A No-go Theorem for Nonabelionic Statistics in Gauged Linear Sigma-models

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 U(r), we give a description of the vortex moduli spaces in terms of a fibration over symmetric products of the base surface Σ, 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 semiclassical approximation of the QFTs via supersymmetric quantum mechanics on their classical moduli spaces.