The Chiral Index of the Fermionic Signature Operator

TL;DR

This paper introduces a new index for the fermionic signature operator on certain spin manifolds, explores its invariance properties, and extends the concept to causal fermion systems with examples of non-trivial indices.

Contribution

It defines a novel index for the fermionic signature operator, studies its invariance, and generalizes it to causal fermion systems with chiral grading.

Findings

Index is non-trivial for specific space-times and Dirac operators.

Index remains invariant under homotopies.

Generalization to causal fermion systems is established.

Abstract

We define an index of the fermionic signature operator on even-dimensional globally hyperbolic spin manifolds of finite lifetime. The invariance of the index under homotopies is studied. The definition is generalized to causal fermion systems with a chiral grading. We give examples of space-times and Dirac operators thereon for which our index is non-trivial.