Geometry and Energy of Non-abelian Vortices

TL;DR

This paper investigates non-abelian vortices derived from Yang--Mills theory on a product of a Riemann surface and a sphere, revealing new topological and energetic properties, and connecting to quiver bundle frameworks.

Contribution

It introduces a special reduction of Yang--Mills with gauge group SU(N)/Z_N yielding a single non-abelian Higgs field and integral vortex numbers, clarifying topological charge relations.

Findings

Integral vortex numbers despite fractional instanton numbers.

Existence of energy bounds proportional to surface area.

Embedded solutions from abelian Higgs model.

Abstract

We study pure Yang--Mills theory on $\Sigma\times S^2$, where $\Sigma$ is a compact Riemann surface, and invariance is assumed under rotations of $S^2$. It is well known that the self-duality equations in this set-up reduce to vortex equations on $\Sigma$. 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 $\SU{N}/\ZZ_N$ 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 non-trivial lower and upper bounds for the energy of the reduced theory on $\Sigma$. These bounds are proportional to the area of $\Sigma$. We give special solutions of the theory on $\Sigma$ 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.