Adiabatic Paths and Pseudoholomorphic Curves

We consider the Taubes correspondencebetween solutions of the Seiberg– Witten equations on a compact 4-dimensional symplectic manifold and pseudoholomorphic curves on this manifold. We compare it with its 3-dimensional analogue in which the roles of Seiberg–Witten equations and pseudoholomorphic curves are played respectively by the Ginzburg–Landau equations and adiabatic paths in the moduli space of static Ginzburg–Landau solutions. C.H.Taubes in his papers [7,8] has established a correspondence between solutions of the Seiberg–Witten equations on a compact 4-dimensional symplectic manifold and pseudoholomorphic curves. The Taubes correspondence involves a certain limiting procedure, called the scaling limit. It turns out that there exists a non-trivial 3-dimensional analogue of this procedure. In the 3-dimensional (or, better to say, in the (2+1)-dimensional) setting the role of Seiberg–Witten equations is played by the hyperbolic Ginzburg–Landau equations and the scaling limit is replaced by the adiabatic limit, involving the introduction of the ”slow” time. The adiabatic limit construction establishes a correspondence between solutions of Ginzburg–Landau equations and certain adiabatic paths in the moduli space of static solutions. The Taubes correspondence can be considered from this point of view as a complex (or, better to say, (2+2)-dimensional) analogue of the adiabatic limit construction in which pseudoholomorphic curves should be considered as complex adiabatic paths. In this paper I present an overview of the results, related to the Taubes correspondence and its 3-dimensional analogue. The work on this subject started when I first visited the Erwin Schrödinger Institute in 1999 and I would like to thank ESI for the hospitality. I. ABELIAN (2+1)-DIMENSIONAL HIGGS MODEL 1.1. Ginzburg–Landau Equations We consider the (2+1)-dimensionalAbelian Higgs model, governed by theGinzburg– Landau action functional S(A,Φ) = ∫ {T (A,Φ)− U(A,Φ)}dt (1.1) 1991 Mathematics Subject Classification. Primary 59E15.