Black hole superpotential as a unifying entropy function and BPS thermodynamics

TL;DR

This paper develops an effective superpotential formalism for five-dimensional gauged supergravity, enabling exact analysis of BPS black hole solutions, their thermodynamics, and a unifying entropy extremization principle.

Contribution

It introduces an exact superpotential for all 1/4 BPS solutions in a specific supergravity sector, unifying entropy extremization methods for various BPS black holes.

Findings

Exact superpotential describes all 1/4 BPS solutions.

Analytic evaluation of on-shell action and conserved charges.

Unified entropy extremization principle encompassing known methods.

Abstract

We develop an effective superpotential formalism for the SU(2)$\times$U(1) invariant sector of $\mathcal{N}=2$ gauged supergravity in five dimensions with a U(1)$^3$ Fayet-Iliopoulos gauging, and determine the exact superpotential that describes all 1/4 BPS solutions in this sector. This includes the Gutowski-Reall black holes, but also a much broader class of solutions with a squashed $S^3$, magnetic flux and vector multiplet sources, as well as complex Euclidean BPS saddles. Some of these solutions are known only numerically, but the exact superpotential allows us to analytically evaluate the on-shell action, holographic one-point functions and conserved charges of all BPS solutions and to study their thermodynamics. In particular, by examining the supersymmetry Ward identities we show that solutions with supersymmetric vector multiplet sources break supersymmetry spontaneously. We also demonstrate the first law for black holes in the SU(2)$\times$U(1) invariant sector and show that the conserved charges of supersymmetric solutions satisfy the generalized BPS relation derived in arXiv:1703.04299, which includes the supersymmetric Casimir energy as a consequence of the anomalous supersymmetry transformation of the $\mathcal{N}=1$ supercurrent at the boundary. Finally, we show that the effective superpotential provides a unifying entropy extremization principle, reproducing Sen's entropy function for near extremal black holes and the Hosseini-Hristov-Zaffaroni functional for complex Euclidean BPS saddles.