Sergeev, A. G. 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. Reviewer: Fortuné Massamba (Gaborone) MSC: 35Q56 Ginzburg-Landau equations 53C27 Spin and Spin\({}^c\) geometry 57R57 Applications of global analysis to structures on manifolds Keywords:adiabatic limit; Ginzburg-Landau equations; Seiberg-Witten equations PDFBibTeX XMLCite \textit{A. G. Sergeev}, Proc. Steklov Inst. Math. 270, 230--239 (2010; Zbl 1236.35180); translation from Trudy Mat. Inst. Steklova 270, 233--242 (2010) Full Text: DOI References: [1] A. Jaffe and C. Taubes, Vortices and Monopoles: Structure of Static Gauge Theories (Birkhäuser, Boston, 1980). · Zbl 0457.53034 [2] D. Salamon, ”Spin Geometry and Seiberg-Witten Invariants,” Preprint (Warwick Univ., 1996). · Zbl 0865.57019 [3] A. G. Sergeev, ”Adiabatic Limit in the Seiberg-Witten Equations,” in Geometry, Topology, and Mathematical Physics (Am. Math. Soc., Providence, RI, 2004), AMS Transl., Ser. 2, 212, pp. 281–295. · Zbl 1075.58009 [4] D. Stuart, ”Dynamics of Abelian Higgs Vortices in the Near Bogomolny Regime,” Commun. Math. Phys. 159, 51–91 (1994). · Zbl 0807.35141 [5] C. H. Taubes, ”SW Gr: From the Seiberg-Witten Equations to Pseudo-holomorphic Curves,” J. Am. Math. Soc. 9, 845–918 (1996). · Zbl 0867.53025 [6] C. H. Taubes, ”Gr SW: From Pseudo-holomorphic Curves to Seiberg-Witten Solutions,” J. Diff. Geom. 51, 203–334 (1999). · Zbl 1036.53066 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.