  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.
