Adiabatic limit in the Ginzburg-Landau and Seiberg-Witten equations

Abstract

We study an adiabatic limit in (2 + 1)-dimensional hyperbolic Ginzburg-Landau equations and 4-dimensional symplectic Seiberg-Witten equations. In dimension 3 = 2+1 the limiting procedure establishes a correspondence between solutions of Ginzburg-Landau equations and adiabatic paths in the moduli space of static solutions, called vortices. The 4-dimensional adiabatic limit may be considered as a complexification of the (2+1)-dimensional procedure with time variable being “complexified.” The adiabatic limit in dimension 4 = 2+2 establishes a correspondence between solutions of Seiberg-Witten equations and pseudoholomorphic paths in the moduli space of vortices.