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

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.