From Ginzburg–Landau vortices to Seiberg–Witten equations

Ginzburg–Landau vortices are the static solutions of Ginzburg–Landau equations arising in the superconductivity theory. They remind the hydrodynamical vortices which explains the origin of their name. If we switch on the time in the considered model then the vortices start to move and may collide with each other. For example, two vortices moving along the straight line towards each other scatter to the right angle. To describe the vortex dynamics it is convenient to use the so called adiabatic limit by tending their velocity to zero. The limiting behavior of vortex trajectories is described by geodesics on the vortex space in the metric determined by the kinetic energy of the model.

It turns out that this model has a nontrivial 4-dimensional analogue described by the Seiberg–Witten equations. These are equations on 4-dimensional Riemannian manifolds being, together with Yang–Mills equations, the limiting case of the supersymmetric Yang–Mills theory. We are most interested in the case of symplectic manifolds since such manifolds have, apart from Riemannian metric, also an almost complex structure compatible with this metric. We plug into the Seiberg–Witten equations a scale parameter and take the adiabatic limit by tending this parameter to infinity. The limiting trajectories are described by the pseudoholomorphic curves which may be considered as complex analogs of Ginzburg–Landau geodesics. Solutions of Seiberg–Witten equations in the adiabatic limit reduce to the families of Ginzburg–Landau vortices in the planes normal to the limiting pseudoholomorphic curve. In this sense the Seiberg–Witten equations may be considered as complex analogs of dynamical Ginzburg–Landau equations in which the role of “time” is played by the parameter running along the limiting pseudoholomorphic curve.