Abstract
In this paper we consider two generalizations of the Skyrme model. One is a variational problem for maps from a compact 3-manifold to a compact Lie group. The other is a variational problem for flat connections. We describe the path components of the configuration spaces of smooth fields for each of the variational problems. We prove that the invariants separating the path components are well-defined for (not necessarily smooth) fields with finite Skyrme energy. We prove that for every possible value of these invariants there exists a minimizer of the Skyrme functional. Throughout the paper we emphasize the importance of holonomy in the Skyrme model. Some of the results may be useful in other contexts. In particular, we define the holonomy of a distributionally flat L 2 loc connection; the local developing maps for such connections need not be continuous.
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References
Adams, J.F.: Lectures on exceptional Lie groups. Chicago and London: The University of Chicago Press, 1996
Bott, R.: On torsion in Lie groups. Proc. Nat. Acad. Sci. U.S.A. 40, 586–588 (1954)
Bröcker, Th., tom Dieck, T.: Representation of compact Lie groups. New York: Springer-Verlag, 1985
Brown, K.S.: Cohomology of groups. New York: Springer-Verlag, 1982
Burstall, F.E.: Harmonic maps of finite energy for non-compact manifolds. J. Lond. Math. Soc. 30, 361–370 (1984)
Donaldson, S.K., Kronheimer, P.B.: The geometry of four-manifolds. New York: Oxford University Press, 1990
Esteban, M.J., Müller, S.: Sobolev maps with integer degree and applications to Skyrme's problem. Proc. R. Soc. Lond. A 436, 197–201 (1992)
Gisiger, T., Paranjape, M.B.: Recent mathematical developments in the Skyrme model. Phys. Rep. 306, 109–211 (1998)
Gross, K.I.: The Plancherel transform on the nilpotent part of G 2 and some applications to the representation theory of G 2. Trans. Am. Math. Soc. 132, 411–446 (1968)
Zahed, I., Brown, G.E.: The Skyrme model. Phys. Rep. 142, 1–102 (1982)
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. 229–242
Kobayashi, S., Nomizu, K.: Foundations of differential geometry, Vols. I, II. New York: John Wiley & Sons, Inc., 1996
Loss, M.: The Skyrme model on Riemannian manifolds. Lett. Math. Phys. 14, 149–156 (1987)
Manton, N.S., Ruback, P.J.: Skyrmions in flat space and curved space. Phys. Lett. B 181, 137–140 (1986)
Moise, E.: Affine structures in 3-manifolds, V: The triangulation theorem and Hauptvermutung. Ann. Math. 56 (2), 96–114 (1952)
Müller, S.: Higher integrability of determinants and weak convergence in L 1. J. reine angew. Math. 412, 20–34 (1990)
Munkres, J.R.: Elementary differential topology. Revised edition. Annals of Mathematics Studies, No. 54 Princeton, NJ: Princeton University Press, 1966
Mimura, M., Toda, H.: Topology of Lie groups, I and II. Transl. Am. Math. Soc., Providence, Rhode Island: Am. Math. Soc., 1991
Robbin, J.W., Rogers, R.C., Temple, B.: On weak continuity and the Hodge decomposition. Trans. AMS 303 (2), 609–618 (1987)
Rolfsen, D.: Knots and links. Mathematics Lecture Series, Vol. 7, Houston, TX: Publish or Perish, 1990
Simon, B.: Representations of finite and compact groups. Graduate Studies in Math., Vol 10, Providence, RI: AMS, 1996
Skyrme, T.H.R.: A non-linear field theory. Proc. R. Soc. Lond. A 260 (1300), 127–138 (1961)
Skyrme, T.H.R.: A unified theory of mesons and baryons. Nucl. Phy. 31, 556–569 (1962)
Skyrme, T.H.R.: The origins of Skyrmions. Int. J. Mod. Phys. A3, 2745–2751 (1988)
Spanier, E.H.: Algebraic topology. New York: Springer, 1966
Šverák, V.: Regularity properties of deformations with finite energy. Arch. Rat. Mech. Anal. 100, 105–127 (1988)
Uhlenbeck, K.K.: The Chern classes of Sobolev connections. Commun. Math. Phys. 101, 449–457 (1985)
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Communicated by L. Takhtajan
The first author was partially supported by NSF grant DMS-0204651.
The second author was partially supported by NSF grants DMS-9970638, and DMS-0200670
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Auckly, D., Kapitanski, L. Holonomy and Skyrme's Model. Commun. Math. Phys. 240, 97–122 (2003). https://doi.org/10.1007/s00220-003-0901-x
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DOI: https://doi.org/10.1007/s00220-003-0901-x