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# Advances in Theoretical and Mathematical Physics

## Volume 21 (2017)

### Number 4

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

Pages: 901 – 920

DOI: http://dx.doi.org/10.4310/ATMP.2017.v21.n4.a2

#### Authors

#### 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 $U(r)$, 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.

#### Keywords

gauged linear sigma-model, vortex equation, nonabelions, Hecke transformation

#### 2010 Mathematics Subject Classification

14D21, 14H81, 58Z05

This work draws on discussions held at the occasion of the program “The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles” at the Institute for Mathematical Sciences, National University of Singapore; the authors would like to thank the organizers and NUS for hospitality. The first-named author is supported by a J. C. Bose Fellowship.