Spectral multiplicity and odd K-theory-II

TL;DR

This paper explores the relationship between spectral multiplicity, spectral flow, and index gerbe in families of unbounded self-adjoint Fredholm operators parametrized by 3-manifolds, using holonomy-based descriptions.

Contribution

It introduces new descriptions of the spectral flow and index gerbe classes in terms of holonomy and analyzes their connection to spectral multiplicity variations.

Findings

Spectral multiplicity influences the spectral flow and gerbe classes.

Holonomy-based descriptions provide new insights into these classes.

The study is focused on families parametrized by closed 3-manifolds.

Abstract

Let {D_x} be a family of unbounded self-adjoint Fredholm operators representing an element of K^1(M). Consider the first two components of the Chern character of the family. It is known that these correspond to the spectral flow of the family and the index gerbe. In this paper we consider descriptions of these classes, both of which are in the spirit of holonomy. These are then studied for families parametrized by a closed 3-manifold. A connection between the multiplicity of the spectrum (and how it varies) and these classes is developed.