The BPS limit of rotating AdS black hole thermodynamics

TL;DR

This work establishes a unified gravity-side derivation of the BPS black hole entropy in AdS across dimensions by demonstrating that the Bekenstein–Hawking entropy is the Legendre transform of a homogeneous entropy function of complex chemical potentials, under a linear complex constraint. By extending Cabo-Bizet’s BPS limit to AdS$_5$ with multiple electric charges and to AdS$_4$, AdS$_6$, and AdS$_7$, the authors show that the supersymmetric on-shell action along a complexified, supersymmetric trajectory matches the proposed entropy functions; the resulting extremization reproduces the correct BPS entropies and nonlinear charge relations. The framework connects black hole thermodynamics with dual SCFT partition functions, via the quantum statistical relation and holographic renormalization, and provides a robust method to derive entropy functions in diverse AdS/CFT settings. These results support a broad, dimension-spanning mechanism for counting microstates of rotating BPS black holes and pave the way for further cross-dimension checks with field theory indices and Cardy-like limits.

Abstract

We consider rotating, electrically charged, supersymmetric AdS black holes in four, five, six and seven dimensions, and provide a derivation of the respective extremization principles stating that the Bekenstein-Hawking entropy is the Legendre transform of a homogeneous function of chemical potentials, subject to a complex constraint. Extending a recently proposed BPS limit, we start from finite temperature and reach extremality following a supersymmetric trajectory in the space of complexified solutions. We show that the entropy function is the supergravity on-shell action in this limit. Chemical potentials satisfying the extremization equations also emerge from the complexified solution.