The Joint Spectral Flow and Localization of the Indices of Elliptic Operators

TL;DR

This paper introduces the joint spectral flow, a new topological invariant generalizing spectral flow, and applies it to localize indices of elliptic operators, connecting analysis with topology.

Contribution

It defines the joint spectral flow using connective K-theory and applies it to index localization, inspired by Witten's deformation techniques.

Findings

Established the notion of joint spectral flow.

Rephrased analytic index localization in topological terms.

Connected spectral flow with K-theory and index theory.

Abstract

We introduce the notion of the joint spectral flow, which is a generalization of the spectral flow, by using Segal's model of the connective $K$-theory spectrum. We apply it for some localization results of indices motivated by Witten's deformation of Dirac operators and rephrase some analytic techniques in terms of topology.