Seiberg-Witten Equations and Pseudoholomorphic Curves

Abstract

We consider the Taubes correspondence between solutions of Seiberg-Witten equations on a compact four-dimensional symplectic manifold and pseudo-holomorphic curves. We start from Kähler surfaces, in which case there is a direct correspondence between solutions of Seiberg-Witten equations and holomorphic curves. The general Taubes correspondence for symplectic four-manifolds involves, in contrast with the Kähler case, a limiting procedure, called the scaling limit. Under this scaling limit solutions of Seiberg-Witten equations reduce to families of solutions of certain vortex equations in the normal bundle of the limiting pseudoholomorphic curve.