Local Index Theory for Lorentzian Manifolds
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866908224200900608 |
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| author | Baer, Christian Strohmaier, Alexander |
| author_facet | Baer, Christian Strohmaier, Alexander |
| contents | Index theory for Lorentzian Dirac operators is a young subject with significant differences to elliptic index theory. It is based on microlocal analysis instead of standard elliptic theory and one of the main features is that a nontrivial index is caused by topologically nontrivial dynamics rather than nontrivial topology of the base manifold. In this paper we establish a local index formula for Lorentzian Dirac-type operators on globally hyperbolic spacetimes. This local formula implies an index theorem for general Dirac-type operators on spatially compact spacetimes with Atiyah-Patodi-Singer boundary conditions on Cauchy hypersurfaces. This is significantly more general than the previously known theorems that require the compatibility of the connection with Clifford multiplication and the spatial Dirac operator on the Cauchy hypersurface to be selfadjoint with respect to a positive definite inner product. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_01364 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Local Index Theory for Lorentzian Manifolds Baer, Christian Strohmaier, Alexander Differential Geometry Mathematical Physics Primary 58J20, 58J45, secondary 35L02, 35L05, 58J32 Index theory for Lorentzian Dirac operators is a young subject with significant differences to elliptic index theory. It is based on microlocal analysis instead of standard elliptic theory and one of the main features is that a nontrivial index is caused by topologically nontrivial dynamics rather than nontrivial topology of the base manifold. In this paper we establish a local index formula for Lorentzian Dirac-type operators on globally hyperbolic spacetimes. This local formula implies an index theorem for general Dirac-type operators on spatially compact spacetimes with Atiyah-Patodi-Singer boundary conditions on Cauchy hypersurfaces. This is significantly more general than the previously known theorems that require the compatibility of the connection with Clifford multiplication and the spatial Dirac operator on the Cauchy hypersurface to be selfadjoint with respect to a positive definite inner product. |
| title | Local Index Theory for Lorentzian Manifolds |
| topic | Differential Geometry Mathematical Physics Primary 58J20, 58J45, secondary 35L02, 35L05, 58J32 |
| url | https://arxiv.org/abs/2012.01364 |