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Harmonic maps into loop spaces of compact Lie groups

  • A. G. Sergeev
Article

Abstract

We study harmonic maps from Riemann surfaces M to the loop spaces ΩG of compact Lie groups G, using the twistor approach. Harmonic maps into loop spaces are of special interest because of their relation to the Yang-Mills equations on ℝ4.

Keywords

Manifold Modulus Space Riemannian Manifold Riemann Surface Twistor Space 
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References

  1. 1.
    M. F. Atiayh, “Instantons in two and four dimensions,” Commun. Math. Phys., 93, 437–451 (1984).CrossRefGoogle Scholar
  2. 2.
    M. F. Atiayh, N. J. Hitchin, and I. M. Singer, “Self-duality in four-dimensional Riemannian geometry,” R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 362, 425–461 (1978).CrossRefGoogle Scholar
  3. 3.
    F. E. Burstall, “A twistor description of harmonic maps of a 2-sphere into a Grassmanian,” Math. Ann., 274, 61–74 (1986).zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    F. E. Burstall and S. Salamon, “Tournaments, flags and harmonic maps,” Math. Ann., 277, 249–265 (1987).zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    J. Davidov and A. G. Sergeev, “Twistor spaces and harmonic maps,” Russian Math. Surveys, 48, No. 3, 1–91 (1993).zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    S. K. Donaldson, “Instantons and geometric invariant theory,” Commun. Math. Phys., 93, 453–460 (1984).zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    J. Eells and S. Salamon, “Twistorial constructions of harmonic maps of surfaces into four-manifolds, ” Ann. Scuola Norm. Super. Pisa, 12, 589–640 (1985).zbMATHMathSciNetGoogle Scholar
  8. 8.
    A. Pressley and G. Segal, Loop Groups, Clarendon Press, Oxford (1986).zbMATHGoogle Scholar
  9. 9.
    A. G. Sergeev, “Harmonic maps into homogeneous Riemannian manifolds: Twistor approach,” Russian Math. Surveys, 386, No. 6, 1181–1203 (1988).MathSciNetGoogle Scholar
  10. 10.
    A. G. Sergeev, Kähler Geometry of Loop Spaces, Moscow Center for Continuous Math. Education, Moscow (2001).Google Scholar
  11. 11.
    J. C. Wood, “The explicit construction and parametrization of all harmonic maps from the two-sphere to a complex Grassmanian,” J. Reine Angew. Math., 386, 1–31 (1988).zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia

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