Thus, if |$K,L\in{\mathcal{K}}^n$| contain the origin and |$\mu $| is a measure on |${\mathbb{R}}^n$| given by |$\textrm{d}\mu (x)=\phi (x)\,\textrm{d} x$|⁠, where |$\phi :{\mathbb{R}}^n\longrightarrow [0,\infty )$| is a radially decreasing function, then \begin{equation*} \frac{\mu\bigl(\textrm{conv}\ (K\cup L)\bigr)}{n+1}\leq\int_0^1\mu\bigl((1-\theta)K+\theta L\bigr)\,\textrm{d}\theta \leq\frac{2^n}{n+1}\dfrac{\textrm{vol}(K)}{\textrm{vol}\bigl(K\cap(-L)\bigr)}\,\sup_{\substack{y\in K\\ \theta\in(0,1]}}\frac{\mu\bigl((1-\theta)y+\theta L\bigr)}{\theta^n}. \end{equation*} As a consequence of Theorem 1.2, we get in Theorem 2.12 below functional versions of both (11) and (12). Regarding another functional version of (10), in the log-concave setting, we refer the reader to [12, Theorem 1.1]. The advantage of the inequality we present here is that, in contrast to the above-mentioned result, inequality (10) may be recovered just by taking |$f=\chi _{_K}$|⁠. We use here the same notation as for Theorem 2.9. Theorem 2.12. Let |$f:{\mathbb{R}}^n\longrightarrow [0,\infty )$| be an integrable quasi-concave function. Let |$\mu $| be a measure on |${\mathbb{R}}^n$| given by |$\textrm{d}\mu (x)=\phi (x)\,\textrm{d} x$|⁠, where |$\phi :{\mathbb{R}}^n\longrightarrow [0,\infty )$| is radially decreasing. Then \begin{equation} \int_0^1\int_{{\mathbb{R}}^n}\Delta_{-\infty,\theta}f(x)\,\textrm{d}\mu(x)\,\textrm{d}\theta \leq\frac{2^n}{n+1}\int_0^{\infty}\sup_{\substack{y\in S_{\geq t}(f)\\\theta\in(0,1]}}\frac{\mu\bigl((1-\theta)y-\theta S_{\geq t}(f)\bigr)}{\theta^n}\,\textrm{d} t \end{equation} (32) and \begin{equation} \int_{{\mathbb{R}}^n}\widetilde{\Delta}_{-\infty}\,f(x)\,\textrm{d}\mu(x)\leq 2^n\int_0^{\infty}\sup_{\substack{y\in S_{\geq t}(f)\\\theta\in(0,1]}}\frac{\mu\bigl((1-\theta)y-\theta S_{\geq t}(f)\bigr)}{\theta^n}\,\textrm{d} t. \end{equation} (33) In particular, by choosing |$\textrm{d}\mu (x)=\textrm{d} x$|⁠, the Lebesgue measure, we get \begin{equation*} \int_0^1\int_{{\mathbb{R}}^n}\Delta_{-\infty,\theta}f(x)\,\textrm{d} x\,\textrm{d}\theta \leq\frac{2^n}{n+1}\int_{{\mathbb{R}}^n}f(x)\,\textrm{d} x \end{equation*} and \begin{equation*} \int_{{\mathbb{R}}^n} \widetilde{\Delta}_{-\infty} f(x)\,\textrm{d} x\leq 2^n \int_{{\mathbb{R}}^n} f(x) \,\textrm{d} x. \end{equation*} Proof. Since |$f$| is quasi-concave and integrable, the closure of the superlevel sets |$S_{\geq t}(f)$| are convex bodies for all |$0<t<\|f\|_\infty $|⁠. Thus, we may apply Theorem 1.2 to |$S_{\geq t}(f)$| (since the boundary of a convex set has null measure) to obtain \begin{equation*} \int_0^1\mu\bigl((1-\theta)S_{>t}(f)-\theta S_{>t}(f)\bigr)\,\textrm{d}\theta \leq \frac{2^n}{n+1}\,\sup_{\substack{y\in S_{\geq t}(f)\\\theta\in(0,1]}}\frac{\mu\bigl((1-\theta)y-\theta S_{\geq t}(f)\bigr)}{\theta^n} \end{equation*} and \begin{equation*} \mu\Bigl(\textrm{conv}\ \bigl(S_{>t}(f)\cup(-S_{>t}(f))\bigr)\Bigr)\leq 2^n\,\sup_{\substack{y\in S_{\geq t}(f)\\\theta\in(0,1]}}\frac{\mu\bigl((1-\theta)y-\theta S_{\geq t}(f)\bigr)}{\theta^n}. \end{equation*} Integrating on |$t\in [0,\infty )$|⁠, (32) and (33) now follow by applying Fubini’s theorem together with (ii) and (iii) in (27), respectively. Finally, if |$\textrm{d}\mu (x)=\,\textrm{d} x$|⁠, then we have \begin{equation*} \sup_{\substack{y\in S_{\geq t}(f)\\\theta\in(0,1]}}\frac{\textrm{vol}\bigl((1-\theta)y-\theta S_{\geq t}(f)\bigr)}{\theta^n}=\textrm{vol}\bigl(S_{\geq t}(f)\bigr). \end{equation*} This concludes the proof. 3 A Projection-Section Inequality for Quasi-Concave Functions We start this section by showing a general result for functions that will be exploited throughout the rest of the paper. Proposition 3.1. Let |$\mu $| be a measure on |${\mathbb{R}}^n$| given by |$\textrm{d}\mu (x)=\phi (x)\,\textrm{d} x$|⁠, where |$\phi :{\mathbb{R}}^n\longrightarrow [0,\infty )$| is quasi-concave and such that |$\|\phi \|_{\infty }=\phi (0)$|⁠. Let |$f:{\mathbb{R}}^n\longrightarrow [0,\infty )$| be a |$p$|-concave function, |$p>0$|⁠, with |$\|f\|_{\infty }=f(0)$|⁠, and let |$g:{\mathbb{R}}^n\longrightarrow [0,\infty )$| be a measurable function. Then \begin{equation} \int_{\textrm{supp}\ f}\int_0^1(1-\theta^{p})^n g\bigl((1-\theta^{p})x\bigl)\,\textrm{d} \theta\,\textrm{d}\mu(x) \leq\dfrac{1}{\|f\|_{\infty}}\int_{\textrm{supp} \ f}g(x)f(x)\,\textrm{d}\mu(x). \end{equation} (34) Moreover, if |$\textrm{supp}\ f$| is bounded, |$g$| is non-zero on |$\textrm{supp}\ f$| and |$\phi $| is continuous at the origin, equality in (34) implies that |$\mu $| is a constant multiple of the Lebesgue measure on |$\textrm{supp}\ f$|⁠. Proof. Since |$f$| is |$p$|-concave, then |${\mathcal{C}}_{\theta }(f)$| is a convex set for every |$\theta \in [0,1]$|⁠. We notice that \begin{equation*} \dfrac{{\mathcal{C}}_{\theta_1}(f)}{1-\theta_1^p}\subset\dfrac{{\mathcal{C}}_{\theta_2}(f)}{1-\theta_2^p} \end{equation*} for |$0\leq \theta _1\leq \theta _2<1$|⁠. In particular, taking |$\theta _1=0$|⁠, we have \begin{equation} \textrm{supp} \ f\subset\frac{1}{1-\theta^p}{\mathcal{C}}_{\theta}(f)\quad\textrm{ for any} \; \theta\in[0,1), \end{equation} (35) and hence \begin{equation} (\textrm{supp} \ f)\cap{\mathcal{C}}_t(\phi)\subset\left(\frac{1}{1-\theta^p}{\mathcal{C}}_{\theta}(f)\right)\cap{\mathcal{C}}_t(\phi) \subset\frac{{\mathcal{C}}_{\theta}(f)\cap{\mathcal{C}}_t(\phi)}{1-\theta^p} \end{equation} (36) for all |$\theta \in [0,1)$| and every |$t\in [0,1]$|⁠. Therefore, \begin{equation*} \bigl(1-\theta^p\bigr)\bigl[(\textrm{supp} \ f)\cap{\mathcal{C}}_t(\phi)\bigr] \subset{\mathcal{C}}_{\theta}(f)\cap{\mathcal{C}}_t(\phi), \end{equation*} which yields \begin{equation} \int_0^1\int_0^1\int_{(1-\theta^p)[(\textrm{supp} \ f)\cap{\mathcal{C}}_t(\phi)]}g(x)\,\textrm{d} x\,\textrm{d}\theta\,\textrm{d} t \leq\int_0^1\int_0^1\int_{{\mathcal{C}}_{\theta}(f)\cap{\mathcal{C}}_t(\phi)}g(x)\,\textrm{d} x\,\textrm{d}\theta\,\textrm{d} t. \end{equation} (37) Now we compute both sides of inequality (37). On the one hand, by Fubini’s theorem and the change of variable |$x=\bigl (1-\theta ^p\bigr )y$|⁠, we get \begin{equation*} \begin{split} \int_0^1\int_0^1\int_{(1-\theta^p)[(\textrm{supp} \ f)\cap{\mathcal{C}}_t(\phi)]}g(x)\,\textrm{d} x\,\textrm{d}\theta\,\textrm{d} t & =\!\!\int_0^1\!\!\int_0^1\!\!\int_{(\textrm{supp} \ f)\cap{\mathcal{C}}_t(\phi)}g\bigl((1-\theta^p)y\bigr) (1-\theta^p)^n\,\textrm{d} y\,\textrm{d}\theta\,\textrm{d} t\\ & =\int_{\textrm{supp} \ f}\int_0^1\!(1\!\!-\theta^p)^ng\bigl((1\!\!-\theta^p)y\bigl) \int_0^1\!\!\chi_{_{{\mathcal{C}}_t(\phi)}}(y)\,\textrm{d} t\,\textrm{d}\theta\,\textrm{d} y\\ & =\int_{\textrm{supp} \ f}\int_0^1(1-\theta^p)^ng\bigl((1-\theta^p)y\bigl) \frac{\phi(y)}{\|\phi\|_{\infty}}\,\textrm{d}\theta\,\textrm{d} y\\ & =\dfrac{1}{\|\phi\|_{\infty}}\int_{\textrm{supp} \ f}\int_0^1(1-\theta^p)^n g\bigl((1-\theta^p)y\bigl)\,\textrm{d}\theta\,\textrm{d}\mu(y). \end{split} \end{equation*} On the other hand, using again Fubini’s theorem, \begin{equation*} \begin{split} \int_0^1\int_0^1\int_{{\mathcal{C}}_{\theta}(f)\cap{\mathcal{C}}_t(\phi)}g(x) \,\textrm{d} x\,\textrm{d}\theta\,\textrm{d} t & =\int_0^1\int_0^1\int_{{\mathbb{R}}^n}g(x)\,\chi_{_{{\mathcal{C}}_{\theta}(f)}}(x)\chi_{_{{\mathcal{C}}_t(\phi)}}(x)\,\textrm{d} x\,\textrm{d}\theta\,\textrm{d} t\\ & =\int_{{\mathbb{R}}^n}g(x)\int_0^1\chi_{_{{\mathcal{C}}_t(\phi)}}(x)\int_0^1\chi_{_{{\mathcal{C}}_{\theta}(f)}}(x)\,\textrm{d}\theta\,\textrm{d} t\,\textrm{d} x\\ & =\int_{\textrm{supp} \ f}g(x)\dfrac{f(x)}{\|f\|_{\infty}}\dfrac{\phi(x)}{\|\phi\|_{\infty}}\,\textrm{d} x \\&=\dfrac{1}{\|f\|_{\infty}\|\phi\|_{\infty}}\int_{\textrm{supp} \ f}g(x)\,f(x)\,\textrm{d}\mu(x). \end{split} \end{equation*} Thus, (34) follows from inequality (37). Now we deal with the equality case. First, we observe that since |$\textrm{supp}\ f$| is a bounded set and |$f$| is |$p$|-concave, then |${\mathcal{C}}_{\theta }(f)$| is a bounded convex set for all |$\theta \in [0,1)$|⁠. Without loss of generality we may assume that |$\phi $| is upper semicontinuous. Indeed, otherwise we would work with its upper closure that is determined via the closure of the superlevel sets of |$\phi $| [33, page 14 and Theorem 1.6] and thus defines the same measure because of Fubini’s theorem together with the facts that all the superlevel sets of |$\phi $| are convex (since it is quasi-concave) and the boundary of a convex set has null (Lebesgue) measure. Then its superlevel sets |${\mathcal{C}}_t(\phi )$| are closed (cf. [33, Theorem 1.6]) for every |$t\in [0,1]$|⁠. In the same way, |$f$| may be assumed to be upper semicontinuous (in fact, it is already continuous in the interior of its support because of the |$p$|-concavity). Moreover, since the definitions of both |${\mathcal{C}}_\theta (f)$| and |${\mathcal{C}}_t(\phi )$| involve the essential supremum, these superlevel sets have positive volume for all |$\theta <1$| and |$t<1$|⁠, and therefore both |${\mathcal{C}}_\theta (f)$| and |${\mathcal{C}}_t(\phi )$| are closed convex sets with nonempty interior, for any |$\theta ,t\in [0,1)$|⁠. From the continuity of |$\phi $| at the origin, we know that |$0\in \operatorname{int}{\mathcal{C}}_t(\phi )$| for all |$t<1$| and then |$0\in{\mathcal{C}}_\theta (f)\cap \operatorname{int}{\mathcal{C}}_t(\phi )$| because |$f(0)=\|f\|_{\infty }$|⁠. Hence, and taking into account that |$\textrm{supp}\ f$| (and thus |${\mathcal{C}}_\theta (f)$| for any |$\theta \in [0,1]$|⁠) is bounded, both |${\mathcal{C}}_\theta (f)\cap (1-\theta ^{p}){\mathcal{C}}_t(\phi )$| and |${\mathcal{C}}_\theta (f)\cap{\mathcal{C}}_t(\phi )$| are convex bodies for all |$\theta ,t\in [0,1)$|⁠. Thus, if equality holds in (34) then, in particular, there is equality in the right-hand inclusion of (36) for almost all |$\theta \in [0,1]$| and almost all |$t\in [0,1]$| because |$g>0$| on |$\textrm{supp}\ f$|⁠. Let us assume that there exists |$x_0\in \textrm{supp}\ f$| such that |$\phi (x_0)<\|\phi \|_{\infty }$|⁠. Taking |$t\in \bigl (\phi (x_0)/\|\phi \|_{\infty },1\bigr ]$|⁠, since |$x_0\not \in{\mathcal{C}}_t(\phi )$| then we have that \begin{equation*} (\textrm{supp} \ f)\cap{\mathcal{C}}_t(\phi)\subsetneq \textrm{supp} \ f. \end{equation*} Let |$x_t\in \textrm{bd} \bigl ((\textrm{supp}\ f)\cap{\mathcal{C}}_t(\phi )\bigr )\backslash \textrm{bd}\ (\textrm{supp}\ f)$|⁠. Since both sets are convex bodies, we can always take |$x_t\neq 0$|⁠. Then for all |$t\in \bigl (\phi (x_0)/\|\phi \|_{\infty },1\bigr ]$|⁠, the continuity of |$f$| on |$\operatorname{int}(\textrm{supp}\ f)$| yields the existence of |$\theta _t\in (0,1)$| such that \begin{equation*} x_t\in{\mathcal{C}}_{\theta}(f)\cap{\mathcal{C}}_t(\phi)\quad\textrm{ for all} \; \theta\in[0,\theta_t). \end{equation*} However, since |$x_t\in \textrm{bd}\ {\mathcal{C}}_t(\phi )$| and |$0\in \operatorname{int}{\mathcal{C}}_t(\phi )$|⁠, \begin{equation*} x_t\not\in{\mathcal{C}}_{\theta}(f)\cap\bigl(1-\theta^p\bigr){\mathcal{C}}_t(\phi). \end{equation*} This contradicts the equality in the right-hand inclusion of (36) for almost every |$\theta \in [0,1]$| and |$t\in [0,1]$|⁠. Therefore, we may conclude that |$\phi (x)\geq \|\phi \|_{\infty }$| for all |$x\in \textrm{supp}\ f$| and thus |$\phi \equiv \|\phi \|_{\infty }$| almost everywhere on |$\textrm{supp}\ f$|⁠. This implies that |$\mu $| is a constant multiple of the Lebesgue measure on |$\textrm{supp}\ f$|⁠. It is an interesting question whether Proposition 3.1 can be adapted to log-concave functions, that is, when |$p=0$|⁠. We notice that the above approach cannot be followed in this case. Indeed, considering, for example, the function |$f:{\mathbb{R}}\longrightarrow [0,\infty )$| given by |$f(x)=e^{-x^2}$|⁠, we have that |$\textrm{supp}\ f={\mathbb{R}}$| whereas |${\mathcal{C}}_{\theta }(f)$| is a convex body for all |$t\in (0,1]$|⁠. Hence, there is no chance to get an inclusion of the type (35), that is, |$\lambda (\theta )\textrm{supp}\ f\subset{\mathcal{C}}_{\theta }(f)$| for any |$\theta \in [0,1]$| and some |$\lambda (\theta )>0$|⁠. In what follows we use Proposition 3.1 to prove several results, including Theorem 1.4. Let us first introduce a helpful family of constants and notice a few facts. We denote by \begin{equation*} \alpha^n_{p,q}=\int_0^1(1-\theta^p)^n\,\theta^{p\,q}\,\textrm{d}\theta =\dfrac{\Gamma\left(\frac{1}{p}+q\right)\Gamma(1+n)}{p\,\Gamma\left(1+n+\frac{1}{p}+q\right)}, \end{equation*} for each |$p,q>0$|⁠. Let us assume that |$g$| is concave. Then \begin{equation*} g\bigl((1-\theta^p)x\bigr)\geq\theta^pg(0)+(1-\theta^p)g(x), \end{equation*} and so, we get from (34) that \begin{equation} \alpha^n_{p,1}\,g(0)\,\mu(\textrm{supp} \ f)+\alpha^{n+1}_{p,0}\int_{\textrm{supp} \ f}g(x)\,\textrm{d}\mu(x)\leq\dfrac{1}{\|f\|_{\infty}}\int_{\textrm{supp} \ f}g(x)\,f(x)\,\textrm{d}\mu(x). \end{equation} (38) Another possibility is assuming that |$g$| is radially decreasing. Then, from (34), we get \begin{equation} \alpha^n_{p,0}\int_{\textrm{supp} \ f}g(x)\,\textrm{d}\mu(x) \leq\dfrac{1}{\|f\|_{\infty}}\int_{\textrm{supp} \ f}g(x)\,f(x)\,\textrm{d}\mu(x). \end{equation} (39) We point out that |$\alpha ^n_{p,0}=\alpha ^n_{p,1}+\alpha ^{n+1}_{p,0}$|⁠, which shows that the expression on the left-hand side of (38) and that of (39) are in a sense “similar”, as shown by considering the constant function |$g(x)=1$|⁠. Indeed, when |$g\equiv 1$|⁠, (39) reads \begin{equation} \alpha^n_{p,0}\,\mu(\textrm{supp} \ f)\leq\dfrac{1}{\|f\|_{\infty}}\int_{\textrm{supp} \ f}f(x)\,\textrm{d}\mu(x). \end{equation} (40) Moreover, it can be proved that (40) remains true even in the more general case when |$\|f\|_{\infty }=f(x_0)$| for an arbitrary |$x_0\in{\mathbb{R}}^n$|⁠, and without the maximality assumption for |$\phi $|⁠. Corollary 3.2. Let |$f:{\mathbb{R}}^n\longrightarrow [0,\infty )$| be a |$p$|-concave function, |$p>0$|⁠, with |$\|f\|_{\infty }=f(x_0)$| for some |$x_0\in{\mathbb{R}}^n$|⁠, and let |$\mu $| be a measure on |${\mathbb{R}}^n$| given by |$\textrm{d}\mu (x)=\phi (x)\,\textrm{d} x$|⁠, where |$\phi :{\mathbb{R}}^n\longrightarrow [0,\infty )$| is a bounded quasi-concave function. Then \begin{equation} \alpha^n_{p,0}\frac{\phi(x_0)}{\|\phi\|_{\infty}}\,\mu(\textrm{supp} \ f)\leq\dfrac{1}{\|f\|_{\infty}}\int_{\textrm{supp} \ f}f(x)\,\textrm{d}\mu(x). \end{equation} (41) Moreover, if |$\textrm{supp}\ f$| is bounded and |$\phi $| is continuous at |$x_0$|⁠, equality in (41) implies that |$\mu $| is a constant multiple of the Lebesgue measure on |$\textrm{supp}\ f$|⁠. Proof. The proof follows similar steps as those of Proposition 3.1 but with some key variations. We will highlight these differences. We consider the function |$\psi :{\mathbb{R}}^n\longrightarrow [0,\infty )$| given by |$\psi (x)=f(x+x_0)$|⁠, which satisfies |$\|\psi \|_{\infty }=\|f\|_{\infty }$| and |$\textrm{supp}\ \psi =(\textrm{supp}\ f)-x_0$|⁠. Then (cf. (35)) \begin{equation} \textrm{supp} \ \psi\subset\frac{1}{1-\theta^p}{\mathcal{C}}_{\theta}(\psi)\quad\textrm{ for all }\; \theta\in[0,1). \end{equation} (42) We observe that |$y\in{\mathcal{C}}_{\theta }(\psi )$| if and only if |$f(y+x_0)\geq \theta \|f\|_{\infty }$|⁠, or equivalently, when |$y+x_0\in{\mathcal{C}}_{\theta }(f)$|⁠. Hence, |${\mathcal{C}}_{\theta }(\psi )+x_0={\mathcal{C}}_{\theta }(f)$|⁠, and thus (42) turns into \begin{equation*} (\textrm{supp} \ f)-x_0\subset\frac{1}{1-\theta^p}\bigl({\mathcal{C}}_{\theta}(f)-x_0\bigr)\quad\textrm{ for all} \; \theta\in[0,1). \end{equation*} Therefore, \begin{equation*} \bigl((\textrm{supp} \ f)-x_0\bigr)\cap\bigl({\mathcal{C}}_t(\phi)-x_0\bigr) \subset\!\left(\!\frac{1}{1-\theta^p}\bigl({\mathcal{C}}_{\theta}(f)-x_0\bigr)\!\right)\cap\bigl({\mathcal{C}}_t(\phi)-x_0\bigr) \subset\frac{1}{1-\theta^p}\Bigl(\bigl[{\mathcal{C}}_{\theta}(f)\cap{\mathcal{C}}_t(\phi)\bigr]-x_0\Bigr) \end{equation*} for all |$\theta \in [0,1)$| and every |$t\in \bigl [0,\phi (x_0)/\|\phi \|_{\infty }\bigr ]$|⁠, where in the last inclusion we have used that |$x_0\in{\mathcal{C}}_t(\phi )$|⁠. Consequently, we obtain \begin{equation} (1-\theta^p)\Bigl(\bigl[(\textrm{supp} \ f)\cap{\mathcal{C}}_t(\phi)\bigr]-x_0\Bigr)\subset\bigl({\mathcal{C}}_{\theta}(f)\cap{\mathcal{C}}_t(\phi)\bigr)-x_0. \end{equation} (43) Next, integrating over |$x\in{\mathbb{R}}^n$| the constant function |$1$|⁠, using (43) and the change of variable |$x=(1-\theta ^p)y$|⁠, we get \begin{equation*} (1-\theta^p)^n\int_{[(\textrm{supp} \ f)\cap{\mathcal{C}}_t(\phi)]-x_0}\textrm{d} y \leq\int_{[{\mathcal{C}}_{\theta}(f)\cap{\mathcal{C}}_t(\phi)]-x_0}\textrm{d} y, \end{equation*} which yields \begin{equation} (1-\theta^p)^n\int_{(\textrm{supp} \ f)\cap{\mathcal{C}}_t(\phi)}\textrm{d} x \leq\int_{{\mathcal{C}}_{\theta}(f)\cap{\mathcal{C}}_t(\phi)}\textrm{d} x. \end{equation} (44) Now, computing the left-hand side in (41), we get \begin{equation*} \begin{split} \alpha^n_{p,0}\frac{\phi(x_0)}{\|\phi\|_{\infty}}\,\mu(\textrm{supp} \ f) & =\alpha^n_{p,0}\|\phi\|_{\infty}\int_{\textrm{supp} \ f}\frac{\phi(x_0)}{\|\phi\|_{\infty}}\,\frac{\phi(x)}{\|\phi\|_{\infty}}\,\textrm{d} x\\ & \leq\|\phi\|_{\infty}\int_0^1(1-\theta^p)^n\,\textrm{d}\theta \int_{\textrm{supp} \ f}\min\left\{\frac{\phi(x)}{\|\phi\|_{\infty}},\frac{\phi(x_0)}{\|\phi\|_{\infty}}\right\}\,\textrm{d} x\\ & =\|\phi\|_{\infty}\int_0^1\int_0^{\frac{\phi(x_0)}{\|\phi\|_{\infty}}} (1-\theta^p)^n\int_{(\textrm{supp} \ f)\cap{\mathcal{C}}_t(\phi)} \textrm{d} x\,\textrm{d} t\,\textrm{d}\theta. \end{split} \end{equation*} Applying (44) we obtain the desired inequality. Indeed from the above computation we get \begin{equation*} \begin{split} \alpha^n_{p,0}\frac{\phi(x_0)}{\|\phi\|_{\infty}}\,\mu(\textrm{supp} \ f) & \leq\|\phi\|_{\infty}\int_0^1\int_0^{\frac{\phi(x_0)}{\|\phi\|_{\infty}}}\int_{{\mathcal{C}}_{\theta}(f)\cap{\mathcal{C}}_t(\phi)}\textrm{d} x\,\textrm{d} t\,\textrm{d} \theta \\&=\frac{\|\phi\|_{\infty}}{\|f\|_{\infty}}\int_{\textrm{supp} \ f}f(x)\int_0^{\frac{\phi(x_0)}{\|\phi\|_{\infty}}}\chi_{_{{\mathcal{C}}_t(\phi)}}(x)\,\textrm{d} t\,\textrm{d} x\\ & \leq\frac{\|\phi\|_{\infty}}{\|f\|_{\infty}}\int_{\textrm{supp} \ f}f(x)\int_0^1\chi_{_{{\mathcal{C}}_t(\phi)}}(x)\,\textrm{d} t\,\textrm{d} x =\frac{1}{\|f\|_{\infty}}\int_{\textrm{supp} \ f}f(x)\,\textrm{d} \mu(x). \end{split} \end{equation*} For the proof of the equality case we observe, on the one hand, that if equality holds in (41) then, in particular, \begin{equation*} \int_{\textrm{supp} \ f}f(x)\int_{\frac{\phi(x_0)}{\|\phi\|_{\infty}}}^1\chi_{_{{\mathcal{C}}_t(\phi)}}(x)\,\textrm{d} t\,\textrm{d} x=0, \end{equation*} which yields |$\phi (x_0)=\textrm{ess}\,\sup_{x\in \textrm{supp}\ f}\phi (x)$|⁠. On the other hand, we may replace |$\|\phi \|_{\infty }$| by |$\textrm{ess}\,\sup_{x\in \textrm{supp}\ f}\phi (x)$| in the above argument to get also \begin{equation*} \alpha_{p,0}^n\frac{\phi(x_0)}{\textrm{ess}\,\sup_{x\in \textrm{supp} \ f}\phi(x)}\,\mu(\textrm{supp} \ f) \leq\frac{1}{\|f\|_{\infty}}\int_{\textrm{supp} \ f}f(x)\,\textrm{d} \mu(x), \end{equation*} and since \begin{equation*} \alpha^n_{p,0}\frac{\phi(x_0)}{\|\phi\|_{\infty}}\,\mu(\textrm{supp} \ f)\leq\alpha_{p,0}^n\,\frac{\phi(x_0)}{\textrm{ess}\,\sup_{x\in \textrm{supp} \ f}\phi(x)}\,\mu(\textrm{supp} \ f)=\alpha_{p,0}^n\,\mu(\textrm{supp} \ f), \end{equation*} equality in (41) implies that |$\phi (x_0)=\|\phi \|_{\infty }$|⁠. Finally, due to the fact that |$\phi (x_0)=\|\phi \|_{\infty }$|⁠, the rest of the proof of the equality case is entirely analogous to the one in Proposition 3.1, and we do not repeat it here. As an application of Proposition 3.1, and the above-mentioned consequences of it, we show Theorem 1.4. Proof of Theorem 1.4. For all |$t\in [0,1]$|⁠, the function |$\varphi _t:P_H{\mathcal{C}}_t(f)\longrightarrow [0,\infty )$| given by \begin{equation*} \varphi_t(x)=\textrm{vol}_k\bigl({\mathcal{C}}_t(f)\cap(x+H^{\bot})\bigr) \end{equation*} is (⁠|$1/k$|⁠)-concave, because of the Brunn–Minkowski inequality (1), and |$\textrm{supp}\ \varphi _t=P_H{\mathcal{C}}_t(f)$|⁠. By hypothesis we have |$\|\varphi _t\|_{\infty }=\varphi _t(0)$|⁠. Then, by applying (39) to |$\varphi _t$|⁠, we get \begin{equation} \alpha_{1/k,0}^{n-k}\int_{P_H{\mathcal{C}}_t(f)}g(x)\,\textrm{d} x \leq\dfrac{1}{\|\varphi_t\|_{\infty}}\int_{H}g(x)\,\varphi_t(x)\,\textrm{d} x \end{equation} (45) and hence, integrating each side of inequality (45) over |$t\in [0,1]$| and noticing that |$\alpha _{1/k,0}^{n-k}= {{n}\choose{k}}^{-1} $|⁠, it follows that \begin{equation} \int_0^1\int_{P_H{\mathcal{C}}_t(f)}g(x)\,\textrm{d} x \int_{H^{\bot}}\chi_{_{{\mathcal{C}}_t(f)}}(y)\textrm{d} y\,\textrm{d} t \leq\binom{n}{k}\int_0^1\int_{H}g(x)\int_{x+H^{\bot}}\chi_{_{{\mathcal{C}}_t(f)}}(y)\textrm{d} y\,\textrm{d} x\,\textrm{d} t. \end{equation} (46) On the one hand, by Fubini’s theorem and noticing that \begin{equation*} P_H{\mathcal{C}}_t(f)\supset P_H\Bigl(\bigl\{x\in{\mathbb{R}}^n: f(x)> t\|f\|_{\infty}\bigr\}\Bigr)=\bigl\{x\in H: P_Hf(x)> t\|f\|_{\infty}\bigr\}, \end{equation*} we obtain \begin{equation} \begin{split} \int_0^1\int_{H}g(x)\chi_{_{P_H{\mathcal{C}}_t(f)}}(x)\,\textrm{d} x \int_{H^{\bot}} \chi_{_{{\mathcal{C}}_t(f)}}(y)\,\textrm{d} y\,\textrm{d} t & =\int_{H}\int_{H^{\bot}}g(x)\int_0^1\chi_{_{P_H{\mathcal{C}}_t(f)}}(x)\chi_{_{{\mathcal{C}}_t(f)}}(y)\,\textrm{d} t\,\textrm{d} y\,\textrm{d} x\\ & \geq\int_{H}\int_{H^{\bot}}g(x)\min\left\{\frac{P_Hf(x)}{\|f\|_{\infty}},\frac{f(y)}{\|f\|_{\infty}}\right\}\,\textrm{d} y\,\textrm{d} x\\ & \geq\int_{H}\int_{H^{\bot}}g(x)\frac{P_Hf(x)}{\|f\|_{\infty}}\frac{f(y)}{\|f\|_{\infty}}\,\textrm{d} y\,\textrm{d} x\\ & =\int_{H}g(x)\frac{P_Hf(x)}{\|f\|_{\infty}}\,\textrm{d} x\int_{H^{\bot}}\frac{f(y)}{\|f\|_{\infty}}\,\textrm{d} y. \end{split} \end{equation} (47) On the other hand, Fubini’s theorem yields \begin{equation} \begin{split} \int_0^1\int_{H}g(x)\int_{x+H^{\bot}}\chi_{_{{\mathcal{C}}_t(f)}}(y)\textrm{d} y\,\textrm{d} x\,\textrm{d} t & =\int_{H}g(x)\int_{x+H^{\bot}}\int_0^1\chi_{_{{\mathcal{C}}_t(f)}}(y)\textrm{d} t\,\textrm{d} y\,\textrm{d} x\\ & =\int_{H}\int_{x+H^{\bot}}g(x)\frac{f(y)}{\|f\|_{\infty}}\,\textrm{d} y\,\textrm{d} x =\int_{{\mathbb{R}}^n}g(P_Hz)\frac{f(z)}{\|f\|_{\infty}}\,\textrm{d} z. \end{split} \end{equation} (48) Therefore, from (46), (47), and (48) we obtain \begin{equation*} \int_{H}g(x)P_Hf(x)\,\textrm{d} x\int_{H^{\bot}}f(y)\,\textrm{d} y \leq\binom{n}{k}\|f\|_{\infty}\int_{{\mathbb{R}}^n}g(P_Hx)f(x)\,\textrm{d} x. \end{equation*} This concludes the proof. With the above approach, but using (40) instead of (39), we notice that the maximality assumption at the origin can be relaxed to get the following result, which has been recently obtained in the setting of a log-concave integrable function in [1, Theorem 1.1]. Corollary 3.3. Let |$k\in \{1,\dots ,n-1\}$| and |$H\in \textrm{G}(n,n-k)$|⁠. Let |$f:{\mathbb{R}}^n\longrightarrow [0,\infty )$| be a quasi-concave function such that \begin{equation*} \sup_{x\in H}\textrm{vol}_k\bigl({\mathcal{C}}_t(f)\cap\bigl(x+H^{\bot}\bigr)\bigr) \end{equation*} is attained for all |$t\in (0,1)$|⁠. Then \begin{equation} \int_{H}P_Hf(x)\,\textrm{d} x\max_{x_0\in H}\int_{x_0+H^{\bot}}f(y)\,\textrm{d} y \leq\binom{n}{k}\|f\|_{\infty}\int_{{\mathbb{R}}^n}f(x)\,\textrm{d} x. \end{equation} (49) We point out that, in the case of an integrable function |$f$| whose restriction to its support is continuous, the above assumption on the volume of the sections of |${\mathcal{C}}_t(f)$| trivially holds, since |${\mathcal{C}}_t(f)$| is compact for every |$t\in (0,1)$|⁠. Notice also that, when dealing with certain classes of functions with a more restrictive concavity (such as log-concave ones), continuity on the interior of their support is already guaranteed. 4 Rogers–Shephard Type Inequalities for Measures with Quasi-Concave Densities As a direct application of Corollary 3.3 we obtain the following result. Theorem 4.1. Let |$k\in \{1,\dots ,n-1\}$| and |$H\in \textrm{G}(n,n-k)$|⁠. Let |$\phi _i:{\mathbb{R}}^i\longrightarrow [0,\infty )$|⁠, |$i=n-k,k$|⁠, be functions with |$\|\phi _i\|_{\infty }=\phi _i(0)$|⁠, and such that the function |$\phi :{\mathbb{R}}^n\longrightarrow [0,\infty )$| given by |$\phi (x,y)=\phi _{n-k}(x)\phi _k(y)$|⁠, |$x\in{\mathbb{R}}^{n-k}$|⁠, |$y\in{\mathbb{R}}^k$|⁠, is quasi-concave. Let |$\mu _n=\mu _{n-k}\times \mu _{k}$| be the product measure on |${\mathbb{R}}^n$| given by |$\textrm{d}\mu _{n-k}(x)=\phi _{n-k}(x)\,\textrm{d} x$| and |$\textrm{d}\mu _{k}(y)=\phi _k(y)\,\textrm{d} y$|⁠. Let |$K\in{\mathcal{K}}^n$| with |$P_HK\subset K$| and so that |$\textrm{vol}\bigl ({\mathcal{C}}_t(\phi )\cap K\cap (x+H^{\bot })\bigr )$| attains its maximum for all |$t\in (0,1)$|⁠. Then \begin{equation} \mu_{n-k}\bigl(P_HK\bigr)\max_{x_0\in H}\left[\frac{\phi_{n-k}(x_0)}{\|\phi_{n-k}\|_{\infty}}\mu_k\bigl(K\cap(x_0+H^{\bot})\bigr)\right]\leq\binom{n}{k}\mu_n(K). \end{equation} (50) Proof. It is a straightforward consequence of (49) applied to the function |$f:{\mathbb{R}}^n\longrightarrow [0,\infty )$| given by |$f(x,y)=\phi _{n-k}(x)\phi _k(y)\chi _{_K}(x,y)$|⁠. Indeed, since |$P_HK\subset K$| then \begin{equation*} P_Hf(x)=\sup_{y\in H^{\bot}}\phi_{n-k}(x)\phi_k(y)\chi_{_K}(x,y)=\phi_{n-k}(x)\phi_{k}(0)\chi_{_{P_HK}}(x) \end{equation*} and |$\|f\|_{\infty }=\phi _{n-k}(0)\phi _{k}(0)$|⁠. We point out that the assumption |$P_HK\subset K$| is needed in order to conclude the above Rogers–Shephard type inequality (as well as Theorem 1.3). Example 4.2. Let |$\mu _1$| be the measure on |${\mathbb{R}}$| given by |$\textrm{d}\mu _1(x)=e^{-x^2}\,\textrm{d} x$| and let |$\mu _2=\mu _{1}\times \mu _{1}$|⁠, that is, |$\textrm{d}\mu _2(x)=e^{-|x|^2}\,\textrm{d} x$|⁠. Let |$H=\bigl \{(x,y)\in{\mathbb{R}}^2:y=0\bigr \}$| and, for a given |$0<\alpha <\pi /2$|⁠, let |$K_{\alpha }$| be the centrally symmetric parallelogram |$K_{\alpha }=\textrm{conv} \bigl \{(1,\tan \alpha \pm 1),(-1,-\tan \alpha \pm 1)\bigr \}$|⁠. On the one hand, |$K_{\alpha }(0)=\bigl [(0,1),(0,-1)\bigr ]$| is the “maximal” section of |$K_{\alpha }$| (with respect to |$\mu _1$|⁠) and |$P_HK_{\alpha }=\bigl [(-1,0),(1,0)\bigr ]$|⁠. On the other hand, since |$K_{\alpha }$| is contained in the infinite strip |$S_{\alpha }$| determined by the straight lines |$y=(\tan \alpha ) x\pm 1$|⁠, and |$\mu _2$| is rotationally invariant, we have that \begin{equation*} \mu_2(K_{\alpha})\leq\mu_2(S_{\alpha})=\sqrt{2\pi}\,\mu_1(I_{\alpha}), \end{equation*} where |$I_{\alpha }$| denotes the line segment centered at the origin and with length the width of |$S_{\alpha }$|⁠. Hence, |$\mu _1(I_{\alpha })$|⁠, and so |$\mu _2(K_{\alpha })$|⁠, can be made arbitrarily small when |$\alpha \rightarrow \pi /2$|⁠. However, the term |$\mu _1\bigl (P_HK_{\alpha }\bigr )\mu _1\bigl (K_{\alpha }(0)\bigr )=\mu _1\bigl ([(-1,0),(1,0)]\bigr )^2$| is a fixed positive constant. This shows the necessity of assuming |$P_HK\subset K$| in order to derive both (50) and (14). In order to avoid the assumption |$P_HK\subset K$|⁠, one may exchange the orthogonal projection by the corresponding maximal section. To this end, first we fix some notation: given a measure |$\mu $| in |${\mathbb{R}}^n$| with density |$\phi $|⁠, we will denote by |$\mu _i$|⁠, |$i=1,\dots ,n-1$|⁠, the marginal of |$\mu $| in the corresponding |$i$|-dimensional affine subspace, that is, for given |$M\subset z+H$| with |$H\in \textrm{G}(n,i)$| and |$z\in H^{\bot }$|⁠, \begin{equation*} \mu_i(M)=\int_H\chi_{_{M}}(x,z)\phi(x,z)\,\textrm{d} x. \end{equation*} Taking the function |$f:{\mathbb{R}}^n\longrightarrow [0,\infty )$| given by |$f(x,y)=\phi (x,y)\chi _{_K}(x,y)$|⁠, |$x\in H$|⁠, |$y\in H^{\bot }$|⁠, since \begin{equation*} P_Hf(x)=\sup_{y\in H^{\bot}}\phi(x,y)\chi_{_K}(x,y)\geq\phi(x,y)\chi_{_K}(x,y)=f(x,y), \end{equation*} we get the following result, as direct consequence of (49). Corollary 4.3. Let |$k\in \{1,\dots ,n-1\}$| and |$H\in \textrm{G}(n,n-k)$|⁠. Let |$\mu $| be a measure on |${\mathbb{R}}^n$| given by |$\textrm{d}\mu (x)=\phi (x)\,\textrm{d} x$|⁠, where |$\phi :{\mathbb{R}}^n\longrightarrow [0,\infty )$| is a quasi-concave function with |$\|\phi \|_{\infty }=\phi (0)$|⁠. Let |$K\in{\mathcal{K}}^n$| be such that there exists the maximum of |$\textrm{vol}\bigl ({\mathcal{C}}_t(\phi )\cap K\cap (x+H^{\bot })\bigr )$| for all |$t\in (0,1)$|⁠. Then \begin{equation} \max_{y\in H}\mu_{n-k}\bigl(K\cap (y+H)\bigr)\max_{x_0\in H}\mu_k\bigl(K\cap(x_0+H^{\bot})\bigr)\leq\binom{n}{k}\|\phi\|_{\infty}\mu(K). \end{equation} (51) We notice that, from (50), \begin{equation} \mu_{n-k}\bigl(P_HK\bigr)\mu_k\bigl(K\cap H^{\bot}\bigr)\leq\binom{n}{k}\mu_n(K) \end{equation} (52) holds provided that the density of |$\mu _n$|⁠, |$\phi (x,y)=\phi _{n-k}(x)\phi _k(y)$|⁠, is quasi-concave. Although the latter implies that both |$\phi _{n-k}$| and |$\phi _k$| are quasi-concave, the converse is, in general, not true. In the following we exploit the approach followed in the previous section in order to derive (52) for the more general case of measures |$\mu _{n-k}$| and |$\mu _{k}$|⁠, with radially decreasing and quasi-concave densities, respectively, and their product |$\mu _n=\mu _{n-k}\times \mu _{k}$|⁠, provided that the maximality assumption \begin{equation*} \max_{x\in P_HK}\textrm{vol}_k\bigl({\mathcal{C}}_t(\phi_k)\cap K(x)\bigr) =\textrm{vol}_k\bigl({\mathcal{C}}_t(\phi_k)\cap K(0)\bigr) \end{equation*} holds. Again, we need to assume the condition |$P_HK\subset K$|⁠. Proof of Theorem 1.3. By an appropriate choice of the coordinate axes, we may assume that |$H=\{x_{n-k+1}=\dots =x_n=0\}$|⁠. For every |$t\in [0,1]$|⁠, and |$x\in P_H K$|⁠, we consider the set \begin{equation*} {\mathcal{C}}_{x,t}=\Bigl(\{0\}\times{\mathcal{C}}_t(\phi_k)\Bigr)\cap K(x) \end{equation*} and the function |$\varphi _t:P_HK\longrightarrow [0,\infty )$| given by \begin{equation*} \varphi_t(x)=\textrm{vol}_k\bigl({\mathcal{C}}_{x,t}\bigr). \end{equation*} Since |$P_HK\subset K$| and |$\phi _k$| is continuous at the origin (which implies that |$0\in \operatorname{int}{\mathcal{C}}_t(\phi _k)$| for all |$t<1$|⁠), we may assure that, for every |$t<1$|⁠, |$\varphi _t(x)>0$| for any |$x$| in the (relative) interior of |$P_HK$| and hence |$\textrm{supp}\ \varphi _t=P_HK$|⁠. Moreover, |$\varphi _t$| is |$(1/k)$|-concave by (1) and, by hypothesis, we have |$\|\varphi _t\|_{\infty }=\varphi _t(0)$|⁠. Then, applying (39), with |$p=1/k$|⁠, to the function |$g:P_HK\longrightarrow [0,\infty )$| given by |$g(x,0)=\phi _{n-k}(x)$|⁠, |$x\in{\mathbb{R}}^{n-k}$|⁠, we get \begin{equation} \int_{P_HK}\phi_{n-k}(x)\,\textrm{d} x \leq\binom{n}{k}\dfrac{1}{\|\varphi_t\|_{\infty}}\int_{P_HK}\phi_{n-k}(x)\,\varphi_t(x)\,\textrm{d} x, \end{equation} (53) and hence, integrating (53) over |$t\in [0,1]$|⁠, we obtain \begin{equation*} \int_0^1\int_{P_HK}\phi_{n-k}(x)\,\textrm{d} x\int_{{\mathbb{R}}^k} \chi_{_{{\mathcal{C}}_{0,t}}}(y)\,\textrm{d} y\,\textrm{d} t\leq\binom{n}{k}\int_0^1\int_{P_HK}\phi_{n-k}(x)\int_{{\mathbb{R}}^k}\chi_{_{{\mathcal{C}}_{x,t}}}(y) \,\textrm{d} y\,\textrm{d} x\,\textrm{d} t. \end{equation*} Therefore, by Fubini’s theorem we have \begin{equation*} \begin{split} \mu_{n-k}\bigl(P_HK\bigr)\mu_k\bigl(K\cap H^{\bot}\bigr) & =\|\phi_k\|_{\infty}\int_{P_HK}\phi_{n-k}(x)\,\textrm{d} x\int_{K(0)}\int_0^1 \chi_{_{{\mathcal{C}}_t(\phi_k)}}(y)\,\textrm{d} t\,\textrm{d} y\\ & =\|\phi_k\|_{\infty}\int_0^1\int_{P_HK}\phi_{n-k}(x)\,\textrm{d} x\int_{{\mathbb{R}}^k} \chi_{_{{\mathcal{C}}_{0,t}}}(y)\,\textrm{d} y\,\textrm{d} t\\ & \leq\binom{n}{k}\|\phi_k\|_{\infty}\int_0^1\int_{P_HK}\phi_{n-k}(x)\int_{{\mathbb{R}}^k}\chi_{_{{\mathcal{C}}_{x,t}}}(y)\,\textrm{d} y\,\textrm{d} x\,\textrm{d} t\\ & =\binom{n}{k}\|\phi_k\|_{\infty}\int_{P_HK}\phi_{n-k}(x)\int_{K(x)}\int_0^1\chi_{_{{\mathcal{C}}_t(\phi_k)}}(y)\,\textrm{d} t\,\textrm{d} y\,\textrm{d} x\\ & =\binom{n}{k}\int_{P_HK}\phi_{n-k}(x)\mu_k\bigl(K(x)\bigr)\textrm{d} x=\binom{n}{k}\mu_n(K). \end{split} \end{equation*} This concludes the proof. Next we show an extension of the above Rogers–Shephard type inequalities involving maximal sections of convex bodies (cf. (51)) in the spirit of [1, Lemma 4.1]. Corollary 4.4. Let |$i,j\in \{2,\dots ,n-1\}$|⁠, |$i+j\geq n+1$|⁠, and let |$E\in \textrm{G}(n,i)$|⁠, |$H\in \textrm{G}(n,j)$| be such that |$E^{\bot }\subset H$|⁠. Let |$\phi :{\mathbb{R}}^n\longrightarrow [0,\infty )$| be a |$(-1/n)$|-concave function and let |$\mu $| be the measure on |${\mathbb{R}}^n$| given by |$\textrm{d}\mu (x)=\phi (x)\,\textrm{d} x$|⁠. Then, for every |$K\in{\mathcal{K}}^n$|⁠, if |$F=E\cap H$|⁠, \begin{equation} \sup_{x\in E^{\bot}}\mu_i\bigl(K\cap (x+E)\bigr)\sup_{y\in H^{\bot}}\mu_j\bigl(K\cap (y+H)\bigr) \leq\binom{n-k}{n-i}\sup_{x\in{\mathbb{R}}^n}\mu_k\bigl(K\cap (x+F)\bigr)\mu(K). \end{equation} (54) Proof. Let |$f:F^{\bot }\longrightarrow [0,\infty )$| be the function given by \begin{equation*} f(x,y)=\int_{{\mathbb{R}}^k}\phi(x,y,z)\chi_{_{K}}(x,y,z) \,\textrm{d} z. \end{equation*} The Borell–Brascamp–Lieb inequality (see, e.g., [16, Theorem 10.1]) implies that |$f$| is quasi-concave and, in particular, |${\mathcal{C}}_t(f)$| is a convex body. Then, we may apply Corollary 3.3 to obtain \begin{equation*} \begin{split} \int_{E^{\bot}}\sup_{y\in H^{\bot}}&\int_{{\mathbb{R}}^k} \phi(x,y,z) \chi_{_{K}}(x,y,z) \,\textrm{d} z\,\textrm{d} x \sup_{x\in E^{\bot}}\int_{H^{\bot}}\int_{{\mathbb{R}}^k}\phi(x,y,z)\chi_{_{K}}(x,y,z) \,\textrm{d} z\,\textrm{d} y\\ & \leq \binom{n-k}{n-i}\sup_{(x,y)\in F^{\bot}}\int_{{\mathbb{R}}^k}\phi(x,y,z)\chi_{_{K}}(x,y,z) \,\textrm{d} z \int_{F^\bot}\int_{{\mathbb{R}}^k}\phi(x,y,z)\chi_{_{K}}(x,y,z) \,\textrm{d} z\,\textrm{d} x\,\textrm{d} y \end{split} \end{equation*} and thus, in particular, for every |$y_0\in H^{\bot }$| we have \begin{equation*} \begin{split} \int_{E^{\bot}}\int_{{\mathbb{R}}^k}\phi(x,y_0,z)\chi_{_{K}}(x,y_0,z) \,\textrm{d} z\,\textrm{d} x & \sup_{x\in E^{\bot}}\int_{H^{\bot}}\int_{{\mathbb{R}}^k}\phi(x,y,z)\chi_{_{K}}(x,y,z) \,\textrm{d} z\,\textrm{d} y\\ & \leq \binom{n-k}{n-i}\sup_{(x,y)\in F^{\bot}}\mu_k\Bigl(K\cap\bigl((x,y)+F\bigr)\Bigr)\mu(K). \end{split} \end{equation*} Hence, for every |$y_0\in H^{\bot }$|⁠, we get \begin{equation*} \mu_j\bigl(K\cap (y_0+ H)\bigr)\sup_{x\in E^{\bot}}\mu_i\bigl(K\cap (x+E)\bigr)\leq \binom{n-k}{n-i}\sup_{x\in{\mathbb{R}}^n}\mu_k\bigl(K\cap (x+F)\bigr)\mu(K), \end{equation*} which implies (54). Next we show how one may exploit the approach we are following in this section to obtain an analogous result to Proposition 2.8, in the setting of quasi-concave densities which are not necessarily continuous. Notice that whereas the right-hand side in (55) is smaller than the right-hand side in (24), the constants |$c(\omega )$| and |$\phi (\omega )/\|\phi \|_{\infty }$| are not comparable in general. Theorem 4.5. Let |$K\in{\mathcal{K}}^n$| and let |$\mu $| be a measure on |${\mathbb{R}}^n$| given by |$\textrm{d}\mu (x)=\phi (x)\,\textrm{d} x$|⁠, where |$\phi :{\mathbb{R}}^n\longrightarrow [0,\infty )$| is a bounded quasi-concave function. Then, for every |$\omega \in{\mathbb{R}}^n$|⁠, \begin{equation} \frac{\phi(\omega)}{\|\phi\|_{\infty}}\mu(K-K+\omega) \leq\binom{2n}{n}\min\left\{\sup_{y\in K}\mu(y+\omega-K), \sup_{y\in K}\mu(-y+\omega+K)\right\}. \end{equation} (55) Moreover, if |$\phi $| is continuous at |$\omega _0$|⁠, for some |$\omega _0\in{\mathbb{R}}^n$|⁠, then equality holds in (55) (for such |$\omega _0$|⁠) if and only if |$\mu $| is a constant multiple of the Lebesgue measure on |$K-K+\omega _0$|⁠, |$\phi (\omega _0)=\|\phi \|_{\infty }$| and |$K$| is a simplex. Proof. Let |$\omega \in{\mathbb{R}}^n$| and consider the function |$f_\omega :K-K+\omega \longrightarrow [0,\infty )$| given by \begin{equation*} f_\omega(x)=\textrm{vol}\bigl(K\cap(x-\omega+K)\bigr). \end{equation*} Notice that, |$f_\omega $| is |$(1/n)$|-concave by (1), |$\textrm{supp}\ f_\omega =K-K+\omega $| and, moreover, that |$\|f_\omega \|_{\infty }=f_\omega (\omega )=\textrm{vol}(K)$|⁠. Then, using (41), we get \begin{equation*} \begin{split} \frac{\phi(\omega)}{\|\phi\|_{\infty}}\mu(K-K+\omega) & \leq\binom{2n}{n}\frac{1}{\textrm{vol}(K)}\int_{{\mathbb{R}}^n}\textrm{vol}\bigl(K\cap(x-\omega+K)\bigr)\,\textrm{d} \mu(x)\\ & =\binom{2n}{n}\frac{1}{\textrm{vol}(K)}\int_{{\mathbb{R}}^n}\phi(x)\int_{{\mathbb{R}}^n}\chi_{_K}(y)\chi_{_{y+\omega-K}}(x)\,\textrm{d} y\,\textrm{d} x\\ & =\binom{2n}{n}\frac{1}{\textrm{vol}(K)}\int_{K}\mu(y+\omega-K)\,\textrm{d} y\leq\binom{2n}{n}\sup_{y\in K}\mu(y+\omega-K). \end{split} \end{equation*} Therefore, exchanging the roles of |$K$| and |$-K$|⁠, (55) infers. Finally, if equality holds in (55) for some |$\omega _0\in{\mathbb{R}}^n$| then, by Corollary 3.2, |$\mu $| is a constant multiple of the Lebesgue measure on |$K-K+\omega _0$| and |$\phi (\omega _0)=\|\phi \|_{\infty }$|⁠. Now, from the equality case of Theorem A, |$K$| must be a simplex. The converse is immediate from Theorem A. We conclude this section by noticing that, from the proof of the previous result, one may also obtain (4) in the slightly less general setting of quasi-concave densities with maximum at the origin. We include it here for the sake of completeness. Corollary 4.6. Let |$K\in{\mathcal{K}}^n$| and let |$\mu $| be the measure on |${\mathbb{R}}^n$| given by |$\textrm{d}\mu (x)=\phi (x)\,\textrm{d} x$|⁠, where |$\phi :{\mathbb{R}}^n\longrightarrow [0,\infty )$| is a quasi-concave function with |$\|\phi \|_{\infty }=\phi (0)$|⁠. Then \begin{equation*} \mu(K-K)\leq \binom{2n}{n}\min\bigl\{\overline{\mu}(K),\overline{\mu}(-K)\bigr\}. \end{equation*} Moreover, if |$\phi $| is continuous at the origin then equality holds if and only if |$\mu $| is a constant multiple of the Lebesgue measure on |$K-K$| and |$K$| is a simplex. 5 A Remark for Measures with |$p$|-Concave Densities, |$p>0$| As we have shown in Example 4.2, the assumption |$P_HK\subset K$| on Theorems 1.3 and 4.1 is necessary. However, when dealing with measures associated with |$p$|-concave densities, |$p>0$|⁠, an inequality in the spirit of (13) can be obtained for an arbitrary |$K\in{\mathcal{K}}^n$|⁠, by setting a binomial coefficient according to the concavity nature of the density. This is the content of the following result. Theorem 5.1. Let |$k\in \{1,\dots ,n-1\}$|⁠, |$r\in{\mathbb{N}}$| and |$H\in \textrm{G}(n,n-k)$|⁠. Given a |$(1/r)$|-concave function |$\phi _k:{\mathbb{R}}^k\longrightarrow [0,\infty )$| and a radially decreasing function |$\phi _{n-k}:{\mathbb{R}}^{n-k}\longrightarrow [0,\infty )$|⁠, let |$\mu _n=\mu _{n-k}\times \mu _{k}$| be the product measure on |${\mathbb{R}}^n$| given by |$\textrm{d}\mu _{n-k}(x)=\phi _{n-k}(x)\,\textrm{d} x$| and |$\textrm{d}\mu _{k}(y)=\phi _k(y)\,\textrm{d} y$|⁠. Let |$K\in{\mathcal{K}}^n$| be such that |$\max _{x\in H} \mu _k\left (K\cap \bigl (x+H^{\bot }\bigr )\right )=\mu _k\left (K\cap H^{\bot }\right )$|⁠. Then \begin{equation*} \mu_{n-k}\bigl(P_HK\bigr)\mu_k\bigl(K\cap H^{\bot}\bigr)\leq\binom{n+r}{n-k}\mu_n(K). \end{equation*} Proof. Consider the function |$f:H\longrightarrow{\mathbb{R}}$| given by \begin{equation*} f(x)=\mu_k\left(K\cap\bigl(x+H^{\bot}\bigr)\right), \end{equation*} which satisfies |$\textrm{supp}\ f=P_HK$|⁠. Now, the Borell–Brascamp–Lieb inequality [16, Theorem 10.1] implies that |$\mu _k$| is (⁠|$1/(k+r)$|⁠)-concave which, together with the convexity of |$K$|⁠, yields that |$f$| is (⁠|$1/(k+r)$|⁠)-concave. Furthermore, by assumption we have that |$\|f\|_{\infty }=f(0)$|⁠. Thus, using (39) for |$g=\phi _{n-k}$|⁠, we obtain \begin{equation*} \alpha^{n-k}_{1/(k+r),0}\int_{P_HK}\phi_{n-k}(x)\,\textrm{d} x\leq\dfrac{1}{\mu_k\bigl(K\cap H^{\bot}\bigr)} \int_{P_HK} \mu_k\left(K\cap\bigl(x+H^{\bot}\bigr)\right)\,\phi_{n-k}(x)\textrm{d} x \end{equation*} and hence \begin{equation*} \mu_{n-k}\bigl(P_HK\bigr)\mu_k\bigl(K\cap H^{\bot}\bigr)\leq\binom{n+r}{n-k}\mu_n(K), \end{equation*} as desired. The latter result can be stated for any positive real number |$r$|⁠, just replacing |$\binom{n+r}{n-k}$| by the suitable constant. We notice that the above inequality includes (13) as a special case, since the constant density (of the Lebesgue measure) is |$\infty $|-concave, and thus |$r=0$|⁠. Funding This work is supported by the DGA E26_17R and MINECO/FEDER MTM2016-77710-P projects and Instituto Universitario de Matemáticas y Aplicaciones (IUMA) [to D.A.G.]; MINECO/FEDER project MTM2015-65430-P and “Programa de Ayudas a Grupos de Excelencia de la Región de Murcia”, Fundación Séneca, 19901/GERM/15 [to M.A.H.C. and J.Y.N.]; U.S. National Science Foundation [DMS-1101636 to M.R. and A.Z]; and Comue Université Paris-Est [to A.Z.]. Acknowledgments We thank the referees for many valuable suggestions and remarks, which have allowed us to considerably improve the manuscript. Communicated by Prof. Igor Rivin References [1] Alonso-Gutiérrez , D. , S. Artstein-Avidan , B. González Merino , C. H. Jiménez , and R. Villa . “ Rogers–Shephard and local Loomis–Whitney type inequalities .” Preprint arXiv:1706.01499v2 . [2] Alonso-Gutiérrez , D. , B. González Merino , C. H. Jiménez , and R. Villa . “ Rogers–Shephard inequality for log-concave functions .” J. Funct. 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Schnitzer,, Jonas;Vitagliano,, Luca

doi: 10.1093/imrn/rnz009pmid: N/A

