Generalised Dirac-Schrödinger operators and the Callias Theorem

Dungen, Koen van den 

Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e11

Résumé

We consider generalised Dirac-Schrödinger operators, consisting of a self-adjoint elliptic first-order differential operator $\mathcal {D}$ 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 $\mathcal {D}$. Our main result in this paper is a generalisation of the Callias Theorem: the index of the Dirac-Schrödinger 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.

Détail

DOI : 10.1017/fms.2024.157

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