Killing-Yano tensors, non-standard supersymmetries and an index theorem

TL;DR

This paper explores the role of Killing-Yano tensors in space-times, revealing new supersymmetries for Dirac particles, and establishes a duality relating space-times with and without torsion through index theorems.

Contribution

It introduces a geometrical duality connecting space-times with and without torsion and relates Dirac operator indices across these dual geometries.

Findings

Dirac particles exhibit new fermionic constants of motion.

A duality links space-times with different torsion properties.

Index relations allow expressing torsion space-time indices via torsion-free counterparts.

Abstract

The existence of Killing-Yano tensors on space-times can be probed by spinning particles. Specifically, Dirac particles possess new fermionic constants of motion corresponding to non-standard supersymmetries on the particle worldline. A geometrical duality connects space-times with Killing-Yano structure, but without torsion, to other space-times with Killing-Yano structure and torsion. A relation between the indices of the Dirac operators on the dual space-times allows to express the index on the space-time with torsion in terms of that of the space-time without torsion.