An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary

TL;DR

This paper establishes an index theorem for the Dirac operator on compact Lorentzian manifolds with spacelike Cauchy boundary, extending the Atiyah-Patodi-Singer index formula to a Lorentzian setting.

Contribution

It proves the Dirac operator is Fredholm under boundary conditions and derives an index formula analogous to the Riemannian case, including a construction of a Feynman parametrix.

Findings

Dirac operator is Fredholm with appropriate boundary conditions

Index formula matches Atiyah-Patodi-Singer for Riemannian manifolds

Constructs Feynman parametrix for product-type metrics

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.