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An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 141, Number 5, October 2019
- pp. 1421-1455
- 10.1353/ajm.2019.0037
- Article
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abstract:
We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same expression as in the index formula of Atiyah-Patodi-Singer for Riemannian manifolds with boundary. The index is also shown to equal that of a certain operator constructed from the evolution operator and a spectral projection on the boundary. In case the metric is of product type near the boundary a Feynman parametrix is constructed.