Generalized gravitational entropy without replica symmetry

TL;DR

The paper extends the Lewkowycz–Maldacena construction of generalized gravitational entropy by allowing replica-symmetry breaking in the bulk and by incorporating a perturbative Einstein–Gauss–Bonnet correction. It introduces a generalized near-surface metric ansatz and boundary conditions that admit replica-symmetry breaking, derives the surface equations of motion by demanding cancellation of leading divergences to order $(n-1)$, and shows that the entangling surface must extremize the Jacobson–Myers entropy $S_{JM}$ in both GR and perturbative GB gravity. The results demonstrate that replica-symmetry breaking terms need not vanish and do not alter the JM-extremality condition, thereby supporting evaluating holographic entanglement entropy on $S_{JM}$-extremal surfaces for higher-curvature theories. This framework paves the way for extensions to Lovelock gravity and raises questions about which replica-breaking saddles dominate the gravitational path integral in holographic contexts.

Abstract

We explore several extensions of the generalized entropy construction of Lewkowycz and Maldacena, including a formulation that does not rely on preserving replica symmetry in the bulk. We show that an appropriately general ansatz for the analytically continued replica metric gives us the flexibility needed to solve the gravitational field equations beyond general relativity. As an application of this observation we study Einstein-Gauss-Bonnet gravity with a small Gauss-Bonnet coupling and derive the condition that the holographic entanglement entropy must be evaluated on a surface which extremizes the Jacobson-Myers entropy. We find that in both general relativity and Einstein-Gauss-Bonnet gravity replica symmetry breaking terms are permitted by the field equations, suggesting that they do not generically vanish.

Figures (3)

• Figure 1: A sketch of the Ryu--Takayanagi surface $\Sigma$ associated with some boundary region $A$. $\Gamma$ is a codimension-one surface satisfying $\partial \Gamma = \Sigma \cup A$.
• Figure 2: A sketch of the $n=3$ replica manifold. The three solid black lines represents the $\tau$ circle of the boundary manifold $B^3$ and the dashed lines represent cuts at $\tau = 2\pi k$ for integer $k$. The gray line is a closed curve in the bulk $M^3$ which illustrates how the three slices are glued together along the cuts. The path integral on $B^3$ computes $\mathop{\mathrm{Tr}}\nolimits[\rho^3]$ and provides a geometric realization of the formula \ref{['eq:rhotothen']}. This path integral can also be expressed as the action associated with the metric $g^{(3)}$, a smooth metric that solves the gravitational field equation on $M^3$, as in \ref{['eq:saddlepoint']}. Note that even if the state $\rho^3$ is replica symmetric, $g^{(3)}$ is not simply three copies of $g^{(1)}$ glued together, as the latter metric would not be smooth.
• Figure 3: A sketch of the coordinates used in the text. $\Sigma$ is the codimension-two entropy surface. In our coordinates, $\Sigma$ is located at $x^1 = 0 =x^2$ and points on its surface are described by the $D-2$ coordinates $\sigma^i$.