Generalized Gravitational Entropy for Warped Anti-de Sitter Space

TL;DR

This paper extends the Lewkowycz-Maldacena holographic entanglement entropy method to non-AAdS spacetimes, specifically warped AdS3, revealing that the entropy can be computed via geodesic lengths in an auxiliary AdS3, supporting WAdS3/CFT2 duality.

Contribution

It adapts the holographic entanglement entropy procedure for non-AAdS spacetimes, highlighting the importance of boundary conditions and presymplectic structures, with explicit calculations in WAdS3.

Findings

Holographic entanglement entropy in WAdS3 matches CFT results.

Entropy can be computed via geodesic length in auxiliary AdS3.

Boundary conditions are crucial for extending the replica trick.

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.