# The BPS limit of rotating AdS black hole thermodynamics

## TL;DR

This paper derives extremization principles for the entropy of supersymmetric AdS black holes across multiple dimensions, linking the BPS limit to supergravity on-shell actions and complexified solutions.

## Contribution

It extends the BPS limit framework to higher dimensions and demonstrates that the entropy function corresponds to the supergravity on-shell action in this limit.

## Key findings

- Entropy is the Legendre transform of a homogeneous function of chemical potentials.
- Extremization equations determine chemical potentials consistent with supersymmetry.
- The entropy function matches the supergravity on-shell action in the BPS limit.

## 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.