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Seiberg-Witten equations and complex Abrikosov strings. (English. Russian original) Zbl 1017.53072

Proc. Steklov Inst. Math. 235, 215-250 (2001); translation from Tr. Mat. Inst. Steklova 235, 224-261 (2001).
The main goal of the paper under review is to present a unified approach to two basic constructions in the theory of Seiberg-Witten equations in 4-dimensional topology, 4-dimensional symplectic geometry, or mathematical physics, respectively. Both constructions, namely the transition to the so-called adiabatic limit in the \((2+1)\)-dimensional Higgs model and the reduction of the 4-dimensional Seiberg-Witten equations to pseudoholomorphic curves, allow the approximate description of the solutions to the Seiberg-Witten equations for a 4-dimensional symplectic manifold in terms of curves in the moduli space of the reduced equations. In the three chapters of this paper, the author analyzes these two approaches in a general framework, together with their physical background and meaning. One of the crucial steps in the author’s analysis is the reduction of the problem of the approximate solutions of the corresponding Euler-Lagrange equations to the solution of the so-called Abrikosov equations on the moduli space of solutions of vortex equations (in the Ginzburg-Landau model). These adiabatic solutions in the case of a dynamical Higgs model describe the scattering of slowly moving vortices, whereas, in the case of its Euclidean analogue, they describe the so-called Abrikosov strings.
For the entire collection see [Zbl 0992.00076].

MSC:

53D35 Global theory of symplectic and contact manifolds
57R57 Applications of global analysis to structures on manifolds
32Q65 Pseudoholomorphic curves
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
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