The stable homotopy theory of vortices on Riemann surfaces

The purpose of these notes is to show that the methods introduced by Bauer and Furuta, see [5, 6, 7], in order to refine the Seiberg-Witten invariants of smooth 4-dimensional manifolds can also be used to obtain stable homotopy classes from 2-dimensional manifolds, using the vortex equations on the latter. So far these notes contain barely more than the necessary analytic estimates to prove this. The implications and applications will be added as time permits. In this section we review the necessary background material on connections on line bundles on Riemann surfaces. Let X be a closed oriented connected surface with a Riemann metric. The Hodge star operator on 1-forms induces a complex structure on X. This structure is inte-grable, so that X is a complex curve. The metric on X is automatically Kähler, and the Kähler form ω agrees with the volume form.