On the space of connections having non-trivial twisted harmonic spinors

TL;DR

This paper investigates the structure of the space of connections on twisting bundles over odd-dimensional spin manifolds, showing that under certain conditions, this space has infinitely many connected components.

Contribution

It demonstrates that the space of connections producing invertible twisted Dirac operators has infinitely many connected components when the untwisted operator is invertible and the bundle dimension is large.

Findings

Infinite connected components in the space of connections

Conditions for invertibility of twisted Dirac operators

Dependence on bundle dimension and untwisted operator invertibility

Abstract

We consider Dirac operators on odd-dimensional compact spin manifolds which are twisted by a product bundle. We show that the space of connections on the twisting bundle which yield an invertible operator has infinitely many connected components if the untwisted Dirac operator is invertible and the dimension of the twisting bundle is sufficiently large.