Quantum black hole entropy from 4d supersymmetric Cardy formula

Abstract

We study supersymmetric index of 4d SU (N ) N = 4 super Yang-Mills theory on S 1 × M 3 . We compute asymptotic behavior of the index in the limit of shrinking S 1 for arbitrary N by a refinement of supersymmetric Cardy formula. The asymptotic behavior for the superconformal index case (M 3 = S 3 ) at large N agrees with the Bekenstein-Hawking entropy of rotating electrically charged BPS black hole in AdS 5 via a Legendre transformation as recently shown in literature. We also find that the agreement formally persists for finite N if we slightly modify the AdS/CFT dictionary between Newton constant and N . This implies an existence of non-renormalization property of the quantum black hole entropy. We also study the cases with other gauge groups and additional matters, and the orbifold N = 4 super Yang-Mills theory. It turns out that the entropies of all the CFT examples in this paper are given by 2π Q 1 Q 2 + Q 1 Q 3 + Q 2 Q 3 − 2c(J 1 + J 2 ) with charges Q 1,2,3 , angular momenta J 1,2 and central charge c. The results for other M 3 make predictions to the gravity side. * mh974ATdamtp.cam.ac.uk 1 See also [10] for another embedding.

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