Abstract Generalized contact bundles are odd-dimensional analogues of generalized complex manifolds. They have been introduced recently and very little is known about them. In this paper we study their local structure. Specifically, we prove a local splitting theorem similar to those appearing in Poisson geometry. In particular, in a neighborhood of a regular point, a generalized contact bundle is either the product of a contact and a complex manifold or the product of a symplectic manifold and a manifold equipped with an integrable complex structure on the gauge algebroid of the trivial line bundle. Introduction Generalized complex manifolds have been introduced by Hitchin in [21] and further investigated by Gualtieri in [20], and the literature about them is now rather wide. Generalized complex manifolds are necessarily even dimensional and they encompass symplectic and complex manifolds as extreme cases. A natural question is what is the odd-dimensional analogue of a generalized complex manifold. Several answers to this question appeared already in the literature but the works on generalized geometry in odd dimensions are still sporadic [2, 22, 30, 32, 35, 41]. Recently, Wade and the 2nd author proposed a partially new definition of an odd-dimensional analogue of a generalized complex manifold called a generalized contact bundle [38]. Generalized contact bundles are a slight generalization of Iglesias–Wade integrable generalized almost contact structures [22] to the realm of (generically nontrivial) line bundles, and encompass not necessarily coorientable contact manifolds as an extreme case. At the other extreme they encompass line bundles equipped with an integrable complex structure on their gauge algebroid. In turn, such line bundles are intrinsic models for so-called normal almost contact manifolds [7]. In our opinion, generalized contact bundles have an advantage over previous proposals of a generalized contact geometry; they have a firm conceptual basis in the so-called homogenization scheme [40], which is, in essence, a dictionary from contact and related geometries to symplectic and related geometries. In principle, applying the dictionary is straightforward; it is enough to replace functions on a manifold |$M$| with sections of a line bundle |$L \to M$|⁠, vector fields over |$M$| with derivations of |$L$|⁠, etc. In practice, applying the dictionary can be actually challenging and may lead to interesting new features [9, 19, 24, 25, 33, 34, 36, 37, 39, 40]. In [38] the authors define generalized contact bundles and study their structure equations, showing, in particular, that every generalized contact bundle is a Jacobi bundle [23, 26, 29]. This puts odd-dimensional generalized geometry in the framework of Jacobi geometry. In this paper we begin a systematic study of generalized contact bundles by studying their local structure. Our main results are two splitting theorems. In this introduction we provide for them rough statements to be better explained and made precise in the bulk of the paper. Theorem (A). Let |$M$| be a manifold equipped with a generalized contact bundle, and let |$x_0 \in M$| be a point in an odd-dimensional characteristic leaf of |$M$|⁠. Then, locally around |$x_0$|⁠, |$M$| is isomorphic, up to a |$B$|-field transformation, to the product of a contact manifold and a homogeneous generalized complex manifold whose homogeneous Poisson structure vanishes at a point. Theorem (B). Let |$M$| be a manifold equipped with a generalized contact bundle, and let |$x_0 \in M$| be a point in an even-dimensional characteristic leaf of |$M$|⁠. Then, locally around |$x_0$|⁠, |$M$| is isomorphic, up to a |$B$|-field transformation, to the product of a symplectic manifold and a manifold with a generalized contact bundle whose Jacobi structure vanishes at a point. We also explicitly discuss the local structure of a generalized contact bundle in a neighborhood of a regular point, proving the following two local normal form theorems. Theorem (C). Let |$M$| be a |$(2n + 2d + 1)$|-dimensional manifold equipped with a generalized contact bundle, and let |$x_0 \in M$| be a point in a |$(2d +1)$|-dimensional characteristic leaf of |$M$|⁠. If |$x_0$| is a regular point, then, locally around |$x_0$|⁠, |$M$| is isomorphic, up to a |$B$|-field transformation, to the product of the standard |$(2d +1)$|-dimensional contact manifold |$(\mathbb{R}^{2d+1}, \theta _{\textit{can}})$| and the standard complex space |$\mathbb{C}^n$|⁠. Theorem (D). Let |$M$| be a |$(2n + 2d + 1)$|-dimensional manifold equipped with a generalized contact bundle, and let |$x_0 \in M$| be a point in a |$2d$|-dimensional characteristic leaf of |$M$|⁠. If |$x_0$| is a regular point, then, locally around |$x_0$|⁠, |$M$| is isomorphic, up to a |$B$|-field transformation, to the product of the standard |$2d$|-dimensional symplectic space |$(\mathbb{R}^{2d}, \Omega _{\textit{can}})$| and the cylinder |$\mathbb{R} \times \mathbb{C}^n$| equipped with the canonical complex structure on the gauge algebroid of the trivial line bundle |$(\mathbb{R} \times \mathbb{C}^n) \times \mathbb{R} \to \mathbb{R} \times \mathbb{C}^n$|⁠. We stress that the word “product” in the statements of Theorems (A)–(D) does not exactly refer to the (cartesian) product of manifolds, but rather to a certain technical notion of product of line bundles over possibly different base manifolds that we explain in Section 2.2.5 below. Here we only mention that if |$L_1 \to M_1$| and |$L_2 \to M_2$| are line bundles, then a product of them in our sense is a line bundle over |$M_1 \times M_2$| equipped with additional structure (see diagram (2.11)). We also mention here that the proof of Theorem (D) requires proving that certain Dolbeault-like cohomologies associated with a complex structure on the gauge algebroid of a line bundle are locally trivial. For our proofs, we use an adaptation of the methods of Bursztyn et al. [11]. This suggests that the Splitting Theorems (A) and (B) are a manifestation of more general normal form theorems around appropriate transversals to a characteristic leaf (see [11] for more details). This is indeed the case, as showed by the 1st author [31]. Even more, it is showed in [31] that there are normal forms of Jacobi (and, more generally, Dirac–Jacobi) structures around appropriate transversals, extending the splitting theorems of Dazord et al. [14], and paralleling similar normal forms in Poisson (and Dirac) Geometry (see [11], see also [18]). In this paper, we prefer to stick on a less general formulation. Our choice is mainly dictated by space reasons; we think the paper contains already enough material, including several generalities with a certain interest, besides our main results. We actually hope that this paper could also serve as one possible reference for (local aspects of) Generalized Contact Geometry in future works. The paper is organized as follows. In Section 1 we collect the necessary preliminaries on gauge algebroids and Jacobi structures. In Section 2 we recall the notions of generalized contact bundles [38] and complex Dirac–Jacobi structures [37]. In this section we also discuss in detail symmetries of the omni-Lie algebroid, which plays for generalized contact and Dirac–Jacobi bundles a similar role as the generalized tangent bundle plays for generalized complex and Dirac manifolds. Finally, we discuss homogeneous generalized complex structures and a suitable notion of product of Dirac–Jacobi bundles, which appears to be unavoidable in a precise formulation of our splitting theorems. In Section 3 we describe in detail the structures induced on the characteristic leaves of a generalized contact structure, and on their transversals. Section 4 contains our main results, the splitting theorems around a point in a contact and around a point in a locally conformal symplectic (lcs) leaf. In the last Section 5 we prove, as corollaries, local normal form theorems around a regular point. Finally, in Appendix A, we discuss a very special class of generalized contact structures, complex structures on the gauge algebroid of a line bundle. We prove a local normal form theorem analogous to the Newlander–Nirenberg theorem and the local vanishing of an associated Dolbeault-like cohomology. Both are consequences of their standard even-dimensional counterparts. We assume the reader is familiar with the fundamentals of Lie algebroids, Dirac manifolds, and generalized complex structures. 1 Preliminaries 1.1 The gauge algebroid A derivation of a vector bundle |$E \to M$| is an |$\mathbb{R}$|-linear operator \begin{equation*} \Delta: \Gamma (E) \to \Gamma (E) \end{equation*} satisfying the following Leibniz rule \begin{equation*} \Delta (f \varepsilon) = X(f) \varepsilon + f \Delta \varepsilon, \quad f \in C^\infty (M), \quad \varepsilon \in \Gamma (E), \end{equation*} for a, necessarily unique, vector field |$X \in \mathfrak X (M)$|⁠, called the symbol of |$\Delta$| and denoted by |$\sigma (\Delta )$|⁠. Derivations are sections of a Lie algebroid |$DE \rightarrow M$|⁠, called the gauge algebroid of |$E$|⁠, whose anchor is the symbol and whose bracket is the commutator of derivations [27]. The fiber |$D_x E$| of |$DE$| over a point |$x \in M$| consists of |$\mathbb{R}$|-linear maps |$\Delta : \Gamma (E) \to E_x$| satisfying the Leibniz rule |$\Delta (f \varepsilon ) = v(f) \varepsilon _x + f(x) \Delta \varepsilon$| for some tangent vector |$v \in T_x M$|⁠, the symbol of |$\Delta$|⁠. The correspondence |$E \mapsto DE$| is functorial, in the following sense. Let |$F \to N$| and |$E \to M$| be two vector bundles, and let |$\Phi : F \to E$| be a regular vector bundle map, that is, a bundle map, covering a smooth map |$\phi : N \to M$|⁠, which is an isomorphism on fibers. Then |$\Phi$| induces a (generically non-regular) vector bundle map |$D \Phi : D F \to D E$| via formula \begin{equation*} D \Phi (\Delta) \varepsilon = (\Phi \circ \Delta) ( \Phi^\ast \varepsilon), \end{equation*} for all |$\Delta \in D F$|⁠, and |$\varepsilon \in \Gamma (E)$|⁠. Here |$\Phi ^\ast \varepsilon$| is the pull-back of |$\varepsilon$| along |$\Phi$|⁠, that is, it is the section of |$F$| given by |$(\Phi ^\ast \varepsilon )_y = \Phi |_{F_y}^{-1} (\varepsilon _{\phi (y)})$|⁠, |$y \in N$|⁠. The vector bundle map |$D \Phi$| will be sometimes denoted by |$\Phi _\ast$| if there is no risk of confusion. Correspondence |$\Phi \mapsto D \Phi$| preserves identity and compositions. Derivations of a vector bundle |$E$| can be seen as linear vector fields on |$E$|⁠, that is, vector fields generating a flow by vector bundle automorphisms. Namely, for every derivation |$\Delta$| of |$E$|⁠, there exists a unique flow |$\{ \Phi _t \}$| by vector bundle automorphisms |$\Phi _t: E \to E$| such that \begin{equation*} \Delta \varepsilon = \frac{d}{dt} |_{t = 0} \Phi_t^\ast \varepsilon \end{equation*} for all |$\varepsilon \in \Gamma (E)$|⁠. The gauge algebroid acts tautologically on the vector bundle |$E$|⁠. Accordingly, there is a de Rham complex of |$DE$| with coefficients in |$E$|⁠, denoted |$(\Omega ^{\bullet }_E, d_D)$|⁠. Cochains in |$(\Omega ^{\bullet }_E, d_D)$| will be referred to as Atiyah forms. They are vector bundle maps |$\wedge ^{\bullet } DE \to E$|⁠. The differential |$d_D$| is given by the usual formula. Atiyah forms can be pulled back along regular vector bundle maps. Namely, let |$F \to N$| and |$E \to M$| be vector bundle, and let |$\Phi : F \to E$| be a regular vector bundle map covering a smooth map |$\phi : M \to N$|⁠. For |$\omega \in \Omega ^k_E$|⁠, we define |$\Phi ^\ast \omega \in \Omega ^k_F$| via \begin{equation*} (\Phi^\ast \omega)_{y} (\Delta_1, \ldots, \Delta_k) = \Phi|_{F_y}^{-1} \circ \omega_{\phi (y)} (\Phi_\ast \Delta_1, \ldots, \Phi_\ast \Delta_k) \end{equation*} for all |$y \in N$|⁠, and |$\Delta _1, \ldots , \Delta _k \in D_{y} F$|⁠. One can also take the Lie derivative |$\mathcal{L}_\Delta := [ \iota _\Delta , d_D ]$| of Atiyah forms along a derivation |$\Delta$|⁠, and all these operators satisfy the usual Cartan calculus identities. Additionally, there is a distinguished derivation, namely the identical one: 𝟙|$: \Gamma (E) \to \Gamma (E)$|⁠, |$\varepsilon \mapsto \varepsilon$|⁠, and the contraction |$\iota$|𝟙 of Atiyah forms with 𝟙 is a contracting homotopy for |$(\Omega ^{\bullet }_E, d_D)$|⁠. In particular, |$(\Omega ^{\bullet }_E, d_D)$| is acyclic. In the case of a line bundle |$L \to M$|⁠, every 1st order differential operator |$\Gamma (L) \to \Gamma (L)$| is a derivation. Consequently, there are vector bundle isomorphisms |$DL \cong \textsf{Hom} (J^1 L, L)$|⁠, and |$J^1 L \cong \textsf{Hom} (D L, L)$|⁠, and a non-degenerate pairing |$\langle -, - \rangle : J^1 L \otimes DL \to L$|⁠, where |$J^1 L \to M$| is the 1st jet bundle of |$L$|⁠. In this case, the identical derivation 𝟙 spans the kernel of the symbol and there is a short exact sequence \begin{equation} 0 \longrightarrow \mathbb{R}_M \longrightarrow DL \overset{\sigma}{\longrightarrow} TM \longrightarrow 0, \end{equation} (1.1) where |$\mathbb{R}_M = M \times \mathbb{R}$| is the trivial line bundle over |$M$|⁠. Dually, there is a short exact sequence \begin{equation} 0 \longleftarrow L \longleftarrow J^1 L \longleftarrow T^\ast M \otimes L \longleftarrow 0. \end{equation} (1.2) The embedding |$T^\ast M \otimes L \hookrightarrow J^1 L$| extends to an embedding \begin{equation*} \Omega^{\bullet} (M, L) \hookrightarrow \Omega^{\bullet}_L, \end{equation*} of |$L$|-valued forms on |$M$| into Atiyah forms on |$L$|⁠, consisting in composing with the symbol |$\sigma : DL \to TM$|⁠. We will often interpret |$\Omega ^{\bullet } (M, L)$| as a subspace in |$\Omega ^{\bullet }_L$| without further comments. Notice that |$\Omega ^1_L = \Gamma (J^1 L)$|⁠, and the 1st differential |$d_D: \Gamma (L) \to \Omega ^1_L$| agrees with the 1st jet prolongation |$j^1: \Gamma (L) \to \Gamma (J^1 L)$|⁠. Remark 1.1.1 (Atiyah forms on the trivial line bundle). When |$L = \mathbb{R}_M$| is the trivial line bundle, then sections of |$L$| are just functions on |$M$|⁠, both sequences (1.1) and (1.2) splits canonically via the standard flat connection in |$\mathbb{R}_M$|⁠, and we have \begin{equation*} \begin{aligned} D \mathbb{R}_M & = TM \oplus \mathbb{R}_M, \\ J^1 \mathbb{R}_M & = T^\ast M \oplus \mathbb{R}_M. \end{aligned} \end{equation*} In this case, a generic derivation is of the form |$X + f,$| where |$X$| is a vector field and |$f$| is a function. Similarly a generic section of |$J^1 \mathbb{R}_M$| is of the form |$\eta + g \cdot j^1 1$|⁠, where |$\eta$| is a |$1$|-form, |$g$| is a function, and |$j^1 1$| is the 1st jet prolongation of the constant function |$1 \in C^\infty (M)$|⁠. In the following, we will denote |$\mathfrak j = j^1 1$|⁠. Then we have \begin{equation*} j^1 f = df + f \cdot \mathfrak j, \quad f \in C^\infty (M). \end{equation*} More generally, any Atiyah form |$\omega \in \Omega ^{\bullet }_{\mathbb{R}_M}$| can be uniquely written as \begin{equation} \omega = \omega_0 + \omega_1 \wedge \mathfrak j, \end{equation} (1.3) with |$\omega _0, \omega _1 \in \Omega ^{\bullet } (M)$|⁠, and we used the symbol to give Atiyah forms the structure of a graded |$\Omega ^{\bullet } (M)$|-module. The correspondence |$\omega \mapsto (\omega _0, \omega _1)$| establishes an isomorphism of graded |$\Omega ^{\bullet } (M)$|-modules \begin{equation*} \Omega^{\bullet}_{\mathbb{R}_M} \cong \Omega^{\bullet} (M) \oplus \Omega^{\bullet -1} (M). \end{equation*} In terms of the decomposition (1.3) the natural operations on Atiyah forms read as follows: \begin{equation*} \begin{aligned} d_D \omega & = d \omega_0 + \left(d\omega_1 + (-)^{|\omega_0|} \omega_0\right) \wedge \mathfrak j \\ \iota_{X + f} \omega & = \iota_X \omega_0 + (-)^{|\omega_1|} f \omega_1 + \iota_X \omega_1 \wedge \mathfrak j \\ \mathcal L_{X + f} \omega& = \mathcal L_X \omega_0 + f \omega_0 + \omega_1 \wedge df + \left(\mathcal L_X \omega_1 + f \omega_1\right) \wedge \mathfrak j \end{aligned} \end{equation*} for all |$X + f \in \Gamma (D \mathbb{R}_M)$|⁠, where the bars “|$|-|$|” denote the degree. 1.2 Jacobi bundles and their characteristic foliations Jacobi manifolds were introduced by Kirillov [23] and, independently, Lichnerowicz [26], as generalizations of Poisson manifolds. Here we adopt to Jacobi manifolds the slightly more intrinsic approach via Jacobi bundles [29] (see also [33]). Jacobi bundles encompass (not necessarily coorientable) contact manifolds as nondegenerate instances. Let |$L \to M$| be a line bundle. A Jacobi structure on |$L$| is a Lie bracket \begin{equation*} \{-,-\}: \Gamma (L)\times \Gamma (L) \to \Gamma (L), \end{equation*} which is also a 1st-order bi-differential operator or, equivalently, a bi-derivation. The bracket |$\{-,-\}$| is also called the Jacobi bracket. A Jacobi bundle is a line bundle equipped with a Jacobi structure. A Jacobi bracket |$\{-,-\}$| can be regarded as a |$2$|-form \begin{equation*} J: \wedge^2 J^1 L \to L \end{equation*} satisfying an additional integrability condition, and in the following we will often take this point of view. Example 1.2.1. Every contact manifold is canonically equipped with a Jacobi bundle containing a full information on the contact structure. Indeed, let |$(M, H)$| be a contact manifold, that is, |$H \subset TM$| is a maximally non-integrable hyperplane distribution, and consider the normal line bundle |$L:= TM / H$|⁠. The distribution |$H$| can be equivalently encoded in a line bundle valued |$1$|-form |$\theta \in \Omega ^1 (M, L)$|⁠: the canonical projection |$\theta : TM \to L$|⁠. In its turn |$\theta$| can be seen as an Atiyah |$1$|-form on |$L$|⁠. One can prove that |$\omega := d_D \theta \in \Omega ^2_L$| is a nondegenerate (and closed) Atiyah |$2$|-form (see, for example, [37]). Here the nondegeneracy means that the induced vector bundle map \begin{equation*} \omega_\flat: DL \to J^1 L \end{equation*} is invertible. Its inverse |$\omega _\flat ^{-1}$| is the sharp map |$J^\sharp : J^1 L \to DL$| of a (unique, nondegenerate) Jacobi structure |$J:= \omega ^{-1}: \wedge ^2 J^1 L \to L$|⁠. Conversely, every nondegenerate Jacobi structure on a line bundle |$L \to M$| determines a contact structure |$H \subset TM$| on |$M$|⁠, with |$TM/H = L$|⁠. For some more details, see the discussion at the beginning of Section 3.1. Example 1.2.2. Every lcs manifold is canonically equipped with a Jacobi bundle containing a full information on the lcs structure. We adopt a slightly more intrinsic approach to lcs manifolds. Namely, in this paper, an lcs structure on a line bundle |$L \to M$| is a pair |$(\Omega , \nabla )$|⁠, where |$\nabla$| is a flat connection in |$L$|⁠, and |$\Omega$| is an |$L$|-valued |$2$|-form on |$M$|⁠, which is (1) nondegenerate and (2) closed with respect to the connection differential |$d^\nabla : \Omega ^{\bullet } (M, L) \to \Omega ^{\bullet +1} (M, L)$|⁠. When |$L = \mathbb{R}_M$| is the trivial line bundle we recover the usual definition. So let |$L \to M$| be a line bundle equipped with an lcs symplectic structure |$(\nabla , \Omega )$|⁠. The bracket \begin{equation*} \{ -, -\}: \Gamma (L) \times \Gamma (L) \to \Gamma (L), \quad (\lambda, \mu) \mapsto \Omega^{-1}(d^\nabla \lambda, d^\nabla \mu) \end{equation*} is a Jacobi bracket. If we interpret it as a |$2$|-form |$J: \wedge ^2 J^1 L \to L$|⁠, it is easy to see that the rank of |$J$| is |$\dim M$|⁠. Conversely, every Jacobi structure |$J$| on a line bundle |$L \to M$| such that |$\operatorname{rank}\ J = \dim \ M$| determines an lcs structure on |$L$|⁠. For some more details, see the discussion at the beginning of Section 3.2. Similarly as a Poisson manifold, a manifold |$M$| equipped with a Jacobi bundle |$(L, J)$| possesses a canonical (generically singular) foliation, called the characteristic foliation and defined as follows. Consider the sharp map associated with |$J$|⁠, |$J^\sharp : J^1 L \to DL$|⁠. Composing with the symbol we get a map |$\sigma J^\sharp : J^1 L \to TM$|⁠, whose image is an involutive distribution on |$M$|⁠. The integral foliation |$\mathcal F$| of |$\operatorname{im} \sigma J^\sharp$| is the characteristic foliation of the Jacobi bundle |$(L, J)$|⁠, and its leaves are the characteristic leaves. Odd-dimensional leaves of |$\mathcal F$| are naturally contact manifolds, while even-dimensional leaves are lcs manifolds. For more details about properties of the characteristic leaves in Jacobi geometry see, for example, [33] (see also Section 3). When |$L = \mathbb{R}_{M} \to M$| is the trivial line bundle, a Jacobi bracket |$\{-,-\}$| on |$L$| is equivalent to a Jacobi pair, that is, a pair |$(\Lambda , E)$|⁠, consisting of a bi-vector |$\Lambda \in \mathfrak X^2 (M)$| and a vector field |$E \in \mathfrak X (M)$| such that \begin{equation*} [\Lambda, \Lambda]^{SN} = 2 E \wedge \Lambda, \quad \textrm{and} \quad [E, \Lambda]^{SN} = 0, \end{equation*} where |$[-,-]^{SN}$| is the Schouten–Nijenhuis bracket of multivectors. The equivalence is provided by the following formula: \begin{equation*} \{ f, g \} = \Lambda (f, g) + E(f) g - f E(g), \quad f,g \in C^\infty (M). \end{equation*} Example 1.2.3. On |$\mathbb{R}^{2d +1}$|⁠, with coordinates |$(x^1, \ldots , x^d, p_1, \ldots , p_d, u)$|⁠, there is a canonical Jacobi pair |$(\Lambda _{\textit{can}}, E_{\textit{can}})$| given by \begin{equation*} \Lambda_{\textit{can}}:= \frac{\partial}{\partial p_i} \wedge \left( \frac{\partial}{\partial x^i} + p_i \frac{\partial}{\partial u} \right), \quad \textrm{and} \quad E_{\textit{can}} = \frac{\partial}{\partial u}. \end{equation*} We denote by |$J_{\textit{can}}$| the Jacobi structure corresponding to the Jacobi pair |$(\Lambda _{\textit{can}}, E_{\textit{can}})$|⁠. 2 Generalized Contact and Dirac–Jacobi Geometry 2.1 The omni-Lie algebroid and its symmetries The natural arena for generalized geometry in odd dimensions is the omni-Lie algebroid|$\mathbb{D} L$| of a line bundle |$L \to M$| [12]. Recall that |$\mathbb{D} L = DL \oplus J^1 L$|⁠, where |$DL \to M$| is the gauge algebroid. The omni-Lie algebroid possesses the following structures: |$\triangleright$| a natural projection \begin{equation} \operatorname{pr}_{D}: {{\mathbb{D}}} L \longrightarrow D L; \end{equation} (2.1) |$\triangleright$| a nondegenerate, symmetric, split signature |$L$|-valued |$2$|-form \begin{equation*} \langle \langle -, - \rangle \rangle: {{\mathbb{D}}} L \otimes{{\mathbb{D}}} L \to L \end{equation*} given by \begin{equation*} \langle \langle (\Delta, \psi), (\square, \chi) \rangle \rangle:= \langle \chi,\Delta \rangle + \langle \psi, \square \rangle; \end{equation*} |$\triangleright$| a (non-skew symmetric, Dorfman-like) bracket \begin{equation*} [ \! [ -,- ] \! ]: \Gamma ({{\mathbb{D}}} L) \times \Gamma ({{\mathbb{D}}} L) \to \Gamma ({{\mathbb{D}}} L) \end{equation*} given by \begin{equation} [ \! [ (\Delta, \psi), (\square, \chi) ] \! ]:= \left([\Delta, \square], {{\mathcal{L}}}_\Delta \chi - \iota_\square d_{D} \psi \right) \end{equation} (2.2) for all |$\Delta , \square \in D L$|⁠, and all |$\psi , \chi \in \Gamma (J^1 L)$|⁠. These structures satisfy certain identities that we do not report here (for more details see, for example, [37]). Most of them are just the obvious analogues of those holding for the standard Courant algebroid, the generalized tangent bundle|$\mathbb{T} M = TM \oplus T^\ast M$|⁠. Accordingly, the rest of this subsection is just an adaptation from similar features of |$\mathbb{T} M$|⁠. We now describe symmetries of the omni-Lie algebroid |$\mathbb{D} L$|⁠. First of all, for a vector bundle |$E \to M$|⁠, we denote by |$\textrm{Aut} (E)$| the group of its automorphisms. Definition 2.1.1. A Courant-Jacobi automorphism of |$\mathbb{D} L$| is a pair (, |$\Phi )$| consisting of an automorphism of the vector bundle |$\mathbb{D} L$| and an automorphism |$\Phi$| of |$L$|⁠, such that and |$\Phi$| cover the same diffeomorphism |$\phi : M \to M$|⁠, and, additionally, for all |$\alpha , \beta \in \Gamma (\mathbb{D} L)$|⁠. The group of Courant-Jacobi automorphisms is denoted by |$\textrm{Aut}_{CJ} (\mathbb{D} L)$|⁠. Example 2.1.2. Let |$B$| be a closed Atiyah |$2$|-form, that is, |$B \in \Omega ^2_L$| and |$d_D B = 0$| (in particular, |$B$| is exact). Denote by |$e^B: \mathbb{D}L \to \mathbb{D}L$| the vector bundle automorphism defined by \begin{equation*} e^B (\Delta, \psi):= (\Delta, \psi + \iota_\Delta B), \quad (\Delta, \psi) \in \mathbb{D} L. \end{equation*} Using the decomposition |$\mathbb{D} L = DL \oplus J^1 L$| we can write |$e^B$| in matrix form \begin{equation} e^B = \left( \begin{array}{cc} \textrm{id} & 0 \\ B_\flat & \textrm{id} \end{array} \right). \end{equation} (2.3) An easy computation shows that |$(e^B, \textrm{id})$| is a Courant-Jacobi automorphism. We will refer to it as a |$B$|-field transformation, adopting the same terminology as for Courant automorphisms of the generalized tangent bundle. Clearly |$e^0 = \textrm{id}$|⁠, |$e^{B_1} \circ e^{B_2} = e^{B_1 + B_2}$|⁠, and |$(e^B)^{-1} = e^{-B}$|⁠, for all closed Atiyah |$2$|-forms |$B, B_1, B_2$|⁠, showing that |$B$|-field transformations form a(n abelian) subgroup of |$\textrm{Aut}_{CJ} (\mathbb{D} L)$| isomorphic to |$Z^2_L$|⁠, the group of |$2$|-cocycles in |$(\Omega ^{\bullet }_L, d_L)$|⁠. Example 2.1.3. Let |$\Phi : L \to L$| be a vector bundle automorphism covering a diffeomorphism |$\phi : M \to M$|⁠. Define a vector bundle automorphism |$\mathbb{D} \Phi : \mathbb{D} L \to \mathbb{D} L$| via \begin{equation*} \mathbb{D} \Phi (\Delta, \psi):= (\Phi_\ast (\Delta), (\Phi^{-1})^\ast \psi), \quad (\Delta, \psi) \in \mathbb{D} L. \end{equation*} It is easy to see that |$(\mathbb{D} \Phi , \Phi )$| is a Courant-Jacobi automorphism. Additionally |$\mathbb{D} \textrm{id} = \textrm{id}$|⁠, |$\mathbb{D} \Phi _1 \circ \mathbb{D} \Phi _2 = \mathbb{D} (\Phi _1 \circ \Phi _2)$|⁠, and |$(\mathbb{D} \Phi )^{-1} = \mathbb{D} (\Phi ^{-1})$|⁠, for all |$\Phi , \Phi _1, \Phi _2 \in \textrm{Aut} (L)$|⁠, showing that Courant-Jacobi automorphisms of the form |$\mathbb{D} \Phi$| form a subgroup of |$\textrm{Aut}_{CJ} \mathbb{D} L$| isomorphic to |$\textrm{Aut} (L)$|⁠. Finally, let |$B \in Z^2_L$| and |$\Phi \in \textrm{Aut}(L)$|⁠. Then \begin{equation*} e^{ B} \circ \mathbb{D} \Phi = \mathbb{D} \Phi\circ e^{\Phi^\ast B}. \end{equation*} In particular we see that |$B$|-field transformations and automorphisms of |$L$| generate a subgroup in |$\textrm{Aut}_{CJ} (\mathbb{D} L)$| isomorphic to \begin{equation*} Z^2_L \rtimes \textrm{Aut} (L), \end{equation*} where |$\textrm{Aut} (L)$| acts on |$Z^2_L$| (from the right) via pull-backs. Actually, exactly as for the generalized tangent bundle, |$B$|-field transformations and automorphisms of |$L$| generate the full group of Courant-Jacobi automorphisms, according to the following proposition which we report here for completeness. Proposition 2.1.4. Let |$L \to M$| be a line bundle. Then \begin{equation*} \textrm{Aut}_{CJ}(\mathbb{D} L) \cong Z^2_L \rtimes \textrm{Aut} (L). \end{equation*} Proof. The proof follows exactly the same lines as in standard Dirac geometry (see, e.g., [20, Proposition 2.5]) and we omit it. We now pass to infinitesimal symmetries of |$\mathbb{D} L$|⁠. First of all, for a vector bundle |$E \to M$|⁠, denote by |$\mathfrak{aut} (E)$| the Lie algebra of its infinitesimal automorphisms. As already remarked, |$\mathfrak{aut} (E)$| is canonically isomorphic to the Lie algebra |$\Gamma (DE)$| of derivations |$E$|⁠. Definition 2.1.5. An infinitesimal Courant-Jacobi automorphism of |$\mathbb{D} L$| is a pair (, |$\Delta )$| consisting of a derivation of |$\mathbb{D} L$| and a derivation |$\Delta$| of |$L$|⁠, such that and |$\Delta$| have the same symbol, and, additionally \begin{equation*} \end{equation*} for all |$\alpha , \beta \in \Gamma (\mathbb{D} L)$|⁠. Equivalently (, |$\Delta )$| generates a flow by Courant-Jacobi automorphisms of |$\mathbb{D} L$|⁠. The Lie algebra of infinitesimal Courant-Jacobi automorphisms is denoted by |$\mathfrak{aut}_{CJ} (\mathbb{D} L)$|⁠. Example 2.1.6. Let |$B$| be a closed Atiyah |$2$|-form, that is, |$B \in Z^2_L$|⁠. Denote by |$\overline B$| the endomorphism of |$\mathbb{D} L$| given by \begin{equation} \overline B:= \left( \begin{array}{cc} 0 & 0 \\ B_\flat & 0 \end{array} \right). \end{equation} (2.4) Then |$(\overline B, 0)$| is an infinitesimal Courant-Jacobi automorphism, exponentiating to the |$B$|-field transformation corresponding to |$B$|⁠. Example 2.1.7. Let |$\square$| be a derivation of |$L$|⁠. Define a derivation |$\mathcal L_\square$| of |$\mathbb{D} L$| via \begin{equation*} \mathcal L_\square (\Delta, \psi):= ([\square, \Delta], \mathcal L_\square \psi), \quad (\Delta, \psi) \in \Gamma (\mathbb{D} L). \end{equation*} It is easy to see that |$(\mathcal L_\square , \square )$| is an infinitesimal Courant-Jacobi automorphism. Infinitesimal automorphisms of the form |$(\mathcal L_\square , \square )$| together with those of the form |$(\overline B, 0)$| from the previous example generate a Lie subalgebra in |$\mathfrak{aut}_{CJ} (\mathbb{D} L)$| isomorphic to \begin{equation*} \mathfrak{aut} (L) \ltimes Z^2_L \end{equation*} in the obvious way. Here |$\mathfrak{aut} (L)$| acts on |$Z^2_L$| via Lie derivatives. Proposition 2.1.8. Let |$L \to M$| be a line bundle. Then \begin{equation*} \mathfrak{aut}_{CJ}(\mathbb{D} L) \cong \mathfrak{aut} (L) \ltimes Z^2_L \end{equation*} Proof. The proof is similar to that of Proposition 2.1.4, and it is left to the reader. Remark 2.1.9. Let |$(B, \square ) \in \mathfrak{aut} (L) \ltimes Z^2_L$|⁠, and let |$(\mathcal L_\square + \overline B, \square )$| be the corresponding infinitesimal Courant-Jacobi automorphism. If |$\square$| generates the flow |$\{ \Phi _t \}$| by vector bundle automorphism of |$L$|⁠, then |$(\mathcal L_\square + \overline B, \square )$| generates the flow \begin{equation*} \{ (e^{C_t} \circ \mathbb{D} \Phi_t, \Phi_t) \} \end{equation*} by Courant-Jacobi automorphisms corresponding to |$\{(C_t, \Phi _t)\} \subset Z^2_L \rtimes \textrm{Aut} (L),$| where \begin{equation*} C_t:= - \int_0^t (\Phi_{-\epsilon}^\ast B) d \epsilon. \end{equation*} 2.2 Generalized contact and Dirac–Jacobi bundles 2.2.1 Generalized contact bundles A generalized contact bundle [38] is a line bundle |$L \to M$| equipped with a generalized contact structure, that is, and endomorphism |$\mathbb{K}: \mathbb{D} L \to \mathbb{D} L$| of the omni-Lie algebroid such that |$\triangleright$||$\mathbb{K}$| is almost complex, that is, |$\mathbb{K}^2 = - \textrm{id}$|⁠, |$\triangleright$||$\mathbb{K}$| is skew symmetric, that is, |$\langle\!\!\langle \mathbb{K} \alpha , \beta \rangle\!\!\rangle + \langle\!\!\langle \alpha , \mathbb{K} \beta \rangle\!\!\rangle = 0$| for all |$\alpha , \beta \in \mathbb{D} L$|⁠, and |$\triangleright$||$\mathbb{K}$| is integrable, that is, |$[ \! [ \mathbb{K} \alpha , \mathbb{K} \beta ] \! ] - [ \! [ \alpha , \beta ] \! ] - \mathbb{K} [ \! [ \mathbb{K} \alpha , \beta ] \! ] - \mathbb{K} [ \! [ \alpha , \mathbb{K} \beta ] \! ] = 0$| for all |$\alpha , \beta \in \Gamma (\mathbb{D} L)$|⁠. Let |$(L \to M, \mathbb{K})$| be a generalized contact bundle. Then |$M$| is odd dimensional. Actually, generalized contact bundles are odd-dimensional analogues of generalized complex manifolds, and they encompass contact manifolds and complex structures on the gauge algebroid of |$L$| as extreme cases. To see this, use the direct sum decomposition |$\mathbb{D} L = D L \oplus J^1 L$| to present |$\mathbb{K}$| in the form \begin{equation} {{\mathbb{K}}} = \left( \begin{array}{cc} \varphi & J^\sharp \\ \omega_\flat & -\varphi^\dagger \end{array} \right). \end{equation} (2.5) Then |$\triangleright$||$\varphi : D L \to D L$| is a vector bundle endomorphism, |$\triangleright$||$\varphi ^\dagger : J^1 L \to J^1 L$| is its adjoint, that is, |$\langle \varphi ^\dagger \psi , \Delta \rangle = \langle \psi , \varphi \Delta \rangle$|⁠, |$(\Delta , \psi ) \in \mathbb{D} L$|⁠, |$\triangleright$||$J: \wedge ^2 J^1 L \to L$| is a |$2$|-form with sharp map |$J^\sharp : J^1 L \to D L$|⁠, and |$\triangleright$||$\omega : \wedge ^2 D L \to L$| is an Atiyah |$2$|-form with flat map |$\omega _\flat : D L \to J^1 L$|⁠. Additionally, |$\varphi , J, \omega$| satisfy some identities [38]. In particular, |$J$| is a Jacobi bracket, so that |$(L, J)$| is a Jacobi bundle [29]. When |$\varphi = 0$|⁠, then |$\omega _\flat ^{-1} = - J^\sharp$| and |$J$| is the Jacobi bracket of a (unique) contact structure |$H \subset M$| such that |$TM/H = L$| (see Example 1.2.1). When |$J = \omega = 0$|⁠, then |$\varphi$| is an integrable complex structure on the gauge algebroid |$DL$| (see Appendix A). Remark 2.2.1. Let |$\mathbb{K}$| be a generalized contact structure on |$L$|⁠, and let (, |$\Phi )$| be a Courant-Jacobi automorphism of |$\mathbb{D} L$|⁠. Then |$\circ \mathbb{K} \circ$||$^{-1}$| is a generalized contact structure as well. In particular, for (, |$\Phi ) = (e^B, \textrm{id})$|⁠, the |$B$|-field transformation corresponding to a closed Atiyah |$2$|-form |$B$|⁠, we obtain that |$e^B \circ \mathbb{K} \circ e^{-B}$| is a generalized contact structure. The latter will be denoted by |$\mathbb{K}^{B}$|⁠. 2.2.2 Dirac–Jacobi bundles Similarly as for generalized complex structures, generalized contact structures can be seen as (particularly nice) complex Dirac–Jacobi structures, that is, complex Dirac structures in the omni-Lie algebroid. A Dirac–Jacobi structure on |$L$| [37] (see also [12, 13]) is a vector subbundle |$\mathfrak L \subset \mathbb{D} L$| such that |$\triangleright$||$\mathfrak L$| is maximally isotropic with respect to |$\langle\!\!\langle -, - \rangle\!\!\rangle$|⁠; |$\triangleright$||$\mathfrak L$| is involutive, that is, |$[ \! [ \Gamma (\mathfrak L), \Gamma (\mathfrak L) ] \! ] \subset \Gamma (\mathfrak L)$|⁠. Remark 2.2.2. Recall that there is an alternative formulation of (standard) Dirac structures (in particular, generalized complex structures) via the Clifford algebra of the generalized tangent bundle and pure spinors. There is a similar formulation for Dirac–Jacobi bundles, but it is much more involved. Actually, it exploits the technology of quadratic modules and their (even) Clifford algebras (see [6], see also [4] and references therein). We mean to develop this line of thoughts in a future project. Example 2.2.3. |$\triangleright$| Let |$L \to M$| be a line bundle and let |$J: \wedge ^2 J^1 L \to L$| be a bi-differential operator on |$\Gamma (L)$|⁠. Then |$\operatorname{\textsf{graph}} J:= \{(J^\sharp \psi , \psi ): \psi \in J^1 L \}\subset \mathbb{D} L$| is a maximally isotropic subbundle, and it is a Dirac–Jacobi structure if and only if |$J$| is a Jacobi structure. |$\triangleright$| Let |$L \to M$| be a line bundle, and let |$\omega \in \Omega ^2_L$| be an Atiyah |$2$|-form on |$L$|⁠. Then |$\operatorname{\textsf{graph}} \omega := \{(\Delta , \iota _\Delta \omega ): \Delta \in DL \}\subset \mathbb{D} L$| is a maximally isotropic subbundle, and it is a Dirac–Jacobi structure if and only if |$d_D \omega = 0$|⁠. Now, let |$\mathbb{K}$| be a generalized contact structure on |$L$|⁠. Consider the complexified omni-Lie algebroid |$\mathbb{D} L \otimes \mathbb{C}$|⁠, and let |$\mathfrak L_{\mathbb{K}} \subset \mathbb{D}L \otimes \mathbb{C}$| be the |$+\textrm{i}$|-eigenbundle of |$\mathbb{K}$|⁠. Then |$\mathfrak L_{\mathbb{K}}$| is a (complex) Dirac–Jacobi structure such that |$\mathfrak L_{\mathbb{K}} \cap \overline{\mathfrak L}_{\mathbb{K}} = 0$|⁠, in particular |$\mathbb{D} L \otimes \mathbb{C}= \mathfrak L_{\mathbb{K}} \oplus \overline{\mathfrak L}_{\mathbb{K}}$|⁠. Additionally, |$\mathfrak L_{\mathbb{K}}$| contains the full information about |$\mathbb{K}$|⁠. Finally, all Dirac–Jacobi structures |$\mathfrak L \subset \mathbb{D} L \otimes \mathbb{C}$| such that |$\mathfrak L \cap \overline{\mathfrak L} = 0$| arise in this way, and we will call them complex Dirac–Jacobi structures of generalized contact type. Remark 2.2.4. Let |$\mathfrak L$| be a Dirac–Jacobi structure on |$L$|⁠, and let (, |$\Phi )$| be a Courant–Jacobi automorphism of |$\mathbb{D} L$|⁠. Then |$(\mathfrak L)$| is a Dirac–Jacobi structure as well. In particular, |$e^B (\mathfrak L)$| is a Dirac–Jacobi structure, denoted by |$\mathfrak L^B$|⁠, for every closed Atiyah |$2$|-form |$B$|⁠. If |$\mathfrak L = \mathfrak L_{\mathbb{K}}$| is the |$+\textrm{i}$|-eigenbundle of a generalized contact structure |$\mathbb{K}$|⁠, then |$e^B (\mathfrak L) = \mathfrak L_{\mathbb{K}}^B =\mathfrak L_{\mathbb{K}^{B}}$|⁠. We stress that, in general, |$\mathbb{K}^B = e^B \circ \mathbb{K} \circ e^{-B}$| is not an honest generalized contact structure, unless |$B$| is a real Atiyah form (see, e.g., Remark 4.1.1). Lemma 2.2.5. Let |$\mathbb{K}$| be a generalized contact structure as in (2.5), and let |$\mathfrak L = \mathfrak L_{\mathbb{K}} \subset \mathbb{D}L \otimes \mathbb{C}$| be its |$+\textrm{i}$|-eigenbundle. Then \begin{equation*} p_D \mathfrak L \cap p_D \overline{\mathfrak L} = \operatorname{im} J^\sharp \otimes \mathbb{C}. \end{equation*} Proof. The proof follows the same lines as in standard generalized complex geometry [20, Proposition 3.24 and Corollary 3.25]. 2.2.3 The |$2$|-form of a complex Dirac–Jacobi structure Let |$L \to M$| be a line bundle, and let |$\mathfrak L \subset \mathbb{D} L\otimes \mathbb{C}$| be a complex Dirac–Jacobi structure on |$L$|⁠. There is a canonical skew symmetric, |$L \otimes \mathbb{C}$|-valued bilinear map |$\varpi$| defined pointwise on the smooth, but not necessarily regular, subbundle |$p_D \mathfrak L$| as follows: \begin{equation} \varpi: \wedge^2 p_D \mathfrak L \to L \otimes \mathbb{C}, \quad (\Delta, \nabla) \mapsto \langle \psi, \nabla \rangle, \end{equation} (2.6) here |$\psi \in J^1 L \otimes \mathbb{C}$| is any |$1$|-jet such that |$(\Delta , \psi ) \in \mathfrak L$|⁠. It immediately follows from the definition of |$\varpi$| that \begin{equation} \mathfrak L = \left\{ (\Delta, \psi) \in \mathbb{D} L \otimes \mathbb{C}: \langle \psi, \nabla \rangle = \varpi (\Delta, \nabla) \textrm{ for all} \nabla \in p_D \mathfrak L \right\}. \end{equation} (2.7) Similarly as in generalized complex geometry [1], when |$\mathfrak L$| is of generalized contact type, we can relate |$\varpi$| to the corresponding generalized contact structure |$\mathbb{K}$|⁠. First consider the complex conjugate form \begin{equation*} \overline \varpi: \wedge^2 p_D \overline{\mathfrak L} \to L \otimes \mathbb{C}, \quad (\Delta, \nabla) \mapsto \overline{\varpi (\overline \Delta, \overline \nabla)}. \end{equation*} The real and imaginary parts of |$\varpi$|⁠: \begin{equation*} \operatorname{Re} \varpi:= \frac{1}{2}(\varpi + \overline \varpi) \quad \textrm{and} \quad \operatorname{Im} \varpi:= \frac{1}{2\textrm{i}}(\varpi - \overline \varpi) \end{equation*} are only defined on the intersection |$p_D \mathfrak L \cap p_D \overline{\mathfrak L} = \operatorname{im} J^\sharp \otimes \mathbb{C}$|⁠, and we have the following. Lemma 2.2.6. Let |$\mathbb{K}$| be a generalized contact structure as in (2.5), let |$\mathfrak L = \mathfrak L_{\mathbb{K}}$| be its |$+\textrm{i}$|-eigenbundle, and let |$\varpi$| be the canonical |$2$|-form on |$p_D \mathfrak L$|⁠. Then \begin{equation} - J ( \psi, \psi^{\prime}) = \operatorname{Im} \varpi (J^\sharp \psi, J^\sharp \psi^{\prime}) \end{equation} (2.8) for all |$\psi , \psi ^{\prime} \in J^1 L \otimes \mathbb{C}$|⁠. Proof. The proof follows the same lines as in generalized complex geometry (see, e.g. [1, Lemmas 3.1, 3.2, and 3.3]). Remark 2.2.7. As recalled in Example 2.2.3, every Jacobi structure |$J$| on a line bundle |$L \to M$| determines a real Dirac–Jacobi structure: |$\mathfrak L_J:= \operatorname{\textsf{graph}} J \subset DL \oplus J^1 L = \mathbb{D} L$|⁠, and we have |$p_D \mathfrak L_J = \operatorname{im} J^\sharp$|⁠. In turn, for every real Dirac–Jacobi structure |$\mathfrak L$| on |$L \to M$|⁠, there is a canonical |$L$|-valued |$2$|-form |$\wedge ^2 p_D \mathfrak L \to L$|⁠, defined by the same formula (2.6) as |$\varpi$|⁠. Formula (2.8) then states that the imaginary part of |$\varpi$| agrees with the (complexification of the) |$2$|-form |$\omega _J: \wedge ^2 \operatorname{im} J^\sharp \to L$| induced by the Jacobi structure |$J$| underlying the generalized contact structure |$\mathbb{K}$|⁠: \begin{equation*} \operatorname{Im} \varpi = \omega_J. \end{equation*} Notice that |$\omega _J$|⁠, hence |$\operatorname{Im} \varpi\!$|⁠, is (pointwise) nondegenerate. 2.2.4 Backward images of (complex) Dirac–Jacobi structures Let |$(L \to M, \mathbb{K})$| be a generalized contact bundle, and let |$\mathfrak L = \mathfrak L_{\mathbb{K}} \subset \mathbb{D} L\otimes \mathbb{C}$| be its |$+\textrm{i}$|-eigenbundle. Like in generalized complex geometry, not all submanifolds of |$M$| inherits from |$\mathbb{K}$| a generalized contact bundle structure. However, all submanifolds of |$M$| inherit from |$\mathfrak L$| a complex Dirac–Jacobi structure (up to regularity issues), via the backward image construction that we now recall for later use. We describe backward images for real Dirac–Jacobi structures. The following considerations extend straightforwardly to complex Dirac–Jacobi structures. So let |$L \to M$| be a line bundle equipped with a Dirac–Jacobi structure |$\mathfrak L \subset \mathbb{D} L$|⁠. Consider another line bundle |$L_N \to N$| together with a regular vector bundle map |$\Phi : L_N \to L$| covering a smooth map |$\phi : N \to M$|⁠. We recall from Section 1.1 that, by a regular vector bundle map, we mean a fiber-wise invertible vector bundle map. Now, define a subbundle |$\Phi ^! \mathfrak L \subset \mathbb{D} L_N$| via \begin{equation*} \Phi^! \mathfrak L:= \left\{(\Delta, \Phi^\ast \psi) \in \mathbb{D} L_N: (\Phi_\ast \Delta, \psi) \in \mathfrak L \right\}. \end{equation*} The bundle |$\Phi ^! \mathfrak L$| is always maximal isotropic; hence, it has constant rank, but it needs not to be smooth. Nonetheless, if it is smooth, it is an honest vector subbundle and a Dirac–Jacobi structure on |$L_N$|⁠, called the backward image of |$\mathfrak L$| along |$\Phi$|⁠. There is a simple sufficient condition for smoothness, sometimes referred to as the clean intersection condition [10, 37]. For the purposes of this paper, we only need to know the following: |$\triangleright$| if |$\phi$| is a submersion, the clean intersection condition is automatically satisfied; hence, |$\Phi ^!\! \mathfrak L$| is a Dirac–Jacobi structure; |$\triangleright$| if |$\phi$| is the immersion of a(n immersed) submanifold |$S \hookrightarrow M$|⁠, the clean intersection condition boils down to \begin{equation} \operatorname{rank} \left(p_D \mathfrak L|_S + D (L|_S)\right) = \textrm{constant}. \end{equation} (2.9) We refer to [10, 37] for more details. 2.2.5 Products of Dirac–Jacobi structures Let |$(M_1, \mathfrak L_1)$| and |$(M_2, \mathfrak L_2)$| be manifolds equipped with (standard) Dirac structures, that is, maximally isotropic, and involutive subbundles |$\mathfrak L_i$| of the generalized tangent bundles|$\mathbb{T} M_i = TM_i \oplus T^\ast M_i, i =1,2$|⁠. Then |$\mathfrak L_1 \times \mathfrak L_2 \subset \mathbb{T} M_1 \times \mathbb{T} M_2 = \mathbb{T} (M_1 \times M_2)$| is a Dirac structure on the product |$M_1 \times M_2$|⁠, called the product of |$\mathfrak L_1$| and |$\mathfrak L_2$|. It is not immediately obvious how to extend this simple construction to line bundles and Dirac–Jacobi structures. In this section we propose such an extension. The splitting theorems of Section 4 will be formulated in terms of the product of Dirac–Jacobi structures as defined here. Begin with two Dirac–Jacobi structures |$\mathfrak L_1, \mathfrak L_2 \subset \mathbb{D} L$| on the same line bundle |$L \to M$|⁠. Let |$\mathfrak L_1 \star \mathfrak L_2 \subset \mathbb{D} L$| be the (not necessarily regular) subbundle defined by \begin{equation} \mathfrak L_1 \star \mathfrak L_2:= \left\{ (\Delta, \psi_1 + \psi_2): (\Delta, \psi_i) \in \mathfrak L_i, \, i = 1,2 \right\}. \end{equation} (2.10) A similar construction for Dirac structures appeared probably in [20] for the 1st time (see also [3]). Lemma 2.2.8. If |$\mathfrak L_1 \star \mathfrak L_2 \subset \mathbb{D} L$| is a smooth subbundle, then it is a Dirac–Jacobi structure. Proof. The lemma can be proven, for example, by adapting the analogous proof for Dirac structures in [28] in the obvious way. Remark 2.2.9. Actually, |$B$|-field transformations are special instances of the construction (2.10). Namely, let |$B$| be a closed Atiyah |$2$|-form. Then \begin{equation*} \mathfrak L_B = \{ (\Delta,\iota_\Delta B) : \Delta\in DL\} \subset \mathbb{D} L \end{equation*} is a Dirac–Jacobi structure, and, for every other Dirac–Jacobi structure |$\mathfrak L$|⁠, we have \begin{equation*} \mathfrak L \star \mathfrak L_B = \mathfrak L^B. \end{equation*} In particular, the graph |$DL \subset \mathbb{D} L$| of the null Atiyah |$2$|-form acts as an identity with respect to the product |$\star$|⁠. We are now ready to define a notion of product of Dirac–Jacobi structures. So let |$L_i \to M_i$| be line bundles equipped with Dirac–Jacobi structures |$\mathfrak L_i \subset \mathbb{D} L_i$|⁠, |$i= 1,2$|⁠. We assume we have an additional datum, namely a line bundle |$L \to M_1 \times M_2$| over the product, together with regular vector bundle maps |$P_i: L \to L_i$| covering the canonical projections |$p_i: M_1 \times M_2 \to M_i$|⁠, |$i = 1,2$|⁠: $$ $$ (2.11) In this situation we can consider backward images, |$P_1^! \mathfrak L_1, P_2^! \mathfrak L_2 \subset \mathbb{D} L$|⁠, and they are regular because the |$p_i$| are submersions, |$i = 1, 2$|⁠. Finally, consider \begin{equation*} P_1^! \mathfrak L_1 \star P_2^! \mathfrak L_2. \end{equation*} If it is regular, it is a Dirac–Jacobi structure on |$L$|⁠. Now notice that, in view of diagram (2.11), |$L$| comes with (partial) connections |$D_i$|⁠, along |$\ker p_i$|⁠, |$i = 1, 2$|⁠, and we can define a genuine connection |$D^\times$| in |$L$|⁠, by putting \begin{equation*} D^\times_{X_1 + X_2} \lambda = (D_1)_{X_1} \lambda + (D_2)_{X_2}\lambda, \quad X_i \in \ker p_i,\quad i = 1,2. \end{equation*} Lemma 2.2.10. The following conditions are equivalent: The connection |$D^\times$| is flat. Around every point of |$M_1 \times M_2$| there is a nowhere vanishing local section |$\lambda \in \Gamma (L)$| such that |$\lambda = P_1^\ast \lambda _1 = P_2^\ast \lambda _2$| for some local sections |$\lambda _i \in \Gamma (L_i)$|⁠, |$i = 1,2$|⁠. For every |$(\bar x_1, \bar x_2) \in M_1 \times M_2$|⁠, there are local trivializations |$L_i \cong \mathbb{R}_{M_i}$|⁠, around |$\bar x_i$|⁠, |$i = 1,2$|⁠, and |$L \cong \mathbb{R}_{M_1 \times M_2}$|⁠, around |$(\bar x_1, \bar x_2)$|⁠, such that the |$P_i: L \to L_i$| identify with the projections |$\mathbb{R}_{M_1 \times M_2} \to \mathbb{R}_{M_i}$|⁠, |$(x_1, x_2; r) \mapsto (x_i ; r)$|⁠, where |$(x_1, x_2) \in M_1 \times M_2$|⁠, and |$r \in \mathbb{R}$|⁠. Proof. (1) |$\Longrightarrow$| (2). Choose as |$\lambda$| a nowhere vanishing flat section with respect to |$D^\times$|⁠. (2) |$\Longrightarrow$| (3). Choose the (local) trivializations |$L \cong \mathbb{R}_{M_1 \times M_2}$|⁠, and |$L_i \cong \mathbb{R}_{M_i}$|⁠, that identify |$\lambda$|⁠, and |$\lambda _i$|⁠, with the constant functions |$1$|⁠, |$i = 1,2$|⁠. (3) |$\Longrightarrow$| (1). Obvious. When one, hence all three, of the conditions in Lemma 2.2.10 hold, we say that the product (2.11) is flat. If, additionally, |$P_1^! \mathfrak L_1 \star P_2^! \mathfrak L_2$| is regular, we call it the (flat) product of |$\mathfrak L_1$| and |$\mathfrak L_2$| (with respect to |$P_1, P_2$|⁠) and denote it by \begin{equation*} \mathfrak L_1 \times^! \mathfrak L_2. \end{equation*} We will provide examples of products of Dirac–Jacobi structures later on. For now we only remark that, if we apply an analogous construction to a pair of Dirac structures, we get exactly their standard product. Remark 2.2.11. The above discussion applies to complex Dirac–Jacobi structures without modifications. Remark 2.2.12. Let |$\mathfrak L_i$| be Dirac–Jacobi structures on the line bundles |$L_i \to M_i$|⁠, |$i = 1,2$|⁠, and let |$\mathfrak L_1 \times ^! \mathfrak L_2$| be a flat product of them with respect to some projections |$P_1, P_2$| as in diagram (2.11). Finally, let |$B$| be a closed Atiyah |$2$|-form on |$L_1$|⁠. It is easy to see that \begin{equation*} (\mathfrak L_1 \times^! \mathfrak L_2)^{P_1^\ast B} = \mathfrak L_1^B \times^! \mathfrak L_2. \end{equation*} Remark 2.2.13. By now, it should be clear that, when working with Dirac–Jacobi structures, we are working in the category of line bundles and regular vector bundle maps between them. So, no surprise that the appropriate notion of product in this setting includes a line bundle on a product manifold, and regular vector bundle maps onto the factors. Nonetheless, in what follows, as we are only interested in local properties, we will use Lemma 2.2.10, and we will mainly consider the case when the line bundles |$L_i \to M_i$| and |$L \to M_1 \times M_2$| are trivial and the projections |$P_i: L \to L_i$| are the obvious ones, |$i = 1,2$|⁠. 