Equivariant reduction of Yang-Mills theory over the fuzzy sphere and the emergent vortices

Abstract

We consider a U(2) Yang-Mills theory on M x S(F)(2) where M is a Riemannian manifold and S(F)(2) is the fuzzy sphere. Using essentially the representation theory of SU(2) we determine the most general SU(2)equivariant gauge field on M x S(F)(2). This allows us to reduce the Yang-Mills theory on M x S(F)(2) down to an Abelian Higgs-type model over M. Depending on the enforcement (or non-enforcement) of a "constraint" term, the latter may (or may not) lead to the standard critically-coupled Abelian Higgs model in the comutative limit, S(F)(2) -> S(2). For M = R(2), we find that the Abelian Higgs-type model admits vortex solutions corresponding to instantons in the original Yang-Mills theory. Vortices are in general no longer BPS, but may attract or repel according to the values of parameters. (C) 2009 Elsevier B.V. All rights reserved.