Communications in Mathematical Physics

, Volume 296, Issue 2, pp 429–446 | Cite as

Reducible Connections and Non-local Symmetries of the Self-dual Yang–Mills Equations

  • James D. E. GrantEmail author


We construct the most general reducible connection that satisfies the self-dual Yang–Mills equations on a simply-connected, open subset of flat \({\mathbb{R}^4}\). We show how all such connections lie in the orbit of the flat connection on \({\mathbb{R}^4}\) under the action of non-local symmetries of the self-dual Yang–Mills equations. Such connections fit naturally inside a larger class of solutions to the self-dual Yang–Mills equations that are analogous to harmonic maps of finite type.


Modulus Space Open Subset Loop Group Coadjoint Orbit Laurent Expansion 
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  1. 1.
    Atiyah, M.F.: Geometry on Yang–Mills Fields, Scuola Normale Superiore Pisa, Pisa, 1979Google Scholar
  2. 2.
    Atiyah M.F., Hitchin N.J., Drinfel′d V.G., Manin Y.I.: Construction of instantons. Phys. Lett. A 65, 185–187 (1978)CrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Atiyah M.F., Hitchin N.J., Singer I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. London Ser. A 362, 425–461 (1978)zbMATHCrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Bergvelt M.J., Guest M.A.: Actions of loop groups on harmonic maps. Trans. Amer. Math. Soc. 326, 861–886 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Burstall, F.E., Pedit, F.: Harmonic maps via Adler-Kostant-Symes theory. In: Harmonic Maps and Integrable Systems, Aspects Math., E23, Braunschweig: Vieweg, 1994, pp. 221–272Google Scholar
  6. 6.
    Burstall F.E., Pedit F.: Dressing orbits of harmonic maps. Duke Math. J. 80, 353–382 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Chau L.L., Ge M.L., Sinha A., Wu Y.S.: Hidden-symmetry algebra for the self-dual Yang–Mills equation. Phys. Lett. B 121, 391–396 (1983)CrossRefMathSciNetADSGoogle Scholar
  8. 8.
    Chau L.L., Ge M.L., Wu Y.S.: Kac–Moody algebra in the self-dual Yang-Mills equation. Phys. Rev. D (3) 25, 1086–1094 (1982)CrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Chau L.-L., Wu Y.S.: More about hidden-symmetry algebra for the self-dual Yang–Mills system. Phys. Rev. D (3) 26, 3581–3592 (1982)CrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Crane L.: Action of the loop group on the self-dual Yang–Mills equation. Commun. Math. Phys. 110, 391–414 (1987)zbMATHCrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Donaldson S.K.: An application of gauge theory to four-dimensional topology. J. Diff. Geom. 18, 279–315 (1983)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Freed, D.S., Uhlenbeck, K.K.: Instantons and Four-Manifolds, Vol. 1 of Mathematical Sciences Research Institute Publications, New York: Springer-Verlag, Second ed., 1991Google Scholar
  13. 13.
    Grant, J.D.E.: The ADHM construction and non-local symmetries of the self-dual Yang–Mills equations. Commun. Math. phys. doi: 10.1007/s00220-010-1024-9
  14. 14.
    Guest M.A.: Harmonic Maps, Loop Groups, and Integrable Systems, Vol. 38 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge (1997)Google Scholar
  15. 15.
    Guest M.A., Ohnita Y.: Group actions and deformations for harmonic maps. J. Math. Soc. Japan 45, 671–704 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hitchin N.J.: Linear field equations on self-dual spaces. Proc. Roy. Soc. London Ser. A 370, 173–191 (1980)zbMATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Ivanova T.A.: On infinite-dimensional algebras of symmetries of the self-dual Yang–Mills equations. J. Math. Phys. 39, 79–87 (1998)zbMATHCrossRefMathSciNetADSGoogle Scholar
  18. 18.
    Ivanova T.A.: On infinitesimal symmetries of the self-dual Yang–Mills equations. J. Nonlinear Math. Phys. 5, 396–404 (1998)zbMATHCrossRefMathSciNetADSGoogle Scholar
  19. 19.
    Mason, L.J., Woodhouse, N.M.J.: Integrability, Self-Duality, and Twistor Theory, Vol. 15 of London Mathematical Society Monographs. New Series, New York: The Clarendon Press/Oxford University Press, 1996Google Scholar
  20. 20.
    Park Q.-H.: 2D sigma model approach to 4D instantons. Int. J. Mod. Phys. A 7, 1415–1447 (1992)CrossRefADSGoogle Scholar
  21. 21.
    Plebański J.F.: Some solutions of complex Einstein equations. J. Math. Phys. 16, 2395–2402 (1975)CrossRefADSGoogle Scholar
  22. 22.
    Popov A.D.: Self-dual Yang–Mills: symmetries and moduli space. Rev. Math. Phys. 11, 1091–1149 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Pressley A., Segal G.: Loop Groups, Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, New York (1986)Google Scholar
  24. 24.
    Takasaki K.: A new approach to the self-dual Yang–Mills equations. Commun. Math. Phys. 94, 35–59 (1984)zbMATHCrossRefMathSciNetADSGoogle Scholar
  25. 25.
    Uhlenbeck K.: Harmonic maps into Lie groups: classical solutions of the chiral model. J. Diff. Geom. 30, 1–50 (1989)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Uhlenbeck K.K.: Removable singularities in Yang–Mills fields. Commun. Math. Phys. 83, 11–29 (1982)zbMATHCrossRefMathSciNetADSGoogle Scholar
  27. 27.
    Ward R.S.: On self-dual gauge fields. Phys. Lett. A 61, 81–82 (1977)CrossRefMathSciNetADSGoogle Scholar

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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität WienWienAustria

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