  Gauged linear sigma-models at critical coupling on Riemann surfaces yield
self-dual field theories, their classical vacua being described by the vortex
equations. For local models with structure group ${\rm U}(r)$, we give a
description of the vortex moduli spaces in terms of a fibration over symmetric
products of the base surface $\Sigma$, which we assume to be compact. Then we
show that all these fibrations induce isomorphisms of fundamental groups. A
consequence is that all the moduli spaces of multivortices in this class of
models have abelian fundamental groups. We give an interpretation of this fact
as a no-go theorem for the realization of nonabelions through the ground states
of a supersymmetric version (topological via an A-twist) of these gauged
sigma-models. This analysis is based on a semi-classical approximation of the
QFTs via supersymmetric quantum mechanics on their classical moduli spaces.
