  It is shown that $\Vert A \Vert_{L^d}^2 \ge \frac{d}{d-2}\, S_d$ is a
necessary condition for the existence of a nontrivial solution of the Dirac
equation $\gamma \cdot (-i\nabla -A)\psi = 0$ in $d$ dimensions. Here, $S_d$ is
the sharp Sobolev constant. If $d$ is odd and $\Vert A \Vert_{L^d}^2=
\frac{d}{d-2}\, S_d$, then there exist vector potentials that allow for zero
modes. A complete classification of these vector potentials and their
corresponding zero modes is given.
