Communications in Mathematical Physics

, Volume 256, Issue 3, pp 611–620 | Cite as

Analysis of S2-Valued Maps and Faddeev’s Model

  • Dave Auckly
  • Lev Kapitanski


In this paper we consider a generalization of the Faddeev model for the maps from a closed three-manifold into the two-sphere. We give a novel representation of smooth S2-valued maps based on flat connections. This representation allows us to obtain an analytic description of the homotopy classes of S2-valued maps that generalizes to Sobolev maps. It also leads to a new proof of an old theorem of Pontrjagin. For the generalized Faddeev model, we prove the existence of minimizers in every homotopy class.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Analytic Description 
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  1. 1.
    Auckly, D., Kapitanski, L.: Holonomy and Skyrme’s model. Commun. Math. Phys. 240, 97–122 (2003)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Auckly, D., Speight, M.: Fermionic quantization and configuration spaces for the Skyrme and Faddeev-Hopf models.
  3. 3.
    Bott, R., Tu, L.W.: Differential forms in algebraic topology. New York-Berlin: Springer-Verlag, 1982Google Scholar
  4. 4.
    Bredon, G.E.: Introduction to compact transformation groups. New York-London: Academic Press, 1972Google Scholar
  5. 5.
    Faddeev, L.D.: Quantization of solitons. Preprint IAS print-75-QS70 (1975)Google Scholar
  6. 6.
    Faddeev, L.D.: Knotted solitons and their physical applications. Phil. Trans. R. Soc. Lond. A 359, 1399–1403 (2001)Google Scholar
  7. 7.
    Faddeev, L.D., Niemi, A.J.: Stable knot-like structures in classical field theory. Nature 387, 58–61 (1997)CrossRefGoogle Scholar
  8. 8.
    Hopf, H.: Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche. Math. Annalen 104, 637–665 (1931)CrossRefGoogle Scholar
  9. 9.
    Kapitanski, L.: On Skyrme’s model. In: Nonlinear Problems in Mathematical Physics and Related Topics II: In Honor of Professor O. A. Ladyzhenskaya, Birman et al., (eds.), Dordrecht: Kluwer, 2002, pp. 229–242Google Scholar
  10. 10.
    Kobayashi, S., Nomizu, K.: Foundations of differential geometry, Vols. I, II. New York: John Wiley & Sons, Inc., 1996Google Scholar
  11. 11.
    Munkres, J.R.: Elementary differential topology. Revised edition. Annals of Mathematics Studies, No. 54, Princeton, N.J.: Princeton University Press, 1966Google Scholar
  12. 12.
    Pontrjagin, L.: A classification of mappings of the three-dimensional complex into the two- dimensional sphere Rec. Math. [Mat. Sbornik] N. S. 9(51), 331–363 (1941)Google Scholar
  13. 13.
    Robbin, J. W., Rogers, R. C., Temple, B.: On weak continuity and the Hodge decomposition. Trans. AMS 303(2), 609–618 (1987)Google Scholar
  14. 14.
    Skyrme, T.H.R.: A non-linear field theory. Proc. R. Soc. London A 260(1300), 127–138 (1961)Google Scholar
  15. 15.
    Skyrme, T.H.R.: A unified theory of mesons and baryons. Nucl. Phys. 31, 556–569 (1962)CrossRefGoogle Scholar
  16. 16.
    Skyrme, T.H.R.: The origins of Skyrmions. Int. J. Mod. Phys. A3, 2745–2751 (1988)Google Scholar
  17. 17.
    Spanier, E.H.: Algebraic topology. New York: Springer, 1966Google Scholar
  18. 18.
    Vakulenko, A.F., Kapitanski, L.: On S2-nonlinear σ-model. Dokl. Acad. Nauk SSSR 248, 810–814 (1979)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Dave Auckly
    • 1
  • Lev Kapitanski
    • 2
  1. 1.Department of MathematicsKansas State UniversityManhattanUSA
  2. 2.Department of MathematicsUniversity of MiamiCoral GablesUSA

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