The holographic shape of entanglement and Einstein’s equations

Published for SISSA by Springer Received: April 7, 2018 Accepted: May 15, 2018 Published: May 23, 2018 Aitor Lewkowycza and Onkar Parrikarb a Stanford Institute for Theoretical Physics, Deptartment of Physics, Stanford University, Stanford, CA 94305, U.S.A. b David Rittenhouse Laboratory, University of Pennsylvania, 209 S. 33rd Street, Philadelphia, PA 19104, U.S.A. E-mail: , Abstract: We study shape-deformations of the entanglement entropy and the modular Hamiltonian for an arbitrary subregion and state (with a smooth dual geometry) in a holographic conformal field theory. More precisely, we study a double-deformation comprising of a shape deformation together with a state deformation, where the latter corresponds to a small change in the bulk geometry. Using a purely gravitational identity from the Hollands-Iyer-Wald formalism together with the assumption of equality between bulk and boundary modular flows for the original, undeformed state and subregion, we rewrite a purely CFT expression for this double deformation of the entropy in terms of bulk gravitational variables and show that it precisely agrees with the Ryu-Takayanagi formula including quantum corrections. As a corollary, this gives a novel, CFT derivation of the JLMS formula for arbitrary subregions in the vacuum, without using the replica trick. Finally, we use our results to give an argument that if a general, asymptotically AdS spacetime satisfies the Ryu-Takayanagi formula for arbitrary subregions, then it must necessarily satisfy the non-linear Einstein equation. Keywords: AdS-CFT Correspondence, Conformal Field Theory ArXiv ePrint: 1802.10103 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP05(2018)147 JHEP05(2018)147 The holographic shape of entanglement and Einstein’s equations Contents 1 2 Preliminaries 2.1 Perturbative approach to entanglement 2.2 Shape deformations 2.3 Hollands-Iyer-Wald formalism 4 4 6 12 3 Integrating in the modular hamiltonian 3.1 Calculation 3.2 Results 13 14 18 4 Discussion 19 A Gaussian normal coordinates 22 B Extremality condition in Gaussian normal coordinates 23 C Details of the symplectic 2-form and boundary terms 26 1 Introduction A lot of progress has been made in recent years in understanding the gravity dual of entanglement entropy in holographic conformal field theories [1–7]. So far, much of this work has focussed on using the replica trick (except for [3], which leverages the symmetries of ball-shaped regions in the vacuum), which in quantum field theories requires putting the theory on a different background with a conical deficit at the entangling surface, together with other subtle operations such as analytic continuation in the replica index. It is desirable to gain further understanding of holographic entanglement entropy using more direct techniques, given that it should be computable directly within the original Hilbert space. There are several motivations for this — firstly, it could potentially provide a clearer understanding of the meaning of subregions in quantum gravity (in AdS) and could provide further insight into the microscopic origin of the Bekenstein-Hawking entropy, perhaps in terms of counting of edge modes [8, 9]. Another motivation would be to give a more direct derivation of the Ryu-Takayanagi (RT) formula, without using the replica trick. Finally, there has been much work in recent years suggesting a deep connection between the emergence of spacetime geometry and entanglement in the AdS/CFT correspondence [10–13]. For instance, it was shown in these papers that any asymptotically AdS spacetime which computes the entanglement entropies for ball-shaped regions in the CFT using the RyuTakayanagi formula for up to first order state deformations around the vacuum, necessarily –1– JHEP05(2018)147 1 Introduction where ∂RB is a small (Euclidean) tube of radius B which surrounds the entangling surface, and V is the vector field parametrizing the shape deformation. This formula essentially follows from the setup in [22] and will be explained in more detail in section 2, but at this point we would like to highlight a few of its salient properties. Firstly, it contains is/2π the evolution operator ρR involving the density matrix of ψ reduced over R, which generates what is commonly called modular flow ; this is crucial for the right hand side to have a non-trivial limit as B → 0. Secondly, it depends on the integral of the stress tensor on a co-dimension one surface ∂RB (which naively becomes co-dimension 2 as B → 0), which greatly facilitates rewriting it in terms of bulk gravitational variables. Finally, equation (1.1) provides a purely field-theoretic constraint on a particular deformation of the entropy (on the left hand side) in terms of the stress tensor expectation value (on the right hand side), which we will see has an interesting manifestation in the bulk. Indeed, for holographic theories dual to Einstein gravity, we expect this doubledeformation of the entanglement entropy to be computed by the change in the area of –2– JHEP05(2018)147 satisfies the linearized Einstein equation around AdS. It is likely that understanding this connection further will involve essentially new techniques. One approach along these lines is entanglement (or modular) perturbation theory, where one studies the entanglement entropy (or correlation functions of the modular Hamiltonian) perturbatively around a background state, for small deformations in the state or shape of the subregion. This approach was first explored in [14], and later improved upon in [15]. Since then, there have been many advances, especially when the perturbations are shape deformations [16–29]. In addition to being computationally useful, entanglement perturbation theory has found several interesting applications. For instance, [22] derived the averaged null energy condition (ANEC) in Minkowski spacetime from the monotonicity of relative entropy together with entanglement perturbation theory for shape deformations (see also [30] for another proof of the ANEC using OPE techniques). Entanglement perturbation theory was also used [31, 32] to derive the gravitational equations of motion from entanglement to second order around AdS. Importantly, much of this work so far has focussed on special symmetric situations, such as for instance, deformations around ballshaped regions in the CFT vacuum, which in the holographic context only probes small deformations around AdS spacetime. Further progress necessitates moving away from such special cases. In this paper, we will take the first steps in this direction by studying the shape deformations of entanglement entropy for a general region R and a general state ψ (with a smooth AAdS dual geometry g) in a holographic conformal field theory with Einstein gravity dual (although our techniques can also be applied to higher-curvature theories). More precisely, we will be interested in a (...truncated)