Adiabatic limit in the Ginzburg-Landau and Seiberg-Witten equations (Q5894164)

The author studies an adiabatic limit of the \((2+1)\)-dimensional hyperbolic Ginzburg-Landau and \(4\)-dimensional symplectic Seiberg-Witten equations. The author proves that, for the hyperbolic Ginzburg-Landau equations, there is an adiabatic limit procedure, establishing a relation between solutions of these equations and adiabatic paths in the moduli space of static solutions, called vortices. Adiabatic paths coincide with the geodesics in the moduli space of vortices with respect to a natural metric given by kinetic energy. In dimension \(4= 2+ 2\), the adiabatic limit establishes a correspondence between the solutions of the Seiberg-Witten equations and the pseudoholomorphic paths in the moduli space of vortices.