# Quantum Black Hole Entropy from 4d Supersymmetric Cardy formula

## TL;DR

This paper derives a refined asymptotic formula for the supersymmetric index of 4d $ ext{SU}(N)$ $ ext{N}=4$ super Yang-Mills theory, connecting it to black hole entropy in AdS$_5$ and exploring non-renormalization properties.

## Contribution

It introduces a refined supersymmetric Cardy formula for arbitrary $N$ and demonstrates its agreement with black hole entropy, including finite $N$ modifications and extensions to various gauge groups and geometries.

## Key findings

- Asymptotic index matches Bekenstein-Hawking entropy for large $N$.
- Finite $N$ modifications suggest non-renormalization of black hole entropy.
- Universal entropy formula involving charges and angular momenta.

## Abstract

We study supersymmetric index of 4d $SU(N)$ $\mathcal{N}=4$ super Yang-Mills theory on $S^1 \times M_3$. 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 $\mathcal{N}=4$ super Yang-Mills theory. It turns out that the entropies of all the CFT examples in this paper are given by $2\pi \sqrt{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.