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Adiabatic limit in the Ginzburg-Landau and Seiberg-Witten equations. (English. Russian original) Zbl 1236.35180

Proc. Steklov Inst. Math. 270, 230-239 (2010); translation from Trudy Mat. Inst. Steklova 270, 233-242 (2010).
The author studies an adiabatic limit of the \((2+1)\)-dimensional hyperbolic Ginzburg-Landau and \(4\)-dimensional symplectic Seiberg-Witten equations. The author proves that, for the hyperbolic Ginzburg-Landau equations, there is an adiabatic limit procedure, establishing a relation between solutions of these equations and adiabatic paths in the moduli space of static solutions, called vortices. Adiabatic paths coincide with the geodesics in the moduli space of vortices with respect to a natural metric given by kinetic energy. In dimension \(4= 2+ 2\), the adiabatic limit establishes a correspondence between the solutions of the Seiberg-Witten equations and the pseudoholomorphic paths in the moduli space of vortices.

MSC:

35Q56 Ginzburg-Landau equations
53C27 Spin and Spin\({}^c\) geometry
57R57 Applications of global analysis to structures on manifolds
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References:

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