The Holographic Shape of Entanglement and Einstein's Equations

TL;DR

The paper develops a framework to study shape and state deformations of entanglement in holographic CFTs by translating boundary modular-flow data into bulk gravitational information via the Hollands-Iyer-Wald formalism. It derives a bulk dual for the boundary double deformation, showing the result reproduces the Ryu-Takayanagi formula with quantum corrections and yields a CFT-based derivation of the JLMS formula for arbitrary subregions in the vacuum. A key outcome is that, under reasonable assumptions, RT for arbitrary subregions implies the full (nonlinear) Einstein equations, linking entanglement structure directly to bulk dynamics. These results extend entanglement perturbation theory beyond symmetric setups and sharpen connections between subregion entanglement, bulk geometry, and gravitational 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.

Figures (3)

• Figure 1: An illustration of the Euclidean path integral representation of the reduced density matrix. The solid blue line denotes the region $R$, the dashed blue line is $R^c$, and the solid black dot is $\partial R$. Also shown are the cylindrical tube surrounding the entangling surface $\partial R_B$, and the cut at $\theta = 0$.
• Figure 2: In the left panel is an illustration of the setup for constructing the bulk-dual of $\delta_VH_R$. The cylindrical tube $\Sigma$ surrounds the extremal surface $\mathcal{S}$ (red curve) in the bulk. The solid blue line is the subregion $R$ in the boundary. The dashed blue line is the cut along $\tau = 0$ on $\Sigma$; its upper and lower boundary is $\Sigma_{+}$ and $\Sigma_-$ respectively as shown in the magnified picture in right panel.
• Figure 3: The strip $0 \leq Im(s) \leq 2\pi$ in the complex $s$-plane. The contour $C$ is shown in blue. The black dots are the poles of the kernel $\sinh^{-2}\left(\frac{s+i\epsilon}{2}\right)$.