$L^2$-Gamma index theorem for spacetimes

TL;DR

This paper proves an $L^2$-Gamma index theorem for the Dirac operator on certain non-compact spacetimes, linking spectral flow to geometric expressions and extending previous results to broader settings.

Contribution

It extends the $L^2$-Gamma index theorem to non-compact Cauchy hypersurfaces, connecting spectral flow with geometric data for Dirac operators on globally hyperbolic manifolds.

Findings

Established $L^2$-Gamma index theorem for non-compact hypersurfaces

Connected spectral flow to geometric expressions

Extended previous index theorem results to broader classes of manifolds

Abstract

We establish an $L^2$-Gamma index theorem for the Dirac operator on a globally hyperbolic manifold $M$ with Cauchy hypersurface $\Sigma$ being a Galois covering of a compact smooth manifold with Galois group $\Gamma$. Our argument rewrites the $L^2$-Gamma index in terms of the spectral flow, which is then connected to the usual geometric expressions. This extends the work of B\"ar and Strohmaier to some non-compact Cauchy hypersurfaces. The analysis here is based on intermediate results by the first author on $L^2$-Gamma Fredholm properties of the Dirac operator.