Geometry and energy of non-Abelian vortices

Abstract

We study pure Yang–Mills theory on Σ × S2, where Σ is a compact Riemann surface, and invariance is assumed under rotations of S2. It is well known that the self-duality equations in this setup reduce to vortex equations on Σ. If the Yang–Mills gauge group is SU(2), the Bogomolny vortex equations of the Abelian Higgs model are obtained. For larger gauge groups, one generally finds vortex equations involving several matrix-valued Higgs fields. Here we focus on Yang–Mills theory with gauge group \documentclass[12pt]{minimal}\begin{document}$\mathrm{SU}(N)/\mathbb {Z}_N$\end{document} SU (N)/ZN and a special reduction which yields only one non-Abelian Higgs field. One of the new features of this reduction is the fact that while the instanton number of the theory in four dimensions is generally fractional with denominator N, we still obtain an integral vortex number in the reduced theory. We clarify the relation between these two topological charges at a bundle geometric level. Another striking feature is the emergence of nontrivial lower and upper bounds for the energy of the reduced theory on Σ. These bounds are proportional to the area of Σ. We give special solutions of the theory on Σ by embedding solutions of the Abelian Higgs model into the non-Abelian theory, and we relate our work to the language of quiver bundles, which has recently proved fruitful in the study of dimensional reduction of Yang–Mills theory.