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.
Highlights
Satisfies the linearized Einstein equation around AdS
We study shape-deformations of the entanglement entropy and the modular Hamiltonian for an arbitrary subregion and state in a holographic conformal field theory
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 in a holographic conformal field theory with Einstein gravity dual
Summary
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. 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, but at this point we would like to highlight a few of its salient properties. 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 the bulk extremal surface: δδV S
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
