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

We consider a U(2) Yang-Mills theory on M×SF2 where M is a Riemannian manifold and SF2 is the fuzzy sphere. Using essentially the representation theory of SU(2) we determine the most general SU(2)-equivariant gauge field on M×SF2. This allows us to reduce the Yang-Mills theory on M×SF2 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 commutative limit, SF2→S2. For M=R2, 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.