Thermodynamics of Dual CFTs for Kerr-AdS Black Holes

Abstract

Recently Gibbons {\it et al.} in hep-th/0408217 defined a set of conserved quantities for Kerr-AdS black holes with the maximal number of rotation parameters in arbitrary dimension. This set of conserved quantities is defined with respect to a frame which is non-rotating at infinity. On the other hand, there is another set of conserved quantities for Kerr-AdS black holes, defined by Hawking {\it et al.} in hep-th/9811056, which is measured relative to a frame rotating at infinity. Gibbons {\it et al.} explicitly showed that the quantities defined by them satisfy the first law of black hole thermodynamics, while those quantities defined by Hawking {\it et al.} do not obey the first law. In this paper we discuss thermodynamics of dual CFTs to the Kerr-AdS black holes by mapping the bulk thermodynamic quantities to the boundary of the AdS space. We find that thermodynamic quantities of dual CFTs satisfy the first law of thermodynamics and Cardy-Verlinde formula only when these thermodynamic quantities result from the set of bulk quantities given by Hawking {\it et al.}. We discuss the implication of our results.