Tunneling topological vacua via extended operators: (Spin-)TQFT spectra and boundary deconfinement in various dimensions

Tunneling topological vacua via extended operators: (Spin-)TQFT spectra and boundary... Prog. Theor. Exp. Phys. 2018, 053A01 (54 pages) DOI: 10.1093/ptep/pty051 Tunneling topological vacua via extended operators: (Spin-)TQFT spectra and boundary deconfinement in various dimensions 1,2,∗ 1 1 3 4 Juven Wang , Kantaro Ohmori , Pavel Putrov , Yunqin Zheng , Zheyan Wan , 5 2,5,6,7 5 2,5,6,7 Meng Guo , Hai Lin , Peng Gao , and Shing-Tung Yau School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, USA Department of Physics, Princeton University, Princeton, NJ 08540, USA School of Mathematical Sciences, USTC, Hefei 230026, China Department of Mathematics, Harvard University, Cambridge, MA 02138, USA Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China Department of Physics, Harvard University, Cambridge, MA 02138, USA E-mail: juven@ias.edu Received January 19, 2018; Revised April 3, 2018; Accepted April 4, 2018; Published May 30, 2018 ................................................................................................................... Distinct quantum vacua of topologically ordered states can be tunneled into each other via extended operators. The possible applications include condensed matter and quantum cosmology. We present a straightforward approach to calculate the partition function on various manifolds and ground state degeneracy (GSD), mainly based on continuum/cochain topological quantum field theories (TQFTs), in any dimension. This information can be related to the counting of extended operators of bosonic/fermionic TQFTs. On the lattice scale, anyonic particles/strings live at the ends of line/surface operators. Certain systems in different dimensions are related to each other through dimensional reduction schemes, analogous to (de)categorification. Examples include spin TQFTs derived from gauging the interacting fermionic symmetry-protected topological 2 2 states (with fermion parity Z ) of symmetry groups Z ×Z and (Z ) in 3+1D, also Z and (Z ) 4 2 4 2 2 in 2+1D. Gauging the last three cases begets non-Abelian spin TQFTs (fermionic topological order). We consider situations where a TQFT lives on (1) a closed spacetime or (2) a spacetime with a boundary, such that the bulk and boundary are fully gapped and short- or long-range entangled (SRE/LRE). Anyonic excitations can be deconfined on the boundary. We introduce new exotic topological interfaces on which neither particle nor string excitations alone condense, but only fuzzy-composite objects of extended operators can end (e.g., a string-like composite object formed by a set of particles can end on a special 2+1D boundary of 3+1D bulk). We explore the relations between group extension constructions and partially breaking constructions (e.g., 0- form/higher-form/“composite” breaking) of topological boundaries, after gauging. We comment on the implications of entanglement entropy for some such LRE systems. ................................................................................................................... Subject Index A13, A63, B00, B37, I46 1. Introduction and summary Many-body quantum systems can possess entanglement structures—the entanglement between either neighboring or long-distance quantum degrees of freedom, the properties of which have been pon- dered by many physicists since Einstein–Podolsky–Rosen’s work [1]. Roughly speaking, there can be short-range or long-range entanglements (see the recent review in Ref. [2]). Within the concept of the locality in space (or spacetime) and the short-distance cutoff lattice regularization, the short- range entangled (SRE) state can be deformed to a trivial product state (a trivial vacuum) through © The Author(s) 2018. Published by Oxford University Press on behalf of the Physical Society of Japan. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. local unitary transformations on local sites by series of local quantum circuits. The long-range entanglement (LRE) is, however, much richer. Long-range entangled states cannot be deformed to a trivial gapped vacuum through local unitary transformations on local sites by series of local quantum circuits. Some important signatures of long-range entanglements contain a subset or the full set of the following: 1. Fractionalized excitations and fractionalized quantum statistics: Anyonic particles in 2+1D (see Refs. [3–7] and references therein) and anyonic strings in 3+1D (see Refs. [8–10], as well as Ref. [7] and references therein). 2. Topological degeneracy: In d + 1D spacetime dimensions, the number of (approximate) degen- d d d−1 erate ground states on a closed space M or an open space M with a boundary  (denoted d−1 d as  = ∂M ) can depend on the spatial topology. This is the so-called topological ground state degeneracy (GSD) of zero energy modes. Although in general, for the quantum many- body system, both the gapless and gapped system can have topological degeneracy, it is easier to extract that for the gapped system. The low-energy sector of the gapped system can be approximated by a topological quantum field theory (TQFT) [11] (see further discussion in Ref. [7]), and one can compute the GSD from the partition function Z of the TQFT as d 1 Z (M × S ) = dim H d ≡ GSD, (1.1) 1 2 where S is a compact time circle. 3. Emergent gauge structure: Gauge theory (see Refs. [13,14] and references therein). Such long-range entangled states are usually termed as intrinsic topological orders [15]. The three particular signatures outlined above are actually closely related. For example, the first two signatures must require LRE topological orders (see, e.g., Ref. [16]). Other more detailed phenomena were recently reviewed in Ref. [2]. d 1 In this work, we plan to systematically compute the path integral Z, namely GSD = Z (M × S ) for various TQFTs in diverse dimensions. These GSD computations have merits and applications to distinguish the underlying LRE topological phases in condensed matter systems, including quantum Hall states [17] and quantum spin liquids [18]. On the other hand, these GSDs are quantized numbers d 1 obtained by putting a TQFT on a spacetime manifold M × S . So they are also mathematically rigorous invariants for topological manifolds. Normally, one defines a GSD by putting a TQFT on a closed spatial manifold without a boundary. However, recent developments in physics suggest that one can also define a GSD by putting a TQFT on an open spatial manifold with a boundary (possibly with multiple components) [19–22]. To distinguish the two, the former, on a closed spacetime, is We denote the spacetime dimensions as d + 1D. 2 (G) One can also consider a generalization of this relation by turning on a background flat connection A for a global symmetry G. First, nontrivial holonomies along 1-cycles of M will result in the replacement of H d tw 1 by the corresponding twisted Hilbert space H . Second, a nontrivial holonomy g ∈ G along the time S will result in the insertion of ρ(g) into the trace, where ρ is the representation of G on the Hilbert space: d 1 (G) Z (M × S ; A ) = Tr tw ρ(g). (1.2) In condensed matter, this is related to the symmetry twist inserted on M to probe the symmetry- protected/enriched topological states (SPTs/SETs) [2,12]. Instead, in this work, we mainly focus on Eq. (1.1). 2/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. named bulk topological degeneracy; the latter, on an open spacetime, is called boundary topological degeneracy [19]. For the case with a boundary the GSD is evaluated as d 1 Z (M × S ) = dim H | ≡ GSD. (1.3) d d−1 d d d−1 ∂M = M ∂M = As already emphasized in Ref. [19], this boundary GSD encodes both the bulk TQFT data as well as the gapped topological boundary conditions data [22–25]. These gapped topological boundary conditions can also be viewed as: (d − 1)D defect lines/domain walls in dD space, or dD defect surfaces in (d + 1)D spacetime. These topological boundaries/domain walls/interfaces are codimension-1 objects with respect to space (in the Hamiltonian picture) or spacetime (in the path integral picture). We will especially implement the unifying boundary conditions of symmetry extension and sym- metry breaking (of gauge symmetries) developed recently by Ref. [26], and will compute GSDs on manifolds with boundaries. In Ref. [26], the computation of the path integral is mostly based on discrete cocycle/cochain data of group cohomology on the spacetime lattice; here we will approach it from the continuum TQFT viewpoint. Following the setup in Ref. [7], the systems and QFTs of our concern are: (1) unitary; (2) emergent as infrared (IR) low-energy physics from fully regularized quantum mechanical systems with an ultraviolet (UV) high-energy lattice cutoff (this setup is suitable for condensed matter or quantum information/code); and (3) anomaly-free for the full d + 1D. However, the dD boundary of our QFTs on the open manifold can be anomalous, with gauge or gravitational ’t Hooft anomalies (see, e.g., Ref. [27]). 1.1. Tunneling topological vacua, counting GSD, and extended operators Using these GSDs, one can characterize and count the discrete vacuum sectors of QFTs and gauge theories. In 2+1D or higher dimensions, the distinct vacuum sectors for topological order are robustly separated against local perturbations. Distinct vacuum sectors cannot be tunneled into each other by local operator probes. In other words, the correlators of local probes should be zero or exponentially decaying: O (x )O (x )| =g.s.|O (x )O (x )|g.s.| 0. (1.4) 1 1 2 2 |x −x |→∞ 1 1 2 2 |x −x |→∞ 1 2 1 2 Here |g.s. means one of the ground states, sometimes denoted as |g.s.=|0. However, distinct vacuum sectors can be unitarily deformed into each other only through extended operators W (line and surface operators, etc.) winding nontrivial cycles (1-cycle, 2-cycle, etc.) along compact directions of space. In the case in which the extended operator W is a line operator, the insertion of W can be understood as the process of creation and annihilation of a pair of anyonic Here a boundary generically means the interface between the nontrivial sector (TQFT and topological order) and the trivial vacuum (gapped insulator). A domain wall means the interface between two nontrivial sectors (two different TQFTs). We will use domain walls and interfaces interchangeably. However, we will only consider the boundaries, and not more general domain walls, since the domain walls are related to boundaries by the famous folding trick. 3/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. Fig. 1. We show the quantum energy spectrum as several discrete energy levels in terms of light gray horizontal dashed lines. The approximate semiclassical energy potential is drawn in terms of the continuous solid black curve. The vertical axis shows the energy value E. The horizontal axis illustrates their different quantum numbers, which can be, e.g., (1) different eigenvectors spanning different subspaces in the Hilbert space; or (2) different spin/angular/spacetime momenta, etc. This figure shows 3 topological degenerate ground states |g.s. , |g.s. , and |g.s.  with the dark gray horizontal dashed lines for their energy levels: Their energy levels 1 2 3 −#V only need to be approximately the same (within the order of e , where V is the system size), but they remain topologically robust. Namely, only via the insertion of the extended operator shown in Eq. (1.5) winding around a non-contractible cycle can the |g.s.  tunnel to the other sectors, even though their energy levels are nearly the same. The energy barrier is proportional to the cost of creating two anyonic excitations at the end of the extended operators W in Eq. (1.5). This energy barrier  naively seems to be infinite in TQFTs, but it is actually of a finite order 4J , where J is the lattice coupling constant in the UV complete lattice (e.g., in Kitaev’s toric code [28] or more general twisted quantum double models [29,30]). In reality, as an example 2 1 in 2+1D, the 3 topological degenerate ground states on a T × S can be induced from the filling fraction space time ν = -Laughlin fractional quantum Hall states from electrons, or a U (1) -Chern–Simons theory at deep IR. Further illustration is shown in Fig. 2. excitations. Namely, a certain well designed extended operator W can indeed connect two different ground states/vacua, |g.s.  and |g.s. , inducing nontrivial correlators: α β g.s. |W (γ )|g.s. → finite = 0. (1.5) α β Again, |g.s.  means the ground state α among the total GSD sector, and γ is a nontrivial cycle in the space. Therefore, computing GSDs also provides important data for counting extended operators, thus counting distinct types of anyonic particles or anyonic strings, etc. Different degenerate ground states can also be regarded as different approximate vacuum sectors in particle physics or in cosmology; see Fig. 1 for further explanations and analogies. Therefore, in summary, our results might be of general interest to the condensed matter, mathematical physics, high-energy particle theory, and quantum gravity/cosmology communities. 1.2. The plan of the article and a short summary First, in Sect. 2, we describe how the formal mathematical idea of decategorification can be helpful to organize the topological data. In down-to-earth terms, we can decompose GSD data read from d + 1D into a direct sum of several sub-dimensional GSD sectors in dD, by compactifying one of the spatial dimensions on a small circle. Then in Sects. 3 and 4, we start from more familiar discrete gauge theories, e.g., the Z gauge theory [31,32]. More generally, we can consider the twisted discrete gauge theories, known as 4/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. Dijkgraaf–Witten (DW) gauge theories [33]. These are bosonic TQFTs that can be (1) realized at the UV lattice cutoff through purely bosonic degrees of freedom, and (2) defined on both non-spin and spin manifolds. We will study the GSD for these bosonic TQFTs. There has been a lot of recent progress on understanding bosonic Dijkgraaf–Witten gauge theories in terms of continuum TQFTs. However, to the best of our knowledge, so far there have been no explicit calculations of GSDs from the continuum field theories for the proposed non-Abelian DW gauge theories. Our work will fill in this gap for better analytical understanding, by computing non- Abelian GSDs using continuum TQFTs that precisely match the predictions of GSDs computed from the original Dijkgraaf–Witten group cohomology data: discrete cocycle path integrals. We present these results in Sects. 3 and 4. By non-Abelian topological orders, we mean that some of the following properties are matched: d 1 d The GSD = Z (S × S ; σ , σ , σ , ...) = dim H computed on a sphere S with 1 2 3 S ;σ ,σ ,σ ,... 1 2 3 operator insertions (or the insertions of anyonic particle/string excitations on S )havethe d 1 following behavior: (1) Z (S × S ; σ , σ , σ , ...) = dim H will grow exponen- 1 2 3 S ;σ ,σ ,σ ,... 1 2 3 tially as k for a certain set of large-n number of insertions, for some number k. The anyonic particle that causes this behavior is called a non-Abelian anyon or non-Abelian particle. The anyonic string that causes this behavior can be called a non-Abelian string [10,36]. (2) d 1 Z (S × S ; σ , σ , σ ) = dim H > 1 for a certain set of three insertions. 1 2 3 S ;σ ,σ ,σ 1 2 3 The Lie algebra of the underlying Chern–Simons theory is non-Abelian, if such a Chern–Simons theory exists. d 1 d The GSD Z (T × S ) = dim H d for a discrete gauge theory of a gauge group G on a T spatial torus behaves as GSD < |G| , i.e., reduced to a smaller number than the Abelian GSD. This criterion, however, only works for 1-form gauge theory. d 1 d Demonstrating that Z (M ×S ) effectively counts the dimensions of Hilbert space on M provides a more convincing quantum mechanical understanding of continuum/cochain TQFTs. By computing the following data independently, without using particular triangulations of spacetime, 1. GSD data, counting dimensions of Hilbert space, 2. Various braiding statistics and link invariants derived in Ref. [7], for Abelian or non-Abelian cases, we solidify and justify their continuum/cochain field theory descriptions of both Abelian or non-Abelian Dijkgraaf–Witten theories, as we show that the data are By non-Abelian DW gauge theories, we do not mean that the gauge group is non-Abelian. Some non- Abelian DW theories can be obtained from certain Abelian gauge groups with additional cocycle twists. In an unpublished article (J. Wang et al., unpublished material) in 2015, some of the current authors computed these non-Abelian GSDs. Part of the current work is based on an extension of that work. We wish to thank Edward Witten for first suggesting this continuum QFT method for computing non-Abelian GSDs in June 2015. In contrast, for the computation of GSDs for Abelian TQFTs, it has been done in Ref. [34] and other related work. 5 I I The simplest continuum bosonic TQFTs of discrete gauge theories have the following form: B dA + 2π N N ···N p 1 2 n 1 2 n A A ··· A . See details in later sections: we will show their GSD computations in Sect. 3 for 1+1D n−1 (2π) N 123···n to 3+1D, and Sect. 4 for any dimension. For all the N = 1, the GSD = 1 has been computed earlier in Ref. [35]. Also, there is only a trivial ground state, so it is thus suitable for describing symmetry-protected topological states (SPTs) without intrinsic topological order. We will see that examples like higher-form gauge theories, e.g., BdA+BB, have GSDs reduced compared to |G| , but they are still Abelian in the sense that they are free theories (have quadratic action). In addition, 3 1 its GSD = Z (S × S ; σ , σ , σ , ...) = dim H d = 1, which means an Abelian topological order. 1 2 3 S ;σ ,σ ,σ ,... 