Index Theory and Supersymmetry of 5D Horizons

TL;DR

This paper proves that near-horizon geometries in minimal gauged five-dimensional supergravity preserve at least half of the supersymmetry, with higher symmetry implying a specific AdS structure, using index theory and Dirac operators.

Contribution

It establishes supersymmetry preservation bounds for 5D horizons and characterizes geometries with enhanced symmetry using index theorems and Dirac operators.

Findings

Near-horizon geometries preserve at least half of the supersymmetry.

Enhanced supersymmetry implies geometries are locally isometric to AdS_5.

Half-supersymmetric horizons admit an sl(2,R) symmetry subalgebra.

Abstract

We prove that the near-horizon geometries of minimal gauged five-dimensional supergravity preserve at least half of the supersymmetry. If the near-horizon geometries preserve a larger fraction, then they are locally isometric to $AdS_5$. Our proof is based on Lichnerowicz type theorems for two horizon Dirac operators constructed from the supercovariant connection restricted to the horizon sections, and on an application of the index theorem. An application is that all half-supersymmetric five-dimensional horizons admit an $\mathfrak{sl}(2, {\mathbb{R}})$ symmetry subalgebra.