On the uniqueness of supersymmetric AdS$_5$ black holes with toric symmetry

TL;DR

This paper classifies supersymmetric AdS5 black holes with toric symmetry, showing that known solutions are unique under certain conditions and deriving the general form of the symplectic potential near horizons.

Contribution

It proves the uniqueness of the CCLP black hole solution among supersymmetric, toric, timelike solutions with Calabi-type Kähler bases.

Findings

The symplectic potential near horizons is explicitly characterized.

The CCLP solution has a remarkably simple symplectic potential.

Any solution with the specified properties must be the CCLP black hole or its near-horizon geometry.

Abstract

We consider the classification of supersymmetric AdS$_5$ black hole solutions to minimal gauged supergravity that admit a torus symmetry. This problem reduces to finding a class of toric K\"ahler metrics on the base space, which in symplectic coordinates are determined by a symplectic potential. We derive the general form of the symplectic potential near any component of the horizon or axis of symmetry, which determines its singular part for any black hole solution in this class, including possible new solutions such as black lenses and multi-black holes. We find that the most general known black hole solution in this context, found by Chong, Cvetic, L\"u and Pope (CCLP), is described by a remarkably simple symplectic potential. We prove that any supersymmetric and toric solution that is timelike outside a smooth horizon, with a K\"ahler base metric of Calabi type, must be the CCLP black hole solution or its near-horizon geometry.