Dirac–Schrödinger operators, index theory and spectral flow

In this article, we study generalised Dirac–Schrödinger 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–Schrödinger 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–Schrödinger 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 non-compact 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.