1 2 3 5/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. matched with the calculations based on triangulations [10,37]. Our present results combined together with Ref. [7] positively support the previous attempts based on continuum TQFTs [12,35,37–50]. Various data derived from continuum TQFTs can be checked and compared through the discrete cocycle and lattice formulations [8–10,29,30,36,51–56]. In Sect. 5, we study fermionic TQFTs (the so-called spin TQFTs) and their GSDs. These fermionic spin TQFTs are much subtler. They are obtained from dynamically gauging the global symmetry of fermionic SPTs [7]. Although the original fermionic SPTs and the gauged fermionic spin TQFTs have UV completion on the lattice, the effective IR field theory may not necessarily guarantee good local action descriptions. These somehow non-local topological invariants include, e.g., Arf–Brown– Kervaire (ABK) and η invariants, intrinsic to the fermionic nature of the systems. Nevertheless, there are still well defined partition functions/path integrals and we can compute explicit physical observables. Our examples include intrinsically interacting 3+1D and 2+1D fermionic SPTs (fSPTs) as short-range entangled (SRE) states, and their dynamically gauged spin TQFTs as long-range entangled (LRE) states. Recently, Refs. [57–59] and L. Fidkowski et al. (manuscript in preparation) also explored the related interacting 3+1D fSPTs protected by the symmetry of finite groups. In Sect. 5, we will briefly comment on the relations between our work and that in Refs. [57–59] and L. Fidkowski et al. (manuscript in preparation). In Sect. 6, we explore dimensional reduction scheme of partition functions. This section is based on the abstract and general thinking in Sect. 2 on (de)categorification. We implement it on explicit examples, in Sects. 3 and 4 on bosonic TQFTs and in Sect. 5 on fermionic TQFTs. In Sect. 7, we mainly consider the long-range entangled (LRE) topologically ordered bulk and boundary systems, denoted as LRE/LRE bulk/boundary for brevity. The LRE/LRE bulk/boundary systems can be obtained from dynamically gauging the bulk and unifying boundary conditions of symmetry extension and symmetry breaking introduced in Ref. [26]. In contrast, we will also compare the systems of LRE/LRE bulk/boundary to those of SRE/SRE bulk/boundary and SRE/LRE bulk/boundary. In Sect. 8, we conclude with various remarks on long-range entanglements and entanglement entropy, and implications for the studied systems in various dimensions. 1.3. Topological boundary conditions: Old anyonic condensation versus new condensation of composite extended operators Section 7 offers a mysterious and exotic new topological boundary mechanism, worthwhile enough for us to summarize its message in the introduction first. An important feature of LRE/LRE bulk/boundary is that both the bulk and boundary can have deconfined anyonic excitations. The anyonic excitations are 0D particles, 1D strings, etc., which can be regarded as energetic excitations at the ends of extended operators supported on 1D lines, 2D surfaces, etc. In contrast to conventional wisdom, which suggests that the LRE topological gapped boundary is defined through the condensation of certain anyonic excitations, we emphasize that there are some additional subtleties and modifications needed. The previously established lore suggests that the topological gapped boundary conditions are given by anyon condensation (see Refs. [21,25,60–64]as well as the recent review [65] and references therein), Lagrangian subgroups, or their generalization For LRE/LRE bulk/boundary topologically ordered systems, symmetry breaking/extension really means the gauge symmetry breaking/extension. The symmetry usage here is slightly abused to include the gauge symmetry. 6/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. 2 3 [19,22,24,66–68]. For example, in a 1+1D boundary of 2+1D bulk (say  = ∂M is the boundary), the condensation of anyons suggests that their line operators can end on the boundary  . Formally, we have boundary conditions of the following type: q A = 0, (1.6) i i or similar, i.e., certain linear combinations of line operators (with coefficients q ) can end on  . Here and below, A denote 1-form gauge fields. For example, if we consider a Z gauge theory of action B ∧ dA (i.e., Z toric code/topological N N 2π d+1 order) on any d +1D M , we can determine two types of conventional topological gapped boundary d d+1 8 conditions on  = ∂M : 1. By condensations of Z charge (i.e., the electric e particle attached to the ends of the Z Wilson N N worldline A of a 1-form gauge field), set by: A| d = 0, as Z charge e condensed on  . (1.7) 2. By condensations of Z flux (i.e., the magnetic m flux attached to the ends of the Z ’t Hooft N N worldvolume B of a (d − 1)-form gauge field), with the m-condensed boundary set by: B| = 0, as Z flux m condensed on  . (1.8) The UV lattice realization of the above two boundary conditions is constructed in Kitaev’s toric code [69] as well as the Levin–Wen string-net [25]. The two boundary conditions in Eqs. (1.7) and (1.8) are incompatible. Namely, each given physical boundary segment can choose either one of them, either e or m condensed, but not the other. However, in Sect. 7, we find that the usual anyon condensations like q A = 0 (including i i d Eqs. (1.7) and (1.8)) are not sufficient. We find that there are certain exotic, unfamiliar, new topological boundary conditions on the 2+1D boundary of 3+1D bulk, such that neither A | = 0 nor A = 0, i j 3 but only the composite of extended operators can end on the boundary: A ∪ A = 0. (1.9) i j 3 Here ∪ is a cup product. Heuristically, we interpret these types of topological boundary conditions as the condensation of composite objects of extended operators. Here on a 2+1D boundary of a certain 8 N In 2+1D, given the Z gauge bulk theory as B ∧ dA, we can gap the boundary by a cosine term of the 2π vortex field φ of A, via g dtdx cos(N φ ) 1 1 at strong g coupling, which corresponds to the A = 0 boundary condition [19]. We can also gap the boundary by another cosine term of the vortex field φ of B, via g dtdx cos(N φ ) 2 2 at strong g coupling, which corresponds to the B = 0 boundary condition [19]. These two bound- aries correspond to the rough e and the smooth m boundaries in the lattice Hamiltonian formulation of Bravyi–Kitaev [69]. 7/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. 3+1D bulk, we have a string-like composite object formed by a set of particles. The 1D string-like composite object is at the ends of a 2D worldsheet A ∪ A . The set of 0D particles that we refer to i j is the ends of 1D worldlines A and A . The boundary condition A ∪ A = 0 is achieved neither i j i j 2 by intrinsic 0D particle nor by intrinsic 1D string excitation condensation alone. We suggest that this exotic topological deconfined boundary condition may be interpreted as condensing a certain composite 1D string formed by 0D particles. In summary, in Sect. 7, we find that gauge symmetry-breaking boundary conditions are indeed related to the usual anyon condensation of particles/strings/etc. The gauge symmetry extension of LRE/LRE bulk/boundary in Ref. [26] can sometimes be reduced to the usual anyon condensation story (e.g., for 2+1D bulk), while, at other times, instead of the condensations of a set of anyonic excitations, one has to consider condensations of certain composite objects of extended operators (e.g., for certain 3+1D bulk). 1.4. Tunneling topological vacua in LRE/LRE bulk/boundary/interface systems We offer one last remark before moving on to the main text in Sect. 2. Similarly to Eq. (1.5), we can also interpret switching the topological sectors of the gapped boundary/interface systems of Sect. 7 in terms of tunneling topological vacua by using extended operators W . Equation (1.5) still holds d−2 d−2 when the operator W has a support with two boundary components γ and γ that, in turn, 1 2 d−1 support (d − 2)D operators L /L and lie in two different boundary components/interfaces 1 2 d−1 and  of the spatial manifold: g.s. |L (γ )W (γ , γ )L (γ )|g.s.  → finite = 0. (1.10) d−2 d−1 1 1 1 2 2 2 α β γ ⊂ j j Here the open spacetime manifold ∂M has two or more boundary components d d−1 d−1 ∂M =     ··· . 1 2 As usual,  means a disjoint union. Physically, by moving certain (anyonic) excitations of either the usual extended operators or the composite extended operators, from one boundary component d−1 d−1 to another boundary component  , we have switched the ground state between |g.s.  and 1 2 |g.s. , as Eq. (1.10) suggests, shown in Fig. 2. This idea is deeply related to Laughlin’s thought experiment in condensed matter [70]: Adia- batically dragging a fractionalized quasiparticle between two edges of the annulus by threading a background magnetic flux through the hole of the annulus would change the ground state sector. This also lays the foundation for Kitaev’s fault-tolerant quantum computation in 2+1D by anyons [28]. Various applications can be found in Refs. [19,21,71,72] and references therein. In our work, we generalize the idea to any dimensions. This idea in some sense also helps us with the counting of GSDs and extended operators for LRE/LRE bulk/boundary systems. 9 d−2 More generally, one can consider a configuration where the support of W has a boundary γ (possibly d−1 with multiple connected components) that coincides with a nontrivial cycle in the boundary  (also possibly d d−2 with multiple components) of the spatial manifold M . Each connected component of γ supports a certain (d − 1)D operator. 8/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. (a) (b) Fig. 2. Illustration of tunneling between topological vacua, from |g.s.  to |g.s. , via an extended operator β α W . In (a), we see the topological vacuum in an original ground state |g.s. , where the spatial manifold M is shown. On top of M , there are LRE/LRE bulk/boundary with topological orders (TQFTs). In (b), after d−1 d−1 inserting a certain extended operator W connecting two boundary components ( and  ), usually by an 1 2 adiabatic process, we switch or tunnel to another topological vacuum |g.s. . In the case of a closed manifold, the extended W goes along a non-contractible cycle (representing a nontrivial element of the homology group of M ). 2. Strategy: (De)categorification, dimensional decomposition, and the intuitive physical picture In this section, we address physical ideas of dimensional reduction/extension of partition functions and topological vacua (GSD), and their relations to formal mathematical ideas of decategorifi- cation/categorification. These ideas are actually relevant to physical phenomena measurable in a laboratory; see Fig. 3. In condensed matter, the related idea of dimensional reduction was first stud- ied in Refs. [73] and [10] for 3+1D bulk theories. Here we apply the idea to an arbitrary dimension. Later, gathering the concrete calculations in Sects. 3, 4, and 5, we will implement the strategy outlined here on those examples in Sect. 6. Figure 3 shows how the energy spectrum of a topologically ordered sample (shown as a gray cuboid) effectively described by an underlying TQFT gets affected by the system size and by the holonomies of gauge fields through compact circles. The topologically ordered cuboid is displayed in real space. The energy eigenstates live in the Hilbert space. The energy spectrum can be solved from a quantum mechanical Hamiltonian system. Figure 3(a) shows the system at a large or infinite size limit in real space (in the case when the spatial d d 1 manifold is M = T , d = 3 with every S circle size →∞), when the topological degeneracy of zero energy modes becomes almost exact. The zero modes are separated from higher excitations by a finite energy gap  . 9/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. (a)(b)(c) Fig. 3. Relating the dimensional reduction and (de)categorification to measurable physical quantum phenom- ena in the laboratory. The top part of the subfigure (a) shows the bulk energy spectrum E with energy gap , in the large 3+1D size limit. The bottom part shows in gray a 3D spatial sample on a T torus with large compact circles in all x, y, z directions. The degenerate zero modes in the energy spectrum are due to the nontrivial topological order (described by a TQFT) of the quantum system. The subfigure (b) shows that the energy spectrum splits slightly due to the finite size effect, but its approximate GSD is still topologically robust. The subfigure (c) shows the system on a T torus in the limit of a small circle in the z direction. The energy spectrum forms several sectors, which can be labeled by a quantum number b associated with the holonomy A of a gauge field A along z (or a background flux through the compact circle) as b ∼ A. See the main text for a more detailed explanation. Figure 3(b) shows the system at finite size in real space. The GSD becomes approximate but still topologically robust. d d−1 1 d 3 Figure 3(c) shows that, when M = M × S (in the case M = T ) and the compact z direction’s S circle becomes small, the approximate zero energy modes form several sectors, labeled by a quantum number b associated with the holonomy b ∼ A of a gauge field A along z (or a background flux threading via the compact circle). In d + 1 dimensions, this means that GSD d = GSD d−1  . (2.1) T , d+1D-TQFT T , (d−1)+1D-TQFT (b) The energy levels within each sector of GSD  are approximately grouped together. T ,(d−1)+1D-TQFT (b) However, energy levels of different sectors, labeled by different b, can be shifted upward/downward differently by tuning the quantum number b ∼ A. This energy level shifting is due to anAharonov– Bohm type of effect. This provides the physical and experimental meanings of this decomposition. More generally, one can consider the decomposition of the (zero mode part of the) Hilbert space d−1 1 d of a d + 1D TQFT on M × S into the Hilbert spaces of (d − 1) + 1D TQFTs on M : d−1 1 d−1 H (M × S ) = H  (M ). (2.2) d+1D-TQFT (d−1)+1D-TQFT (b) Note that Eq. (2.2) in principle contains more information than just the decomposition of the GSDs d−1 d−1 (as in Eq. (2.1) for M = T ). This is because the Hilbert space H d of a d + 1D TQFT d d d on M forms a representation of the mapping class group of M , MCG(M ). Therefore Eq. (2.2) 10/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. d−1 should be understood as the direct sum decomposition of representations of MCG(M ). This generalizes the relation of MCG to the dimensional decomposition scheme proposed in Ref. [74]. Examples in Refs. [10,73] show that for a 3+1D to 2+1D decomposition, we indeed have the modular xy,2D n xy,3D S and T representation of MCG(T ) = SL(n, Z) data decomposition: S = S and xy,2D xy,3D 2 T = T on a 2D spatial torus T . b xy The statement can be made more precise in the case when the d + 1D TQFT is realized by gauging a certain SPT with finite Abelian (0-form) symmetry G. Suppose that other (ungauged) symmetries of the theory are contained in H , an extension of the SO (or O when there is time-reversal symmetry) structure group of a spacetime manifold. For example, when there is Z fermionic parity, one considers manifolds with spin structure. The corresponding SPT states are then classified by d+1 H (BG) := Hom( (BG), U (1)), (2.3) H d+1 where  (BG) is the bordism group of manifolds with H structure (e.g., H = spin) [75–77] and d+1 equipped with maps to BG (the classifying space of G). Then, for an SPT state corresponding to a choice d+1 μ ∈  (BG), (2.4) the partition function of the corresponding gauged theory on a closed d + 1-manifold is given by d+1 d+1 Z (M ) := μ([(M , a )]) (2.5) μ d+1 d+1 |π (M )| |G| 1 d+1 a ∈H (M ,G) d+1 d+1 where the pair (M , a ),a (d +1)-manifold with H structure and a map M → BG, represents d+1 d+1 an element in  (BG). Here and below we use one-to-one correspondence between homotopy d+1 d+1 1 d+1 classes of maps M → BG and elements of H (M , G). Note that the choice of μ can be under- d+1 d+1 iS(M ,a ) d+1 stood as the choice of the action for finite group gauge theory: μ([(M , a)]) ≡ e ∈ U (1). d 1 d 1 Suppose that H structures on M and S define H structure on M × S (this is true for the H = spin example). Then one can consider the following map for a given element b ∈ G: H H φ :  (BG) −→  (BG) d d+1 d d 1 (2.6) [(M , a )] −→ [(M × S , a ⊕ (b ⊕ ... ⊕ b))] d d |π (M )| 1 d 1 1 d |π (M )| ∼ 0 where we have used H (M × S ) H (M ) ⊕ G . It is easy to see that the map above is well defined, i.e., the image of [(M , a )] does not depend on the choice of the representative d H (M , a ) in  (BG). From the definition of gauged SPTs (2.5), it follows that for connected closed manifolds we have the following relation: d 1 d Z (M × S ) = Z ∗ (M ). (2.7) φ (μ) d b |π (M )|=1 b∈G 10 H Suppose for simplicity that  (BG) contains only torsion elements Tor. Otherwise we redefine it by d+1 H H replacing  (BG) with Tor  (BG) in the formulas below. Throughout this work, we focus on the torsion d+1 d+1 Tor part. 11/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. d+1 That is, the partition function of the gauged d + 1D SPT labeled by μ ∈  (BG) is given by the ∗ d sum of gauged dD SPTs labeled by φ (μ) ∈  (BG). b H Similarly, for Hilbert spaces of the corresponding TQFTs we have d−1 1 d−1 H (M × S ) = H (M ). (2.8) μ φ (μ) d−1 b |π (M )|=1 b∈G d−1 d−1 d d For a connected bordism N , ∂N =  (−M )   M we then have i j i j d 1 d Z (N × S ) = i ◦ Z (N ) ◦ pr (2.9) μ diag φ (μ) diag d b |π (N )|=1 b∈G where d−1 d−1 d 1 Z (N × S ) : ⊗ H (M ) −→ ⊗ H (M ) μ i μ j μ i j (2.10) d−1 d−1 d 1 ∗ ∗ ∗ Z (N × S ) : ⊗ H (M ) −→ ⊗ H (M ) i j φ (μ) φ (μ) φ (μ) i j b b b and d−1 d−1 d−1 ˜ ˜ ˜ ∗ ∗ i : H (M ) −→ H (M ) = H (M ) diag φ (μ) φ (μ) μ j j j b b j j j b∈G b ∈G d−1 d−1 d−1 ∗ ∗ pr : H (M ) = H (M ) −→ H (M ) diag μ φ (μ) φ (μ) i i i b b i i b ∈G b∈G i (2.11) are inclusions of the diagonal and projection onto the diagonal. Let us denote the TQFT functor (in Atiyah’s usual meaning) for the (d + 1)D gauged SPT labeled dD by μ as C (so that its value on objects is given by H (•) and its value on morphisms is Z (•)). μ μ Then throughout the paper we will often write simply (d−1)D dD C = C (2.12) φ (μ) b∈G by which we actually mean that the functors satisfy relations (2.7), (2.8), (2.9). We thus decompose a (d + 1)D-TQFT to many sectors of ((d − 1) + 1)D-TQFT labeled by b, in the topological vacua subspace within the nearby lowest-energy Hilbert space. We mark that the related ideas of the dimensional decomposition scheme are explored in Refs. [78–81]. Relation (2.7) can also be interpreted as follows: d d 1 Z (M ) = Z (M × S ) = Tr 1. (2.13) φ (μ) μ H (M ) That is, the trace over the Hilbert space of d + 1D TQFT can be interpreted as a sum over the partition functions of dD TQFTs. This is an example of the general notion of decategorification in mathematics, where the vector spaces are replaced by numbers. The inverse, i.e., a increase of numbers to vector spaces is known as categorification. Note that even though the partition function of a single dD TQFT in the sum above cannot be categorified (i.e., interpreted as a trace over some diag 11 |G| |G| Relation (2.7) can be understood as a special case of Eq. (2.9) with i : C → C, pr : C → C . diag diag 12/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. Hilbert space), a particular sum of them can be. The notion of (de)categorification can be extended to the level of the extended TQFT functors. In particular, Eq. (2.8) can be interpreted as 0 d−1 d−1 K (BCond (M )) = H (M ) (2.14) μ φ (μ) b∈G d−1 where BCond (M ) is the category of boundary conditions of the (d + 1)D TQFT (obtained by d−1 0 gauging SPT labeled by μ)on M and K is the Grothendieck group. Note that in the case of fermionic theories the Hilbert spaces in Eq. (2.2) have an additional structure: Z grading (see Sect. 5 for details). 