Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS$_{\bf 5}$ black holes

TL;DR

The paper tackles the microscopic origin of the Bekenstein-Hawking entropy for supersymmetric AdS$_5$ black holes by formulating a BPS limit through complexified, supersymmetric bulk solutions and showing the entropy arises from a constrained Legendre transform of the on-shell action, with three chemical potentials linked by SUSY regularity. On the field theory side, it uses exact localization to compute the dual $\mathcal N=1$ SCFT partition function on a twisted $S^1\times S^3$, producing a factorization into a generalized supersymmetric Casimir energy $\mathcal F$ and a Hamiltonian index $\mathcal I$, where $\mathcal F$ scales as $\propto c\,\varphi^3/(\omega_1\omega_2)$ at large $N$, reproducing the black hole entropy via the constrained Legendre transform. The results establish a precise gravity–field theory dictionary for BPS AdS$_5$ black holes, show the entropy function arises from a supersymmetric Euclidean gravity action, and connect the index to the Hamiltonian counting of BPS states, revealing a cardy-like structure in four dimensions. They also provide a general framework that extends to theories with flavor symmetries and to more general AdS/CFT setups, offering insight into the role of anomalies and extremization in holographic entropy.

Abstract

We present a holographic derivation of the entropy of supersymmetric asymptotically AdS$_5$ black holes. We define a BPS limit of black hole thermodynamics by first focussing on a supersymmetric family of complexified solutions and then reaching extremality. We show that in this limit the black hole entropy is the Legendre transform of the on-shell gravitational action with respect to three chemical potentials subject to a constraint. This constraint follows from supersymmetry and regularity in the Euclidean bulk geometry. Further, we calculate, using localization, the exact partition function of the dual $\mathcal{N}=1$ SCFT on a twisted $S^1\times S^3$ with complexified chemical potentials obeying this constraint. This defines a generalization of the supersymmetric Casimir energy, whose Legendre transform at large $N$ exactly reproduces the Bekenstein-Hawking entropy of the black hole.