2.2.6 Homogeneous generalized complex structures As already mentioned, unlike Poisson manifolds, manifolds |$M$| with a Jacobi bundle |$(L \to M, J)$| possess two kinds of characteristic leaves. Odd-dimensional ones inherit from |$J$| a canonical contact structure, and we call them contact leaves. Even-dimensional leaves inherit from |$J$| an lcs structure, and we call them lcs leaves. Let |$\mathcal O$| be a leaf and |$x_0 \in \mathcal O$|⁠. By a transversal to |$\mathcal O$| at |$x_0$|⁠, we mean a submanifold |$N$| such that |$x_0 \in N$|⁠, and |$T_{x_0} M = T_{x_0} N \oplus T_{x_0} \mathcal O$|⁠. It turns out that transversals to lcs leaves, with the restricted line bundle, possess a canonical Jacobi structure around |$x_0$|⁠. Additionally, this Jacobi structure vanishes at |$x_0$|⁠. On the other hand, transversals to contact leaves possess a canonical homogeneous Poisson structure (up to the choice of a nowhere vanishing section of |$L$|⁠) around |$x_0$|⁠. The homogeneous Poisson structure vanishes at |$x_0$|⁠. Recall that a homogeneous Poisson structure on a manifold |$M$| is a pair |$(\pi , Z)$| where |$\pi$| is a Poisson bi-vector, and |$Z$| is a vector field, called the homogeneity vector field, such that |$\mathcal L_Z \pi = - \pi$|⁠. Example 2.2.14. On |$\mathbb{R}^{2d}$|⁠, with coordinates |$(x^1, \ldots , x^d, p_1, \ldots , p_d)$|⁠, there is a canonical homogeneous Poisson structure |$(\pi _{\textit{can}}, Z_{\textit{can}})$| given by \begin{equation*} \pi_{\textit{can}} = \frac{\partial}{\partial p_i} \wedge \frac{\partial}{\partial x^i}, \quad \textrm{and} \quad Z_{\textit{can}} = p_i \frac{\partial}{\partial p_i}. \end{equation*} The theory of Jacobi structures is strongly related to that of homogeneous Poisson structures, as the example of transversals to contact leaves shows (see also [14]). In a similar way generalized contact geometry is strongly related to the theory of homogeneous generalized complex structures that we define now. Let |$M$| be a manifold. Definition 2.2.15. A homogeneous generalized complex structure on |$M$| is a pair |$(\mathbb{J}, \mathbb{Z})$|⁠, where \begin{equation*} \mathbb{J} = \left( \begin{array}{cc} A & \pi^\sharp \\ \sigma_\flat & -A^\ast \end{array} \right) \in \textrm{End} (\mathbb{T} M) \end{equation*} is a generalized complex structure, and |$\mathbb{Z} = (Z, \zeta )$| is a section of the generalized tangent bundle |$\mathbb{T}M$| such that |$\triangleright$||$\mathcal L_Z A = \pi ^\sharp \circ (d \zeta )_\flat$|⁠, |$\triangleright$||$\mathcal L_Z \pi = - \pi$| (in particular |$(\pi , Z)$| is a homogeneous Poisson structure), |$\triangleright$||$\mathcal L_Z \sigma = \sigma - \iota _A d \zeta$|⁠, where |$\iota _A d \zeta$| is the |$2$|-form defined by \begin{equation*} (\iota_A d \zeta)(X, Y) = d \zeta (A X, Y) + d \zeta (X, A Y), \end{equation*} for all |$X, Y \in \mathfrak X (M)$|⁠. Remark 2.2.16. The main motivation for this definition is that the transversal to a contact leaf in the base of a generalized contact bundle is a homogeneous generalized complex manifold, as we will show in Section 3.1. Another motivation is that a generalized contact structure on a line bundle |$L \to M$| is equivalent to a homogeneous generalized complex structure on the frame bundle |$\widetilde M = L^\ast \smallsetminus 0$| of |$L$|⁠. In that case, the section |$\mathbb{Z}$| is of the special form |$(\mathcal E, 0)$|⁠, where |$\mathcal E$| is the Euler vector field on |$\widetilde M$| [38, Remark 3.6 in the arXiv version]. The 2nd motivation suggests the following problem: let |$(\mathbb{J}, \mathbb{Z})$| be a homogeneous generalized complex structure on |$M$| with |$\mathbb{Z} = (Z, \zeta )$|⁠. Is it possible to find a closed |$2$|-form |$B$| on |$M$|⁠, equivalently, a |$B$|-field transformation, such that |$(\mathbb{J}^B, (Z, 0))$| is a homogeneous generalized complex structure? Answering this question goes beyond the scope of the present paper. Definition 2.2.15 can be rephrased in terms of the complex Dirac structure associated to |$\mathbb{J}$|⁠, that is, the |$+\textrm{i}$|-eigenbundle |$\mathfrak L_{\mathbb{J}}$| of |$\mathbb{J}$| in the complexified generalized tangent bundle |$\mathbb{T} M \otimes \mathbb{C}$|⁠. Namely, we have the following. Proposition 2.2.17. Let \begin{equation*} \mathbb{J} = \left( \begin{array}{cc} A & \pi^\sharp \\ \sigma_\flat & -A^\ast \end{array} \right) \end{equation*} be a generalized complex structure on |$M$|⁠, and let |$\mathbb{Z} = (Z, \zeta )$| be a section of the generalized tangent bundle |$\mathbb{T} M$|⁠. Then the following conditions are equivalent: |$(\mathbb{J}, \mathbb{Z})$| is a homogeneous generalized complex structure; |$([Z, X], \mathcal L_Z \eta - \eta + \iota _X d \zeta ) \in \Gamma (\mathfrak L_{\mathbb{J}})$| for all |$(X, \eta ) \in \Gamma (\mathfrak L_{\mathbb{J}})$|⁠; |$([Z, X] + X, \mathcal L_Z \eta + \iota _X d \zeta ) \in \Gamma (\mathfrak L_{\mathbb{J}})$| for all |$(X, \eta ) \in \Gamma (\mathfrak L_{\mathbb{J}})$|⁠. Proof. It is clear that (2) and (3) are equivalent. It remains to prove that (1) |$\Leftrightarrow$| (3). Assume first that |$(\mathbb{J}, \mathbb{Z})$| is a homogeneous generalized complex structure; let |$\alpha = (X, \eta ) \in \Gamma (\mathfrak L_{\mathbb{J}})$| and compute \begin{equation} \begin{aligned} \mathbb{J} \left( \begin{array}{c} [Z, X] + X \\ \mathcal L_Z \eta + \iota_X d \zeta \end{array} \right) & = \left( \begin{array}{cc} A & \pi^\sharp \\ \sigma_\flat & -A^\ast \end{array} \right) \left( \begin{array}{c} [Z, X] + X \\ \mathcal L_Z \eta + \iota_X d \zeta \end{array} \right) \\ & = \left( \begin{array}{c} A[Z, X] + AX + \pi^\sharp \mathcal L_Z \eta + \pi^\sharp \iota_X d \zeta \\ \sigma_\flat [Z, X] + \sigma_\flat X - A^\ast \mathcal L_Z \eta - A^\ast \iota_X d \zeta \end{array} \right). \end{aligned} \end{equation} (2.12) The 1st entry is \begin{equation*} \begin{aligned} & A[Z, X] + AX + \pi^\sharp \mathcal L_Z \eta + \pi^\sharp \iota_X d \zeta \\ & = [Z, AX] - (\mathcal L_Z A)X + AX + [Z, \pi^\sharp \eta] - (\mathcal L_Z \pi)^\sharp \eta + (\pi^\sharp \circ (d \zeta)_\flat) X \\ & = [Z, AX + \pi^\sharp \eta] + AX + \pi^\sharp \eta \\ & = \textrm{i} ([Z, X] + X), \end{aligned} \end{equation*} where we used that |$AX + \pi ^\sharp \eta$| is the 1st entry of |$\mathbb{J} \alpha$|⁠. Similarly, the 2nd entry in (2.12) is \begin{equation*} \begin{aligned} & \sigma_\flat [Z, X] + \sigma_\flat X - A^\ast \mathcal L_Z \eta - A^\ast \iota_X d \zeta \\ & = \mathcal L_Z (\sigma_\flat X) - (\mathcal L_Z \sigma)_\flat X + \sigma_\flat X - \mathcal L_Z (A^\ast \eta) + (\mathcal L_Z A)^\ast \eta - (A^\ast \circ (d \zeta)_\flat) X \\ & = \mathcal L_Z (\sigma_\flat X-A^\ast \eta) + (d \zeta)_\flat ( A X + \pi^\sharp \eta) \\ & = \textrm{i} (\mathcal L_Z \eta + \iota_X d \zeta), \end{aligned} \end{equation*} showing that |$([Z, X] + X, \mathcal L_Z \eta + \iota _X d \zeta )$| is a |$+\textrm{i}$|-eigensection of |$\mathbb{J}$|⁠. Conversely, let |$([Z, X] + X, \mathcal L_Z \eta + \iota _X d \zeta )$| be a |$+\textrm{i}$|-eigensection of |$\mathbb{J}$| for all |$+\textrm{i}$|-eigensections |$\alpha = (X, \eta )$|⁠. One can show that |$(\mathbb{J}, \mathbb{Z})$| is a homogeneous generalized complex structure with a similar computation as above (but in the reverse order). Every homogeneous generalized complex structure |$(\mathbb{J}, \mathbb{Z})$| determines a complex Dirac–Jacobi structure on the trivial line bundle |$\mathbb{R}_M:= M \times \mathbb{R} \to M$| according to the following. Proposition 2.2.18. Let |$(\mathbb{J}, \mathbb{Z})$| be a homogeneous generalized complex structure on |$M$|⁠, with |$\mathbb{Z} = (Z, \zeta )$|⁠. In |$\mathbb{D} \mathbb{R}_M \otimes \mathbb{C}$| consider the subbundle |$\mathfrak L_{(\mathbb{J}, \mathbb{Z})}$| spanned over |$\mathbb{C}$| as follows: \begin{equation} \mathfrak L_{(\mathbb{J}, \mathbb{Z})}:= \left\langle \left(1- Z, \zeta + \zeta(Z) \cdot \mathfrak j \right), \left(X, \eta + (\eta (Z) - \zeta (X)) \cdot \mathfrak j \right): (X, \eta) \in \mathfrak L_{\mathbb{J}} \right\rangle \end{equation} (2.13) (where we use the same notations as in Remark 1.1.1). Then |$\mathfrak L_{(\mathbb{J}, \mathbb{Z})}$| is a (complex) Dirac–Jacobi structure. Proof. A direct computation with the generators shows that |$\mathfrak L_{(\mathbb{J}, \mathbb{Z})}$| is isotropic. As its rank is |$\dim M +1$|⁠, it is also maximal isotropic. For the involutivity, we will show that the trilinear form \begin{equation*} \Upsilon: \wedge^3 \mathfrak L_{(\mathbb{J}, \mathbb{Z})} \to \mathbb{R}_M, \quad (\alpha, \beta, \gamma) \mapsto \langle \langle \alpha, [ \! [ \beta, \gamma ] \! ] \rangle \rangle \end{equation*} vanishes on generators. To do this, first denote by |$\langle\!\!\langle -,- \rangle\!\!\rangle _{\mathbb{T} M}$| and |$[ \! [ -, - ] \! ]_{\mathbb{T} M}$| the bilinear form and the Dorfman bracket in the generalized tangent bundle, and notice that, for all \begin{equation*} (X_i + f_i, \eta_i + g_i \cdot \mathfrak j) \in \Gamma (\mathbb{D} \mathbb{R}_M\otimes \mathbb{C}), \end{equation*} with |$X_i \in \mathfrak X(M)$|⁠, |$\eta _i \in \Omega ^1 (M)$|⁠, and |$f_i, g_i \in C^\infty (M)$|⁠, |$i = 1, 2$|⁠, we have \begin{equation} \langle \langle (X_1 + f_1, \eta_1 + g_1 \cdot \mathfrak j), (X_2 + f_2, \eta_2 + g_2 \cdot \mathfrak j) \rangle \rangle = \langle \langle (X_1, \eta_1), (X_2, \eta_2) \rangle \rangle_{\mathbb{T} M} + f_1 g_2 + f_2 g_1, \end{equation} (2.14) and \begin{equation} \begin{aligned} & [ \! [ (X_1 + f_1, \eta_1 + g_1 \cdot \mathfrak j), (X_2 + f_2, \eta_2 + g_2 \cdot \mathfrak j) ] \! ] = [ \! [ (X_1, \eta_1), (X_2, \eta_2) ] \! ]_{\mathbb{T} M} \\ & \quad + \Big(X_1 (f_1) - X_2 (f_1), g_2 df_1 + g_1 df_2 + (X_1 (g_2) - X_2 (g_1) + g_2 f_1 + \eta_1 (X_2))\cdot \mathfrak j \Big). \end{aligned} \end{equation} (2.15) Now, let |$\alpha = (1-Z, \zeta + \zeta (Z) \cdot \mathfrak j)$| and for |$(X_i, \eta _i) \in \Gamma (\mathfrak L_{\mathbb{J}})$|⁠, let \begin{equation*} \beta_i = \Big(X_i, \eta_i + (\eta_i (Z)- \zeta (X_i))\cdot \mathfrak j\Big) \end{equation*} be the corresponding generator of |$\Gamma (\mathfrak L_{(\mathbb{J}, \mathbb{Z})})$|⁠, |$i = 1, 2, 3$|⁠. A straightforward computation exploiting (2.14) and (2.15) shows that \begin{equation*} \Upsilon (\beta_1, \beta_2, \alpha) = \langle \langle (X_1, \eta_1), ([Z, X_2], \mathcal L_Z \eta_2 - \eta_2 + \iota_{X_2} d \zeta)\rangle \rangle_{\mathbb{T} M} \end{equation*} and the right hand side vanishes in view of Proposition 2.2.17. Finally, again from (2.14) and (2.15) we get \begin{equation*} \Upsilon (\beta_1, \beta_2, \beta_3) = \langle \langle (X_1, \eta_1), [ \! [ (X_2, \eta_2), (X_3, \beta_3) ] \! ] \rangle \rangle_{\mathbb{T} M} = 0 \end{equation*} and this concludes the proof. Complex Dirac–Jacobi structures on |$\mathbb{R}_M$|⁠, of the form |$\mathfrak L_{(\mathbb{J}, \mathbb{Z})}$| for some homogeneous generalized complex structure |$(\mathbb{J}, \mathbb{Z})$|⁠, can be characterized as follows. First of all, denote by |$p_{\mathbb{R}}: D\mathbb{R}_M = TM \oplus \mathbb{R}_M \to \mathbb{R}_M$| the natural projection. Proposition 2.2.19. A complex Dirac–Jacobi structure |$\mathfrak L \subset \mathbb{D} \mathbb{R}_M\otimes \mathbb{C}$| is of the form |$\mathfrak L_{(\mathbb{J}, \mathbb{Z})}$| for some homogeneous generalized complex structure |$(\mathbb{J}, \mathbb{Z})$| if and only if it satisfies the following conditions: |$\operatorname{rank}_{\mathbb{C}} (\mathfrak L \cap \overline{\mathfrak L}) = 1$|⁠, |$p_D \mathfrak L + p_D \overline{\mathfrak L} = D \mathbb{R}_M \otimes \mathbb{C}$|⁠, |$p_{\mathbb{R}} \circ p_D: \mathfrak L \cap \overline{\mathfrak L} \to M \times \mathbb{C}$| is surjective (hence, an isomorphism). Proof. Begin with a homogeneous generalized complex structure |$(\mathbb{J}, \mathbb{Z})$|⁠, |$\mathbb{Z} = (Z, \zeta )$|⁠, and the associated complex Dirac–Jacobi structure |$\mathfrak L = \mathfrak L_{(\mathbb{J}, \mathbb{Z})}$| as in (2.13). It is easy to see that |$\mathfrak L\cap \overline{\mathfrak L}$| is spanned by |$(1-Z, \zeta + \zeta (Z) \cdot \mathfrak j)$|⁠, in particular |$\mathfrak L$| satisfies property (1) in the statement. For property (2) notice that |$p_D \mathfrak L + p_D \overline{\mathfrak L}$| is spanned by |$1- Z$| and \begin{equation*} p_T \mathfrak L_{\mathbb{J}} + p_T \overline{\mathfrak L}_{\mathbb{J}} = p_T (\mathfrak L_{\mathbb{J}} + \overline{\mathfrak L}_{\mathbb{J}}) = TM \otimes \mathbb{C}, \end{equation*} where we denoted by |$p_T: \mathbb{T}M \to TM$| the projection. So |$\mathfrak L$| satisfies also (2). Property (3) now follows from the fact that \begin{equation*} p_{\mathbb{R}} (1 - Z) = 1 \neq 0. \end{equation*} This concludes the “only if” part of the proof. For the “if” part, let |$\mathfrak L \subset \mathbb{D} \mathbb{R}_M\otimes \mathbb{C}$| be a complex Dirac–Jacobi structure satisfying properties (1)–(3) in the statement. It follows from (1) and (3) that there exists a unique, necessarily real, section |$\alpha$| of |$\mathfrak L \cap \overline{\mathfrak L}$| such that |$(p_{\mathbb{R}} \circ p_D) \alpha = 1$|⁠. In particular, |$\alpha$| is of the form |$(1 - Z, \zeta + g \cdot \mathfrak j)$|⁠, for a real vector field |$Z$|⁠, a real |$1$|-form |$\zeta$|⁠, and a real function |$g$|⁠. From isotropy, |$g = \zeta (Z)$|⁠, so \begin{equation*} \alpha = (1 - Z, \zeta + \zeta (Z) \cdot \mathfrak j), \quad Z \in \mathfrak X(M), \quad \zeta \in \Omega^1 (M). \end{equation*} We put |$\mathbb{Z}:= (Z, \zeta )$|⁠. Next we want to construct a generalized complex structure |$\mathbb{J}: \mathbb{T}M \to \mathbb{T}M$|⁠. To do this, we first define \begin{equation*} \mathfrak L_{\mathbb{T}}:= \left\{(X, \eta) \in \mathbb{T} M\otimes \mathbb{C}: (X, \eta + (\eta (Z) - \zeta (X)) \cdot \mathfrak j) \in \mathfrak L \right\}. \end{equation*} We claim that |$\mathfrak L_{\mathbb{T}}$| is a complex Dirac structure such that |$\mathfrak L_{\mathbb{T}} \cap \overline{\mathfrak L}_{\mathbb{T}} = 0$|⁠. From (2.14), |$\mathfrak L_{\mathbb{T}}$| is (pointwise) maximal isotropic. So it is a regular vector subbundle provided only it is the image of a vector bundle map. Our next aim is constructing such a map. First of all, consider the endomorphism \begin{equation*} F: \mathbb{D} \mathbb{R}_M \otimes \mathbb{C}\to \mathbb{D} \mathbb{R}_M\otimes \mathbb{C}, \quad (X + f, \eta + g \cdot \mathfrak j) \mapsto (X + fZ + f, \eta -f \zeta + g \cdot \mathfrak j) \end{equation*} and the natural projection \begin{equation*} p_{\mathbb{T}}: \mathbb{D} \mathbb{R}_M\otimes \mathbb{C} \to \mathbb{T} M \otimes \mathbb{C}, \quad (X + f, \eta + g \cdot \mathfrak j) \mapsto (X, \eta). \end{equation*} We want to show that \begin{equation*} \mathfrak L_{\mathbb{T}} = (p_{\mathbb{T}} \circ F) \mathfrak L. \end{equation*} As |$F$| fixes elements of the form |$(X, \eta + g \cdot \mathfrak j)$|⁠, it is clear that |$\mathfrak L_{\mathbb{T}} \subset (p_{\mathbb{T}} \circ F) \mathfrak L$|⁠. In order to check the reverse inclusion, begin with |$\beta = (X + f, \eta + g \cdot \mathfrak j) \in \mathfrak L$|⁠. It follows from isotropy that |$g = \eta (Z) - \zeta (X) - f \zeta (Z)$|⁠. Now compute \begin{equation*} (\tilde X, \tilde \eta):= (p_{\mathbb{T}} \circ F) \beta = (X + fZ, \eta - f \zeta). \end{equation*} But |$(\tilde X, \tilde \eta ) \in \mathfrak L_{\mathbb{T}}$|⁠, indeed \begin{equation*} \begin{aligned} & (\tilde X, \tilde \eta + (\tilde \eta (Z) - \zeta(\tilde X)) \cdot \mathfrak j)\\ & = (X + fZ, \eta - fZ + (\eta(Z) - \zeta (X) - 2f\zeta(Z)) \cdot \mathfrak j) \\ & = \beta - f \alpha, \end{aligned} \end{equation*} which belongs to |$\mathfrak L$|⁠. So |$\mathfrak L_{\mathbb{T}}$| is a regular maximal isotropic subbundle of |$\mathbb{T} M \otimes \mathbb{C}$|⁠. Involutivity follows from (2.15) and the involutivity of |$\mathfrak L$|⁠. Next we check |$\mathfrak L_{\mathbb{T}} \cap \overline{\mathfrak L}_{\mathbb{T}} = 0$|⁠. So let |$(X, \eta ) \in \mathfrak L_{\mathbb{T}} \cap \overline{\mathfrak L}_{\mathbb{T}}$|⁠. This means that \begin{equation*} (X, \eta + (\eta(Z) - \zeta (X)) \cdot \mathfrak j) \in \mathfrak L \cap \overline{\mathfrak L}. \end{equation*} As |$p_{\mathbb{R}} X = 0$| this can only be if |$(X, \eta ) = 0$|⁠. We conclude that |$\mathfrak L_{\mathbb{T}}$| is the |$+\textrm{i}$|-eigenbundle of a generalized complex structure |$\mathbb{J}$| on |$M$|⁠. Using (2.15) again, and Proposition 2.2.17, it is easy to see that |$(\mathbb{J}, \mathbb{Z})$| is a homogeneous generalized complex structure in a similar way as in the proof of Proposition 2.2.18. Finally, it is obvious that |$\mathfrak L_{(\mathbb{J}, \mathbb{Z})} \subset \mathfrak L$|⁠. As they are both maximal isotropic, they actually coincide. This concludes the proof. Notice that conditions (1) and (2) in Proposition 2.2.19 make sense for every complex Dirac–Jacobi structure. So we give the following. Definition 2.2.20. A complex Dirac–Jacobi structure |$\mathfrak L \subset \mathbb{D} L\otimes \mathbb{C}$| on a line bundle |$L \to M$| is of homogeneous generalized complex type if |$\operatorname{rank}_{\mathbb{C}} (\mathfrak L \cap \overline{\mathfrak L}) = 1$|⁠, |$p_D \mathfrak L + p_D \overline{\mathfrak L} = D L\otimes \mathbb{C}$|⁠. The above definition is motivated by the following. Proposition 2.2.21. Let |$\mathfrak L \subset \mathbb{D} L\otimes \mathbb{C}$| be a complex Dirac–Jacobi structure of homogeneous generalized complex type on a line bundle |$L \to M$|⁠. Then, locally, around every point of |$M$|⁠, there exists a trivialization |$L \cong \mathbb{R}_M$| identifying |$\mathfrak L$| with the complex Dirac–Jacobi structure |$\mathfrak L_{(\mathbb{J}, \mathbb{Z})} \subset \mathbb{D} \mathbb{R}_M\otimes \mathbb{C}$| induced by a homogeneous generalized complex structure |$(\mathbb{J}, \mathbb{Z})$|⁠. Proof. Let |$\mathfrak L$| be as in the statement, and let |$x_0 \in M$|⁠. Choose a nowhere vanishing local section |$\alpha = (\Delta , \psi ) \in \Gamma (\mathfrak L \cap \overline{\mathfrak L})$| around |$x_0$|⁠. We can choose |$(\Delta , \psi )$| to be real. Then we have |$\Delta \neq 0$|⁠. Indeed, if |$\Delta _{x} = 0$| for some |$x$|⁠, then \begin{equation*} 0 \neq \psi_{x} \in \mathfrak L \cap J^1 L \subset \textsf{Ann} (p_D \mathfrak L) \cap \textsf{Ann} (p_D \overline{\mathfrak L}) = \textsf{Ann} (p_D \mathfrak L + p_D \overline{\mathfrak L}) = \textsf{Ann} (D L \otimes \mathbb{C}) = 0, \end{equation*} a contradiction. So |$\Delta$| is a nonvanishing (local) derivation. It is easy to see that, for a nonvanishing derivation of |$L$|⁠, locally, around every point, there always exists a trivialization |$L \simeq \mathbb{R}_M$| identifying |$\Delta$| with a derivation of the form |$f (1 - Z)$| with |$f$| a nowhere vanishing function. As |$p_{\mathbb{R}} (f(1 -Z)) = f \neq 0$|⁠, this is the trivialization we where looking for. Remark 2.2.22. Let |$\mathfrak L$|⁠, |$x_0$| and |$\Delta$| be as in the proof of Proposition 10. If |$\Delta$| can be chosen so that |$\Delta_{x_0} =\ $|𝟙|$_{x_0}$|⁠, then every local trivialization |$L \simeq \mathbb{R}_M$| around |$x_0$| identifies |$\mathfrak L$| with the complex Dirac–Jacobi structure induced by a homogeneous generalized complex structure. 3 The Transversal to a Leaf Let |$(L \to M, \mathbb{K})$| be a generalized contact bundle with \begin{equation*} \mathbb{K} = \left( \begin{array}{cc} \varphi & J^\sharp \\ \omega_\flat & - \varphi^\dagger \end{array} \right), \end{equation*} and let |$\mathfrak L$| be its |$+\textrm{i}$|-eigenbundle. In this sections, as a preparation for the splitting theorems, we study special classes of submanifolds of |$M$|⁠. Specifically, characteristic leaves of the underlying Jacobi structure |$J$| and their transversals. As we already outlined, in this paper, by a transversal to a leaf |$\mathcal O$| at a point |$x_0 \in \mathcal O$|⁠, we will always understand a minimal dimension transversal, that is, a submanifold |$N$| through |$x_0$| such that |$T_{x_0} M = T_{x_0} N \oplus T_{x_0} \mathcal O$|⁠. We begin with contact leaves. 3.1 Contact leaves and their transversals Recall that an odd-dimensional characteristic leaf |$\mathcal O$| of |$J$| possesses a canonical contact structure |$H \subset T \mathcal O$|⁠. This can be seen as follows. First of all, |$J$| restricts to a Jacobi structure |$J_{\mathcal O}$| on the restricted line bundle |$L_{\mathcal O}:= L{|_\mathcal O} \to \mathcal O$|⁠. Now |$\sigma J_{\mathcal O}^\sharp : J^1 L_{\mathcal O} \to T\mathcal O$| is surjective (by definition of characteristic leaf), and it follows from |$\dim \mathcal O = \textrm{odd}$| that |$J_{\mathcal O}^\sharp : J^1 L_{\mathcal O} \to D L_{\mathcal O}$| is surjective, hence an isomorphism. Let |$\omega _{\mathcal O} = J_{\mathcal O}^{-1} \in \Omega ^2_{L_{\mathcal O}}$| be the Atiyah |$2$|-form inverting |$J_{\mathcal O}$|⁠, that is, |$(\omega _{\mathcal O})_\flat := (J_{\mathcal O}^\sharp )^{-1}$|⁠. Notice that |$DL_{\mathcal O} = (\operatorname{im} J^\sharp ) |_{\mathcal O}$| and |$\omega _{\mathcal O}$| agrees with the pointwise restriction to |$\mathcal O$| of the |$2$|-form |$\omega_{J} : \wedge ^2 \operatorname{im} J^\sharp \to L$| from Remark 2.2.7. Now, the integrability condition for |$J_{\mathcal O}$| is equivalent to |$d_D \omega _{\mathcal O} = 0$|⁠, and |$\iota$|𝟙|$\ \omega _{\mathcal O} \in \Omega ^1_{L_{\mathcal O}}$| is necessarily of the form |$\iota$|𝟙|$\ \omega _{\mathcal O} = \theta _{\mathcal O} \circ \sigma$| for a unique |$L_{\mathcal O}$|-valued |$1$|-form |$\theta _{\mathcal O}: T\mathcal O \to L_{\mathcal O}$|⁠. The kernel |$H$| of |$\theta _{\mathcal O}$| is a contact structure containing the full information on |$J_{\mathcal O}$|⁠. This contact structure can be equivalently encoded in a generalized contact structure \begin{equation*} \mathbb{K}_{\mathcal O} = \left( \begin{array}{cc} 0& J_{\mathcal O}^\sharp \\ - (J^{-1}_{\mathcal O})_\flat & 0 \end{array} \right) \end{equation*} on |$L_{\mathcal O}$|⁠. Let |$\mathfrak L_{\mathcal O} \subset \mathbb{D} L_{\mathcal O}\otimes \mathbb{C}$| be the |$+\textrm{i}$|-eigenbundle of |$\mathbb{K}_{\mathcal O}$|⁠. We have \begin{equation} \mathfrak L_{\mathcal O}:= \left\{ (J_{\mathcal O}^\sharp (\psi), \textrm{i} \psi) \in \mathbb{D} L_{\mathcal O}\otimes \mathbb{C}: \psi \in J^1 L_{\mathcal O} \otimes \mathbb{C} \right\}. \end{equation} (3.1) In the following, for an (immersed) submanifold |$S \hookrightarrow M$|⁠, we simply denote by |$L_S$| the restricted line bundle |$L|_S$|⁠, and by |$I_{S}: L|_{S} \hookrightarrow L$| the natural (injective) immersion. It is a regular vector bundle map covering the injective immersion |$i_{S}: S \hookrightarrow M$|⁠. Proposition 3.1.1. Let |$\mathcal O$| be an odd-dimensional leaf of |$J$|⁠. The backward image of the complex Dirac–Jacobi structure |$\mathfrak L$| along the immersion |$I_{\mathcal O}: L_{\mathcal O} \hookrightarrow L$| is a Dirac–Jacobi structure of generalized contact type on |$L_{\mathcal O}$|⁠, denoted |$I^!_{\mathcal O} \mathfrak L$|⁠. Additionally, it is a |$B$|-field transformation of |$\mathfrak L_{\mathcal O}$|⁠: \begin{equation*} I^!_{\mathcal O} \mathfrak L = \mathfrak L_{\mathcal O}^B \end{equation*} for some |$B \in Z^2_{L_{\mathcal O}}$|⁠. Proof. We divide the proof in several steps. First we prove that |$I^!_{\mathcal O} \mathfrak L \subset \mathbb{D}L_{\mathcal O}\otimes \mathbb{C}$| is a regular subbundle, checking the clean intersection condition (2.9), \begin{equation*} \operatorname{rank}_{\mathbb{C}} \left(p_D \mathfrak L|_{\mathcal O} + DL_{\mathcal O} \otimes \mathbb{C}\right) = \textrm{constant}. \end{equation*} To do this notice that \begin{equation*} D L_{\mathcal O} \otimes \mathbb{C} = \operatorname{im} J^\sharp|_{\mathcal O} \otimes \mathbb{C} = p_D \mathfrak L|_{\mathcal O} \cap p_D\overline{ \mathfrak L}|_{\mathcal O} \subset p_D \mathfrak L|_{\mathcal O}. \end{equation*} Hence, \begin{equation*} p_D \mathfrak L|_{\mathcal O} + DL_{\mathcal O} \otimes \mathbb{C} = p_D \mathfrak L|_{\mathcal O}, \end{equation*} which is constant rank because of the following: |$p_D \mathfrak L|_{\mathcal O} + p_D\overline{ \mathfrak L}|_{\mathcal O} = (DL)|_{\mathcal O} \otimes \mathbb{C}$| is constant rank, |$p_D \mathfrak L|_{\mathcal O} \cap p_D\overline{ \mathfrak L}|_{\mathcal O} = D L_{\mathcal O} \otimes \mathbb{C}$| is constant rank, and |$p_D \mathfrak L|_{\mathcal O}$| and |$p_D\overline{ \mathfrak L}|_{\mathcal O}$| have the same rank. This proves the 1st part of the statement. Next we show that |$I^!_{\mathcal O} \mathfrak L$| is a Dirac–Jacobi structure of generalized contact type. For this, it is enough to check that \begin{equation*} I^!_{\mathcal O} \mathfrak L \cap \overline{I^!_{\mathcal O} \mathfrak L} = I^!_{\mathcal O} \mathfrak L \cap I^!_{\mathcal O} \overline{\mathfrak L} = 0. \end{equation*} So let |$(\Delta , \psi ) \in (I^!_{\mathcal O} \mathfrak L \cap I^!_{\mathcal O} \overline{\mathfrak L})_x$| for some |$x \in \mathcal O$|⁠. This means that, there exist |$\chi ^{\prime}, \chi ^{\prime\prime} \in J^1_x L \otimes \mathbb{C}$| such that |$\psi = I^\ast _{\mathcal O} \chi ^{\prime} = I^\ast _{\mathcal O} \chi ^{\prime\prime}$|⁠, and, additionally, |$(\Delta , \chi ^{\prime}) \in \mathfrak L_x$|⁠, and |$(\Delta , \chi ^{\prime\prime}) \in \overline{\mathfrak L}_x$|⁠. Now, define |$\chi \in J^1_x L \otimes \mathbb{C}$| by putting \begin{equation*} \langle \chi, \nabla \rangle = \left\{ \begin{array}{cl} \langle \chi^{\prime}, \nabla \rangle & \text{if }\nabla \in p_D \mathfrak L_x \\ \langle \chi^{\prime\prime}, \nabla \rangle & \text{if }\nabla \in p_D \overline{\mathfrak L}_x \end{array} \right.. \end{equation*} As both |$\chi ^{\prime}$| and |$\chi ^{\prime\prime}$| agree with |$\psi$| on |$(p_D \mathfrak L \cap p_D \overline{\mathfrak L})_x = (\operatorname{im} J^\sharp )_x \otimes \mathbb{C} = D_x L_{\mathcal O}\otimes \mathbb{C}$|⁠, then |$\chi$| is well defined. It immediately follows from (2.7) that |$(\Delta , \chi ) \in (\mathfrak L \cap \overline{\mathfrak L})_x = 0$|⁠. So |$(\Delta , \chi ) = 0$|⁠, hence |$(\Delta , \psi ) = 0$|⁠. We conclude that |$I^!_{\mathcal O} \mathfrak L$| is a Dirac–Jacobi structure of generalized contact type. In particular, there is an underlying Jacobi structure |$\tilde J$| on |$L_{\mathcal O}$|⁠. As a 3rd step, we prove that the Jacobi structure underlying |$I^!_{\mathcal O} \mathfrak L$| is precisely |$J_{\mathcal O}$|⁠: the restriction to |$\mathcal O$| of the Jacobi structure |$J$|⁠. In other words, |$\tilde J = J_{\mathcal O}$|⁠. First of all, \begin{equation*} p_{D} I^!_{\mathcal O} \mathfrak L = p_D \mathfrak L \cap (D L_{\mathcal O} \otimes \mathbb{C}) = D L_{\mathcal O} \otimes \mathbb{C}. \end{equation*} In particular the |$L_{\mathcal O}$|-valued |$2$|-form |$\varpi _{\mathcal O}$| induced by |$I^!_{\mathcal O} \mathfrak L$| on |$p_{D} I^!_{\mathcal O} \mathfrak L$| is a genuine (complex) Atiyah |$2$|-form on |$L_{\mathcal O}$|⁠. Notice that |$\varpi _{\mathcal O}$| actually agrees with |$\varpi$| on |$D L_{\mathcal O}$|⁠. Indeed let |$\Delta , \nabla \in DL_{\mathcal O}$|⁠. There is |$\psi \in J^1 L$| such that |$(\Delta , I_{\mathcal O}^\ast \psi ) \in I^!_{\mathcal O} \mathfrak L$|⁠. Compute \begin{equation*} \varpi_{\mathcal O} (\Delta, \nabla) = \langle I_{\mathcal O}^\ast \psi, \nabla \rangle = \langle \psi, \nabla \rangle = \varpi (\Delta, \nabla). \end{equation*} Now, let |$\psi \in J^1 L_{\mathcal O}$|⁠, and let |$\Psi \in J^1 L$| be such that |$I^\ast _{\mathcal O} \Psi = \psi$|⁠. We want to compare |$J_{\mathcal O}^\sharp \psi$| and |$\tilde{J}^{\sharp} \psi$|⁠. To do this pick |$\nabla \in D L_{\mathcal O}$| and compute \begin{gather*} \operatorname{Im} \varpi_{\mathcal O} (\tilde J{}^\sharp \psi, \nabla) = \langle \psi, \nabla \rangle = \langle I_{\mathcal O}^\ast \Psi, \nabla \rangle = \langle \Psi, \nabla \rangle \\ = \operatorname{Im} \varpi (J^\sharp \Psi, \nabla) = \operatorname{Im} \varpi (J^\sharp_{\mathcal O} \psi, \nabla) = \operatorname{Im} \varpi_{\mathcal O} (J^\sharp_{\mathcal O} \psi, \nabla). \end{gather*} But |$\operatorname{Im} \varpi _{\mathcal O}$| is nondegenerate (Remark 2.2.7) so |$\tilde J^\sharp \psi = J^\sharp _{\mathcal O} \psi$|⁠. In particular |$\operatorname{Im} \varpi _{\mathcal O} = J^{-1}_{\mathcal O}$|⁠. Finally, we prove that |$I^!_{\mathcal O} \mathfrak L$| is a |$B$|-field transformation of |$\mathfrak L_{\mathcal O}$|⁠. From |$p_D I^!_{\mathcal O} \mathfrak L = D L_{\mathcal O} \otimes \mathbb{C}$|⁠, we have \begin{equation} I^!_{\mathcal O} \mathfrak L = \left\{(\Delta, \iota_\Delta \varpi_{\mathcal O}): \Delta \in DL_{\mathcal O} \right\} = \operatorname{\textsf{graph}} (\varpi_{\mathcal O})_\flat \subset \mathbb{D} L_{\mathcal O}\otimes \mathbb{C}. \end{equation} (3.2) Let |$B = \operatorname{Re} \varpi _{\mathcal O} \in \Omega ^2_{L_{\mathcal O}}$|⁠. From (3.2) and the involutivity of |$I^{!}_{\mathcal O} \mathfrak L$| we have |$d_D \varpi _{\mathcal O} = 0$|⁠; hence, |$d_D B = 0$|⁠. Finally, compute \begin{equation*} (I^!_{\mathcal O} \mathfrak L)^{- B} = \operatorname{\textsf{graph}} (\varpi_{\mathcal O} - \operatorname{Re} \varpi_{\mathcal O}) = \operatorname{\textsf{graph}} (\textrm{i} \operatorname{Im} \varpi_{\mathcal O}) = \mathfrak L_{\mathcal O}, \end{equation*} where we used (3.1) and the fact that |$\operatorname{Im} \varpi _{\mathcal O} = J^{-1}_{\mathcal O}$|⁠. We now pass to transversals. A transversal to a characteristic leaf of a generalized complex manifold inherits a generalized complex structure, at least around the intersection point with the leaf. The precise analogue cannot be true for contact leaves of a generalized contact structure, simply because, in this case, transversals are even dimensional. Proposition 3.1.2. Let |$N$| be a transversal at |$x_0 \in \mathcal O$| to an odd-dimensional leaf |$\mathcal O$| of |$J$|⁠. Around |$x_0$|⁠, the backward image of the complex Dirac–Jacobi structure |$\mathfrak L$| along the embedding |$I_N: L_N \hookrightarrow L$| is a complex Dirac–Jacobi structure |$I^!_N \mathfrak L$| of homogeneous generalized complex type such that \begin{equation*} \left(I^!_N \mathfrak L \cap \overline{I^!_N \mathfrak L}\right)_{x_0} \end{equation*} is spanned by a vector of the form (𝟙|$_{x_0}, \psi )$|⁠. In particular, any local trivialization |$L_N \cong \mathbb{R}_N$| around |$x_0$| identifies |$I^!_N \mathfrak L$| with the complex Dirac–Jacobi structure corresponding to a homogeneous generalized complex structure. Proof. First of all we prove that |$I^{!}_N \mathfrak L \subset \mathbb{D} L_N$| is a regular subbundle, hence a Dirac–Jacobi structure on |$L_N$|⁠, checking the clean intersection condition (2.9), \begin{equation*} \operatorname{rank}_{\mathbb{C}}(p_D \mathfrak L|_N + DL_N \otimes \mathbb{C}) = \textrm{constant}. \end{equation*} We have \begin{equation*} p_D \mathfrak L|_N \supset p_D \mathfrak L|_N \cap p_D \overline{\mathfrak L}|_N = \operatorname{im} J^\sharp |_N \otimes \mathbb{C}. \end{equation*} At the point |$x_0$| we have |$(\operatorname{im} J^\sharp )_{x_0} \otimes \mathbb{C} = D_{x_0} L_{\mathcal O}$|⁠, so \begin{equation*} p_D \mathfrak L_{x_0} + D_{x_0}L_N \otimes \mathbb{C} \supset D_{x_0} L_{\mathcal O} + D_{x_0} L_N = D_{x_0} L. \end{equation*} But |$p_D \mathfrak L|_N + DL_N \otimes \mathbb{C} \subset (DL)|_N$| is a smooth, possibly non-regular subbundle; hence, its rank can only increase around |$x_0$|⁠, and we conclude that \begin{equation} p_D \mathfrak L|_N + DL_N \otimes \mathbb{C} = (DL)|_N \otimes \mathbb{C} \end{equation} (3.3) in a whole neighborhood of |$x_0$|⁠. In particular, the left-hand side has constant rank. Next we show that |$\operatorname{rank}_{\mathbb{C}}(I^!_N \mathfrak L \cap I^!_N \overline{\mathfrak L}) = 1$| around |$x_0$|⁠. Denote by |$\nu N = TM|_N / TN$| and |$\nu ^\ast N = \textsf{Ann}(TN)\subset T^\ast M|_N$| the normal and the conormal bundle to |$N$|⁠, respectively. It is useful to consider the following skew-symmetric bilinear map: \begin{equation*} \mu: \wedge^2 \left(\nu^\ast N \otimes L_N\right) \to L_N, \quad (\eta, \theta) \mapsto \langle J^\sharp\eta, \theta \rangle, \end{equation*} and the associated vector bundle map |$\mu ^\sharp : \nu ^\ast N \otimes L_N \to \nu N$| implicitly defined by \begin{equation*} \langle \theta, \mu^\sharp \eta\rangle = \mu (\eta, \theta), \quad \eta, \theta \in \nu^\ast N \otimes L_N. \end{equation*} In other words |$\mu ^\sharp$| is the composition \begin{equation*} \nu^\ast N \otimes L_N \overset{J^\sharp}{\longrightarrow} (D L)|_N \overset{\sigma}{\longrightarrow} TM|_N \longrightarrow \nu N, \end{equation*} where the last arrow is the natural projection (with kernel |$TN$|⁠). We want to show that |$\mu$| has maximal rank around |$x_0$|⁠: that is, |$\operatorname{rank} \mu = \operatorname{rank}(\nu N) - 1 = \dim \mathcal O -1 = \textrm{even}$|⁠. To do this it is enough to show that |$\operatorname{rank}_{x_0} \mu = \dim \mathcal O -1$|⁠, in other words |$\dim (\ker \mu ^\sharp _{x_0}) = 1$|⁠. So compute \begin{equation*} \ker \mu^\sharp_{x_0} = \left\{\eta \in \nu^\ast_{x_0} N \otimes L_{x_0}: J_{x_0}^\sharp \eta \in DL_N \right\}. \end{equation*} But |$\nu ^\ast _{x_0} N \otimes L_{x_0} = T^\ast _{x_0} \mathcal O \otimes L_{x_0}$|⁠, and |$(\operatorname{im} J^\sharp )_{x_0} = D_{x_0} L_{\mathcal O}$|⁠, so we find \begin{equation} \end{equation} (3.4) Now, we go back to |$I^!_N \mathfrak L$| and consider the real (a priori not necessarily regular) subbundle \begin{equation*} R:= \left\{\operatorname{Re} \alpha: \alpha \in I^!_N \mathfrak L \cap I^!_N \overline{\mathfrak L} \right\} \subset I^!_N \mathfrak L \cap I^!_N \overline{\mathfrak L}. \end{equation*} Clearly |$I^!_N \mathfrak L \cap I^!_N \overline{\mathfrak L}$| is (canonically isomorphic to) the complexification of |$R$|⁠. We want to show that there is an (pointwise) exact sequence \begin{equation} 0 \longrightarrow R \overset{\kappa}{\longrightarrow} \nu^\ast N \otimes L_N \overset{\mu^\sharp}{\longrightarrow} \nu N, \end{equation} (3.5) proving that, around |$x_0$|⁠, |$\operatorname{rank}_{\mathbb{R}} R = \operatorname{rank}_{\mathbb{C}}(I^!_N \mathfrak L \cap I^!_N \overline{\mathfrak L}) = 1$| as claimed. Define |$\kappa : R \to \nu ^\ast N \otimes L_N$| as follows. Take |$\alpha = (\Delta , \psi ) \in R$|⁠. This means that |$(\Delta , \psi ) \in \mathbb{D} L_N$| is such that there exists |$\chi \in J^1 L \otimes \mathbb{C}$| with |$(\Delta , \chi ) \in \mathfrak L|_N$| (hence, |$(\Delta , \overline \chi ) \in \overline{\mathfrak L}|_N$|⁠) and |$I^\ast _N \chi = \psi$|⁠. Actually, |$\chi$| is unique. Indeed, let |$\chi ^{\prime} \in J^1 L \otimes \mathbb{C}$| be such that |$(\Delta , \chi ^{\prime}) \in \mathfrak L|_N$| and |$I^\ast _N \chi ^{\prime} = \psi$|⁠. Then, on one side \begin{equation*} \chi - \chi^{\prime} \in \left((J^1 L)|_N \otimes \mathbb{C} \right)\cap \mathfrak L|_N = \textsf{Ann} \left(p_D \mathfrak L|_N\right). \end{equation*} On the other side, |$I_N^\ast (\chi - \chi ^{\prime}) = 0$|⁠, that is, \begin{equation*} \chi - \chi^{\prime} \in \textsf{Ann} \left( D L_N \otimes \mathbb{C}\right), \end{equation*} so \begin{equation*} \chi - \chi^{\prime} \in \textsf{Ann} \left( p_D \mathfrak L|_N + D L_N \otimes \mathbb{C} \right) = 0, \end{equation*} where we used (3.3) (which holds true around |$x_0$|⁠). So if we work around |$x_0$|⁠, |$\chi = \chi ^{\prime}$|⁠, we put \begin{equation*} \kappa (\Delta, \psi):= \operatorname{Im} \chi, \end{equation*} and, from |$\psi = I^\ast _N \chi = I^\ast _N \overline \chi$|⁠, it belongs to |$\nu ^\ast N \otimes L_N = \textsf{Ann} (D L_N) \subset (J^1 L)|_N$|⁠. Before proving that the sequence (3.5) is exact, the following remark is useful. Let |$(\Delta ,\psi ) \in R$| and let |$\chi$| be as above. Then \begin{equation} \Delta = J^\sharp (\operatorname{Im} \chi). \end{equation} (3.6) Indeed, from |$\mathbb{K} (\Delta , \chi ) = \textrm{i} (\Delta , \chi )$|⁠, we find |$\textrm{i} \Delta = \varphi \Delta + J^\sharp \chi$| (take just the 1st component). Similarly, from |$\mathbb{K} (\Delta , \overline \chi ) = -\textrm{i} (\Delta , \overline \chi )$|⁠, we find |$\textrm{i} \Delta = - \varphi \Delta - J^\sharp \overline \chi$|⁠. So |$\Delta = J^\sharp (\chi - \overline \chi )/2\textrm{i} = J^\sharp (\operatorname{Im} \chi )$|⁠. Now, we prove that (3.5) is exact. First of all |$\kappa$| is injective. Indeed, if |$\kappa (\Delta , \psi ) = \operatorname{Im} \chi = 0$|⁠, then |$\chi = \overline \chi$| and |$(\Delta , \chi ) \in \mathfrak L \cap \overline{\mathfrak L} = 0$|⁠, so |$\chi = 0$|⁠, and, from (3.6), |$(\Delta ,\psi ) = (0, I_N^\ast \chi ) = 0$|⁠. It remains to show that |$\ker \mu ^\sharp = \operatorname{im} \kappa$|⁠. So let |$(\Delta , \psi ) \in R$|⁠, let |$\chi$| be as above, and let |$\eta \in \nu ^\ast N \otimes L_N$|⁠. Compute \begin{equation*} \langle \mu^\sharp (\operatorname{Im} \chi), \eta \rangle = \langle J^\sharp (\operatorname{Im} \chi), \eta \rangle = \langle \Delta, \eta \rangle = 0, \end{equation*} where we used (3.6) again and the fact that |$\Delta \in DL_N$|⁠. So |$\ker \ \mu ^\sharp \subset \operatorname{im} \ \kappa$|⁠. Finally, let |$\eta \in \nu ^\ast N \otimes L_N$| be such that |$\mu ^\sharp \eta = 0$|⁠. This means that |$\Delta := J^\sharp \eta \in D L_N$|⁠. Put \begin{equation*} \alpha:= (\Delta, -I^\ast_N (\varphi^\dagger \eta)). \end{equation*} We claim that |$\alpha \in R$|⁠, and |$\eta = \kappa (\alpha )$|⁠. To see this notice that \begin{equation*} (\Delta, \textrm{i} \eta - \varphi^\dagger \eta) = \textrm{i} \left(\textrm{id} - \textrm{i} \mathbb{K} \right)(0, \eta) \in \mathfrak L; \end{equation*} hence, |$(\Delta , I^\ast _N (\textrm{i} \eta - \varphi ^\dagger \eta )) = \alpha \in R$|⁠. Additionally \begin{equation*} \kappa (\alpha) = \operatorname{Im} (\textrm{i} \eta - \varphi^\dagger \eta) = \eta. \end{equation*} We conclude that |$\ker \mu ^\sharp = \operatorname{im} \kappa$|⁠, and |$\operatorname{rank}_{\mathbb{C}} (I^!_N \mathfrak L \cap I^!_N \overline{\mathfrak L}) = 1$| as claimed. To prove that |$I^!_N \mathfrak L$| is a complex Dirac–Jacobi structure of homogeneous generalized complex type, it remains to show that |$p_D I^!_N \mathfrak L + p_D I^!_N \overline{\mathfrak L} = D L_N$|⁠. To do this we compute \begin{equation*} \textsf{Ann} \left( p_D I^!_N \mathfrak L + p_D I^!_N \overline{\mathfrak L} \right) = \textsf{Ann} \left( p_D I^!_N \mathfrak L \right)\cap \textsf{Ann} \left(p_D I^!_N \overline{\mathfrak L} \right) = \left(J^1 L_N \otimes \mathbb{C}\right) \cap I^!_N \mathfrak L \cap I^!_N \overline{\mathfrak L}. \end{equation*} But the above discussion, together with formula (3.4), reveals that, at the point |$x_0$|⁠, |$R$|⁠, hence |$I^!_N \mathfrak L \cap I^!_N \overline{\mathfrak L}$|⁠, is spanned by an element of the form (𝟙|$_{x_0}, \zeta )$|⁠. In particular, \begin{equation*} \left(J^1 L_N \otimes \mathbb{C}\right) \cap I^!_N \mathfrak L \cap I^!_N \overline{\mathfrak L} = 0 \end{equation*} at the point |$x_0$|⁠, hence in a whole neighborhood of |$x_0$|⁠. This concludes the proof. 3.2 Lcs leaves and their transversals We now pass to lcs leaves. As already mentioned, an even-dimensional characteristic leaf of |$J$| possesses a canonical lcs structure. To see this one can argue as follows. As before, |$J$| restricts to a Jacobi structure |$J_{\mathcal O}$| on the restricted line bundle |$L_{\mathcal O} \to \mathcal O$|⁠. Now |$\sigma J_{\mathcal O}^\sharp : J^1 L_{\mathcal O} \to T\mathcal O$| is surjective again, and, as |$\dim \mathcal O = \textrm{even}$|⁠, then |$J_{\mathcal O}^\sharp : J^1 L_{\mathcal O} \to D L_{\mathcal O}$| takes values in a |$\dim \mathcal O$|-dimensional subbundle |$C_{\mathcal O} \subset D L_{\mathcal O}$| transversal to |$\mathbb{R}_{\mathcal O} \subset D L_{\mathcal O}$|⁠. In other words, |$C_{\mathcal O}$| is the image of a linear connection |$\nabla : T \mathcal O \to DL_{\mathcal O}$|⁠. Additionally, |$J_{\mathcal O}$| induce a nondegenerate |$L_{\mathcal O}$|-valued |$2$|-form on |$C_{\mathcal O}$|⁠, hence on |$T \mathcal O$|⁠. So we get a nondegenerate |$\Omega _{\mathcal O} \in \Omega ^2 (\mathcal O, L_{\mathcal O})$|⁠. Notice that |$C_{\mathcal O} = \operatorname{im} J^\sharp{}|_{\mathcal O}$|⁠, and |$\Omega _{\mathcal O}$| (viewed as a |$2$|-form on |$C_{\mathcal O}$|⁠) agrees with the pointwise restriction to |$\mathcal O$| of the |$2$|-form |$\omega _J: \wedge ^2 \operatorname{im} J^\sharp \to L$| from Remark 2.2.7. Now the integrability condition for |$J_{\mathcal O}$| is equivalent to |$\nabla$| being flat and |$d^\nabla \Omega _{\mathcal O} = 0$|⁠. So |$(\Omega _\mathcal O, \nabla )$| is an lcs structure on |$L_{\mathcal O}$| and it contains the full information on |$J_{\mathcal O}$|⁠. This lcs structure |$(\Omega _{\mathcal O}, \nabla )$| can be equivalently encoded in a complex Dirac–Jacobi structure |$\mathfrak L_{\mathcal O}$| given by the same formula (3.1) as before. Remark 3.2.1. Let |$\mathcal O$| be an even-dimensional leaf of |$J$|⁠. Then the subbundle |$\mathfrak L_{\mathcal O} \subset \mathbb{D} L_{\mathcal O}\otimes \mathbb{C}$| given by formula (3.1) is a complex Dirac–Jacobi structure such that \begin{equation*} p_D \mathfrak L_{\mathcal O} = p_D \overline{\mathfrak L}_{\mathcal O} = C_{\mathcal O} \otimes \mathbb{C} \quad \textrm{and} \quad \mathfrak L_{\mathcal O} \cap \overline{\mathfrak L}_{\mathcal O} = \textsf{Ann} (C_{\mathcal O} \otimes \mathbb{C}). \end{equation*} Proposition 3.2.2. Let |$\mathcal O$| be an-even dimensional leaf of |$J$|⁠. The backward image of the complex Dirac–Jacobi structure |$\mathfrak L$| along the immersion |$I_{\mathcal O}: L_{\mathcal O} \hookrightarrow L$| is a Dirac–Jacobi structure on |$L_{\mathcal O}$|⁠, denoted |$I^!_{\mathcal O} \mathfrak L$|⁠, such that |$\operatorname{rank}_{\mathbb{C}} (I^!_{\mathcal O}\mathfrak L \cap I^!_{\mathcal O}\overline{\mathfrak L}) = 1$|⁠, 𝟙 |$\notin p_D I^!_{\mathcal O} \mathfrak L + p_D I^!_{\mathcal O}\overline{\mathfrak L}$|⁠, and |$(p_D I^!_{\mathcal O}\mathfrak L + p_D I^!_{\mathcal O}\overline{\mathfrak L}) \oplus \langle$| 𝟙 |$\rangle = DL_{\mathcal O} \otimes \mathbb{C}$|⁠. Additionally, it is locally a |$B$|-field transformation of |$\mathfrak L_{\mathcal O}$|⁠: \begin{equation*} I^!_{\mathcal O} \mathfrak L = \mathfrak L_{\mathcal O}^B \end{equation*} for some |$B \in Z^2_{L_{\mathcal O}}$|⁠. Proof. First of all, we prove that |$I^!_{\mathcal O} \mathfrak L \subset \mathbb{D} L_{\mathcal O}\otimes \mathbb{C}$| is a regular subbundle. As usual, we check the clean intersection condition \begin{equation*} \operatorname{rank}_{\mathbb{C}} \left(p_D \mathfrak L|_{\mathcal O} + DL_{\mathcal O} \otimes \mathbb{C} \right) = \textrm{constant}. \end{equation*} So notice that, in this case However, 𝟙 |$\notin p_D \mathfrak L|_{\mathcal O}$|⁠, otherwise, from , and |$\operatorname{im} J^\sharp \otimes \mathbb{C} = p_D \mathfrak L \cap p_D \overline{\mathfrak L}$| we would get 𝟙 |$\in C_{\mathcal O}$|⁠, which is not the case. We conclude that which is constant rank in the same way as for contact leaves (see the proof of Proposition 3.1.1). So |$I^!_{\mathcal O} \mathfrak L$| is a Dirac–Jacobi structure on |$L_{\mathcal O}$|⁠. Next we show that \begin{equation} p_D I^!_{\mathcal O} \mathfrak L = p_D I^!_{\mathcal O} \overline{\mathfrak L} = C_{\mathcal O} \otimes \mathbb{C}. \end{equation} (3.7) We will get, in particular, properties (2) and (3) in the statement. From |$p_D I^!_{\mathcal O} \mathfrak L = D L_{\mathcal O} \cap p_D \mathfrak L$|⁠, and |$p_D \mathfrak L \cap p_D \overline{\mathfrak L} = \operatorname{im} J^\sharp \otimes \mathbb{C}$| we get |$C_{\mathcal O} \otimes \mathbb{C} \subset p_D I^!_{\mathcal O} \mathfrak L \cap p_D I^!_{\mathcal O} \overline{\mathfrak L}$|⁠. Now let |$(\Delta , \psi ) \in I^!_{\mathcal O} \mathfrak L$|⁠, so that |$\Delta \in p_D I^!_{\mathcal O}\mathfrak L$|⁠. In particular, |$\Delta \in D L_{\mathcal O} \otimes \mathbb{C}$|⁠, meaning that |$\Delta = \Delta _0 + z \cdot$| 𝟙 for some |$\Delta _0 \in C_{\mathcal O}\otimes \mathbb{C}$|⁠, and some |$z \in \mathbb{C}$|⁠. It follows that |$\Delta - \Delta _0 \in p_D \mathfrak L|_{\mathcal O}$|⁠. As 𝟙 |$\notin p_D \mathfrak L|_{\mathcal O}$|⁠, we have |$z = 0$|⁠, and |$\Delta = \Delta _0 \in C_{\mathcal O}\otimes \mathbb{C}$|⁠. So |$p_D I^!_{\mathcal O} \mathfrak L \subset C_{\mathcal O}\otimes \mathbb{C}$|⁠, and, similarly, |$p_D I^!_{\mathcal O} \overline{\mathfrak L} \subset C_{\mathcal O}\otimes \mathbb{C}$|⁠. Now we show that \begin{equation*} I^!_{\mathcal O} \mathfrak L \cap I^!_{\mathcal O} \overline{\mathfrak L} = \textsf{Ann} (C_{\mathcal O} \otimes \mathbb{C}). \end{equation*} We will get, in particular, property (1) in the statement. So let |$(\Delta , \psi ) \in (I^!_{\mathcal O} \mathfrak L \cap I^!_{\mathcal O} \overline{\mathfrak L} )_x$| for some |$x \in \mathcal O$|⁠. This means that there exist |$\chi ^{\prime}, \chi ^{\prime\prime}$| as in the proof of Proposition (3.1.1), and we can even construct |$\chi$| exactly as there. As |$\mathfrak L$| is of generalized contact type, actually |$(\Delta , \chi ) = 0$|⁠, that is, |$\Delta = 0$|⁠, and |$\psi \in \textsf{Ann} (C_{\mathcal O} \otimes \mathbb{C})$|⁠. This shows that |$I^!_{\mathcal O} \mathfrak L \cap I^!_{\mathcal O} \overline{\mathfrak L} \subset \textsf{Ann} (C_{\mathcal O} \otimes \mathbb{C}) \subset J^1 L_{\mathcal O} \otimes \mathbb{C}$|⁠. The reverse inclusion |$\textsf{Ann} (C_{\mathcal O} \otimes \mathbb{C}) \subset I^!_{\mathcal O} \mathfrak L \cap I^!_{\mathcal O} \overline{\mathfrak L}$| immediately follows from (3.7). It remains to show that, locally, |$I^!_{\mathcal O} \mathfrak L$| is a |$B$|-field transformation of \begin{equation} \mathfrak L_{\mathcal O}= \left\{ (J^\sharp_{\mathcal O} (\psi), \textrm{i} \psi) \in \mathbb{D} L_{\mathcal O} \otimes \mathbb{C}: \psi \in J^1 L_{\mathcal O} \otimes \mathbb{C} \right\}. \end{equation} (3.8) To do this, denote by |$\varpi _{\mathcal O}$| the |$L_{\mathcal O}$|-valued |$2$|-form induced by |$I^!_{\mathcal O} \mathfrak L$| on |$p_D I^!_{\mathcal O} \mathfrak L = C_{\mathcal O} \otimes \mathbb{C}$|⁠. We can extend |$\varpi _{\mathcal O}$| to a genuine Atiyah |$2$|-form on |$L_{\mathcal O}$|⁠, by putting |$\iota$|𝟙|$\varpi _{\mathcal O} = 0$|⁠. Similarly as in the case of a contact leaf, |$\varpi _{\mathcal O}$| actually agrees with |$\varpi$| on |$C_{\mathcal O}$|⁠. It follows that the imaginary part |$\operatorname{Im} \varpi _{\mathcal O}$| agrees with the lcs form |$\Omega _{\mathcal O}$|⁠. From (2.7) we get \begin{equation} I^!