Boundary Conditions for NHEK through Effective Action Approach

We study the asymptotic symmetry group (ASG) of the near horizon geometry of extreme Kerr black hole through the effective action approach developed by Porfyriadis and Wilczek (arXiv:1007.1031v1gr-qc). By requiring a finite boundary effective action, we derive a new set of asymptotic Killing vectors and boundary conditions, which are much more relaxed than the ones proposed by Matsuo Y et al. Nucl. Phys. B 825 (2010) 231, and still allow a copy of a conformal group as its ASG. In the covariant formalism, the asymptotic charges are finite, with the corresponding central charge vanishing. By using the quasi-local charge and introducing a plausible cut-off, we find that the higher order terms of the asymptotic Killing vectors, which could not be determined through the effective action approach, contribute to the central charge as well. We also show that the boundary conditions suggested by Guica et al. Phys. Rev. D 80 (2009) 124008 lead to a divergent first-order boundary effective action.