An index of strongly Callias operators on Lorentzian manifolds with non-compact boundary

TL;DR

This paper extends the index theory of hyperbolic Dirac-type operators with growing potentials to non-compact Lorentzian manifolds, providing a Fredholm boundary value problem and an index formula involving local integrals and eta-invariants.

Contribution

It introduces an index formula for strongly Callias operators on non-compact Lorentzian manifolds with boundary, expanding previous results to non-compact settings.

Findings

The boundary value problem for the operator is Fredholm.

An explicit index formula involving local integrals and eta-invariants is derived.

Extension of index theory to non-compact Lorentzian manifolds with boundary.

Abstract

We consider a hyperbolic Dirac-type operator with growing potential on a a spatially non-compact globally hyperbolic manifold. We show that the Atiyah-Patodi-Singer boundary value problem for such operator is Fredholm and obtain a formula for this index in terms of the local integrals and the relative eta-invariant introduced by Braverman and Shi. This extends recent results of B\"ar and Strohmaier, who studied the index of a hyperbolic Dirac operator on a spatially compact globally hyperbolic manifold.