Callias-type operators associated to spectral triples

TL;DR

This paper introduces Callias-type operators linked to spectral triples, computes their indices via index pairing, and interprets the results as a non-commutative spectral flow index theorem, covering both even and odd cases.

Contribution

It extends the theory of Callias-type operators to the setting of spectral triples, providing a new index theorem in non-commutative geometry.

Findings

Computed indices of Callias-type operators using spectral triples

Established an index theorem as a non-commutative spectral flow

Provided examples in both commutative and non-commutative contexts

Abstract

Callias-type (or Dirac-Schr\"odinger) operators associated to abstract semifinite spectral triples are introduced and their indices are computed in terms of an associated index pairing derived from the spectral triple. The result is then interpreted as an index theorem for a non-commutative analogue of spectral flow. Both even and odd spectral triples are considered, and both commutative and non-commutative examples are given.