_{\mathcal O} \mathfrak L = \left\{ (\Delta, \iota_\Delta \varpi_{\mathcal O} + A): \Delta \in C_{\mathcal O} \otimes \mathbb{C} \textrm{ and}\ A \in \textsf{Ann} C_{\mathcal O} \right\}. \end{equation} (3.9) Using |$C_{\mathcal O} \cong T\mathcal O$|⁠, we can also think of |$\varpi _{\mathcal O}$| as an |$L_{\mathcal O} \otimes \mathbb{C}$|-valued |$2$|-form on |$\mathcal O$|⁠. Then, if we denote by |$\nabla : T \mathcal O \to D L_{\mathcal O}$| the flat connection in |$L_{\mathcal O}$| whose image is |$C_{\mathcal O}$|⁠, from (3.9) and involutivity, we get |$d^\nabla \varpi _{\mathcal O} = 0$|⁠. In particular, |$d^\nabla \operatorname{Re} \varpi _{\mathcal O} = 0$|⁠, and, locally, |$\operatorname{Re} \varpi _{\mathcal O} = d^\nabla \eta$|⁠, for some |$\eta \in \Omega ^1 (\mathcal O, L_{\mathcal O}) \subset \Gamma (J^1 L_{\mathcal O})$|⁠. Put |$B:= d_D \eta$|⁠. An easy computation shows that \begin{equation*} B = \operatorname{Re} \varpi_{\mathcal O} + \mathcal C \wedge \eta, \end{equation*} where |$\mathcal C: DL_{\mathcal O} \to \mathbb{R}_{\mathcal O}$| is the unique |$1$|-form with kernel |$C_{\mathcal O}$|⁠, and such that |$\langle \mathcal C,$| 𝟙 |$\rangle = 1$|⁠. Hence, \begin{equation*} (I^!_{\mathcal O} \mathfrak L)^{-B} = \left\{ (\Delta, \textrm{i} \cdot \iota_\Delta \operatorname{Im} \varpi_{\mathcal O} + A ): \Delta \in C_{\mathcal O} \otimes \mathbb{C} \textrm{ and}\ A \in \textsf{Ann} C_{\mathcal O} \right\} = \mathfrak L_{\mathcal O}. \end{equation*} For the very last step we used (3.8), and the fact that |$\operatorname{Im} \varpi _{\mathcal O} = \Omega _{\mathcal O}$| (together with the relationship between |$\Omega _\mathcal O$| and |$J_{\mathcal O}$| discussed at the beginning of this subsection). Proposition 3.2.3. Let |$N$| be a transversal at |$x_0 \in \mathcal O$| to an even-dimensional leaf |$\mathcal O$| of |$J$|⁠. Around |$x_0$|⁠, the backward image of the complex Dirac–Jacobi structure |$\mathfrak L$| along the embedding |$I_N: L_N \hookrightarrow L$| is a complex Dirac–Jacobi structure |$I^!_N \mathfrak L$| of generalized contact type. Proof. One can prove that |$I^!_N \mathfrak L \subset \mathbb{D} L_N$| is a regular subbundle in a very similar way as for the transversal to a contact leaf (proof of Proposition 3.1.2) and we leave it to the reader to take care of the obvious adaptations. Now, we show that \begin{equation*} \left( I^!_N \mathfrak L \cap I^!_N \overline{\mathfrak L} \right)_{x_0} = 0. \end{equation*} It will follow that |$I^!_N \mathfrak L \cap I^!_N \overline{\mathfrak L} = 0$| in a whole neighborhood of |$x_0$|⁠. So let |$(\Delta , \psi ) \in (I^!_N \mathfrak L \cap I^!_N \overline{\mathfrak L})_{x_0}$|⁠. Then \begin{equation*} \Delta \in p_D \mathfrak L_{x_0} \cap p_D \overline{\mathfrak L}_{x_0} \cap D_{x_0} L_N \otimes \mathbb{C}= ((C_{\mathcal O})_{x_0} \cap D_{x_0} L_N) \otimes \mathbb{C} = 0, \end{equation*} and we find |$\chi ^{\prime}, \chi ^{\prime\prime} \in J_{x_0}^1 L \otimes \mathbb{C}$|⁠, such that |$(0, \chi ^{\prime}) \in \mathfrak L_{x_0}$| (i.e., |$\chi ^{\prime} \in \textsf{Ann} (p_D \mathfrak L_{x_0})$|⁠), |$(0, \chi ^{\prime\prime}) \in \overline{\mathfrak L}_{x_0}$| (i.e., |$\chi ^{\prime\prime} \in \textsf{Ann} (p_D \overline{\mathfrak L}_{x_0})$|⁠), and, additionally, |$\psi = I^\ast _N \chi ^{\prime} = I^\ast _N \chi ^{\prime\prime}$|⁠. Hence, \begin{equation*} \begin{aligned} \chi^{\prime} - \chi^{\prime\prime} & \in \textsf{Ann} \left(p_D \mathfrak L_{x_0} \cap p_D \overline{\mathfrak L}_{x_0}\right) \cap \textsf{Ann} \left(D_{x_0} L_N \otimes \mathbb{C}\right) \\ & = \textsf{Ann} \left( (C_{\mathcal O})_{x_0} + D_{x_0} L_N\right) \otimes \mathbb{C} = 0. \end{aligned} \end{equation*} It follows that |$(0, \chi ^{\prime}) = (0, \chi ^{\prime\prime}) \in \mathfrak L_{x_0} \cap \overline{\mathfrak L}_{x_0} = 0$|⁠, so that |$\psi = 0$| as well. This concludes the proof. 4 Splitting Theorems In this section we prove a local splitting theorem for generalized contact bundles analogous to Weinstein splitting theorem for Poisson structures [42], and similar splitting theorems in the following Poisson-related geometries: Jacobi geometry [14], Dirac geometry [8] (see also [16]), Lie algebroid geometry [15, 17, 43], and generalized complex geometry [1] (see also [5] for an important refinement of Abouzaid–Boyarchenko result). As expected, our splitting theorem is similar to that for generalized complex manifolds on one side, and to that for Jacobi bundles on the other side. In particular, we actually prove two splitting theorems: one about the local structure around a point in a contact leaf and one about the local structure around a point in a lcs leaf. Our proof is different in spirit from that of Abouzaid and Boyarchenko, and it is rather inspired by the recent work of Bursztyn et al. [11], who provided a unified approach to splitting theorems in Poisson (and related) geometries. We begin recalling the splitting theorems of Dazord et al. [14] for Jacobi bundles. Theorem 4.0.1. Let |$(L \to M, J)$| be a Jacobi bundle, and let |$N$| be a sufficiently small transversal at |$x_0 \in \mathcal O$| to a |$(2d + 1)$|-dimensional characteristic leaf |$\mathcal O$| of |$J$|⁠. Then there is |$\triangleright$| a homogenous Poisson structure |$(\pi _N, Z_N)$| on |$N$|⁠, |$\triangleright$| an open neighborhood |$V$| of |$0$| in |$\mathbb{R}^{2d +1}$|⁠, and |$\triangleright$| a line bundle isomorphism |$\Phi : L \to \mathbb{R}_{N \times V}$|⁠, covering a diffeomorphism |$\phi : M \to N \times V$|⁠, locally defined around |$x_0$|⁠, such that |$\phi$| identifies |$N$| with |$N \times \{0\}$|⁠, and (a neighborhood of |$x_0$| in) |$\mathcal O$| with |$\{x_0\} \times V$|⁠, |$\Phi$| identifies |$J$| with the Jacobi structure |$J^\times$| corresponding to the Jacobi pair |$(\Lambda ^\times , E^\times )$| given by \begin{equation} \Lambda^\times = \Lambda_{\textit{can}} + \pi_N - E_{\textit{can}} \wedge Z_N, \quad \textrm{and} \quad E^\times = E_{\textit{can}}, \end{equation} (4.1) where |$(\Lambda _{\textit{can}}, E_{\textit{can}})$| is the Jacobi pair from Example 1.2.3. Remark 4.0.2. Formula (4.1) has a nice interpretation in terms of Dirac–Jacobi structures. Namely, let |$\mathfrak L_{J_{\textit{can}}} = \operatorname{\textsf{graph}} J_{\textit{can}} \subset \mathbb{D} \mathbb{R}_V$| be the Dirac–Jacobi structure induced by |$J_{\textit{can}}$| on the trivial line bundle, and let |$\mathfrak L_N \subset \mathbb{D} \mathbb{R}_{N}$| be the Dirac–Jacobi structure spanned as follows: \begin{equation*} \mathfrak L_N = \left\langle (1-Z_N, 0), (\pi_N^\sharp \eta, \eta + \eta (Z_N) \cdot \mathfrak j ): \eta \in T^\ast N \right\rangle. \end{equation*} Additionally, let |$\mathfrak L^\times = \operatorname{\textsf{graph}} J^\times \subset \mathbb{D} \mathbb{R}_{N \times V}$|⁠. Then |$\mathfrak L^\times$| is the flat product of |$\mathfrak L_N$| and |$\mathfrak L_{J_{\textit{can}}}$| with respect to the standard projections |$\mathbb{R}_{N \times V} \to \mathbb{R}_N$|⁠, and |$\mathbb{R}_{N \times V} \to \mathbb{R}_V$| \begin{equation*} \mathfrak L^\times = \mathfrak L_N \times^! \mathfrak L_{J_{\textit{can}}}. \end{equation*} Notice that |$\mathfrak{L}_N=\phi |_N^!\mathfrak{L}^\times$|⁠. We stress that it is not a coincidence that |$\phi |_N^!\mathfrak{L}^\times$| is a Dirac structure coming from a homogeneous Poisson structure. It is proven in [14] that every transversal to a contact leaf of a Jacobi structure possesses, at least locally, a homogeneous Poisson structure and that different transversals possess isomorphic homogeneous Poisson structures. A similar statement holds for Dirac–Jacobi structures [37, Proposition 6.9]. In a similar way, every transversal to an lcs leaf possesses a Jacobi structure (see [14] again, and [37, Proposition 6.9] for the Dirac–Jacobi case). Theorem 4.0.3. Let |$(L \to M, J)$| be a Jacobi bundle, and let |$N$| be a sufficiently small transversal at |$x_0 \in \mathcal O$| to a |$2d$|-dimensional characteristic leaf |$\mathcal O$| of |$J$|⁠. Then there is |$\triangleright$| a Jacobi pair |$(\Lambda _N, E_N)$| on |$N$|⁠, |$\triangleright$| an open neighborhood |$V$| of |$0$| in |$\mathbb{R}^{2d}$|⁠, and |$\triangleright$| a line bundle isomorphism |$\Phi : L \to \mathbb{R}_{N \times V}$|⁠, covering a diffeomorphism |$\phi : M \to N \times V$|⁠, locally defined around |$x_0$|⁠, such that |$\phi$| identifies |$N$| with |$N \times \{0\}$|⁠, and (a neighborhood of |$x_0$| in) |$\mathcal O$| with |$\{x_0\} \times V$|⁠, |$\Phi$| identifies |$J$| with the Jacobi structure |$J^\times$| corresponding to the Jacobi pair |$(\Lambda ^\times , E^\times )$| given by \begin{equation} \Lambda^\times = \Lambda_{N} + \pi_{\textit{can}} - E_{N} \wedge Z_{\textit{can}}, \quad \textrm{and} \quad E^\times = E_{N}, \end{equation} (4.2) where |$(\pi _{\textit{can}}, Z_{\textit{can}})$| is the homogeneous Poisson structure from Example 2.2.14. Remark 4.0.4. Again, formula (4.2) has an interpretation in terms of Dirac–Jacobi structures. Namely, let |$\mathfrak L_{(\pi _{\textit{can}}, Z_{\textit{can}})} \subset \mathbb{D} \mathbb{R}_{V}$| be the Dirac–Jacobi structure spanned as follows: \begin{equation*} \mathfrak L_{(\pi_{\textit{can}}, Z_{\textit{can}})} = \left\langle (1-Z_{\textit{can}}, 0), (\pi_{\textit{can}}^\sharp \eta, \eta + \eta (Z_{\textit{can}}) \cdot \mathfrak j ): \eta \in T^\ast V \right\rangle \end{equation*} and let |$\mathfrak L_{N} = \operatorname{\textsf{graph}} J_{N} \subset \mathbb{D} \mathbb{R}_N$| be the Dirac–Jacobi structure induced by |$J_{N}$|⁠. Finally, let |$\mathfrak L^\times = \operatorname{\textsf{graph}} J^\times \subset \mathbb{D} \mathbb{R}_{N \times V}$|⁠. Then |$\mathfrak L^\times$| is the flat product of |$\mathfrak L_{(\pi _{\textit{can}}, Z_{\textit{can}})}$| and |$\mathfrak L_N$| with respect to the standard projections |$\mathbb{R}_{N \times V} \to \mathbb{R}_N$|⁠, and |$\mathbb{R}_{N \times V} \to \mathbb{R}_V$|⁠: \begin{equation*} \mathfrak L^\times = \mathfrak L_{(\pi_{\textit{can}}, Z_{\textit{can}})}\times^! \mathfrak L_N. \end{equation*} 4.1 Splitting around a contact point We are finally ready to prove our main results. We begin with a remark. Remark 4.1.1. The Jacobi structure |$J_{\textit{can}}$| from Example 1.2.3 is nondegenerate. Hence, it corresponds to a contact structure |$H_{\textit{can}}$|⁠. Namely, let |$\omega _{\textit{can}} = J_{\textit{can}}^{-1}$| be the Atiyah |$2$|-form inverting |$J_{\textit{can}}$|⁠. Then \begin{equation*} \omega_{\textit{can}} = dx^i \wedge dp_i - (du - p_i dx^i) \wedge \mathfrak j, \end{equation*} and |$\iota$|𝟙|$\omega _{\textit{can}}$| agrees with \begin{equation*} \theta_{\textit{can}} = du - p_i dx^i, \end{equation*} the canonical contact |$1$|-form on |$\mathbb{R}^{2d +1}$|⁠, and |$H_{\textit{can}} = \ker \theta _{\textit{can}}$| is the canonical contact structure on |$\mathbb{R}^{2d +1}$|⁠. The latter can be equivalently encoded in a generalized contact structure \begin{equation} \left( \begin{array}{cc} 0 & J_{\textit{can}}^\sharp \\ -(\omega_{\textit{can}})_\flat & 0 \end{array} \right) \end{equation} (4.3) whose |$+\textrm{i}$|-eigenbundle is \begin{equation*} \mathfrak L_{\textit{can}}^{\textit{odd}} = \left\{ (J^\sharp_{\textit{can}} (\psi), \textrm{i} \psi): \psi \in J^1 \mathbb{R}_{\mathbb{R}^{2d + 1}} \otimes \mathbb{C} \right\}. \end{equation*} Clearly, we also have \begin{equation} \mathfrak L_{\textit{can}}^{\textit{odd}} = \left\{ (\Delta, \textrm{i} \cdot \iota_{\Delta}\omega_{\textit{can}}): \Delta \in D\mathbb{R}_{\mathbb{R}^{2d + 1}} \otimes \mathbb{C}\right\} = (D\mathbb{R}_{\mathbb{R}^{2d + 1}} \otimes \mathbb{C})^{\textrm{i} \omega_{\textit{can}}}, \end{equation} (4.4) that is, |$\mathfrak L_{\textit{can}}^{\textit{odd}}$| can be seen as the complex|$B$|-field transformation of the complex Dirac–Jacobi structure |$D\mathbb{R}_{\mathbb{R}^{2d + 1}} \otimes \mathbb{C} \subset \mathbb{D} \mathbb{R}_{\mathbb{R}^{2d + 1}}\otimes \mathbb{C}$| by means of the closed complex Atiyah |$2$|-form |$\textrm{i} \omega _{\textit{can}}$|⁠. This simple remark will be useful below. Actually, similar considerations hold for any nondegenerate Jacobi structure. Notice, however, that (4.4) does not mean that there is a Courant–Jacobi automorphism intertwining (4.3) with some other generalized contact structure. Yet in other words, |$D\mathbb{R}_{\mathbb{R}^{2d + 1}} \otimes \mathbb{C}$| is not a complex Dirac–Jacobi structure of generalized contact type, and the obvious reason is that only real|$B$|-field transformations are Courant–Jacobi automorphisms, while |$\textrm{i} \omega _{\textit{can}}$| is a purely imaginary Atiyah |$2$|-form. Theorem 4.1.2. Let |$(L \to M, \mathbb{K})$| be a generalized contact bundle, let |$\mathfrak L \subset \mathbb{D} L\otimes \mathbb{C}$| be the |$+\textrm{i}$|-eigenbundle of |$\mathbb{K}$|⁠, and let |$N$| be a sufficiently small transversal at |$x_0 \in \mathcal O$| to a |$(2d + 1)$|-dimensional characteristic leaf |$\mathcal O$|⁠. Then there is |$\triangleright$| an open neighborhood |$U$| of |$0$| in |$\mathbb{R}^{2d +1}$|⁠, |$\triangleright$| a line bundle isomorphism |$\Phi : L \to \mathbb{R}_{N \times U}$|⁠, covering a diffeomorphism |$\phi : M \to N \times U$|⁠, locally defined around |$x_0$|⁠, and |$\triangleright$| a closed Atiyah |$2$|-form |$B$| on |$\mathbb{R}_{N \times U}$| such that |$\phi$| identifies |$N$| with |$N \times \{0\}$|⁠, and (a neighborhood of |$x_0$| in) |$\mathcal O$| with |$\{x_0\} \times U$|⁠, the Courant–Jacobi automorphism |$e^B \circ \mathbb{D} \Phi$| identifies |$\mathfrak L$| with \begin{equation} \mathfrak L_N \times^! \mathfrak L_{\textit{can}}^{\textit{odd}}, \end{equation} (4.5) the flat product of |$\mathfrak L_N$| and |$\mathfrak L_{\textit{can}}^{\textit{odd}}$| with respect to the standard projections |$P_N: \mathbb{R}_{N \times U} \to \mathbb{R}_N$|⁠, and |$P_U: \mathbb{R}_{N \times U} \to \mathbb{R}_U$|⁠. Here |$\mathfrak L_N = I_N^! \mathfrak L$| is the complex Dirac–Jacobi structure of homogeneous generalized complex type induced by |$\mathfrak L$| on |$N$| (see Proposition 3.1.2), and |$\mathfrak L_{\textit{can}}^{\textit{odd}}$| is the complex Dirac–Jacobi structure of generalized contact type from Remark 4.1.1. Proof. The present proof and, similarly, the proof of Theorem 4.2.2 below are inspired by a general technique recently proposed by Bursztyn et al. to prove splitting theorems in Poisson and related geometries. Without loss of generality, we can assume that |$M = N \times V$|⁠, |$L = \mathbb{R}_{N \times V}$| is the trivial line bundle, and the Jacobi structure underlying |$\mathbb{K}$| is |$J^\times$|⁠, where |$V$|⁠, |$N$| and |$J^\times$| are as in Theorem 4.0.1. Now let |$\psi \in \Gamma (J^1 \mathbb{R}_{N \times V})$| be given by \begin{equation*} \psi = x^i dp_i - p_i dx^i + \left(x^i p_i -u \right) \cdot \mathfrak j. \end{equation*} Put |$\mathcal E:= J^\sharp \psi$|⁠. Then \begin{equation} \mathcal E = x^i \frac{\partial}{\partial x^i} + u \frac{\partial}{\partial u} + p_i \frac{\partial}{\partial p_i} \end{equation} (4.6) is the Euler vector field on |$V$|⁠. More precisely, it is the covariant derivative along the Euler vector field with respect to the canonical flat connections in |$\mathbb{R}_{N \times V}$|⁠. By changing |$\psi$| into |$f \psi$| with |$f \in C^\infty (V \times N)$| a suitable bump function equal to |$1$| around |$N$|⁠, we can arrange that |$\mathcal E$| is complete, while (4.6) still holds around |$N$|⁠. Denote by |$\{ \Phi _t \}$| the flow of |$\mathcal E$| on |$\mathbb{R}_{N \times U}$|⁠, and let |$\{ \phi _t \}$| be its projection to |$N \times V$|⁠. Then, for all |$t \leq 0$| we have \begin{equation*} \Phi_t (x, v\,; r) = (x, \textrm{exp} (t) \cdot v\,; r), \quad (x, v\,; r) \in N \times V \times \mathbb{R}, \end{equation*} at least when |$v$| is small enough. Put \begin{equation*} U:= \left\{ v \in V: \lim_{t \to - \infty} \phi_t (x, v) \in N \times \{0\} \text{ for all }x \in N\right\}. \end{equation*} Then |$U \subset V$| is an open subset and |$\mathcal E$| remains complete when restricted to |$U$|⁠. Additionally, the family of maps \begin{equation*} K_s:= \Phi_{\log (s)}: \mathbb{R}_{N \times U} \to \mathbb{R}_{N \times U} \end{equation*} extends smoothly to |$s = 0$|⁠, and |$K_0 = I_N \circ P_N$|⁠, where |$P_N: \mathbb{R}_{N \times U} \to \mathbb{R}_N$| is the canonical projection and |$I_N: \mathbb{R}_N \to \mathbb{R}_{N \times U}$| is the embedding at |$v = 0 \in U$|⁠. Now consider the |$+\textrm{i}$|-eigensection |$\alpha$| of |$\mathbb{K}$| given by \begin{equation} \textrm{i} (0, \psi) + \mathbb{K} (0, \psi) = (\mathcal E, \chi) \in \Gamma (\mathfrak L), \end{equation} (4.7) where we put |$\chi := \textrm{i} \psi - \varphi ^\dagger \psi$|⁠. Consider also the infinitesimal Courant–Jacobi automorphism \begin{equation} (\mathcal L_{\mathcal E} - \overline{d_D \chi}, \mathcal E) = \left([ \! [ (\mathcal E, \chi), - ] \! ], \mathcal E \right). \end{equation} (4.8) From (4.7), (4.8), and involutivity, the flow of |$(\mathcal L_{\mathcal E} - \overline{d_D \chi }, \mathcal E)$| preserves |$\mathfrak L$|⁠. From Remark 2.1.9 this flow is \begin{equation*} \{ (e^{C_t} \circ \mathbb{D} \Phi_t, \Phi_t) \}, \quad \textrm{where}\ C_t = \int_{0}^t \Phi_{-\epsilon}^\ast (d_D \chi) d \epsilon. \end{equation*} In particular \begin{equation} \mathfrak L = (\mathbb{D}\Phi_{- \log (s)} \mathfrak L)^{C_{- \log (s)}} = (K_s^! \mathfrak L)^{C_{-\log (s)}}, \end{equation} (4.9) for all |$s> 0$|⁠. Put |$B_s:= C_{- \log (s)}$| and compute \begin{equation*} \begin{aligned} B_s & = \int_{0}^{- \log (s)} \Phi_{- \epsilon}^\ast (d_D \chi) d \epsilon = \int_s^1 \tau^{-1} K_\tau^\ast (d_D \chi) d\tau \\ & = \textrm{i} \int_s^1 \tau^{-1} K_\tau^\ast (d_D \psi) d\tau - \int_s^1 \tau^{-1} K_\tau^\ast (d_D \varphi^\dagger \psi) d\tau. \end{aligned} \end{equation*} In a possibly smaller neighborhood of |$N \times \{0\}$| we have \begin{equation*} K_\tau^\ast (d_D \psi) = K_\tau^\ast \left(2 dx^i \wedge dp_i + (2p_i dx^i - du) \wedge \mathfrak j \right) = 2\tau^2 dx^i \wedge dp_i + (2\tau^2 p_i dx^i - \tau du) \wedge \mathfrak j, \end{equation*} for all |$\tau \in [0,1]$|⁠. Hence, for all |$s \in (0, 1]$|⁠, \begin{equation*} \int_s^1 \tau^{-1} K_\tau^\ast (d_D \psi) d\tau = (1-s) \left((1+s)dx^i \wedge dp_i - (du - (1+s)p_i dx^i) \wedge \mathfrak j \right), \end{equation*} which extends to |$s = 0$|⁠. We conclude that, in a possibly smaller neighborhood of |$N \times \{0\}$|⁠, |$B_0$| is well defined, and, more precisely, \begin{equation*} B_0 = B + \textrm{i} \omega_{\textit{can}}, \end{equation*} where |$B$| is a certain real closed Atiyah |$2$|-form. Finally, from (4.9), by continuity, we get, in a neighborhood of |$N \times \{0\}$| \begin{equation*} \begin{aligned} \mathfrak L & = (K_0^! \mathfrak L)^{B_0} \\ & = (P_N^! I_N^! \mathfrak L)^{B + \textrm{i} \omega_{\textit{can}}} \\ & = ( P_N^! \mathfrak L_N \star D \mathbb{R}_{N \times U} \otimes \mathbb{C})^{B + \textrm{i} \omega_{\textit{can}}} \quad \text{(Remark 2.2.9)} \\ & = (P_N^! \mathfrak L_N \star P_U^! (D \mathbb{R}_{U} \otimes \mathbb{C}))^{B + \textrm{i} \omega_{\textit{can}}} \\ & = (\mathfrak L_N \times^! D \mathbb{R}_{U}\otimes \mathbb{C})^{B + \textrm{i} \omega_{\textit{can}}} \\ & = (\mathfrak L_N \times^! (D \mathbb{R}_{U}\otimes \mathbb{C})^{\textrm{i} \omega_{\textit{can}}})^B \quad \text{(Remark 2.2.12)}\\ & = (\mathfrak L_N \times^! \mathfrak L_{\textit{can}}^{\textit{odd}})^B \quad \text{(Equation 4.4)}. \end{aligned} \end{equation*} 4.2 Splitting around an lcs point Remark 4.2.1. Consider the homogeneous Poisson structure |$(\pi _{\textit{can}}, Z_{\textit{can}})$| from Example 2.2.14. The Poisson structure |$\pi _{\textit{can}}$| is nondegenerate and its inverse is |$\Omega _{\textit{can}} = dx^i \wedge dp_i$|⁠, the canonical symplectic structure on |$\mathbb{R}^{2d}$|⁠. In its turn |$\Omega _{\textit{can}} = - d \Theta _{\textit{can}}$|⁠, where \begin{equation*} \Theta_{\textit{can}} = p_i dx^i \end{equation*} is the Liouville |$1$|-form. The pair |$(\Omega _{\textit{can}}, Z_{\textit{can}})$| is a homogeneous symplectic structure in the sense that |$\mathcal L_{Z_{\textit{can}}}\Omega _{\textit{can}} = \Omega _{\textit{can}}$|⁠, and we can encode it in a complex Dirac–Jacobi structure of homogeneous generalized complex type \begin{equation*} \mathfrak L_{\textit{can}}^{\textit{ev}}:= \left\langle (1-Z_{\textit{can}}, 0), (\pi_{\textit{can}}^\sharp \eta, \textrm{i} \cdot \eta): \eta \in T^\ast \mathbb{R}^{2d} \otimes \mathbb{C} \right\rangle. \end{equation*} Now, consider the exact Atiyah |$2$|-form |$\xi _{\textit{can}} = - d_D \Theta _{\textit{can}}$|⁠. It is easy to see that \begin{equation*} \mathfrak L_{\textit{can}}^{\textit{ev}} = \left\{ (\Delta, \textrm{i} \cdot \iota_\Delta \xi_{\textit{can}}): \Delta \in D \mathbb{R}_{\mathbb{R}^{2d}} \otimes \mathbb{C} \right\} = (D\mathbb{R}_{\mathbb{R}^{2d}} \otimes \mathbb{C})^{\textrm{i} \xi_{\textit{can}}}, \end{equation*} that is, |$\mathfrak L_{\textit{can}}^{\textit{ev}}$| is the complex |$B$|-field transformation of the complex Dirac–Jacobi structure |$D\mathbb{R}_{\mathbb{R}^{2d}} \otimes \mathbb{C} \subset \mathbb{D} \mathbb{R}_{\mathbb{R}^{2d}}\otimes \mathbb{C}$| by means of the complex closed Atiyah |$2$|-form |$\textrm{i} \xi _{\textit{can}}$|⁠. Similar considerations hold for any homogeneous Poisson structure |$(\pi , Z)$| such that |$\pi$| is nondegenerate. We leave the simple details to the reader. Theorem 4.2.2. Let |$(L \to M, \mathbb{K})$| be a generalized contact bundle, let |$\mathfrak L \subset \mathbb{D} L\otimes \mathbb{C}$| be the |$+\textrm{i}$|-eigenbundle of |$\mathbb{K}$|⁠, and let |$N$| be a sufficiently small transversal at |$x_0 \in \mathcal O$| to a |$2d$|-dimensional characteristic leaf |$\mathcal O$|⁠. Then there is |$\triangleright$| an open neighborhood |$U$| of |$0$| in |$\mathbb{R}^{2d}$|⁠, |$\triangleright$| a line bundle isomorphism |$\Phi : L \to \mathbb{R}_{N \times U}$|⁠, covering a diffeomorphism |$\phi : M \to N \times U$|⁠, locally defined around |$x_0$|⁠, and |$\triangleright$| a closed Atiyah |$2$|-form |$B$| on |$\mathbb{R}_{N \times U}$| such that |$\phi$| identifies |$N$| with |$N \times \{0\}$|⁠, and |$\mathcal O$| with |$\{x_0\} \times U$|⁠, the Courant–Jacobi automorphism |$e^B \circ \mathbb{D} \Phi$| identifies |$\mathfrak L$| with \begin{equation} \mathfrak L_N \times^! \mathfrak L_{\textit{can}}^{\textit{ev}}, \end{equation} (4.10) the flat product of |$\mathfrak L_N$| and |$\mathfrak L_{\textit{can}}^{\textit{ev}}$| with respect to the standard projections |$P_N: \mathbb{R}_{N \times U} \to \mathbb{R}_N$|⁠, and |$P_U: \mathbb{R}_{N \times U} \to \mathbb{R}_U$|⁠. Here |$\mathfrak L_N = I_N^! \mathfrak L$| is the complex Dirac–Jacobi bundle structure of generalized contact type induced by |$\mathfrak L$| on |$N$| (see Proposition 3.2.3), and |$\mathfrak L_{\textit{can}}^{\textit{ev}}$| is the complex Dirac–Jacobi structure of homogeneous generalized complex type from Remark 4.2.1. Proof. Without loss of generality, we assume that |$M = N \times V$|⁠, |$L = \mathbb{R}_{N \times V}$| is the trivial line bundle, and the Jacobi structure underlying |$\mathbb{K}$| is |$J^\times$|⁠, where |$V$|⁠, |$N$| and |$J^\times$| are as in Theorem 4.0.3. Let |$\psi \in \Gamma (J^1 \mathbb{R}_{N \times V})$| be given by \begin{equation*} \psi = x^i dp_i - p_i dx^i + \left(x^i p_i \right) \cdot \mathfrak j. \end{equation*} So that \begin{equation*} \mathcal E:= J^\sharp \psi = x^i \frac{\partial}{\partial x^i} + p_i \frac{\partial}{\partial p_i} \end{equation*} is the Euler vector field on |$V$|⁠. We define |$U$|⁠, |$K_0$|⁠, |$B_0$| exactly as in the proof of Theorem 4.1.2. A direct computation then shows that |$B_0$| is well defined around |$N \times \{0\}$| and it is given by \begin{equation*} B_0 = B + \textrm{i} \xi_{\textit{can}} \end{equation*} for some real closed Atiyah |$2$|-form |$B$|⁠. Exactly as in the proof of Theorem 4.1.2 we now get \begin{equation*} \mathfrak L = (K_0^! \mathfrak L)^{B_0} = (\mathfrak L_N \times^! \mathfrak L_{\textit{can}}^{ev})^B. \end{equation*} 5 The Regular Case Let |$(L \to M, \mathbb{K})$| be a generalized contact bundle, and let |$J$| be the Jacobi structure underlying |$\mathbb{K}$|⁠. A point |$x_0 \in M$| is a regular point for |$\mathbb{K}$| if the characteristic leaves of |$J$| has constant dimension around |$x_0$|⁠. Similarly as in the generalized complex case [20], when |$x_0 \in M$| is a regular point, the Splitting Theorems 4.1.2 and 4.2.2 simplify and we get honest local normal form theorems around |$x_0$|⁠. 5.1 Local normal form around a regular contact point Remark 5.1.1. Denote by |$A_{\textit{can}}$| the standard complex structure on |$\mathbb{C}^n$|⁠. It can be encoded in a generalized complex structure \begin{equation} \left( \begin{array}{cc} A_{\textit{can}} & 0 \\ 0 & - A_{\textit{can}}^\ast \end{array} \right) \end{equation} (5.1) whose |$+\textrm{i}$|-eigenbundle is |$T^{1,0} \mathbb{C}^n \oplus (T^{0,1} \mathbb{C}^n)^\ast$|⁠. The generalized complex structure (5.1) is homogeneous with respect to the zero section |$(0,0) \in \Gamma (\mathbb{T} \mathbb{C}^n)$|⁠, and we get the following complex Dirac–Jacobi structure of homogeneous generalized complex type on |$\mathbb{R}_{\mathbb{C}^n}$|⁠: \begin{equation*} \mathfrak L_{\mathbb{C}^n}:= \left\langle (1, 0), (X, \eta): X \in T^{1,0} \mathbb{C}^n, \textrm{ and} \ \eta \in (T^{0,1} \mathbb{C}^n)^\ast \right\rangle. \end{equation*} Theorem 5.1.2. Let |$(L \to M, \mathbb{K})$| be a generalized contact bundle with |$\dim M = 2 (n+d) +1$|⁠. Let |$\mathfrak L \subset \mathbb{D} L\otimes \mathbb{C}$| be the |$+\textrm{i}$|-eigenbundle of |$\mathbb{K}$|⁠, and let |$x_0 \in M$| be a regular point in a |$(2d + 1)$|-dimensional characteristic leaf. Then, locally, around |$x_0$|⁠, |$\mathfrak L$| is isomorphic to the flat product \begin{equation*} \mathfrak L_{\mathbb{C}^n} \times^! \mathfrak L_{\textit{can}}^{\textit{odd}} \end{equation*} with respect to the standard projections |$\mathbb{R}_{\mathbb{C}^n \times \mathbb{R}^{2d +1}} \to \mathbb{R}_{\mathbb{C}^n}$|⁠, and |$\mathbb{R}_{\mathbb{C}^n \times \mathbb{R}^{2d +1}} \to \mathbb{R}_{\mathbb{R}^{2d +1}}$|⁠, up to a |$B$|-field transformation. Proof. Let |$N$| be a sufficiently small transversal to the characteristic leaf through |$x_0$|⁠. From Theorem 4.1.2, it is enough to show that the induced Dirac–Jacobi structure |$\mathfrak L_N = I^!_N \mathfrak L$| on |$N$| is isomorphic to |$\mathfrak L_{\mathbb{C}^n}$| around |$x_0$| up to a |$B$|-field transformation. From the proof of Proposition 3.1.2 and the fact that |$x_0$| is a regular point, it easily follows that |$\mathfrak L_N \cap \overline{\mathfrak L_N}$| is (everywhere, not only at |$x_0$|⁠) spanned by a section of the form (𝟙, |$\zeta )$|⁠, with |$\zeta \in T^\ast N \otimes L_N$|⁠, and, from the proof of Proposition 2.2.19, |$\mathfrak L_N$| is isomorphic to a Dirac–Jacobi structure of generalized complex type of the form |$\mathfrak L_{(\mathbb{J}, \mathbb{Z})}$| (see (2.13)) with |$Z = 0$|⁠. In particular, |$\pi = 0$|⁠, |$A$| is a complex structure on |$N$|⁠, and |$\sigma = \iota _A d \zeta$| (see Definition 2.2.15). A direct computation exploiting (1) and (3) shows that, after the |$B$|-field transformation |$(e^{d_D \zeta }, \textrm{id})$|⁠, we achieve |$\zeta = \sigma = 0$|⁠. Finally, with a diffeomorphism, we achieve |$A = A_{\textit{can}}$|⁠, showing that |$\mathfrak L_N$| is isomorphic to |$\mathfrak L_{\mathbb{C}^n}$| up to a |$B$|-field transformation. 5.2 Local normal form around a regular lcs point Remark 5.2.1. Consider the standard complex structure |$\varphi _{\textit{can}}$| on the gauge algebroid of the trivial line bundle over the cylinder |$\mathbb{R} \times \mathbb{C}^n$| from Example A.0.1 in the Appendix. It can be encoded in a generalized contact structure \begin{equation} \left( \begin{array}{cc} \varphi_{\textit{can}} & 0 \\ 0 & - \varphi_{\textit{can}}^\dagger \end{array} \right) \end{equation} (5.2) whose |$+\textrm{i}$|-eigenbundle is \begin{equation*} \mathfrak L_{\mathbb{R} \times \mathbb{C}^n} = D^{1,0} \mathbb{R}_{\mathbb{R} \times \mathbb{C}^n} \oplus (D^{0,1} \mathbb{R}_{\mathbb{R} \times \mathbb{C}^n})^\ast \end{equation*} (see Appendix A for more details). Theorem 5.2.2. Let |$(L \to M, \mathbb{K})$| be a generalized contact bundle with |$\dim M = 2 (n+d) +1$|⁠. Let |$\mathfrak L \subset \mathbb{D} L\otimes \mathbb{C}$| be the |$+\textrm{i}$|-eigenbundle of |$\mathbb{K}$|⁠, and let |$x_0 \in M$| be a regular point in a |$2d$|-dimensional characteristic leaf. Then, locally, around |$x_0$|⁠, |$\mathfrak L$| is isomorphic to the flat product \begin{equation*} \mathfrak L_{\mathbb{R} \times \mathbb{C}^n} \times^! \mathfrak L_{\textit{can}}^{\textit{ev}} \end{equation*} with respect to the standard projections |$\mathbb{R}_{(\mathbb{R} \times \mathbb{C}^n) \times \mathbb{R}^{2d}} \to \mathbb{R}_{\mathbb{R} \times \mathbb{C}^n}$|⁠, and |$\mathbb{R}_{(\mathbb{R} \times \mathbb{C}^n) \times \mathbb{R}^{2d}} \to \mathbb{R}_{\mathbb{R}^{2d}}$|⁠, up to a |$B$|-field transformation. Proof. Let |$N$| be a sufficiently small transversal to the characteristic leaf through |$x_0$|⁠. From Theorem 4.2.2, it is enough to show that the induced Dirac–Jacobi structure |$\mathfrak L_N = I^!_N \mathfrak L$| on |$N$| is isomorphic to |$\mathfrak L_{\mathbb{R} \times \mathbb{C}^n}$| around |$x_0$| up to a |$B$|-field transformation. As |$x_0$| is a regular point, the characteristic foliation |$\mathcal F$| is a regular lcs foliation around |$x_0$|⁠. In particular |$p_D \mathfrak L \cap p_D \overline{\mathfrak L} = \operatorname{im} \nabla _{\mathcal F}$|⁠, where |$\nabla _{\mathcal F}$| is a flat leaf-wise connection along |$\mathcal F$| in |$L$|⁠. Hence, |$p_D \mathfrak L_N \cap p_D \overline{\mathfrak L_N} = p_D \mathfrak L \cap p_D \overline{\mathfrak L} \cap D L_N = \operatorname{im} \nabla _{\mathcal F} \cap D L_N = 0$|⁠. This means that |$\mathfrak L_N$| is the |$+\textrm{i}$|-eigenbundle of a generalized contact structure |$\mathbb{K}_N$| of the form \begin{equation} \mathbb{K}_N = \left( \begin{array}{cc} \varphi_N & 0 \\ (\omega_N)_\flat & - \varphi_N^\dagger \end{array} \right). \end{equation} (5.3) In particular |$\varphi _N$| is an integrable complex structure on the Atiyah algebroid |$DL_N$|⁠. In the following we refer to the Appendix for notation and the main properties of such a complex structure. First of all, notice that from (5.3) we have |$p_D \mathfrak L_N = D^{(1,0)} L_N$| (recall from the Appendix that |$D^{(1,0)} L_N$| denotes the |$+\textrm{i}$|-eigenbundle of |$\varphi _N$|⁠). Define a complex Atiyah |$2$|-form |$\gamma \in \Omega ^{(2,0)}_{L_N} \otimes \mathbb{C}$| by putting \begin{equation*} \gamma (\Delta, \nabla) = \langle \psi, \nabla \rangle, \quad \Delta, \nabla \in D^{(1,0)}L_N, \end{equation*} where |$\psi \in J^1 L \otimes \mathbb{C}$| is any element such that |$(\Delta , \psi ) \in \mathfrak L_N$|⁠. A straightforward computation using the involutivity of |$\mathfrak L_N$| shows that |$\partial _D \gamma = 0$|⁠, and, from Remark A.2.2, locally around |$x_0$|⁠, there is |$\rho \in \Omega ^{(1,0)}_{L_N}$| such that |$\gamma = \partial _D \rho$|⁠. It is easy to see that \begin{equation*} B:= - 2 \operatorname{Re} \left(\gamma + \overline \partial_D \rho \right) \end{equation*} is a (real) closed Atiyah |$2$|-form. We claim that \begin{equation} \mathfrak L_N^B = D^{(1,0)} L_N \oplus \textsf{Hom}(D^{(0,1)} L_N, L_N). \end{equation} (5.4) This follows, after a simple computation, from the remark that \begin{equation*} \mathfrak L_N = \operatorname{\textsf{graph}} \gamma \oplus \textsf{Hom}(D^{(0,1)} L_N, L_N) \end{equation*} and the fact that |$(\gamma + B)_\flat$| takes values in |$\textsf{Hom}(D^{(0,1)} L_N, L_N)$|⁠. Notice that (5.4) means that |$\mathfrak L_N^B$| is the |$+\textrm{i}$|-eigenbundle of the generalized contact structure \begin{equation*} \left( \begin{array}{cc} \varphi_N & 0 \\ 0 & - \varphi_N^\dagger \end{array} \right). \end{equation*} Finally, in view of Theorem A.1.1, with a line bundle isomorphism we can achieve |$\varphi _N = \varphi _{\textit{can}}$|⁠, and this concludes the proof. Appendix A. Complex Structures on the Gauge Algebroid Let |$L \to M$| be a line bundle. In this appendix we study the local properties of a generalized contact structure of complex type, that is, a generalized contact structure |$\mathbb{K}$| on |$L$|⁠, of the form \begin{equation} \mathbb{K} = \left( \begin{array}{cc} \varphi & 0 \\ 0 & -\varphi^\dagger \end{array} \right). \end{equation} (A.1) In this case |$\varphi : DL \to DL$| is a(n integrable) complex structure on the gauge algebroid |$DL$|⁠, that is, |$\triangleright$||$\varphi$| is almost complex, that is, |$\varphi ^2 = - \textrm{id}$|⁠, |$\triangleright$||$\varphi$| is integrable, that is, its Lie algebroid Nijenhuis torsion|$\mathcal N_\varphi$| vanishes. Here |$\mathcal N_\varphi : \wedge ^2 DL \to DL$| is the skew-symmetric bilinear map defined by \begin{equation*} \mathcal N_\varphi (\Delta, \nabla) = [\varphi \Delta, \varphi \nabla] - [\Delta, \nabla] - \varphi \left( [\varphi \Delta, \nabla] + [\Delta, \varphi \nabla] \right), \quad \Delta, \nabla \in \Gamma(DL). \end{equation*} Conversely, given a complex structure on |$DL$|⁠, (A.1) defines a generalized contact structure. Example A.0.1. Consider the cylinder |$\mathbb{R} \times \mathbb{C}^n$| over the standard complex space |$\mathbb{C}^n$|⁠. Let |$u$| be the standard real coordinate on the 1st factor, and let |$z^i = x^i + \textrm{i} y^i$|⁠, |$i = 1, \ldots , n$|⁠, be the standard complex coordinates on the 2nd factor. There is a canonical integrable complex structure |$\varphi _{\textit{can}}$| on the gauge algebroid of the trivial line bundle |$\mathbb{R}_{\mathbb{R} \times \mathbb{C}^n}$| defined by Example A.0.2 (Normal almost contact structures). Our main reference for this example is [22], where the reader will find basically all the proofs. We will see in this example and Lemma A.0.3 that almost contact structures (resp. normal almost contact structures) are locally the same as almost complex structures (resp. integrable almost complex structures) on the gauge algebroid of a trivial line bundle |$\mathbb{R}_M \to M$|⁠. Recall that an almost contact structure on a manifold |$M$| is a triple |$(\Phi , \xi , \eta )$|⁠, where |$\Phi : TM \to TM$| is a |$(1,1)$|-tensor, |$\xi$| is a vector field, and |$\eta$| is a |$1$|-form on |$M$| such that \begin{equation*} \Phi^2=-\textrm{id} +\eta\otimes \xi, \quad \Phi(\xi)=0,\quad \eta\circ\Phi=0, \quad \textrm{and} \quad \eta(\xi)=1. \end{equation*} See, for example, [7] for more details. The idea behind this definition is that an almost contact structure is the odd-dimensional analogue of an almost complex structure. We believe that the use of line bundles and their gauge algebroids makes the analogy much more transparent. Namely, recall that the gauge algebroid of the trivial line bundle is |$D \mathbb{R}_M \cong TM \oplus \mathbb{R}_M$|⁠. Now take a triple |$(\Phi , \xi , \eta )$| consisting of an |$(1,1)$|-tensor, a vector field and a |$1$|-form on |$M$|⁠, and let |$\varphi : D\mathbb{R}_M \to D\mathbb{R}_M$| be the endomorphism given by \begin{equation} \varphi (X,r) = (\Phi(X)-r\xi, \eta(X)). \end{equation} (A.2) Then |$(\Phi , \xi , \eta )$| is an almost contact structure if and only if |$\phi$| is an almost complex structure, that is, |$\varphi ^2=-\textrm{id}$|⁠. Additionally, |$\varphi$| is integrable if and only if \begin{equation} \mathcal N_\Phi+d\eta\otimes\xi=0,\quad d\eta(\Phi-,-)+d\eta(-,\Phi-)=0,\quad \mathcal{L}_\xi \Phi=0, \quad \textrm{and} \quad \mathcal{L}_{\xi}\eta=0, \end{equation} (A.3) where |$\mathcal N_\Phi$| is the Nijenhuis torsion of |$\Phi$| [22]. One can actually show that the 1st condition in (A.3) implies the other ones [7, Section 6.1] (see also [22]). An almost contact structure |$(\Phi ,\xi ,\eta )$| such that |$\mathcal N_\Phi +d\eta \otimes \xi = 0$| is called normal [7]. So normal almost contact structures provide examples of complex structures on the Atiyah algebroid (of the trivial line bundle), and, in turn, of generalized contact structures of complex type. It turns out that, locally, every generalized contact structure of complex type is of this form (see Lemma A.0.3 below). Example A.0.2 is special in view of the following. Lemma A.0.3. Let |$L\to M$| be a line bundle, and let |$\varphi : DL \to DL$| be an integrable complex structure. Then around every point of |$M$|⁠, there is a trivialization |$L \cong \mathbb{R}_M$| identifying |$\varphi$| with a complex structure of the form (A.2) for some normal almost contact structure |$(\Phi ,\xi ,\eta )$|⁠. Proof. Without loss of generality, we can assume |$L = \mathbb{R}_M$|⁠, so that |$DL \cong TM \oplus \mathbb{R}_M$|⁠. It is clear that, under this identification, |$\varphi$| is necessarily of the form \begin{equation} \varphi(X,r)=(\Phi(X)-r\xi,\eta(X)+ gr) \end{equation} (A.4) for some quadruple |$(\Phi , \xi , \eta , g)$|⁠, where |$\Phi$| is a |$(1,1)$|-tensor, |$\xi$| is a vector field, |$\eta$| is a |$1$|-form, and |$g$| is a smooth function on |$M$|⁠. Locally, we can achieve |$g = 0$| as follows. First of all, let |$f \in C^\infty (M)$|⁠. A straightforward computation shows that, under the line bundle automorphism |$\mathbb{R}_M \to \mathbb{R}_M$|⁠, |$(x, r) \mapsto e^{-f(x)} r$|⁠, the quadruple |$(\Phi , \xi , \eta , g)$| changes into \begin{equation*} (\Phi +df\otimes \xi,\ \xi,\ \eta + df\circ \Phi+(\xi(f) -g)df,\ g-\xi(f)) \end{equation*} Now, from |$\varphi ^2 = - \textrm{id}$|⁠, we easily find that |$\xi$| is everywhere nonzero. Hence, locally, around every point, there exists a function |$f$| such that |$\xi (f) = g$|⁠. This concludes the proof. Remark A.0.4. Not all integrable complex structures on |$DL$| are globally of the form (A.2), in general, not even when |$L = \mathbb{R}_M$| is the trivial line bundle. To see this, let |$M$| be a manifold such that |$\textrm{H}_{\textrm{dR}}^1(M) \neq 0$|⁠, and let |$(\Phi ^{\prime},\xi ^{\prime},\eta ^{\prime})$| be a normal almost contact structure on |$M$| (such manifolds exist, and the one-dimensional sphere provides the simplest possible example). Now, pick a closed, but not exact, |$1$|-form |$\alpha$| on |$M$|⁠, and put \begin{equation} \Phi = \Phi^{\prime} +\alpha\otimes \xi^{\prime}, \quad \xi = \xi^{\prime},\quad \eta = \eta^{\prime}+\alpha\circ\Phi^{\prime} +\alpha(\xi^{\prime})\alpha, \quad g = -\alpha(\xi^{\prime}). \end{equation} (A.5) Then the endomorphism |$\varphi : D \mathbb{R}_M \to D \mathbb{R}_M$| given by (A.4) is an integrable complex structure that cannot be put in the form (A.2) by a global line bundle automorphism |$\mathbb{R}_M \to \mathbb{R}_M$|⁠. A.1 Local normal form Theorem A.1.1. Let |$L \to M$| be a line bundle equipped with a complex structure |$\varphi : DL \to DL$| on the gauge algebroid. Then, locally, around every point of |$M$|⁠, there are |$\triangleright$| coordinates |$(u, x^1, \ldots , x^n, y^1, \ldots , y^n)$| on M, and |$\triangleright$| a flat connection |$\nabla$| in |$L$|⁠, such that \begin{equation} \end{equation} (A.6) In other words, locally, around every point of |$M$|⁠, there is trivialization |$L \cong \mathbb{R}_{ \mathbb{R} \times \mathbb{C}^n}$| identifying |$\varphi$| with |$\varphi _{\textit{can}}$| from Example A.0.1. The proof will essentially follow from the Newlander–Nirenberg theorem after applying the homogenization trick [40], which we now recall. First of all, consider the frame bundle |$\widetilde M = L^\ast \smallsetminus 0 \to M$| of |$L$|⁠: it is a principal |$\mathbb{R}^\times$|-bundle. We denote by |$h: \mathbb{R}^\times \times \widetilde M \to \widetilde M$|⁠, |$(s, \varepsilon ) \mapsto h_s (\varepsilon )$| the group action, and by |$\mathcal E = \frac{d}{dt}|_{t = 0} h_{\textrm{exp} (t)}$| the restriction to |$\widetilde M$| of the Euler vector field. A section |$\lambda$| of |$L$| corresponds to a linear function on |$L^\ast$|⁠, and, by restriction, to a homogeneous function|$\widetilde \lambda$| on |$\widetilde M$|⁠, where, by “homogeneous”, we mean that |$h^\ast _s (\widetilde \lambda ) = s \widetilde \lambda$|⁠, for all |$s \in \mathbb{R}^\times$|⁠. In particular, |$\mathcal E (\widetilde \lambda ) = \widetilde \lambda$|⁠. Every homogeneous function on |$\widetilde M$| arises in this way. Secondly, let |$\Delta$| be a derivation of |$L$|⁠. Then there exists a unique vector field |$\widetilde \Delta$| on |$\widetilde M$| such that \begin{equation*} \widetilde \Delta (\widetilde \lambda) = \widetilde{\Delta \lambda}, \quad \text{for all }\lambda \in \Gamma (L). \end{equation*} Vector field |$\widetilde \Delta$| is homogeneous in the sense that |$h_s^\ast (\widetilde \Delta ) = \widetilde \Delta$|⁠, for all |$s \in \mathbb{R}^\times$|⁠. In particular, |$\widetilde \Delta$| commutes with |$\mathcal E$| and it is projectable onto |$M$| with projection |$\sigma (\Delta )$|⁠. Every homogeneous vector field on |$\widetilde M$| arises in this way. Notice that . Thirdly, let |$\varphi : DL \to DL$| be a vector bundle endomorphism. Then there exists a unique |$(1,1)$|-tensor |$\widetilde \varphi : T \widetilde M \to T \widetilde M$| such that \begin{equation} \widetilde \varphi \widetilde \Delta = \widetilde{\varphi \Delta}, \quad \text{for all }\Delta \in \Gamma (DL). \end{equation} (A.7) The |$(1,1)$|-tensor |$\widetilde \varphi$| is homogeneous in the sense that |$h^\ast _s (\widetilde \varphi ) = \widetilde \varphi$|⁠, for all |$s \in \mathbb{R}^\times$|⁠. In particular, the Lie derivative |$\mathcal L_{\mathcal E} \widetilde \varphi$| vanishes. Every homogeneous |$(1,1)$|-tensor on |$\widetilde M$| arises in this way. Additionally, |$\varphi$| is an integrable complex structure if and only if |$\widetilde \varphi$| is a complex structure on |$\widetilde M$|⁠. Example A.1.2. An immediate consequence of the homogenization construction described above is that all odd-dimensional real projective spaces possess a(n integrable) complex structure on the gauge algebroid of the dual of their tautological bundle. Indeed, let |$k$| be a positive integer, and let |$L$| be the dual of the tautological bundle on the projective space |$\mathbb{RP}^{2k-1}$|⁠. The total space of the frame bundle of |$L$| identifies canonically with |$\mathbb{R}^{2k} \smallsetminus \{0\}$|⁠, and the action |$h$| of |$\mathbb{R}^\times$| consists of homotheties. The standard complex structure on |$\mathbb{R}^{2k} = \mathbb{C}^k$| is homogeneous; hence, |$DL$| is equipped with an integrable complex structure. Viewing the sphere as a double cover of the projective space, we also conclude that there is a canonical integrable complex structure on the Atiyah algebroid of the trivial line bundle over any odd-dimensional sphere, and one can show that this complex structure does actually correspond to the normal almost contact structure underlying the well-known canonical Sasaki structure. Proof. of Theorem A.1.1 Let |$\varphi : DL \to DL$| be an integrable complex structure. Consider |$\widetilde \varphi$|⁠. It is a complex structure on |$\widetilde M$|⁠. As |$\mathcal E$| is nowhere vanishing, it can be locally completed to a holonomic complex frame, that is, locally, around every point of |$\widetilde M$|⁠, there are coordinates |$(T, U, X^1, \ldots , X^n, Y^1, \ldots , Y^n)$| such that \begin{equation*} \mathcal E = \frac{\partial}{\partial T}, \quad \widetilde \varphi \mathcal E = \frac{\partial}{\partial U}, \quad \textrm{and} \quad \widetilde \varphi \frac{\partial}{\partial X^i} = \frac{\partial}{\partial Y^i}. \end{equation*} As all coordinate vector fields commute with |$\mathcal E$|⁠, they all come from (commuting) derivations of |$L$|⁠. In particular |$\triangleright$||$(U, X^1, \ldots , X^n, Y^1, \ldots , Y^n)$|⁠, are pullbacks via the projection |$\widetilde M \to M$| of uniquely defined coordinates |$(u, x^1, \ldots , x^n, y^1, \ldots , y^n)$| on |$M$|⁠, and |$\triangleright$| there exists a unique flat connection |$\nabla$| in |$L$| such that \begin{equation*} \frac{\partial}{\partial U} = \nabla_{\partial / \partial u}, \ \ldots\, \ \frac{\partial}{\partial X^i} = \nabla_{\partial / \partial x^i}, \ \ldots\,\ \frac{\partial}{\partial Y^i} = \nabla_{\partial / \partial y^i},\ \ldots \end{equation*} From (A.7), the coordinates |$(u, x^1, \ldots , x^n, y^1, \ldots , y^n)$| on |$M$| and the flat connection |$\nabla$| possess all the required properties. As an immediate corollary of Theorem A.1.1 and Lemma A.0.3 we get a local normal form for normal almost contact structures. Corollary A.1.3. Let |$(\Phi ,\xi ,\eta )$| be a normal almost contact structure on a manifold |$M$|⁠. Then, around every point, there exist local coordinates |$(u,x^i,y^i)$| and a local function |$f$|⁠, such that: |$\xi =\frac{\partial } {\partial u}$|⁠, |$\eta =du+\frac{\partial f}{\partial y^i}dx^i- \frac{\partial f}{\partial x^i}dy^i$|⁠, |$\Phi = dx^i\otimes \frac{\partial }{\partial y^i}- dy^i\otimes \frac{\partial }{\partial x^i} + df \otimes \frac{\partial } {\partial u}$|⁠. A.2 Dolbeault–Atiyah cohomology Let |$L \to M$| be a line bundle, and let |$\varphi : DL \to DL$| be an integrable complex structure on the gauge algebroid of |$L$|⁠. Similarly as in the case of a complex manifold, there is a cohomology theory attached to |$\varphi$|⁠. Namely, consider the complexification |$DL \otimes \mathbb{C}$| of the gauge algebroid and denote by |$D^{(1,0)} L$| and |$D^{(0,1)} L$| the |$+\textrm{i}$| and the |$-\textrm{i}$|-eigenbundles of |$\varphi$|⁠, respectively, so that \begin{equation*} DL \otimes \mathbb{C} = D^{(1,0)} L \oplus D^{(0,1)} L, \end{equation*} and complex Atiyah forms |$\Omega _L^{\bullet } \otimes \mathbb{C}$| splits as \begin{equation*} \Omega_L^{\bullet} \otimes \mathbb{C} = \bigoplus_{r,s} \Omega^{(r,s)}_L, \end{equation*} where we denoted by |$\Omega ^{(r,s)}_L$| the sections of the (complex) vector bundle \begin{equation*} \wedge^r (D^{(1,0)}L)^\ast \otimes \wedge^s (D^{(0,1)} L)^\ast \otimes L. \end{equation*} The de Rham differential |$d_D$| splits, in the obvious way, as |$d_D = \partial _D + \overline \partial _D$|⁠, where \begin{equation*} \partial_D: \Omega^{(\bullet, \bullet)}_L \to \Omega^{(\bullet + 1, \bullet)}_L, \quad \textrm{and} \quad \overline \partial_D: \Omega^{(\bullet, \bullet)}_L \to \Omega^{(\bullet, \bullet + 1)}_L, \end{equation*} and the integrability of |$\varphi$| is equivalent to \begin{equation*} \partial_D^2 = \overline \partial{}^2_D = \partial_D \overline \partial_D + \overline \partial_D \partial_D = 0. \end{equation*} We call the cohomology of |$\overline{\partial }_D$| the Dolbeault–Atiyah cohomology. Theorem A.2.1. The Dolbeault–Atiyah cohomology vanishes locally. Proof. In view of Theorem A.1.1, it is enough to work in the case when |$M = \mathbb{R} \times \mathbb{C}^n$|⁠. Let |$u$| be the standard (real) coordinate on the 1st factor and let |$z^i = x^i + \textrm{i} y^i$|⁠, |$i = 1, \ldots , n$|⁠, be the standard complex coordinates on the 2nd factor. We can also assume that |$L = \mathbb{R}_M$| is the trivial line bundle and (A.6) holds with |$\nabla$| being the canonical flat connection on |$\mathbb{R}_M$|⁠. In this case |$D^{(1,0)} L$| is spanned by the complex derivations \begin{equation} \end{equation} (A.8) It is easy to see that every complex Atiyah form |$\omega$| on |$\mathbb{R}_M$| can be uniquely written as \begin{equation*} \omega = \omega_0 + \omega_1 \wedge \mathfrak k, \end{equation*} where |$\omega _0, \omega _1$| are standard complex forms on |$M$| and \begin{equation*} \mathfrak k = \mathfrak j + \textrm{i} \cdot du. \end{equation*} A long but straightforward computation exploiting (A.8) shows that \begin{equation*} \overline \partial_D \omega = \overline \partial \omega_0 + \left(\overline \partial \omega_1+(-)^{|\omega_0|} \left(\omega_0 + \mathcal L_Y \omega_0 \right)\right) \wedge \mathfrak k, \end{equation*} where \begin{equation*} Y:= \sigma (\overline \square) = \frac{\textrm{i}}{2} \frac{\partial}{\partial u}, \end{equation*} and |$\overline \partial$| is the standard Dolbeault differential on |$\mathbb{C}^n$| (acting on forms on |$\mathbb{R} \times \mathbb{C}^n$| in the obvious way). So |$\omega$| is |$\overline \partial _D$|-closed iff \begin{equation*} \overline \partial \omega_0 = \overline \partial \omega_1+(-)^{|\omega_0|} \left(\omega_0 + \mathcal L_Y \omega_0 \right) = 0. \end{equation*} In this case, use the vanishing of standard Dolbeault cohomology (with a real parameter |$u$|⁠), to choose a form |$\rho _0$| such that |$\overline \partial \rho _0 = \omega _0$|⁠. As the Lie derivative along |$Y$| commutes with |$\overline \partial$| we find \begin{equation*} \overline \partial \left( \omega_1 - (-)^{|\rho_0|} \left(\rho_0 + \mathcal L_Y \rho_0 \right)\right) = 0, \end{equation*} and we can choose |$\rho _1$| such that |$\overline \partial \rho _1 = \omega _1 - (-)^{|\rho _0|} \left (\rho _0 + \mathcal L_Y \rho _0 \right )$|⁠. It is now easy to see that \begin{equation*} \overline \partial_D \left (\rho_0 + \rho_1 \wedge \mathfrak k \right) = \omega. \end{equation*} This concludes the proof. Remark A.2.2. 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This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

