Generalised Dirac-Schrödinger operators and the Callias Theorem
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866917903759048704 |
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| author | Dungen, Koen van den |
| author_facet | Dungen, Koen van den |
| contents | We consider generalised Dirac--Schrödinger 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--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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_17600 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generalised Dirac-Schrödinger operators and the Callias Theorem Dungen, Koen van den K-Theory and Homology 19K35, 19K56, 58J20 We consider generalised Dirac--Schrödinger 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--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. |
| title | Generalised Dirac-Schrödinger operators and the Callias Theorem |
| topic | K-Theory and Homology 19K35, 19K56, 58J20 |
| url | https://arxiv.org/abs/2312.17600 |