Black hole superpotential as a unifying entropy function and BPS thermodynamics

Abstract

We develop an effective superpotential formalism for the SU(2)×U(1) invariant sector of $$ \mathcal{N} $$ 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 S3, 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)×U(1) invariant sector and show that the conserved charges of supersymmetric solutions satisfy the generalized BPS relation derived in [1], which includes the supersymmetric Casimir energy as a consequence of the anomalous supersymmetry transformation of the $$ \mathcal{N} $$ 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.

On the uniqueness of supersymmetric AdS(5) black holes with toric symmetry

“…It is interesting that for supersymmetric black holes with a smooth horizon the near-horizon geometry determines the full solution uniquely and does not allow for more general conformal boundary metrics. While it is known that supersymmetry constrains the boundary geometry [38][39][40][41][42], our uniqueness theorem shows that for supersymmetric solutions with smooth horizons the boundary geometry is even more constrained (at least for toric Calabi type Käher bases). It would be interesting to understand this phenomenon from a holographic perspective.…”

“…In this section we will determine the general form of the symplectic potential near any connected component of a supersymmetric horizon. The strategy is to start with the near-horizon geometry (42), find the coordinate change between GNC and symplectic coordinates and then match to the general form for a supersymmetric toric solution (22).…”

Thermodynamics of BPS and Near-BPS AdS6 Black Holes

“…From various perspectives these are the most natural black holes to study, but the solutions are so complicated that they remain poorly explored. The study of the type presented above should be manageable and would illuminate further aspects of the much discussed five-dimensional AdS black holes [58][59][60].…”

Thermodynamics of BPS and near-BPS AdS6 black holes

“…From various perspectives these are the most natural black holes to study, but the solutions are so complicated that they remain poorly explored. The study of the type presented above should be manageable and would illuminate further aspects of the much discussed five-dimensional AdS black holes [59][60][61].…”