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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 in 1007.1031. 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 in 0907.0303, and still allow a copy of 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 in 0809.4266 lead to a divergent first order boundary effective action.