Adiabatic limit in the Ginzburg-Landau and Seiberg-Witten equations
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.
KeywordsModulus Space Line Bundle STEKLOV Institute Dirac Operator Compact Riemann Surface
Unable to display preview. Download preview PDF.
- 9.D. Kotschick, “The Seiberg-Witten invariants of symplectic four-manifolds (after C.H. Taubes),” in Séminaire Burbaki, Volume 1995/96, Exposés 805–819 (Soc. Math. France, Paris, 1997), Exp. 812, Astérisque 241, pp. 195–220.Google Scholar
- 16.R. V. Pal’velev, “Justification of the adiabatic principle in the Abelian Higgs model,” Tr. Mosk. Mat. Obshch. 72(2), 281–314 (2011) [Trans. Moscow Math. Soc. 2011, 219–244 (2011)].Google Scholar
- 18.D. Salamon, “Spin geometry and Seiberg-Witten invariants,” Preprint (Warwick Univ., 1996).Google Scholar
- 21.A. G. Sergeev, Vortices and Seiberg-Witten Equations (Nagoya Univ., Nagoya, 2002).Google Scholar