Boundary Conditions of Warped AdS$_3$ in New Massive Gravity

TL;DR

This work implements boundary conditions that realize Warped AdS$_3$ within three-dimensional New Massive Gravity and derives the full solution space, residual symmetries, and surface charges. The asymptotic algebra forms a direct sum of a Virasoro (Witt) and a $u(1)$ Kac‑Moody sector, with charges initially non-integrable but made integrable by restricting to a sub-sector, yielding a centrally extended $Vir\oplus u(1)$ algebra. The central charges $c$ and level $k$ are explicit functions of the theory’s parameters, and special limits (null warped, CSS) are analyzed. Importantly, the warped Cardy formula reproduces the bulk entropy of Warped BTZ black holes, reinforcing the WCFT/WAdS$_3$ holographic correspondence in $NMG$, and suggesting avenues for extending the analysis to other 3D massive gravities and gauge choices.

Abstract

To satisfy the Cardy formula for Warped AdS$_3$ (WAdS$_{3}$) solutions in a quadratic ensemble, a specific set of boundary conditions has been proposed in \cite{Aggarwal:2020igb}. In this paper, these boundary conditions have been investigated in the three-dimensional new massive gravity (NMG) framework. The associated solution space, asymptotic symmetries, and charge algebra have been extracted. It has been shown that the surface charges are finite, but not integrable, and the integrability of the charges is obtained after restricting to a sub-sector of the original solution space. Finally, we have proved that the Cardy formula reproduces the thermodynamic entropy of a warped BTZ black hole.