A Lorentzian Equivariant Index Theorem

TL;DR

This paper derives a Lorentzian equivariant index theorem for twisted Dirac operators on globally hyperbolic spacetimes with boundary, extending Riemannian index formulas to a Lorentzian setting with boundary conditions.

Contribution

It introduces a Lorentzian equivariant index formula that parallels the Riemannian case, including boundary terms, and simplifies the reduction from equivariant to non-equivariant cases.

Findings

The index formula matches the Riemannian case with boundary terms.

A simple reduction technique from equivariant to non-equivariant regimes.

An equivariant Lorentzian index = spectral flow formula established.

Abstract

We develop a formula for the equivariant index of a twisted Dirac operator on a compact globally hyperbolic spacetime with timelike boundary on which a group acts isometrically, subject to APS boundary conditions. The formula is the same as in the Riemannian case: the equivariant index for a group element is an integral over the fixed point set of that element plus some boundary terms. The proof uses a surprisingly simple technique for reducing from the equivariant to the non-equivariant regime in order to show an equivariant version of the Lorentzian "index $=$ spectral flow" formula.