The Holographic Shape of Entanglement and Einstein's Equations

TL;DR

This paper explores how shape and state deformations in holographic conformal field theories relate to bulk geometry changes, deriving key formulas and showing that Einstein's equations follow from entanglement properties.

Contribution

It provides a CFT derivation of the JLMS formula and demonstrates that the Ryu-Takayanagi formula implies Einstein's equations for arbitrary subregions.

Findings

Derived a CFT expression for double deformations of entanglement entropy.

Showed the Ryu-Takayanagi formula with quantum corrections matches bulk gravitational variables.

Argued that satisfying the Ryu-Takayanagi formula for all subregions implies 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.