# An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary

#### Autoren: Christian Bär, Alexander Strohmaier (2019)

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.

Zeitschrift:
Amer. J. Math.
Verlag:
Johns Hopkins Univ. Press
Seiten:
1421-1455
Band:
141 (5)

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