Galilean Conformal Algebra in Two Dimensions and Cosmological Topologically Massive Gravity

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.

15 pages, no figure; minor changes, a reference added, version to be published in Nucl. Phys. B

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