Quantum black hole entropy from 4d supersymmetric Cardy formula

We study the canonical AdS/CFT correspondence between 4d SU(N) N = 4 super Yang-Mills theory (SYM) and type IIB superstring theory on AdS(5) x S-5. We analyze the supersymmetric index of the N = 4 SYM on S-1 x M-3 which counts supersymmetric states with fixed quantum numbers. We compute an asymptotic behavior of the index in the limit of shrinking S-1 for any N by a refinement of the 4d 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 a rotating electrically charged Bogomolnyi-Prasad-Sommerfeld (BPS) black hole in AdS(5) via a Legendre transformation as recently shown in the literature. We also find that the agreement formally persists for finite N if we slightly modify the AdS/CFT dictionary between the Newton constant and N. This implies the existence of a nonrenormalization property of the black hole entropy against quantum corrections. We also study the cases with other gauge groups and additional matter and the orbifold N = 4 SYM. It turns out that the entropies of all the CFT examples in this paper are universally given by 2 pi root 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.