Galilean Conformal Algebra in Two Dimensions and Cosmological Topologically Massive Gravity

• Published in: arXiv.org 2010-06
• Main Authors: Hotta, Kyosuke, Kubota, Takahiro, Nishinaka, Takahiro
• Format: Article
• Language: English
• Subjects: Algebra; Black holes; Cosmological constant; Field theory; Gravitation theory; Relativism; Relativistic effects
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1003.1203

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.