Generalized Gravitational Entropy for Warped Anti–de Sitter Space

For spacetimes that are not asymptotic to anti–de Sitter (non AAdS) space, 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 (${\mathrm{WAdS}}_3$) in a consistent truncation of string theory, the so-called $S$-dual dipole theory. It turns out that the generalized gravitational entropy in ${\mathrm{WAdS}}_3$ is captured by the least action of a charged particle in ${\mathrm{WAdS}}_3$ space, or equivalently, by the geodesic length in an auxiliary ${\mathrm{AdS}}_3$. Consequently, the bulk calculation agrees with the CFT results, providing another piece of evidence for the ${\mathrm{WAdS}}_3/{\mathrm{CFT}}_2$ correspondence.