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Title: The holographic shape of entanglement and Einstein’s equations

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.

Authors:
 [1];  [2]
  1. Stanford Univ., CA (United States). Stanford Inst. for Theoretical Physics, Dept. of Physics
  2. Univ. of Pennsylvania, Philadelphia, PA (United States). David Rittenhouse Lab.
Publication Date:
Research Org.:
Duke Univ., Durham, NC (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1507703
Grant/Contract Number:  
FG02-05ER41367
Resource Type:
Accepted Manuscript
Journal Name:
Journal of High Energy Physics (Online)
Additional Journal Information:
Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2018; Journal Issue: 5; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; AdS-CFT Correspondence; Conformal Field Theory

Citation Formats

Lewkowycz, Aitor, and Parrikar, Onkar. The holographic shape of entanglement and Einstein’s equations. United States: N. p., 2018. Web. doi:10.1007/jhep05(2018)147.
Lewkowycz, Aitor, & Parrikar, Onkar. The holographic shape of entanglement and Einstein’s equations. United States. doi:10.1007/jhep05(2018)147.
Lewkowycz, Aitor, and Parrikar, Onkar. Wed . "The holographic shape of entanglement and Einstein’s equations". United States. doi:10.1007/jhep05(2018)147. https://www.osti.gov/servlets/purl/1507703.
@article{osti_1507703,
title = {The holographic shape of entanglement and Einstein’s equations},
author = {Lewkowycz, Aitor and Parrikar, Onkar},
abstractNote = {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.},
doi = {10.1007/jhep05(2018)147},
journal = {Journal of High Energy Physics (Online)},
number = 5,
volume = 2018,
place = {United States},
year = {2018},
month = {5}
}

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