The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT

TL;DR

This work develops a holographic framework to compute entanglement Renyi entropies for 1+1D CFTs with gravity duals by mapping the n-sheeted replica surface to a bulk handlebody in AdS3 using Schottky uniformization. Through a monodromy/ accessory-parameter approach applied to an associated ODE, the authors construct replica-symmetric bulk saddles and relate their on-shell actions to boundary EREs, reproducing the RT entanglement entropy in the n→1 limit. They provide analytic results for the n=2 case, and a numerical strategy for higher n, demonstrating consistency with known RT predictions and the vacuum purity constraints, while discussing the role and potential limitations of missing saddles that may break replica symmetry. The paper also connects bulk actions to Liouville/ZT formalisms and outlines two complementary routes to compute the bulk action, validating the holographic prescription for disjoint intervals and the mutual Renyi information. Overall, it advances a concrete holographic program to derive RT-like results from gravitational saddles in AdS3 for multi-interval entanglement structures.

Abstract

We study entanglement Renyi entropies (EREs) of 1+1 dimensional CFTs with classical gravity duals. Using the replica trick the EREs can be related to a partition function of n copies of the CFT glued together in a particular way along the intervals. In the case of two intervals this procedure defines a genus n-1 surface and our goal is to find smooth three dimensional gravitational solutions with this surface living at the boundary. We find two families of handlebody solutions labelled by the replica index n. These particular bulk solutions are distinguished by the fact that they do not spontaneously break the replica symmetries of the boundary surface. We show that the regularized classical action of these solutions is given in terms of a simple numerical prescription. If we assume that they give the dominant contribution to the gravity partition function we can relate this classical action to the EREs at leading order in G_N. We argue that the prescription can be formulated for non-integer n. Upon taking the limit n -> 1 the classical action reproduces the predictions of the Ryu-Takayanagi formula for the entanglement entropy.

Figures (10)

• Figure 1: The RT prescription for computing the EE in 1+1 dimensional CFTs for 3 disjoint intervals. The CFT spatial direction is $\sigma$ and $r$ is the radial direction of the dual $AdS_3$. The minimal surfaces are simply geodesics connecting the ends of the intervals. The sum of the regularized lengths of these geodesics computes the EE. There is more than one minimal set of such geodesics and one is instructed to find the global minimum. We have shown only 2 cases out of a total of 5.
• Figure 3: A picture of the correspondence between boundary cycles for a fixed configuration $\Gamma_\gamma \in \mathcal{T}_3$ and the bulk geodesics of the RT formula (green curves hanging down from the boundary). The geodesics connect points defined by $P_\gamma$ in \ref{['pgama']}. Notice that in this picture the cycles in $\Gamma_\gamma$ are contractable in the bulk without crossing the geodesics.
• Figure 4: The case for two intervals. The black sold curves show the monodromy cycles $\Gamma \in \mathcal{T}_2$. There are two sets of cycles which we label $\alpha,\beta$. The dashed curve on the right figure is due to the fact that this curve actually moves through the branch cut into the last ($n$'th) replica. We also define here the specific cycles $C_\alpha, C_\beta$ for later reference.
• Figure 5: Calculated contributions to the Mutual Renyi Information (MRI) in holographic CFTs. We show in the left panel a subtracted version of the MRI as defined in \ref{['subtracted']}. Both saddles $\Gamma_\alpha$ and $\Gamma_\beta$ are important and dominate for $x<1/2$ and $x>1/2$ respectively. In the right we show the dependence of the MRI on $n$ for fixed $x$ (after analytically continuing from integer $n$). The $n=1$ limit for $x \leq 1/2$ is zero as predicted by the RT formula. Numerically it was more convenient to use \ref{['master']} to find this right plot.
• Figure 6: The Schottky fundamental domain for a genus $2$ surface with $2$ generators $L_1,L_2$. The shaded region is the domain (continued to infinity.) Note that typically one normalizes the generators using the freedom to conjugate by $PSL(2,\mathbb{C})$ such that one of the circles surrounds $w = \infty$ which is then absent from the domain. In the above picture we have not done this, since this will be convenient for us later.