Nonlinear Gravity from Entanglement in Conformal Field Theories

TL;DR

This work derives nonlinear gravitational dynamics from entanglement in a broad class of CFTs by linking ball-entanglement entropies, computed via the HRRT surface, to a perturbative AdS geometry. Through a perturbative state construction using Euclidean sources for scalar primaries and the stress tensor, the authors express second-order relative entropy in terms of bulk canonical energy and boundary terms, demonstrating that the emergent geometry satisfies Einstein's equations with matter sourced by the CFT data when $ ilde C_T = a^*$. The analysis leverages Hollands-Wald gauge, covariant phase space, and symplectic methods to connect CFT entanglement to bulk dynamics, offering a nontrivial consistency check for AdS/CFT and suggesting a path to generalizations to higher-curvature theories via appropriate entanglement functionals. The results underscore the central role of entanglement structure in spacetime emergence and provide a framework for exploring gravity from quantum information in conformal field theories.

Abstract

In this paper, we demonstrate the emergence of nonlinear gravitational equations directly from the physics of a broad class of conformal field theories. We consider CFT excited states defined by adding sources for scalar primary or stress tensor operators to the Euclidean path integral defining the vacuum state. For these states, we show that up to second order in the sources, the entanglement entropy for all ball-shaped regions can always be represented geometrically (via the Ryu-Takayanagi formula) by an asymptotically AdS geometry. We show that such a geometry necessarily satisfies Einstein's equations perturbatively up to second order, with a stress energy tensor arising from matter fields associated with the sourced primary operators. We make no assumptions about AdS/CFT duality, so our work serves as both a consistency check for the AdS/CFT correspondence and a direct demonstration that spacetime and gravitational physics can emerge from the description of entanglement in conformal field theories.

Figures (8)

• Figure 1: AdS-Rindler patch associated with a ball $A$ on a spatial slice of the boundary. Solid blue paths indicate the boundary flow associated with $H_A$ and the conformal Killing vector $\zeta_A$. Dashed red paths indicate the action of the Killing vector $\xi_A$.
• Figure 2: The flow generated by the modular Hamiltonian of the region $A$ (in blue). We start by working in Euclidean time $x^0_E$, depicting the vector field $\partial_\tau$ on the left. The calculation then introduces an imaginary shift in $\tau$, which results in a displacement of the operator insertion into real time $t$. The picture of the real time vector field $\partial_s=\frac{1}{2\pi} \zeta_A$ is shown on the right with the causal structure indicated in gray. The two plotted planes can be thought of as orthogonal to each other and sewn together along the region $A$.
• Figure 3: The two-point function can be written in terms of the symplectic flux integrated over the domain of dependence $\mathcal{D}(A)$ (the shaded region) of $A$ (blue line) on a radial slice of an auxiliary $AdS$ spacetime, close to the asymptotic boundary.
• Figure 4: The asymptotic symplectic flux evaluated on $\mathcal{D}(A)$ at $r_B \to \infty$ (dashed blue line) is equal to the symplectic flux at the horizon $r_B \to 1$ (solid blue line).
• Figure 5: The null singularities of the Wightman propagators. (a) The $s_B$ integral is over the blue portion; the dashed blue line is the spacelike region which we can include for free. (b) The $s$ integral is over the blue portion, and once again we can include the dashed blue portion for free.