Pan,, Xuanyu

doi: 10.1093/imrn/rnz026pmid: N/A

Abstract In this paper, we prove that the moduli space of low degree rational curves on a low-degree complete intersection passing several suitable points is a complete intersection. The proof involves some relationships among some divisors on the Kontsevich spaces, advanced techniques of deformations of rational curves, and classical projective geometry. 1 Introduction Rationally connected varieties have been central objects of study in both algebraic geometry and arithmetic geometry. This notion can be seen as an algebraic analog of path connectedness in topology. Rational simple connectedness, which can also be thought of as an analog of simple connectedness in topology, is a notion proposed by B. Mazur and systematically developed by de Jong, Harris, and Starr. It has been sucessfully used by de Jong–He–Starr to finish a proof of Serre’s conjecture II in [8] and a new proof of de Jong’s period–index theorem in [28]. One thing that we have learned from [8] is that a good understanding of the geometry of the moduli space of rational curves on a rationally connected variety is essential for applications in arithmetic. Another example of such application is a beautiful theorem of B. Hassett using rational simple connectedness to prove weak approximation. For more arithmetic applications, see [19], [29], and [7]. On the other hand, a good understanding of these moduli spaces could also have applications in algebraic cycles on Fano manifolds, see [30] and [24]. Given the importance of understanding the geometry of the moduli spaces of rational curves, it is somewhat unfortunate that we do not know much beyond the cases of homogeneous spaces [21, 31] or spaces with a large group action. For example, not much is known for these moduli spaces of rational curves on the complete intersections in the projective space except for their irreducibility and dimensions, see [27] and [3]. In this paper, we explore the moduli spaces of rational curves on complete intersections and fill in the gap when the complete intersections are of low degree. In general, these moduli spaces are relatively easy to define as functors, but very difficult to understand concretely. We combine classical projective geometry with advanced techniques of deformations of rational curves to give some concrete descriptions of these moduli spaces. Before giving these descriptions, we fix some notations as follows. For the sake of simplicity, all schemes in this paper are taken over the field of complex numbers |$\mathbb{C}$| if not stated otherwise. However, many proofs in this paper hold in arbitrary characteristic and we will highlight which results make essential use of the characteristic |$0$| hypothesis. Let |$X$| be a projective variety in |${\mathbb{P}}^n_{\mathbb{C}}$| with an effective curve class |$\beta \in H_2(X,{\mathbb Z})$|⁠. The Kontsevich moduli space |$\overline{{\mathcal{M}}}_{0,m}(X,\beta )$| parametrizes data |$(C, f, x_1,\dots ,x_m)$| of (1) a proper, connected, at-worst-nodal, arithmetic genus |$0$| curve |$C$|⁠, (2) an ordered collection |$x_1,\dots ,x_m$| of distinct smooth points of |$C$|⁠, and (3) a morphism |$f:C\rightarrow X$| with |$f_*([C])=\beta $| such that |$(C,f,x_1,\dots ,x_m)$| has only finitely many automorphisms. There is an evaluation morphism \begin{equation*} \textrm{ev}_{\beta}:\overline{{\mathcal{M}}}_{0,m}(X,\beta) \rightarrow X^m, \ \ (C,f,x_1,\dots,x_m) \mapsto (f(x_1),\dots,f(x_m)). \end{equation*} The space |$\overline{{\mathcal{M}}}_{0,m}(X,\beta )$| is a Deligne–Mumford stack, see [6] for the construction. Definition 1.1. Let |$s$| and |$d$| be natural numbers. We say that a family of homogeneous polynomials on |${\mathbb{P}}^n$| is of the type|$T_1(d,s)$| if this family consists of one polynomial of degree |$d$| and |$s$| polynomials of degree |$i$| for |$i=2,3,\ldots ,d-1$|⁠. We also write this family of homogeneous polynomials as \begin{equation*} T_1(d,s)=\left( \begin{array}{ccccc} 2 & 3 & \cdots & d-1 & \\ \vdots &\vdots & & \vdots & d \\ 2 & 3 & \cdots & d-1 & \\ \end{array} \right). \end{equation*} We say that a family of homogeneous polynomials on |${\mathbb{P}}^n$| is of the type|$T_2(d,s)$| if this family consists of one polynomial of degree |$d$| and |$s$| polynomials of degree |$i$| for |$i=1,2,3,\ldots ,d-1$|⁠. Write this family of homogeneous polynomials as \begin{equation*} T_2(d,s)=\left( \begin{array}{ccccc} 1 & 2 & \cdots & d-1 & \\ \vdots &\vdots & & \vdots & d \\ 1 & 2 & \cdots & d-1 & \\ \end{array} \right). \end{equation*} In the following, we say that a complete intersection of codimension |$k$| is of type |$(c_1,c_2,\ldots ,c_k)$| if it is defined by |$k$| homogeneous polynomials |$F_j$| of degree |$c_j$| for |$j=1,\ldots ,k$|⁠. When |$T_i(d,s)$| appears in such expression, it is the obvious juxtaposition. For instance, |$(T_1(d_1,s),T_2(d_2,s))$| is of the type \begin{equation*} \left( \begin{array}{cccccccccc} 2 & 3 & \cdots & d_1-1 & & 1 &2 &\ldots &d_2-1\\ \vdots &\vdots & & \vdots & d_1 &\vdots &\vdots & &\vdots &d_2 \\ 2 & 3 & \cdots & d_1-1 & &1 &2 &\ldots &d_2-1\\ \end{array} \right).\end{equation*} Condition 1.2. Throughout this paper, we always assume the following. The smooth complete intersection |$X$| in |${\mathbb{P}}^n_{{\mathbb{C}}}$| is of the type |$(d_1,\ldots ,d_c)$| with |$c\leq n$|⁠, |$d_i\geq 2$|⁠, and |$(c,d_1,\ldots ,d_c)\neq (1,2)$| satisfying the “Fano bound” \begin{equation*}n+m\left(c-\sum\limits_{i=1}^c d_i\right)-c\geq 2,\end{equation*} and let |$m$| be an integer with |$3\leq m\leq n+1$|⁠, and let \begin{equation*}\underline{P}=( p_1, \ldots, p_m)\in X^m\end{equation*} be a point of |$X^m$|⁠, where |$\{ p_1,\ldots ,p_m \}$| are |$m$| general points of |$X$|⁠. Denote by |${\mathcal{F}}$| the general fiber of |$\textrm{ev}_{m\alpha }$| over |$\underline{P}$|⁠, where |$\alpha $| is the homology class of a line. Condition 1.2 implies that |$X$| is a Fano manifold and |${\mathcal{F}}_t$| is non-empty, see Proposition 5.6 and Lemma 2.5. Let |${\mathbb{P}}(V)/{\mathbb{P}}(W)$| be the projective space |${\mathbb{P}}(V/W)$| for a flag |$(W\subseteq V)$| of a vector space |$V$|⁠, and let |$\operatorname{Span}(D)$| be the smallest projective subspace containing the algebraic set |$D(\subseteq{\mathbb{P}}^n)$|⁠. We have a map \begin{equation*} \Phi: {\mathcal{F}} \dashrightarrow{\mathbb{P}}^{n-m}={\mathbb{P}}^n/\operatorname{Span}(p_1,\ldots,p_m) \end{equation*} associating to a point |$(f,C,x_1,\ldots ,x_m)\in{\mathcal{F}} $| the point |$\operatorname{Span}(f(C)) \in{\mathbb{P}}^{n-m}$|⁠. This map is introduced in the proof of [9, Lemma 6.4]. Now, we are able to state the main theorem of this paper. Theorem 1.3. Assume that Condition 1.2 hold. Let |${\mathcal{F}}_t$| be the fiber of the forgetful map |$F:{\mathcal{F}} \rightarrow \overline{{\mathcal{M}}}_{0,m}$| (see [14]) over a general point |$t$| of |$\overline{{\mathcal{M}}}_{0,m}$|⁠. Then the map |$\Phi $| is well-defined on |${\mathcal{F}}_t$| and the general fiber |${\mathcal{F}}_t$| is a smooth complete intersection in |${\mathbb{P}}^{n-m(c-1)}$| of the type \begin{equation*}\left(T_1(d_1,m), T_1(d_2,m),\ldots, T_{1}(d_{c-1},m),T_1(d_c,m) \right)\end{equation*} via the complete linear system |$|\Phi _t^*{\mathcal{O}}(1)|:{\mathcal{F}}_t \hookrightarrow{\mathbb{P}}^{n-m(c-1)}$|⁠, where |$\Phi _t$| is the restriction of |$\Phi $| to |${\mathcal{F}}_t$|⁠. In [25], we have studied the fibers |${\mathcal{F}}$| for conics on complete intersections (⁠|$m=2$|⁠) and a similar result is obtained. Namely, the moduli spaces of conics through two general points in complete intersections are complete intersections, see [25, Theorem 1.1]. This result for conics strengthens some key results in [9] that is used to prove the (strongly) rational simple connectedness of low-degree complete intersections, see [9, Theorem 1.2] for details. It also leads to a proof of strong approximation, which is considered to be harder to prove than weak approximation, for low-degree affine complete intersections over function fields by Q. Chen and Y. Zhu [7]. Our main theorem has some geometric applications instantly. For example, we show the rational connectedness of |${\mathcal{F}}$|⁠, see Proposition 8.1, which is related to the rational simple connectedness of |$X$|⁠; we reprove the formula in enumerative geometry for counting the number of twisted cubics passing through three general points on complete intersections due to A. Beauville, see Proposition 8.3; and we also find a new formula for crossing conics, see Remark 8.4. In addition, Theorem 1.3 implies that the Picard group of |${\mathcal{F}}$| is finitely generated, see Proposition 8.8. Lastly, we want to mention that Theorem 1.3 answers a question for searching new |$2$|-Fano manifolds. In [10], de Jong and Starr generalize Fano varieties to higher Fano varieties, namely, |$2$|-Fano varieties. Unfortunately, apart from low-degree complete intersections in weighted projective spaces, some Grassmannians, and some hypersurfaces in some Grassmannians, one does not know other examples of 2-Fano varieties. On the other hand, [1, Theorem 1.4] predicts that moduli spaces of rational curves on |$n$|-Fano complete intersections should be |$(n-1)$|-Fano. Therefore, people expect that one can find some new |$2$|-Fano manifolds among these “abstract” moduli spaces |${\mathcal{F}}$| and |${\mathcal{F}}_t$|⁠. Some moduli spaces |${\mathcal{F}}$| are indeed |$2$|-Fano, for example, when |$m$| equals |$3$| and |$X$| is of low degree. Contrary to the expectation, the varieties |${\mathcal{F}}$| do not give new Fano manifolds: whenever |${\mathcal{F}}$| is Fano, it is, in fact, a complete intersection. Organization of Paper. In Section 2, we collect and prove some useful facts about the smoothness and the dimension of the moduli space |${\mathcal{F}}$|⁠. In Section 3, we analyze the possible degeneration types of the maps parametrized by |${\mathcal{F}}_t$| and show that |$\Phi $| is well-defined on |${\mathcal{F}}_t$|⁠, see Proposition 3.6 and Lemma 3.8. Roughly speaking, the image of a stable map parametrized by |${\mathcal{F}}_t$| is a union of a rational normal curve and lines. We also introduce some morphisms |$\{\pi _{p_i}\}_{i=1}^m$| on |${\mathcal{F}}_t$|⁠, where the morphism |$\pi _{p_i}$| associates to a stable map a projective space spanned by the marked points and the tangent at the point |$p_i$|⁠. We show that each of these morphisms gives a sublinear system of the complete linear system |$|\lambda |_{{\mathcal{F}}_t}|:=|\Phi _t^*{\mathcal{O}}(1)|$|⁠, see Proposition 5.1. We use these sublinear systems to show that |$|\lambda |_{{\mathcal{F}}_t}|$| separates the points of |${\mathcal{F}}_t$| in Section 5. Then we prove that |${\mathcal{F}}_t$| contains a subvariety |$Y$| parametrizing the stable maps whose image is a union of lines. We also show that |$Y$| is a smooth complete intersection in a projective space, see Corollary 5.8. In Section 6, we use the deformation theory of stable maps and the morphisms |$\{\pi _{p_i}\}_{i=1}^m$| to show that |$|\lambda |_{{\mathcal{F}}_t}|$| gives an embedding of |${\mathcal{F}}_t$| into a projective space, see Proposition 6.3. In Section 7, we apply a criterion (Proposition 7.1) to the pair |$({\mathcal{F}}_t,Y)$| to show by induction that |${\mathcal{F}}_t$| is also a complete intersection in a projective space. In Section 8, we give some applications of our main theorem to the rational connectedness of moduli spaces of rational curves, the enumerative geometry, and the Picard group of |${\mathcal{F}}$|⁠. 2 Preliminary Let us recall some general facts about the moduli spaces of stable maps and the deformation theory of rational curves. The main references for this section are [9, Sections 5 and 6], [22, Chapter II], and [14]. Lemmas 2.2, 2.3, and 2.5 hold only for characteristic zero since they follow from the generic smoothness theorem. Let |$Y$| be a smooth projective variety. Lemma 2.1. |$($|[22], [9, Lemma 3.1], and [14]|$)$| Let |$f:C\rightarrow Y$| be a stable map of arithmetic genus zero. (1) If every component of |$C$| is contracted or free [22, Chapter 2], then deformations of |$f$| are unobstructed. Moreover, there exist deformations of |$(C,f)$| smoothing all the nodes of |$C$|⁠. A general such deformation is free. (2) If the stable map |$f$| is parametrized by |$\overline{{\mathcal{M}}}_{0,k}(Y,\beta )$| and deformations of |$f$| are unobstructed, then |$\overline{{\mathcal{M}}}_{0,k}(Y,\beta )$| is smooth at the point |$\zeta = [(C,f,x_1,\ldots ,x_k)]$| and its dimension at the point |$\zeta $| is \begin{equation*}\dim_{\zeta} \left(\overline{{\mathcal{M}}}_{0,k}(Y,\beta)\right)=\dim (Y)+\int\limits_{\beta} c_1(T_Y)+k-3.\end{equation*} Lemma 2.2. |$($|[9, Lemma 5.1]|$)$| With the same notation as above, if every point in a general fiber of the evaluation map |$\textrm{ev}_{\beta }:\overline{{\mathcal{M}}}_{0,m}(Y,\beta )\rightarrow Y^m$| parametrizes a curve whose irreducible components are all free, then a (non-empty) general fiber of |$\textrm{ev}_{\beta }$| is smooth of the expected dimension \begin{equation*} \int\limits_{\beta} c_1(T_Y)-(m-1)\dim (Y)+m-3\end{equation*} and the intersection with the boundary is a simple normal crossing divisor. Lemma 2.3. With Condition 1.2, the general fiber |${\mathcal{F}}$| is a smooth projective variety of the expected dimension |$\left (c+2-\sum \limits _{i=1}^c d_i\right )m+n-c-3$| and its boundary is a simple normal crossing divisor. Proof. Because of Condition 1.2, the methods of [9] apply, see [9, Lemma 5.3]. Lemma 2.4. [9, Lemma 6.4] The rational map |$\Phi : {\mathcal{F}} \dashrightarrow{\mathbb{P}}^{n-m}={\mathbb{P}}^n/\operatorname{Span}(p_1,\ldots ,p_m)$| is a morphism except when |$c=2, d_1=d_2=2, m\geq 6$|⁠, or |$c=1, d_1=3, m\geq 5$|⁠. Proof. This lemma follows from the proof of [9, Lemma 6.4]. Lemma 2.5. Let |$(C,f,x_1,\ldots ,x_m)$| be a stable map parametrized by a point in |${\mathcal{F}}$|⁠. Suppose that the domain |$C$| consists of a comb with |$m$| rational teeth (see [22, Page 156 and Definition 7.7]). If the map |$f$| contracts the handle and maps the teeth to the lines meeting at a point in |$X$|⁠, then the forgetful morphism |$F:{\mathcal{F}}\rightarrow \overline{{\mathcal{M}}}_{0,m}$| has a section |$\sigma $| and the fiber |${\mathcal{F}}_t$| is a smooth projective variety. Definition 2.6. We say that such map |$f$| in Lemma 2.5 is of maximal degeneration type. Remark 2.7. In Proposition 5.6, we show that |${\mathcal{F}}$| contains a point parametrizing a map of maximal degeneration type, in particular, |${\mathcal{F}}_t$| is a smooth projective variety. Proof. It follows from the assumption that there are |$m$| lines |$l_1,\ldots ,l_m$| in |$X$| passing through |$m$| general points |$p_1,\ldots ,p_m$| and intersecting at a different point |$q\in X$|⁠. Suppose that |$p_i$| is contained in |$l_i$|⁠. Let |$(C,x_1,\ldots ,x_m)$| be a point in |$\overline{{\mathcal{M}}}_{0,m}$|⁠. By [6, Theorem 3.6], there exists a stable map \begin{equation*} f: C\cup l_1\cup \ldots \cup l_m \rightarrow X,\end{equation*} such that the domain |$C\cup l_1\cup \ldots \cup l_m$| is the union of |$C$| and lines |$l_i$|⁠, |$p_i$| on |$l_i$| is the marked point |$x_i$|⁠, the map |$f$| is the identity on each |$l_i$|⁠, and the map |$f$| contracts |$C$| to the point |$q$|⁠. Therefore, we have a section |$\sigma $| of the map |$F:{\mathcal{F}}\rightarrow \overline{{\mathcal{M}}}_{0,m}$| as follows. The section |$\sigma $| associates to a point |$(C,x_1,\ldots ,x_m)\in \overline{{\mathcal{M}}}_{0,m}$| the point \begin{equation*} (C\cup l_1\cup \ldots \cup l_m,f,x_1,\ldots,x_m) \in{\mathcal{F}}. \end{equation*} By the generic smoothness theorem and [9, Lemma 3.2], the general fiber |${\mathcal{F}}_t$| is smooth and irreducible. 3 Classification of Degeneration Type of Stable Maps In this section, we show that the image of a stable map parametrized by |${\mathcal{F}}_t$| is a union of lines and a rational normal curve, see Proposition 3.6. This gives a classification of degeneration types of stable maps parametrized by |${\mathcal{F}}_t$|⁠. By this classification, we are able to show that the map |$\Phi $| is well-defined on |${\mathcal{F}}_t$|⁠, see Lemma 3.7. At the end of this section, we introduce some morphisms |$\pi _{p_i}$| on |${\mathcal{F}}_t$|⁠, which will be used later. The proofs in this section are just dimension counts with some tedious analysis of degrees of rational curves in a projective space, we suggest the reader to skip their proofs in the 1st reading. We start with a lemma that goes back to Del Pezzo. Lemma 3.1. [13, Theorem 1] With the notations as above, the degree of a reduced and irreducible curve |$C$| in |${\mathbb{P}}^n$| passing through |$\{p_1,\ldots ,p_m\}$| is at least |$m-1$|⁠. The degree of |$C$| is |$m-1$| if and only if the curve |$C$| is a rational normal curve in |$\operatorname{Span}(p_1,\ldots ,p_m)={\mathbb{P}}^{m-1}$|⁠. Let |$(C,f,x_1,\ldots ,x_m)$| be a stable map parametrized by |${\mathcal{F}}$|⁠. Assume that the image |$f(C)$| in |$X$| consists of |$k$| irreducible components |$C_1,\ldots ,C_k$|⁠. Denote by |$P$| the set |$\{p_1,\ldots ,p_m\}$| and by |$P(C_i)$| the subset |$P\cap C_i$| of |$P$|⁠. Definition 3.2. We say that |$f$| is a pseudo-embedding if the restriction of |$f:C\rightarrow X$| to each irreducible component of |$C$| is a contraction or an embedding. Lemma 3.3. With the notations as above, a rational curve on |$X$| passing through |$l$| general points of |$X$| has degree at least |$l$|⁠. In particular, each irreducible component |$C_i$| of |$f(C)$| has degree at least |$\#(P(C_i))$|⁠. Proof. This follows from [9, Lemma 5.5]. Lemma 3.4. With the notations as above, for a stable map |$(C,f,x_1,\ldots ,x_m)$|⁠, if the dimension of |$\operatorname{Span}(f(C))$| equals |$m$|⁠, then it is a pseudo-embedding map and the image |$f(C)(\subseteq X)$| is a union |$C_1 \cup \ldots \cup C_k$| of rational normal curves with |$P=\bigcup \limits _{i=1}^k P(C_i)$| in which |$P(C_i)\cap P(C_j)=\emptyset $| for |$i\neq j$| and |$\deg (C_i)= \#P(C_i)$|⁠. Proof. By Lemma 3.3, we can assume that |$f(C)$| is a union |$C_1\cup \ldots \cup C_k$| of curves |$C_1,\ldots ,C_k$|⁠, where each |$C_i$| is a reduced and irreducible curve on |$X$| passing through |$P(C_i)$| and the degree |$\deg (C_i)$| of |$C_i$| is at least |$\#(P(C_i))$|⁠. It implies the inequalities \begin{equation} m=\deg(f_*([C]))\geq \deg(f(C)) = \sum\limits_{i=1}^k \deg(C_i)\geq \sum\limits_{i=1}^k \#(P(C_i)), \end{equation} (3.1) where the 1st inequality is an equality if and only if |$f$| is a contraction or a one-to-one map on each irreducible component of |$C$|⁠, and the 2nd inequality is an equality if and only if |$\deg (C_i)= \#P(C_i)$|⁠. On the other hand, it is clear that \begin{equation} P=\bigcup\limits_{i=1}^k P(C_i)\qquad \text{~and~}\qquad m=\#(P)\leq \sum\limits_{i=1}^k \#(P(C_i)), \end{equation} (3.2) where the inequality is an equality if and only if the family of sets |$P(C_i)$| is a partition of |$P$|⁠, that is, |$P(C_i)\cap P(C_j)=\emptyset $| for |$i\neq j$|⁠. From the argument above, all the inequalities in (3.1) and (3.2) are equalities. If all the curves |$\{C_i\}$| are smooth rational normal curves, then |$f$| is a pseudo-embedding map and the proposition follows. Indeed, note that the irreducible and reduced curve |$C_i$| passes through |$P(C_i)$| that are linearly nondegenerate and the degree of |$C_i$| is |$\#(P(C_i))$|⁠. It follows that |$\operatorname{Span}(C_i)$| is a projective space of dimension at most |$\deg (C_i)$|⁠. If the dimension of |$\operatorname{Span}(C_i)$| attains |$\deg (C_i)$|⁠, then |$C_i$| is a nondegenerate curve in |$\operatorname{Span}(C_i)$| and it is a smooth rational normal curve by the argument in [16, Page 179]. If the dimension of |$\operatorname{Span}(C_j)$| does not attain |$\deg (C_j)$| for some |$j$|⁠, then the dimension of |$\operatorname{Span}(f(C))=\operatorname{Span}(C_1\cup \ldots \cup C_k)$| does not attain the maximal value |$\sum \limits _{i=1}^k \deg (C_i)=m$|⁠, which is a contradiction. Lemma 3.5. Let |$t\in \overline{{\mathcal{M}}}_{0,m} $| be a general point, and let |${\mathcal{F}}_t$| be the fiber of the forgetful map |$ {\mathcal{F}} \rightarrow \overline{{\mathcal{M}}}_{0,m}$| over |$t$|⁠. If a stable map |$(C,f,x_1,\ldots ,x_m)$| is parametrized by |${\mathcal{F}}_t$|⁠, then projective space |$\operatorname{Span}(f(C))$| has dimension |$m$|⁠. Proof. It follows from Lemma 2.4 that it suffices to show that the dimension of the locus in |${\mathcal{F}}$| parametrizing the stable maps into |$\operatorname{Span}(p_1,\ldots , p_m)$| is at most |$m-4~(=\dim (\overline{{\mathcal{M}}}_{0,m})-1)$| for |$(c,d_1,\ldots ,d_c)=(1,3)$| and |$(2,2,2)$|⁠. We follow the argument in [9, Page 31]. Let |$Z$| be the intersection of |$X$| with a general |${\mathbb{P}}^{m-1}$| in |${\mathbb{P}}^n$|⁠. Then we have \begin{equation*}\dim(Z)=\dim(X)+m-h^0(X,{\mathcal{O}}_X(1))\end{equation*} and (by the adjunction formula) \begin{equation*}\int\limits_{\alpha}c_1(T_Z)=\int\limits_{\alpha}c_1(T_X)+m-h^0(X,{\mathcal{O}}_X(1)).\end{equation*} We may assume that the general |${\mathbb{P}}^{m-1}$| is |$\operatorname{Span}(p_1,\ldots ,p_m),$| where |$\{p_1,\ldots ,p_m\}$| are general points of |$Z$| and |$X$|⁠. We can assume that the map \begin{equation*}\textrm{ev}_Z:\overline{{\mathcal{M}}}_{0,m}(Z,m\alpha)\rightarrow Z^m\end{equation*} is dominant, otherwise there are no stable maps parametrized by |${\mathcal{F}}$| and mapped into |$\operatorname{Span}(p_1,\ldots ,p_m)$|⁠, which implies the lemma. Moreover, by [9, Lemma 5.5], the class |$m\alpha $| is |$m$|-minimal-dominant for |$X$| in the sense of [9, Definition 5.2], and the general fiber |${\mathcal{F}}_Z$| of |$\textrm{ev}_Z$| over |$\{p_1,\ldots ,p_m\}$| is contained in |${\mathcal{F}}$|⁠. It follows that the class |$m\alpha $| is |$m$|-minimal-dominant for |$Z$|⁠. Therefore, we can apply [9, Lemma 5.3] and Lemma 2.2 to |$Z$|⁠. It follows that the dimension of the general fiber |${\mathcal{F}}_Z$| is given by \begin{align*} &\int\limits_{m\alpha} c_1(T_Z)-(m-1)\dim(Z)+m-3 \\ =&m(n+1-\sum\limits_{i=1}^c d_i)+(m-h^0(X,{\mathcal{O}}_X(1)))-(m-1)\dim(X)+m-3. \end{align*} A direct calculation shows that |$\dim ({\mathcal{F}}_Z)$| is at most |$m-4$| for |$(c,d_1,\ldots ,d_c)=(1,3)$| and |$(2,2,2)$|⁠. It is clear that |${\mathcal{F}}_Z$| is the locus in |${\mathcal{F}}$| parametrizing the stable maps mapping into |$\operatorname{Span}(p_1,\ldots , p_m)$|⁠. By the lemmas above, we are able to classify the degeneration types of stable maps parametrized by |${\mathcal{F}}_t$|⁠. Proposition 3.6. Let |$t\in \overline{{\mathcal{M}}}_{0,m} $| be a general point, and let |$(C,f,x_1,\ldots ,x_m)$| be a stable map parametrized by |${\mathcal{F}}_t$|⁠. The curve |$C$| consists of |$C_1$| and |$l_1,\ldots ,l_k$| such that |$C$| is a comb with handle |$C_1$| and teeth |$\{l_i\}$|⁠. Moreover, there is a unique marked point |$x_{j_i}$| on |$\{l_i\}$| and the rest marked points among the points {|$x_1,\ldots ,x_m$|} are on |$C_1$|⁠. The map |$f$| is a pseudo-embedding. The image |$f(l_i)$| of |$l_i$| is a line in |$X$| passing through |$p_{j_i}$| and |$f(C_1)$| is a rational normal curve of degree |$m-k$| for |$0\leq k \leq m$|⁠. In particular, if |$k\neq m$|⁠, then |$(C,f,x_1,\ldots ,x_m)\in{\mathcal{F}}_t$| is an embedded curve. Remark 3.7. In the case when |$k$| equals |$m$|⁠, the map |$f$| contracts the handle |$C_1$| to a point |$p$| and maps the tooth |$l_i$| isomorphically to a line passing through |$p$| and |$p_i$|⁠. Proof. Suppose that |$C$| consists of rational curves |$C_1,l_1,l_2,\ldots ,l_s$|⁠. Since the point |$t\in \overline{{\mathcal{M}}}_{0,m}$| is general, the image |$F([C,f,x_1,\ldots ,x_m])$| parametrizes a smooth rational curve with |$m$| marked points. There is a unique component of |$C$| that is not contracted by the stabilization process of the forgetful map |$F$|⁠, say |$C_1$|⁠. Let |$D$| be a connected component of the union of |$l_1,\ldots ,l_s$|⁠. We claim that there is a unique marked point on |$D$|⁠. In fact, note that the stabilization process of |$F$| contracts |$D$| to a point. It follows that |$D$| has at most one marked point. If |$D$| does not have any marked point, then the map |$f$| contracts |$D$| to a point by Lemmas 3.4 and 3.5. Since the stabilization process of |$F$| contracts |$D$|⁠, the curve |$C$| has infinitely many automorphisms mapping |$D$| onto |$D$| and fixing |$C-D$|⁠. Therefore, these automorphisms of |$C$| induce infinitely many automorphisms of the stable map |$f:C\rightarrow X$|⁠, which contradicts with the stability of |$f$|⁠. We have proved the claim. A similar argument shows that |$D$| has a unique irreducible component, that is, |$D$| consists of exactly one of |$l_1, \ldots , l_s.$| In fact, suppose that |$D$| has at least two irreducible components. By the claim above, there is one irreducible component |$l$| of |$D$| that does not contain any marked point. It follows from Lemma 3.4 that the map |$f$| contracts |$l$| to a point. Since the stabilization process of |$F$| contracts |$l$|⁠, the curve |$C$| has infinitely many automorphisms mapping |$l$| onto |$l$| and fixing |$C-l$|⁠. As above, it follows that there are infinitely many automorphisms of |$f:C\rightarrow X$| induced by these automorphisms of |$C$|⁠, which contradicts with the stability of |$f$|⁠. We have shown the 1st assertion. The 2nd assertion follows from the 1st assertion, Lemmas 3.5 and 3.4. In the next lemma, we show that the rational map |$\Phi $| is well-defined on |${\mathcal{F}}_t$|⁠. Lemma 3.8. Let |$(C,f,x_1,\ldots ,x_m)$| be a stable map parametrized by |${\mathcal{F}}_t$|⁠. Then |$f(C)$| is smooth at |$p_i$| and the tangent direction |$T_{f(C),p_i}$| points out of |$\operatorname{Span}(p_1,\ldots ,p_m)$| (i.e., |$ T_{f(C),p_i}$| is not contained in |$\operatorname{Span}(p_1,\ldots ,p_m)$|⁠). In particular, the restriction |$\Phi |_{{\mathcal{F}}_t}$| of the map |$\Phi $| is a morphism. Proof. We follow the proof of [9, Lemma 6.4] and use the notations of Proposition 3.6. If |$f(C_1)$| is in |$\operatorname{Span}(p_1,\ldots , p_m)$|⁠, then each |$f(l_i)$| is in |$\operatorname{Span}(p_1,\ldots , p_m)$|⁠. Therefore, |$f(C)$| is in |$\operatorname{Span}(p_1,\ldots , p_m)$| that contradicts with Lemma 3.5. If |$f(l_1)$| is in |$\operatorname{Span}(p_1,\ldots , p_m)$|⁠, then the intersection of |$\operatorname{Span}(p_1,\ldots , p_m)$| and |$f(C_1)$| contains |$m-k$| marked points plus |$f(l_1\cap C_1)$|⁠. It follows that |$f(C_1)$| is in |$\operatorname{Span}(p_1,\ldots , p_m)$| by the fact that |$f(C_1)$| is a rational normal curve of degree |$m-k$|⁠, see Proposition 3.6. As before, it is also a contradiction. In short, no non-contracted irreducible component of |$C$| is mapped into the linear subspace |$\operatorname{Span}(p_1, \ldots , p_m)$|⁠. By Proposition 3.6, if the tangent direction |$T_{f(C),p_i}$| is in |$\operatorname{Span}(p_1,\ldots ,p_m)$|⁠, then the component of |$f(C)$| passing through |$p_i$| is in |$\operatorname{Span}(p_1,\ldots ,p_m)$|⁠, which is a contradiction. Let |$U_{\underline{P}}$| be the maximal open substack of the corresponding fiber of \begin{equation*} ev:\overline{{\mathcal{M}}}_{0,m}({\mathbb{P}}^n,m)\rightarrow ({\mathbb{P}}^n)^m \end{equation*} parametrizing stable maps for which no irreducible component is mapped into the linear subspace |$\operatorname{Span} (p_1,\ldots , p_m)$| (see [9, Definition 6.1]). It is clear that |$U_{\underline{P}}$| contains |$[(C,f,x_1,\ldots ,x_m)]$|⁠; hence, |${\mathcal{F}}_t\subseteq U_{\underline{P}}$|⁠. By the proof of [9, Lemma 6.4], it implies that the map |$\Phi $| is well-defined on |${\mathcal{F}}_t$|⁠. In the following, we define some maps on |${\mathcal{F}}$| and |${\mathcal{F}}_t$| that will be used in the next section. Let |$(\pi : {\mathcal{U}}\rightarrow{\mathcal{F}}, f_{{\mathcal{U}}}:{\mathcal{U}}\rightarrow X, \sigma _1, \ldots , \sigma _m)\!$||$\left (\textrm{resp}. (\pi _t: {\mathcal{U}}_t\rightarrow{\mathcal{F}}, f_{{\mathcal{U}}_t}:{\mathcal{U}}_t\rightarrow X, \sigma _1, \ldots , \sigma _m)\right )$| be the universal family of |${\mathcal{F}}$| (resp. |${\mathcal{F}}_t$|⁠), where the morphism |$\sigma _i$| is the universal (disjoint) section of |$\pi $| induced by the |$i$|-th marked point. So we have a map |$f_{{\mathcal{U}}}^*\Omega _X^1\rightarrow \Omega _{{\mathcal{U}}}^1$|⁠. Since there is a canonical map |$\Omega ^1_{{\mathcal{U}}}\rightarrow \Omega _{{\mathcal{U}}/{\mathcal{F}}}^1$|⁠, the composition of these two maps gives rise to |$f_{{\mathcal{U}}}^*\Omega ^1_X\rightarrow \Omega _{{\mathcal{U}}/{\mathcal{F}}}^1$| and induces a morphism \begin{equation*}\sigma_i^*f_{{\mathcal{U}}}^*\Omega_X^1\rightarrow\sigma_i^*\Omega_{{\mathcal{U}}/{\mathcal{F}}}^1.\end{equation*} Note that the composition |$f_{{\mathcal{U}}}\circ \sigma _i$| is a constant map with value |$p_i$| and the image of |$\sigma _i$| is in the locus of the smooth point of |$\pi $|⁠. It gives rise to a map \begin{equation} h:(T_{p_i}X)^{\vee}\otimes_{\mathbb{C}}{\mathcal{O}}_{\mathcal{F}} \rightarrow \sigma_i^*\omega_{{\mathcal{U}}/{\mathcal{F}}}, \end{equation} (3.3) where |$\omega _{{\mathcal{U}}/{\mathcal{F}}}$| is the dualizing sheaf of |$\pi $|⁠. At a point |$[(C,f,x_1,\ldots ,x_m)]\in{\mathcal{F}}$|⁠, the map |$h$| is surjective if and only if |$df_*(T_{C,x_i})$| is not 0. Definition 3.9. We define a map |$\pi _{p_i}:{\mathcal{F}} \dashrightarrow{\mathbb{P}}(T_{p_i}X)={\mathbb{P}}^{n-c-1}$| associating to a point |$(C,f, x_1,\ldots ,x_m)\in{\mathcal{F}}$| (⁠|$p_i=f(x_i)$|⁠) the tangent direction \begin{equation*} T_{f(C),p_i}=df_*(T_{C,x_i}) \in{\mathbb{P}}(T_{p_i}X). \end{equation*} Lemma 3.10. With the notations as above, the rational map |$\pi _{p_i}$| is well defined on |${\mathcal{F}}_t$|⁠. Proof. The locus of indeterminacy of |$\pi _{p_i}$| parametrizes stable maps |$(C,f, x_1,\ldots ,x_m)$| such that |$df_*(T_{C,x_i})$| is 0. We have a map \begin{equation*}\widetilde{h}:(T_{p_i}X)^{\vee}\otimes_{\mathbb{C}}{\mathcal{O}}_{{\mathcal{F}}_t} \rightarrow \sigma_i^*\omega_{{\mathcal{U}}_t/{\mathcal{F}}_t}\end{equation*} that is similar as the map |$h$| (3.3). By Proposition 3.6, the map |$\widetilde{h}$| is surjective and induces a morphism |$\pi _{p_i}|_{{\mathcal{F}}_t}$| that is the restriction of the map |$\pi _{p_i}$| to |${\mathcal{F}}_t$|⁠. In particular, the rational map |$\pi _{p_i}$| is well defined on |${\mathcal{F}}_t$|⁠. Lemma 3.11. Let |$(\pi _{p_i})_t$| (resp. |$\Phi _t$|⁠) be the restriction of the map |$\pi _{p_i}$| (resp. |$\Phi $|⁠) to |${\mathcal{F}}_t$|⁠. The line bundle |$(\pi _{p_i})_t^*({\mathcal{O}}_{{\mathbb{P}}^{n-c-1}}(1))$| is isomorphic to |$\Phi _t^*({\mathcal{O}}_{{\mathbb{P}}^{n-m}}(1)).$| Proof. The lemma is clear from the following commutative diagram: |$ $| (3.4) where |$L$| is the projection map |$L:{\mathbb{P}}(T_{p_i}X) \dashrightarrow{\mathbb{P}}^n/\operatorname{Span}(p_1,\ldots ,p_m)$| associating to a point |$[\overrightarrow{v}]\in{\mathbb{P}}(T_{p_i}X)$| the point |$\operatorname{Span}(\overrightarrow{v},p_1,\ldots ,p_m)/\operatorname{Span}(p_1,\ldots ,p_m)$|⁠. The indeterminacy locus of |$L$| consists of the points in |${\mathbb{P}}(T_{p_i}X)$| that can be represented by a nonzero vector |$\overrightarrow{v}\in T_{p_i}X$| lying in |$\operatorname{Span}(p_1,\ldots ,p_m)$|⁠. Hence, the image of |$\pi _{p_i}$| is outside of the indeterminacy locus of |$L$| by Lemma 3.8. The restriction of the composition |$L\circ \pi _{p_i}$| to |${\mathcal{F}}_t$| is just the morphism |$\Phi |_{{\mathcal{F}}_t}$|⁠. It implies the lemma. 4 Geometry of Rational Normal Curves In this section, we analyze rational normal curves in a projective space. We show that two rational normal curves of degree |$n$| coincide if they pass through the same |$n$| linearly nondegenerate points with the same tangent directions at these points. A way of showing this property is to put these two rational normal curves in a smooth surface and calculate their intersection, see Proposition 4.5. We also show that the degeneration types (cf. Definition 4.8) of two stable maps parametrized by |${\mathcal{F}}_t$| are the same if they have the same image for all the map |$\pi _{p_i}$|⁠, see Lemma 4.9. Since the proofs of Lemmas 4.7 and 4.9 are just some tedious analysis of degrees of rational curves in a projective space, we suggest the reader to skip their proofs in the 1st reading. Definition 4.1. Let |$C$| be a smooth rational curve in |${\mathbb{P}}^n$| with normal bundle \begin{equation*}N_{C/{\mathbb{P}}^n}=\bigoplus\limits ^{n-1}_{i=1} {\mathcal{O}}_C(a_i) ~~~(a_1\leq a_2 \leq \ldots \leq a_{n-1}).\end{equation*} We say that the rational curve |$C$| is almost balanced if |$a_{n-1}-a_1\leq 1$|⁠. In [26], Z. Ran gives a careful analysis about the balanced property of rational curves in projective spaces. We cite one result from [26]. Theorem 4.2. [26, Theorem 6.1 (Sacchiero)] A general rational curve of degree |$d$| at least |$n$| in |${\mathbb{P}}^n$| is almost balanced. Remark 4.3. This theorem that we cite here is proved only for characteristic zero. However, the following corollary is exactly what we use in this paper and we do not know whether it holds for any characteristic. A standard direct normal bundle calculation gives the following corollary. Corollary 4.4. Let |$C$| be a rational normal curve in |${\mathbb{P}}^n$|⁠. The normal bundle |$N_{C/{\mathbb{P}}^n}$| is |$\bigoplus \limits ^{n-1}_{i=1} {\mathcal{O}}_C(n+2).$| If the curve |$C$| is of degree |$m$| and contained in the complete intersection |$X$| passing through |$p_1,\ldots p_m$|⁠, then |$H^0\left (C, N_{C/X}\left (\sum \limits _{i=1}^m -2p_i\right )\right )=0$|⁠. Proposition 4.5. Let |$C$| and |$C^{\prime}$| be rational normal curves of degree |$m$| in |${\mathbb{P}}^m$|⁠. Let |$\{p_1,\ldots ,p_m\}$| be |$m$| points of |${\mathbb{P}}^m$| in general position and let |$T_{p_i}C$| (resp. |$T_{p_i}C^{\prime}$|⁠) be the tangent line to |$C$| (resp. |$C^{\prime}$|⁠) at the point |$p_i$|⁠. If |$C$| and |$C^{\prime}$| satisfy the following assumptions both curves |$C$| and |$C^{\prime}$| pass through the points |$\{p_1,\ldots ,p_m\}$|⁠, and |$T_{p_i}C$| coincides with |$T_{p_i}C^{\prime}$| for all |$i$|⁠, then |$C$| coincides with |$C^{\prime}$|⁠. Proof. We show the proposition by induction on |$m$|⁠. In the case when |$m$| equals |$3$|⁠, we project |$C$| and |$C^{\prime}$| from |$p_1$| to a plane |${\mathbb{P}}^2$| by a projection |$\pi $|⁠. The image |$\pi (C)$| and |$\pi (C^{\prime})$| are conics passing through the point |$Q=\pi (p_1)=T_{p_1}C \cap{\mathbb{P}}^2$|⁠. The conics are tangent to two distinct lines |$l_1=\pi (T_{p_2}C)$| and |$l_2=\pi (T_{p_3}C)$|⁠. The point |$Q$| is not on these two lines. By an elementary calculation, there is a unique conic |$S$| passing through |$\{Q,\pi (p_2),\pi (p_3) \}$| with tangent lines |$T_{S,\pi (p_2)}=l_1$| and |$T_{S,\pi (p_3)}=l_2$|⁠. Denote by |$\operatorname{Cone} (S)$| the projective cone of |$S$| with the vertex |$p_1$|⁠. The curves |$C$| and |$C^{\prime}$| are contained in a singular quadric surface |$\operatorname{Cone} (S)$|⁠. For the point |$p_2$|⁠, the same argument implies that |$C$| and |$C^{\prime}$| are in another (distinct) singular quadric surface |$\operatorname{Cone} (S^{\prime})$| with the vertex point |$p_2$|⁠. Thus, the union |$C\cup C^{\prime}$| of curves is contained in the intersection |$\operatorname{Cone} (S^{\prime})\cap \operatorname{Cone} (S)$| of two singular quadric surfaces. Note that the intersection of |$\operatorname{Cone} (S^{\prime})$| and |$\operatorname{Cone} (S)$| is a curve of degree |$4$|⁠. Therefore, |$C$| coincides with |$C^{\prime}$|⁠. Suppose that the proposition holds for the case when |$m$| equals |$s$|⁠. We consider the case when |$m$| is |$s+1$|⁠. As above, we project |$C$| and |$C^{\prime}$| from the point |$p_1$| to a hyperplane |${\mathbb{P}}^{s}$| by a projection |$\pi $|⁠. The rational normal curves |$\pi (C)$| and |$\pi (C^{\prime})$||$(\subseteq{\mathbb{P}}^s)$| are of degree |$s$| with the same tangent line |$\pi (T_{p_i}C)(=\pi (T_{p_i}C^{\prime}))$| at |$\pi (p_i)$| for |$i=2,3,\ldots ,s+1$|⁠. By induction, we conclude that the curve |$\pi (C)$| coincides with |$\pi (C^{\prime})$|⁠. We denote the curve |$\pi (C)$| by |$D$|⁠. Hence, both |$C$| and |$C^{\prime}$| are in the projective cone |$\operatorname{Cone} (D)$| joining |$p_1$| and |$D$|⁠. We blow up the unique singular point |$p_1$| of |$\operatorname{Cone} (D)$| that gives the proper transform |$\widetilde{C}$| (resp. |$\widetilde{C^{\prime}}$|⁠) of |$C$| (resp. |$C^{\prime}$|⁠). A standard calculation of the divisors on a Hirzebruch surface shows that the intersection number of |$\widetilde{C}$| and |$\widetilde{C^{\prime}}$| equals |$s+2$|⁠. Note that |$\widetilde{C}$| and |$\widetilde{C^{\prime}}$| intersect at |$s+1$| distinct point. At |$s$| points (⁠|$\{p_2,\ldots ,p_{s+1}\}$|⁠) among these |$s+1$| points, the curves |$\widetilde{C}$| and |$\widetilde{C^{\prime}}$| are tangent. If |$\widetilde{C}$| does not coincide with |$\widetilde{C^{\prime}}$|⁠, then this intersection number is at least |$2s+1$|⁠, which gives a contradiction. In the rest of this section, we will show that the degeneration types (cf. Definition 4.8) of two stable maps parametrized by |${\mathcal{F}}_t$| are the same (Definition 4.8) if they have the same image for all the map |$\pi _{p_i}$|⁠, see Lemma 4.9. Lemma 4.6. Let |$C$| be a rational normal curve of degree |$m$| in |${\mathbb{P}}^m$|⁠. Then any |$m+1$| distinct points on |$C$| are linearly nondegenerate. Proof. It is a direct calculation on the standard rational normal curve, see [17, Chapter I]. Lemma 4.7. If |$C$| is a rational normal curve of degree |$m$| in |${\mathbb{P}}^m$| passing through m distinct points |$p_1,\ldots ,p_m$|⁠, then the projective space |$\operatorname{Span} (p_1,\ldots ,p_m,T_{p_i}C)$| has dimension |$m$|⁠. If the space |$\operatorname{Span} (Q_1,\ldots ,Q_k, T_{Q_1}C,\ldots ,T_{Q_k}C)$| has dimension |$l$| for distinct |$k$| (⁠|$1<k<m$|⁠) points |$Q_1,\ldots ,Q_k$| on |$C$|⁠, then |$l$| is greater than |$k$|⁠. Proof. Since the space |$\operatorname{Span}(C)$| has dimension |$m$| and the space |$\operatorname{Span}(p_1,\ldots ,p_m,T_{p_i}C)$| is tangent to |$C$|⁠, the degree |$\deg \left (C\cap \operatorname{Span}(p_1,\ldots ,p_m,T_{p_i}C)\right )$| is greater than |$m$|⁠. If the space |$\operatorname{Span}(p_1,\ldots ,p_m,T_{p_i}C)$| has dimension |$m-1$|⁠, then it would contradict with the fact that the degree |$\deg C$| of |$C$| equals |$m$|⁠. The 1st assertion follows. Since the space |$\operatorname{Span} (Q_1,\ldots ,Q_k, T_{Q_1}C,\ldots ,T_{Q_k}C)$| contains |$k$| points of |$C$| and at least one tangent line to |$C$|⁠, similar as above, the space |$\operatorname{Span} (Q_1,\ldots ,Q_k, T_{Q_1}C,\ldots ,T_{Q_k}C)$| is just some projective space |${\mathbb{P}}^l$| with |$l$| at least |$k$|⁠. We exclude the case when |$l$| equals |$k$|⁠. Suppose to the contrary that |$\operatorname{Span} (Q_1,\ldots ,Q_k, T_{Q_1}C,\ldots ,T_{Q_k}C)$| is a projective space |${\mathbb{P}}^k$| of dimension |$k$|⁠. We can pick up |$m-k-1$| points on |$C$| but not in |$\operatorname{Span} (Q_1,\ldots ,Q_k, T_{Q_1}C,\ldots ,T_{Q_k}C)(={\mathbb{P}}^k)$|⁠. By Lemma 4.5, the space |$\operatorname{Span} (Q_1,\ldots ,Q_k, T_{Q_1}C,\ldots ,T_{Q_k}C)$| with these |$m-k-1$| points on |$C$| spans a projective space |${\mathbb{P}}^{m-1}$| of dimension |$m-1$|⁠. Hence, we have the following inequalities: \begin{equation*}\deg({\mathbb{P}}^{m-1}\cap C)\geq m-k-1+2k=m+k-1>m,\end{equation*} which is impossible. It follows that |$l$| is greater than |$k$|⁠. Definition 4.8. Two stable maps |$(C,f,x_1,\ldots ,x_m)$| and |$(C^{\prime},f^{\prime},x^{\prime}_1,\ldots ,x^{\prime}_m)$| parametrized by |${\mathcal{F}}_t$| are said to have the same degeneration type if the following condition is satisfied: the marked point |$x_i$| is on the handle of |$C$| if and only if the marked point |$x^{\prime}_i$| is on the handle of |$C^{\prime}$| for all |$i=1,\ldots m$|⁠. (The handle of a stable map parametrized by |${\mathcal{F}}_t$| is described in Proposition 3.6.) Lemma 4.9. Let |$(C,f,x_1,\ldots ,x_m)$| and |$(C^{\prime},f^{\prime},x^{\prime}_1,\ldots ,x^{\prime}_m)$| be two stable maps parametrized by |${\mathcal{F}}_t$| with |$m$| at least |$3$|⁠. If the point |$\pi _{p_i}([(C,f,x_1,\ldots ,x_m)])$| coincides with the point |$\pi _{p_i}([(C^{\prime},f^{\prime},x^{\prime}_1,\ldots ,x^{\prime}_m)])$| for all i, then the stable maps have the same degeneration type. Proof. From the assumption, we know that |$T_{p_i}f(C)$| coincides with |$T_{p_i}f^{\prime}(C^{\prime})$|⁠, where |$T_{p_i}f(C)$| is the tangent line to |$f(C)$| at the point |$p_i$| and similarly for |$T_{p_i}f^{\prime}(C^{\prime})$|⁠, see Lemma 3.8. By Proposition 3.6, we may assume that the curve |$f(C)$| is a union of |$C_1, L_1\ldots $| and |$L_k$|⁠; and |$f^{\prime}(C^{\prime})$| is a union of |$C^{\prime}_1, L^{\prime}_1\ldots $| and |$L^{\prime}_s$| with |$0\leq k$| and |$s\leq m$|⁠, the components |$C_1$| and |$C^{\prime}_1$| are rational normal curves and the components |$\{L_i\}$| and |$\{L^{\prime}_i\}$| are lines. We hope it will cause no confusion if we sometimes also denote |$f(C)$| (resp. |$f^{\prime}(C^{\prime})$|⁠) by |$C$| (resp. |$C^{\prime}$|⁠). In the maximal degeneration case (cf. Remark 3.7), the curve |$C_1$| or |$C_2$| is collapsed to a point. The lemma in this case is trivial. Suppose that the curve |$C$| is smooth. By the 2nd assertion of Lemma 4.7, the curve |$C^{\prime}$| is smooth, that is, |$s=0$|⁠. Hence, we may assume that the numbers |$k$| and |$s$| are at least |$1$|⁠. If both lines |$L_1$| and |$L^{\prime}_1$| (after relabeling the teeth) pass through the same marked point |$p_1$| with the same tangent lines at the point |$p_1$| (i.e., |$T_{p_1}L_1=T_{p_1}L^{\prime}_1$|⁠), then the line |$L_1$| coincides with the line |$L^{\prime}_1.$| We take out |$L_1$| and |$L^{\prime}_1$| from these two stable maps. By induction on the number of teeth (⁠|$k$| and |$s$|⁠), we show the assertion. Therefore, we may assume that the curve |$C^{\prime}_1$| contains the points \begin{equation*}p_1(\in L_1),\ldots,p_k(\in L_k).