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

Understand

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 commutative limit, S_F^2 -> S^2.