# Supersymmetric near-horizon geometry and Einstein-Cartan-Weyl spaces

## TL;DR

This paper demonstrates that supersymmetric black hole horizons in five-dimensional gauged supergravity are described by a specific Einstein-Cartan-Weyl structure in three dimensions, linking to known geometries in the ungauged limit.

## Contribution

It identifies the precise geometric structure of supersymmetric black hole horizons in gauged supergravity, extending previous results to include torsion and nonmetricity.

## Key findings

- Horizon geometry characterized by Einstein-Cartan-Weyl structure with torsion and nonmetricity.
- Reduction to hyper-CR Einstein-Weyl structure in the zero cosmological constant limit.
- Consistent geometric description bridging gauged and ungauged supergravity cases.

## Abstract

We show that the horizon geometry for supersymmetric black hole solutions of minimal five-dimensional gauged supergravity is that of a particular Einstein-Cartan-Weyl (ECW) structure in three dimensions, involving the trace and traceless part of both torsion and nonmetricity, and obeying some precise constraints. In the limit of zero cosmological constant, the set of nonlinear partial differential equations characterizing this ECW structure reduces correctly to that of a hyper-CR Einstein-Weyl structure in the Gauduchon gauge, which was shown by Dunajski, Gutowski and Sabra to be the horizon geometry in the ungauged BPS case.