Galilean Conformal Algebra in Two Dimensions and Cosmological Topologically Massive Gravity

TL;DR

This paper explores the realization of the two-dimensional Galilean conformal algebra on the boundary of a three-dimensional gravity theory, deriving a new non-relativistic entropy formula from black hole solutions.

Contribution

It demonstrates how the GCA emerges from AdS boundary conditions in CTMG and introduces a novel entropy formula for Galilean field theories.

Findings

GCA is obtained from Virasoro algebra via a scaling limit.

A new non-relativistic entropy formula analogous to Cardy’s formula is proposed.

The analysis connects boundary GCA properties with bulk black hole entropy.

Abstract

We consider a realization of the Galilean conformal algebra (GCA) in two dimensional space-time on the AdS boundary of a particular three dimensional gravity theory, the so-called cosmological topologically massive gravity (CTMG), which includes the gravitational Chern-Simons term and the negative cosmological constant. The infinite dimensional GCA in two dimensions is obtained from the Virasoro algebra for the relativistic CFT by taking a scaling limit $t\to t$, $x\to\epsilon x$ with $\epsilon\to 0$. The parent relativistic CFT should have left and right central charges of order $\mathcal{O}(1/\epsilon)$ but opposite in sign in the limit $\epsilon\to 0$. On the other hand, by Brown-Henneaux's analysis the Virasoro algebra is realized on the boundary of AdS$_3$, but the left and right central charges are asymmetric only by the factor of the gravitational Chern-Simons coupling $1/\mu$. If $\mu$ behaves as of order $\mathcal{O}(\epsilon)$ under the corresponding limit, we have the GCA with non-trivial centers on AdS boundary of the bulk CTMG. Then we present a new entropy formula for the Galilean field theory from the bulk black hole entropy, which is a non-relativistic counterpart of the Cardy formula. It is also discussed whether it can be reproduced by the microstate counting.