  We construct actions for (p,0)- and (p,1)- supersymmetric, 1 <= p <= 4,
two-dimensional gauge theories coupled to non-linear sigma model matter with a
Wess-Zumino term. We derive the scalar potential for a large class of these
models. We then show that the Euclidean actions of the (2,0) and
(4,0)-supersymmetric models without Wess-Zumino terms are bounded by
topological charges which involve the equivariant extensions of the Kahler
forms of the sigma model target spaces evaluated on the two-dimensional
spacetime. We give similar bounds for Euclidean actions of appropriate gauge
theories coupled to non-linear sigma model matter in higher spacetime
dimensions which now involve the equivariant extensions of the Kahler forms of
the sigma model target spaces and the second Chern character of gauge fields.
The BPS configurations are generalisations of abelian and non-abelian vortices.
