Replica R\'enyi Wormholes and Generalised Modular Entropy in JT Gravity

TL;DR

This paper explores the computation of Rènyi entropies in JT gravity with nontrivial backreaction, introducing generalized modular entropy and analyzing replica wormholes, including explicit solutions for the double trumpet geometry.

Contribution

It extends the understanding of Rènyi entropies in JT gravity by incorporating backreaction effects and generalizing the quantum extremal surface condition for arbitrary replica number n.

Findings

Derived the QES condition involving generalized modular entropy.

Explicitly constructed the n=2 double trumpet wormhole geometry.

Calculated n-dependent Page times for high-temperature black holes.

Abstract

We consider the problem of computing semi-classical R\'enyi entropies of CFT on AdS$_2$ backgrounds in JT gravity with nongravitating baths, for general replica number $n$. Away from the $n\to 1$ limit, the backreaction of the CFT twist fields on the geometry is nontrivial. For one twist field insertion and general $n$, we show that the quantum extremal surface (QES) condition involves extremisation of the generalised modular entropy, consistent with Dong's generalisation of the Ryu-Takayanagi formula for general $n$. For multiple QES we describe replica wormhole geometries using the theory of Fuchsian uniformisation, explicitly working out the analytically tractable case of the $n=2$ double trumpet wormhole geometry. We determine the off-shell dependence of the gravitational action on the QES locations and boundary map. In a factorisation limit, corresponding to late times, we are able to relate this action functional to area terms given by the value of the JT dilaton at the (off-shell) QES locations, with computable corrections. Applied to the two-sided eternal black hole, we find the $n$-dependent Page times for R\'enyi enropies in the high temperature limit.