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Progress of Theoretical and Experimental Physics
, Volume 2018 (3) – Mar 1, 2018

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/lp/ou_press/flow-equation-conformal-symmetry-and-anti-de-sitter-geometry-ScHNSLDKii

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- Oxford University Press
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- © The Author(s) 2018. Published by Oxford University Press on behalf of the Physical Society of Japan.
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- 2050-3911
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- 2050-3911
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- 10.1093/ptep/pty013
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Prog. Theor. Exp. Phys. 2018, 031B01 (8 pages) DOI: 10.1093/ptep/pty013 Letter Flow equation, conformal symmetry, and anti-de Sitter geometry ∗ ∗ Sinya Aoki and Shuichi Yokoyama Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo-Ku, Kyoto, Japan E-mail: saoki@yukawa.kyoto-u.ac.jp, shuichi.yokoyama@yukawa.kyoto-u.ac.jp Received January 8, 2018; Revised January 29, 2018; Accepted January 29, 2018; Published March 5, 2018 ................................................................................................................... We argue that the anti-de Sitter (AdS) geometry in d + 1 dimensions naturally emerges from an arbitrary conformal ﬁeld theory in d dimensions using the free ﬂow equation. We ﬁrst show that an induced metric deﬁned from the ﬂowed ﬁeld generally corresponds to the quantum information metric, called the Bures or Helstrom metric, if the ﬂowed ﬁeld is normalized appropriately. We next verify that the induced metric computed explicitly with the free ﬂow equation always becomes the AdS metric when the theory is conformal. We ﬁnally prove that the conformal symmetry in d dimensions converts to the AdS isometry in d + 1 dimensions after d-dimensional quantum averaging. This guarantees the emergence of AdS geometry without explicit calculation. ................................................................................................................... Subject Index B21 1. Introduction The anti-de Sitter/conformal ﬁeld theory (AdS/CFT) (or gravity/gauge theory) correspondence [1] is a promising tool to crack a hard problem in strongly coupled gauge theories (see Refs. [2–4] for some reviews), but is still mysterious even after many pieces of evidence and application appeared after the ﬁrst proposal. Although the correspondence may be a manifestation of the closed string/open string duality, an alternative understanding might exist due to its holographic property. One important mystery is the precise mechanism by which the AdS radial direction emerges from CFT. It may be common sense that the AdS radial direction is emergent as a renormalization scale of dual CFT as both behave similarly under dilatational symmetry [1], and this viewpoint works for renormalization group ﬂows triggered by relevant deformations [5,6]. However, a direct approach to search for the Wilsonian cutoff corresponding to the sharp cutoff on the AdS radial direction [7] is still far from a clear answer, since the ordinary Wilsonian renormalization gives rise to non-local interaction in the bulk interpretation (see also Ref. [8]). While several approaches have been developed to grab the tail of the AdS radial direction and construct bulk dynamics from CFT [10–13] (see also Refs. [14,15]), one of the present authors, together with his collaborators, has proposed an alternative method to deﬁne a geometry from a quantum ﬁeld theory and explicitly calculated the metric from several quantum ﬁeld theories [16–18]. In Ref. [16], the method was proposed and applied to the O(n) nonlinear sigma model in This situation may drastically change in the AdS /CFT correspondence. A clear dictionary was proposed 3 2 in Ref. [9]. © 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/2018/3/031B01/4922029 Funded by SCOAP by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 031B01 S. Aoki and S. Yokoyama 2 dimensions, and the 3D metric in the large-n limit was shown to describe an AdS space in the massless limit. In Ref. [17], it was shown in the large-n limit that the induced metric