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

  • A. G. Sergeev
Article

Abstract

Hyperbolic Ginzburg-Landau equations arise in gauge field theory as the Euler-Lagrange equations for the (2 + 1)-dimensional Abelian Higgs model. The moduli space of their static solutions, called vortices, was described by Taubes; however, little is known about the moduli space of dynamic solutions. Manton proposed to study dynamic solutions with small kinetic energy with the help of the adiabatic limit by introducing the “slow time” on solution trajectories. In this limit the dynamic solutions converge to geodesics in the space of vortices with respect to the metric generated by the kinetic energy functional. So, the original equations reduce to Euler geodesic equations, and by solving them one can describe the behavior of slowly moving dynamic solutions. It turns out that this procedure has a 4-dimensional analog. Namely, for the Seiberg-Witten equations on 4-dimensional symplectic manifolds it is possible to introduce an analog of the adiabatic limit. In this limit, solutions of the Seiberg-Witten equations reduce to families of vortices in normal planes to pseudoholomorphic curves, which can be considered as complex analogs of geodesics parameterized by “complex time.” The study of the adiabatic limit for the equations indicated in the title is the main content of this paper.

Keywords

Modulus Space Line Bundle STEKLOV Institute Dirac Operator Compact Riemann Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    M. F. Atiyah and N. Hitchin, The Geometry and Dynamics of Magnetic Monopoles (Princeton Univ. Press, Princeton, NJ, 1988).CrossRefzbMATHGoogle Scholar
  2. 2.
    S. B. Bradlow, “Vortices in holomorphic line bundles over closed Kähler manifolds,” Commun. Math. Phys. 135, 1–17 (1990).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    A. V. Domrin, “Ginzburg-Landau vortex analogues,” Teor. Mat. Fiz. 124(1), 18–35 (2000) [Theor. Math. Phys. 124, 872–886 (2000)].MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    S. K. Donaldson, “An application of gauge theory to four dimensional topology,” J. Diff. Geom. 18, 279–315 (1983).MathSciNetzbMATHGoogle Scholar
  5. 5.
    O. García-Prada, “A direct existence proof for the vortex equations over a compact Riemann surface,” Bull. London Math. Soc. 26, 88–96 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    M. Gromov, “Pseudo holomorphic curves in symplectic manifolds,” Invent. Math. 82, 307–347 (1985).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    A. Jaffe and C. Taubes, Vortices and Monopoles: Structure of Static Gauge Theories (Birkhäuser, Boston, 1980).zbMATHGoogle Scholar
  8. 8.
    J. L. Kazdan and F. W. Warner, “Curvature functions for compact 2-manifolds,” Ann. Math., Ser. 2, 99, 14–47 (1974).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    D. Kotschick, “The Seiberg-Witten invariants of symplectic four-manifolds (after C.H. Taubes),” in Séminaire Burbaki, Volume 1995/96, Exposés 805–819 (Soc. Math. France, Paris, 1997), Exp. 812, Astérisque 241, pp. 195–220.Google Scholar
  10. 10.
    P. B. Kronheimer and T. S. Mrowka, “The genus of embedded surfaces in the projective plane,” Math. Res. Lett. 1, 797–808 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 2: Theory of the Condensed State (Pergamon Press, London, 1980).zbMATHGoogle Scholar
  12. 12.
    H. B. Lawson, Jr. and M.-L. Michelsohn, Spin Geometry (Princeton Univ. Press, Princeton, NJ, 1989).zbMATHGoogle Scholar
  13. 13.
    N. S. Manton, “A remark on the scattering of BPS monopoles,” Phys. Lett. B 110, 54–56 (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    J. W. Morgan, The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds (Princeton Univ. Press, Princeton, NJ, 1996).zbMATHGoogle Scholar
  15. 15.
    R. V. Palvelev, “Scattering of vortices in the Abelian Higgs model,” Teor. Mat. Fiz. 156(1), 77–91 (2008) [Theor. Math. Phys. 156, 1028–1040 (2008)].MathSciNetCrossRefGoogle Scholar
  16. 16.
    R. V. Pal’velev, “Justification of the adiabatic principle in the Abelian Higgs model,” Tr. Mosk. Mat. Obshch. 72(2), 281–314 (2011) [Trans. Moscow Math. Soc. 2011, 219–244 (2011)].Google Scholar
  17. 17.
    R. V. Palvelev and A. G. Sergeev, “Justification of the adiabatic principle for hyperbolic Ginzburg-Landau equations,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 277, 199–214 (2012) [Proc. Steklov Inst. Math. 277, 191–205 (2012)].MathSciNetGoogle Scholar
  18. 18.
    D. Salamon, “Spin geometry and Seiberg-Witten invariants,” Preprint (Warwick Univ., 1996).Google Scholar
  19. 19.
    N. Seiberg and E. Witten, “Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory,” Nucl. Phys. B 426, 19–52 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    N. Seiberg and E. Witten, “Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD,” Nucl. Phys. B 431, 484–550 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    A. G. Sergeev, Vortices and Seiberg-Witten Equations (Nagoya Univ., Nagoya, 2002).Google Scholar
  22. 22.
    A. G. Sergeev and S. V. Chechin, “Scattering of slowly moving vortices in the Abelian (2 + 1)-dimensional Higgs model,” Teor. Mat. Fiz. 85(3), 397–411 (1990) [Theor. Math. Phys. 85, 1289–1299 (1990)].MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    D. M. A. Stuart, “Periodic solutions of the Abelian Higgs model and rigid rotation of vortices,” Geom. Funct. Anal. 9, 568–595 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    C. H. Taubes, “Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations,” Commun. Math. Phys. 72, 277–292 (1980).MathSciNetCrossRefGoogle Scholar
  25. 25.
    C. H. Taubes, “On the equivalence of the first and second order equations for gauge theories,” Commun. Math. Phys. 75, 207–227 (1980).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    C. H. Taubes, “The Seiberg-Witten invariants and symplectic forms,” Math. Res. Lett. 1, 809–822 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    C. H. Taubes, “The Seiberg-Witten and Gromov invariants,” Math. Res. Lett. 2, 221–238 (1995).MathSciNetCrossRefGoogle Scholar
  28. 28.
    C. H. Taubes, “SW ⇒ Gr: From the Seiberg-Witten equations to pseudo-holomorphic curves,” J. Am. Math. Soc. 9, 845–918 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    C. H. Taubes, “Counting pseudo-holomorphic submanifolds in dimension 4,” J. Diff. Geom. 44, 818–893 (1996).MathSciNetzbMATHGoogle Scholar
  30. 30.
    C. H. Taubes, “Gr ⇒ SW: From pseudo-holomorphic curves to Seiberg-Witten solutions,” J. Diff. Geom. 51, 203–334 (1999).MathSciNetzbMATHGoogle Scholar
  31. 31.
    C. H. Taubes, “Gr = SW: Counting curves and connections,” J. Diff. Geom. 52, 453–609 (1999).MathSciNetzbMATHGoogle Scholar
  32. 32.
    E. Witten, “Monopoles and four-manifolds,” Math. Res. Lett. 1, 769–796 (1994).MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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