The kernel of Dirac operators on ${\mathrm{\S }}^3$ and ${\mathbb{R}}^3$

In this paper we describe an intrinsically geometric way of producing magnetic fields on ${\mathrm{\S }}^3$ and ${\mathbb{R}}^3$ 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).