Science China Mathematics

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

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

Articles
  • 58 Downloads

Abstract

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.

Keywords

Ginzburg-Landau equations vortices Seiberg-Witten equations 

MSC(2010)

53D42 53Z05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This work was supported by Russian Foundation of Basic Research (Grants Nos. 16-01-00117 and 16-52-12012), the Program of support of Leading Scientific Schools (Grants No. NSh-9110.2016.1) and the Program of Presidium of Russian Academy of Sciences “Nonlinear dynamics”.

References

  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

Personalised recommendations