Proceedings of the Steklov Institute of Mathematics

, Volume 270, Issue 1, pp 230–239 | Cite as

Adiabatic limit in the Ginzburg-Landau and Seiberg-Witten equations

  • A. G. Sergeev


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.


Modulus Space STEKLOV Institute Landau Equation Dynamic Solution Adiabatic Limit 
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  1. 1.
    A. Jaffe and C. Taubes, Vortices and Monopoles: Structure of Static Gauge Theories (Birkhäuser, Boston, 1980).zbMATHGoogle Scholar
  2. 2.
    D. Salamon, “Spin Geometry and Seiberg-Witten Invariants,” Preprint (Warwick Univ., 1996).Google Scholar
  3. 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.Google Scholar
  4. 4.
    D. Stuart, “Dynamics of Abelian Higgs Vortices in the Near Bogomolny Regime,” Commun. Math. Phys. 159, 51–91 (1994).zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    C. H. Taubes, “SW ⇒ Gr: From the Seiberg-Witten Equations to Pseudo-holomorphic Curves,” J. Am. Math. Soc. 9, 845–918 (1996).zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    C. H. Taubes, “Gr ⇒ SW: From Pseudo-holomorphic Curves to Seiberg-Witten Solutions,” J. Diff. Geom. 51, 203–334 (1999).zbMATHMathSciNetGoogle Scholar

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© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Steklov Institute of MathematicsRussian Academy of SciencesMoscowRussia

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