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 × S 2 F where M is a Riemannian manifold and S 2 F is the fuzzy sphere. Using essentially the representation theory of SU(2) we determine the most general SU(2)-equivariant gauge field on M×S 2 F . This allows us to reduce the Yang-Mills theory on M × S 2 F 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 criticallycoupled abelian Higgs model in the commutative limit, S 2 F → 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.

Equivariant reduction ofU(4)gauge theory overSF2×SF2Kürkçüoğlu1 2012Phys. Rev. D7110View full textAdd to dashboardCite

“…Such a numerical computation was given in [17], for the case of Uð1Þ vortices emerging from the equivariant reduction of a Uð2Þ theory over M Â S 2 F . We will not go into numerical calculations in this article.…”

“…The construction of the most general SUð2Þ-equivariant gauge field on S 2 F can be performed as follows [17]:…”

“…Starting with the article [17], we have initiated investigations on the equivariant reduction of gauge theories over fuzzy extra dimensions. In [17] the most general SUð2Þ-equivariant Uð2Þ gauge field over M Â S 2 F has been found, and it was utilized to perform the dimensional reduction over S 2 F .…”

“…In [17] the most general SUð2Þ-equivariant Uð2Þ gauge field over M Â S 2 F has been found, and it was utilized to perform the dimensional reduction over S 2 F . It was shown that for M ¼ R 2 the emergent theory is an Abelian Higgs type model which has non-BPS vortex solutions corresponding to the instantons in the original theory.…”

“…The V L 2 and the V R 2 serve as constraint terms to suppress the normal components of the gauge fields on each of the fuzzy spheres, in a similar manner as discussed for the case of a single fuzzy sphere in [2,17].…”