The kernel of Dirac operators on $\S^3$ and $\R^3$

TL;DR

This paper presents a geometric method to construct magnetic fields on three-dimensional spheres and Euclidean space that yield Dirac operators with non-trivial kernels, including examples with arbitrarily specified kernel dimensions.

Contribution

It introduces an intrinsic geometric approach to generate magnetic fields on  and  with Dirac operators having prescribed kernel dimensions, extending previous examples.

Findings

Explicit construction of magnetic fields with non-trivial Dirac kernels.

Ability to produce examples with any desired kernel dimension.

Generalization of earlier work by Loss and Yau.

Abstract

In this paper we describe an intrinsically geometric way of producing magnetic fields on $\S^3$ and $\R^3$ for which the corresponding Dirac operators have a non-trivial kernel. In many cases we are able to compute the dimension of the kernel. In particular we can give examples where the kernel has any given dimension. This generalizes the examples of Loss and Yau (Commun. Math. Phys. 104 (1986) 283-290).