A no-go theorem for nonabelionic statistics in gauged linear sigma-models

TL;DR

This paper proves that in certain supersymmetric gauged sigma-models, the vortex moduli spaces have abelian fundamental groups, preventing the realization of nonabelionic statistics in their ground states.

Contribution

It establishes a no-go theorem showing that these models cannot support nonabelionic statistics due to their topological properties.

Findings

Vortex moduli spaces have abelian fundamental groups.

Fibrations over symmetric products induce isomorphisms of fundamental groups.

Implication: nonabelions cannot arise from these models' ground states.

Abstract

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.