describes an AdS space with d ≥ 3 in the UV limit and with d ≥ 1 in the IR limit, if the method is applied d+1 to the massive O(n)ϕ model and an appropriate normalization is introduced for the ﬂowed ﬁeld. In Ref. [18], the large-n expansion was performed for the massless O(n)ϕ model in 3 dimensions. While the induced metric describes the AdS space at the leading order, the next-leading-order corrections make the space asymptotically AdS only in the UV and IR limits with different radii. By observing that the induced metric always gives the AdS metric when the theory is conformal, it is natural to expect that symmetry plays a key role behind these results. In this letter, we investigate a direct relation between CFT and the AdS metric in this framework. The goal of this letter is to generalize the previous results to an arbitrary conformal ﬁeld theory incorporating the symmetry argument. The rest of this letter is organized as follows. In Sect. 1, after a brief explanation of the proposal of Ref. [16], we ﬁrst show that our metric corresponds to the information metric, stressing the importance of the ﬁeld normalization introduced in Ref. [17]. In Sect. 2, we explicitly derive the AdS metric directly from the CFT. We then prove that the induced metric in d + 1 dimensions possesses the isometry of the AdS space as a consequence of the conformal symmetry in d dimensions. Section 6 is devoted to the summary and discussion. 2. Gradient ﬂow and information metric In this section, we brieﬂy review the proposal in Refs. [16,17] to deﬁne a d + 1-dimensional induced metric from a d-dimensional quantum ﬁeld theory. We also show that the metric deﬁned in this way with appropriate normalization can be interpreted as a quantum information metric, called the Bures or Helstrom metric. Gradient ﬂow and induced metric We consider an n real component scalar ﬁeld ϕ(x) in d dimensions, whose quantum dynamics is controlled by the action functional S(ϕ). The ﬂowed ﬁeld φ is deﬁned from ϕ with the initial condition φ(x;0) = ϕ(x) through the ﬂow equation as δS (ϕ) ∂φ (x; t) =− , (1) ∂t δϕ (x) ϕ(x)→φ(x;t) where the ﬂow time t has the (length) dimension, x is the d-dimensional coordinate system, a = 1, 2, ... , n labels a component of the scalar ﬁeld, and S (ϕ) is an appropriate action for ϕ, which is not necessarily related to the original action S(ϕ) in general. In particular, when they coincide, the ﬂow is called the gradient ﬂow [19–22]. In the case of the free ﬂow (i.e., S is the free action), the ﬂow equation becomes the heat equation. Thus the ﬂow equation deﬁnes a procedure to smear the original ﬁeld ϕ into a smeared ﬁeld φ, the correlation functions of which are all ﬁnite at t > 0. A d + 1-dimensional metric operator is given by a a ∂σ (x; t) ∂σ (x; t) g ˆ (x; t) := R , (2) MN M N ∂z ∂z a=1 M μ where R is a constant with the length dimension, and z = (x , τ) with τ = 2dt, which is regarded as the d + 1-dimensional coordinates after d-dimensional quantum averaging, and σ (x; t) is the 2/8 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/031B01/4922029 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 031B01 S. Aoki and S. Yokoyama (dimensionless) normalized ﬂowed ﬁeld deﬁned as φ (x; t) σ (x; t) := . (3) a 2 φ (x; t) a=1 Here O(ϕ) denotes the quantum average with the d-dimensional action S, given by −S(ϕ) −S(ϕ) O(ϕ) := Dϕ O(ϕ) e , Z = Dϕ e . (4) a a Since σ (x; t) always satisﬁes σ (x; t)σ (x; t) = 1, we call this normalization condition the a=1 nonlinear sigma model (NLSM) normalization, the importance of which will become clear in the next subsection. The vacuum expectation value (VEV) of the metric is denoted as g (z) :=gˆ (x; t) , MN MN S the ﬂuctuations of which are suppressed in the large-n limit (see Refs. [16–18] for more details). Information metric In this subsection, we show that g (z) deﬁned in the previous subsection MN is equivalent to a quantum information metric, called the Bures (or Helstrom) metric. The NLSM normalization is important to show this. The Bures metric for the density matrix is deﬁned from the inﬁnitesimal distance between two density matrices ρ and ρ + dρ as D(ρ, ρ + dρ) = tr(dρ G), (5) where G is the Hermitian 1-form operator implicitly given by ρ G + G ρ = dρ. In particular, for the density