3. Bosonic TQFTs and ground state degeneracy In this section we compute the ground state degeneracy (GSD, or, equivalently, the vacuum degen- eracy) of some topological field theories, using a strategy and setup similar to those in Ref. [7]. We will consider TQFTs with a continuum field description in terms of n-form gauge fields. The level-quantization constraint for such theories is derived and given in Ref. [12]. Below we compute the GSD on a spatial manifold M via the absolute value of the partition function Z on a spacetime d 1 manifold M × S based on its relation to the dimension of Hilbert space H: d 1 GSD d = dim H d = Z (M × S ). M M As a warm-up, we start with (untwisted) Z gauge theory [31], also known as Z spin liquid [82], N N Z topological order [83], or Z toric code [28]. Then we proceed to more general twisted discrete N N gauge theories: bosonic Dijkgraaf–Witten (DW) gauge theories. In most of the cases we consider the torus as the spatial manifold for simplicity: d+1 GSD = dim H =|Z (T )|. d d T T Below we use the notation N ≡ gcd(N , N , N , ...). We will always use A to denote a 1-form ijk··· i j k gauge field, while B can be a higher-form gauge field. In most cases, without introducing ambiguity, we omit the explicit wedge product ∧ between differential forms. We will also often omit the explicit summations over the indices I , J , K , ... in the formulas. We note that related calculations of bosonic GSD are also derived based on independent and different methods in Refs. [10,37,84]. Some of the main results of this section are briefly summarized in Table 1. 3.1. BdA in any dimension To warm up, we evaluate the ground state degeneracy of the untwisted Z gauge theory in d + 1D d d+1 d+1 on torus T as the partition function on M = T spacetime in two different ways. In the first approach, we integrate out a (d − 1)-form B field that yields a condition of A being flat together with quantization of its holonomies. We evaluate iN GSD = [DB][DA] exp[ B ∧ dA] (3.1) d 1 2π T ×S Sometimes we may find the wedge product (∧) implicitly without writing it down. 13/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. Table 1. Table of TQFTs and GSDs. For the twisted gauge theories of Dijkgraaf–Witten (DW) theory, we will sometimes restrict ourselves to the case Z = Z = Z = Z = ··· ≡ Z where N is prime. Here p is N N N N N 1 2 3 4 nontrivial and gcd(p, N ) = 1 for those non-Abelian theories within DW theories; G denotes the total finite gauge group in the DW setup. Our derivations are based on continuum field descriptions. These results can be independently compared with the discrete cocycle/cochain lattice path integral method in Refs. [10,37]. Action Dim gauge group G GSD GSD d d−2 1 T S ×S (local) I I I d any D Z B dA |G| – I I 2π n I J IJ 2+1D U (1) (level K ) A dA | det K | – 4π N p I I I IJ I J 2 2+1D Z B dA + A dA |G| – I 2π 4π N N N N p I I I 1 2 3 1 2 3 4 3 2+1D Z B dA + A A A N + N − N – N 2 2π (2π) N I =1,2,3 N N N p I I I J K 3 I I J IJK 3+1D Z B dA + A A dA |G| |G| I 2π (2π) N IJ I =1,2,3 10 9 8 7 N + N + N − N N N N N N p I I I 1 2 3 4 1 2 3 4 3+1D Z B dA + A A A A |G| N 3 6 5 3 2π (2π) N 1234 −N − N + N I =1,2,3,4 N p N N I I I J 3 I IJ I J 3+1D Z B dA + B B gcd(p, N ) gcd(p, N ) I 2π 4πN IJ N N N ···N p I I I 1 2 5 1 2 3 4 5 4+1D Z B dA + A A A A A Eq. (4.4), Eq. (4.5) |G| I 2π (2π) N I =1,...,5 N N N ···N p I I I 1 2 d 1 d dD Z B dA + A ··· A Eq. (4.7) |G| d−1 I 2π (2π) N 1···d I =1,...,d −1 = [DA] 1| 2πn = N 1 dA=0, A= , n ∈Z μ N 1 d+1 S ⊂T N 1 d+1 a∈H (M ,G) 1 d+1 d+1 |H (M , G)| N −1 1 d+1 d = N ·|H (M , G)|= = = N . 0 d+1 |H (M , G)| N −1 −1 0 d+1 The G = Z is the gauge group. The N =|H (M , G)| is the normalization factor that takes into account the gauge redundancy of the 1-form gauge field. In the second approach, we integrate out a 1-form A field that yields a flat B condition together d−1 d+1 with quantization of its flux through any codimension-2 cycle M ⊂ T . We evaluate iN GSD = [DB][DA] exp[ B ∧ dA] (3.2) d+1 2π = [DB] 1| 2πn dB=0, B= , n∈Z d−1 N M N d−2 (−1) −1 d−1 d+1 d−2−j d+1 −1 d−1 d+1 = N ·|H (M , G)|= ( |H (M , G)| ) |H (M , G)| j=0 d+1 d+1 d+1 d+1 d+1 (−) − d ( ) ( ) ( ) ( ) 0 d−3 d−2 d−1 = (N ··· N N )N = N . −1 The N factor again takes into account the gauge redundancy of the (d − 1)-form gauge field B. (d−2) (d−2) The gauge transformation of B → B + dλ contains the (d − 2)-form gauge parameter λ , (d−2) (d−2) (d−3) whose gauge transformation allows a λ → λ + dλ change with further lower-form −1 redundancy. Considering the gauge redundancy layer by layer, we obtain the N factor in the third d+1 d+1 d+1 d+1 d+1 2 line in the above equation. The last equality uses (1 − s) = 1 − s + s + d+1 d d−1 14/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. d+1 d+1 d+1 ··· + (−) s with s = 1. The results of the above first and second approaches match well: d 13 GSD d =|G| . d−1 3.2. K A dA , BdA+AdA in 2+1D, BdA+AAdA in 3+1D, and BdA + A dA in IJ I J any dimension 3 p N N N I I 1 2 3 123 1 2 First we compute the GSD of B ∧ dA +c A ∧ A ∧ dA (where c = , 123 123 I =1 2π (2π) N p ∈ Z) theory on a torus. Other details of the theory are studied in Ref. [7], with the level- quantization constraint derived/given in Ref. [12]: iN I I 1 2 3 GSD 3 = [DB][DA] exp[ B ∧ dA +ic A ∧ A ∧ dA ] (3.3) 3 1 2π T ×S 1 2 3 = [DA] exp[ ic A ∧ A ∧ dA ]| 2πn I I I dA =0, A = , n ∈Z 1 I N 3 1 N S I T ×S I 3 3 = 1 = (N N N ) =|G| . 1 2 3 n ,n ,n ∈Z I ,x I ,y I ,z N We have used the concept that A satisfies the flatness condition in the second line, so all the configu- 1 2 3 rations weigh with exp[ ic A ∧ A ∧ dA ]= 1. To sum over [DA] in the partition function, we 2πn I I simply need to sum over all the possible holonomies A = around every non-contractible S N 2 N I I 1 2 2 direction. Similarly, for B ∧ dA +c A ∧ A ∧ dA theory, from the flatness condition I =1 2π on A on the torus it follows that the partition function is given by 3 3 GSD 3 = 1 = (N N ) =|G| . (3.4) 1 2 n ,n ,n ∈Z I ,x I ,y I ,z N N I I I J In 2+1D, the same strategy allows us to evaluate the GSD for B ∧ dA + c A ∧ dA IJ I 2π IJ IJ ( c = , p ∈ Z) theory on a torus: IJ IJ 4π iN I I I J GSD 2 = [DB][DA] exp[ B ∧ dA +ic A ∧ dA ] (3.5) IJ 2 1 2π T ×S I J = [DA] exp[ ic A ∧ dA ]| 2πn IJ I I I dA =0, A = , n ∈Z 1 I N 2 1 S N I T ×S I 2 2 = 1 = (N ) =|G| . n ,n ∈Z I I ,x I ,y N The result can be interpreted as the volume of a rectangular polyhedron with edges of size N (each appearing twice). More generally, for an Abelian Chern–Simons theory with matrix level K [85], i.e., 14 I J J IJ with the action A ∧ dA , the flatness condition is modified to K dA = 0. The result is IJ 4π then given by the volume of the polyhedron with edges given by column vectors of the matrix K : GSD 2 =| det K |. (3.6) The partition functions for Z gauge theory with 1-form and d − 1-form gauge fields in d + 1D match d+1 χ(M ) only up to the gravitational counter-term N , where χ is the Euler number, if we use the normalization d+1 1 d factors N explained in the main text. However, when M = S ×M , χ = 0 and the partition function agree, which is consistent with the fact that the GSD itself is an observable quantity. See Eq. (B.23) of Ref. [34]. If there is an odd entry along the diagonal of K , then it requires a spin structure, otherwise it is non-spin. II 15/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. The calculation above can be easily generalized to the case of d + 1D theory with the action of the d−1 d form BdA + A dA. The result is GSD =|G| . This is in line with the fact that these theories are of Abelian nature. One can also obtain the GSD of the above theories based on the cochain path integral; see Ref. [37] on these Abelian TQFTs. 3.3. BdA + BB in 3+1D 3.3.1. Twisted Z theory with a B ∧ B term Np We first consider a 3+1D action B ∧ dA+ B ∧ B. This theory has been considered in detail 2π 2π in Ref. [34] (Appendix B). In the action above we chose a less refined level quantization, which is valid for any manifold possibly without a spin structure. For a non-spin bosonic TQFT, the level quantization can be easily derived based on Ref. [12]. For a spin fermionic TQFT, Ref. [34] provides a refined level quantization on a spin manifold, where the p can take half-integer values; namely Np we can redefine p = p /2 with an integer p . In short, we get B ∧ dA+ B ∧ B where now 2π 4π p ∈ Z. This is a spin TQFT when both N and p are odd. The gauge transformation is B → B + dλ, A → A − 2pλ + dg = A − p λ + dg. Using an approach similar to that in the second part of Sect. 3.1, we can evaluate its GSD on a 3-torus: iN iNp GSD 3 = [DB][DA] exp[ B ∧ dA+ B ∧ B] (3.7) 3 1 2π 2π T ×S iNp = [DB] exp[ B ∧ B]| 2πn dB=0, B= , n∈Z 2 N M N 3 1 2π T ×S 2π(2p) −1 = N exp[i (n n − n n + n n )] xy zt xz yt yz xt n ∈Z αβ N 2π(2p/ gcd(2p, N )) −1 = N exp[i (n n − n n + n n )] xy zt xz yt yz xt (N / gcd(2p, N )) n ∈Z αβ N N · N N N · N −1 3 −1 3 3  3 = N ( ) = ( ) ( ) = gcd(2p, N ) = gcd(p , N ) , (N / gcd(2p, N )) N (N / gcd(2p, N )) where n are fluxes of the field B through 2-tori in the directions α, β. The sum over n factorizes αβ αβ into the product of sums over the pairs n , n ; n , n ; and n , n . These sums can be interpreted xy zt xz yt yz xt as sums over integral points inside squares of size N . Each square has N · N area. We divide this area by (N / gcd(p , N )), since a summation of an (N / gcd(p , N )) number of exponential factors gives −1 one. We have used the fact that (p / gcd(p , N )) and (N / gcd(p , N )) are relatively prime. The N factor is derived from dividing by the number of 1-form gauge symmetries, |H (M , G)|, which is 4 3 1 0 equal to N on the T × S , and then multiplying by the order of the gauge group, |H (M , G)|= N . −1 4 3 This gives the normalization factor N = 1/(N /N ) = 1/N , which accounts for the redundancy of “gauge symmetries” and “gauge symmetries of gauge symmetries”. Overall, we obtain GSD 3 = gcd(p , N ) , which is consistent with Ref. [34]. See Ref. [34] for the evaluation of the partition function on other manifolds. We can also use another independent argument based on Ref. [34] to verify the GSD obtained Np above. In Ref. [34], it was found that B ∧ dA+ B ∧ B theory has a similar GSD to Z gcd(N ,p ) 2π 4π gauge theory at low energy. First, we know that the commutator between the conjugate field and 16/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. i2π momentum operators is [A(x), B(x )]= δ(x − x ).At p = 0, there is a 2-form global symmetry Z and a 1-form global symmetry Z generated by: N N i A i B U = e , V = e . At p = 0, the symmetry transformation gives −1 B → UBU = B − ξ , (3.8) −1 A → VAV = A + ζ , (3.9) N N where ξ and ζ are flat and satisfy ξ = 2π and ζ = 2π , so that U = V = 1 and B A B A i2π UV = e VU . The operators U and V can be referred to as the clock and the shift operators (like the angle and angular momentum operators). They generate N distinct ground states along each non-contractible loop. On the other hand, when p = 0, we can consider an open cylindrical surface () operator with two ends on closed loops γ and γ : W = exp[i A + ip B − i A]. γ  γ The boundary components of  are γ and γ , which makes the operator gauge invariant under the gauge transformation. The closed line operator with exp[i A] can be defined whenever the contribution from the open surface part becomes trivial. Let  and  be two distinct surfaces 1 2 bounded by γ and γ (i.e., ∂ = ∂ = γ ∪ (−γ ) where the minus sign indicates the opposite 1 2 1 2 2π orientation), then  −  is a closed surface, and we have B − B = n with some n ∈ Z , 1 2 N 1 2 2π N since B = n. The minimum integer I enforcing Ip B = 2π is I = . This −  − N gcd(p ,N ) 1 2 1 2 means that exp[i(I A + Ip B − I A)] does not depend on the choice of the open surface , γ  γ and we can view exp[i(I A + Ip B − I A)] formally as two deconfined line operators as γ  γ exp[i(I A − I A)]. Thus we can define the line operator alone as: γ γ i A gcd(p ,N ) gcd(p ,N ) U = exp[iI A]= e , with U = 1. (3.10) 2π 2πn The reasoning is, again, that since A = n with some n ∈ Z , then we have U = exp[i ] γ N gcd(p ,N ) gcd(p ,N ) satisfying U = 1. The closed surface operator alone can be defined as: N p gcd(p ,N ) V = exp[i B], while V = 1 and V = 1, so V = 1. (3.11) 2π Here V = 1 is due to B = n. On the other hand, we can close the open surface by letting two closed curves γ and γ coincide, then the open surface operator exp[i A + ip B − i A] γ  γ becomes the surface operator exp[ip B]. But the original open surface operator must be trivial (inside correlation functions) because the theory describes topological and gapped systems. This Recall that in general a generator of q-form symmetry is realized by an operator supported on a submanifold codimension q + 1 (i.e., of dimension d − q for a d + 1D spacetime). 17/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. p N implies W = 1 and thus exp[ip B]= 1 ⇒ V = 1. The superposed conditions of V = 1 and p gcd(p ,N ) V = 1 give the final finest constraint V = 1. Finally we obtain: i2π N i2π gcd(p ,N ) UV = e VU , because of [ A(x), B(x )]= δ(x − x ). gcd(p , N ) gcd(p , N ) Thus the new clock and shift operators generate gcd(p , N ) distinct ground states along each non- contractible loop. For GSD 3 on a T spatial torus with three spatial non-contractible loops, we obtain GSD 3 = gcd(p , N ) as in Ref. [34]. 3.3.2. More general theory 2 N p N N I I 12 1 2 1 2 We can also consider a more generic action B ∧ dA + B ∧ B where N ≡ I =1 2π 2πN gcd(N , N ) and p can be a half-integer. Again we choose a less refined level quantization, which 1 2 12 is true for any generic manifold without a spin structure. The gauge transformation is the following: N N 1 2 I I I 1 1 1 2 2 2 B → B + dλ , A → A − p λ + dg , A → A − p λ + dg . (3.12) 12 12 N N 12 12 Again, we derive GSD 3 on a T spatial torus: iN ip N N I 12 1 2 I I 1 2 GSD = [DB][DA] exp[ B ∧ dA + B ∧ B ] (3.13) 3 1 2π 2πN T ×S 12 ip N N 12 1 2 1 2 = [DB] exp[ + B ∧ B ]| 2πn I I I dB =0, B = , n ∈Z 2 I N 3 1 2πN M N T ×S I 2p 2π( ) gcd(2p ,N ) 12 12 −1 1 2 1 2 1 2 1 2 1 2 1 2 = N exp[i (n n − n n + n n + n n − n n + n n )] xy zt xz yt yz xt zt xy yt xz xt yz ( ) gcd(2p ,N ) 12 12 n ∈Z αβ I N · N |H (M , G)| N · N 1 2 1 2 −1 6 −1 6 = N ( ) = ( ) · ( ) (N / gcd(2p , N )) |H (M , G)| (N / gcd(2p , N )) 12 12 12 12 12 12 N N N · N lcm(N , N ) 1 2 1 2 1 2 6 3 6 = · ( ) = ( ) gcd(2p , N ) . 12 12 4 4 (N / gcd(2p , N )) gcd(N , N ) N N 12 12 12 1 2 1 2 −1 Similarly to Sect. 3.3.1, the N factor is derived from dividing by the number of 1-form gauge 1 0 symmetries, |H (M , G)|, and then multiplying by the order of the gauge group, |H (M , G)|. This accounts for the redundancy of “gauge symmetries” and “gauge symmetries of gauge symmetries”. 3.4. BdA + AA in 1+1D N p N N I I 12 1 2 1 2 We consider the 1+1D TQFT with the action B ∧ dA + A ∧ A . Locally B is a 0-form 2π 2π N field and A is a 1-form field. The level quantization is described in Refs. [7,12]. This theory can be obtained by dynamically gauging an SPT with the symmetry group G = Z [35]. Its s N I =1 I dimension of Hilbert space on S is computed as a discrete sum, after integrating out the B field: 2π −1 GSD = N exp[−ip det(n  , n  )]. (3.14) 12 1 2 n  ∈Z 18/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. Consider a specific example N = N = N , which is a prime number, so that gcd(p , N ) = 1. In 1 2 12 this case, 2π −1 GSD 1 = N exp[−ip det(n  , n  )]= δ(n )δ(n ) = 1. (3.15) 12 1 2 2,x 2,t n ,n ∈Z 2,x 2,t N n  ∈Z There is a unique ground state degeneracy without robust topological order in this case. −1 12 For more generic N and N , the normalization factor N is . We can rewrite as 1 2 N N N 1 2 12 p /gcd(N ,p ) 12 12 12 for generic non-coprime N and p . A direct computation shows 12 12 N /gcd(N ,p ) 12 12 12 1 2π 1 N N 1 2 GSD 1 = exp[−ip det(n  , n  )]= ( ) 12 1 2 N N N N N N /gcd(N , p ) 1 2 12 1 2 12 12 12 n  ∈Z lcm(N , N ) 1 2 = gcd(N , p ) . (3.16) 12 12 gcd(N , N ) 1 2 The GSD depends on the level/class index p . Note that gcd(N , p ) = gcd(N , N , p ). 12 12 12 1 2 12 Some numerical evidence, such as the tensor network renormalization group method [86], suggests that there is no robust intrinsic topological order in 1+1D. We can show that “no robust topological order in 1+1D” can already be seen in terms of the fact that a local non-extended operator, such as the 0D vortex operator B, can lift the degeneracy. Thus this GSD is accidentally degenerate, not topologically robust. From Eq. (3.16) it follows that for the level-1 action (i.e., N = 1) we have GSD = 1 and no intrinsic topological order, which is consistent with the use of this level-1 theory for SPTs [35]. 