Geodesic
Adiabatic limit in the Ginzburg--Landau and Seiberg--Witten equations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 270 (2010), pp. 233-242.

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We study an adiabatic limit in $(2+1)$-dimensional hyperbolic Ginzburg–Landau equations and 4-dimensional symplectic Seiberg–Witten equations. In dimension $3=2+1$ the limiting procedure establishes a correspondence between solutions of Ginzburg–Landau equations and adiabatic paths in the moduli space of static solutions, called vortices. The 4-dimensional adiabatic limit may be considered as a complexification of the $(2+1)$-dimensional procedure with time variable being “complexified.” The adiabatic limit in dimension $4=2+2$ establishes a correspondence between solutions of Seiberg–Witten equations and pseudoholomorphic paths in the moduli space of vortices. 
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     author = {A. G. Sergeev},
     title = {Adiabatic limit in the {Ginzburg--Landau} and {Seiberg--Witten} equations},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {233--242},
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     url = {http://geodesic.mathdoc.fr/item/TM_2010_270_a17/}
}
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A. G. Sergeev. Adiabatic limit in the Ginzburg--Landau and Seiberg--Witten equations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 270 (2010), pp. 233-242. http://geodesic.mathdoc.fr/item/TM_2010_270_a17/

[1] Jaffe A., Taubes C., Vortices and monopoles: Structure of static gauge theories, Birkhäuser, Boston, 1980 | MR | Zbl

[2] Salamon D., Spin geometry and Seiberg–Witten invariants, Preprint, Warwick Univ., 1996 | MR

[3] Sergeev A.G., “Adiabatic limit in the Seiberg–Witten equations”, Geometry, topology, and mathematical physics, AMS Transl. Ser. 2, 212, Amer. Math. Soc., Providence (RI), 2004, 281–295 | MR | Zbl

[4] Stuart D., “Dynamics of Abelian Higgs vortices in the near Bogomolny regime”, Commun. Math. Phys., 159 (1994), 51–91 | DOI | MR | Zbl

[5] Taubes C.H., “SW $\Rightarrow$ Gr: From the Seiberg–Witten equations to pseudo-holomorphic curves”, J. Amer. Math. Soc., 9 (1996), 845–918 | DOI | MR | Zbl

[6] Taubes C.H., “Gr $\Rightarrow$ SW: From pseudo-holomorphic curves to Seiberg–Witten solutions”, J. Diff. Geom., 51 (1999), 203–334 | MR | Zbl