\end{equation*} We will get a contradiction as follows. Note that the intersection |$C_1\cap C^{\prime}_1$| consists of |$(m-s)-k(\geq 0)$| points that are in general position. Suppose that the intersection |$C_1\cap C^{\prime}_1$| consists of the points \begin{equation*}\{p^{\prime}_1,p^{\prime}_2,\ldots,p^{\prime}_{m-s-k}\}(\subseteq \{p_1,\ldots,p_m\}).\end{equation*} By Proposition 3.6, the space |$\operatorname{Span} C_1$| is |${\mathbb{P}}^{m-k}$| and the space |$\operatorname{Span} C^{\prime}_1$| is |${\mathbb{P}}^{m-s}$|⁠. Note that the curve |$C^{\prime}_1$| contains the points {|$p_1,\ldots ,p_k$|}. It follows that the space |$\operatorname{Span}(C_1\cup C^{\prime}_1)$| is |${\mathbb{P}}^m.$| Therefore, we conclude that the space |$\operatorname{Span} C_1 \cap \operatorname{Span} C^{\prime}_1$| is |${\mathbb{P}}^{(m-s)+(m-k)-m}(={\mathbb{P}}^{m-s-k}).$| By Lemma 4.7, if |$m-s-k$| is at least |$2$|⁠, then the space |$\operatorname{Span}(p^{\prime}_1,\ldots ,p^{\prime}_{m-s-k}, T_{p^{\prime}_1}C,\ldots ,T_{p^{\prime}_{m-s-k}}C )$| is |${\mathbb{P}}^l$| and |$l$| is greater than |$m-s-k$|⁠. However, the space |$\operatorname{Span}(p^{\prime}_1,\ldots ,p^{\prime}_{m-s-k}, T_{p^{\prime}_1}C,\ldots ,T_{p^{\prime}_{m-s-k}}C )$| is contained in the space |$\operatorname{Span} C^{\prime}_1\cap \operatorname{Span} C_1={\mathbb{P}}^{m-s-k}.$| It is a contradiction. It follows that |$m-s-k$| is at most |$1$|⁠. If |$m-s-k$| equals |$0$|⁠, then the intersection |$\operatorname{Span} C_1\cap \operatorname{Span} C^{\prime}_1$| is a point. Since the curve |$C_1$| contains the points {|$p_1,\ldots ,p_k$|}, the space |$\operatorname{Span} C$| contains |$L_1,\ldots , L_k$|⁠. Note that the lines |$L_1,\ldots , L_k$| intersect |$C_1$| at |$k$| points. It follows that |$k$| equals |$1$|⁠. By symmetry, we also have that |$s$| equals |$1$| as well. Therefore, we conclude that |$m$| equals |$2$|⁠. It is a contradiction. If |$m-s-k$| equals |$1$|⁠, then the intersection |$\operatorname{Span} C_1 \cap \operatorname{Span} C^{\prime}_1$| is |${\mathbb{P}}^1$|⁠. It also implies that the intersection |$C_1\cap C^{\prime}_1$| is a point. We claim that the intersection of |$\operatorname{Span} C^{\prime}_1$| and |$C_1$| consists of two points and one of them is |$C_1\cap C^{\prime}_1$|⁠. Suppose that the intersection of |$\operatorname{Span} C^{\prime}_1$| and |$C_1$| consists of at least three points. By Lemma 4.6, the intersection |$\operatorname{Span} C_1 \cap \operatorname{Span} C^{\prime}_1$| contains a projective plane |${\mathbb{P}}^2$|⁠. This contradiction proves the claim. Note that the lines |$L_1,\ldots ,L_k (\subseteq \operatorname{Span} C^{\prime}_1)$| intersect |$C_1$| at |$k$| distinct points |$\{c_1,\ldots ,c_k\}$|⁠. Since the intersection |$C_1\cap C^{\prime}_1 \cap \{c_1,\ldots ,c_k\}$| is empty, we have the set |$\{c_1,\ldots ,c_k\}$| of points is just |$\operatorname{Span} C^{\prime}_1\cap C_1-C_1\cap C^{\prime}_1$|⁠. It follows that |$k$| equals |$1$|⁠. By symmetry, it implies that |$s$| equals |$1$|⁠. Therefore, we conclude that |$m$| equals to |$1+k+s$||$(=3)$|⁠. In this case, by some elementary analysis of the possible degeneration types, we conclude that the stable maps have the same degeneration type. 5 Cycle Relations In this section, we give a cycle relation in Proposition 5.6, from which it follows that each irreducible component (say |$\Delta _{i,t}$|⁠) of the boundary divisor |$\Delta _t$| of |${\mathcal{F}}_t$| is smooth and gives an ample divisor. The idea of the proof of Proposition 5.6 is to show that the line bundle |${\mathcal{O}}_{{\mathcal{F}}_t}(\Delta _{i,t})$| coincides with the line bundle |$\sigma _i^*\omega _{{\mathcal{U}}_t/{\mathcal{F}}_t}$| numerically (Lemma 5.5) that is isomorphic to an ample line bundle |$\lambda |_{{\mathcal{F}}_t}$| (see Proposition 5.1 and Lemma 5.4). We have shown in Lemma 3.8 that the restriction |$\Phi _t:{\mathcal{F}}_t\rightarrow{\mathbb{P}}^{n-m}$| is a morphism. Denote |$\Phi _t ^*({\mathcal{O}}_{{\mathbb{P}}^{n-m}}(1))$| by |$\lambda |_{{\mathcal{F}}_t}$|⁠. We first use the results in Section 4 to show that the complete linear system |$|\lambda |_{{\mathcal{F}}_t}|$| separates points. Proposition 5.1. Let |$(C,f,x_1,\ldots ,x_m)$| and |$(C^{\prime},f^{\prime},x^{\prime}_1,\ldots ,x^{\prime}_m)$| be two stable maps parametrized by |${\mathcal{F}}_t$|⁠. If the point |$\pi _{p_i}([(C,f,x_1,\ldots ,x_m)])$| coincides with the point |$\pi _{p_i}([(C^{\prime},f^{\prime},x^{\prime}_1,\ldots ,x^{\prime}_m)])$| for all |$i$|⁠, then these two stable maps are the same. In particular, the complete linear system |$|\lambda |_{{\mathcal{F}}_t}|$| on |${\mathcal{F}}_t$| separates points, in other words, the morphism |$|\lambda |_{{\mathcal{F}}_t}|:{\mathcal{F}}_t\rightarrow{\mathbb{P}}^N$| induced by |$|\lambda |_{{\mathcal{F}}_t}|$| is injective and the line bundle |$\lambda |_{{\mathcal{F}}_t}$| is ample. Proof. From Lemma 3.11, we see that the maps |$\pi _{p_1}|_{{\mathcal{F}}_t},\pi _{p_2}|_{{\mathcal{F}}_t}\ldots ,\pi _{p_m}|_{{\mathcal{F}}_t}$| are induced by sections of |$\lambda |_{{\mathcal{F}}_t}$|⁠. Therefore, the 2nd assertion follows from the 1st assertion. The stable maps parametrized by |$(C,f,x_1,\ldots ,x_m)$| and |$(C^{\prime},f^{\prime},x^{\prime}_1,\ldots ,x^{\prime}_m)$| have the same degeneration type by Lemma 4.9. So it follows from Proposition 4.5 that the stable maps |$(C,f,x_1,\ldots ,x_m)$| and |$(C^{\prime},f^{\prime},x^{\prime}_1,\ldots ,x^{\prime}_m)$| are the same if the curve |$C$| or |$C^{\prime}$| is smooth. In general, by Proposition 3.6, each of the image curves |$f(C)$| and |$f^{\prime}(C^{\prime})$| is a union of a rational normal curve and several lines. The configuration of these lines is uniquely determined by the tangent directions at the marked points and the degeneration type of the stable map. Therefore, the 1st assertion follows by induction on the number of components of |$C$| and |$C^{\prime}$|⁠. Definition 5.2. Let |$\Delta $| be the boundary divisor of |${\mathcal{F}}$|⁠. Denote |${\mathcal{F}}_t \cap \Delta $| by |$\Delta _t$|⁠. For |$i\in \{1,2\ldots ,m\}$|⁠, we have a divisor |$\Delta _{i,t} (\subseteq \Delta _t)$| whose general points are parametrizing the stable maps whose image is a union of a line containing |$p_i$| and a rational normal curve, see Proposition 3.6. For the sake of simplicity, we sometimes denote by |$\Delta _i$| the divisor |$\Delta _{i,t}$|⁠. Lemma 5.3. The divisor |$\Delta _t$| is a simple normal crossing divisor on |${\mathcal{F}}_t$|⁠. Proof. Recall that the boundary divisor |$\Delta $| of |${\mathcal{F}}$| is a simple normal crossing divisor in |${\mathcal{F}}$|⁠, see Lemma 2.3. Since |$t\in \overline{{\mathcal{M}}}_{0,m}$| is a general point, the lemma follows from Lemma 2.3 and the generic smoothness theorem. Recall that |${\mathcal{F}}$| (resp. |${\mathcal{F}}_t$|⁠) has the universal family |$(\pi : {\mathcal{U}}\rightarrow{\mathcal{F}}, f_{{\mathcal{U}}}:{\mathcal{U}}\rightarrow X, \sigma _1, \ldots , \sigma _m)$||$\left (resp. (\pi _t: {\mathcal{U}}_t\rightarrow{\mathcal{F}}, f_{{\mathcal{U}}_t}:{\mathcal{U}}_t\rightarrow X, \sigma _1, \ldots , \sigma _m)\right )$|⁠, where |$\sigma _i$| are disjoint sections of |$\pi $|⁠. The proof of our main theorem uses the key fact that the line bundle associated with the divisor |$\Delta _{i,t}$| is isomorphic to |$\sigma _i^*\omega _{{\mathcal{U}}_t/{\mathcal{F}}_t}$|⁠, which is Proposition 5.6. As preparation, we show in Lemma 5.5 that the isomorphism |$\Delta _{i,t}\cong \sigma _i^*\omega _{{\mathcal{U}}_t/{\mathcal{F}}_t}$| holds numerically. The idea of the proof of Lemma 5.5 is to use testing curves, namely, we can reduce the proof to the simple case in which |${\mathcal{F}}_t$| is a curve and |${\mathcal{U}}_t$| is a blow-up surface of |${\mathcal{F}}_t\times{\mathbb{P}}^1$|⁠. Lemma 5.4. The line bundle |$\lambda |_{{\mathcal{F}}_t}$| is isomorphic to the line bundle |$\sigma _i^*\omega _{{\mathcal{U}}_t/{\mathcal{F}}_t}$|⁠. Proof. Since the image of |$\sigma _i$| is in the smooth locus of |$\pi _t$|⁠, we have \begin{equation*}\sigma _i^*\omega_{{\mathcal{U}}_t/{\mathcal{F}}_t}^{\vee}=\sigma _i^*(\Omega^1_{{\mathcal{U}}_t/{\mathcal{F}}_t})^{\vee}=\sigma _i^*T_{{\mathcal{U}}_t/{\mathcal{F}}_t}.\end{equation*} From Definition 3.9, the line bundle |$\sigma _i^*T_{{\mathcal{U}}_t/{\mathcal{F}}_t}$| is isomorphic to |$(\pi _{p_i}|_{{\mathcal{F}}_t})^*{\mathcal{O}}_{{\mathbb{P}}(T_{p_i}X)}(-1)$|⁠. It follows from Lemma 3.11 that |$\sigma _i^* T_{{\mathcal{U}}_t/{\mathcal{F}}_t}$| is isomorphic to |$-\lambda |_{{\mathcal{F}}_t}$|⁠. Lemma 5.5. The line bundle |$\sigma _i^*(\omega _{{\mathcal{U}}_t/{\mathcal{F}}_t})\otimes{\mathcal{O}}_{{\mathcal{F}}_t}(-\Delta _{i,t})$| on |${\mathcal{F}}_t$| is numerically trivial. In particular, the divisor |$\Delta _{i,t}$| is an ample divisor on |${\mathcal{F}}_t$|⁠. Proof. Let |$B$| be a smooth projective curve on |${\mathcal{F}}_t$| such that |$B$| and |$\Delta _i$| intersect transversely at general points of |$\Delta _i$| for |$i=1,\ldots ,m$|⁠. Denote the restriction of |${\mathcal{U}}_t\rightarrow{\mathcal{F}}_t$| to |$B$| by |$C\rightarrow B$|⁠. We have a Cartesian diagram as follows: where the section |$\widehat{\sigma _k}$| of |$\pi _C$| is the pullback of the section |$\sigma _k$| of |$\pi _t$| with |$g\circ \widehat{\sigma _k}=\sigma _k\circ h$|⁠. We first show that the line bundle |$h^*\left (\sigma _i^*(\omega _{{\mathcal{U}}_t/{\mathcal{F}}_t})\right )$| is the same as |$h^*\left ({\mathcal{O}}_{{\mathcal{F}}_t}( \Delta _i)\right ).$| We claim that |$C$| is a smooth surface. From Lemma 5.3, |$\Delta _t$| is a simple normal crossing divisor on |${\mathcal{F}}_t$|⁠. It is clear that |${\mathcal{U}}_t$| is a family of semistable curves over |${\mathcal{F}}_t$| with discriminant locus |$\Delta _t$|⁠. Note that |$B$| and |$\Delta _t(= \cup \Delta _{i,t})$| intersect transversely. Therefore, around a nodal point of some fiber |$C_b$| of |$\pi _C$| over |$b\in B$|⁠, |$C$| is defined by the equation |$xy=s$| (in |$\mathbb{C}^2\times B$|⁠), where |$s$| is a uniformizer of |$B$| at |$b\in B\cap \Delta _t$|⁠. A local calculation shows that the surface |$C$| is smooth. Recall from Definition 5.2 that general points of |$\Delta _i$| parametrize the stable maps whose image is a union of a line containing |$p_i$| and a rational normal curve. Note that |$B$| intersects with |$\Delta _i$| at general points of |$\Delta _i$|⁠. Therefore, the fibers of |$\pi _C$| are either a smooth rational curve or a union of two smooth rational curves intersecting transversely at a point. Namely, the fiber |$C_s$| of |$\pi _C$| over |$s\in \Delta _t \cap B$| has two components |$L_s$| and |$Q_s$|⁠, where |$L_s$| is the unique component of the stable map parametrized by |$s$| whose image in |$X$| is a line and |$Q_s$| is the remaining component. Moreover, we have \begin{equation*}C_s=L_s+Q_s, C_s\cdot L_s=0\qquad \text{~and~}\qquad L_s\cdot Q_s=1.\end{equation*} It follows that |$L_s$| is a |$(-1)$|-curve. By Castelnuovo’s contraction theorem, we have a map |$\phi :C \rightarrow C^{\prime}$| contracting |$(-1)$|-curves |$L_s$| (⁠|$s\in \Delta _t\cap B$|⁠) and the map |$\pi _C$| factors through |$\phi $|⁠, that is, we have that the map |$\pi _C$| coincides with the composition |$q \circ \phi : C \xrightarrow{\phi } C^{\prime} \xrightarrow{q} B$|⁠. The maps |$\phi \circ \widehat{\sigma _k}$| (⁠|$k=1,\ldots ,m$|⁠) give at least three disjoint sections of the |${\mathbb{P}}^1$|-bundle |$q:C^{\prime}\rightarrow B$|⁠. It follows that the |${\mathbb{P}}^1$|-bundle |$C^{\prime}/B$| is trivial, that is, |$C^{\prime}\cong B\times{\mathbb{P}}^1$|⁠. Now, we are able to express the dualizing sheaf |$\omega _{C/B}$| in an explicit way: \begin{align*} \omega_{C/B}=K_C\otimes \pi_C^*K_B^{-1}&=\left(\phi^*K_{C^{\prime}}\otimes{\mathcal{O}}_C(\sum\limits_{i=1}^m\sum\limits_{s\in \Delta_i\cap B} L_s)\right)\otimes \pi_C^*K_B^{-1}\\ &=\phi^*\omega_{C^{\prime}/B}\otimes{\mathcal{O}}_C(\sum\limits_{i=1}^m\sum\limits_{s\in \Delta_i\cap B} L_s), \end{align*} where the 2nd equality in the 1st line is due to the blow-up formula for the canonical bundle. Note that the composition |$B\xrightarrow{\widehat{\sigma _k}} C \xrightarrow{\phi } C^{\prime}= B\times{\mathbb{P}}^1 \xrightarrow{pr_2} {\mathbb{P}}^1$| is a constant morphism and |$\omega _{B\times{\mathbb{P}}^1/B}=pr_2^*(\omega _{{\mathbb{P}}^1})$|⁠. It follows that \begin{align*} \widehat{\sigma_k}^* \omega_{C/B}&=\widehat{\sigma_k}^*\left(\phi^* pr_2^*(\omega_{{\mathbb{P}}^1})\right)\otimes \widehat{\sigma_k}^* \left( {\mathcal{O}}_C \left(\sum\limits_{i=1}^m\sum\limits_{s\in \Delta_i\cap B} L_s\right) \right)\\ &={\mathcal{O}}_B\left(\sum\limits_{s\in \Delta_k\cap B} L_s \cdot \widehat{\sigma_k}(B) \right)\\ &={\mathcal{O}}_B\left(\sum\limits_{s\in \Delta_k\cap B} s\right)=h^*\left({\mathcal{O}}_{{\mathcal{F}}_t}( \Delta_k)\right), \end{align*} where the equalities in the 2nd and 3rd lines are due to the fact that |$\widehat{\sigma }_k(B)$| does not intersect |$L_s$| for |$s \not \in \Delta _k\cap B$| and intersect |$L_s$| transversely for |$s \in \Delta _k\cap B$|⁠. On the other hand, from the base change property of dualizing sheaves, we have \begin{equation*}h^*\left(\sigma_k^*(\omega_{{\mathcal{U}}_t/{\mathcal{F}}_t})\right)=\widehat{\sigma_k}^*g^*\omega_{{\mathcal{U}}_t/{\mathcal{F}}_t}=\widehat{\sigma_k}^* \omega_{C/B}.\end{equation*} Therefore, we conclude that |$h^*\left (\sigma _k^*(\omega _{{\mathcal{U}}_t/{\mathcal{F}}_t})\right )$| is the same as |$h^*\left ({\mathcal{O}}_{{\mathcal{F}}_t}( \Delta _k)\right ).$| In general, to show the proposition, it suffices to prove that the line bundle |$\sigma _i^*(\omega _{{\mathcal{U}}_t/{\mathcal{F}}_t})\otimes{\mathcal{O}}_{{\mathcal{F}}_t}(-\Delta _i)$| is numerically trivial on the curves given by complete intersections of a very ample divisor, cf. [12, Page 69, Chapter 3 and Remark 3.8]. By the theorem of Bertini, this can be reduced to the case for which the curve is smooth and intersects with |$\Delta _i$| at general points of |$\Delta _i$| transversely, which we have already proved. It follows from Proposition 5.1 and Lemma 5.4 that the line bundle |$\sigma _i^*\omega _{{\mathcal{U}}_t/{\mathcal{F}}_t}$| is ample. By Kleiman’s ampleness criterion, we conclude that the divisor |$\Delta _i$| is ample. Proposition 5.6. Suppose that |$n-m(\sum \limits _{i=1}^c d_i-c)-c\geq 2$|⁠. Then |${\mathcal{F}}_t$| is nonempty and the line bundle |${\mathcal{O}}_{{\mathcal{F}}_t}(\Delta _{i,t})$| is the restriction |$\lambda |_{{\mathcal{F}}_t}$| of |$\lambda $| to |${\mathcal{F}}_t$|⁠. Remark 5.7. We learn from the anonymous referee that Proposition 5.6 can be also deduced from the 2nd formula in [11, Lemma 5.8]. Proof. Let |$D$| be the space of |$m$| lines in |$X$| such that each line contains exactly one point among the points |$p_1,\ldots ,p_m$| and these |$m$| lines intersect at a point. The space |$D$| is a non-singular complete intersection in |${\mathbb{P}}^n$| of type \begin{equation} (T_2(d_1,m),(T_2(d_2,m), \ldots, T_2(d_c,m)), \end{equation} (5.1) see [8, Page 83 (2)]. Hence, the dimension of |$D$| is |$n-m(\sum \limits _{i=1}^c d_i-c)-c$| that is at least |$2$|⁠. Let |$t \in \overline{{\mathcal{M}}}_{0,m}$| be a general point parametrizing a smooth rational curve with |$m$| marked points. Denote this rational curve by |$(R,y_1,\ldots , y_m)$|⁠. Let |$u\in D$| be the point representing the union of lines |$l_1\cup l_2\cup \ldots \cup {l\!\!^{\tiny{/}}}_m$| with |$p_i\in l_i$| and |$\cap _{i=1}^ml_i=\{Q\}$|⁠. We can canonically associate to |$u$| the stable map |$(C,f,x_1,\ldots ,x_m)$| of maximal degeneration type (see Lemma 2.5) in |${\mathcal{F}}_t$| as follows: the domain |$C$| is |$R\coprod l_1\coprod l_2 \ldots \coprod {l\!\!^{\tiny{/}}}_m/$||$\left (\textrm{identifying}\ Q(\in l_i)\ \textrm{with}\ y_i \right )$|⁠, and the map |$f$| maps |$l_i$| identically to the line |$l_i \subseteq X$| with |$f(x_i)=p_i$| and contracts |$R$| to the point |$Q$|⁠. This gives rise to a morphism |$ $| (5.2) which is a bijection, and hence an isomorphism by the smoothness and irreducibility of |$D$| and Lemma 5.3. Note that the dimension of |${\mathcal{F}}_t$| is \begin{align*} \dim({\mathcal{F}}_t)&=\dim({\mathcal{F}})-\dim(\overline{{\mathcal{M}}}_{0,m})= (c+1-\sum\limits_{i=1}^c d_i)m+n-c \end{align*} by Lemma 2.3. It follows that the dimension |$\dim (Y)$| equals |$\dim ({\mathcal{F}}_t)-m$|⁠. Therefore, by Lemma 2.3, the intersections |$\bigcap \limits _{i=1}^s\Delta _i$| are smooth and connected for |$s=1,\ldots ,m$|⁠. Note that |$D$| is a smooth complete intersection of dimension at least |$2$| in a projective space. It follows from the Lefschetz hyperplane theorem that |$H^2(Y,{\mathbb Z})$| is torsion-free and |$H^1(Y,{\mathcal{O}}_Y)$| is |$0$|⁠. By the Lefschetz hyperplane theorem and induction on s, we have that |$H^1({\mathcal{F}}_t,{\mathcal{O}}_{{\mathcal{F}}_t})$| is |$0$|⁠, |$H^2({\mathcal{F}}_t,{\mathbb Z})$| is torsion-free, and the map of the 1st Chern class |$c_1:\operatorname{Pic}({\mathcal{F}}_t)\rightarrow H^2({\mathcal{F}}_t,{\mathbb Z})$| is injective. It follows from Lemma 5.5 and the Poincaré duality theorem that |${\mathcal{O}}_{{\mathcal{F}}_t}(\Delta _{i,t})$| is |$\lambda |_{{\mathcal{F}}_t}$|⁠. Finally, the proposition follows from Lemma 5.4. From the proof of Proposition 5.6, we have the following corollary. Corollary 5.8. Let |$Y$| be the locus in |${\mathcal{F}}_t$| parametrizing the stable maps of maximal degeneration type, that is, the image of the stable map is a union of |$m$| lines that intersect at a point. Assume that |$n+m(c-\sum \limits _{i=1}^c d_i)-c\geq 2$|⁠. Then, (1) The intersection |$\bigcap \limits _{i=1}^{s}\Delta _{i,t}$| is smooth and connected for |$s=1,\ldots m$|⁠. The locus |$Y$| is |$\bigcap \limits _{i=1}^{m}\Delta _{i,t}$| and of dimension |$n+m(c-\sum \limits _{i=1}^c d_i)-c$|⁠. (2) Via the map |$\Phi |_{Y}:Y \hookrightarrow{\mathbb{P}}^{n-m}$|⁠, the variety |$Y$| is a complete intersection in |${\mathbb{P}}^{n-m}$| defined by a (disjoint) union of homogeneous polynomials of type (see Definition 1.1) \begin{equation*}(T_1(d_1,m), T_2(d_2,m),\ldots, T_2(d_c,m)).\end{equation*} Proof. The 1st assertion follows from the proof of Proposition 5.6. With the same notation as in the proof of Proposition 5.6, we recall that |$D$| is a smooth complete intersection in |${\mathbb{P}}^n$| of the type as in (5.1). To show the 2nd assertion, we consider the map where |$\iota $| is the natural inclusion and |$pr$| is the projection from the projective space |${\mathbb{P}}^{m-1}=\operatorname{Span}(p_1,\ldots ,p_m)$| to a projective subspace |${\mathbb{P}}^{n-m}$| of |${\mathbb{P}}^n$|⁠. We can choose the projective space |${\mathbb{P}}^{n-m}$| to be the intersection of |$m$| hyperplanes defined by linear forms in |$T_2(d_1,m)$|⁠. To be precise, these |$m$| hyperplanes are projective hyperplanes tangent to the hypersurface |$F_1=0$| at |$p_1,\ldots ,p_m$|⁠, respectively, where |$F_1$| is a homogeneous polynomial of degree |$d_1$| for defining |$X$| as a complete intersection |$X=V(F_1,\ldots , F_c)$|⁠. We claim that the composition |$pr\circ \iota $| is the morphism |$\Phi |_Y$|⁠, from which the 2nd assertion follows. In fact, let |$(C,f,x_1,\dots , x_m)$| be a stable map parametrized by |$Y$|⁠. Suppose that |$f(C)$| is a union |$l_1\cup l_2\cup \ldots \cup l_m$| of lines that intersect at |$Q$|⁠. The projective space |$\operatorname{Span}(f(C))$| is given by |$\operatorname{Span}(p_1, \ldots , p_m, Q)$|⁠. Note that |$D$| is the intersection |$\bigcap \limits ^m_{i=1}L_i$|⁠, where |$L_i$| is a union of lines on X passing through |$p_i$|⁠. Therefore, the point |$Q$| is on |$D$| and |$\psi (Q)=[(C,f,x_1,\dots , x_m)]$|⁠, see (5.2). It follows that the point |$pr\circ \iota \left ([(C,f,x_1,\dots , x_m)] \right )$| is \begin{align*} pr (Q)&=\operatorname{Span}(p_1,\ldots,p_m,Q)\cap{\mathbb{P}}^{n-m}\\ &=\operatorname{Span}(f(C))\cap{\mathbb{P}}^{n-m}. \end{align*} On the other hand, we can identify |${\mathbb{P}}^n/\operatorname{Span}(p_1,\ldots ,p_m)$| with |${\mathbb{P}}^{n-m}$| by sending a point |$P\in{\mathbb{P}}^n/\operatorname{Span}(p_1,\ldots ,p_m)$| parametrizing a projective space |$P$| of dimension |$m$| and containing |$\operatorname{Span}(p_1,\ldots ,p_m)$| to the point |$P\cap{\mathbb{P}}^{n-m}$|⁠. Under this identification, we conclude that |$pr\circ \iota \left ([(C,f,x_1,\dots , x_m)]\right )$| is |$\Phi |_Y\left ([(C,f,x_1,\dots , x_m)] \right ).$| 6 Embedding Maps In this section, we show that the map induced by |$|\lambda |_{{\mathcal{F}}_t}|$| is an embedding, see Proposition 6.2. The following lemma helps us to prove that |$|\lambda |_{{\mathcal{F}}_t}|$| separates tangents. Lemma 6.1. Let |$Y$| be a smooth projective manifold with a line bundle |$L$| and irreducible divisors |$D_i$| for |$i=1,\ldots k$|⁠. If the following conditions are satisfied: (1) the complete linear system of |$L$| defines a morphism \begin{equation*}h:Y\rightarrow{\mathbb{P}}(H^0(Y,L)^{\vee})={\mathbb{P}}^N\end{equation*} with |$D_i\in |L|$|⁠, and the divisors |$D_i$| are reduced, (2) the differential |$Dh:T_Y\rightarrow h^*T_{{\mathbb{P}}^N}$| of |$h$| is injective over |$Y-\left (\bigcup \limits _{i=1}^m D_i\right )$|⁠, (3) the restriction |$Dh|_{T_{D_i,s_i}}: T_{D_i,s_i}\rightarrow T_{{\mathbb{P}}^N,h(s_i)}$| of |$Dh$| to a general point |$s_i$| of |$D_i$| for each |$i$| is injective, then the morphism |$h$| is unramified (i.e., the differential map |$dh$| is injective). Proof. Let |$g:Y\rightarrow Z$| be a map of complex manifolds. Recall that the ramified locus of |$g$| on |$Y$| is empty, or of codimension one or |$Y$|⁠. Therefore, by the 2nd condition, it suffices to show that the differential |$Dh|_{T_{Y,s}}:T_{Y,s}\rightarrow T_{{\mathbb{P}}^N,s}$| is injective at a general point |$s$| of |$D_i$| for each |$i$|⁠. Suppose that there is a nonzero vector |$v\in T_{Y,s}$| with |$Dh(v)=0$|⁠. Since |$D_i$| is generically smooth, the 3rd condition implies that |$v$| and |$T_{D_i,s}$| intersect transversally. We choose a small disk |$\mathbb{D}(\subseteq{\mathbb{C}})$| in |$Y$| such that its tangent vector at |$0\in \mathbb{D}$| is |$v$|⁠, that is, we have an embedding |$\iota :\mathbb{D} \hookrightarrow Y$| with |$\iota (0)=s$| such that |$v$| is given by |$D\iota (w)$| for a nonzero tangent vector |$w$| to |$\mathbb{D}$| at |$0\in \mathbb{D}$|⁠. The disk |$\mathbb{D}$| and |$D_i$| intersect transversally at |$0\in \mathbb{D}$|⁠. In other words, the function |$\iota ^*(G)$| has only simple zero at |$0\in \mathbb{D,}$| where |$G$| is the local function around |$s\in Y$| defining |$D_i$|⁠. On the other hand, let |$H$| be the hyperplane in |${\mathbb{P}}^N$| with |$h^{-1}(H)=D_i$|⁠. Let |$L=0$| be a local equation defining |$H$| around the point |$h(s)$|⁠. The equation |$h^*(L)=0$| defines |$D_i$| locally. We may assume that |$h^*(L)$| coincides with |$G$|⁠. Since we have |$D(h\circ \iota )(w)=Dh(v)=0$|⁠, the order of the zero of the function |$\iota ^*(h^*(L))=\iota ^*(G)$| is at least two at |$0\in \mathbb{D}$|⁠. This is a contradiction. Remark 6.2. The anonymous referee points out that a similar proof works for arbitrary characteristic by using the Purity Theorem. We now turn to the deformation theory of stable maps. The general reference for the deformation problems of maps is [20]. Specializing to the case of stable maps, we refer to [15, Page 61], [5], and [4]. Let |$f:C\rightarrow X$| be an unmarked stable map. By [5] and [4], the space of 1st order deformations and the obstruction group are given by the hypercohomology groups \begin{align*} \operatorname{Def}(f)={\mathbb H}^1(C,\mathbb{R}\operatorname{Hom}_{{\mathcal{O}}_C}(\Omega^{.}_f,{\mathcal{O}}_C)),\\ \operatorname{Obs}(f)={\mathbb H}^2(C,\mathbb{R}\operatorname{Hom}_{{\mathcal{O}}_C}(\Omega^{.}_f,{\mathcal{O}}_C)), \end{align*} where |$\Omega ^{.}_f$| is the complex . If |$f$| is an embedding, then \begin{equation*} \mathbb{R}\operatorname{Hom}_{{\mathcal{O}}_C}(\Omega^{.}_f,{\mathcal{O}}_C)\cong N_f[-1], \end{equation*} where |$N_f$| is the normal bundle of |$f$|⁠. Therefore, the space of the 1st order deformations of a stable map |$(C,f,x_1,\ldots ,x_m)$| such that |$f$| is an embedding is |$H^0( C, N_f).$| If the deformations fix the points |$\{x_i\}_{i=1}^m$|⁠, then the space of the 1st order deformations is given by \begin{equation} H^0\left(C,N_f\left(-\sum\limits_{i=1}^m x_i\right)\right). \end{equation} (6.1) Proposition 6.2. With the same notation as in Proposition 5.1, the complete linear system of |$\lambda |_{{\mathcal{F}}_t}$| separates tangent vectors of |${\mathcal{F}}_t$|⁠, that is, the differential of the map induced by the complete linear system |$|\lambda |_{{\mathcal{F}}_t}|$| is injective. In particular, together with Proposition 5.1, the morphism |$|\lambda |_{{\mathcal{F}}_t}|$| is a closed embedding. Proof. We apply Lemma 6.1 to |$Y={\mathcal{F}}_t$|⁠, |$D_i=\Delta _i$|⁠, and |$L=\lambda |_{{\mathcal{F}}_t}$|⁠, where |$\{\Delta _i\}$| are irreducible components of |$\Delta _t$|⁠. It suffices to check the conditions of Lemma 6.1. The 1st condition is verified by Lemma 3.8, Corollary 5.8, and Proposition 5.6. For the 2nd condition, we claim that the morphism |$h$| induced by |$|\lambda |_{{\mathcal{F}}_t}|$| is unramified on |$U={\mathcal{F}}_t-(\bigcup \limits _{i=0}^m \Delta _i)$|⁠. In fact, we have |$\pi _{p_i}^*({\mathcal{O}}_{{\mathbb{P}}(T_{p_i}X)}(1))=\lambda |_{{\mathcal{F}}_t}$| by Lemma 3.11, it suffices to show that the morphism |$\pi _{p_1}\times \pi _{p_2}\times \ldots ,\times \pi _{p_m}$| \begin{equation*} {\mathcal{F}}_t \longrightarrow{\mathbb{P}}(T_{p_1}X)\times\ldots\times{\mathbb{P}}(T_{p_m}X) \end{equation*} is unramified on |$U$|⁠. In fact, the kernel of the differential |$D(\pi _{p_1}\times \pi _{p_2}\times \ldots ,\times \pi _{p_m})$| at the point |$(C,f,x_1,\ldots ,x_m)\in{\mathcal{F}}_t$| is the vector space |$\widetilde{\operatorname{Def}}_{(C,f,x_1,\ldots ,x_m)} $| parametrizing the 1st order deformations of a stable map |$(C,f,x_1,\ldots ,x_m)$| fixing both the points |$\{x_i\}_{i=1}^m$| and the tangent directions at |$\{x_i\}$|⁠. Similar to (6.1), we have \begin{equation} \widetilde{\operatorname{Def}}_{(C,f,x_1,\ldots,x_m)} =H^0\left(C, N_{C/X}\left(\sum\limits_{i=1}^m -2x_i\right)\right) \end{equation} (6.2) if |$f$| is an embedding, where |$N_{C/X}$| is the normal bundle |$N_f$| of |$f:C\hookrightarrow X$|⁠. By Corollary 4.4, we have |$H^0\left (C, N_{C/X}\left (\sum \limits _{i=1}^m -2x_i\right )\right )=0$| if |$m$| is at least |$3$| and |$C$| is smooth. Denote by |$h$| the morphism |$|\lambda |_{{\mathcal{F}}_t}|$|⁠. It is obvious that |$h|_{\Delta _i}$| is generically unramified since the map |$h$| is injective by Proposition 5.1. The 3rd condition of Lemma 6.1 is satisfied. Remark 6. In the proof above, we use the fact that the restriction |$h|_{\Delta _i}$| is generically unramified that is due to the hypothesis of characteristic zero. Unfortunately, we do not know how to show this fact by a deformation argument (a simple calculation does not work out directly). 7 Proof of the Main Theorem In this section, we prove Theorem 1.3. An important ingredient is a criterion given in Proposition 7.1 for characterizing when a smooth projective variety is a complete intersection in a projective space. Note that |${\mathcal{F}}_t$| contains a subvariety |$Y$| that is a complete intersection |$\bigcap \limits _{i=1}^m\Delta _i$| in |${\mathcal{F}}_t$|⁠. Note also that |$\Delta _i$| are very ample divisors and |$Y$| is a complete intersection in a projective space, see Corollary 5.8. Theorem 1.3 then follows from Corollary 5.8 and Proposition 7.1. Proposition 7.1. |$($|[25, Proposition 7.3]|$)$| Let |$\Delta $| and |${\mathcal{F}}$| be smooth projective varieties in |${\mathbb{P}}^N$|⁠. Assume that the variety |$\Delta $| is a smooth divisor of |${\mathcal{F}}$|⁠, the dimension of |$\Delta $| is at least |$1$|⁠, the divisor |$\Delta $| is a complete intersection in |$\mathbb{P}^N$| of type |$(d_1,\ldots ,d_c)$| with |$d_i\geq 1$|⁠, the divisor |$\Delta $| is the schematic intersection of |${\mathcal{F}}$| and a hypersurface of degree |$d_1$| in |${\mathbb{P}}^N$|⁠. Then the smooth variety |${\mathcal{F}}$| is a complete intersection of type |$(d_2,\ldots , d_c)$| in |$\mathbb{P}^N$|⁠. Using Proposition 7.1 above, we are able to show that |${\mathcal{F}}_t$| is a complete intersection in |${\mathbb{P}}^{N}$| and |$N$| is |$n-m(c-1)$|⁠. That is the following proposition. Proposition 7.2. With the same notations as in Corollary 5.8, if the following inequality holds: \begin{equation*}n+m(c-\sum\limits_{i=1}^c d_i)-c\geq 2,\end{equation*} then the map |$|\lambda |_{{\mathcal{F}}_t}|:{\mathcal{F}}_t \rightarrow{\mathbb{P}}^N$| is an embedding and the dimension |$\dim H^0(Y,{\mathcal{O}}_Y(1))$| is |$n-mc$|⁠, where |$N$| is the dimension of the complete linear system |$|\lambda |_{{\mathcal{F}}_t}|$|⁠. Moreover, the varieties |${\mathcal{F}}_t$| and |$\bigcap \limits _{i=1}^k\Delta _{i,t}$| for |$1\leq k\leq m$| are smooth complete intersections via this embedding. Proof. By Corollary 5.8, we have an embedding |$\Phi |_Y:Y\hookrightarrow{\mathbb{P}}^{n-m}.$| The image of this embedding is in a projective space |${\mathbb{P}}^{n-mc}$|⁠. Moreover, |$Y$| is a complete intersection in |${\mathbb{P}}^{n-mc}$| of type \begin{equation} \left(T_1(d_1,m), T_1(d_2,m),\ldots, T_1(d_c,m)\right). \end{equation} (7.1) It follows that dim |$H^0(Y,{\mathcal{O}}_Y(1))=n-mc$|⁠. Consider the map |$|\lambda |_{{\mathcal{F}}_t}|:Y \rightarrow{\mathbb{P}}^N.$| Via this map, the space |$Y$| is a complete intersection in |${\mathbb{P}}^N$| defined by a union of some hyperplanes and polynomials of type (7.1). Note that |$Y$| is the intersection |$\bigcap \limits _{i=1}^{m} \Delta _{i,t}$| and |$n+m(c-\sum \limits _{i=1}^{c} d_i)-c$| is at least |$2.$| It follows that |$\dim Y$| is at least |$1$|⁠. By Corollary 5.8, we note that the intersection |$\bigcap \limits _{i=1}^{k}\Delta _{i,t}$| is a smooth projective variety for |$1\leq k \leq m$|⁠. Hence, by Proposition 5.6 and Proposition 6.3, we can apply Proposition 7.1 to |$\bigcap \limits _{i=1}^{k}\Delta _{i,t}$| inductively, from |$k=m$| to |$k=0$|⁠, where |$\bigcap \limits _{i=1}^{k}\Delta _{i,t}$| is |${\mathcal{F}}_t$| for |$k=0$|⁠. It follows that |$\bigcap \limits _{i=1}^{k}\Delta _{i,t}$| and |${\mathcal{F}}_t$| are complete intersections in |${\mathbb{P}}^N$|⁠. They are defined by a union of polynomials of type (7.1) and some linear forms. Proposition 7.3. With the same conditions as in Proposition 7.2, we have \begin{equation*}N=\dim |\lambda|_{{\mathcal{F}}_t}|=n-m(c-1).\end{equation*} Proof. Denote |$\bigcap \limits _{i=1}^{k}\Delta _{i,t}$| by |$Y_k$|⁠, and let |$Y_0$| be |${\mathcal{F}}_t$|⁠. Note that |$Y_k$| is a smooth projective variety with the embedding |$|\lambda |_{{\mathcal{F}}_t}|:Y_0\rightarrow{\mathbb{P}}^N$| by Corollary 5.8 and Proposition 7.2. Note also that |${\mathcal{O}}_{{\mathcal{F}}_t}(\Delta _{i,t})$| is |$\lambda |_{{\mathcal{F}}_t}(={\mathcal{O}}_{Y_0}(1))$| by Proposition 5.6. We have the short exact sequence \begin{equation*}0\rightarrow{\mathcal{O}}_{Y_k}(-\lambda|_{Y_k})\rightarrow{\mathcal{O}}_{Y_k}\rightarrow j_*{\mathcal{O}}_{Y_{k+1}}\rightarrow 0,\end{equation*} where |$j$| is the inclusion |$Y_{k+1}\subseteq Y_k$|⁠. Note that |$Y_k$| is a complete intersection of dimension |$>1$| for |$k\leq m-1$|⁠. It follows that |$H^1(Y_k,{\mathcal{O}}_{Y_k})=0$|⁠. Tensor the short exact sequence above with |${\mathcal{O}}_{Y_k}(1)$| and apply |$H^i(\_)$|⁠. It gives rise to the long sequence \begin{equation*}0\rightarrow H^0(Y_k,{\mathcal{O}}_{Y_k})\rightarrow H^0(Y_k, {\mathcal{O}}_{Y_k}(1))\rightarrow H^0(Y_{k+1},{\mathcal{O}}_{Y_{k+1}}(1))\rightarrow H^1(Y_k,{\mathcal{O}}_{Y_k})=0.\end{equation*} Therefore, the dimension |$\dim |\lambda |_{{\mathcal{F}}_t}|$| equals \begin{equation*}\dim H^0({\mathcal{F}}_t,{\mathcal{O}}_{{\mathcal{F}}_t}(1)))=\dim H^0(Y_m,{\mathcal{O}}_{Y_m}(1))+m=n-mc+m,\end{equation*} where the last equality follows from Proposition 7.2. We are now able to give a proof of Theorem 1.3. Proof. With the notations as above, the general fiber |$Y_0={\mathcal{F}}_t\subseteq{\mathbb{P}}^N$| is a complete intersection in |${\mathbb{P}}^N$| defined by homogeneous polynomials of type (7.1) with |$s$| linear forms (see the end of the proof of Proposition 7.2). Therefore, the dimension of |${\mathcal{F}}_t$| is |$N-s- \left (m(\sum \limits _{i=1}^{c} d_i -2c)+c\right )$|⁠. On the other hand, the dimension of |${\mathcal{F}}_t$| is \begin{equation*} \dim{\mathcal{F}}_t=\dim{\mathcal{F}}-\dim\overline{{\mathcal{M}}}_{0,m}=(c+1-\sum\limits_{i=1}^{c} d_i)m+n-c. \end{equation*} It follows from Proposition 7.3 that |$s$| equals |$N-\dim{\mathcal{F}}_t-m(\sum \limits _{i=1}^{c} d_i -2c)-c$| that is just 0. This completes the proof of the theorem. 8 Applications We learn from [8] that the rational connectedness of some moduli spaces of rational curves on a rationally connected variety could imply the weak approximation and the existence of rational points on varieties over function fields. It motivates us to consider the rational connectedness of |${\mathcal{F}}$|⁠. As an application of Thereom 1.3, we show that |${\mathcal{F}}$| is rationally connected if |$X$| is a complete intersection of low degree. The 2nd application of Thereom 1.3 is inclined to enumerative geometry. We give a proof of a classical formula to count the number of twisted cubics on a complete intersection (see [2] and [14]). Moreover, we provide a new formula for counting the number of two crossing conics on a complete intersection. The 3rd one is to show that the Picard group of |${\mathcal{F}}$| is finitely generated. The rational connectedness of moduli spaces Proposition 8.1. With the same conditions as in Theorem 1.3, if \begin{equation*}m\left(\sum\limits_{i=1}^c\frac{d_i(d_i-1)}{2}-1\right)+\sum\limits_{i=1}^c d_i \leq n,\end{equation*} then |${\mathcal{F}}$| is rationally connected. Proof. With the same notation as in Theorem 1.3, the canonical bundle of the fiber |${\mathcal{F}}_t$| is given by \begin{equation*} K_{{\mathcal{F}}_t}={\mathcal{O}}_{{\mathbb{P}}^{N}}\left(-N-1+m\left(\sum\limits_{i=1}^c(\frac{d_i(d_i-1)}{2}-1)\right)+\sum\limits_{i=1}^c d_i\right)|_{{\mathcal{F}}_t},\end{equation*} where |$t\in \overline{{\mathcal{M}}}_{0,m}$| is a general point as in Theorem 1.3 and |$N=n-m(c-1)$|⁠. The inequality in the hypothesis of the proposition is equivalent to say that |$K_{{\mathcal{F}}_t}$| is anti-ample. In particular, the fiber |${\mathcal{F}}_t$| is a smooth projective Fano variety, hence, it is rationally connected, see [22, Chapter V]. We claim that the fiber |${\mathcal{F}}$| is rationally connected. Let |$p$| and |$q$| be two general points of |${\mathcal{F}}$| such that |$F(p)$| and |$F(q)$| (see Theorem 1.3 for the map |$F$|⁠) are general points in |$\overline{{\mathcal{M}}}_{0,m}$|⁠. Since |$\overline{{\mathcal{M}}}_{0,m}$| is a smooth projective and rational variety, there is a rational curve |$D$| in |$\overline{{\mathcal{M}}}_{0,m}$| connecting |$F(p)$| and |$F(q)$| such that the fiber of |$F$| over a general point |$D$| is rationally connected. Therefore, by Corollary |$1.3$| in [15], the general fiber |${\mathcal{F}}$| is rationally connected. Remark 8.2. With [9, Lemma 6.5], it is not hard to prove that the canonical bundle |$K_{\mathcal{F}}$| is trivial on some rational curves sitting inside the maximal degeneration locus if |$m$| is at least |$4$|⁠. Therefore, |${\mathcal{F}}$| is not Fano for |$m\geq 4$|⁠.Enumerative geometry Let |$X^{\prime}$| be a complete intersection that is cut out from a complete intersection |$X$| by |$n-s$| general hyperplanes. Assume that |$m$| is |$3$|⁠. Then |$\overline{{\mathcal{M}}}_{0,m}$| is just a point. Let |${\mathcal{F}}^{\prime}$| be the general fiber of the evaluation map corresponding to |$X^{\prime}$|⁠. It is clear that |${\mathcal{F}}^{\prime}$| is cut out from the general fiber |${\mathcal{F}}$| (corresponding to |$X$|⁠) by |$n-s$| very ample divisors in |$|\Phi ^*({\mathcal{O}}(1))|$|⁠, as shown in the diagram |$ $| (8.1) Roughly speaking, if |${\mathcal{F}}^{\prime}$| consists of discrete points, that is, |$\dim{\mathcal{F}}^{\prime}$|=0, then the number of these points is the number of the rational curves of degree |$m$| passing through |$m$| general points on |$X^{\prime}$|⁠. The following proposition gives a precise statement. Proposition 8.3. |$($|[2, Corollary, Page 9]|$)$| Let |$X$| be a smooth complete intersection of degree |$(d_1,\ldots ,d_r)$| in |$\mathbb{P}^{n+r}$| with |$n=3\sum \limits _{i=1}^r(d_i-1)-3$|⁠. Then the number of twisted cubics in |$X$| passing through three general points |$(p,q,r)$| is |$\frac{1}{d^2}\prod \limits _{i=1}^r (d_i!)^3$|⁠, where d is the degree of |$X$|⁠. Proof. We may assume that the variety |$X$| is cut out by hyperplanes from a smooth complete intersection |$Y$| of type |$(d_1,\ldots ,d_r)$| and |$Y\subseteq{\mathbb{P}}^e$| has sufficiently large dimension. The degree of the general fiber |${\mathcal{F}}\subseteq \mathbb{P}^{e}$| corresponding to |$Y$| has an enumerative geometrical interpretation. In fact, a point in the |$\mathbb{P}^{e-3}={\mathbb{P}}^e/\operatorname{Span}(p,q,r)$| parametrizes a 3-plane |${\mathbb{P}}^3$| in |${\mathbb{P}}^e$| containing the 2-plane |${\mathbb{P}}^2=\operatorname{Span}(p,q,r)$|⁠. More generally, a sub-projective space |$\mathbb{P}^k(\subseteq \mathbb{P}^{e-3})$| corresponds to a |$(k+3)$|-plane |$\mathbb{P}^{k+3}$| in |${\mathbb{P}}^e$| (containing |$\operatorname{Span}(p,q,r)$|⁠). So if we take an |$l$|-plane |$\mathbb{P}^l\subseteq{\mathbb{P}}^e$| such that |$l+\textrm{dim}({\mathcal{F}})=e$|⁠, then |$\#({\mathbb{P}}^l\cap{\mathcal{F}})$| is the number of twisted cubics in |$X=\mathbb{P}^{l+3}\cap Y$| passing through |$p, q$|⁠, and |$r$|⁠. In other words, the degree of |${\mathcal{F}}$| equals # {twisted cubics in |$X$| passing through |$p, q,$| and |$r$|}. By Theorem 1.3, the degree of |${\mathcal{F}}$| is |$\frac{1}{d^2}\prod \limits _{i=1}^r (d_i!)^3$|⁠. Remark 8.4. One can get a similar formula for crossing conics on |$X$| passing through |$4$| general points. More precisely, the number of these crossing conics is |$\frac{1}{d^3}\prod \limits _{i=1}^r (d_i!)^4$|⁠, where |$d$| is the degree of |$X$|⁠, where |$X$| is a smooth complete intersection of degree |$(d_1,\ldots ,d_r)$| in |$\mathbb{P}^{n+r}$| with |$n=4\sum \limits _{i=1}^r(d_i-1)-4$|⁠. The Picard group of moduli spaces The following lemma is standard and easy to prove. The proof is left as an exercise for the reader. Lemma 8.5. Consider a morphism |$h:A\rightarrow B$| between two smooth varieties |$A$| and |$B$| over |$\mathbb{C}$|⁠. Suppose that the morphism |$h$| is proper and dominant. Let |$K$| be the function field of |$B$|⁠. If the following two conditions are satisfied: the generic fiber |${A_K}$| of |$h$| is geometrical connected, the Picard groups |$\operatorname{Pic}({A_K})$| and |$\operatorname{Pic}(B)$| are finitely generated, then the Picard group |$\operatorname{Pic}(A)$| is finitely generated. Lemma 8.6. Let |$U$| be a variety, and let |$Y$| be a subscheme of |${\mathbb{P}}^{n_U}$| such that |$Y\xrightarrow{g} U$| is flat. We assume that the fiber |$Y_s$| over any |$s\in U(\mathbb{C})$| is a complete intersection of type |$(d_1,\ldots ,d_c)$| in |${\mathbb{P}}^n_{s}$|⁠. If |$K$| is the function field of |$U$|⁠, then the generic fiber |${Y_K}$| of |$g$| is a complete intersection of type |$(d_1,\ldots ,d_c)$| in |${\mathbb{P}}^{n_K}$|⁠. Proof. If a projective variety |$T$| in |${\mathbb{P}}^n$| is a complete intersection of type |$(d_1,\dots , d_g)$| defined by homogenous polynomials |$(F_1,F_2,\ldots ,F_g)$|⁠, then by Hilbert’s theory (see [18, Section 7 of Chapter I]), the polynomials |$(F_1,F_2,\ldots ,F_g)$| are minimal generators of the ideal sheaf |$I_T$| and the dimension of |$H^0({\mathbb{P}}^n,I_T(m))$| depends only on the type |$(d_1,\dots , d_g)$|⁠. Therefore, for any |$m$|⁠, the dimension function \begin{equation*}s\mapsto \dim H^0({\mathbb{P}}^n_s,I_{Y_s}(m))\end{equation*} is a constant function for |$s\in U(\mathbb{C})$|⁠. It follows from [23, Page 48, Corollary 2] that |$\dim H^0({\mathbb{P}}^n_K,I_{Y_K}(m))$| equals to |$\dim H^0({\mathbb{P}}^n_s,I_{Y_s}(m)).$| This equality implies that we can choose homogenous polynomials |$(F_1,F_2,\ldots ,F_c)$| defined over |$K$| such that, on the domain |$V(\subseteq U)$| of the coefficients of |$F_i$|⁠, the fiber |$Y_s$| of |$g$| over a point |$s\in V(\mathbb{C})$| is a complete intersection in |${\mathbb{P}}_s^n$| defined by the equations \begin{equation*}(F_1=0,F_2=0,\ldots,F_c=0)|_{{\mathbb{P}}^n_s}.\end{equation*} It follows that the projective variety |$Y_K$| is a complete intersection in |${\mathbb{P}}^n_K$| defined by |$(F_1,F_2,\ldots ,F_c)$|⁠. Lemma 8.7. Consider a projective morphism |$h$| from a scheme |$A$| to |$\operatorname{Spec}(K)$| for a field |$K$|⁠. If the following two conditions hold: the morphism |$h$| has a section |$\sigma $|⁠, |$\operatorname{Pic}(A_{{\overline K}}$|⁠) equals |${\mathbb Z}[L|_{A_{{\overline K}}}]$|=|${\mathbb Z}$|⁠, where |$L$| is a line bundle on |$A$| and |${\overline K}$| is the algebraic closure of |$K$|⁠, then |$\operatorname{Pic}(A)$| is |${\mathbb Z}$| with a generator |$L$|⁠. Proof. The 1st condition ensures that the Picard functor is representable by the Picard scheme |$\operatorname{Pic}_{A/K}$|⁠. Therefore, the Picard group |$\operatorname{Pic}(A)$| (resp. |$\operatorname{Pic}(A_{{\overline K}})$|⁠) is just the |$K$|-points (resp. |${\overline K}$|-points) of |$\operatorname{Pic}_{A/K}$|⁠. In other words, we have that |$\operatorname{Pic}(A)$| equals |$\operatorname{Pic}_{A/K}(K)$| and |$\operatorname{Pic} (A_{{\overline K}})$| equals |$\operatorname{Pic}_{A/K}({\overline K}).$| It follows that |$\operatorname{Pic} (A)$| equals \begin{equation*} \operatorname{Pic}(A_{{\overline K}})^{\operatorname{Gal}({\overline K}/K)}={\mathbb Z}[L|_{A_{{\overline K}}}]^{\operatorname{Gal}({\overline K}/K)}={\mathbb Z}[L], \end{equation*} where the last equality follows from the fact that the line bundle |$L|_{A_{{\overline K}}}$| is |$\operatorname{Gal}({\overline K}/K)$| invariant. Proposition 8.8. With the same conditions as in Theorem 1.3, the Picard group |$\operatorname{Pic}({\mathcal{F}})$| is finitely generated. Proof. Let |$K$| be the function field of |$\overline{{\mathcal{M}}}_{0,m}$|⁠. It follows from Theorem 1.3 and Lemma 8.6 that the generic fiber |${\mathcal{F}}_K$| of the forgetful map |$F$| (Theorem 1.3) is a complete intersection. The conditions in Theorem 1.3 implies that |$\dim{\mathcal{F}}_K$| is at least |$3$|⁠. It follows from the equality |$\operatorname{Pic}({\mathcal{F}}_{{\overline K}})={\mathbb Z}={\mathbb Z}[{\mathcal{O}}_{{\mathbb{P}}^N_{{\overline K}}}(1)]={\mathbb Z}[\lambda _{{\overline K}}]$| and Lemma 8.7 that |$\operatorname{Pic}({\mathcal{F}}_K)$| is |${\mathbb Z}[\lambda _K]={\mathbb Z}$|⁠. If we take |${\mathcal{F}}=A$| and |$\overline{{\mathcal{M}}}_{0,m}=B$| in Lemma 8.5, then we conclude that |$\operatorname{Pic}({\mathcal{F}})$| is finitely generated. Funding Research partially supported by NSFC [No. 11688101]. Acknowledgments The author is very grateful to his advisor A. J. de Jong for teaching him moduli techniques. The author is also grateful to J. Starr, B. Hassett, C. Liu, and M. Fedorchuk for reading some parts of this paper and giving some suggestions and to the anonymous referee for providing many useful suggestions for simplifying and improving some proofs. Some parts of this paper were written when the author was a graduate student in the Department of Mathematics of Columbia University, a post doctoral in Washington University in St. Louis and Max Planck Institute for Mathematics. The author is very grateful to each of these departments for providing comfortable environment for study and work. Communicated by Prof. Enrico Arbarello References [1] Araujo , C. and Castravet , A. . “ Polarized minimal families of rational curves and higher Fano manifolds .” Amer. J. Math. 134 , no. 1 ( 2012 ): 87 – 107 . Google Scholar Crossref Search ADS [2] Beauville , A. “ Quantum cohomology of complete intersections .” Mat. Fiz. Anal. Geom. 2 , nos. 3–4 ( 1995 ): 384 – 398 . [3] Beheshti , R. and Kumar , N. . “ Spaces of rational curves on complete intersections .” Compos. Math. 149 , no. 6 ( 2013 ): 1041 – 1060 . Google Scholar Crossref Search ADS [4] Behrend , K. “ Gromov–Witten invariants in algebraic geometry .” Invent. 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Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

Kalinov,, Daniil

doi: 10.1093/imrn/rnz005pmid: N/A