matrix ρ of a pure state, the Hermitian operator is determined as G = dρ since ρ = ρ. In order to apply to our case, we consider an eigenstate of the position operator as well as the ﬂow time one denoted by |z=|(x, τ) and deﬁne the inner product of the state as a a z|w := σ (x; t)σ (y; s) , w = (y, 2ds). (6) a=1 Notice that the NLSM normalization guarantees z|z= 1. Then the information metric for this pure state is computed as 2 2 M N R D(ρ , ρ ) = g (z)dz dz , (7) z z+dz MN where ρ =|zz|, and we used σ(x; t) · ∂ σ(x; t) = 0 and z M S σ(x; t) · ∂ ∂ σ(x; t) =−∂ σ(x; t) · ∂ σ(x; t) M N S M N S a a with A · B := A B . Our metric g (z) deﬁnes a distance in the space of the density matrices MN a=1 made of the pure states (in units of R). In particular, when the action S of the original theory has a 2 O(n) symmetry, such pure states deﬁned above in an abstract way may be given by σ (x; t)|0. 2 a a In the case of the O(n) invariant mixed state ρ := σ (x; t)|00|σ (x; t),wehave G = ndρ from z z ρ = ρ /n. 3/8 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/031B01/4922029 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 031B01 S. Aoki and S. Yokoyama 3. AdS isometry from conformal symmetry In this section, we directly relate arbitrary con- formal ﬁeld theory in the ﬂat d-dimensional space-time with the AdS metric in d + 1 dimensions. We here assume that the CFT contains a real scalar primary operator ϕ(x) of a general conformal dimension without specifying any concrete models for CFT. In this section, we are interested in the VEV of the metric operator, which determines the classical geometry induced from this primary operator, though its “quantum” ﬂuctuations around the VEV exist for n = 1 in this case. AdS metric from CFT We start with the free generally massive ﬂow equation with n = 1: ∂φ(x; t) 2 2 = (∂ − m )φ (x; t), (8) ∂t 2 μν where ∂ = η ∂ ∂ . This is easily solved as μ ν 2 2 t(∂ −m ) φ(x; t) = e ϕ(x). (9) The two-point function of φ is evaluated as G (x; t|y; s) :=φ(x; t)φ (y; s) 0 CFT 2 2 2 −(t+s)m (t∂ +s∂ ) x y = e e ϕ(x)ϕ(y) , (10) CFT where we have used the subscript CFT instead of S, since the action is not speciﬁed. The Poincaré invariance and the scale transformation of ϕ with ϕ(λx) = λ ϕ(x) ﬁx the form of G such that −(t+s)m 2 e (x − y) G (x; t|y; s) = F , (11) 0 0 (t + s) t + s where F is a certain smooth function. Furthermore, the ﬂow equation (8) implies 2 2 G (x; t|y; s) = (∂ − m )G (x; t|y; s), (12) 0 0 ∂t which leads to F (0) =−2dF (0). Thus the two-point function of the normalized ﬁeld σ becomes m independent as G(x; t|y; s) :=σ(x; t)σ (y; s) CFT 2 ts (x − y) = F , (13) t + s t + s where F (x) ≡ F (x)/F (0). Hence F (0) = 1 and 2dF (0) =− . Taking the d + 1-dimensional 0 0 coordinates as z = (x, τ = 2dt), the vacuum expectation value of the induced metric g ˆ is MN g (z) = R ∂ σ(x; t)∂ σ(x; t) . (14) MN M N CFT The argument and calculation below hold just by changing the signature suitably when we consider the Euclidean ﬂat space. Explicitly, F (x) is computed as (d/2) −1 d/2− −1 −xv/4 F (x) = dv v (1 − v) e . ( ) (d/2 − ) 4/8 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/031B01/4922029 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 031B01 S. Aoki and S. Yokoyama This is computed as (g (z) = g (z) = 0) μτ τν 2 2 R R g (z) = η , g (z) = , (15) μν μν ττ 2 2 τ τ which is nothing but the AdS metric with R as its radius. Isometry from conformal transformation In the previous subsection, we have explicitly calcu- lated the induced metric g from CFT, and have shown that it becomes the AdS metric in the MN Poincaré patch. In this subsection, we discuss the relation between the AdS metric and the CFT only from the symmetry. Namely, we show that the induced metric g (z) = g ˆ (x; t) possesses MN MN CFT the isometry of the AdS space. This is necessary and sufﬁcient since AdS is a maximally symmetric space, so that the metric is completely ﬁxed by the isometry group SO(2, d) up to an overall constant. We ﬁrst relate the conformal transformation to the isometry of AdS. The inﬁnitesimal conformal transformation and the response of the primary scalar operator are given by μ μ μ ν μ μ 2 μ ν δx = a + ω x + λx + b x − 2x (b x ), ν ν conf μ μ δ ϕ(x) =−δx ∂ ϕ(x) − (∂ δx )ϕ(x). (16) μ μ μ μν μ Here a , ω , λ, b are parameters of the transformation. Since the inﬁnitesimal conformal transformation is quadratic in the coordinate x, the normalized ﬁeld σ , given in terms of ϕ by 2t t∂ σ(x; t) = √ e ϕ(x), (17) F (0) is transformed as conf 2 μ 2 ν ρ μ ν μ μ δ σ(x; t) =− t(∂ δx ) + 2t (∂ ∂ δx )∂ ∂ + 2t(∂ δx )∂ + δx ∂ σ(x; t) ν ρ ν μ ν μ μ − 2t(∂ ∂ δx )∂ + (∂ δx ) σ(x; t). (18) μ ν μ Plugging Eq. (16) into Eq. (18), we obtain conf μ μ μ 2 μ δ σ(x; t) =−{2λ − 4(b x )}t∂ σ(x; t) −{δx + 2t(d − 2 − )b }∂ σ(x; t) + 4t b ∂ ∂ σ(x; t), μ t μ μ t (19) where we have used ∂ σ(x; t) = ( + ∂ )σ (x; t). Notice that a special conformal transformation of 2t the normalized ﬂow operator contains a higher-derivative term as the last term in Eq. (19), which cannot be interpreted as an inﬁnitesimal diffeomorphism in the bulk. To deal with this we rewrite This question was addressed in the early stage after Maldacena’s proposal in Ref. [23], in which was studied the mechanism of how the special conformal transformation of the adjoint scalar ﬁelds in N = 4 super Yang–Mills theory was metamorphosed into the corresponding isometry transformation in the bulk. We would like to thank T. Yoneya for his valuable comment given in the KEK Theory workshop 2017 “East Asia Joint Workshop on Fields and Strings 2017”. Difﬁculty in the bulk interpretation of the special conformal transformation of a key operator was also observed in the bilocal ﬁeld approach by using a vector model [24], where the special conformal transformation of the collective ﬁeld mixes up ﬁelds with different spin. The authors speculated on the necessity of a suitable ﬁeld redeﬁnition to resolve this issue, though our answer may be different in the ﬂow ﬁeld approach, as shown below. We would like to thank S. Das for his valuable comment given in the KIAS-YITP joint workshop 2017 “Strings, Gravity and Cosmology”. 5/8 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/031B01/4922029 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 031B01 S. Aoki and S. Yokoyama Eq. (19)as conf diff extra δ σ(x; t) = δ σ(x; t) + δ σ(x; t), (20) + 2 diff μ extra 2 ν ¯ ¯ δ σ(x; t) =−(δt∂ + δx ∂ )σ (x; t), δ σ(x; t) = 4t b ∂ ∂ + σ(x; t), (21) t μ ν t 2t μ μ μ μ 2 diff ¯ ¯ with δx = δx + 2dtb , δt = (2λ − 4(b x ))t. By setting τ = 2dt, the transformation δ can be rewritten as μ μ 2 μ μ ¯ ¯ δx = δx + τ b , δτ = (λ − 2(b x ))τ. (22) This is nothing but the isometry transformation of the AdS space whose metric is given by ds ∝ AdS 2 μ2 dτ + dx The conformal transformation of the induced metric operator is computed as ∂ ∂ conf diff 2 extra extra δ g ˆ (x; t) = δ g ˆ (x; t)+R lim δ σ(x; t)σ (y; s) + σ(x; t)δ σ(y; s) . MN MN M N (y;s)→(x;t) ∂z ∂w (23) The ﬁrst term is diff K K K ¯ ¯ ¯ δ g ˆ (x; t) =−δz ∂ g ˆ (x; t) − ∂ δz g ˆ (x; t) − ∂ δz g ˆ (x; t), (24) MN K MN M KN N MK which is nothing but the diffeomorphism of the metric tensor in d + 1 dimensions. Thus our task is to show that the second term in Eq. (23) vanishes in the vacuum expectation value. By using Eqs. (13) and (21), the term is computed as extra extra δ σ(x; t)σ (y; s) + σ(x; t)δ σ(y; s) ( 4ts) (x − y) μ 2 =−8 (t − s)b (x − y) (x − y) F . (25) (t + s) t + s Then it is easy to see ∂ ∂ extra extra lim δ σ(x; t)σ (y; s) + σ(x; t)δ σ(y; s) = 0. (26) M N (y;s)→(x;t) ∂z ∂w We stress that this happens only when the conformal transformation is decomposed as Eq. (21). Note that the quantum averaging and the differentiation commute since all correlation functions of σ are ﬁnite as long as the ﬂow time is non-zero. Therefore we obtain conf diff δ g ˆ (x; t) = δ g ˆ (x; t) . (27) MN MN Since the conformal invariance of the two-point function of the primary scalar operator implies conf diff that δ g ˆ (x; t) = 0, it follows that δ g ˆ (x; t) = 0. This means that the induced metric MN MN g (z) satisﬁes the Killing equation of the AdS space as MN diff K K K ¯ ¯ ¯ δ g (z) =−δz ∂ g (z) − ∂ δz g (z) − ∂ δz g (z) = 0, (28) MN K MN M KN N MK which implies that the induced metric must be the AdS metric up to an overall constant. This completes the proof of our claim. 