An equivariant index theorem on Lorentzian manifolds

For differential operators preserved by the action of a group $G$, the notion of index generalises to the $G$-index. A prototype arises for Dirac operators on a compact Riemannian spin manifold admitting an isometry group $G$ and the resulting theorem is known as the Atiyah-Segal-Singer index theorem. In this talk, I shall discuss a generalization of this theorem for a $G$-equivariant Dirac operator on a globally hyperbolic spin spacetime with spacelike boundaries subject to the Atiyah-Patodi-Singer boundary condition. Our analysis is based on the singularity structure of Feynman parametrices instead of the heat-kernel proof due to Berlin and Vergne. (Joint work with C. Bär and L. Ronge).