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

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