Galilean Conformal Algebra in Two Dimensions and Cosmological Topologically Massive Gravity

TL;DR

The paper shows that the Galilean conformal algebra (GCA) in two dimensions emerges as a contraction of the Virasoro algebra realized on the AdS$_3$ boundary of cosmological topologically massive gravity (CTMG). By taking a non-relativistic scaling limit and letting the CTMG Chern-Simons coupling scale as $\mu\to\epsilon\mu$, the authors obtain finite GCA central charges $C_1$ and $C_2$ and finite scaling data $\Delta$ and $\xi$, enabling a gravity dual for the GCA. They derive a non-relativistic Cardy-like entropy $S_{\text{GCA}}=\pi\left(C_1\sqrt{\frac{2\xi}{C_2}}+\Delta\sqrt{\frac{2C_2}{\xi}}\right)$ from the BTZ black hole entropy and discuss modular-inspired counting of GCA states, with connections to logarithmic CFT structures. The work provides a concrete holographic realization of the GCA in CTMG and motivates further exploration of non-relativistic holography and microstate counting for Galilean field theories.

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 $t\to t$, $x\toεx$ with $ε\to 0$. The parent relativistic CFT should have left and right central charges of order $\mathcal{O}(1/ε)$ but opposite in sign in the limit $ε\to 0$. 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 $1/μ$. If $μ$ behaves as of order $\mathcal{O}(ε)$ 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.