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arXiv · 2312.17600

Generalised Dirac-Schr\"odinger operators and the Callias Theorem

Koen van den Dungen
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Abstract

We consider generalised Dirac--Schr\"odinger operators, consisting of a self-adjoint elliptic first-order differential operator D with a skew-adjoint 'potential' given by a (suitable) family of unbounded operators. The index of such an operator represents the pairing (Kasparov product) of the K-theory class of the potential with the K-homology class of D. Our main result in this paper is a generalisation of the Callias Theorem: the index of the Dirac--Schr\"odinger operator can be computed on a suitable compact hypersurface. Our theorem simultaneously generalises (and is inspired by) the well-known result that the spectral flow of a path of relatively compact perturbations depends only on the endpoints.

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Koen van den Dungen. 2023-12-29. Generalised Dirac-Schr\"odinger operators and the Callias Theorem. https://doi.org/10.1017/fms.2024.157

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