Dirac-Schrödinger operators, index theory, and spectral flow

Abstract: 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 Theorem, which shows that the index (or the spectral flow, or abstractly the K-theory class) of Dirac-Schr\"odinger operators can be computed on a suitable 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 operators. Combining both results, we obtain an index (or spectral flow) equality relating Toeplitz operators on the noncompact manifold to Toeplitz operators on the compact hypersurface. Our results generalise various known results from the literature, while presenting these results in a common unified framework.