HyperKahler manifolds and Seiberg-Witten equations

In this talk, I will discuss the so called nonlinear gauged
sigma model in dimension four. An important element of the construction is
a nonlinear generalization of the Dirac operator on a 4-manifold such that
the fiber of the spinor vector bundle, a copy of quaternions H, is replaced
by a hyperKahler manifold admitting certain symmetries. This Dirac operator
is used to define a generalization of the Seiberg-Witten equations.
However, the candidate hyperKahler manifolds are neither compact nor
complete and therefore the compactification of the moduli space of
solutions poses a challenge. I will discuss some progress made in this
direction. I will also describe a ``dimensional reduction" of the above
theory to two-dimensions which produce a Higgs-field and the equations we
obtain generalize the Symplectic Vortex equations.