Advertisement

Science in China Series A: Mathematics

, Volume 51, Issue 4, pp 695–706 | Cite as

Harmonic maps into loop spaces of compact Lie groups

  • Armen Glebovich 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. We conjecture that harmonic maps of the Riemann sphere ℂℙ1 into ΩG are related to Yang-Mills G-fields on ℝ4.

Keywords

harmonic maps Yang-Mills fields twistor bundles 

MSC(2000)

58E20 53C28 32L25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Atiayh M F. Instantons in two and four dimensions. Comm Math Phys, 93: 437–451 (1984)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Davidov J, Sergeev A G. Twistor spaces and harmonic maps. Russian Math Surveys, 48(3): 1–91 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Sergeev A G. Harmonic maps into homogeneous Riemannian manifolds: twistor approach. Russian Math Surveys, 386(6): 1181–1203 (1988)MathSciNetGoogle Scholar
  4. 4.
    Atiayh M F, Hitchin N J, Singer I M. Self-duality in four-dimensional Riemannian geometry. Proc Roy Soc London, 362: 425–461 (1978)CrossRefGoogle Scholar
  5. 5.
    Eells J, Salamon S. Twistorial constructions of harmonic maps of surfaces into four-manifolds. Ann Scuola Norm Super Pisa, 12: 589–640 (1985)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Burstall F E, Salamon S. Tournaments, flags and harmonic maps. Math Ann, 277: 249–265 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Burstall F E. A twistor description of harmonic maps of a 2-sphere into a Grassmanian. Math Ann, 274:61–74 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Wood J C. 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
  9. 9.
    Sergeev A G. Kähler geometry of loop spaces. In: Proceedings of RCME Moscow Centre for Continuous Math Education, Moscow, 2001, 127Google Scholar
  10. 10.
    Pressley A, Segal G. Loop Groups. Oxford: Clarendon Press, 1986zbMATHGoogle Scholar
  11. 11.
    Donaldson S K. Instantons and geometric invariant theory. Comm Math Phys, 93: 453–460 (1984)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Science Press 2008

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

Personalised recommendations