Quantum black hole entropy from 4d supersymmetric Cardy formula

Abstract

We study the canonical $\mathrm{AdS}/\mathrm{CFT}$ correspondence between 4d $SU(N)$ $\mathcal{N}=4$ super Yang-Mills theory (SYM) and type IIB superstring theory on ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$. We analyze the supersymmetric index of the $\mathcal{N}=4$ SYM on ${S}^{1}\ifmmode\times\else\texttimes\fi{}{M}_{3}$ 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 ${\mathrm{AdS}}_{5}$ 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 $\mathrm{AdS}/\mathrm{CFT}$ 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 $\mathcal{N}=4$ SYM. It turns out that the entropies of all the CFT examples in this paper are universally given by $2\ensuremath{\pi}\sqrt{{Q}_{1}{Q}_{2}+{Q}_{1}{Q}_{3}+{Q}_{2}{Q}_{3}\ensuremath{-}2c({J}_{1}+{J}_{2})}$ 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.