3.5. BdA + AAA in 2+1D I I I 1 2 3 We can also consider the 2+1D TQFT with the action B ∧ dA + c A ∧ A ∧ A (where 2π p N N N 123 1 2 3 c = , p ∈ Z). This can be obtained from dynamically gauging some SPTs with 123 2 123 (2π) N the symmetry group G = Z [35]. The level quantization is discussed in Refs. [7,12]. One s N I =1 I can confirm that it is equivalent to Dijkgraaf–Witten topological gauge theory with the gauge group G = Z with type-III cocycle twists by computing its dimension of Hilbert space on a torus. I =1 In the first step, we integrate out B to get a flat A constraint and obtain the following expression for GSD : iN I I 1 2 3 [DB][DA] exp[ B ∧ dA + ic A ∧ A ∧ A ] 2 1 2π T ×S (3.17) 2π −1 = N exp[ip det(n  , n  , n  )]. 123 1 2 3 n  ∈Z The above formula is general but we take a specific example N = N = N = N where N is a 1 2 3 prime number, so that gcd(p , N ) = 1 below. The calculation of GSD reduces to a calculation of the following discrete Fourier sum: 2π −1 GSD 2 = N exp[ip det(n  , n  , n  )]= δ(det(minor(n  , n  ) )). (3.18) 123 1 2 3 2 3 1,j 3 3 n  ∈Z n  ,n  ∈Z I 2 3 N N We first sum over the vector n  , and this gives us the product of discrete delta functions of the determinants of the minors minor(n  , n  ) . Case by case, there are a few choices of n  , n  when 2 3 1,j 2 3 19/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. the delta function is non-zero: (1) n  is a zero vector, then n  can be arbitrary. Each of these choices 2 3 gives one distinct ground state configuration for GSD 2. We have in total 1 · N such choices. (2) n  is not a zero vector, then as long as n  is parallel to the n  , namely n  = C n  (mod N ) for 2 3 2 2 3 some factor C, the determinants of the minor matrices are zero. The number of such configurations is (N − 1) · N . The total ground state sectors are the sum of the contributions from (1) and (2): 3 3 4 3 GSD 2 = 1 · N + (N − 1) · N = N + N − N . (3.19) Our continuum field theory derivation here independently reproduces the result from the discrete spacetime lattice formulation of 2+1D Dijkgraaf–Witten topological gauge theory computed in Sect. IV C of Ref. [10] and Ref. [38]. The agreement of the Hilbert space dimension (thus GSD) together with the braiding statistics/link invariants [7,36] confirms that the field theory can be regarded as the low-energy long-wavelength continuous field description of Dijkgraaf–Witten theory with the gauge group G = Z with type-III 3-cocycle twists. I =1 I 3.6. BdA + AAAA in 3+1D I I I 1 2 3 4 Below we consider the 3+1D TQFT action B ∧ dA + c A ∧ A ∧ A ∧ A (where c = 1234 1234 2π p N N N N 1234 1 2 3 4 , p ∈ Z) obtained from dynamically gauging some SPTs with the symmetry group (2π) N G = Z . See the level quantization in Refs. [7,12]. This is equivalent to Dijkgraaf–Witten s N I =1 topological gauge theory at low energy of the gauge group G = Z with type-IV 4-cocycle I =1 twists [12]. First, we verify it by computing its dimension of Hilbert space on a torus: iN I I 1 2 3 4 GSD 3 = [DB][DA] exp[ B ∧ dA + ic A ∧ A ∧ A ∧ A ] (3.20) 3 1 2π T ×S 1 2 3 4 = [DA] exp[ ic A ∧ A ∧ A ∧ A ]| 2πn 1234 I I I dA =0, A = , n ∈Z 1 I N 3 1 S N I T ×S I 2π −1 = N exp[ip det(n  , n  , n  , n  )]. 1234 1 2 3 4 n  ∈Z 1 3 1 Here we assume that the four non-contractible S in T × S have coordinates x, y, z, t: n n n n 1,x 1,y 1,z 1,t n n n n 2,x 2,y 2,z 2,t 1+j det(n  , n  , n  , n  ) ≡ = (−1) n · det(minor(n  , n  , n  ) ). (3.21) 1 2 3 4 1,j 2 3 4 1,j n n n n 3,x 3,y 3,z 3,t n n n n 4,x 4,y 4,z 4,t Here the minor submatrix minor(n  , n  , n  ) of the remaining vectors n  , n  , n  excludes the row 2 3 4 1,j 2 3 4 −1 and the column of n . Also, N is the proper normalization factor that takes into account the 1,j −1 −1 gauge redundancy. Namely, N =|G| is the inverse of the order of the gauge group so that |Z | have proper integer values. Without losing the generality of our approach, we take a specific example N = N = N = N = N where N is a prime number. Hence we use the fact that gcd(p , N ) = 1 1 2 3 4 1234 below. The calculation of GSD 3 reduces to a calculation of the discrete Fourier summation: 2π −1 GSD = N exp[ip det(n  , n  , n  , n  )]= δ(det(minor(n  , n  , n  ) )). 1234 1 2 3 4 2 3 4 1,j 4 4 n  ∈Z n  ,n  ,n  ∈Z I 2 3 4 N N We first sum over the vector n  , and this gives us discrete delta functions on the minor(n  , n  , n  ) . 1 2 3 4 1,j Case by case, there are a few choices when the product of delta functions does not vanish: (1) n  is a 20/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. zero vector, then n  , n  can be arbitrary. Each of these gives us a distinct ground state configuration 3 4 4 4 for GSD 3 with weight one. Altogether this contributes 1 · N · N . (2) n  is not a zero vector, then as long as n  is parallel to n  , namely n   n and n  = C n  (mod N ), for some factor C, then 3 2 2 3 2 3 the product of the determinants of the minor matrices is zero. Here n  can be arbitrary. This gives 4 4 (N − 1) · N · N distinct ground state configurations. (3) n  is not a zero vector, and n  is not parallel 2 3 to n  , namely n  = C n  (mod N ) for any C, then the determinant of the minor matrices is zero if 2 2 3 n  is a linear combination of n  and n  . Namely, n  = C n  + C n  for some integers C , C ∈ Z . 4 2 3 4 1 2 2 3 1 2 N 4 4 This gives (N − 1) · (N − N ) · N · N distinct ground state configurations. The total number of topological vacua is the sum of the contributions from (1), (2), and (3): 4 4 4 4 4 4 GSD 3 = 1 · N · N + (N − 1) · N · N + (N − 1) · (N − N ) · N · N (3.22) 8 9 5 10 7 6 3 = N + N − N + N − N − N + N . Our continuous field theory derivation here independently reproduces the result from the discrete spacetime lattice formulation of DW topological gauge theory computed in Sect. IV C of Ref. [10]. The agreement of the Hilbert space dimension (thus GSD) together with the braiding statistics/link invariants [7,36] imply that the field theory can be regarded as the low-energy long-wavelength continuum field description of DW theory of the gauge group G = Z with type-IV 4-cocycle I I twists. 4. Higher-dimensional non-Abelian TQFTs 4.1. BdA + A in 4+1D Consider continuum field theory, which describes twisted Dijkgraaf–Witten (DW) theory with the 5 I I I 1 gauge group G = Z with type-V 5-cocycle twists in 4 + 1 dimensions: B ∧ dA + c A ∧ 2π pN N N N N 2 3 4 5 1 2 3 4 5 A ∧ A ∧ A ∧ A (where c = , p ∈ Z). The level quantization is described in (2π) N I I Ref. [12]. We would like to compute the GSD on a torus. Integrating over B restricts A to being flat, and the only degree of freedom is the holonomies around cycles of the spacetime torus. We I μ 4 1 denote the holonomy of A around the cycle γ (which wraps around the μ direction of T × S ) as A = 2πn /N , μ = 0, 1, 2, 3, 4. Following the method in Sect. 3.6, the partition function μ i γ i reduces to N −1 4 1 2πp μ ρ μνρσ λ ν σ λ GSD = exp i  n n n n n . (4.1) T 2 4 5 1 3 N N μ μ μ μ μ μ,ν,ρ,σ ,λ=0 n ,n ,n ,n ,n =0 1 2 3 4 5 We further sum over n using the discrete Fourier transformation, i2πp α n N exp = N δ α = 0 mod , (4.2) N gcd(N , p) n∈Z which yields 4 4 μνρσ λ ν σ λ GSD 4 = δ  n n n n = 0 mod . (4.3) T 2 3 4 5 gcd(N , p) μ=0 ν,ρ,σ ,λ=0 n ∈Z The product of the delta functions imposes the constraint that n  , n  , n  , n  are linearly independent 2 3 4 5 mod and the partition function counts the number of configurations that satisfy such a gcd(N ,p) constraint. There are a few cases: (1) We first consider the case when p = 1. If n  = 0 mod N , 21/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. 5 5 5 the other vectors n  , n  , n  can be chosen at will. Hence there are 1 · N · N · N configurations 3 4 5 in this case. (2) If n  = 0 mod N and n  = Cn  , the other vectors n  , n  can be chosen at will. 2 3 2 4 5 5 5 There are (N − 1) choices of n  , N choices of n  , and N choices of n  and n  separately. Hence 2 3 4 5 5 5 5 there are (N − 1) · N · N · N configurations in this case. (3) If n  = 0 mod N , n  = Cn  , and 2 3 2 5 5 n  = C n  + C n  , n  can be chosen at will. There are N − 1 choices of n  , N − N choices 4 1 2 2 3 5 2 of n  , N · N choices of n  (there are N choices of C and C respectively), and N choices of 3 4 1 2 5 5 5 n  . Hence there are (N − 1) · (N − N ) · (N · N ) · N configurations in this case. (4) If n  = 0 5 2 mod N , n  = Cn  , n  = C n  + C n  , and n  = C n  + C n  + C n  , there are N − 1 choices 3 2 4 1 2 2 3 5 3 2 4 3 5 4 5 5 2 of n  , N − N choices of n  , N − N choices of n  , and N · N · N choices of n  . Hence there are 2 3 4 5 5 5 5 2 (N − 1) · (N − N ) · (N − N ) · (N · N · N ) configurations in this case. In summary, the GSD with the gcd(N , p) = 1is 6 2 3 4 6 7 8 10 11 12 GSD 4 | = N − 1 + N + N + N − N − 2N − N + N + N + N . gcd(N ,p)=1 (4.4) For a generic level p, the configurations for each n split into gcd(N , p) sectors, and we need to sum over all the sectors in the partition function. For instance, when n  = 0 mod , there gcd(N ,p) 5 5 5 are (gcd(N , p)) choices of n  , and N choices of n  , n  , n  separately. Hence there are gcd(N , p) · 2 3 4 5 5 5 5 N · N · N configurations in this case. It is clear that this result can be obtained from the p = 1 case by replacing N with , and multiplying by the number of sectors gcd(N , p) for each n  . gcd(N ,p) 4 5 5 5 5 5 5 5 N N Specifically, gcd(N , p) ·N ·N ·N can be rewritten as gcd(N , p) · · · gcd(N ,p) gcd(N ,p) . For the other cases, we can count similarly. Generalizing the ground state degeneracy gcd(N ,p) to generic p, one obtains the following expression: 4 6 2 3 N N N GSD = gcd(N , p) − 1 + + gcd(N , p) gcd(N , p) gcd(N , p) 4 6 7 8 N N N N (4.5) + − − 2 − gcd(N , p) gcd(N , p) gcd(N , p) gcd(N , p) 10 11 12 N N N + + + . gcd(N , p) gcd(N , p) gcd(N , p) 4 1 In particular, when p = 0, gcd(N,0) = N , the partition function is reduced to Z (T × S ) = p=0 5 5 4 4 gcd(N , p) = (N ) =|G| as expected. 4.2. Counting vacua in any dimension for non-Abelian BdA + A We can discuss such non-Abelian TQFTs in any general dimensions. We first consider p = 1 theories, and the pattern is obvious: d d d d d d d 2 d d GSD | = 1 · N ··· N +(N − 1) · N · N ··· N +(N − 1) · (N − N ) · N · N ··· N d−1 p=1 d−2 d−3 d−4 d d d d−3 d−2 + ··· + (N − 1) · (N − N ) ··· (N − N ) · N (4.6) d−3 k d d−2 d i k+1 d d−(k+2)−1 = (N ) + (N − N )N (N ) . k=0 i=0 22/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. For general p, the pattern can be generalized; we have d(d−2) d−1 d−1 1 d GSD = Z (T × S ) = gcd(N , p) d−1 gcd(N , p) (4.7) !  2 d−3 k d i d −(k+3)d+k+1 N N N + − . gcd(N , p) gcd(N , p) gcd(N , p) k=0 i=0 d−1 1 d−1 When p = 0, we have Z (T × S ) =|G| as expected. p=0 All these examples, including Sects. 3.5, 3.6, and 4 of BdA + A type, are non-Abelian TQFTs d−1 due to the GSD reduction from |G| to a smaller value. This can be understood as the statement that the quantum dimensions d of some anyonic excitations are not equal to but greater than 1, i.e., d > 1[10]. d−1 By the same calculation, we obtain that the GSD on T of the theory with the action in Eq. (3.22) with N = 1isGSD = 1 (no intrinsic topological order in this case). 5. Fermionic spin TQFTs from gauged fermionic SPTs and ground state degeneracy The gapped theories in d + 1 dimensions with fermionic degrees of freedom can be effectively described in terms of d + 1D spin TQFTs. Unlike in the bosonic case, the partition function of a spin d+1 d+1 TQFT on a (d + 1)-manifold M depends not just on the topology of M , but also on a choice 1 d+1 of spin structure. If a spin structure exists, there are H (M , Z ) different choices. Similarly, the Hilbert space H d depends on the choice of spin structure on the spatial manifold M . Moreover, H d can be decomposed into fermionic (f ) and bosonic (b) parts: H = H ⊕ H . (5.1) M d M M Equivalently, H is a Z -graded vector space. When we give results about H in particular d d M M examples we will use the following condensed notation: GSD = n f + n b, n ∈ Z (5.2) f b f ,b + f ,b where n ≡ dim H . In general the fermionic and bosonic GSDs n and n can be determined f ,b f b from the following partition functions of the spin TQFT: d 1 Z (M × S ) = Tr 1 = n + n , (5.3) H b f A d d 1 F Z (M × S ) = Tr (−1) = n − n , (5.4) b f P d where A/P denote anti-periodic/periodic boundary conditions on fermions along the time circle S (i.e., even/odd spin structure on S ). f f 5.1. Examples of fermionic SPTs and spin TQFTs: Z × Z and (Z ) × Z in 2+1D, 2 2 2 2 f f Z × Z × Z and (Z ) × Z in 3+1D 4 2 4 2 2 In this section, we consider fermionic spin TQFTs arising from gauging a unitary global symmetry of fermionic SPTs (fSPTs), set up in Ref. [7] (with some corrections and improvements, the basic idea remaining the same). More 2+1D/3+1D spin TQFTs are given later in Sect. 6.2. A systematic study (using the cobordism approach) of fermionic SPTs with finite group symmetries and the 23/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. Table 2. Table of d + 1D (d = 2 or 3) spin TQFTs, and their corresponding GSDs. The expressions in the 1 d+1 fourth column are actions for cochain spin TQFTs with a ∈ H (M , Z ). Note that there seem to be no i n n 4 n purely fermionic SPTs in (3+1) dimensions with just Z global symmetry, due to the fact that  (BZ ) = 2 spin 2 4 n 4 n (BZ ) = H (BZ , U (1)). This also suggests that there is no gauged version of intrinsic fermionic spin SO 2 2 TQFTs of a gauge group Z ; thus their gauged TQFTs are Dijkgraaf–Witten theories and their continuum bosonic TQFTs discussed in Sect. 3. In contrast, in Ref. [87], we obtain torsion parts of the above cobordism (BZ ) Z 2 8 ⎪ spin ⎨ 3 2 2 (B(Z )) Z × Z spin 2 8 groups as , which introduce additional new intrinsic interacting fermionic ⎪  (B(Z × Z )) Z × Z 2 4 2 4 ⎪ spin 2 2 (B(Z )) Z × Z spin 4 4 SPTs, beyond group cohomology and bosonic SPTs. The fourth column shows their topological terms (SPT invariants) that generate the intrinsic interacting fermionic SPTs. We gauge their global symmetries (leaving Z remained ungauged) to study their fermionic spin TQFTs, and GSDs in the last three columns. In particular, the odd generators for the 1), 2), and 4) gauged spin TQFTs are indeed non-Abelian fermionic topological orders. The reason for this is that Abelian topological orders, say 3), yield GSD d =|G| . However, non-Abelian topological orders yield the reduction GSD d < |G| ; see Sect. 1.2. In Ref. [87], we also compute cobordism 4 2 4 2 4 3 groups of  (B(Z × Z )),  (B(Z × Z )), and  (B(Z )), etc., which suggest the classification of 4 2 spin spin spin 2 4 4 fSPTs and new additional topological terms. Spin TQFTs d d Dim Group from gauging fSPTs: T (all P) T (other) RP Action (formal notation) 1) 2+1D Z × Z a ∪ ABK 3f 3b – 2 π 2) 2+1D (Z ) × Z a ∪ a ∪˜ η 6f + 1b 7b – 2 1 2 3) 3+1D Z × Z × Z (a mod 2) ∪ (a ∪ ABK) 512b 512b 3b 4 2 2 1 2 4) 3+1D (Z ) × Z π (a mod 2) ∪ (a mod 2) ∪ Arf 64 · (42f + 1b) 64 · 43b 4b 4 1 2 corresponding fermionic gauged theories will be given in Ref. [87]; here we will just use some of the results from that work. Previously, Refs. [88,89] (and references therein) studied the classifications of 2+1 interacting fSPTs involving finite groups. Recently, Refs. [57–59] and L. Fidkowski et al. (manuscript in prepara- tion) studied pertinent issues of 3+1 interacting fSPTs. Reference [57] provides explicit 3+1D fSPTs and their bosonized TQFTs. The bosonization is performed by dynamically gauging the fermion parity Z , which results in bosonic TQFTs (or non-spin TQFTs). In contrast, in our work, we only dynamically gauge the (finite unitary onsite) symmetry group but leave the Z global symmetry intact, which results in fermionic spin TQFTs. Reference [58] provides fixed-point fSPT wavefunctions for 3+1D interacting fermion systems and generalized group super-cohomology theory. Reference [59] uses the gauged fSPTs and their braiding statistics to detect underlying nontrivial fSPTs and propose their tentative classifications. L. Fidkowski et al. (manuscript in preparation) studies the surface TQFTs for 3+1D fSPTs. We briefly summarize the results for the examples considered in the current paper in Table 2. The precise meaning of the expressions for the actions is explained below (see Ref. [87] for details). Note that the theories considered here do not have time-reversal symmetry, so the spacetime manifold d+1 M is considered to be oriented. 1) a ∪ ABK ≡ ABK[PD(a)]∈ Z (5.5) 24/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. where PD(a) is a smooth, possibly non-orientable submanifold in M representing the Poincaré dual 1 3 2 3 to a ∈ H (M , Z ) (it always exists in the codimension-1 case). Given a spin structure on M , the − 16 submanifold PD(a) can be given a natural induced Pin structure; see Ref. [90]. ABK denotes a Z -valued Arf–Brown–Kervaire invariant of surfaces with Pin structure (the invariant that provides Pin the explicit isomorphism  = Z ). 2) a ∪ a ∪˜ η ≡˜ η(PD(a ) ∩ PD(a )) ≡ f 3 (a , a ) ∈ Z . (5.6) 1 2 1 2 1 2 4 As before, by PD(a ) we mean a smooth submanifold representing the Poincaré dual to a .By η ˜ we i i denote a Z -valued invariant associated with a 1D submanifold equipped with an additional structure (induced by spin structure on M as well as by embedded surfaces PD(a )). Its value, f 3 (a , a ), i 1 2 with 1 3 1 3 f 3 : H (M , Z ) × H (M , Z ) → Z (5.7) 2 2 4 can be concretely defined as follows [90]. Take a ∈ H (M , Z ). As has already been discussed 1,2 3 2 − − in case 1), spin structure on M induces Pin structures q on PD(a ). Pin structures on Riemann 3 i i surfaces can be understood as quadratic enhancements of intersection form on H (PD(a ), Z ): 1 i 2 q : H (PD(a ), Z ) −→ Z . (5.8) i 1 i 2 4 Then f : (a , a ) −→ q ([PD(a ) ∩ PD(a )]). (5.9) 1 2 1 1 2 Note that in fact f is symmetric under a ↔ a . More geometrically the value f (a , a ) can be 1 2 1 2 understood as follows. Choose a trivialization of the normal bundle to PD(a ) ∩ PD(a ) such that 1 2 the induced spin structure makes this 1D manifold a spin boundary. Then f (a , a ) is the number 1 2 of half-twists modulo 4 (only the mod 4 value is independent of the choices made) of the section of the normal bundle inward to PD(a ) or PD(a ). Note that 1 2 f 3 (a , a ) = a ∪ a ∪ a = a ∪ a ∪ a mod 2. (5.10) 1 2 1 1 2 1 2 2 3 3 M M 3) (a mod 2)∪(a ∪ABK) ≡ a | ∪ABK mod 4 ∈ Z (5.11) 1 2 2 PD(a mod 2) 4 M PD(a mod 2) 16 2 3 3 The idea is that the normal bundle to the submanifold PD(a) ≡ N ⊂ M for oriented M can be realized 2 3 2 2 as a determinant line bundle det TN , so that TM | 2 = TN ⊕ det TN . For a general vector bundle V , there is a natural bijection between Pin structures on V and spin structures on V ⊕ det V . 