Generalized Gravitational Entropy for Warped Anti-de Sitter Space

TL;DR

This work extends holographic entanglement entropy to non-AAdS spacetimes by adapting the Lewkowycz–Maldacena replica trick. It identifies three key modifications—compatibility between near-boundary and near-surface expansions, periodicity constraints on phase-space coordinates rather than all fields, and a presymplectic-boundary-term evaluation of the entropy—to define a generalized entropy for non-AAdS backgrounds. Focusing on warped AdS$_3$ in the S-dual dipole truncation, the authors show that the generalized entropy is captured by the length of a geodesic in an auxiliary AdS$_3$ (or equivalently by the world-line action of a charged particle in $WAdS_3$), with explicit expressions in terms of BTZ-like data and a Cardy-like logarithmic form. The results align with CFT$_2$ expectations and provide a concrete bulk prescription that extends holographic entanglement entropy beyond AdS, reinforcing the $WAdS_3$/CFT$_2$ correspondence and its relation to WCFT-like structures.

Abstract

For spacetimes that are not asymptotic to anti-de Sitter Space (non AAdS), we adapt the Lewkowycz-Maldacena procedure to find the holographic entanglement entropy. The key observation, which to our knowledge is not very well appreciated, is that asymptotic boundary conditions play an essential role on extending the replica trick to the bulk. For non AAdS, we expect the following three main modifications: (1) the expansion near the special surface has to be compatible with the asymptotic expansion; (2) periodic conditions are imposed to coordinates on the phase space with diagonalized symplectic structure, not to all fields appearing in the action; (3) evaluating the entanglement functional using the boundary term method amounts to evaluating the presymplectic structure at the special surface, where some additional exact form may contribute. An explicit calculation is carried out for three-dimensional warped anti-de Sitter spacetime (WAdS3) in a consistent truncation of string theory, the so-called S-dual dipole theory. It turns out that the generalized gravitational entropy in WAdS3 is captured by the least action of a charged particle in WAdS3 space, or equivalently, by the geodesic length in an auxiliary AdS3. Consequently, the bulk calculation agrees with the CFT results, providing another piece of evidence for the WAdS3/CFT2 correspondence.