Science China Mathematics

, Volume 60, Issue 6, pp 1089–1100 | Cite as

Seiberg-Witten theory as a complex version of Abelian Higgs model



The adiabatic limit procedure associates with every solution of Abelian Higgs model in (2 + 1) dimensions a geodesic in the moduli space of static solutions. We show that the same procedure for Seiberg-Witten equations on 4-dimensional symplectic manifolds introduced by Taubes may be considered as a complex (2+2)-dimensional version of the (2 + 1)-dimensional picture. More precisely, the adiabatic limit procedure in the 4-dimensional case associates with a solution of Seiberg-Witten equations a pseudoholomorphic divisor which may be treated as a complex version of a geodesic in (2+1)-dimensional case.


Ginzburg-Landau equations vortices Seiberg-Witten equations 


53D42 53Z05 


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  1. 1.
    Donaldson S K. The Seiberg-Witten equations and 4-manifold topology. Bull Amer Math Soc, 1996, 33: 45–50MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Jaffe A, Taubes C H. Vortices and Monopoles. Boston: Birkhäuser, 1980MATHGoogle Scholar
  3. 3.
    Manton N S. A remark on the scattering of BPS monopoles. Phys Lett Ser B, 1982, 110: 54–56MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Palvelev R V. Justification of the adiabatic principle in the Abelian Higgs model. Trans Moscow Math Soc, 2011, 72: 219–244MathSciNetCrossRefGoogle Scholar
  5. 5.
    Palvelev R V, Sergeev A G. Justification of the adiabatic principle for hyperbolic Ginzburg-Landau equations. Proc Steklov Inst Math, 2012, 277: 191–205MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Salamon D. Spin Geometry and Seiberg-Witten Invariants. Warwick: University of Warwick, 1996MATHGoogle Scholar
  7. 7.
    Seiberg N, Witten E. Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory. Nuclear Phys B, 1994, 426: 19–22; Erratum: Nuclear Phys B, 1994, 430: 485–486MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Seiberg N, Witten E. Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD. Nuclear Phys B, 1994, 431: 484–550MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Sergeev A G. Adiabatic limit in Ginzburg-Landau and Seiberg-Witten equations. Proc Steklov Inst Math, 2015, 289: 227–285MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Taubes C H. SWGr: From the Seiberg-Witten equations to pseudoholomorphic curves. J Amer Math Soc, 1996, 9: 845–918MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Witten E. Monopoles and four-manifolds. Math Res Lett, 1994, 1: 769–796MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia

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