17 3 2 2 The corresponding cobordism group is  (B(Z )) = Z × Z . The presented action corresponds to spin 2 8 the generator of the Z factor. More generally in Ref. [87], we obtain torsion parts of cobordism groups as, 2 3 2 4 3 2 n −n n −3n +2n n +2n +11n −14n 3 n n 2 6 4 n 24 (BZ ) = Z ⊕ Z ⊕ Z and  (BZ ) = Z . The latter coincides with the group spin 4 2 spin 2 2 8 2 cohomology result [12]. 18 4 The corresponding cobordism group is  (B(Z × Z )) = Z × Z . The presented action corresponds 2 4 2 4 spin to the generator of the Z factor. 25/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. where PD(a mod 2) is a 3D smooth submanifold in M representing the Poincaré dual to 1 4 1 4 (a mod 2) where mod 2 : H (M , Z ) → H (M , Z ) is the part of the long exact sequence 1 4 2 induced by the short exact sequence 0 → Z → Z → Z → 0. Note that PD(a mod 2) can be 2 4 2 1 1 4 4 chosen to be orientable, because a mod 2 = 0 implies that a ∈ H (M , Z ) = H (M , Z ) is not 1 1 4 3 4 2-torsion. Moreover, a determines the orientation on PD(a mod 2) via trivialization of the normal 1 1 bundle. In particular, the trivialization of the normal bundle allows spin structure to be induced on PD(a mod 2) from spin structure on M . The formula (5.11) then defines the invariant in terms of the invariant considered in case 1) modulo 4 (only the mod 4 value is invariant under the choice of oriented PD(a mod 2)). Note that, equivalently, (a mod 2) ∪ (a ∪ ABK) = f (a , a ) ∈ Z (5.12) 1 2 PD(a mod 2) 2 2 4 with f defined as in case 2). 4) (a mod 2) ∪ (a mod 2) ∪ Arf ≡ Arf(PD(a mod 2) ∩ PD(a mod 2)) ∈ Z (5.13) 1 2 1 2 2 2 2 where Arf( ) denotes the Z -valued Arf invariant of the spin surface  . The spin structure on 2 4 = PD(a mod 2) ∩ PD(a mod 2) is induced as follows from the spin structure on M .As in 1 2 3 4 case 3), one can first consider a 3D oriented submanifold M = PD(a mod 2) ⊂ M with induced spin structure. By a similar argument  = PD(a | 3 mod 2) is also oriented and gets induced spin structure from M . 5.2. 2+1D spin TQFTs from gauging Ising-Z of Z × Z symmetry 2 2 A 2+1D example is a spin TQFT obtained from gauging unitary Ising-Z of Z × Z symmetric 2 2 fSPTs. Before gauging, this represents a class of 2+1D fermionic topological superconductor with a Z classification. The Z denotes a fermion number parity symmetry. After gauging, the TQFTs are identified in Table 2 of Ref. [7], matching the mathematical classification through the cobordism spin group  (BZ ) = Z in Ref. [76]. The exactly solvable lattice Hamiltonian constructions for 2 8 (un-gauged) SPTs [91] and for gauged theory [92] have been recently explored. Here we follow a TQFT approach following Ref. [7]. Given a class ν ∈ Z , the corresponding spin-TQFT partition function reads πiν 3 ABK[PD(a),s| ] PD(a) Z (M , s) = e . (5.14) 1 3 a ∈H (M ,Z ) The partition function is defined on a closed 3-manifold M with spin structure s ∈ spin(M ) 1 3 with a dynamical Z gauge connection a ∈ H (M , Z ), summed over in the path integral. As 2 2 already mentioned, ABK[···] is the Z -valued Arf–Brown–Kervaire (ABK) invariant of PD(a),a (possibly non-orientable) surface in M that represents a class in the H (M , Z ) Poincaré dual to 3 2 2 1 3 − a ∈ H (M , Z ). The s| is the Pin structure on PD(a) induced by s as described in 1) above. 2 PD(a) Note that there is no good local realization of the ABK invariant via characteristic classes. 19 4 2 2 The corresponding cobordism group is  (B(Z )) = Z × Z . The presented action corresponds to the spin 4 4 generator of the sZ factor. 26/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. 2 3 Fig. 4. Counting GSD on T , i.e., Tr 1. The shaded 2-tori embedded into a T represent Poincaré duals to o 2 1 3 elements of H (M , Z ) = Z . The red letters A/P denote anti-periodic/periodic boundary conditions on the 3 2 embedded 2-tori. To compute GSD on T for a spin TQFT, we have to specify choices of spin structure on the spatial 2-torus T . There are 4 choices corresponding to periodic or anti-periodic (P or A) boundary conditions along each of the two 1-cycles: (P,P), (A,P), (P,A), (A,A). It turns out that Hilbert space only depends on the parity (i.e., the value of the Arf invariant of T ). It is odd for (P,P), and even for (A,P), (P,A), (A,A). This is consistent with the fact that MCG(T ) = SL(2, Z) only permutes spin structures with the same parity. We will denote the corresponding two equivalences classes of 2 2 spin 2-tori as T and T . As described at the beginning of this section, the GSD is determined by o e 3 3 3 2 1 1 the partition function Z (T , s), with M = T = T × S . The time circle S can have P e|o time time or A boundary conditions. Consider, e.g., the choice of odd spin structure on T and anti-periodic boundary condition on S . Then, as shown in Fig. 4, 1 πiν 1 2 1 ABK[PD(a)] πiνArf[PD(a)] Tr 1 = Z (T × S ) = e = e = 2 o A 2 2 3 3 1 3 ∼ 1 3 ∼ a∈H (T ,Z ) Z a∈H (T ,Z ) Z 2 = 2 = 2 2 1 3, ν = 1 mod 2 = (1 + (−1) + 1 + 1 + 1 + 1 + 1 + 1) = (5.15) 2 4, ν = 0 mod 2 where we have used the fact that ABK = 4Arf for oriented surfaces, where Arf is the ordinary Arf invariant of spin 2-manifolds. Similarly, −3, ν = 1 mod 2 F 2 1 ν Tr (−1) = Z (T × S ) = (1 + 7(−1) ) = , (5.16) 2 o P 2 4, ν = 0 mod 2 which means that for ν = 1 mod 2 all states are fermionic. For the even spin structure on T we have: 3, ν = 1 mod 2 2 1 ν Tr 1 = Z (T × S ) = (7 + (−1) ) = (5.17) 2 e A 2 4, ν = 0 mod 2 27/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. 1 3, ν = 1 mod 2 F 2 1 ν Tr (−1) = Z (T × S ) = (7 + (−1) ) = . (5.18) 2 e P 2 4, ν = 0 mod 2 So all states are bosonic. The result for all allowed spin structures can be summarized as follows: 2 2 For odd ν: 3 fermionic states for T and 3 bosonic states for T . o e 2 2 For even ν: 4 states, all bosonic, for both T and T . o e We can implement similar counting for the other 2+1D and 3+1D spin TQFTs given in Ref. [7]. Notice that at least for 2+1D fermionic topological orders/TQFTs up to some finite states of GSD were classified in Ref. [93]. Our GSD counting can be compared with Ref. [93]. So far we have focused on a 2+1D example, but more 2+1D/3+1D spin-TQFT examples are given in Sect. 6.2 using the dimensional reduction scheme for GSD counting. We summarize these fermionic TQFTs and GSD data in Table 2. 6. Dimensional reduction scheme of partition functions and topological vacua 6.1. Bosonic dimensional reduction scheme Here we perform the dimensional reduction of the TQFTs described in Sect. 2 for explicit bosonic examples, and match it to the data computed in Sects. 3 and 4. We write the decomposition in terms dD of Eq. (2.12), but it is also applicable to Eqs. (2.1) and (2.2). The notation C below means a d + 1D TQFT. For a 3+1D Z gauge theory reduction to 2+1D, we can write the continuum field theory form as 3D 2D C = 2C , (6.1) 2 2 BdA BdA 2π 2π 3D 2D or, equivalently, in terms of the cocycle C = 2C . The sub-index 1 means a trivial cocycle. The B 1 1 3D 2D fields represent a 2-form gauge field in C , but represent a 1-form gauge field in C . Here we take the compact z direction (among the x–y–z–t in 3+1D) as the compactification direction, and each sector comes from holonomy around the direction that is A = 0or π (in terms of the 1-cochain field a = 0 or 1) respectively. The resulting sectors of 2+1D Z gauge theories are equivalent in this case. See Fig. 3 for a physical illustration. 2 20 For a 3+1D twisted (Z ) gauge theory reduction to 2+1D, we obtain 3D 2D 2D 2D C = C ⊕ C ⊕ 2C . (6.2) 2 2 2 2 2 2 2 2 B dA +c A A dA B dA ( B dA +A dA ) ( B dA +A dA ) i i i i i i i i 122 1 2 2 2 2 1 2 2π 2π 2π 2π i=1 i=1 i=1 i=1 We can explain this easily by converting those continuous descriptions to the cochain field the- (12) ory description with 4-cocycles and 3-cocycles. Relevant cocycles are a 4-cocycle ω = 4,II (1) a ∪a ∪a ∪a a ∪a ∪a (a ) 1 2 2 2 1 1 1 1 (−1) in 3+1D; and also 3-cocycles 1, ω = (−1) ≡ (−1) , and 3,I (12) a ∪a ∪a 1 2 2 ω = (−1) in 2+1D. Here all the a (say a , a , etc.) are the Z -valued 1-cochain i 1 2 2 3,II Note that, to be precise, expressions like A dA are of a formal nature, since in general, on a manifold i j of nontrivial topology, A is not globally defined (as there can be nontrivial U (1) bundles). One possible way to treat this is to define A locally with possible “jumps” along codimension-1 loci. More rigorously, it can be treated using Deligne–Beilinson cohomology (see, e.g., Ref. [94]). In 3 dimensions (since any 3-manifold is a boundary of some 4-manifold), one can also extend the theory to 4 dimensions with the corresponding term F F , F = dA . i j i i 28/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. field. In this section, all these a , B, and A are dynamical fields, which we also need to sum over dD all configurations in the path integral in C to obtain long-range entangled TQFTs (instead of short-range entangled SPTs). Equation (6.2) above can be derived, effectively, as 3D 2D 2D 2D 2D C = C ⊕ C ⊕ C ⊕ C a ∪a ∪a ∪a 1 a ∪a ∪a (a +a )∪a ∪a 1 2 2 2 (a ) 1 2 2 1 2 2 2 (6.3) 2D 2D 2D = C ⊕ C ⊕ 2C . 1 a ∪a ∪a (a ) 1 2 2 In the first line, we decompose the 3+1D theory with respect to holonomies around the compactifying z direction (among the x–y–z–t in 3+1D), which are ( a , a ) = (0, 0), (1, 0), (0, 1), (1, 1). Then 1 2 z z we obtain the second line by field redefinition, or equivalently a SL(2, Z ) transformation, sending a + a → a in the last sector. 1 2 1 For 3+1D twisted (Z ) gauge theory reduction to 2+1D, we can also use the cochain field expression to ease the calculation: 3D 2D 2D 2D C = C ⊕ 4C ⊕ 6C a ∪a ∪a ∪a 1 a ∪a ∪a a ∪a ∪a +a ∪a ∪a j j j 1 2 3 4 k l k l k 2D 2D ⊕ 4C ⊕ C (6.4) a ∪a ∪a +a ∪a ∪a +a ∪a ∪a a ∪a ∪a +a ∪a ∪a +a ∪a ∪a +a ∪a ∪a j k l j k m j l m 1 2 3 1 2 4 1 3 4 2 3 4 2D 2D 2D = C ⊕ 10 C ⊕ 5 C . 1 a ∪a ∪a a ∪a ∪a +a ∪a ∪a +a ∪a ∪a +a ∪a ∪a j k l 1 2 3 1 2 4 1 3 4 2 3 4 In the first line, we get each sector on the right-hand side from each holonomy ( a , a , a , a ) ∈ Z around the compactifying z direction. We decompose the 16 sectors 1 2 3 4 z z z z 2 into a multiplet with multiplicities (1,4,6,4,1), where the first 1 selects a  = (a , a , a , a ) = 1,z 2,z 3,z 4,z (0, 0, 0, 0). The second 4 selects only one element of a  as 1 as nontrivial, given by the combinatory = 4. The third 16 selects only two elements out of a  as 1 as nontrivial, given by the combinatory 4 4 4 = 6. Similarly, the fourth = 4 and the fifth = 1 are selected. All these indices j, k, l, l , m 2 3 4 given above are dummy but fixed and distinct indices, selected from the set {1, 2, 3, 4}. In the second line of Eq. (6.4), it turns out that we can do an M ∈ SL(4, Z ) transformation in the dimensionally reduced sector, among the a  = (a , a , a , a ), to redefine the fields through M a  = a → a. The 1 2 3 4 second and third sectors turn out to be the same, via M = M . The fourth and fifth sectors turn 2↔3 out to be the same, via M = M : 4↔5 ⎛ ⎞ ⎛ ⎞ 1100 1111 ⎜ ⎟ ⎜ ⎟ 0100 0100 ⎜ ⎟ ⎜ ⎟ M = ⎜ ⎟ , M = ⎜ ⎟ ∈ SL(4, Z ). 2↔3 4↔5 2 ⎝ 0010 ⎠ ⎝ 0010 ⎠ 0001 0001 For example, we see that M can change (a + a ) ∪ a ∪ a to a ∪ a ∪ a ; thus we can combine 23 1 2 3 4 1 3 4 the 6 of the third sectors into the 4 of the second sectors. Overall, similar forms of M = M do the 2↔3 2D job to identify these 10 sectors as 10 equivalent copies of a TQFT written as 10 C . Similarly, a ∪a ∪a j k l we can use similar forms of M = M to identify the last fourth and fifth sectors, obtaining 5 4↔5 2D copies of a TQFT, written as 5 C . a ∪a ∪a +a ∪a ∪a +a ∪a ∪a +a ∪a ∪a 1 2 3 1 2 4 1 3 4 2 3 4 In terms of continuum gauge field theory, we can rewrite Eq. (6.4)as 3D 2D C = C 4 4 2 1 2 B dA + A A A A B dA i i 1 2 3 4 i i 2π 3 2π i=1 i=1 (6.5) 2D 2D ⊕ 10C ⊕ 5C . 4 4 2 1 2 1 B dA + A A A B dA + (A A A +A A A +A A A +A A A ) i i 1 2 3 i i 1 2 3 1 2 4 1 3 4 2 3 4 2π 2 2π 2 π π i=1 i=1 29/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. In terms of TQFT dimensional decomposition, Eq. (6.4)/Eq. (6.5) give the information that we can obtain based on the field theory actions. What other topological data can we obtain to check the decompositions in Eq. (6.5)? We can consider the following: 1. GSD data on T show that GSD 3 = GSD 2 T ,3+1D-TQFT T ,2+1D-TQFT (b) (6.6) ⇒ GSD = 1576 = 256 + 15 × (2 × 22). 2 1 T , B dA + A A A A i i 1 2 3 4 2π 3 i=1 2D The GSD data only distinguish the b = 0 trivial sector C with GSD 2 = 256 from 4 T B dA i i 2π i=1 the remaining 15 sectors. Each of the remaining 15 sectors has GSD = 88, which is the same as the tensor product (Z gauge theory) ⊗ (a non-Abelian D gauge theory) with a trivial DW 2 4 cocycle in 2+1D [10]. Therefore, GSD cannot distinguish the second and the last sectors of the decomposition (6.5). 2. GSD data on RP show that 3 3 1 1 GSD = 11 = 1 × 1 + 4 × + 6 × + 4 × + 1 × . 2 1 4 4 2 2 RP , B dA + A A A A i i 1 2 3 4 2π 3 i=1 (6.7) 3 1 = 1 × 1 + 10 × + 5 × . 4 2 3 1 2 1 In this case, we first compute Z (RP × S ) = 11 for B dA + A A A A . These data i i 1 2 3 4 2π i=1 match the dimensionally reduced 16 sectors of 2+1D TQFTs in terms of their Z (RP ), which we also compute. In terms of the 2+1D TQFT grouping in Eq. (6.4) as a multiplet (1,4,6,4,1), the first sector contributes Z (RP ) = 1, each of the second (4) and third (6) contributes 3 3 3 1 Z (RP ) = , and each of the fourth (4) and fifth (1) contributes Z (RP ) = . 4 2 3. We can also adopt additional data such as the modular T matrix of the SL(2, Z) representation, measuring the topological spin or the self-statistics of anyonic particle/string excitations, in Ref. [10]. The diagonal T matrix contains only four distinct eigenvalues, (1, −1, i, −i). We can specify a T matrix by a tuple of numbers containing these eigenvalues, as (N , N , N , N ). 1 −1 i −i We find 3D T in terms of (N , N , N , N ) = (836, 580, 80, 80) 1 −1 i −i 2 1 B dA + A A A A i i 1 2 3 4 2π 3 i=1 (6.8) = 1 × (136, 120, 0, 0) + 10 × (48, 32, 4, 4) + 5 × (44, 28, 8, 8). 2D 2D The 10 sectors of T with (N , N , N , N ) = (48, 32, 4, 4) are again the same as T of 1 −1 i −i the (Z gauge theory) ⊗ (a non-Abelian D gauge theory) in 2+1D. The overall structure of 2 4 3D 3 1 T decomposition agrees with Z (RP × S ) decomposition. In summary, Eq. (6.5) suggests that there are at most three distinct classes among the 16 sectors of dimensionally reduced 2+1D TQFTs, and the distinction among the three is guaranteed by the data of Z (RP ) and T matrices. 30/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. For untwisted gauge theories, one can derive: 3D 2D 2D 2D C = 2C ⊕ 2C ⊕ C , (6.9) 3+1D-D gauge 2+1D-D gauge 2+1D-Z gauge 4 4 2+1D-(Z ) gauge 4 3D 2D 2D C = 2C ⊕ 3C . (6.10) 3+1D-Q gauge 2+1D-Q gauge 2+1D-Z gauge 8 8 4 Each conjugacy class of holonomy around the compactifying circle gives the lower-dimensional theory with the maximal subgroup commuting with the holonomy as its subgroup. These results can be checked by the information in Refs. [10,73] and the GSD data in an earlier section. 6.2. Fermionic dimensional reduction scheme Based on the strategy in Sect. 2, we examine the dimensional decomposition of some of the spin TQFTs listed in Sect. 5. We obtain these spin TQFTs from gauging some global symmetries of fermionic symmetry-protected topological states (fSPTs). 6.2.1. 2+1D → 1+1D gauged fSPT reduction: 2+1D Z × Z fSPT and its gauged spin TQFT 2 2 Consider a 2+1D fSPT state with Z × Z symmetry and its partition function 2 2 i a ∪a ∪˜ η 1 2 Z = e (6.11) 2+1D fSPT where the precise definition of the action is spelled out in point 2) at the beginning of Sect. 5. When 3 2 1 implementing the theory on an M = M × S , depending on the spin structure and Z holonomies 1 2 along S , it reduces to a 1+1D fSPT with Z × Z symmetry of one of the 3 following types: 2 2 1. Trivial: Z = 1. (6.12) 1+1D fSPT I 2 1 1 The gauged theory has GSD = 4b (given by the partition function of M = S × S ) independently on the spin structure on S . πi (  α a )∪η ij i j i,j=1 Z = e , (6.13) 1+1D fSPT 1 2 where α ∈ Z are some parameters not all simultaneously zero, a ∈ H (M , Z ) describe i 2 i 2 background Z gauge fields, and  is the standard anti-symmetric tensor. The formal expression 2 ij for the action has the following definition: 1, odd spin structure on PD[a], a ∪ η ≡ η[PD(a)]≡ (6.14) 2 0, even spin structure on PD[a], where the spin structure on 1-manifold PD[a] is induced from the spin structure on M in an obvious way (cf. beginning of Sect. 5). Note that (a + a ) ∪ η = a ∪ η + a ∪ η + a ∪ a . (6.16) 1 2 1 2 1 2 2 2 2 2 M M M M Equivalently, η[PD(a)]= q([PD(a)])/2 (6.15) where q : H (M ) → Z is the quadratic enhancement of the intersection form. 1 4 31/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. II 1 II The gauged theory has GSD = 2 × (1f + 0b) for the odd spin structure on S and GSD = P A 2 × (0f + 1b) for the even spin structure on S . πi a ∪a +(  α a )∪η 1 2 ij i j i,j=1 Z = e . (6.17) 1+1D fSPT III 1 The gauged theory, for any α , has GSD = 1b independently on the spin structure on S . Namely, for the even spin structure and trivial Z holonomies along S , the 2+1D fSPT reduces to trivial (type-I) theory on M . For the odd spin structure and nontrivial Z holonomies, it reduces to a theory of type II with α = a . For the odd spin structure and trivial Z holonomies along i 1 i 2 S , or the even spin structure and nontrivial Z holonomies, it reduces to a theory of type III with α = a . i 1 i 3 3 3 Consider now the gauged 2+1D fSPT on M = T . Let us order the circles in T such that the first one is the time circle and the last one is the S on which we are doing the reduction. The GSD decomposition then reads as follows: III II GSD = GSD + 3 GSD = 1b + 3 × 2 × (1f + 0b) = 6f + 1b, PP P P III II GSD = GSD + 3 GSD = 1b + 3 × 2 × (0f + 1b) = 7b, AP A A (6.18) I III GSD = GSD + 3 GSD = 4b + 3 × 1b = 7b, PA P P I III GSD = GSD + 3 GSD = 4b + 3 × 1b = 7b. AA A A The decompositions of GSDs can be promoted to the decomposition of the spin-TQFT functor. For odd spin structure on S : 2D 1D 1D 1D 1D C = C ⊕ C ⊕ C ⊕ C = a ∪a ∪˜ η π a ∪a π a ∪η π a ∪η π (a +a )∪η 1 2 1 2 1 2 1 2 1D 1D C ⊕ 3C ; (6.19) π a ∪a π a ∪η 1 2 1 for even spin structure on S : 2D 1D 1D 1D 1D C = C ⊕ C ⊕ C ⊕ C = a ∪a ∪˜ η π a ∪a +a ∪η π a ∪a +a ∪η π a ∪a +(a +a )∪η 1 2 1 2 1 1 2 2 1 2 1 2 1D 1D C ⊕ 3C , (6.20) π a ∪a +a ∪η 1 2 1 where we have used field redefinitions to combine equivalent theories together. Note that all the summands except the first one give isomorphic Hilbert spaces on S . This is the reason for the factors of 3 in Eq. (6.18). 