Abstract We classify the “finite-dimensional” irreducible representations of the Yangians |$Y(\mathfrak{g}\mathfrak{l}_t)$| and |$Y(\mathfrak{s}\mathfrak{l}_t)$|⁠. These are associative ind-algebras in the Deligne category |$\textrm{Rep}(GL_t)$|⁠, which generalize the regular Yangians |$Y(\mathfrak{g}\mathfrak{l}_n)$| and |$Y(\mathfrak{s}\mathfrak{l}_n)$| to complex rank. They were first defined in the paper [14]. Here we solve [14, Problem 7.2]. We work with the Deligne category |$\textrm{Rep}(GL_t)$| using the ultraproduct approach introduced in [7] and [16]. Introduction The study of representation theory in complex rank was initiated by Pierre Deligne. In his paper [7], he defined the tensor categories |$\textrm{Rep}(S_t)$|⁠, |$\textrm{Rep}(GL_t)$|⁠, |$\textrm{Rep}(SO_t)$|⁠, and |$\textrm{Rep}(Sp_t)$| interpolating the categories of representations of the corresponding groups in finite rank (actually the category |$\textrm{Rep}(GL_t)$| appears even earlier in the paper [8] by Deligne and Milne and [6] by Deligne). These categories were studied by Deligne himself and many other authors, for example see [5], [4], and [20]. Later, Etingof in his papers [13] and [14] suggested the methods for interpolating the representation theory of many other algebras connected to |$S_n$| or |$GL_n$| to complex rank. These papers included many open problems, some of which were solved, for example see [12], [11], [17], and [25]. In this paper, we study the representation theory of the generalization of the Yangians |$Y(\mathfrak{g}\mathfrak{l}_n)$| and |$Y(\mathfrak{s}\mathfrak{l}_n)$| to complex rank, which were introduced by Etingof in [14, Section 7], thus solving [14, Problem 7.2] in the case of general linear and special linear groups. The Yangian |$Y(\mathfrak{g})$| was introduced by Vladimir Drinfeld in his paper [9] for a general simple Lie algebra |$\mathfrak{g}$|⁠. He later classified the finite-dimensional irreducible representations of |$Y(\mathfrak{g})$| in [10]. The theory of Yangians for |$\mathfrak{g}\mathfrak{l}_n$| and |$\mathfrak{s}\mathfrak{l}_n$| is described in detail also in the textbooks [23] and [3]. These books use somewhat different approaches, the first one highly relying on the Faddeev–Reshetikhin–Takhtajan presentation of |$Y(\mathfrak{g}\mathfrak{l}_n)$|⁠. We will mostly follow the notations and techniques of [23], but we will describe some connections with methods of [3] later on. The main result of this paper is the classification of ”finite-dimensional” irreducible representations of |$Y(\mathfrak{s}\mathfrak{l}_t)$| and |$Y(\mathfrak{g}\mathfrak{l}_t)$| by Drinfeld polynomials, which generalizes the known theorems in the finite rank case. We can not prove this theorem using the same techniques as in the finite rank case, since there is no obvious way to generalize the triangular decomposition of |$Y(\mathfrak{g}\mathfrak{l}_n)$| or |$Y(\mathfrak{s}\mathfrak{l}_n)$| to the complex rank case. Thus, we need to use some instruments to connect results in the finite rank case to the result in complex rank. There are several methods of generalizing results from the representation theory in the finite rank cases to the representation theory in Deligne categories. In this paper, we will use the method of ultraproducts, which was introduced for transcendental rank in [7] and later generalized to algebraic ranks in the work [16] motivated by Deligne’s letter to Ostrik. This method provides a way to generalize results from finite rank to both algebraic and transcendental |$t$| provided that we know enough about the representation theory in positive characteristic. For other methods, see for example [22]. We also would like to note that previously this subject was studied by Léa Bittmann in [1]. There, she has studied the representation of Yangians |$Y(\mathfrak{s}\mathfrak{l}_t)$| and |$Y(\mathfrak{g}\mathfrak{l}_t)$| using the invariant algebra |$Y(\mathfrak{g})^{\mathfrak{g}}$|⁠. The structure of the paper is as follows. In Section 1, we first briefly discuss the construction of the Deligne category |$\textrm{Rep}(GL_t)$| and its properties, which are going to be relevant to us. Later, in the section, we review the theory of ultrafilters and ultraproducts and describe the contruction of |$\textrm{Rep}(GL_t)$| as an ultraproduct of the categories of representations of |$GL_n$|⁠. In Section 2, we recall the Faddeev–Reshetikhin–Takhtajan presentation of |$Y(\mathfrak{g}\mathfrak{l}_t)$| and describe its basic properties, define the subalgebra |$Y(\mathfrak{s}\mathfrak{l}_t)$|⁠, and describe the classification theorem of irreducible finite-dimensional representations of |$Y(\mathfrak{g}\mathfrak{l}_n)$| and |$Y(\mathfrak{s}\mathfrak{l}_n)$|⁠. In Section 3, we generalize some of the results of Section 2 to positive characteristic under some conditions on |$n$| and |$p$|⁠. These results will be crucial for us when we work with |$Y(\mathfrak{g}\mathfrak{l}_t)$| for algebraic |$t$|⁠. In Section 4, we first define |$Y(\mathfrak{g}\mathfrak{l}_t)$| and |$Y(\mathfrak{s}\mathfrak{l}_t)$| and prove some of their properties, using the ultraproduct approach. Later, we study irreducible representations of these algebras and prove the classification theorem for them. 1 Deligne category |$\textrm{Rep} (GL_t)$| 1.1 Construction of |$\textrm{Rep} (GL_t)$| and its main properties Here we will give a definition of the category |$\textrm{Rep}(GL_t)$| for |$t \in \mathbb C$| and state a number of important properties of this category. For a more detailed discussion, see [14] and [5]. First, we define a preliminary category |$\textrm{Rep}_0(GL_t)$|⁠: Definition 1.1.1. |$\textrm{Rep}_0(GL_t)$| is a skeletal symmetric tensor (we will usually assume tensor categories to be rigid) category with objects being given by pairs |$(n,m)$| with |$n,m \in \mathbb Z_{\ge 0}$|⁠, represented as rows consisting of |$n$||$\bullet $|’s and |$m$||$\circ $|’s. |$\textrm{Hom}_{\textrm{Rep}_0(GL_t)}((m,n),(m^{\prime},n^{\prime}))$| is a vector space over |$\mathbb C$| with a basis given by all possible matchings, i.e., ways to pair all elements of two rows corresponding to |$(n,m)$| and |$(n^{\prime},m^{\prime})$| such that |$\bullet $| can be paired with |$\bullet $| or |$\circ $| with |$\circ $| only if they belong to different rows and |$\bullet $| can be paired with |$\circ $| only if they are in the same row. Composition is given by vertical concatenation of diagrams and forgetting the middle row and then deletion of each resulting loop and multiplying the coefficient of the corresponding basis vector by |$t$| for each loop deleted. Tensor product is given by horizontal concatenation of diagrams. Now we can easily construct the Deligne category from this: Definition 1.1.2. The Deligne category |$\textrm{Rep}(GL_t)$| is the Karoubian envelope of the additive envelope of |$\textrm{Rep}_0(GL_t)$|⁠. We will also need auxiliary definitions: Definition 1.1.3. A bipartition |$\lambda $| is a pair of partitions |$\lambda = (\lambda ^{\bullet }, \lambda ^\circ )$|⁠. The size of |$\lambda $| is equal to |$|\lambda | = |\lambda ^{\bullet }| + |\lambda ^\circ |$|⁠. The length of |$\lambda $| is equal to |$l(\lambda ) = l(\lambda ^{\bullet }) + l(\lambda ^\circ ).$| Definition 1.1.4. The object |$V((\square , \emptyset )) = (1,0)$| is called the fundamental representation and is denoted by |$V$|⁠. The object |$V((\emptyset ,\emptyset )) = (0,0)$| is called the trivial representation and is denoted by |$\mathbb C$| (abusing notation). Here are some properties of |$\textrm{Rep}(GL_t)$|⁠: Proposition 1.1.5. ([5, Section 4]) (a) The category |$\textrm{Rep}(GL_t)$| is semisimple for |$t\notin \mathbb Z$|⁠. (b) The irreducible objects of |$\textrm{Rep}(GL_t)$| are in |$1-1$| correspondence with bipartitions of arbitrary size. They are denoted by |$V(\lambda )$|⁠. Moreover, |$V(\lambda )$| is a direct summand in |$(r,s)$|⁠. (c) The dimension of |$V$| is |$t$|⁠, and the dimension of |$\mathbb C$| is |$1$|⁠. (d)|$V((\lambda ^{\bullet },\lambda ^\circ ))^* = V((\lambda ^\circ , \lambda ^{\bullet }))$| In addition, |$\textrm{Rep}(GL_t)$| has an important universal property. Suppose |$\mathcal T$| is a symmetric rigid tensor category. When we have the following proposition ([14, Theorem 2.9(ii)] and [7, Proposition 10.3]): Proposition 1.1.6. (Deligne) Isomorphism classes of (possibly non-faithful) symmetric tensor functors |$\textrm{Rep}(GL_t) \to T$| are in bijection with isomorphism classes of objects |$X \in \mathcal T$| of dimension |$t$|⁠, via |$F \to F(V)$|⁠. In particular, this means that for any |$t$|-dimensional object in |$\mathcal{T}$|⁠, we have a symmetric tensor functor |$\textrm{Rep}(GL_t) \to \mathcal{T}$| sending |$V$| to this object. 1.2 Ultrafilters and ultraproducts It is crucial to us that there is another construction of |$\textrm{Rep}(GL_t)$| as a subcategory in a certain ultraproduct category, which formalizes the fact that |$\textrm{Rep}(GL_t)$| is a “limit” of representation categories of general linear groups. We will quickly define what ultrafilters and ultraproducts are, state their main properties, and give some examples. For more details, see [24]. Definition 1.2.1. An ultrafilter |$\mathcal F$| on a set |$X$| is a subset of |$2^{X}$| satisfying the following properties: |$\bullet $||$X \in \mathcal F$| ; |$\bullet $| If |$A \in \mathcal F$| and |$A \subset B$|⁠, then |$B \in \mathcal F$| ; |$\bullet $| If |$A,B \in \mathcal F$|⁠, then |$A\cap B \in \mathcal F$| ; |$\bullet $| For any |$A\subset X$|⁠, either |$A$| or |$X \backslash A$| belongs to |$A$| but not both. There is an obvious family of examples of ultrafilters: |$\mathcal F_x = \{ A| x \in A \}$| for |$x \in X$|⁠. Such ultrafilters are called principal. Using Zorn’s lemma, one can show that there exist non-principal ultrafilters |$\mathcal F$|⁠. In addition, it follows that all cofinite sets belong to such an |$\mathcal F$| (but not all sets belonging to |$\mathcal F$| are cofinite). From now on, we will denote by |$\mathcal F$| a fixed non-principal ultrafilter on |$\mathbb N$|⁠. In addition, by something being true for “almost all |$n$|”, we will mean that it is true for all |$n$| in some |$A \in \mathcal F$|⁠. Note that by definition of an ultrafilter, if two statements hold for almost all |$n$|⁠, then their conjunction holds for almost all |$n$|⁠. In addition, note that if for almost all |$n$| the disjunction of a finite number of statements holds, then one of them holds for almost all |$n$| (if not, then each of them holds on some subset |$A \notin \mathcal F$| and the union of this subsets is not in |$\mathcal F$|⁠). We will use these elementary observations quite frequently. Let’s now define a notion of an ultraproduct. Definition 1.2.2. Suppose we have a collection of sets |$S_i$| labeled by natural numbers. Suppose that for all |$x\in A$| with |$A \in \mathcal F$|⁠, one has |$S_x \ne \emptyset $|⁠. Then |$\prod _{\mathcal F}S_x$| is the quotient of |$\prod _{x \in A} S_x$| by the following relation: |$\{s_x\} \sim \{s^{\prime}_x\}$| iff |$s_x = s_x^{\prime}$| for almost all |$x$|⁠. If for almost all |$x$|⁠, one has |$S_x = \emptyset $|⁠, then |$\prod _{\mathcal F}S_x = \emptyset $|⁠. The set |$\prod _{\mathcal F}S_x$| is called the ultraproduct of |$S_x$|⁠. Usually, we will denote |$\{ s_x \} \in \prod _{\mathcal F}S_i$| by |$\prod _\mathcal F s_x$|⁠. The most important property of ultraproducts is the following: Theorem 1.2.3. Łoś’s theorem ([24, Theorem 2.3.2]) Suppose we have a collection of sequences of sets |$S^{(k)}_i$| for |$k = 1,\dots ,m$| and a collection of sequences of elements |$f^{(r)}_i$| for |$r = 1,\dots , l$| and a formula of a first-order language |$\phi (x_1,\dots ,x_l, Y_1, \dots , Y_m)$| depending on some parameters |$x_i$| and sets |$Y_j$|⁠. Denote by |$S^{(k)} = \prod _{\mathcal F}S^{(k)}_{n}$| and |$f^{(r)} = \prod _{\mathcal F} f^{(r)}_n$|⁠. Then |$\phi (f^{(1)}_n, \dots , f^{(l)}_n, S^{(1)}_n, \dots S^{(m)}_n)$| is true for almost all |$n$| iff |$\phi (f^{(1)}, \dots , f^{(l)}, S^{(1)}, \dots S^{(m)})$| is true. In plain language, this means that if we have a sequence of collections of sets with some algebraic structure given by maps between them, then, first, we have the corresponding maps between the ultraproducts of these sets. Second, these maps satisfy a given set of axioms or properties for the ultraproducts iff they satisfy these axioms/properties for almost all |$n$|⁠. In addition, frequently, it is useful to think about ultraproducts as a some kind of limits as |$n \mapsto \infty $|⁠. We give a number of examples of such constructions, which are going to be useful to us below: Example 1.2.4. If |$S_i$| is a sequence of monoids/groups/rings/fields then |$\prod _{\mathcal F} S_i$| with operations given by taking the ultraproduct of the operations in the corresponding sets of |$\textrm{Hom}_{Sets}$| gives us a structure of monoid/group/ring/field by Łoś’s theorem. Example 1.2.5. If |$V_i$| are finite-dimensional vector spaces over a field |$k$|⁠, then |$\prod _{\mathcal F} V_i$| is not necessarily a finite-dimensional vector space, since the property of being finite dimensional cannot be written in first-order language. However, if the dimensions of |$V_i$| are bounded, then they are the same for almost all |$i$| and hence |$V$| has the same dimension (for example, because the ultraproduct of bases is a basis). Example 1.2.6. Take the ultraproduct of a countably infinite number of copies of |$\overline{\mathbb Q}$|⁠. By Łoś’s theorem, |$\prod _{\mathcal F} \overline{\mathbb Q}$| is a field, which is algebraically closed. It has characteristic zero since |$\forall k \in \mathbb Z$| such that |$k\ne 0$|⁠; it follows that |$ k = \prod _{\mathcal F} k\ne 0$|⁠. In addition, it is easy to see that its cardinality is continuum. Hence, by Steinitz’s theorem (this theorem tells us that two uncountable algebraically closed fields are isomorphic iff their characteristic and cardinality are the same; it is proven in [26]) |$\prod _{\mathcal F} \overline{\mathbb Q} \simeq \mathbb C$|⁠. Note that there is no canonical isomorphism. Example 1.2.7. Take the ultraproduct of |$\overline{\mathbb F}_{p_n}$| for some sequence of distinct prime numbers |$p_n$|⁠. As before, by Łoś’s theorem, |$\prod _{\mathcal F} \overline{\mathbb F}_{p_n}$| is a field, which is algebraically closed. In addition, as before, it has cardinality continuum. Now |$k = \prod _{\mathcal F} k \ne 0$|⁠, since it is equal to zero for at most one |$k$|⁠. Hence, |$\prod _{\mathcal F} \overline{\mathbb F}_{p_n} \simeq \mathbb C$|⁠, again not in a canonical way. Example 1.2.8. Suppose |$\mathcal C_i$| is a collection of small categories. We can define an ultraproduct category |$\widehat{\mathcal C} = \prod _{\mathcal F} \mathcal C_i$| as a category with objects |$Ob( \widehat{ \mathcal C}) = \prod _{\mathcal F} Ob(\mathcal C_i)$| and |$\textrm{Hom}_{\widehat{\mathcal C}}(\prod _{\mathcal F} X_i,\prod _{\mathcal F} Y_i) = \prod _{\mathcal F} \textrm{Hom}_{\mathcal C_i}(X_i,Y_i)$|⁠; the composition maps are given by the ultraproducts of the composition maps, i.e., |$(\prod _{\mathcal F}f_i) \circ (\prod _{\mathcal F}g_i) = \prod _{\mathcal F} (f_i \circ g_i)$|⁠. By Łoś’s theorem, this data satisfies the axioms of a category. If the categories |$\mathcal C_i$| have some structures, for example the structures of an abelian/monoidal/tensor category, then |$\widehat{\mathcal C}$| also has these structures (but the finite length property, for example, does not survive). Usually, |$\widehat{\mathcal C}$| is too big, and it is interesting to consider some full subcategories |$\mathcal C$| in there or, equivalently, consider ultraproducts only of some sequences of objects of |$\mathcal C_i$|⁠, for example bounded in some sense. This construction obviously extends to essentially small categories. All categories that we will consider are essentially small, so we will not bother mentioning this later. 1.3 Deligne categories as ultraproducts Here, we will show how to construct |$\textrm{Rep}(GL_t)$| using ultraproducts. See [16] and [7]. We want to apply the last example of the previous section to |$\mathcal C_i = \textbf{Rep}(GL_{n_i}, \mathbb K_i)$|—the tensor category of finite-dimensional representations of |$GL_{n_i}$| over |$\mathbb K_i$|⁠. From now on, we will denote by |$\textrm{Rep}(GL_n) = \textrm{Rep}(GL_n,\overline{\mathbb Q})$| and by |$\textrm{Rep}_p(GL_n) = \textrm{Rep}(GL_n, \overline{\mathbb F}_p)$|⁠. We have the following result ([7, Introduction] or [16, Theorem 1.1]): Theorem 1.3.1. (a) Suppose |$t\in \mathbb C$| is transcendental. Consider |$\widehat{\mathcal C} = \prod _{\mathcal{F}} \textrm{Rep}(GL_n)$|⁠. Denote by |$V_i$| the fundamental representation of |$GL_i$| and |$V = \prod _{\mathcal{F}}V_i$|⁠. Fix an isomorphism |$\prod _{\mathcal F}\overline{\mathbb Q}\simeq \mathbb C$| such that |$\prod _{\mathcal F} i = t$|⁠. Then the full subcategory of the |$\prod _{\mathcal F}\overline{\mathbb Q}$|-linear category |$\widehat{\mathcal C}$| generated by |$V$| under taking duals, tensor products, direct sums, and direct summands is equivalent to the |$\mathbb C$|–linear category |$\textrm{Rep}(GL_t)$|⁠, in a way consistent with the above isomorphism |$\prod _{\mathcal F}\overline{\mathbb Q} \simeq \mathbb C$|⁠. (b) Suppose |$t \in \mathbb C$| is algebraic but not integer, with minimal polynomial |$q(x) \in \mathbb Z[x]$|⁠. Fix a sequence of distinct primes |$p_n$| and sequence of integers |$t_n$| tending to infinity (we do not actually need to ask for a sequence |$t_n$| to tend to infinity, since it is satisfied automatically, since any |$q(t)$| has a finite number of prime factors, and so if |$t_n$| is bounded, |$p_n$| is bounded too) such that |$q(t_n) = 0$| in |$\mathbb F_{p_n}$|⁠. Moreover, fix an isomorphism |$\prod _{\mathcal F}\overline{\mathbb F}_{p_n}\simeq \mathbb C$| such that |$\prod _{\mathcal F} t_i = t$|⁠. Set |$\widehat{\mathcal C} = \prod _{\mathcal{F}} \textrm{Rep}_{p_n}(GL_{t_n})$|⁠. Denote by |$V_{t_i}$| the fundamental representation of |$GL_{t_i}$| and set |$V_t = \prod _{\mathcal{F}}V_{t_i}$|⁠. Then the full subcategory of the |$\prod _{\mathcal F}\overline{\mathbb F}_{p_n}$|-linear category |$\widehat{\mathcal C}$| generated by |$V$| under taking duals, tensor products, direct sums and direct summands is equivalent to the |$\mathbb C$|-linear category |$\textrm{Rep}(GL_t)$|⁠, in a way consistent with the above isomorphism |$\prod _{\mathcal F}\overline{\mathbb F}_{p_n}\simeq \mathbb C$|⁠. Proof. (a) First let’s prove that it is indeed possible to fix such an isomorphism. The ultraproduct |$\prod _{\mathcal F}i$| is an element of |$\mathbb C$|⁠. Suppose it is algebraic over |$\mathbb Q$|⁠, then it should satisfy a monic equation |$f$| with coefficients in |$\mathbb Q$|⁠. Then by Łoś’s theorem for almost all |$i$| we have |$f(i) = 0$|⁠, but since this is true for infinite number of distinct |$i$|’s, it follows that |$f = 0$|⁠. Hence, by contradiction, we conclude that |$\prod _{\mathcal F} i$| is transcendental. Now, by fixing an automorphism of |$\mathbb C$| over |$\mathbb Q$|⁠, we may send this transcendental number to |$t$|⁠. So we have a tensor category |$\widehat{\mathcal C}$| linear over |$\mathbb C$|⁠, with an object |$\prod _{\mathcal F}V_i$| of dimension |$t$|⁠. Hence, by Proposition 1.1.6, we obtain a tensor functor |$F:\textrm{Rep}(GL_t) \to \widehat{\mathcal C}$|⁠. Since |$\textrm{Rep}(GL_t)$| is generated by |$V$| under taking duals, tensor products, direct sums, and direct summands, it follows that the image of |$\textrm{Rep}(GL_t)$| under |$F$| is contained in the full subcategory |$\mathcal C$| in |$\widehat{\mathcal C}$| generated by |$V_t$| under taking duals, tensor products, direct sums, and direct summands. So we know that |$F:\textrm{Rep}(GL_t) \to \mathcal C$| is essentially surjective. Now it is enough to prove that it is fully faithful. Note that it is enough to prove that \begin{equation*} \prod_{\mathcal F} \textrm{Hom}_{GL_n}(V_n^{\otimes r} \otimes V_n^{* \otimes s}, V_n^{\otimes p} \otimes V_n^{* \otimes q}) = \textrm{Hom}_{\mathcal C}(V_t^{\otimes r} \otimes V_t^{* \otimes s}, V_t^{\otimes p} \otimes V_t^{* \otimes q}) \ \end{equation*} and that the composition maps are the same. Indeed, both categories can be obtained as the Karoubian envelope of the additive envelope of the categories consisting of all |$V^{\otimes p} \otimes V^{* \otimes q}$| or |$V_t^{\otimes p} \otimes V_t^{* \otimes q}$|⁠, respectively. From Schur–Weyl duality, we know that for a positive integer |$r$|⁠, another positive integer |$n>r$|⁠, the algebra |$\textrm{End}_{GL_n}(V_n^{\otimes r}) = (V_n^{\otimes r} \otimes V_n^{*\otimes r})^{GL_n}$| is naturally isomorphic to |$\overline{\mathbb Q}[S_r]$|⁠. From the invariant theory, we also know that |$(V_n^{\otimes r} \otimes V_n^{*\otimes r})^{GL_n}$| is generated by the elements |$\sum _{i_1,\dots ,i_r} e_{i_1}\otimes \dots \otimes e_{i_r} \otimes \varepsilon _{i_{\sigma (1)}}\otimes \dots \otimes \varepsilon _{i_{\sigma (r)}}$|⁠, where |$e_i$| and |$\varepsilon _j$| are dual bases of |$V_n$| and |$V_n^*$| and |$\sigma \in S_r$|⁠. Since |$\textrm{Hom}_{GL_n}(V_n^{\otimes r} \otimes V_n^{* \otimes s}, V_n^{\otimes p} \otimes V_n^{* \otimes q}) = \left (V_n^{\otimes (r+q)}\otimes V_n^{*\otimes (s+p)}\right )^{GL_n}$|⁠, it follows that it is non-zero only if |$r+q=p+s$| and for sufficently large |$n$| has dimension |$(r+q)!$|⁠. Now rewriting the generating elements above using evaluation and coevaluation maps, we may explicitly describe these elements. It is easy to see that these descriptions can be obtained by looking at the diagram of a fixed matching and interpreting lines connecting different rows as identity arrows, lines connecting elements in the upper row as evaluation maps, and lines connecting elements in the lower row as coevaluation maps. Since the number of matchings is |$(r+q)!$|⁠, it follows that for sufficiently large |$n$| they form a basis. Since they form the basis of |$\textrm{Hom}_{\textrm{Rep}(GL_t)}(V^{\otimes r} \otimes V^{* \otimes s}, V^{\otimes p} \otimes V^{* \otimes q})$| by definition, the equality of |$\textrm{Hom}$| spaces follows (since taking ultraproduct of the same finite-dimensional vector space over |$\overline{\mathbb Q}$| gives you the same vector space tensored with |$\mathbb C$|⁠). It remains to check that the composition maps are the same. In the |$GL_n$| case, they are given also by vertical concatenation of diagrams. The only thing that we need to check is what happens to the loops obtained in this way. Since each loop through the properties of tensor categories can be simplified to |$coev \circ ev$|⁠, it follows that each loop gives us a multiplication by |$n$|⁠; hence, for the composition maps in the ultraproduct, we get the multiplication by |$t$|⁠, so the composition law in |$\mathcal{C}$| and |$\textrm{Rep}(GL_t)$| is the same and we are done. (b) First, again, we need to explain how we can fix such an isomorphism. Let us prove that there is indeed an infinite number of pairs |$t_n$| and |$p_n$| such that |$q(t_n) = 0 \mod p_n$|⁠. It is enough to show that there are infinite numbers of primes dividing the numbers |$q(n)$| (if in this case the sequence |$t_n$| is bounded, it follows that some |$q(t_n)$| is divisible by an infinite number of prime numbers, which is absurd). Suppose it is not so, and there are only |$k$| such primes. Fix |$C$| such that we have |$q(n) < C \cdot n^{\deg (q)}$| for all positive integer |$n$|⁠. Denote by |$Q$| the number of integers of the form |$q(n)$| for |$n \in \mathbb Z_{\ge 0}$| such that |$q(n)<N$|⁠. By the above inequality, this number is at least |$\frac{1}{C}\cdot N^{\frac{1}{\deg (q)}}$|⁠. On the other hand, the number |$P$| of numbers less than |$N$| divisible only by |$k$| fixed primes is less or is equal to |$\log _2(N)^k$|⁠, since each prime number is at least |$2$|⁠. Hence, for big enough |$N$|⁠, we have |$P<Q$|⁠, which contradicts the hypothesis (this proof is also written by Nate Harman in his paper, see the proof of [16, Proposition 2.2]). So we indeed can choose such unbounded sequences |$t_n$| and |$p_n$|⁠. Now, by Łoś’s theorem, it follows that |$\prod _{\mathcal F}t_n$| is a root of |$q$| in |$\mathbb C$|⁠, so by composing with an automorphism of |$\mathbb C$|⁠, we may assume that under an isomorphism |$\mathbb C \simeq \prod _{\mathcal F} \overline{\mathbb F}_{p_n}$|⁠, |$\prod _{\mathcal F}t_n$| maps to |$t$|⁠. The remaining part of the proof is completely the same since the relevant part of Schur–Weyl duality holds over an algebraically closed field of any characteristic. Indeed, the natural isomorphism |$k[S_r] = End_{GL_m}((k^m)^{\otimes r})$| for |$m>r$| holds by the same argument as in the characteristic 0 case if |$k[S_r]$| is semisimple, but this is true for any |$r$| for almost all |$n$|⁠, since |$p_n> r$| for almost all |$n$| (actually, Schur–Weyl duality works in even more general situation [21]). Through this isomorphism, we obtain a connection between irreducible objects in |$\textrm{Rep}(GL_n)$| or |$\textrm{Rep}_p(GL_n)$| and irreducible objects of |$\textrm{Rep}(GL_t)$|⁠. As the 1st ingredient for this, we need to relate bipartitions with partitions. Definition 1.3.2. For a bipartition |$\lambda $| denote by |$\lambda |_n$|⁠, the weight is equal to |$(\lambda |_n)_i = \lambda ^{\bullet }_i - \lambda ^\circ _{n-i+1}$|⁠. In addition, for a complex number |$c\in \mathbb C$| denote by |$c|_n$| elements of |$\overline{\mathbb Q}$| or |$\overline{\mathbb F}_{p_n}$| (which one will be clear in the context) such that |$\prod _{\mathcal F}c|_n = c$|⁠. To show how this works let’s describe the sequence of objects of |$\textrm{Rep}(GL_n)$| corresponding to |$V(\lambda )$|⁠. Fix |$\lambda $|⁠; as we know, |$V(\lambda )$| is a direct summand in |$ V^{\otimes |\lambda ^{\bullet }|} \otimes V^{*\otimes |\lambda ^\circ |}$|⁠. Denote by |$B_{r,s}(t) = \textrm{End}(V^{\otimes r} \otimes V^{*\otimes s})$| the walled Brauer algebra. For |$n$| big enough |$B_{r,s}(n) = \textrm{End}(V_n^{\otimes r} \otimes V_n^{*\otimes s})$| by Łoś’s theorem and proof of Theorem 1.3.1. So we can take the direct summand corresponding to the same idempotent as |$\lambda $|⁠. The corresponding representation is going to be isomorphic to |$V(\lambda |_{n})$|⁠. So it follows that |$\prod _{\mathcal F}V(\lambda |_{n}) = V(\lambda )$|⁠. The same is true for finite characteristic case since, as we show below, for |$n$| big enough |$V(\lambda |_{t_n})$| is going to be irreducible. 2 Yangians in integer rank In the 1st two parts of this section, we will work over an algebraically closed field of characteristic |$0$|⁠, more specifically |$\overline{\mathbb Q}$| or |$\mathbb C$|⁠. Here we will briefly recall the definition and basic properties of the Yangian and then state the classification theorems for finite-dimensional representations of the Yangians for general and special linear groups. For a more detailed study, see [23] or [3] (we will mostly follow [23]). 2.1 The Yangians |$Y(\mathfrak{g}\mathfrak{l}_n)$| and |$Y(\mathfrak{sl}_n)$| Definition 2.1.1. ([23, Sections 1.1 and 1.2]) |$Y(\mathfrak{g}\mathfrak{l}_n)$| is the associative algebra generated by |$t^{(k)}_{ij}$|⁠, where |$1 \le i,j \le n$|⁠, and |$k> 0$| with the following defining relation in |$Y(\mathfrak gl_n) \otimes \textrm{End}( \mathbb C^n) \otimes \textrm{End} (\mathbb C^n)[[u^{-1},v^{-1}]]$|⁠: \begin{equation} (u-v)R(u-v)T^I(u)T^{II}(v) = (u-v)T^{II}(v)T^I(u)R(u-v) \, \end{equation} (1) (here we need to multiply by |$(u-v)$| to cancel the term |$(u-v)^{-1}$| in |$R(u-v)$|⁠, which does not belong to |$Y(\mathfrak gl_n) \otimes \textrm{End}( \mathbb C^n) \otimes \textrm{End} (\mathbb C^n)[[u^{-1},v^{-1}]]$|⁠; it is easy to see that after such a multiplication the expression lies in |$Y(\mathfrak gl_n) \otimes \textrm{End}( \mathbb C^n) \otimes \textrm{End} (\mathbb C^n)[[u^{-1},v^{-1}]]$|⁠) where |$T^{\alpha }(u) = \sum _{i,j} t_{ji}(u)\otimes e^{\alpha }_{ij} \ u^{-k} $|⁠, |$t_{ij}(u) = \delta _{ij} + \sum _{k>0} t_{ij}^{(k)}u^{-k}$|⁠, and |$R(u) = 1 +\frac{\sigma }{u}$| with |$\sigma $| being the operator permuting the 1st copy and 2nd copy of |$\mathbb C^n$|⁠. Remark. Here we are using the definition mentioned in [23, Remark 1.2.3]. That is why our definition of |$R$| has a different sign from [23, (1.12)]. Below, we list properties of this algebra, which are going to be important for us: Proposition 2.1.2. ([23, Sections 1.1–1.7]) (a)|$Y(\mathfrak{g}\mathfrak{l}_n)$| is a Hopf algebra with |$S(T(u)) = T^{-1}(u)$| and |$\Delta (T(u)) = T^I(u)T^{II}(u)$|⁠. (b) There is an algebra homomorphism |$ev:Y(\mathfrak{g}\mathfrak{l}_n)\rightarrow U(\mathfrak{g}\mathfrak{l}_n)$| given by |$T(u)\mapsto R(u)$|⁠. (This definition uses the fact that we can think about |$U(\mathfrak{g}\mathfrak{l}_n)$| as an algebra generated by |$E_{ij}$| such that the polynomial |$1 + u^{-1}(\sum _{imj} E_{ji}\otimes E_{ij})$| satisfies the relation similar to RTT relation of Yangian. In this case, the relation will give us the familiar commutator relation for |$U(\mathfrak{g}\mathfrak{l}_n)$|⁠.) (c) There is a Hopf algebra embedding |$i:U(\mathfrak{g}\mathfrak{l}_n) \rightarrow Y(\mathfrak{g}\mathfrak{l}_n)$| given by |$E_{ij} \mapsto t^{(1)}_{ij}$|⁠. (d) The center |$Z(Y(\mathfrak{g}\mathfrak{l}_n))$| is generated by algebraically independent elements given by the coefficients of a certain series |$\textrm{qdet} \ T(u)$| defined as \begin{equation*} \textrm{qdet} \ T(u) = \sum_{s \in S_n} sgn(s) \cdot t_{1,s(1)}(u-n+1)\dots t_{n,s(n)}(u). \end{equation*} (e) There is a family of automorphisms |$r_f$| of |$Y(\mathfrak{g}\mathfrak{l}_n)$| given by |$T(u) \mapsto f(u)T(u)$| for any |$f(u) \in 1 +u^{-1}\mathbb C[[u^{-1}]]$|⁠. Remark. The formulation of the statement of |$(b)$| is slightly different from [23]. This is because difference in the definition of the Yangian, which is mentioned in the previous remark. However, if we rewrite it in terms of |$t^{(k)}_{ij}$|⁠, it means the same: |$t^{(k)}_{ij} \mapsto e_{ij}$|⁠. From Proposition 2.1.2(b), it follows that any |$\mathfrak{g}\mathfrak{l}_n$| representation has a structure of a |$Y(\mathfrak{g}\mathfrak{l}_n)$| representation, such representations are called evaluation modules. In addition from Proposition 2.1.2(c), it follows conversely that any |$Y(\mathfrak{g}\mathfrak{l}_n)$| representation can be regarded as a |$\mathfrak{g}\mathfrak{l}_n$| representation with respect to the |$t^{(1)}_{ij}$|-action. Now we can define the Yangian of the special linear group following [23, Section 1.8]. Definition 2.1.3. |$Y(\mathfrak{s}\mathfrak{l}_n)$| is the subalgebra of |$Y(\mathfrak{g}\mathfrak{l}_n)$| of elements invariant under all automorphisms specified in Proposition 2.1.2(e). Below are the properties of this algebra: Proposition 2.1.4. (a)|$Y(\mathfrak{s}\mathfrak{l}_n)$| is a Hopf subalgebra in |$Y(\mathfrak{g}\mathfrak{l}_n)$|⁠. (b) There is a Hopf algebra embedding |$i:U(\mathfrak{s}\mathfrak{l}_n) \rightarrow Y(\mathfrak{s}\mathfrak{l}_n)$| given by the restriction of |$i$| from Proposition 2.1.2(c). (c)|$Y(\mathfrak{g}\mathfrak{l}_n) = Y(\mathfrak{s}\mathfrak{l}_n) \otimes Z(Y(\mathfrak{g}\mathfrak{l}_n))$| and thus |$Y(\mathfrak{s}\mathfrak{l}_n) = Y(\mathfrak{g}\mathfrak{l}_n)/(\textrm{qdet} \ T(u)-1)$|⁠. 2.2 Finite-dimensional representation of |$Y(\mathfrak{g}\mathfrak{l}_n)$| and |$Y(\mathfrak{sl}_n)$| One can classify all finite-dimensional irreducible representations of the Yangians defined above. This can be done by using as follows. (We will provide a sketch here; the more detailed discussion of some of the places will appear then we talk about the representation theory in positive characteristic.) First, one needs to consider Verma modules for the Yangian: Definition 2.2.1. ([23, Definition 3.2.3]) For |$\lambda (u) = (\lambda _1(u), \dots , \lambda _n(u)$| and arbitrary tuple of formal series in |$u^{-1}$|⁠, |$M(\lambda (u))$| is a representation of |$Y(\mathfrak{g}\mathfrak{l}_n)$|⁠, which is isomorphic to a quotient of |$Y(\mathfrak{g}\mathfrak{l}_n)$| by the left ideal generated by all the coefficients of |$t_{ij}(u)$| with |$i<j$| and all coefficients of |$t_{ii}(u) - \lambda _i(u)$| for all |$i$|⁠. In addition, one needs to define what the highest weight representation is: Definition 2.2.2. ([23, Proposition 3.2.2]) The highest weight representation of weight |$\lambda (u)$| is a representation |$M$| of |$Y(\mathfrak{g}\mathfrak{l}_n)$| such that there is a vector |$\zeta \in M$|⁠, which generates |$M$|⁠, |$t_{ij}(u)\zeta = 0$| for all |$i<j$| and |$t_{ii}(u)\zeta = \lambda _i(u)\zeta $|⁠. Later, one can prove the following: Theorem 2.2.3. ([23, Theorem 3.2.7]) Any finite-dimensional irreducible representation |$L$| of the Yangian |$Y(\mathfrak{g}\mathfrak{l}_n)$| is the highest weight representation and hence is a simple quotient of Verma module. The next step is to define evaluation modules: introduction of an analog of category |$\mathcal O$| and by reduction to the |$Y(\mathfrak{g}\mathfrak{l}_2)$| case, which can be solved more or less explicitly. More precisely, one can show that each finite-dimensional module has the highest weight vector with a weight given by |$(\lambda _1(u),\dots , \lambda _n(u))$|⁠, which determine the action of |$t_{ii}^{(n)}$| on this vector. The result of this is the classification theorem originally due to Drinfeld. To state it, we need the following definition: Definition 2.2.4. Evaluation modules of |$Y(\mathfrak{g}\mathfrak{l}_n)$| or |$Y(\mathfrak{s}\mathfrak{l}_n)$| are irreducible representations of |$\mathfrak{g}\mathfrak{l}_n$| with a structure of the representation of the Yangian given by the morphism |$ev:Y(\mathfrak{g}\mathfrak{l}_n)\to U(\mathfrak{g}\mathfrak{l}_n)$|⁠. Note that there are slight differences in the definitions of evaluation modules in [3] and [23]. Chari and Pressley construction is as follows: Definition 2.2.5. |$V_z(\mu )$| is a |$Y(\mathfrak{s}\mathfrak{l}_n)$|-module, which is equal to |$V(\mu )$| as an |$\mathfrak{s}\mathfrak{l}_n$|-module. The structure of |$Y(\mathfrak{s}\mathfrak{l}_n)$|-representation is given by the map |$ev_z:Y(\mathfrak{s}\mathfrak{l}_n) \to U(\mathfrak{s}\mathfrak{l}_n)$| given by |$T(u)\mapsto R(u-z)$|⁠. To make things clearer, we would like to establish a connection between this definition and Definition 2.2.4. First, define |$\tau _z$| to be an automorphism of |$Y(\mathfrak{g}\mathfrak{l}_n)$| given by |$T(u) \mapsto T(u-z)$|⁠. (See [23, 1.3]) Let |$j_U$| be the inclusion |$U(\mathfrak{s}\mathfrak{l}_n)\to U(\mathfrak{g}\mathfrak{l}_n)$| and |$j_Y$| be the inclusion |$Y(\mathfrak{s}\mathfrak{l}_n)\to Y(\mathfrak{g}\mathfrak{l}_n)$|⁠. Then we have the following: Lemma 2.2.6. The map |$ev^{\prime}_z = j_U \circ ev_z$| is equal to the map |$ev \circ \tau _z \circ j_Y$|⁠. Proof. Indeed, one map sends |$T(u) \to R(u-z)$| and another map first sends |$T(u) \mapsto T(u-z)$| and then |$T(u) \to R(u)$|⁠, so they are indeed the same. Now fix |$f = (1-\frac{u}{z})$| and consider an automorphism |$r_f$| from Proposition 2.1.2(e). In addition, consider an automorphism |$\psi _z$| of |$U(\mathfrak{g}\mathfrak{l}_n)$|⁠, which sends |$\sigma \to \sigma -z$|⁠. Then we have Lemma 2.2.7. The maps |$ev \circ r_f \circ \tau _z$| and |$\psi _z\circ ev$| are equal. In addition, |$ev^{\prime}_z = \psi _z \circ ev \circ j$|⁠. Proof. Concerning the 1st statement, the 1st map acts |$T(u) \mapsto T(u-z) \mapsto (1-\frac{z}{u})T(u-z) \mapsto (1-\frac{z}{u}) (1 + \frac{\sigma }{u-z})= 1 + \frac{\sigma -z}{u}$|⁠. Moreover, the 2nd map maps |$T(u) \mapsto 1 + \frac{\sigma }{u} \mapsto 1 + \frac{\sigma -z}{u}$|⁠. So they are indeed the same. Now consider |$ev \circ r_f \circ \tau _z \circ j$|⁠, this map is actually equal to |$ev \circ \tau _z \circ j$|⁠, since |$r_f$| acts by |$1$| on elements of |$Y(\mathfrak{s}\mathfrak{l}_n)$|⁠, hence |$ev^{\prime}_z = ev \circ r_f \circ \tau _z \circ j = \psi _z\circ ev \circ j$|⁠. However, this means that to obtain modules |$V_z(\mu )$| with fixed |$\mathfrak{s}\mathfrak{l}_n$|-weight |$\mu $| and different |$z$|⁠, we can equivalently consider all evaluation modules |$V(\lambda )$| in the sense of Definition 2.2.4 with |$\lambda = \mu $| as an |$\mathfrak{s}\mathfrak{l}_n$|-weight, since to take |$ev_z$| is the same as to take |$ev$| and later deform the action of the central element of |$\mathfrak{g}\mathfrak{l}_n$|⁠. So later, we will not mention modules |$V_z(\mathfrak{s}\mathfrak{l}_n)$|⁠. After introducing evaluation modules, one can show that Theorem 2.2.8. ([23, Proposition 3.2.9]) Every simple module over |$Y(\mathfrak{g}\mathfrak{l}_n)$| is a subquotient of a tensor product of evaluation modules. After this, one only needs to examine the theory of |$Y(\mathfrak{g}\mathfrak{l}_2)$| modules more carefully, to arrive at the following theorem ([23, Corollary 3.4.2]): Theorem 2.2.9. Irreducible finite-dimensional representations of |$Y(\mathfrak{g}\mathfrak{l}_n)$| are in |$1-1$| correspondence with tuples |$(f(u), P_1(u), \dots , P_{n-1}(u))$|⁠, where |$P(u)$| are monic polynomials in |$u$| and |$f(u) \in 1 + u^{-1}\mathbb C[[u^{-1}]]$|⁠. Moreover, the highest weight of representation – |$\lambda (u)$| satisfies |$\frac{\lambda _i}{\lambda _{i+1}} = \frac{P_i(u+1)}{P_i(u)}$|⁠. In addition, up to tensoring with one-dimensional representations, each irreducible finite-dimensional representation is a subquotient of the tensor product of evaluation modules. Definition 2.2.10. The polynomials |$P_i$| are called the Drinfeld polynomials. We have an analogous result for |$Y(\mathfrak{s}\mathfrak{l}_n)$| (following [23, Corollary 3.4.8]): Theorem 2.2.11. Finite-dimensional representations of |$Y(\mathfrak{s}\mathfrak{l}_n)$| are in |$1-1$| correspondence with tuples |$(P_1(u), \dots , P_{n-1}(u))$| of Drinfeld polynomials. Moreover, each such representation is a subquotient of a tensor product of evaluation modules. 3 Yangians |$Y(\mathfrak{g}\mathfrak{l}_n)$| and |$Y(\mathfrak{sl}_n)$| and their representations in positive characteristic Note that one can define the Yangians for special and general linear groups over |$\overline{\mathbb F}_p$| exactly in the same way as above. One can use exactly the same arguments to show that all statements in Proposition 2.1.2, except |$(d)$|⁠, hold. The problem is that the center of the Yangian in positive characteristic is bigger than in zero characteristic; it is calculated in [2]. Nevertheless, one can define the Harish-Chandra center |$Z_{HC}(Y(\mathfrak{g}\mathfrak{l}_n))$| as the subalgebra generated by the coefficients of the series |$qdet \ T(u)$| and then for |$p>n$| an analog of Proposition 2.1.3(c) still holds. Indeed, according to [2, Theorem 6.1], we have |$Y(\mathfrak{g}\mathfrak{l}_n) = Y(\mathfrak{s}\mathfrak{l}_n)\otimes Z_{HC}(Y(\mathfrak{g}\mathfrak{l}_n))$|⁠. In order to study representations of the Yangian in complex rank for algebraic |$t$|⁠, according to Theorem 1.3.1(b), we need to know something about the representations of Yangians in positive characteristic for a sufficiently large |$p$|⁠. To do this, we first need some results about |$\mathfrak{g}\mathfrak{l}_n$| representations in positive characteristic. We will discuss this in the following subsection, then in the next subsection we will return to Yangians in positive characteristic. 3.1 Representations of |$\mathfrak{g}\mathfrak{l}_n$| and |$\mathfrak{s}\mathfrak{l}_n$| in positive characteristic Fix an irreducible |$\mathfrak{s}\mathfrak{l}_n$|-representation |$V(\lambda )$| over |$\mathbb C$|⁠. This representation is obtained as a quotient of the Verma module |$M(\lambda )$| by the subrepresentation generated by |$f_i^{\lambda _i+1}v_\lambda $|⁠, where |$v_{\lambda }$| is the highest weight vector of the Verma module. Now, in positive characteristic, we can also consider the Verma module |$M(\lambda ,p)$|⁠, and all the vectors |$f_i^{\lambda _{i}+1}v_\lambda $| are going to be singular because the coefficients for the action of |$e_j$| on these vectors depend only on |$\lambda $| and the structure constants of the Lie algebra, which are all integers. Hence, if the coefficients are zero in characteristic 0, they are going to stay zero after reduction. So we still have quotient modules |$V(\lambda ,p)$| defined in the same way, called the Weyl modules, and the only question is whether they are irreducible. First, consider the set |$Q$| of weights appearing in |$V(\lambda )$|⁠. We want to find a restriction on |$p$| such that no two elements of |$Q$| differ by a linear combination of simple roots with coefficients divisible by |$p$|⁠. Suppose that two weights |$\mu _1,\mu _2$| from |$Q$| differ by a sum |$\sum _i p c_i \alpha _i$|⁠. Since |$Q$| is |$W$|-symmetric, it follows that |$\forall \mu \in Q$|⁠, we have |$\lambda _1 \ge \mu _i \ge \lambda _n$|⁠. Thus |$|(\mu _1)_i - (\mu _2)_i| \le \lambda _1 - \lambda _n$|⁠. Since it is at the same time divisible by |$p$|⁠, we must conclude that for |$p> \lambda _1-\lambda _n+1$|⁠, we have |$\mu _1 = \mu _2$|⁠. Hence, if we consider |$V(\lambda , p)$| for |$p> \lambda _1 - \lambda _n + 1$|⁠, it makes sense to speak about it as the highest weight module, since no two elements of |$Q$| get identified when we pass to positive characteristic and no two elements are connected by a root, which were not connected before passing to positive characteristic. Moreover, we can use the following definition of the weight order: |$\mu < \mu ^{\prime}$| iff |$\exists \mu _i$| such that |$\mu _0 = \mu $|⁠, |$\mu ^{\prime} = \mu _n$|⁠, and |$\mu _i = \mu _{i+1} - \alpha _j$| for some |$j$|⁠. This definition is equivalent to the standard one in characteristic zero. In positive characteristic under this definition, |$\lambda $| is still the maximal weight of |$V(\lambda ,p)$| since all weights |$\lambda + \alpha _j$| are not in |$Q$| even modulo |$p$|⁠. It is known ([18, Theorem 1] or [19, Theorem 2]) that in positive characteristic, one can define a generalization of the Harish-Chandra center and central characters |$\chi _\lambda $| in such a way that the following theorem holds: Theorem 3.1.1. For |$p \ne 2$|⁠, one has |$\chi _\lambda = \chi _\mu $| iff |$w(\lambda + \rho ) = \mu + \rho $| modulo |$p$|⁠, for |$w \in W$|⁠. So we have the following lemma. Lemma 3.1.2. For |$p> \lambda _1 -\lambda _n + n$|⁠, the module |$V(\lambda , p)$| is irreducible. Proof. Suppose we have a submodule |$N\subset V(\lambda ,p)$|⁠. Since already in |$V(\lambda ,p)$| the order < on weights has no cycles for such |$p$|⁠, it follows that it is a well-defined order when restricted to |$N$|⁠; hence, |$N$| has the highest weight vector. Thus |$V(\lambda ,p)$| has a nontrivial singular vector with weight |$\mu $|⁠. However, by the above, |$\mu $| should be equal to |$w(\lambda + \rho ) - \rho $| for some |$w \in W$|⁠. So we only need to prove that there is no element of |$Q$| that differs from |$w(\lambda + \rho ) - \rho $| for some |$w$| by a linear combination of simple roots multiplied by |$p$|⁠. So suppose |$ \lambda \ge \mu \ge w_0(\lambda )$| and |$\mu - w(\lambda + \rho ) - \rho $| is divisible by |$p$|⁠. Write down |$\lambda = (\lambda _1, \dots , \lambda _n)$| and |$w_0(\lambda ) = (\lambda _n, \dots , \lambda _1)$|⁠, where |$\lambda _1 \ge \lambda _2 \ge \dots \ge \lambda _n$|⁠. Now |$w(\lambda + \rho ) - \rho = (\lambda _{w(1)} + e_1, \dots , \lambda _{w(n)} + e_n)$|⁠, where |$e_j$| are some integers |$n> e_j > -n$|⁠. Now, since |$\mu \le \lambda $|⁠, it follows that |$\mu _1 \le \lambda _1$| and |$\mu _n \ge \lambda _n$|⁠. Since |$Q$| is |$W$|-stable, it follows that |$\lambda _1 \ge \mu _i \ge \lambda _n$|⁠. Now |$\mu _i = \lambda _{w(i)} + e_i \ mod \ p$|⁠, in other words, a number between |$\lambda _1$| and |$\lambda _n$| differs from a number between |$\lambda _1 + n-1$| and |$\lambda _n -n+1$| by a multiple of |$p$|⁠, but |$\lambda _1 - \lambda _n +n < p$| and it follows that |$\mu _i = \lambda _{w(i)} + e_i$|⁠. However, this contradicts |$\mu \in Q$|⁠. So we proved that there are no nontrivial singular vectors in |$V(\lambda ,p)$| and, hence, no nontrivial submodules. The same statement of course holds for |$\mathfrak{g}\mathfrak{l}_n$|-modules. 