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

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.