6.2.2. 3+1D → 2+1D gauged fSPT reduction: 3+1D Z × Z fSPT and its gauged spin TQFT 4 2 Consider a 3+1D fSPT state with Z × Z symmetry and its partition function 4 2 π (a mod 2)∪(a mod 2)∪Arf 1 2 Z = e (6.21) 3+1D fSPT 32/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. where the precise definition of the action is spelled out in point 4) at the beginning of Sect. 5. When 4 3 1 1 putting on M = M × S , depending on the spin structure and Z holonomies along S , it reduces 2 22 to a 2+1D fSPT with Z × Z symmetry of one of the following 3 types: 4 2 1. Trivial: Z = 1. (6.22) 2+1D fSPT I 2 The gauged theory has GSD = 256b independently on spin structure on T . πi (  α a mod 2)∪Arf ij i j i,j Z = e , (6.23) 2+1D fSPT 1 3 where α ∈ Z are some parameters not all simultaneously zero and a mod 2 ∈ H (M , Z ) i 2 i 2 II describes background Z gauge fields. The gauged theory has GSD = 16 × 12f for the odd PP 2 II II II spin structure on T and GSD = GSD = GSD = 16 × 12b for an even spin structure PA AP AA on T . πi (  α a mod 2)∪Arf+(a mod 2)∪(a mod 2)∪η ij i j 1 2 i,j Z = e (6.24) 2+1D fSPT III where α ∈ Z are not all simultaneously zero. The gauged theory, for any α , has GSD = i 2 i 16 × 9b independently on spin structure on T . Note that from the definition (5.5), we derive (b + b ) ∪ Arf = b ∪ Arf + b ∪ Arf + b ∪ b ∪ η. (6.25) 1 2 1 2 1 2 3 3 3 3 M M M M πi (a mod 2)∪(a mod 2)∪η 1 2 Z = e . (6.26) 2+1D fSPT IV 2 IV The gauged theory has GSD = 16 × (6f + 1b) for the odd spin structure on T and GSD = PP PA IV IV 2 GSD = GSD = 16 × 7b for an even spin structure on T . AP AA As for 2+1D → 1+1D reduction, for the even spin structure on S and trivial mod 2 holonomies 1 3 along S , the 3+1D fSPT reduces to trivial (type-I) theory on M . For the odd spin structure and nontrivial mod 2 holonomies it reduces to a theory of type II with α = a mod 2. For the odd i 1 i spin structure and trivial mod 2 holonomies along S it reduces to the type-IV theory. For the even spin structure and nontrivial mod 2 holonomies, it reduces to a theory of type III with α = a i 1 i mod 2. 4 4 4 Consider now 3+1D fSPTs on M = T . Let us again order the circles in T such that the first one is the time circle and the last one is the S on which we are doing the reduction. The GSD 22 3 2 2 3 The corresponding classifying cobordism group is  (B(Z )) = Z × Z × Z . Only the generators of spin 4 8 2 Z subgroups will appear in the decomposition below. 33/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. decomposition than reads as follows: IV II GSD = 4GSD + 12 GSD (odd)P (odd) (odd) = 4 × 16 × (6f + 1b) + 12 × 16 × 12f = 64 × (42f + 1b), IV II GSD = 4GSD + 12 GSD = 4 × 16 × 7b + 12 × 16 × 12b = 64 × 43b, (even)P (even) (even) I III GSD = 4GSD + 12 GSD = 4 × 16 × 16b + 12 × 16 × 9b = 64 × 43b, (odd)A (odd) (odd) I III GSD = 4GSD + 12 GSD = 4 × 16 × 16b + 12 × 16 × 9b = 64 × 43b, (6.27) (even)A (even) (even) where (odd) denotes even PP spin structure on T and (even) denotes any of the even spin structures, PA, AP, or AA, on T . The decompositions of GSDs can be promoted to the decomposition of the spin-TQFT functor. For odd spin structure on S , 3D 2D 2D C = 4C ⊕ 12C . (6.28) π (a mod 2)∪(a mod 2)∪Arf π (a mod 2)∪(a mod 2)∪η π (a mod 2)∪Arf 1 2 1 1 1 For even spin structure on S , 3D 2D 2D C = 4C ⊕ 12C . (6.29) π (a mod 2)∪(a mod 2)∪Arf π (a mod 2)∪Arf+(a mod 2)∪Arf 1 2 1 2 We have used field redefinitions to identify equivalent theories. Note that all the summands except the first one give isomorphic Hilbert spaces on T . This is the reason for the factors of 7 in Eq. (6.27). This dimensional decomposition method can be applied to all examples given in Table 2. 7. Long-range entangled bulk/boundary coupled TQFTs Now we consider the bulk/boundary coupled TQFT system. In the work of Ref. [26], for a given bulk d+1 d + 1D G-symmetry-protected phase characterized by a Dijkgraaf–Witten (DW) cocycle ω ∈ d+1 H (BG, U (1)),a dD boundary K gauge theory coupled with the d + 1D bulk SPT is constructed via a so-called group-extension or symmetry-extension scheme. The groups G and K form an exact sequence 1 → K → H → G → 1, ∗ d+1 d+1 ∗ such that r ω = 1 ∈ H (BH , U (1)) where r is the pullback of the homomorphism r. The r is a surjective group homomorphism. The H is a total group associated with the boundary. To help readers to remember the group structure assignment to the bulk/boundary, we can abbreviate the above group extension as 1 → K → H → G → 1. (7.1) boundary boundary bulk This structure is used throughout Sect. 7. In the appendix of Ref. [26], some examples of GSDs are computed for both the bulk SRE (ungauged) case and the bulk LRE (dynamically gauged) case, based on the explicit lattice spacetime path integral. Here we examine some examples given there, and will argue that when the bulk is gauged, some of the boundary degrees of freedom “dissolve” into the bulk. In other words, we will show that after gauging the whole By dissolve, we mean that the boundary operators can move into the bulk, without any energetic penalty. 34/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. (a) (b) Fig. 5. (a) The “re-gauging” of a 1+1D (or 2D) Abelian finite K gauge theory [95,96]. In general, a group surrounded by a circle means a gauged symmetry, and a group surrounded by a square means a global symmetry. There is a global non-anomalous K group acting faithfully on the theory, which is shown by a K surrounded by a square. When K is gauged, the whole system becomes trivial, meaning that the Hilbert space is 1D on any topology. (b) This case is what we focus on in Sect. 7.1. We start from 1+1D K gauge theory coupled with anomalous G symmetry. The anomalous symmetry is thought to be realized as an SPT phase in 2+1D. After gauging the bulk G symmetry, the resulting system would be equivalent to a 2+1D G-theory with some boundary condition breaking G into a subgroup G and without coupling with a 1+1D system. There could possibly be a decoupled K gauge theory on the boundary. In the examples, however, we may regard K as absent. system, a certain group-extension construction in Eq. (7.1) is actually equivalent (dual or indis- tinguishable) to a group-breaking construction also explained in Ref. [26] associated with an inclusion ι: G → G , (7.2) bulk boundary ∗ d+1 where G is a subgroup that G breaks to, and the inclusion should satisfy ι ω = 1 ∈ d+1  ∗ H (BG , U (1)) where ι is its pullback. The annotations below G and G indicate that site/link variables are valued in those groups in boundary and bulk respectively, as was the case in Eq. (7.1). Some heuristic reasoning for this is as follows. This is a generalization of the statement of Refs. [95, 96] that a 1+1D gauge theory, with an Abelian finite gauge group K but without a Dijkgraaf–Witten cocycle twist, has a global symmetry group isomorphic to K , and when the global K is further gauged, the resulting theory is trivial [95,96]. In the setup given by Eq. (7.1), the K gauge theory on the boundary is coupled with anomalous G symmetry. Gauging the bulk G symmetry reduces the boundary degrees of freedom as in the pure 1+1D setup. When K is small enough, the boundary degrees of freedom can even be completely gauged away, and no boundary degrees of freedom remain. Before gauging, the bulk G symmetry has the G-preserving boundary condition and is coupled with the boundary degrees of freedom. However, when the boundary K gauge theory is gauged away by gauging the bulk G, the whole bulk/boundary coupled system should be equivalent to just bulk G symmetry with some boundary condition without being coupled with a 1+1D system, possibly accompanied by a decoupled 1+1D system on the boundary. Namely, we stress the following: There is an equivalence, only after gauging, between “a certain bulk/boundary coupled system” and “the bulk system with only some boundary conditions.” For such a system to be consistent, the boundary condition should break G into some non-anomalous subgroup G . This discussion is summarized in Fig. 5. 35/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. Table 3. We re-examine the d + 1D bulk/dD boundary coupled system based on a group extenssion con- struction, developed in Ref. [26], in terms of more field theoretic understandings in Sect. 7. The system that we analyze the most is system (iii)’s LRE/LRE bulk/boundary TQFT. LRE/SRE stands for long-/short-range entangled. SPT/SET stands for symmetry-protected/enriched topological states. We will especially comment on the gauging process (say, from system (i) to (ii), or (ii) to (iii)), and especially focus on the issue still left open: the boundary conditions and some of their dualities to breaking construction (7.2), after gauging G . bulk d + 1D bulk/dD boundary System Group-extension construction entanglement property global sym global sym global sym System (i) SRE/SRE SPT/symmetry 1 → K → H → G → 1 boundary boundary bulk gauge global sym total System (ii) SRE/LRE SPT/SET(TQFT) 1 → K → H → G → 1 boundary boundary bulk gauge gauge gauge System (iii) LRE/LRE TQFT/TQFT 1 → K → H → G → 1 boundary boundary bulk In the rest of this section, we show how the above scenario occurs in more detail in two examples, 1 1 24 in Sects. 7.1 and 7.2. We will then compute partition functions on the I × S topology in two ways: using the explicit lattice path integral model coming from Ref. [26]’s Eq. (7.1) and using the nontrivial boundary condition on G with no boundary degrees of freedom. Furthermore, we will try to generalize the discussion to a 3+1D/2+1D system, in Sect. 7.3. However, we will see that a certain exotic type of boundary condition of the bulk theory occurs after the bulk gauging. Complete understanding of the higher-dimensional case is left for future work. We clarify that in the discussions of Sect. 7, when we use “breaking” this means breaking the (gauge/global) symmetry with respect to the electric sector (instead of the magnetic sector), and when we use “preserving” this means preserving the (gauge/global) symmetry with respect to the electric sector (instead of the magnetic sector), too. We will also use the language of Ref. [26], summarized in Table 3, throughout Sect. 7. 7.1. 2+1/1+1D LRE/LRE TQFTs: Gauging an extension construction is dual to a gauge-breaking construction Let us start from the easiest case as a warm-up, where the bulk is a 2+1D G = Z gauge theory. Namely, the bulk is the Z gauge theory of field A (represented by a Z -valued 1-cochain) with 2 2 A∪A∪A 25 the unique nontrivial cocycle (−1) . This cocycle can be canceled by a boundary cochain when the boundary has a K = Z gauge theory, as shown in Ref. [26]. In this case the sequence (7.1)is K H G 1 → Z → Z → Z → 1. (7.3) 2 boundary 4 boundary 2 bulk We use the upper indices G to denote the group for the bulk, and the indices K and H to denote the groups for the boundaries following Eq. (7.1) and Ref. [26]. In Ref. [26], the GSD on D is computed 24 1 1 d−1 1 d−2 1 I is an interval. I × T = (I × T ) × S can be regarded as (an annulus or cylinder) × (a torus x time topology) in space, then × (a compact time). Most of the discussion in this subsection does not rely on the bulk Z gauge field having a nontrivial DW action. Here we assume a nontrivial DW action just because nontrivial DW terms will be important in the rest of the section. 36/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. 2 1 K to be Z (D × S ) = 1 when both bulk and boundary are dynamical. This hints that the boundary Z degrees of freedom are actually absent when the bulk Z is gauged. Below we aim to show that, only after gauging Z , this extension construction (7.3) becomes equivalent to the breaking construction (7.2) (also formulated in Ref. [26]) as G G 1 → Z . (7.4) boundary 2 bulk We use the upper indices G to denote the preserved group for the boundary as Eq. (7.2). In brief, it can be explained as follows. On the boundary, there is a vortex operator φ(x) localized K G at a point x. For the boundary Z gauge theory to be coupled with the bulk Z symmetry, the 2 2 operator should be shifted under the bulk Z transformation φ → φ + λ, where λ is a Z -valued 2 2 parameter of the bulk symmetry transformation. Then, we have an operator invariant under the bulk Z transformation exp(iπ( A − φ(x ) + φ(x ))), (7.5) 1 2 G G where A is the bulk Z field, x and x are the boundary points. When the bulk Z is gauged, the 1 2 2 2 boundary vortex operator φ is gauged out; therefore, there is no longer a Z degeneracy on the boundary, and the bulk electric electric Z Wilson line can end on the boundary. Thus, the whole system is indistinguishable from just a bulk Z gauge theory with a boundary condition breaking the electric Z , without any additional degrees of freedom on the boundary. We give a more explicit explanation in the following. Before gauging the bulk Z , the partition 3 G 2 K function of the full spacetime, with bulk M of 2+1D Z SPTs and boundary (∂M ) of 1+1D Z 2 2 gauge theory, is A∪A∪A φδα+α∪A+φA∪A M ∂M Z = (−1) (−1) , (7.6) 1 2 α∈C ((∂M ) ,Z ), 0 2 φ∈C ((∂M ) ,Z ) where φ and α are Z -valued 0-cochain and 1-cochain fields respectively. We denote all such Z - 2 n valued m-cochain fields on the spacetime manifold M in the cochain C (M, Z ). The Z depends on n A the background Z field A, and only φ and α are dynamical here. Under their gauge transformations, A → A + δλ, φ → φ + λ, and α → α + λδλ, with λ being an integral 0-cochain (λ ∈ C (M , Z )), similar to the gauge-invariant calculation done in Ref. [75], we find that the Z is gauge invariant. After gauging the bulk Z , we propose that the partition function of the full spacetime, with bulk 3 G H K M of 2+1D Z gauge theory with a boundary (of 1+1D Z gauge theory also including the Z 2 4 2 gauge sector), in a continuum field description, is 1 1 i( (2BdA+AdA)+ (2φdB+2BA+φdA)) 2π 2π −1 3 ∂M Z = N Z = [DA][DB][Dφ] e . (7.7) Here we use continuum field notations, where A and B are 1-form gauge fields, and φ becomes a 0-form scalar. The whole partition function Z is gauge invariant, under A → A + dη , B → B + dη , A B and φ → φ − η , where η /η are local 0-forms. The α as a 1-cochain field in Eq. (7.6) is related A A B to the B as the continuum 1-form field in Eq. (7.7). We give several remarks in order to explain the 37/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. gauging Z process: 1. SRE/SRE bulk/boundary: Starting from system (i), SRE/SRE bulk/boundary in Table 3,as G H shown in Ref. [26], this is a Z SPT in a bulk, while it has a Z -symmetry extended boundary. 2 4 All global symmetries are preserved and unbroken. 2. SRE/LRE bulk/boundary ⇒ SRE/(SRE+LRO) bulk/boundary: After gauging the Z on the boundary, we arrive at system (ii)’s SRE/LRE bulk/boundary in Table 3, whose partition func- tion is Z in Eq. (7.6). In Sect. 3.3 of Ref. [26], it is found that the two holonomies of Z (or two 2 G ground states on a disk D for this system (ii)) have different Z -symmetry charge. The trivial K G K holonomy of Z has a trivial (no or even) Z charge. The nontrivial holonomy of Z has an odd 2 2 2 G K Z charge. We find that this fact can be understood as Eq. (7.6)’s Z having the Z -holonomy 2 2 α∪A G 26 K ∂M α coupled to the Z -background field in (−1) . Such a Z gauge theory turns out to 2 2 develop Z -spontaneous global symmetry breaking (SSB) long-range order (LRO) [26]. Thus, it turns out that this SRE/LRE bulk/boundary by design turns into an SRE/SRE bulk/boundary, because the Z -SSB boundary has a gapped edge, which has LRO (but no Goldstone modes) but is SRE. G K 3. LRE/LRE bulk/boundary: After gauging the Z in the bulk (the boundary Z is also gauged), 2 2 we arrive at system (iii)’s LRE/LRE bulk/boundary in Table 3, whose partition function we propose as Z in Eq. (7.7). By massaging Eq. (7.7), we obtain 2 1 i( (A+dφ)dB+ (A+dφ)d(A+dφ)) 2π 2π 3 3 M M Z = [DA][DB][Dφ] e . (7.8) From this expression of the partition function we can make several physical observations and predictions listed below. (1) When we gauge the bulk’s Z , A, φ, and α (cochain fields of Eq. (7.6)) become dynamical. This yields A + δφ with no gauge transformation on the boundary; thus B integration implies (A + δφ)| = 0. (7.9) ∂M The dynamical vortex field φ (at the open ends of A) becomes deconfined on the boundary. This can be viewed as the Z electric charge particle (the e anyon) becoming deconfined and condensed on the boundary. By the anyon being condensed on the boundary, we mean that there can be a nontrivial expectation value exp(iφ) = 0 (7.10) for the ground state(s), since the φ are freely popped up and absorbed into the boundary. Thus, G G gauging the bulk’s Z causes the Z gauge symmetry to be broken on the boundary. 2 2 (2) We can (and later will) also read the boundary condition directly from the cochain fields in Eq. (7.6). The Z -SPT partition function indicates the following boundary condition after 26 G G When the bulk-Z is not gauged and therefore treated as an Z -SPT state, the interpretation of the operator 2 2 (7.5) is different. In that case, if the probe operator A ends on the boundary, it changes the boundary vacuum to a different state. Note that the φ is only defined on the boundary ∂M , but an arbitrary extension into the bulk give a unique action. 