The holographic shape of entanglement and Einstein’s equations

• Main Authors: Aitor Lewkowycz, Onkar Parrikar
• Format: Article
• Language: English
• Published: SpringerOpen 2018-05-01
• Series: Journal of High Energy Physics
• Subjects: AdS-CFT Correspondence; Conformal Field Theory
• Online Access: http://link.springer.com/article/10.1007/JHEP05(2018)147

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.