A new look at the vortex equations:
Dimensional reduction without
symmetry
The vortex equations on holomorphic bundles are equations for
special bundle metrics. They are thus similar to the Hermitian-Einstein
equations. Like the latter, they are directly related to a Geometric
Invariant Theory notion of stability, and have interesting moduli spaces
of solutions. In a special case, they have real versions which
correspond to the Seiberg-Witten equations for a four manifold. In this
talk we will describe some of the interesting features of the equations
and their moduli spaces. In particular, we will examine how vortex
equations on bundles over a closed Kähler manifold X are related
via dimensional reduction to Hermitian-Einstein equations on
SU(2)-equivariant bundles over X x
P1. We will discuss how such ideas can be
extended to more general situations where, surprisingly, there is no
longer a symmetry coming from a group action.
Back to
Spring 1997 meeting of the PNGS.
Suggestions or corrections to
Jack Lee <lee@math.washington.edu>.