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

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.