Spectral flow argument localizing an odd index pairing

TL;DR

This paper demonstrates how spectral flow techniques can be used to compute the index pairing of an odd Fredholm module via a finite-dimensional spectral localizer, providing a new computational approach.

Contribution

It introduces a spectral flow argument that localizes the odd index pairing to a finite-dimensional spectral localizer, offering a novel method for index calculation.

Findings

Index can be computed as the signature of the spectral localizer.

Spectral flow provides a localization technique for odd index pairings.

The approach links infinite-dimensional index theory with finite-dimensional matrix signatures.

Abstract

An odd Fredholm module for a given invertible operator on a Hilbert space is specified by an unbounded so-called Dirac operator with compact resolvent and bounded commutator with the given invertible. Associated to this is an index pairing in terms of a Fredholm operator with Noether index. Here it is shown by a spectral flow argument how this index can be calculated as the signature of a finite dimensional matrix called the spectral localizer.