38/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. A∪A∪A Table 4. The GSDs of the LRE/LRE bulk/boundary TQFT theory of dynamically gauged (−1) SPT. The boundary theory is constructed via Eq. (7.3) in Ref. [26]. Our Sect. 7.1 can explain the GSD data in terms of a field theoretic description. LRE/LRE 2+1D bulk/1+1D boundary coupled TQFTs 2 1 3 2 1 1 2 Z (M × S ) Z (T ) Z (D × S ) Z (I × T ) 2 2 1 1 Spatial topology T D I × S GSD 4 1 2 gauging Z : A| = 0, A ∪ A| = 0, ∂M ∂M in terms of cohomology. For example, integrating out α in Eq. (7.6) forces A to be exact on the boundary. The first condition is equivalent to Eq. (7.9), while the second condition automatically holds at the path integral after imposing the first condition. (3) After gauging Z , we expect that all boundary operators can be dissolved into the bulk. This means that the apparent boundary operator (−1) , where C is a 1-cycle in ∂M , should be G G identified with the magnetic Z line operator in the bulk, since the electric Z is broken on the 2 2 boundary, as we saw. Thus we can physically understand the conversion from α (1-cochain field) to B (1-form magnetic Z field), from Eq. (7.6)’s Z to Eq. (7.7)’s Z, only after gauging 2 A G 2 1 the Z . This agrees with the fact that the GSD is Z (D × S ) = 1 in Ref. [26]. 1 1 (4) As an additional check, we consider GSD on I ×S . From Ref. [26], the lattice computation 1 2 shows Z (I × T ) = 2. This is consistent with the alternative description of the symmetry- breaking boundary condition, since the A line can have a nontrivial Z value along the 1 1 interval I , while it cannot have nontrivial holonomy along the spatial S . (This coincides with the gauge symmetry-breaking boundary condition explored in, e.g., Table II of Ref. [19].) See also Table 4. In summary, the LRE/LRE bulk–boundary coupled TQFT systems given by the exact sequence G G (7.3), only when Z is gauged, are equivalent to just the Z -breaking boundary condition, namely, 2 2 double semion condensation (see Ref. [19]), when both bulk and boundary groups are gauged. The gauge symmetry-breaking condition is given in Ref. [26] and Eq. (7.4)as1 → Z . In 2+1D/1+1D LRE/LRE bulk/boundary, the underlying physics of Eqs. (7.3) and (7.4) coincides with double semion condensation. For example, we can write the bulk gauge theory as a twisted Z gauge theory 2 1 or Z double-semion topological order, BdA + AdA, then double semion condensation can 2π 2π be achieved by the A| = 0 boundary condition in Eq. (7.9). We can gap the boundary by turning ∂M on the cosine sine-Gordon term g dtdx cos(2φ) (7.11) at a strong coupling g, where the scalar field φ(x, t) is the same vortex operator mentioned above. 1 20 Under SL(2, Z) field redefinition, we can rewrite the bulk theory as A dA , then we 0 −2 4π I J IJ can gap the boundary by the cosine term of the vortex field φ of A [19]: I I g dtdx cos(2(φ + φ )) (7.12) 1 2 39/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. at a strong coupling g. See the comparison of the physical setup of double semion condensation,or more precisely the condensation of a semion s and an anti-semion s ¯ in Ref. [19]. 7.2. 2+1/1+1D LRE/LRE TQFTs: Gauging an extension construction is dual to a partially gauge-breaking construction We would like to generalize the argument in the previous subsection into a more nontrivial case, namely a 2+1/1+1D coupled system associated with the following exact sequence: K H 3 G 1 → (Z ) → (D × Z ) → (Z ) → 1, (7.13) 2 4 2 boundary boundary bulk where the leftmost (Z ) goes into the order-8 non-Abelian dihedral group D , and the Z in the 2 4 2 3 G middle just become one of the factors of (Z ) . Again the upper index G denotes the group for the bulk, and the indices K and H denote the groups for the boundaries following Eq. (7.1) and Ref. [26]. The bulk Z gauge fields A , A , A have the exponentiated action 1 2 3 A ∪A ∪A 1 2 3 (−1) . (7.14) To cancel the anomaly induced by the bulk, the boundary dynamical Z cochain fields α and φ have the coupling A ∪A ∪A (φδα+α∪A +φA ∪A ) 1 2 3 1 2 3 M ∂M Z = (−1) (−1) . (7.15) 1 2 α∈C ((∂M ) ,Z ), 0 2 φ∈C ((∂M ) ,Z ) This indicates the boundary conditions A = 0, A ∪ A = 0 (7.16) 1 2 3 for the LRE/LRE system. 1 2 Let us compute the partition function on I × T with the boundary condition (7.16), and compare it with the result from the method in the appendix of Ref. [26]. To be precise, the boundary condition “A = 0” means that the field A is an element of the relative cohomology H (M , ∂M ; Z ), while 1 1 2 keeping A inside H (M ; Z ). The second condition of Eq. (7.16) will be imposed by the path 2,3 2 integral, as we will see. Then, the partition function is A ∪A ∪A 1 2 −1 −1 2 1 2 3 I ×T Z (I × T ) = N Z = N (−1) , (7.17) A (A ,A ,A ) 1 2 3 1 1 ⊕2 2 where (A , A , A ) runs through H (M , ∂M ; Z ) ⊕ H (M ; Z ) with M = I × T as said, and 1 2 3 2 2 N is a normalization constant that is to be determined. Note that A ∪ A ∪ A defines an element 1 2 3 of H (M , ∂M ) so that it can be integrated over the fundamental class [M]∈ H (M , ∂M ). In this expression, only the first condition of Eq. (7.16) is imposed by hand, while the summation over A with the sign acts as a projection (times an integer) imposing the second condition. Let us first compute the partition function (7.17) up to the normalization constant N . It is convenient to pack the holonomy data (A , A , A ) into a 3 × 3 matrix 1 2 3 H = A ∈ Z , (7.18) ij j 2 ith direction 40/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. Table 5. The GSDs of the LRE TQFT theory of dynamically gauged Eq. (7.14). The boundary theory is constructed via Eq. (7.13). Our Sect. 7.2 can explain the GSD data in terms of a field theoretic description. LRE/LRE 2+1D bulk/1+1D boundary coupled TQFTs 2 1 3 2 1 1 2 Z (M × S ) Z (T ) Z (D × S ) Z (I × T ) 2 2 1 1 Spatial topology T D I × T GSD 22 1 5 1 1 2 1 2 where the first direction is I and the second and third directions are S in T of M = I × T . The partition function up to the normalization constant can be computed by 1 2 detH N Z (I × T ) = (−1) = 20, (7.19) where the summation is constrained by the conditions A ∈ H (M , ∂M ; Z ) and A , A ∈ 1 2 2 3 1 ⊕2 H (M ; Z ) , which mean H = H = H = H = 0. In the sum of Eq. (7.19), the contribution 2 21 31 12 13 from the configurations that do not satisfy the second equation of Eq. (7.16) on the boundary is automatically canceled. For example, take a configuration given by ⎛ ⎞ H 00 ⎜ ⎟ H = 01 0 . (7.20) ⎝ ⎠ 00 1 With this configuration, A ∪ A = 1, which does not satisfy the second equation of Eq. (7.16). 2 3 However, summation over H yields det diag(H ,1,1) (−1) = 0. (7.21) H =0,1 In this way, the summation over H in Eq. (7.19) is essentially just projecting out the configurations that do not satisfy the second equation of Eq. (7.16), and therefore the partition function Z counts the number of configurations satisfying Eq. (7.16) up to some constant. The normalization constant N should be the number of residual gauge transformations that fixes the cohomology classes (A , A , A ). Such gauge transformations are given by elements of 1 2 3 ⊕2 0 0 ⊕2 H (M , ∂M ; Z ) ⊕ H (M ; Z ) = Z ; therefore the normalization constant N is 4. Then, the 2 2 partition function can be computed to be 1 2 Z (I × T ) = 5. (7.22) Our independent computation matches the lattice model computation exactly [26] based on the exact sequence (7.13); see Table 5. The Z gauge theory with the action is known to be equivalent to the D gauge theory (the order- 8 dihedral group) [38]. The boundary condition (7.16) can be understood as the D -preserving 1 2 boundary condition. Then, the partition function Z (I × T ) = 5 exactly counts the number of D 1 1 1 holonomies up to conjugacy around the S inside a time slice I × S . This observation is in line with a result in Ref. [96]. In Ref. [96], it was shown that when a Z subgroup of Z symmetry with anomaly (7.14) of a 1+1D theory is dynamically gauged, the resulting theory has a non-anomalous 3 G D symmetry. We can divide the bulk (Z ) gauging into two parts, as depicted in Fig. 6; one is the 41/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. Fig. 6. The interpretation of the boundary condition (7.16). The downward arrow is due to Ref. [96]. The 3 G boundary condition (7.16) breaks one of Z ⊂ (Z ) , and preserves (the electric parts of) the other two. On the other hand, we can do the same gauging in two steps. First we gauge one Z only on the boundary, getting the trivial theory coupled with non-anomalous D , and then we gauge every symmetry realized in the system. In this way, we get D gauge theory in the bulk with a D -preserving boundary condition, which should be 4 4 3 G dual to the twisted (Z ) gauge theory in the bulk with a Z -breaking boundary condition. gauging of the Z subgroup on the boundary, and the other is the gauging of the rest of the symmetry. When the first Z ∈ G global symmetry is gauged on the boundary, the Z gauge theory on the boundary is gauged away, and the resulting system is the trivial theory coupled with non-anomalous D symmetry due to the result of Ref. [96]. Then, gauging the rest of the global symmetry merely results in the bulk D gauge theory with a D -preserving boundary condition. This is consistent with 4 4 the fact that the D gauge theory in 2 + 1D is dual to the twisted Z gauge theory [38]. See also Fig. 6. In summary, the group-extension construction (7.13) of the coupled bulk/boundary (after gauging G) system is equivalent/dual to the partially gauge-breaking construction as Eq. (7.2) into 2 G 3 G (Z ) → (Z ) . (7.23) 2 2 boundary bulk 7.3. 3+1/2+1D LRE/LRE TQFTs: Comment on constructions of gauging an extension, and 1-form breaking versus “fuzzy-composite” breaking The system described by the action (7.15) can be generalized to 3 + 1D Z gauge theory whose exponentiated action is A ∪A ∪A ∪A 1 2 3 4 (−1) (7.24) 1 ⊕4 4 where (A , A , A , A ) ∈ H (M ; Z ) are Z cocycle fields. Correspondingly, we consider the 1 2 3 4 4 following new exact sequence, following Eq. (7.1): K 2 H 4 G 1 → (Z ) → (D × Z ) → (Z ) → 1, (7.25) 2 4 2 2 boundary boundary bulk with an order-8 non-Abelian dihedral group D . Namely, the anomaly described by Eq. (7.24) can be canceled by a Z gauge theory on a 2+1D boundary. GSDs of this bulk–boundary coupled system when both bulk and boundary are gauged for some topologies are computed by the lattice model described in Ref. [26]; now we compute the new data and list them in Table 6. For the boundary Z gauge theory to cancel the anomaly (7.24), there should be the following coupling: (α∪A ∪A +β∪A ∪A ) 1 2 3 4 ∂M (−1) , (7.26) 42/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. Table 6. The GSDs of the LRE TQFT theory of dynamically gauged Eq. (7.24) on spatial topologies M , time slice 3 1 which are equal to partition functions Z (M × S ), computed by the lattice path integral construction [26]. 3 K On each boundary of M , there is a Z gauge theory and it is coupled with the bulk through the exact time slice 2 K 2 H 4 G sequence (7.25) via 1 → Z → (D × Z ) → (Z ) → 1. Here both the bulk/boundary are LRE/LRE 2 2 2 coupled TQFTs. Our Sect. 7.3 can explain the GSD data in terms of a field theoretic description. LRE/LRE 3+1D bulk/2+1D boundary coupled TQFTs 3 1 4 3 1 2 2 1 3 Z (M × S ) Z (T ) Z (D × S ) Z (D × T ) Z (I × T ) 3 3 2 1 1 2 Spatial topology T D D × S I × T GSD 1576 1 50 484 where α is the boundary Z gauge field, β its magnetic dual. Integrating α and β out, we get the following boundary conditions: A ∪ A = 0, A ∪ A = 0. (7.27) 1 2 3 4 The partition function on M with this boundary condition would be counted by A ∪A ∪A ∪A 1 2 3 4 Z (M ) = N (M ) (−1) (7.28) (A ,A ,A ,A ) 1 2 3 4 where summation should be taken over some cohomology group that precisely realizes the boundary condition implied by Eq. (7.27), and N (M ) is the normalization factor counting the residual gauge transformations. Unfortunately, the precise cohomology group that actually realizes the above con- dition is harder to determine. We do not have an exact answer for N (M ) yet. Nonetheless, we can at least count the possible Z holonomies consistent with Eq. (7.27). 1 2 1 For M = I × T × S , we can parametrize possible holonomies bya3by4 matrix H = A ∈ Z , (7.29) ij j 2 1 3 where S is the ith 1-cycle of T . We define 2 by 2 submatrices H (a, b; c, d) of H by taking the ath and bth rows and cth and dth columns of H . The boundary condition (7.27) forces that Det(H (a, b; c, d)) = 0 (7.30) for (c, d) = (1, 2) or (c, d) = (3, 4). The number of matrices H with elements 0 or 1 satisfying Eq. (7.30) is 484. Since we are yet to determine the physics affected by the I direction and the nor- malization constant N (M ), this is not a complete computation. However, the fact that the number of possible holonomies around T with the condition (7.27) coincides with the GSD computed by the lattice computation suggests that the condition (7.27) is physically sensible in some way. Similarly, 2 2 2 for the topology M = D × T , we can count the holonomies around T satisfying Eq. (7.27)as 2 2 100, which is different from Z (D × T ) calculated by the lattice model by a factor of two, which is controlled by N (M ). Mysteriously the situation here is not an obvious generalization of what was studied in the previous section. Rather, a straightforward generalization of the situation (7.13) is that the bulk theory has a With condition (7.27) on the boundary, we do not expect that the action (7.24) can become −1. 43/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. (a) (b) 3 G Fig. 7. (a) A straightforward generalization of Fig. 6. In one higher dimension, one Z of (Z ) in the 2+1D/1+1D case is replaced by higher-form Z symmetry here in 3+1D/2+1D; see Eq. (7.33). (b) Currently we do not know precisely how to generalize from the previous Fig. 6 associated with Eq. 7.13 to the present case associated with Eq. (7.25). A possible relationship with the case of (a) is discussed in the main text. The downward arrow should be Z gauging on the boundary in some sense to get a trivial theory out of the Z 2,[1] 2 4 G gauge theory in 2+1D, but the precise relation of Z to the (Z ) is left open for future investigation. 2,[1] Z × Z symmetry, where Z is a 1-form Z symmetry, with an action 2,[1] 2,[1] 2 (2) A ∪A ∪A 4 3 4 (−1) , (7.31) (2) 2 where A is the 2-form gauge field of Z symmetry and a are 1-form gauge fields of Z . See 2,[1] 1,2 (2) Fig. 7(a). A boundary Z gauge field α can be coupled with the field A through (2) α∪A +β∪A ∪A 3 4 ∂M (−1) . (7.32) 2 G The recent paper [97] shows that when Z of Z × (Z ) with an anomaly (7.31) is gauged in 2,[1] 2,[1] 2+1D, the resulting theory has a non-anomalous D symmetry. Therefore the whole picture of Fig. 6 can be lifted in this case, by lifting one Z symmetry into Z . In particular, a breaking boundary 2 2,[1] condition 2 2 G Z → Z × (Z ) (7.33) 2,[1] 2 2 should be dual to the D -preserving boundary condition. What was studied in this subsection is more involved. Instead of Eq. (7.32), we have the coupling (7.26). Still, we can observe a similarity between Eqs. (7.26) and (7.32). Namely, A ∪ A plays 1 2 (2) 2 the role of A in Eq. (7.32). Thus, one might somehow find relations between these Z fields and a “composite” Z field. One might regard the boundary condition (7.27) relating to a boundary 2,[1] condition breaking this “composite” Z . See Fig. 7(b). 2,[1] The boundary condition (7.27) suggests that, on the boundary, some composite strings, composed 4 G of particles charged under the (Z ) symmetries, are condensed, and thus this boundary condition 44/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. might have some novel feature. Since the dimensionality of the composite strings (1+1D) and their composed particles (0+1D) is different, here we introduce the new concept of the “fuzzy composite object” composed by one-lower-dimensional object. We add fuzzy to emphasize the dimensionality differences between the two objects. The condensation of such “fuzzy composite objects” on the boundary can be understood as the “fuzzy composite-breaking” boundary condition. Investigating this boundary condition in detail, in particular constructing a microscopic model (other than the lattice Hamiltonian and lattice path integral given in Ref. [26]) to realize the physical mechanism at the microscopic level on the lattice, will be interesting. 8. Conclusions Below we conclude with remarks on long-range entanglements and entanglement entropy, the gener- alization of topological boundary conditions, and the potential application to strongly coupled gauge theories and quantum cosmology. 