Zero modes of the Weyl-Dirac operator

For a given magnetic potential $A$ one can define the Weyl-Dirac operator $\sigma.(-i\nabla-A)$ on $\mathbb{R}^3$. The question of whether or not this operator possess a zero-energy $L^2$ eigenfunction, or \emph{zero mode}, is quite subtle. We review some known results, including recent work on a conjecture for asymptotics in the strong field (or semi-classical) regime. In particular, we look at cases where the magnetic field is parallel to the generator of a conformal Killing field; such a symmetry allows the zero mode problem to be reduced to the consideration of a family of two-dimensional Dirac-type operators on a suitable quotient space.