An index theorem for non-standard Dirac operators

TL;DR

This paper explores the index theory of non-standard Dirac operators on manifolds with special geometric structures, relating it to standard Dirac operators and analyzing cases with torsion and boundary.

Contribution

It introduces a new index theorem for non-standard Dirac operators associated with manifolds possessing Killing tensors and boundary conditions.

Findings

Established a relation between indices of non-standard and standard Dirac operators.

Summarized recent results on manifolds with torsion and boundary.

Extended index theory to non-traditional geometric settings.

Abstract

On manifolds with non-trivial Killing tensors admitting a square root of the Killing-Yano type one can construct non-standard Dirac operators which differ from, but commute with, the standard Dirac operator. We relate the index problem for the non-standard Dirac operator to that of the standard Dirac operator. This necessitates a study of manifolds with torsion and boundary and we summarize recent results obtained for such manifolds.