Abstract
The phase space of three-dimensional gravity with Compere-Song-Strominger (CSS) boundary conditions is endowed with asymptotic symmetries consisting in the semi-direct product of a Virasoro and a $\hat{u}(1)$ Ka\v{c}-Moody algebra, and contains BTZ black holes whose entropy can be accounted for by the degeneracy of states of a Warped CFT. By embedding these boundary conditions in Topologically Massive Gravity, we observe the existence of two special points in the space of couplings parameterized by the AdS$_3$ radius $\ell$ and the Chern-Simons coupling $\mu$. When $\mu = \pm {1\over \ell}$, the asymptotic symmetries reduce to either a chiral Virasoro algebra or a pure $\hat{u}(1)$ Ka\v{c}-Moody current algebra. At those points, black holes have positive energy while that of linearized excitations are non-negative.
Highlights
Lower-dimensional gravity models have appeared over the years as a fertile playground to address fundamental questions in quantum gravity, such as the black hole entropy problem [1,2] of the black hole information paradox [3,4]
In this work we investigated CSS boundary conditions in topologically massive gravity (TMG) for jμjl ≥ 1 and noticed two special points in the space of couplings
At μl 1⁄4 1: (i) The uð1Þ Kač-Moody charges and level vanish; (ii) BTZ black holes have positive energy, an entropy reproduced by a chiral half of the Cardy formula and lM 1⁄4 −J; (iii) The massive graviton and boundary photon acquire vanishing energy, while the boundary graviton has positive energy
Summary
It was noticed in a series of works that besides the classic Brown-Henneaux boundary conditions with conformal symmetry, a variety of alternative consistent choices could exist [61,62,63,64,65,66,67,68,69] It was observed by Compere, Song and Strominger (CSS) that pure 3d gravity with appropriate boundary conditions (referred to as CSS) has asymptotic symmetries consisting in the semidirect product of a Virasoro and a uð1Þ Kač-Moody algebra. Those symmetries are those of a new type of 2d field theories, Warped CFTs (WCFTs) [70,71]. The main goal of the present work we will be to show that embedding CSS boundary conditions in TMG allows to identify two special points in coupling space where the warped symmetries either reduce to a chiral Virasoro algebra or to a uð1Þ Kač-Moody algebra, thereby possibly providing the simplest example of a gravity theory including black holes
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