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

A. G. Sergeev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected issues of mathematics and mechanics, Tome 289 (2015), pp. 242-303

Résumé

Hyperbolic Ginzburg–Landau equations arise in gauge field theory as the Euler–Lagrange equations for the $(2+1)$-dimensional Abelian Higgs model. The moduli space of their static solutions, called vortices, was described by Taubes; however, little is known about the moduli space of dynamic solutions. Manton proposed to study dynamic solutions with small kinetic energy with the help of the adiabatic limit by introducing the “slow time” on solution trajectories. In this limit the dynamic solutions converge to geodesics in the space of vortices with respect to the metric generated by the kinetic energy functional. So, the original equations reduce to Euler geodesic equations, and by solving them one can describe the behavior of slowly moving dynamic solutions. It turns out that this procedure has a 4-dimensional analog. Namely, for the Seiberg–Witten equations on 4-dimensional symplectic manifolds it is possible to introduce an analog of the adiabatic limit. In this limit, solutions of the Seiberg–Witten equations reduce to families of vortices in normal planes to pseudoholomorphic curves, which can be considered as complex analogs of geodesics parameterized by “complex time.” The study of the adiabatic limit for the equations indicated in the title is the main content of this paper.