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 → t , x → ϵ x with ϵ → 0 . The parent relativistic CFT should have left and right central charges of order O ( 1 / ϵ ) but opposite in sign in the limit ϵ → 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 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.