Quantum Black Hole Entropy from 4d Supersymmetric Cardy formula

TL;DR

This work analyzes the supersymmetric index of 4d $\mathcal{N}=4$ SYM in the Cardy limit to reproduce the Bekenstein-Hawking entropy of rotating AdS$_5$ black holes via a Legendre transform. By employing a refined Cardy formula, the authors identify a dominant saddle at zero holonomy and derive the index asymptotics that scale with the central factor $(N^2-1)$, matching the expected gravitational entropy after a Legendre transformation. They extend the analysis to general manifolds $M_3$, other gauge groups, additional matter in conjugate representations, and orbifold theories, showing a universal entropy structure $S_{\rm QFT}(Q,J)=2\pi\sqrt{Q_1 Q_2 + Q_1 Q_3 + Q_2 Q_3 - 2 c (J_1+J_2)}$ with $c$ tied to the central charge. Finite-$N$ refinements indicate a non-renormalization property of the entropy in the Cardy limit when the AdS/CFT dictionary is adjusted to $4c$, suggesting robustness of the microscopic black hole counting across quantum corrections and providing predictions for gravity in broader holographic settings.

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 $2π\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.

Figures (2)

• Figure 1: $\kappa (x)$ and $\theta (x)$.
• Figure 2: $f(x)$ for $(m_1 ,m_2 )=(0.2, 0.1)$.