Galilean conformal algebra in two dimensions and cosmological topologically massive gravity. (English) Zbl 1206.83100

Hotta, Kyosuke; Kubota, Takahiro; Nishinaka, Takahiro

Nucl. Phys., B 838, No. 3, 358-370 (2010).

Summary: 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 \(t\rightarrow t\), \(x\rightarrow \epsilon x\) with \(\epsilon\rightarrow 0\). The parent relativistic CFT should have left and right central charges of order \({\mathcal O}(1/\epsilon)\) but opposite in sign in the limit \(\epsilon\rightarrow 0\). 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.

MSC:

83C57	Black holes
94A17	Measures of information, entropy
83E30	String and superstring theories in gravitational theory
81R10	Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
81T40	Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
80A10	Classical and relativistic thermodynamics

Keywords:

black hole; entropy; topologically massive gravity; AdS/CFT; infinite-dimensional algebra

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