Spinc Dirac operators over the flat 3-torus

TL;DR

This paper investigates the spectral properties and boundary phenomena of Spinc Dirac operators on a flat 3-torus, providing explicit spectrum calculations and spectral sections for certain families.

Contribution

It offers explicit spectral analysis and constructions of spectral sections for Spinc Dirac operators on a 3-torus boundary, advancing understanding of boundary phenomena in this context.

Findings

Explicit spectrum and eigenbasis for certain families

Construction of spectral sections for small parameters

Description of index in K^1 group

Abstract

The aim of this paper is to study a possible "boundary phenomenon" for Spinc Dirac operators in a special case. If you parametrise Spinc Dirac operators by a family of connections on a Spinc 4-manifold with boundary, this boundary inherits also a family of Spinc Dirac operators which has a spectral section (in the sense of Melrose/Piazza). We want to analyse this situation concretely for a manifold whose boundary is a 3-torus. For some families of Spinc Dirac operators, we calculate spectrum and eigenbasis and give explicit constructions of spectral sections for small disturbance parameters. We also describe the index of these families in $K^1$.