Dirac-Schr\"odinger operators, index theory, and spectral flow

In this article we study generalised Dirac-Schr\"odinger operators in arbitrary signatures (with or without gradings), providing a general KK-theoretic framework for the study of index pairings and spectral flow. We provide a general Callias-type Theorem, which allows to compute the index (or the spectral flow, or abstractly the K-theory class) of Dirac-Schr\"odinger operators from their restriction to a compact hypersurface. Furthermore, if the zero eigenvalue is isolated in the spectrum of the Dirac operator, we relate the index (or spectral flow) of Dirac-Schr\"odinger operators to the index (or spectral flow) of corresponding Toeplitz-type operators. Our results generalise various known results from the literature, while presenting these results in a common unified framework.