8.1. Remarks on long-range entanglement and entanglement entropy with topological boundaries It is well known that the long-range entanglement (LRE) can be partially captured by the topological entanglement entropy (TEE) [98,99]. In 2+1D, the topological entanglement entropy is the constant part of the entanglement entropy (EE), and one can extract the TEE by computing a linear combination of entanglement entropy as suggested in Refs. [98,99]. For discrete gauge theories with gauge group G, when the entanglement cut does not wrap around the spatial cycle, the topological entanglement entropy is − log |G|. For instance, the (Z ) gauge theory has the TEE =−k log n. Notice that the value of the TEE is independent of the twisting parameter (i.e., cocycle) of twisted gauge theories, hence one is not likely to distinguish different Dijkgraaf–Witten (DW) theories with the same gauge group using the TEE of the ground state wavefunctions on a closed manifold. In the following, we consider two generalizations to obtain a richer structure of the entanglement entropy. One generalization is to go to 3 + 1 dimensions and consider the Walker–Wang twisted theory. As was discussed in Refs. [100] and [55], for discrete gauge theories of Walker–Wang type [101] with gauge group Z and twisting parameter p (namely, BF + BB in Sect. 3.3), the topological entan- glement entropy across a torus (which does not wrap around the spatial cycle) is − log gcd(N,2p), which depends on the twisting parameter. Hence the TEE can probe the twisting level p of the Walker–Wang model. This fact can be understood as follows: The genuine line and surface operators of the Walker–Wang model coincide with the line and surface operators of Z gauge theory; gcd(n,2p) see Ref. [34] and Sect. 3.3. The TEE of the Z Walker–Wang model with twists is precisely the TEE of the Z ordinary gauge theory without twists, i.e., − log |G |=− log gcd(N,2p). gcd(N ,2p) eff Another generalization is to consider the entanglement entropy on a spatial manifold with bound- aries where the entanglement cut wraps around spatial cycles. We consider the Z gauge theories (i.e., Z -toric code model and Z -double semion (a twisted Z ) model) on a cylinder geometry with 2 2 2 two boundaries, as shown in Fig. 8. Let us first focus on the left panel and discuss the Z -toric code model. In the toric code model, there are four types of anyons {1, e, m, }. Let us denote |W , T  as x x i A dx i B dx x x x x the eigenstate of the line operators W = e (i.e., e-line in the x direction) and T = e x x (i.e., m-line in the x direction). A generic ground state is a linear combination of |W , T : x x |ψ= c |1+ c |e+ c |m+ c | (8.1) 1 e m 45/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. Fig. 8. In the left panel, the entanglement cuts wrap around the y cycle, which we denote the y cut. In the right panel, the entanglement cuts extend along the x direction, and end at the two boundaries, which we denote the x cut. where we label |1≡|0, 0, |e=|1, 0, |m=|0, 1, |=|1, 1, and the coefficients are properly 2 2 2 2 normalized |c | +|c | +|c | +|c | = 1. Following the computation in Ref. [102], we can 1 e m derive that the entanglement entropy of |ψ  is a linear combination of the entanglement entropy of y 2 2 2 y |1, |e, |m, |, denoted as S (|ψ ) =− |c | log |c | + |c | S (|i) where the i i i i=1,e,m, i=1,e,m, superscript y indicates the direction of the entanglement cut. In the following, we are only interested in the subleading (topological) part of the entanglement entropy, i.e., y y 2 2 2 S (|ψ ) =− |c | log |c | + |c | S (|i). (8.2) i i i topo topo i=1,e,m, i=1,e,m, Because e and m are both self and mutual bosons, they can separately condense on the boundaries. Let us denote a|b as a-condensation on the left boundary and b-condensation on the right boundary. There are 3 types of boundary conditions on a cylinder: e|e, m|m, and e|m. When both boundaries have e-condensation, i.e., the e|e boundary condition, there are two distinct sectors, with odd/even numbers of e-lines across the entanglement cut respectively. The generic ground state is |ψ, e|e= c |1+ c |e (8.3) 1 e where we have explicitly shown the boundary condition. Notice that there are no |m and | in the expansion because the m-particle and -particle cannot end on the boundary. Therefore the m-line and the -line must cross the entanglement cut twice. For each sector, the even/oddness of the e- y y line crossing the entanglement cut is fixed, hence S (|1) = S (|e) =− log 2. According to topo topo Eq. (8.2), we have 2 2 2 2 S (|ψ, e|e) =−|c | log |c | −|c | log |c | − log 2 1 1 e e topo (8.4) 2 2 2 2 =−|c | log |c | − (1 −|c | ) log(1 −|c | ) − log 2. 1 1 1 1 2 2 When |c | =|c | = , the entanglement entropy S (|ψ, e|e) is maximized, 1 e topo 2 2 2 2 S (|ψ, e|e) = 0. When |c | = 0, |c | = 1or |c | = 1, |c | = 0, the entanglement 1 e 1 e MaxES,topo entropy is minimized, S (|ψ, e|e) =− log 2. We can further consider other boundary con- MinES,topo ditions in which the entanglement cuts along the x direction, and the results are summarized in Table 7(a). Furthermore, we also consider the entanglement entropy of the double semion model as shown in Table 7(b). In the double semion model, there are four types of anyons {1, s, s ¯, b}, where the only nontrivial boson is b. Hence there is only one type of boundary condition, i.e., b condensation 46/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. Table 7. (a) Maximal and minimal entanglement entropy of the Z -toric code associated with various boundary conditions. (b) Maximal and minimal entanglement entropy of the Z -double semion model associated with various boundary conditions. b.c. e|em|me|m b.c. b|b y y S 00 − log 2 S 0 MaxES MaxES y y (a) (b) S − log 2 − log 2 − log 2 S − log 2 MinES MinES x x S 000 S 0 MaxES MaxES x x S − log 2 − log 2 0 S − log 2 MinES MinES on both boundaries, which we denote as b|b. From the data in Tables 7(a) and 7(b), we have the following observations: 1. The maximal and minimal entanglement entropies depend on the boundary conditions. In particular, when the types of boundary conditions are the same, S − S = log 2. MaxES MinES However, when the types of boundary conditions are different, S − S = 0. MaxES MinES 2. When the types of boundary conditions differ on two sides, the entanglement entropy is sensitive to whether the cut is in the x direction or the y direction. This enables us to use the entanglement entropy to probe the boundary conditions. We can implement the above approach to other examples studied in Sect. 7. In particular, given a bulk LRE system, we can design various boundary conditions (by group extension or by 0-form/higher- form breaking) on different boundaries. By generalizing the above analysis, we expect that EE is sensitive to not only the bulk but also the boundary/interface conditions. We leave a systematic anal- ysis of the interplay between other boundary/interface conditions and the long-range entanglement for future work. 8.2. More remarks 1. Counting extended (line/surface) operators: For the 3+1D bosonic Abelian-G TQFTs (with or without cocycle twists) that we have studied, the number of distinct types of pure line operators (viewed as the worldline of particle excitations) and the number of distinct types of pure surface operators (viewed as the worldsheet of string/loop excitations) are the same, equal to the number of group elements in G (thus the order of group |G|). By a pure surface operator, we mean that the particular surface operator does not have additional lower-dimensional line operators attached, and vice versa. In a spacetime picture, a pure surface operator is associated with the worldsheet of pure string/loop excitations (of a constant time slice) that does not have additional particles attached. For 3+1D bosonic non-Abelian-G TQFTs, with or without cocycle twists, we, however, expect that the number of distinct line/surface operators is related to the number of representation/conjugacy classes of G. In both cases, we find that the numbers of pure The pure operators here [43,79] are not equivalent to the genuine operators defined in Ref. [34]. The gen- uine operators in d dimensions mean that those operators do not require their attachment to higher-dimensional objects. However, when the same TQFT has two (or more) descriptions of different gauge groups (so-called duality), the counting of electric/magnetic operators can be different. For example, in 2+1D, (1) the (Z ) A ∪A ∪A 1 2 3 gauge theory with type-III cocycle (−1) in Sect. 3.5 is equivalent to (2) the order-8 D gauge theory [12,38]. In terms of (1), there are 8 group elements, thus implying 8 pure electric e-operators of A (the trivial 47/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. line and pure surface operators are equivalent. One simple argument [43,79] is that the number 2 1 of ground states on the spatial manifold S × S in the Hilbert space must be spanned by (1) 1 3 1 the eigenstates obtained from inserting all possible pure line operators along S into D × S or (2) the eigenstates obtained from inserting all possible pure surface operators along S into 2 2 S × D , because TQFT assigns a state vector in a Hilbert space to an open manifold, in the 3 1 2 2 2 1 spacetime path integral picture. Here we use the fact that ∂(D × S ) = ∂(S × D ) = S × S 3 1 2 2 4 and the gluing along their boundary produces (D ×S )(S ×D ) = S [43]. It is obvious that 1 2 the operators along S must be 1-lines, and the operators along S must be 2-surfaces where the loop excitations created by this surface can be shrunk to nothing into the vacuum. Thus, they correspond to pure line/surface operators creating pure particle/string excitations. Because the 2 1 GSD on S × S from two derivations is the same, the numbers of linear-independent pure line operators are equal to those of pure surface operators. For fermionic spin TQFTs (with or without twists from gauging the cobordism topological terms), the line/surface operators in general will have additional labels compared to the bosonic case. In particular, for line operators, one can introduce a 0+1D fermionic SPT (labeled by Z ) supported on the line operator. Note that, as often required in QFT, the line operators should be equipped with framing (trivialization of the normal bundle) in order to be well defined. Then, the spin structure on the spacetime manifold induces a spin structure on the support of the line operator. For a nontrivial choice (i.e., 1 ∈ Z ) of the 0+1D SPT, the expectation value of the line operator will be multiplied by ±1 for even/odd induced spin structure. As in the bosonic case, the surface operators can be understood in terms of drilling out a tabular neighborhood of the operator and considering the theory on the resulting manifold with a boundary with some condition on the holonomies on the boundary. For fermionic theories, one also has to choose a spin structure on this manifold with a boundary. This choice corresponds to the extra label assigned to the surface operator. 2. Fuzzy-composite object/breaking and extended operators: Earlier, in Eqs. (1.9) and (7.27), we introduced a new mechanism to obtain a peculiar gapped topological boundary condition: A ∪ A = 0, which could be viewed as the condensation of a fuzzy-composite string formed i j 3 by two different particles (associated with the ends of two different line operators). We call this the condensation of the fuzzy-composite object associated with the open ends of a set of extended operators. More generally, we could anticipate that in higher spacetime dimensions, line operator 1 included), 8 pure magnetic m-operators of B (the trivial line operator 1 included), and some dyonic operators mixing e and m. In contrast, in (2), as a non-Abelian D , there are 5 pure electric e-operators (the trivial line operator 1 included) related to the representation of D , 5 pure magnetic m-operators of B (the trivial line operator 1 included) related to the conjugacy classes of D , and some dyonic operators mixing e and m. Therefore, two dual descriptions of the same theory may give rise to different countings of e- and m-operators. Nonetheless, the total number of distinct extended operators is the same: There are 22 distinct line operators in both cases [12,38,45]. One can non-canonically identify spin structures on the spacetime manifold M with elements of H (M , Z ). Then, by considering the relevant part of the Mayer–Vietoris sequence 1 1 1 1 1 1 ··· → H (M , Z ) → H (M \ , Z ) ⊕ H (, Z ) → H ( × S , Z ) = H (, Z ) ⊕ Z → ··· , (8.5) 2 2 2 2 2 2 one can see that (for connected ) the set of spin structures on the compliment is given by (the also non- 1 1 canonical) spin(M \ ) = H (M \ , Z ) = H (M , Z ) ⊕ Z . 2 2 2 48/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. d+1 d d+1 say M , there could be other general topological boundary conditions on  = ∂M as A ∪ A ∪ B ∪ ··· = 0, (8.6) i j k d in terms of the condensation of the composite object from a set of extended operators of different dimensionality (1-form, 2-form fields, etc., or 1-cochain, 2-cochain fields, etc.). Further detailed study on this is left for the future. 3. Boundary/interface deconfinement: We discussed in Sect. 7 (see also Ref. [26]) the dynami- cal gauging of the bulk of SRE/LRE bulk/boundary coupled TQFTs to obtain the LRE/LRE bulk/boundary coupled TQFTs. We notice that the former system has an SRE bulk (e.g., an SPT state) and thus naturally has only non-fractionalized excitations in the bulk. The latter system has an LRE bulk (e.g., a topologically ordered state) and thus can also have deconfined fractionalized excitations even in the bulk. However, we stress that the important ingredient, for both cases, is that the deconfined fractionalized excitations happen on the boundary/interface, without much energy penalty. In Sect. 7, we found that the deconfined fractionalized excita- tions indeed condense on the boundary/interface. On the lattice scale, the energy cost for having deconfined excitations in the SRE bulk is E →∞ (i.e., impossible), while that in the LRE bulk costs only E #J (some order of lattice coupling J ; see Fig. 1). But having deconfined excitations on the boundary/interface is E 0, if the ground state is obtained from extended operators ending on the boundary. We note that there has been some recent interest in studying deconfined domain walls [105– 107] (see also footnote 32, and the references in Ref. [103]), where the bulks of systems are, however, confined without fractionalized excitations but only the boundaries harbor deconfined excitations. Our work could potentially help to understand such systems systematically and quantitatively. 4. Besides the generic mixed (gauge/global) symmetry-breaking/extension construction of topo- logical interfaces in Ref. [26], there are many other recent work and applications on related issues. For example, we can study the quantum code or topological quantum computation with a boundary [69,108–113]. One can construct Hamiltonian models for gapped boundaries [114–117]. For LRE/LRE topological bulk/boundary coupled states, there are applications to LRE fractional topological insulators and SETs [118–121]. One can also consider entangle- ment entropy involving the topological interfaces; this has been analyzed recently in the 2+1D case [122]. There are other formal aspects of studying boundaries and surface defects in the categorical setup (see, e.g., Refs. [25,123,124] and references therein). 5. Tunneling between topological quantum vacua: We discussed our interpretations of tunneling between topological quantum vacua in Sects. 1.1 and 1.4. The tunneling rate P is determined by the probability of creating a pair of excitations and anti-excitations out of the vacuum and then winding the pair along a non-contractible spatial cycle. By dimensional analysis, the tunneling rate P is about P ∼ f [(E /), (a/L)]. It is a function f proportional (up to some power) to the energy fluctuation E (quantum or thermal) but anti-proportional to the energy gap  = E −E T n 0 between excited states and ground states. It is also anti-proportional to the system size L over Readers can find many other examples of surface topological orders on the boundary of SRE SPT states in the informative recent review [103] and references therein. The original idea is from Ref. [104]’s observation of quantum disordering of the symmetry defects to restore the broken symmetry as a topologically ordered boundary. Their approach is rather different from our constructions. 49/54 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/5/053A01/5025801 by Ed 'DeepDyve' Gillespie user on 21 June 2018 PTEP 2018, 053A01 J. Wang et al. the lattice constant a (or the Planck scale). In TQFTs, the energy gap  and system size L/a are usually assumed to be sent to the infinite limit. Thus, P ∼ 0; it is not possible simply based on the TQFT alone to obtain further detailed calculations of the tunneling rate. In other words, the P ∼ 0 also guarantees the robustness of fault-tolerant topological quantum computation [28]: The ground state (and data) is topologically robust against local perturbations, unless the artificial manual process drags the excitations along a non-contractible spatial cycle. However, it is possible to consider some lattice models or TQFT coupling to other massive QFT with some physical tunneling rate P. The original motivation for our systems was inspired by long-range entangled condensed matter and strongly correlated electron systems with intrinsic topological orders. These systems are fully quantum and highly entangled. In contrast, in a different discipline, most of the setup and analysis in cosmology on vacuum tunneling is somehow semiclassical, e.g., Coleman’s study on the fate of the false vacuum [125] to a more recent work [126], mostly in a semiclassical theory, and the references in Ref. [127]. We anticipate (or at least speculate on) the potential use of topological quantum vacua tunneling, through extended operators (e.g., in terms of cosmic strings or higher-dimensional analogs), in quantum cosmology. Acknowledgements The authors thank Zhenghan Wang and Edward Witten for conversations. K.O. and J.W. thank Yuji Tachikawa for helpful discussions, especially during the stay at IPMU, and also for conversations about his recent paper. J.W. thanks Meng Cheng, Clay Cordova, and Xueda Wen for information about their recent works, and Ling- Yan Hung for past conversations on entanglements. J.W. gratefully acknowledges the Corning Glass Works Foundation Fellowship and NSF Grant PHY-1314311. Part of the work of J.W. was performed in 2015 at the Center for Mathematical Sciences and Applications at Harvard University. K.O. gratefully acknowledges sup- port from the IAS and NSF Grant PHY-1314311. P.P. gratefully acknowledges the support from the Marvin L. Goldberger Fellowship and the DOE Grant DE-SC0009988. Z.W. gratefully acknowledges support from NSFC grants 11431010, 11571329. M.G. is grateful for the support from the US–Israel Binational Science Founda- tion.Y.Z. thanks H. He and C. von Keyserlingk for collaboration on a related project, and B. A. Bernevig and the Physics Department of Princeton University for support. H.L. thanks H. Wang and K. Zeng for conversations. 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Tunneling topological vacua via extended operators: (Spin-)TQFT spectra and boundary deconfinement in various dimensions

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