Adiabatic limit in the equations of field theory

The notion of adiabatic limit came into mathematics from physics and in last years widely spread in differential geometry, theory of partial differential equations, topology. In our talk we shall speak about the applications of adiabatic limit construction in equations of gauge field theory.
We start from the Ginzburg-Landau equations in dimension 3=1(time)+2(space) arising in the superconductivity theory. In the adiabatic limit these equations convert into the Euler equation for geodesics on the space of vortices (static solutions of Ginzburg–Landau equations) with respect to the metric determined by the kinetic energy.
We turn next to dimension 4 and consider the adiabatic limit in Seiberg–Witten equations on 4-dimensional symplectic manifolds. In the adiabatic limit solutions of these equations converge to families of vortex solutions parameterized by points of pseudoholomorphic curves. Such families satisfy a nonlinear Cauchy–Riemann equation. So the adiabatic limit in Seiberg-Witten equations may be considered as a complex version of the same limit in Ginzburg–Landau equations. Namely, the Euler equation is replaced by the Cauchy–Riemann equation while geodesics on the space of vortices are substituted by the “complex” geodesics in vortex bundles over pseudoholomorphic curves. In other words, dimension 4 in this case may be treated as 4=2(complex time)+2(space).