  Aharonov and Casher showed that Pauli Hamiltonians in two dimensions have
gapless zero modes. We study the adiabatic evolution of these modes under the
slow motion of $N$ fluxons with fluxes $\Phi_a\in\mathbb{R}$. The positions,
$\mathbf{r}_a\in\mathbb{R}^2$, of the fluxons are viewed as controls. We are
interested in the holonomies associated with closed paths in the space of
controls. The holonomies can sometimes be abelian, but in general are not. They
can sometimes be topological, but in general are not. We analyze some of the
special cases and some of the general ones. Our most interesting results
concern the cases where holonomy turns out to be topological which is the case
when all the fluxons are subcritical, $\Phi_a<1$, and the number of zero modes
is $D=N-1$. If $N\ge3$ it is also non-abelian. In the special case that the
fluxons carry identical fluxes the resulting anyons satisfy the Burau
representations of the braid group.
