Maximally Non-Abelian Vortices from Self-dual Yang--Mills Fields

TL;DR

This paper explores a dimensional reduction of SU(2N) Yang--Mills theory leading to vortex equations on a Riemann surface, with solutions that are integrable on the hyperbolic plane, revealing new classes of non-Abelian vortices.

Contribution

It derives a new class of vortex equations from Yang--Mills theory and presents explicit solutions in the hyperbolic plane case, highlighting their integrability.

Findings

Reduction of SU(2N) Yang--Mills to vortex equations on Riemann surfaces

Identification of a subclass of solutions on the hyperbolic plane

Demonstration of formal integrability of these equations

Abstract

A particular dimensional reduction of SU(2N) Yang--Mills theory on $\Sigma \times S^2$, with $\Sigma$ a Riemann surface, yields an $S(U(N) \times U(N))$ gauge theory on $\Sigma$, with a matrix Higgs field. The SU(2N) self-dual Yang--Mills equations reduce to Bogomolny equations for vortices on $\Sigma$. These equations are formally integrable if $\Sigma$ is the hyperbolic plane, and we present a subclass of solutions.