arXiv:1009.3139v1 [hep-th] 16 Sep 2010 Preprint typeset in JHEP style - HYPER VERSION Holographic Dual of Linear Dilaton Black Hole in Einstein-Maxwell-Dilaton-Axion Gravity Ran Li Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, 730000, Gansu, China liran05@lzu.cn Ji-Rong Ren Institute of Theoretical Physics, Lanzhou University, Lanzhou, 730000, Gansu, China renjr@lzu.edu.cn Abstract:Motivated by the recently proposed Kerr/CFT correspondence, we investigate the holographic dual of the extremal and non-extremal rotating linear dilaton black hole in Einstein-Maxwell-Dilaton-Axion Gravity. For the case of extremal black hole, by imposing the appropriate boundary condition at spatial infinity of the near horizon extremal geom- etry, the Virasoro algebra of conserved charges associated with the asymptotic symmetry group is obtained. It is shown that the microscopic entropy of the dual conformal field given by Cardy formula exactly agrees with Bekenstein-Hawking entropy of extremal black hole. Then, by rewriting the wave equation of massless scalar field with sufficient low energy as the SL(2, R)L×SL(2, R)RCasimir operator, we find the hidden conformal symmetry of the non-extremal linear dilaton black hole, which implies that the non-extremal rotating linear dilaton black hole is holographically dual to a two dimensional conformal field theory with the non-zero left and right temperatures. Furthermore, it is shown that the entropy of non-extremal black hole can be reproduced by using Cardy formula.
Contents 1.Introduction1 2.Rotating linear dilaton black hole2 3.Holographic dual of extremal linear dilaton black hole4 4.Hidden conformal symmetry of non-extremal linear dilaton black hole7 5.Conclusion11 1. Introduction The recently proposed extremal Kerr/CFT correspondence [1] states that quantum grav- ity in the region very near the event horizon of an extreme Kerr black hole with proper boundary conditions is holographically dual to a two-dimensional chiral conformal field theory with the central charge proportional to angular momentum. The method employed by Guica, Hartman, Song and Strominger (GHSS) in [1] is very similar to the approach of Brown and Henneaux in [2], where the AdS3background is replaced by the near-horizon extremal Kerr (NHEK) geometry previously obtained in [3].They shown that, by im- posing the appropriate boundary condition at spatial infinity of the NHEK geometry, the conserved charges associated with the asymptotic symmetry group are found to form a copy of Virasoro algebra. If identifying this algebra with the Virasoro algebra of the dual two dimensional conformal field theory, it is shown that the macroscopic Bekenstein-Hawking entropy of extremal Kerr black hole can be reproduced by the microscopic entropy of dual conformal field theory via Cardy formula. This method has been generalized to calculate the entropies of extremal black holes in a lot of theories such as the Einstein theory, string theory, and supergravity theory. Some further studies on the extremal Kerr/CFT dual are listed in [5]-[40]. More recently, Castro, Maloney and Strominger (CMS) in a remarkable paper [41] show that there exists a hidden SL(2, R)L×SL(2, R)Rconformal symmetry for the four dimensional non-extremal Kerr black hole by studying the near-region wave equation of a massless scalar field. Interestingly, this hidden conformal symmetry is not derived from the conformal symmetry of spacetime geometry itself, but probed by the perturbation fields in the near region. It is shown that, for the massive scalar field in the background of Kerr black hole with sufficient low energy, the wave equation can be reproduced by the SL(2, R)L×SL(2, R)RCasimir operator. It is also shown that microscopic entropy computed by Cardy formula agrees exactly with the macroscopic Berenstein-Hawking entropy of non- extremal Kerr black hole. These observations suggest that non-extremal Kerr black hole – 1 –
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