6/8 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/031B01/4922029 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 031B01 S. Aoki and S. Yokoyama 4. Discussion In this letter an induced geometry by a ﬂow equation from a quantum ﬁeld theory was investigated. The induced metric was shown to appear as a quantum information metric, which measures a distance in the space of the pure states constructed by scalar ﬁelds in a general quantum ﬁeld theory. In a conformally symmetric situation, it was shown that the induced metric matches the AdS one when the ﬂow equation is free. This agreement was conﬁrmed only by using symmetry without any explicit computations of the metric. An appearance of the AdS metric from CFT in our method relies on the following two facts. (a) The ﬁeld σ used to deﬁne the metric operator is dimensionless thanks to the NLSM normalization. If φ were used instead, one would not obtain the AdS metric. (b) The VEV of the metric operator is UV ﬁnite thanks to the free ﬂow equation. If the VEV were UV divergent, one would not obtain the AdS metric due to additional dimensionful quantities introduced through renormalization. So far any relation between the induced metric formalism presented in this letter and the other approaches to seeing dual geometry mentioned in the introduction is not known. It may be reasonable to think that there is no relation between them since, e.g., the procedure to renormalize ﬁelds in quantum ﬁeld theory and that to smear operators are generally independent. Still, we expect that the results and technique developed in this letter, particularly the symmetry argument, will become useful in studying the AdS geometry from CFT by other methods. For example, it may be possible to deﬁne an induced metric similarly in the Wilsonian renormalization approach to dual gravity. Then it would be interesting to see whether the metric becomes the AdS one or not. In this letter we assume that the background of quantum ﬁeld theory or conformal ﬁeld theory is ﬂat. It would be interesting to extend the presented calculation to curved backgrounds. In particular, it would be interesting to check whether the induced metric from CFT on a curved space-time is still of AdS form in a different coordinate system from the Poincaré patch. A challenging but important issue is whether this formalism encodes the gravitational dynamics or not. The ﬁrst step toward this goal may be to see how a linearized Einstein gravity is encoded in this formulation, as shown in a different method to derive dual bulk dynamics by using entanglement entropy [25,26]. For this analysis it will be necessary to specify a concrete model to test the proposal such as an O(n) sigma model, since bulk dynamics is dependent on each CFT. Note that the 1/n expansion becomes important to see the dual bulk dynamics beyond the geometry in the AdS/CFT correspondence. A virtue of this formulation is that observables in this formalism are correlation functions of scalar ﬁelds in quantum ﬁeld theory, which admit analytic computation by the ordinary technique of 1/n expansion [18], so that one can proceed by checking one’s guesswork explicitly by hand. We hope to report on resolutions on these issues in the near future. Acknowledgements S.A. would like to thank Dr G. Ishiki for his comment on the information metric. We would also like to thank S. Das and T. Yoneya for valuable comments. S.A. is supported in part by a Grant-in-Aid from the Japanese Ministry of Education, Sciences and Technology, Sports and Culture (MEXT) for Scientiﬁc Research (No. JP16H03978), by a priority issue (Elucidation of the fundamental laws and evolution of the universe) to be tackled by using the Post “K” Computer, and by the Joint Institute for Computational Fundamental Science (JICFuS). This result does not depend on mass m in the ﬂow equation, which violates the conformal symmetry, thanks to the NLSM normalization. 7/8 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/031B01/4922029 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 031B01 S. Aoki and S. Yokoyama Funding Open Access funding: SCOAP . References [1] J. M. Maldacena, Adv. Theor. Math. 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High Energy Phys. 10, 219 (2013) [arXiv:1308.3792 [hep-th]] [Search INSPIRE]. 8/8 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/031B01/4922029 by Ed 'DeepDyve' Gillespie user on 16 March 2018

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