3.2 Representations of |$Y(\mathfrak{g}\mathfrak{l}_n)$| and |$Y(\mathfrak{s}\mathfrak{l}_n)$| in positive characteristic Using the above tools, we can try and repeat some of the arguments from the classification of irreducible representations of the Yangian following [23]. First, we need to understand in which sense we can treat a Yangian representation in positive characteristic as the highest weight representation. Proposition 3.2.1. Consider an irreducible representation L of |$Y(\mathfrak{g}\mathfrak{l}_n)$|⁠, which as a representation of |$\mathfrak{g}\mathfrak{l}_n$| is equal to the sum |$V(\lambda _1,p) \oplus \dots \oplus V(\lambda _k,p)$|⁠. Then for |$p> \max ((\lambda _j)_1-(\lambda _j)_n)+n$|⁠, |$L$| has a unique up to scaling singular vector, whose |$\mathfrak{g}\mathfrak{l}_n$| weight is maximal. Proof. The problem here is to rule out the possibility of |$L$| not having any singular vectors at all. Since we can speak about the underlying |$\mathfrak{g}\mathfrak{l}_n$| representation as the highest weight, we can repeat the argumentin [23, Theorem 3.2.7] without any changes. So it follows that for big enough |$p$|⁠, |$L = L(\lambda (u),p)$| for some |$\lambda (u)$| (where |$L(\lambda (u),p)$| is the irreducible quotient of the Verma module |$M(\lambda (u),p)$|⁠). Now let’s move to the important case |$Y(\mathfrak{g}\mathfrak{l}_2)$|⁠. Proposition 3.2.2. Suppose |$L$| is a finite-dimensional representation of |$Y(\mathfrak{g}\mathfrak{l}_2)$| isomorphic to |$V(\lambda _1,p) \oplus \dots \oplus V(\lambda _k,p)$| as a |$\mathfrak{g}\mathfrak{l}_2$| representation, then for |$p> \max ((\lambda _j)_1 - (\lambda _j)_2)+1$|⁠, we have |$L = L(\lambda (u),p)$| and there is a formal series |$f(u) \in 1 + u^{-1}\overline{\mathbb F}_p[[u^{-1}]]$| such that |$f(u)\lambda _1(u)$| and |$f(u)\lambda _2(u)$| are polynomials in |$u^{-1}$|⁠. Proof. By the previous discussion under our assumption, we can repeat the proof of [23, Proposition 3.3.1] for |$L$| without any change. For |$n \in \mathbb F_p$|⁠, denote by |$[n] \in \mathbb Z$| the minimal element of |$\mathbb Z_{\ge 0}$| such that |$[n] \ mod \ p = n$|⁠. Now for |$\alpha ,\beta \in \overline{\mathbb F}_p$|⁠, denote by |$L(\alpha ,\beta ,p)$| the irreducible quotient of the Verma module |$M(\alpha ,\beta ,p)$| for |$\mathfrak{g}\mathfrak{l}_2$|⁠. By a direct calculation, |$L(\alpha ,\beta ,p)$| has a basis |$f^kv$| for |$k$| from |$0$| to |$l$|⁠, where |$l$| is equal to |$p-1$| if |$\alpha -\beta \notin \mathbb F_p$| and is equal to |$[\alpha -\beta ]$| if |$\alpha -\beta \in \mathbb F_p$|⁠. Before formulating the next proposition, let’s see how |$t_{ij}(u)$| acts on |$L(\alpha ,\beta ,p)$|⁠; by our formulas, we have |$t_{11} = 1 + e_{11}u^{-1}$|⁠, |$t_{22}(u) = 1 + e_{22}u^{-1}$|⁠, |$t_{12}(u) = eu^{-1}$|⁠, and |$t_{21}(u) = fu^{-1}$|⁠. From this, it follows that |$t_{12}(u)$| acts on |$L(\alpha ,\beta ,p)$| nilpotently and operators |$t_{11}(u)$| and |$t_{22}(u)$| act semisimply. Since |$\delta (t_{12}(u)) = t_{11}(u) \otimes t_{12}(u) + t_{12}(u) \otimes t_{22}(u)$|⁠, it follows that this operator acts nilpotently as the sum of the two nilpotent operators. Hence, |$t_{12}(u)$| acts nilpotently on arbitrary tensor products of |$L(\alpha _k,\beta _k,p)$|⁠. Proposition 3.2.3. Given two sequences |$\alpha _i,\beta _i$| of elements of |$\overline{\mathbb F}_p$| for |$i =1,\dots ,k$|⁠, re-numerate them in such a way that |$[\alpha _i-\beta _i]$| is minimal among all |$[\alpha _j-\beta _k]$| for |$i \le j,k$| if defined, and if not defined, then all |$\alpha _j-\beta _k \notin \mathbb F_p$|⁠. Then the representation \begin{equation*} L(\alpha_1,\beta_1,p) \otimes \dots \otimes L(\alpha_k,\beta_k,p) \end{equation*} is irreducible. Proof. We can use the proof of [23, Proposition 3.3.2] with slight changes (in particular, the number |$p$| used in the proof we will denote by |$q$|⁠). First, let’s explain how the re-numeration of |$\alpha _i$|⁠, |$\beta _i$| works. Consider all possible pairs |$i,j$| such that |$[\alpha _i-\beta _j]$| is defined and choose a pair such that |$[\alpha _i-\beta _j]$| is minimal among them. These two elements now will become new |$\alpha _1$| and |$\beta _1$|⁠. Then repeat for all remaining |$\alpha _i$| and |$\beta _j$|⁠. Denote the module in question by |$L$|⁠. Suppose we have a submodule |$N \subset L$|⁠. By the discussion above |$t_{12}(u)$| acts nilpotently on |$L$|⁠. Since |$\Delta (t_{12}(u)) = t_{11}(u)\otimes t_{12}(u) + t_{12}(u) \otimes t_{22}(u)$|⁠, it follows that |$t_{12}(u)$| acts locally nilpotently on |$L$| and |$N$|⁠. Hence, |$N$| has a vector singular with respect to |$t_{12}(u)$|⁠. Thus, if we prove that |$L$| has only one singular vector (the tensor product of singular vectors |$\zeta _i$| of all |$L(\alpha _i,\beta _i,p)$|⁠), it will follow that this module has no nontrivial submodules. We will prove this claim by induction. So suppose we have a singular vector |$\xi = \sum _{r=0}^q e^r\zeta _1 \otimes \xi _r$| and |$\xi _r$| some elements of |$L(\alpha _2,\beta _2,p)\otimes \dots \otimes L(\alpha _k,\beta _k,p)$|⁠. Here |$\zeta _i$| is a singular vector of |$L(\alpha _i,\beta _i,p)$| and |$q$| is an integer less or equal to |$p-1$| or |$[\alpha _i-\beta _i]$| if it is defined. Then repeating all the steps of the proof of [23, Proposition 3.3.2], we obtain the following formula: \begin{equation*} q(\alpha_1-\beta_1-q+1)(\alpha_1-\beta_2-q+1)\dots(\alpha-\beta_k-q+1) = 0. \end{equation*} If |$\alpha _1-\beta _1 \notin \mathbb F_p$|⁠, then all |$\alpha _1-\beta _k \notin \mathbb F_p$|⁠; hence, the equation is satisfied only for |$q=0$|⁠. If |$[\alpha _1-\beta _1] = k \le p-1$|⁠, then |$q \le k$|⁠. Hence |$[\alpha _1-\beta _1]-[q]+1 = k-[q]+1$| lies between |$0$| and |$k+1$|⁠. So it may be equal to zero only if |$k=p-1$| and |$q=0$|⁠. All other |$\alpha _1-\beta _j$| are either not in |$\mathbb F_p$| and hence the corresponding brackets are not zero, or |$[\alpha _1-\beta _j]\ge k$|⁠, and hence |$[\alpha _1-\beta _j]+1-q$| can be zero again only for |$q=0$|⁠. Hence, it follows that |$q = 0$| and the singular vector is equal to |$\zeta _1 \otimes \dots \otimes \zeta _k$| up to scaling. Now the fact that this singular vector generates |$L$| is proved in the same way as in characteristic zero. Hence, |$L$| is irreducible. From Proposition 3.2.2, we know that any finite-dimensional irreducible |$Y(\mathfrak{g}\mathfrak{l}_2)$|-module |$L$| satisfying the condition stated there after tensoring with a one-dimensional representation is isomorphic to |$L^{\prime} = L(\lambda (u),p)$|⁠, where |$\lambda (u)$| is a pair of polynomials in |$u^{-1}$|⁠. Write down |$\lambda _1(u) = (1+\alpha _1u^{-1})\dots (1+\alpha _ku^{-1})$| and |$\lambda _2(u) = (1+\beta _1u^{-1})\dots (1+\beta _ku^{-1})$|⁠. Here |$\alpha _i,\beta _i$| are ordered in a way consistent with Proposition 3.2.3. Now consider |$L(\alpha _1,\beta _1,p)\otimes \dots \otimes L(\alpha _k,\beta _k,p)$|⁠. By the above discussion, it follows that this module is also isomorphic to |$L(\lambda (u),p)$|⁠. But this gives us another condition on |$\alpha _i,\beta _i$|⁠. Since we know that in |$L^{\prime}$| there are no chains of successive weights differing by the action of |$e$| of length |$p-1$|⁠, it follows that no |$L(\alpha _i,\beta _i,p)$| can have dimension |$p-1$|⁠. So all |$\alpha _i-\beta _i \in \mathbb F_p$|⁠. Now we are able to prove the following theorem (for the corresponding theorem in characteristic zero, see [23, Theorem 3.3.3]): Theorem 3.2.4. Suppose |$L$| is a representation satisfying the assumption of Proposition 3.2.2, which is isomorphic to |$L(\lambda (u),p)$|⁠. Then there is a monic polynomial |$P(u)$| in |$u$| such that \begin{equation*} \frac{\lambda_1(u)}{\lambda_2(u)} = \frac{P(u+1)}{P(u)}. \end{equation*} Proof. The proof is easy. By the assumption |$L = L(\nu (u),p)$|⁠. By Proposition 3.2.2, we can make |$\lambda (u) =\nu (u)f(u)$| be a pair of polynomials with roots |$\alpha _i$| and |$\beta _i$|⁠, respectively. By the above, we know that |$\alpha _i-\beta _i \in \mathbb F_p$|⁠; hence, we can take the following: \begin{equation*} P(u) = \prod_{i=1}^k (u+\beta_i)(u+\beta_i+1)\dots(u+\alpha_i-1). \end{equation*} Let’s generalize this to |$Y(\mathfrak{g}\mathfrak{l}_n)$|-modules in positive characteristic. Theorem 3.2.5. Suppose |$L$| is a representation satisfying the assumptions of Proposition 3.2.1, then there are monic polynomials |$P_i(u)$| such that \begin{equation*} \frac{\lambda_i(u)}{\lambda_{i+1}(u)} = \frac{P_i(u+1)}{P_i(u)} \, \end{equation*} Proof. We know that |$L = L(\lambda (u))$|⁠. Now using the inclusion |$Y(\mathfrak{g}\mathfrak{l}_2)\rightarrow Y(\mathfrak{g}\mathfrak{l}_n)$|⁠, where |$t_{i,j} \rightarrow t_{i+k,j+k}$|⁠, we may regard |$L$| as a |$\mathfrak{g}\mathfrak{l}_2$|-module. The |$\mathfrak{s}\mathfrak{l}_2$|-weights with respect to this inclusion lie between |$\lambda _1-\lambda _n$| and |$\lambda _n-\lambda _1$|⁠, so the assumption of Theorem 3.2.4 holds; hence, |$\frac{\lambda _k}{\lambda _{k+1}} = \frac{P_k(u+1)}{P_k(u)}$|⁠, for some |$P_k$|⁠. We also want to be able to construct a finite-dimensional representation with given Drinfeld polynomials. Theorem 3.2.6. Set |$p>2$|⁠. Suppose we have a collection of monic polynomials |$P_1(u), \dots , P_{n-1}(u)$|⁠. Then there is a finite-dimensional representation |$L(\mu (u),p)$| such that \begin{equation*} \frac{\mu_i(u)}{\mu_{i+1}(u)} = \frac{P_i(u+1)}{P_i(u)}. \end{equation*} Proof. The second part of the proof of [23, Theorem 3.4.1] can be repeated without any problem. Indeed, each of |$L(\mu ^{(k)},p)$| has |$\mu ^{(k)}_1 - \mu ^{(k)}_n = 1$|⁠, so they are equal to |$V(\mu ^{(k)},p)$|⁠. Now |$L(\mu ^{(1)},p) \otimes \dots \otimes L(\mu ^{(k)},p)$| has an external grading induced from the characteristic zero case, which is consistent with the |$Y(\mathfrak{g}\mathfrak{l}_n)$|-action. So it follows that |$L(\mu (u),p)$| is a subquotient in this finite-dimensional module. Remark. Note that the weight with maximal |$\lambda _1-\lambda _n$| appearing in |$L(\mu ^{(1)},p) \otimes \dots \otimes L(\mu ^{(k)},p)$| is equal to |$\nu = \sum _k \mu ^{(k)}$|⁠. Moreover, |$\nu _1-\nu _n = \sum _i \deg \ P_i$|⁠. Thus, for |$\sum \deg \ P_i + n < p$| it follows that the corresponding |$L(\mu (u))$| satisfies the conditions of Proposition 3.2.1. Remark. Another difference between the positive characteristic case and zero characteristic case is the fact that |$P_i$| in Theorem 3.2.5 are not unique. Indeed, from |$\frac{Q_i(u+1)}{Q_i(u)} = \frac{P_i(u+1)}{P_i(u)}$|⁠, it follows that |$\frac{P_i}{Q_i} = F_i$| satisfies |$F_i(u)=F_i(u+1)$|⁠. Thus, |$F_i$| is a ratio of products of expressions of the form |$(u+c)(u+1+c) \dots (u+c+p-1) = (u+c)^p-(u+c)$| for some |$c \in \overline{\mathbb F}_p$|⁠. Further, we set |$q_p(u):= u^p-u$|⁠. This also shows the following: Corollary 3.2.7. Suppose |$L$| is a representation satisfying the assumptions of Proposition 3.2.1. Then it is a subquotient of a tensor product of evaluation representations. Proof. Apply Theorem 3.2.5 and then the proof of Theorem 3.2.6. Corollary 3.2.8. Finite-dimensional representations of |$Y(\mathfrak{g}\mathfrak{l}_n)$| satisfying the condition of Proposition 3.2.1 are classified by tuples |$(f(u),[P_1],\dots ,[P_{n-1}])$|⁠, where |$f(u) \in 1 + u^{-1}\overline{\mathbb F}_p[[u^{-1}]]$|⁠, |$\sum \deg \ P_i + n < p$|⁠, and |$[ \ ]$| denote the equivalence classes generated by relation |$P_i \equiv Q_i$| if |$Q_i = P_iq_p(u+c)$|⁠. Corollary 3.2.9. Finite-dimensional representations of |$Y(\mathfrak{s}\mathfrak{l}_n)$| satisfying the condition of Proposition 3.2.1 are classified by tuples |$[P_1],\dots ,[P_{n-1}]$|⁠, where |$\sum _i \deg (P_i) + n < p$| and |$[ \ ]$| denote the equivalence classes generated by relation |$P_i \equiv Q_i$| if |$Q_i = P_iq_p(u+c)$|⁠. 4 Yangians in complex rank 4.1 Yangians |$Y(\mathfrak{g}\mathfrak{l}_t)$| and |$Y(\mathfrak{s}\mathfrak{l}_t)$| For this subsection, fix |$t \in \mathbb C \backslash \mathbb Z$|⁠. Below, |$t_n$| are the same as in Theorem 1.3.1(b) for algebraic |$t$| and |$t_n = n$| for transcendental |$t$|⁠. Similarly, |$p_n$| are as in Theorem 1.3.1(b) for algebraic |$t$| and |$p_n = 0$| for transcendental |$t$|⁠. In addition, |$\overline{\mathbb F}_0:= \overline{\mathbb Q}$|⁠. To define |$Y(\mathfrak{g}\mathfrak{l}_t)$|⁠, we will mimic the Faddeev–Reshetikhin–Takhtajan presentation of |$Y(\mathfrak{g}\mathfrak{l}_n)$| in the following way (following [14, Section 7]). For |$i \in \mathbb Z_{> 0}$|⁠, denote by |$V_i\simeq V$| a collection of copies of |$V \in \textrm{Rep} (GL_t)$|—fundamental representation of |$GL_t$|⁠. Consider the tensor algebra |$A = T(\bigoplus _{i=1}^{\infty }V_i\otimes V_i^*)$|⁠. This is an ind-object of |$\textrm{Rep} (GL_t)$|⁠. Obviously, it is generated as an algebra by the images of the inclusion maps |$T_i: V \otimes V^* \rightarrow A$| sending |$V\otimes V^*$| to |$V_i\otimes V_i^*$|⁠. Using this, we can define a formal power series in |$u^{-1}$| with coefficients in |$\textrm{Hom}(V\otimes V^*, A)$|⁠, |$T(u) = 1 + \sum _{i>0} T_iu^{-i}$|⁠. In addition, by composing with certain evaluation and coevaluation maps, we can regard |$T(u)$| as an element of |$\textrm{Hom}(V,V\otimes A)[[u^{-1}]]$|⁠. Now define |$R(u)\in \textrm{Hom}(V\otimes V, V \otimes V)[[u^{-1}]]$| in the same way as for integer rank; we set |$R(u)=1 +\frac{\sigma }{u}$|⁠, where |$\sigma $| interchanges factors in the tensor product. Consider an element of |$\textrm{Hom}(V_I \otimes V_{II}, V_I\otimes V_{II} \otimes A)[[u^{-1},v^{-1}]]$| (here |$V_I \simeq V_{II} \simeq V$|⁠) given by \begin{equation} Q(u,v) = (u-v)(R(u-v)T^I(u)T^{II}(v) - T^{II}(v)T^I(u)R(u-v)) \, \end{equation} (2) where |$i$| in |$T^i$| specifies on which |$V_i$| it acts and |$R$| in the first summand acts on |$V_I\otimes V_{II} \otimes A$| by |$R \otimes \textrm{Id}$|⁠. Again using the evaluation and coevaluation maps, we can think of |$Q$| as an element of |$\textrm{Hom}(V\otimes V^* \otimes V\otimes V^*, A)[[u^{-1},v^{-1}]]$|⁠. Now write |$Q(u,v) = \sum _{i,j}Q_{i,j} u^{-i}v^{-j}$|⁠. We are ready to define the Yangian. Definition 4.1.1. The Yangian of |$\mathfrak{g}\mathfrak{l}_t$|⁠, |$Y(\mathfrak{g}\mathfrak{l}_t)$|⁠, is the algebra obtained as the cokernel of the following map of ind-objects: \begin{equation*} \bigoplus_{i,j} Q_{i,j}: \bigoplus_{i,j} V\otimes V^* \otimes V\otimes V^* \rightarrow A \, \end{equation*} or, equivalently, is the quotient of |$A$| by the quadratic relations given by |$Q_{i,j}$|⁠. Remark. To specify an algebra homomorphism from |$A$|⁠, it is sufficient to give a bunch of morphisms from |$V\otimes V^*$|⁠, since |$T_i$| freely generate |$A$|⁠. To specify an algebra homomorphism from |$Y(\mathfrak{g}\mathfrak{l}_t)$|⁠, it is sufficient to give a bunch of maps from |$V \otimes V^*$| such that the quadratic relations |$Q_{i,j}$| are satisfied, since |$T_i$| generate the Yangian. The next step is to generalize some important properties by direct methods, but we will rather use the methods of ultraproducts and Łoś’s theorem since these are the main tools of this paper. The main fact that we are going to use is that |$\prod _{\mathcal F} Y(\mathfrak{g}\mathfrak{l}_{t_n})$| is equal to |$Y(\mathfrak{g}\mathfrak{l}_t)$|⁠, which follows from the definition, since it uses exactly the same spaces and maps as the finite rank definition (meaning that they are ultraproducts of the spaces/maps in the finite case). These properties were stated in [14, Section 7]. Here we provide their proof: Proposition 4.1.2. (a) There is a Hopf algebra structure on |$Y(\mathfrak{g}\mathfrak{l}_t)$| given by |$\Delta (T(u)) = T^I(u)T^{II}(u)$| and |$S(T(u)) = T(u)^{-1}$|⁠. (b) There is an algebra homomorphism |$i:U(\mathfrak{g}\mathfrak{l}_t) \rightarrow Y(\mathfrak{g}\mathfrak{l}_t)$|⁠, which on |$V\otimes V^* \subset U(\mathfrak{g}\mathfrak{l}_t)$| acts as |$T_1$|⁠. (c) There is a homomorphism |$ev:Y(\mathfrak{g}\mathfrak{l}_t)\rightarrow U(\mathfrak{g}\mathfrak{l}_t)$| given by |$T(u) \mapsto R(u)$|⁠. Proof. The general strategy is the following. Prove that we have some maps by giving an element-free construction for them (this guarantees that our maps are ultraproducts of finite rank maps), which generalizes the finite rank case and then apply Łoś’s theorem to prove that these maps satisfy the required properties. We will spell out the proof of |$a)$| in more detail to show how this strategy works. (a) First we define the map |$\Delta ^{\prime}:A \rightarrow A\otimes A$|⁠, using the collection of maps |$ \Delta ^{\prime}: V\otimes V^* \rightarrow A \otimes A$|⁠, which is equal to the sum of maps |$T_i^{I}: V \otimes V^* \to Im T^I_i \otimes 1$|⁠, |$(T_{i-j}^I\otimes T^{II}_j)\circ coev: V \otimes V^* \to Im T^I_{i-j} \otimes T^{II}_j$| for |$1 \le j \le i-1$| and |$T^{II}_i: V\otimes V^* \rightarrow 1 \otimes Im T^{II}_i$| (this is just the formula |$\Delta (T(u)) = T^{I}(u)T^{II}(u)$| written explicitly). Now since |$Y(\mathfrak{g}\mathfrak{l}_t)$| is the quotient of |$A$|⁠, we have a map |$A \rightarrow Y(\mathfrak{g}\mathfrak{l}_t) \otimes Y(\mathfrak{g}\mathfrak{l}_t)$|⁠. However, we know that for integer rank Yangians, this map factors through the relations |$Q_{i,j}$|⁠; hence, by Łoś’s theorem, it factors through them for rank |$t$| (because “factors through” means that |$\Delta ^{\prime}$| satisfies a collection of equations). Hence, we have a map |$\Delta $|⁠. The map |$S$| is constructed in the same way. We explicitly define a map |$A \rightarrow Y(\mathfrak{g}\mathfrak{l}_t)$| as in the finite rank case using |$T(u)^{-1}$| and then argue that it gives us a map for |$Y(\mathfrak{g}\mathfrak{l}_t)$| for the same reason. Finally, the fact that |$\Delta $| and |$S$| define a Hopf algebra structure on |$Y(\mathfrak{g}\mathfrak{l}_t)$| is equivalent to saying that these maps satisfy some equations. However, they satisfy these equations for integer rank; hence, by Łoś’s theorem, they satisfy them for rank |$t$|⁠. (b) We can consider a homomorphism |$T(V\otimes V^*) \rightarrow Y(\mathfrak{g}\mathfrak{l}_t)$| given by |$T_1:V \otimes V^* \to Im T_1$|⁠. By the same argument as before this map factors through |$U(\mathfrak{g}\mathfrak{l}_t)$|⁠. (c) The formula |$T(u) \mapsto R(u)$| gives as an algebra homomorphism from |$A$| to |$U(\mathfrak{g}\mathfrak{l}_t)$|⁠. By the same logic it extends to a map: |$Y(\mathfrak{g}\mathfrak{l}_t) \rightarrow U(\mathfrak{g}\mathfrak{l}_t)$|⁠. Now, we can also define the Yangian of the special linear Lie algebra in complex rank. We know that in the finite rank case, |$Y(\mathfrak{s}\mathfrak{l}_{t_n})$| is defined to be the subalgebra of the Yangian for |$\mathfrak{g}\mathfrak{l}_{t_n}$| invariant under automorphisms |$T(u) \mapsto f(u)T(u)$| for |$f \in 1 + u^{-1} \mathbb K[[u^{-1}]]$| (Definition 2.1.2). To mimic this definition in the rank |$t$| case, we need to understand how the ultraproduct of a collection of such automorphisms looks like: Lemma 4.1.3. Consider a collection of |$f^{(n)} \in 1 + u^{-1} \overline{\mathbb F}_{p_n}[[u^{-1}]]$|⁠. The ultraproduct of the automorphisms of the Yangians given by such a collection is the automorphism of |$Y(\mathfrak{g}\mathfrak{l}_t)$| given by some |$f \in 1 + u^{-1}\mathbb C[[u^{-1}]]$|⁠, denoted by |$r_f$|⁠. Proof. Let |$f^{(n)} = 1 + \sum f^{(n)}_i u^{-i}$|⁠. The automorphisms are given explicitly as |$T_i \mapsto T_i + f^{(n)}_1T_{i-1} + \dots + f^{(n)}_i$|⁠. Now the important fact, which we are going to use here, is the isomorphism |$\prod _{\mathcal F} \overline{\mathbb F}_{p_n} \simeq \mathbb C$|⁠. Since this isomorphism is fixed, it follows that the ultraproduct of a collection of maps as above with |$f^{(n)}_i \in \overline{\mathbb F}_{p_n}$| will give us a similar map with |$f_i \in \mathbb C$|⁠. Thus, we obtain |$f \in 1 + u^{-1}\mathbb C[[u^{-1}]]$| and automorphism |$r_f$| sending |$T(u)$| to |$f(u)T(u)$|⁠. Definition 4.1.4. The Yangian |$Y(\mathfrak{s}\mathfrak{l}_t)$| is defined as the subalgebra of |$Y(\mathfrak{g}\mathfrak{l}_t)$| of invariant “elements” under all automorphisms |$r_f$|⁠. Remark. Here, by “elements” of |$Y(\mathfrak{g}\mathfrak{l}_t)$|⁠, we mean the following. Suppose |$X$| is an object of a tensor category |$\mathcal C$|⁠. In addition, suppose that we have an action of group |$G$| on |$X$|⁠, i.e., a homomorphism |$G \to \textrm{Aut}(X)$|⁠. Then we can define |$X^G$| as an intersection |$\cap _{g\in G} \ker (1-g)$|⁠. So if |$G$| is a group of automorphisms |$r_f$|⁠, then we define |$Y(\mathfrak{s}\mathfrak{l}_t) = Y(\mathfrak{g}\mathfrak{l}_t)^G$|⁠. Now it is easy to see that |$Y(\mathfrak{s}\mathfrak{l}_t)$| defined in such a way is the ultraproduct of |$Y(\mathfrak{s}\mathfrak{l}_{t_n})$|⁠. Indeed, by Lemma 4.1.3, |$Y(\mathfrak{s}\mathfrak{l}_t)$| is the subalgebra in the ultraproduct of Yangians of invariants under all possible collections of automorphisms of |$Y(\mathfrak{g}\mathfrak{l}_{t_n})$| given by |$f^{(n)}$|⁠, hence the ultraproduct of |$Y(\mathfrak{s}\mathfrak{l}_{t_n})$|⁠. In addition, we have the following corollary: Corollary 4.1.5. (a) The subalgebra |$Y(\mathfrak{s}\mathfrak{l}_t)$| is a Hopf subalgebra of |$Y(\mathfrak{g}\mathfrak{l}_t)$|⁠. (b) The map |$i$| restricts to the map |$i:U(\mathfrak{s}\mathfrak{l}_t) \rightarrow Y(\mathfrak{s}\mathfrak{l}_t)$|⁠. (c) We have |$Y(\mathfrak{g}\mathfrak{l}_t) = Y(\mathfrak{s}\mathfrak{l}_t) \otimes \mathbb C[z_1,z_2,\dots ]$|⁠. There, the 2nd factor is the sum of trivial representations as an object of |$\textrm{Rep}(GL_t)$|⁠. Proof. (a)We know that this proposition holds for integer rank. Hence, we know that the image of invariant elements under the maps |$\Delta $| and |$S$| is invariant for integer rank; hence, it is invariant for complex rank by Łoś’s theorem. (b) This is obvious since |$r_f$| restricted to |$U(\mathfrak{g}\mathfrak{l}_t)$| is just the automorphism, which sends |$V\otimes V^*$| into itself via |$\textrm{Id} + f_1$|⁠. (Here, by |$f_1$|⁠, we mean |$f_1$| times the projector |$coev \circ ev$|⁠.) Hence, the invariants in |$U(\mathfrak{g}\mathfrak{l}_t)$| are exactly |$U(\mathfrak{s}\mathfrak{l}_t)$|⁠. (c) From Proposition 2.1.3(c) and discussion in the beginning of Section 3, it follows that for almost all |$n$|⁠, we have |$Y(\mathfrak{g}\mathfrak{l}_{t_n}) = Y(\mathfrak{s}\mathfrak{l}_{t_n})\otimes Z_{HC}(Y(\mathfrak{g}\mathfrak{l}_{t_n}))$|⁠, where |$Z_{HC}(Y(\mathfrak{g}\mathfrak{l}_t)) = \overline{\mathbb F}_{p_n}[z_1,z_2,\dots , ]$|⁠. This decomposition respects the action of |$\mathfrak{g}\mathfrak{l}_{t_n}$| on |$Y(\mathfrak{g}\mathfrak{l}_{t_n})$|⁠. So it follows that |$Y(\mathfrak{g}\mathfrak{l}_t) = \prod _{\mathcal F}Y(\mathfrak{g}\mathfrak{l}_{t_n}) = [\prod _{\mathcal F}Y(\mathfrak{s}\mathfrak{l}_{t_n})]\otimes [\prod _{\mathcal F} \overline{\mathbb F}_{p_n}[z_1,z_2,\dots ]] = Y(\mathfrak{s}\mathfrak{l}_t)\otimes \mathbb C[z_1,z_2,\dots ]$|⁠. Remark. Note that Corollary 4.1.5(c) gives us an alternative way to define |$Y(\mathfrak{s}\mathfrak{l}_{t})$|⁠, clarifying the concept of “invariant elements” mentioned before. 4.2 Finite-length representations of |$Y(\mathfrak{g}\mathfrak{l}_t)$|⁠, |$Y(\mathfrak{s}\mathfrak{l}_t)$| Here we again set |$t \in \mathbb C \backslash \mathbb Z$|⁠. First, we define the category of |$Y(\mathfrak{g}\mathfrak{l}_t)$|-modules, which we are interested in. Definition 4.2.1. Denote by |$\textrm{Rep}_0(Y(\mathfrak{g}\mathfrak{l}_t))$| the category with objects being objects |$M \in \textrm{Rep} (GL_t)$| together with an element |$\mu _M \in \textrm{Hom}(Y(\mathfrak{g}\mathfrak{l}_t)\otimes M,M)$| such that (a)|$M$| is a representation of |$Y(\mathfrak{g}\mathfrak{l}_t)$|⁠, i.e., |$\mu _M \circ (1 \otimes \mu _M) = \mu _M \circ (\mu \otimes 1)$| as elements of |$\textrm{Hom}(Y(\mathfrak{g}\mathfrak{l}_t)\otimes Y(\mathfrak{g}\mathfrak{l}_t)\otimes M,M)$|⁠, where |$\mu $| is the product map of the Yangian. (b) The map |$\mu _M \circ (i \otimes 1)$| gives the standard structure of a |$\mathfrak{g}\mathfrak{l}_t$| representation on |$M$|⁠. The morphisms in |$\textrm{Rep}_0(Y(\mathfrak{g}\mathfrak{l}_t))$| are morphisms of |$\textrm{Rep}(GL_t)$|⁠, which commute with the representation structure. In addition, we will denote by |$\textrm{Rep}_0^{\prime}(Y(\mathfrak{g}\mathfrak{l} _t))$| the category defined in the similar way, but we require only the |$\mathfrak{s}\mathfrak{l}_t$|-action to be standard in |$(b)$|⁠. Note that we consider only honest objects of |$\textrm{Rep}(GL_t)$| and not ind-objects. Similarly, one can define the category |$\textrm{Rep}_0(Y(\mathfrak{s}\mathfrak{l}_t))$|(using Corollary 4.1.5). We have the following result connecting these categories to the categories of finite-dimensional representations of finite rank Yangians over |$\overline{\mathbb F}_{p_n}$| denoted by |$\textrm{Rep}_0(Y(\mathfrak{g}\mathfrak{l}_{t_n}),p_n)$| and |$\textrm{Rep}_0(Y(\mathfrak{s}\mathfrak{l}_{t_n}),p_n)$|⁠: Lemma 4.2.2. The category |$\textrm{Rep}_0(Y(\mathfrak{g}\mathfrak{l}_t))$| is the full subcategory of |$\prod _{\mathcal{F}} \textrm{Rep}_0(Y(\mathfrak{g}\mathfrak{l}_{t_n}),p_n)$| given by the objects whose image in |$\prod _{\mathcal{F}} \textbf{Rep}_{p_n}(GL_{t_n})$| is contained in |$\textrm{Rep}(GL_t)$|⁠. Moreover, the irreducible representations of |$Y(\mathfrak{g}\mathfrak{l}_t)$| correspond to collections of representations of |$Y(\mathfrak{g}\mathfrak{l}_{t_n})$|⁠, such that almost all of them are irreducible. So |$\textrm{Irr}(\textrm{Rep}_0(Y(\mathfrak{g}\mathfrak{l}_t)) = \prod _{\mathcal F} \textrm{Irr}(\textrm{Rep}_0(Y(\mathfrak{g}\mathfrak{l}_{t_n}),p_n))$|⁠. Proof. Take |$M \in \textrm{Rep}_0(Y(\mathfrak{g}\mathfrak{l}_t))$|⁠. As we know, |$\textrm{Rep}(GL_t)$| is a full subcategory of the category |$\prod _{\mathcal F} \textrm{Rep}_{p_n}(GL_{t_n})$|⁠. Hence, because |$M$| is an object of |$\textrm{Rep}(GL_t)$|⁠, it follows that we have a corresponding collection of objects |$M_n \in \textrm{Rep}_{p_n}(GL_{t_n})$|⁠. Now, since we have a map |$\mu _M \in \textrm{Hom}(Y(\mathfrak{g}\mathfrak{l}_t)\otimes M,M)$|⁠, it follows that we have a collection of maps |$\mu _{M_n} \in \textrm{Hom}(Y(\mathfrak{g}\mathfrak{l}_{t_n})\otimes M_n,M_n)$|⁠, and, by Łoś’s theorem, it follows that almost all |$\mu _{M_n}$| satisfy the equation |$\mu _{M_n} \circ (1 \otimes \mu _{M_n}) = \mu _{M_n} \circ (\mu \otimes 1)$| giving a structure of a |$Y(\mathfrak{g}\mathfrak{l}_{t_n})$|-module to |$M_n$|⁠. In addition, since |$\mu _M \circ (i \otimes 1)$| gives a standard structure of a |$\mathfrak{g}\mathfrak{l}_t$|-module to |$M$|⁠, by Łoś’s theorem, almost all |$M_n$| have a standard structure of a |$\mathfrak{g}\mathfrak{l}_{t_n}$|-module from |$\mu _{M_n} \circ (i \otimes 1)$|⁠. So objects of |$\textrm{Rep}_0(Y(\mathfrak{g}\mathfrak{l}_t))$| indeed correspond to some objects of |$\prod _{\mathcal F} \textrm{Rep}_0(Y(\mathfrak{g}\mathfrak{l}_{t_n}))$|⁠. Now to show that this is indeed a full subcategory, we need to look at morphisms. Since |$\textrm{Rep}(GL_t)$| is a full subcategory in |$\prod _{\mathcal F}\textrm{Rep}_{p_n}(GL_{t_n})$|⁠, it follows that each morphism in |$\textrm{Rep}_0(Y(\mathfrak{g}\mathfrak{l}_t))$| gives us a unique sequence of morphisms in |$\textrm{Rep}_{p_n}(GL_{t_n})$|⁠, which commute with the |$Y(\mathfrak{g}\mathfrak{l}_{t_n})$|-action for almost all |$n$| by Łoś’s theorem. Hence, this is indeed a full subcategory. Now, in the other direction, if we have an object of the intersection of |$\prod _{\mathcal F} \textrm{Rep}_0(Y(\mathfrak{g}\mathfrak{l}_{t_n}))$| with |$\textrm{Rep} (GL_t)$|⁠, i.e., a sequence of |$M_n$| with a structure of |$Y(\mathfrak{g}\mathfrak{l}_{t_n})$|-modules such that (s.t.) |$M=\prod _{\mathcal F}M_n$| lies in |$\textrm{Rep}(GL_t)$|⁠, it follows by Łoś’s theorem that |$M$| has a required structure of a |$Y(\mathfrak{g}\mathfrak{l}_t)$|-representation. Let us prove the 2nd statement. Suppose that |$M$| is reducible, then we have a non-zero injective morphism from |$N \ne M$| to |$M$|⁠. Thus, we have a sequence of |$N_n$| and a sequence of maps |$N_n \rightarrow M_n$| such that for almost all |$n$| these maps are |$Y(\mathfrak{g}\mathfrak{l}_{t_n})$|-module maps, |$N_n \ne M_n$| and the maps are injective and non-zero. Hence, almost all |$M_n$| are reducible. Moreover, vice versa, if almost all |$M_n$| are reducible, we have a collection of |$Y(\mathfrak{g}\mathfrak{l}_{t_n})$|-modules |$N_n$| and a collection of injective maps |$N_n \rightarrow M_n$|⁠. Since |$N_n$| is a subobject of |$M_n$| as objects of |$\textrm{Rep}_{p_n}(GL_{t_n})$|⁠, it follows that |$N = \prod _{\mathcal F}N_n$| lies in |$\textrm{Rep}(GL_t)$|⁠. In addition, by the above, it lies in |$\textrm{Rep}_0(Y(\mathfrak{g}\mathfrak{l}_t))$|⁠. Finally, by Łoś’s theorem, the resulting map |$N \rightarrow M$| is injective non-zero and |$N\ne M$|⁠; hence, |$M$| is reducible. The same lemma can be stated for |$Y(\mathfrak{s}\mathfrak{l}_t)$|⁠. Lemma 4.2.3. The category |$\textrm{Rep}_0(Y(\mathfrak{s}\mathfrak{l}_t))$| is a full subcategory of |$\prod _{\mathcal F} \textrm{Rep}_0(Y(\mathfrak{s}\mathfrak{l}_{t_n}))$| given by the objects whose image in |$\prod _{\mathcal F} \textbf{Rep}_{p_n}(GL_{t_n})$| is contained in |$\textrm{Rep}(GL_t)$|⁠. Moreover, the irreducible representations of |$Y(\mathfrak{s}\mathfrak{l}_t)$| correspond to collections of representations of |$Y(\mathfrak{s}\mathfrak{l}_{t_n})$|⁠, such that almost all of them are irreducible. Proof. The proof is exactly the same. Note that above prove holds in a more general situation. Corollary 4.2.4. Suppose that |$\mathcal A_n$| is a series of (ind-)algebras in |$\textbf{Rep}_{p_n}(GL_{t_n})$| and |$\textbf{Rep}(\mathcal A_n,p_n)$| are the categories of finite-dimensional representations of |$\mathcal A_n$| in |$\textbf{Rep}_{p_n}(GL_{t_n})$|⁠. Suppose also that |$\mathcal A = \prod _{\mathcal F}\mathcal A_n$| is an (ind-)algebra in |$\textrm{Rep}(GL_t)$| and |$\textrm{Rep}(\mathcal A)$| is the category of representations of |$\textrm{Rep}(GL_t)$| in the sense of Definition 4.2.1 but without |$(c)$|⁠. Then |$\textrm{Rep}(\mathcal A)$| is the full subcategory of |$\prod _{\mathcal F} \textbf{Rep}(\mathcal A_n,p_n)$| given by objects whose image in |$\prod _{\mathcal F}\textbf{Rep}_{p_n}(GL_{t_n})$| is contained in |$\textrm{Rep}(GL_t)$|⁠. Moreover, |$Irr(\textrm{Rep}(\mathcal A)) = \prod _{\mathcal F} Irr(\textbf{Rep}(\mathcal A_n, p_n))$|⁠. Recall the classification theorem of finite-dimensional irreducible |$Y(\mathfrak{s}\mathfrak{l}_n)$|-modules. The important step in the proof of this classification is to show that all such modules are subquotients in the tensor product of evaluation modules (we will use |$\sqsubset $| to denote subquotients). We want to prove the same thing in our case. To do this, first, we define evaluation modules. By |$\mathbb C(c)$|⁠, for a complex number |$c$|⁠, we will denote a representation of |$\mathfrak{g}\mathfrak{l}_t$| where it acts through its projection to |$\mathbb C$| and then by multiplication by |$c$|⁠. Definition 4.2.5. For a fixed bipartition |$\lambda $| and complex number |$c$| denote by |$L(\lambda +c)$|⁠, an object |$\textrm{Rep}^{\prime}_0(Y(\mathfrak{g}\mathfrak{l}_t))$|⁠, which is equal to |$V(\lambda )\otimes \mathbb C(c)$| as an object of |$\textrm{Rep}(GL_t)$|⁠, and the |$Y(\mathfrak{g}\mathfrak{l}_t)$|-module structure is obtained from |$\mathfrak{g}\mathfrak{l}_t$|-module structure by |$ev: Y(\mathfrak{g}\mathfrak{l}_t) \rightarrow U(\mathfrak{g}\mathfrak{l}_t)$|(see Proposition 4.1.2 (c)). We will call such modules evaluation modules. Since |$Y(\mathfrak{g}\mathfrak{l}_t)$| is a Hopf algebra, we can freely take tensor products of such modules. In addition, the structure of a |$Y(\mathfrak{g}\mathfrak{l}_t)$| representation gives them a structure of a |$Y(\mathfrak{s}\mathfrak{l}_t)$| representation as well. We can now generalize the subquotient statement for complex rank. We will need the following lemma: Lemma 4.2.6. Fix |$c \in \mathbb Z$|⁠, then in the case of algebraic |$t$| for almost all |$n$|⁠, we have |$p_n-t_n> c$|⁠. Proof. Indeed, suppose |$p_n-t_n < c$| for almost all |$n$|⁠. Then since there is a finite number of possibilities |$p_n-t_n=d$| for some integer |$d$|⁠. This means that |$q(t_n) = q(-d) \ mod \ p_n$|⁠, so |$q(-d)$| has an infinite number of prime divisors, which is absurd. Proposition 4.2.7. If |$M \in \textrm{Rep}_0(Y(\mathfrak{s}\mathfrak{l}_t))$| is irreducible, then there exist nonempty bipartitions |$\eta _1, \dots , \eta _k$| and complex numbers |$c_1,\dots ,c_k$| such that M is a subquotient of |$L(\eta _1+c_1)\otimes \dots \otimes L(\eta _k+c_k)$|⁠. Proof. Consider the collection of |$Y(\mathfrak{s}\mathfrak{l}_{t_n})$|-modules |$M_n$| corresponding to |$M$|⁠. As we know, for almost all |$n$| these modules are irreducible. |$M$| as an object of |$\textrm{Rep}(GL_t)$| is equal to a finite sum of |$V(\mu _j)$|—the simple objects corresponding to bipartitions |$\mu _j$|⁠. Thus, for almost all |$n$|⁠, we have |$M_n = \oplus _{j=1}^{K} V((\mu _j)|_{t_n},p)$|⁠. Since for big enough |$n$| we have |$((\mu _j)|_{t_n})_1 - ((\mu _j)|_{t_n})_{t_n} = (\mu _j^{\bullet })_1 + (\mu _j^\circ )_1$|⁠, it follows that this parameter (⁠|$((\mu _j)|_{t_n})_1 - ((\mu _j)|_{t_n})_{t_n}$|⁠) does not depend on |$n$|⁠. Hence, by Lemma 4.2.6, it follows that for almost all |$n$|⁠, |$M_n$| satisfies the condition of Proposition 3.2.1, and hence, by Cor. 3.2.7, we deduce that |$M_n$| is a subquotient of evaluation modules |$M_n \sqsubset L(\lambda ^{(n)}_1) \otimes \dots \otimes L(\lambda ^{(n)}_{k_n})$|⁠, where |$\lambda ^{(n)}_i$| is a dominant weight. Note that this means that the unique highest weight vector of |$M_n$| with respect to the |$\mathfrak{s}\mathfrak{l}_n$| action (up to scaling) has weight |$\lambda ^{(n)}_1 + \dots + \lambda ^{(n)}_{k_n}$|⁠. Fix |$L = \max (l(\mu _j))$|⁠. From now on, we will consider only |$t_n>|L|$|⁠, which is okay since all cofinite sets belong to |$\mathcal F$|⁠. So for almost all |$n$|⁠, |$M_n$| as a representation of |$\mathfrak{s}\mathfrak{l}_n$| is equal to |$\bigoplus V(\mu _j|_{t_n})$|⁠, where |$\mu _j|_{t_n}$| is a weight equal to |$(\mu _j|_{t_n})_i = (\mu _j^{\bullet })_i - (\mu _j^\circ )_{n+1-i}$|⁠. Since |$M_n$| must have a unique highest weight vector, it follows that its weight is one of the |$\mu _j|_{t_n}$|⁠. But if |$\mu _j|_{t_n}$| is the unique highest weight for some |$t_n>|L|$|⁠, then it is the unique highest weight for all such |$n$|⁠, since this property depends only on the regions where |$\mu _j|_{t_n}$| is non-zero, which do not change. So denote by |$\mu $| the bipartition corresponding to this unique highest weight. We have |$\mu |_{t_n} = \lambda ^{(n)}_1 + \dots + \lambda ^{(n)}_{k_n}$|⁠. For the next step, note that each |$\lambda ^{(n)}_i$| can be represented as |$\chi ^{(n)}_i + d^{(n)}_i$|⁠, where |$\chi ^{(n)}$| is a partition with |$\chi ^{(n)}_n =0$|⁠. So we have |$\mu |_{t_n} - \sum d_i^{(n)}= \eta ^{(n)}_1+\dots + \eta ^{(n)}_{k_n}$|⁠, where both the left and the right parts are partitions with |$n$|-th term being zero. Here without loss of generality (wlog) we assume that |$\eta _i \ne 0$|⁠, since if it happens to be so, the corresponding evaluation module is trivial as a |$Y(\mathfrak{s}\mathfrak{l}_n)$|-module and hence we can ignore it. The left part is a partition with rows of lengths equal to lengths of the rows of |$\mu ^{\bullet }$| and |$t_n-a_j$|⁠, where |$a_j$| are lengths of the rows of |$\mu ^{\circ }$|⁠. Now each |$\eta ^{(n)}_{k_n}$| must be equal to the sum of several rows of this partition; hence, overall there is a finite number of ways to choose |$\eta ^{(n)}_j$|⁠. Thus, for almost all |$n$|⁠, |$\eta ^{(n)}_j$| correspond to the same rows of |$\mu $|⁠. This means that for almost all |$n$|⁠, we have |$\eta ^{(n)}_j= \eta _j|_{t_n} - (\eta _j|_{t_n})_n$| for a fixed bipartition |$\eta _j$| corresponding to the same collection of rows of |$\mu $|⁠. Hence, for almost all |$n$|⁠, we have \begin{equation*} M_n \sqsubset L(\eta_1|_{t_n}+c_1^{(n)}) \otimes \dots \otimes L(\eta_k|_{t_n} + c_k^{(n)}), \end{equation*} for some other |$c^{(n)}_i \in \overline{\mathbb F}_{p_n}$|⁠. But the ultraproduct of |$\overline{\mathbb F}_{p_n}(c_i^{(n)})$| is equal to |$\prod _{\mathcal F}\overline{\mathbb F}_{p_n}(c_i^{(n)}) = \mathbb C(c_i)$|⁠, for some |$c_i \in \mathbb C$|⁠, and we can obtain |$\mathbb C(c_i)$| with any |$c_i$| in this way. Thus from the ultraproduct construction we have \begin{equation} M \sqsubset L(\eta_1 + c_1) \otimes \dots \otimes L(\eta_k+c_k). \end{equation} (3) Note that since |$\eta _1 + \dots + \eta _k = \mu $|⁠, it follows that |$\sum c_k = 0$| and hence the action of |$\mathfrak{g}\mathfrak{l}_t$| on the tensor product is standard. Corollary 4.2.8. An irreducible module |$M$| is completely determined by the collection of bipartitions |$\lambda _i$| and complex numbers |$c_i$|⁠. Proof. Indeed, if two irreducible modules through the construction above are subquotients of the same tensor product of evaluation modules, it means that the corresponding evaluation modules for finite rank Yangians are isomorphic for almost all |$n$|⁠; hence, the modules themselves are isomorphic. Now we can generalize the parametrization of irreducible modules by the highest weight: Definition 4.2.9. Consider the realization of |$M$| as a subquotient of a tensor product of evaluation modules defined in (3). Then the highest weight of |$M$| is a pair of sequences of elements of |$\mathbb C[[u^{-1}]]$| equals to \begin{equation*} \lambda^{\bullet}_i(u) = \prod_k(1 +[(\eta^{\bullet}_k)_i+c_k]u^{-1}) \textrm{and} \lambda^\circ_i(u) = \prod_k(1 +[(\eta^\circ_k)_i - c_k]u^{-1}) \end{equation*} and an element |$\lambda ^m = \prod _k(1 +c_ku^{-1})$|⁠, defined up to a simultaneous multiplication with any |$f \in \mathbb C[[u^{-1}]]$|⁠. In addition, denote |$l(\lambda ) = \max \ l(\eta _i)$|⁠, |$l(\lambda ^{\bullet }) = \max \ l(\eta ^{\bullet }_i)$|⁠, |$l(\lambda ^\circ ) = \max \ l(\eta ^\circ _i)$|⁠. Remark. Since there is a finite number of |$\eta _k$|⁠, it follows that sequences |$\lambda ^{\bullet }(u)$| and |$\lambda ^\circ (u)$| stabilize at infinity for any |$M$|⁠. In addition, note that for |$i> l(\lambda ^{\bullet })$|⁠, it holds that |$\lambda ^{\bullet }_i = \lambda ^m$| and that for |$i> l(\lambda ^\circ )$| it holds that |$\lambda _i^\circ (u)= \lambda ^m(-u)$|⁠. This is well defined since if two pairs |$\lambda $| and |$\mu $| are the highest weights of the same module, their restriction to finite rank with |$n> |\lambda |$| equals to |$(\lambda |_{t_n})_i = \lambda ^{\bullet }_i(u)$| for |$i \le |\lambda ^{\bullet }|$|⁠, |$(\lambda |_{t_n})_{n-i+1} = \lambda ^\circ _i(-u)$| for |$i \le |\lambda ^\circ |$|⁠, and |$(\lambda |_{t_n})_i = \lambda ^m(u)$| for the rest, where the coefficients are also reduced to |$\overline{\mathbb F}_{p_n}$| through a fixed isomorphism of ultraproducts, are the same up to multiplication by |$f$| for almost all |$n$|⁠, hence the same up to multiplication by the ultraproduct of |$f$|’s for complex rank. We can also generalize the classification of irreducible modules by collections of Drinfeld polynomials. To do this, we will need the following lemma: Lemma 4.2.10. Suppose |$\lambda _1 = \prod _i (u+l^{(1)}_i + c_i)$| and |$\lambda _2 = \prod _i(u+l^{(2)}_i + c_i)$|⁠, where |$l_i$| are integers and |$c_i \in \overline{\mathbb F_p}$|⁠, satisfy |$\frac{\lambda _1(u)}{\lambda _2(u)} = \frac{P(u+1)}{P(u)}$| for some monic |$P(u)$|⁠. Suppose also that |$p> \deg P(u) + 1$|⁠. Then for |$\mu _1(u) = \prod _i(u+l^{(1)}_i + d_i)$| and |$\mu _2(u) = \prod _i(u+l^{(2)}_i+d_i)$|⁠, with arbitrary |$d_i\in \overline{\mathbb F_p}$|⁠, there exists a monic polynomial |$Q(u)$| of the same degree as |$P$| s.t. |$\frac{\mu _1(u)}{\mu _2(u)} = \frac{Q(u+1)}{Q(u)}$|⁠. Proof. Let’s group the roots of |$P$| into a group of “strings” in the following way. Consider the divisor |$D$| of zeroes of |$P$|⁠. Pick a point |$x_0$|⁠. Next find a point |$x$| that lies in the same class in |$\overline{\mathbb F}_p/\mathbb F_p$|⁠, such that it lies in |$D$|⁠, but |$x-1$| does not lie in |$D$| (this is always possible since the number of zeroes is less than |$x-1$|⁠). Now denote by |$s_1$| the set |$\{x,x+1,x+2,\dots ,x+\rho \}$| for maximal |$\rho $| such that |$s_1 \subset D$|⁠. Now take |$D - s_1$| and repeat. Then |$D = \bigcup _i s_i$|⁠. It follows that if we denote by |$r_i$| the minimal element in |$s_i$| and by |$t_i$| the maximal element, then |$\frac{P(u+1)}{P(u)} = \prod \frac{u-r_i+1}{u-t_i}$|⁠. Now if we group |$c_i$| in groups by their image in |$\mathbb C/\mathbb Z$|⁠, it follows that after renumbering inside the groups (which changes nothing), we can assume that |$l^{(1)}_i+c_i = -s_i+1$| and |$l^{(2)}_i+c_i = -t_i$|⁠. Now if we move to |$\mu $|⁠, it follows that we can just consider |$Q$| defined by the divisor |$D_i^{\prime} = \bigcup (s_i -c_i +d_i)$|⁠, where if |$s_i = \{ x_i, x_i + 1, \dots , x_i + \rho _i\}$|⁠, then |$s_i -c_i+d_i = \{x_i-c_i+d_i, \dots , x_i-c_i+d_i+\rho _i\}$|⁠. We are ready to prove the main classification theorem: Theorem 4.2.11. For every irreducible |$M\in \textrm{Rep}_0(Y(\mathfrak{s}\mathfrak{l}_t))$|⁠, there exists a pair of sequences of monic polynomials in |$\mathbb C[u]$|⁠, denoted by |$P(u) =(P^{\bullet }_i(u), P^\circ _i(u))$|⁠, such that the corresponding highest weight satisfies \begin{equation} \frac{\lambda^{\bullet}_i(u)}{\lambda^{\bullet}_{i+1}(u)} = \frac{P^{\bullet}_i(u+1)}{P^{\bullet}_i(u)} \, \ \ \frac{\lambda^\circ_i(-u)}{\lambda_{i-1}^\circ(-u)}=\frac{P^\circ_{i-1}(u+1)}{P^\circ_{i-1}(u)} \, \end{equation} (4) and both sequences of polynomials stabilize and equal to |$1$| for sufficiently large |$i$|⁠. Moreover, for any such |$P(u)$|⁠, there is |$M$| with the highest weight satisfying (4). This also gives a |$1-1$| correspondence between irreducible modules and sequences of polynomials |$P(u)$|⁠. Definition 4.2.12. We will call the polynomials belonging to the sequence |$P(u)$| as above Drinfeld polynomials. Proof. For the 1st part, consider the corresponding sequence of |$M_n \in \textrm{Rep}_0(Y(\mathfrak{s}\mathfrak{l}_n))$| almost all of which are irreducible with the highest weight equals to |$\lambda |_{t_n}(u)$|⁠, where |$\lambda (u)$| is the highest weight of |$M$|⁠. Now, for almost all |$n$|⁠, we have |$\frac{(\lambda |_{t_n})_i(u)}{(\lambda |_{t_n})_{i+1}(u)} = \frac{P^{(n)}_i(u+1)}{P^{(n)}_i(u)}$|⁠. For |$n$| big enough (⁠|$p_n$| bigger than |$l(\lambda )$|⁠) this means |$\frac{(\lambda ^{\bullet }_i)|_{t_n}(u)}{(\lambda ^{\bullet }_{i+1})|_{t_n}(u)} = \frac{P^{(n)}_i(u+1)}{P^{(n)}_i(u)}$| and |$\frac{(\lambda ^\circ _{i+1})|_{t_n}(-u)}{(\lambda ^\circ _{i})|_{t_n}(-u)} = \frac{P^{(n)}_{n-i}(u+1)}{P^{(n)}_{n-i}(u)}$|⁠, for some |$P^{(n)}_i \in \overline{\mathbb F}_{p_n}[u]$|⁠. In addition, note that for big enough |$n$| for |$i \ge |\lambda ^{\bullet }|$| and |$i \le n - 1-|\lambda ^\circ |$|⁠, we have |$P^{(n)}_i=1$|⁠. From Lemma 4.2.10, it follows that all |$P^{(n)}_i$| have the same degree not depending on |$n$| for almost all |$n$|⁠, since |$\lambda |_{t_n}$| satisfy the condition. Introduce the notation |$P^{\bullet }_i = \prod _{\mathcal F} P^{(n)}_i$|⁠, which are well defined since the degree is constant. In addition, set |$P^\circ _i = \prod _{\mathcal F} P^{(n)}_{n-i}$|⁠. Now, by Łoś’s theorem, it follows that these ultraproducts satisfy the required equations, since |$\lambda ^{\bullet }_i $| and |$\lambda ^\circ _i$| are also ultraproducts. Moreover, the equation holds for almost all |$n$|⁠. Fix a pair |$P(u)$|⁠. Denote by |$n_{\bullet }$| the number of polynomials in |$P^{\bullet }$| and by |$n_{\circ }$| the same for |$P^\circ $|⁠. To prove the converse statement, consider the highest weight: \begin{align} & \mu_i^{\bullet}(u) = u^{-k}P^{\bullet}_1(u) \dots P^{\bullet}_{i-1}(u)P^{\bullet}_{i}(u+1) \dots P^{\bullet}_{n_{\bullet}}(u+1)P^\circ_{n_\circ}(u+1) \dots P^\circ_1(u+1) \end{align} (5) \begin{align} & \mu^m (u)= u^{-k}P^{\bullet}_1(u)P^{\bullet}_{n_{\bullet}}(u)P^\circ_{n_\circ}(u+1) \dots P^\circ_1(u+1) \end{align} (6) \begin{align} & \mu_i^\circ(-u) = u^{-k}P^{\bullet}_1(u)P^{\bullet}_{n_{\bullet}}(u)P^\circ_{n_\circ}(u) \dots P^\circ_{i}(u)P^\circ_{i-1}(u+1) \dots \dots P^\circ_1(u+1). \end{align} (7) This highest weight obviously satisfies (4). In addition, the corresponding irreducible finite rank modules are finite dimensional, and since as |$\mathfrak{s}\mathfrak{l}_{t_n}$|-modules, they are subquotients of a fixed tensor product of simple modules, it follows that there is a finite number of possibilities for their structure as an |$\mathfrak{s}\mathfrak{l}_{t_n}$|-module, so for almost all |$n$|⁠, they are the same and the ultraproduct of these modules is well defined. Hence, we have an irreducible |$M \in \textrm{Rep}_0(Y(\mathfrak{s}\mathfrak{l}_t)) $| with this highest weight. (Here we also use Lemma 4.2.6 and Theorem 3.2.3.) Let us prove the last statement. Since the highest weight of |$M$| is unique up to multiplication by |$f$|⁠, it follows that the ratios in (4) are determined uniquely for a fixed |$M$|⁠; hence, the Drinfeld polynomials are determined uniquely. Indeed, this is obvious for transcendental |$t$|⁠, and for algebraic |$t$|⁠, we just need to take the Polynomials, which are not divisible by any |$q_{p_n}(u+c)$| for almost all |$n$|⁠, since any other choice will not lead us to a well-defined ultraproduct (the degrees would increase to infinity). If two irreducible modules |$M$| and |$N$| have the same Drinfeld polynomials, it means that the corresponding |$M_n$| and |$N_n$| have the same Drinfeld polynomials and hence are isomorphic for almost all |$n$|⁠, thus |$M\simeq N$|⁠. To finish, let’s discuss the classification of |$Y(\mathfrak{g}\mathfrak{l}_t)$|-modules. Since we already know that |$Y(\mathfrak{g}\mathfrak{l}_t) = Y(\mathfrak{s}\mathfrak{l}_t)\otimes \mathbb C[z_1,z_2,\dots ]$| by Corollary 4.1.5(c), it follows that to fix an irreducible representation of |$Y(\mathfrak{g}\mathfrak{l}_t)$|⁠, it is enough to fix an irreducible representation of |$Y(\mathfrak{s}\mathfrak{l}_t)$| and a series of complex numbers. Now two such representations corresponding to the same irreducible representations of |$Y(\mathfrak{s}\mathfrak{l}_t)$| differ by multiplication by a one-dimensional representation of |$Y(\mathfrak{g}\mathfrak{l}_t)$|⁠. The only difference is that |$f(u)$| defining the structure of |$Y(\mathfrak{g}\mathfrak{l}_t)$| representation on |$\mathbb C$| should be of the form |$f(u) = 1 + u^{-2}\mathbb C[u]$|⁠, since we want the action of |$\mathfrak{g}\mathfrak{l}_t$| on our modules to be standard. So we have the following corollary: Corollary 4.2.13. Irreducible objects of |$\textrm{Rep}_0(Y(\mathfrak{g}\mathfrak{l}_t))$| are in |$1-1$| correspondence with tuples |$(P(u),f(u))$|⁠, where |$P(u)$| is a pair of sequences of Drinfeld polynomials and |$f(u) \in 1 + u^{-2}\mathbb C[[u^{-1}]]$|⁠. Remark. If we are interested in irreducible objects of |$\textrm{Rep}_0^{\prime}(Y(\mathfrak{g}\mathfrak{l}_t))$|⁠, we should drop the requirement of linear term in |$f(u)$| to be zero and let |$f(u)$| run over |$1 + u^{-1}\mathbb C[[u^{-1}]]$|⁠. Remark. In a similar way, one can define the Yangians and twisted Yangians of other classical Lie algebras (for definitions of Yangians for over classical algebras see [3], for definition if twisted Yangians see [23]) in complex rank(see [14, Section 7]). One can also extend in a similar manner the classification theorems of irreducible finite-dimensional representations of these algebras to complex rank. The new |$RTT$|-presentation for Yangians |$Y(\mathfrak{s}\mathfrak{p}_n)$| and |$Y(\mathfrak{s}\mathfrak{o}_n)$| appearing in [15] might be very useful for this purpose. Funding This work was supported by the Russian Science Foundation [16-12-10151]. 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