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

A. G. Sergeev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 270 (2010), pp. 233-242

Résumé

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.