Abstract
The fundamental group and rational cohomology of the configuration spaces of the Skyrme and Faddeev-Hopf models are computed. Physical space is taken to be a compact oriented 3-manifold, either with or without a marked point representing an end at infinity. For the Skyrme model, the codomain is any Lie group, while for the Faddeev-Hopf model it is S 2. It is determined when the topology of configuration space permits fermionic and isospinorial quantization of the solitons of the model within generalizations of the frameworks of Finkelstein-Rubinstein and Sorkin. Fermionic quantization of Skyrmions is possible only if the target group contains a symplectic or special unitary factor, while fermionic quantization of Hopfions is always possible. Geometric interpretations of the results are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aitchison, I.J.R.: Effective Lagrangians and soliton physics I: derivative expansion, and decoupling. Acta Phys. Polon. B18:3, 191–205 (1987)
Aitchison, I.J.R.: Berry phases, magnetic monopoles, and Wess-Zumino terms or how the Skyrmion got its spin. Acta Phys. Polon. B18:3, 207–235 (1987)
Adams, J.F.: Lectures on exceptional Lie groups. Chicago and London: The University of Chicago Press, 1996
Auckly, D., Kapitanski, L.: Holonomy and Skyrme's Model. Commun. Math. Phys. 240, 97–122 (2003)
Auckly, D., Kapitanski, L.: Analysis of the Faddeev Model. Commun. Math. Phys. 256, 611–620 (2005)
Auckly, D., Kapitanski, L.: The Pontrjagin-Hopf invariants for Sobolev maps. In progress
Balachandran, A., Marmo, G., Skagerstam, B., Stern, A.: Classical topology and quantum states. New Jersey: World Scientific, 1991
Balachandran, A., Marmo, G., Simoni, A., Sparno, G.: Quantum bundles and their symmetries. Int. J. Mod. Phys. A 7:8, 1641–1667 (1982)
Birrell, N., Davies, P.: Quantum fields in curved space. Cambridge: Cambridge University Press, 1982
Bopp, F., Haag, Z.: Über die möglichkeit von Spinmodellen. Zeits. fur Natur. 5a, 644 (1950)
Bröcker, Th., tom Dieck, T.: Representation of compact Lie groups. New York: Springer-Verlag, 1985
Dirac, P.: Quantized singularities in the electromagnetic field. Proc. Roy. Soc. London A133, 60–72 (1931)
Dirac, P.: The theory of magnetic poles. Phys. Rev. 74, 817–830 (1948)
Federer, H: A study of function spaces by spectral sequences. Ann. of Math. 61, 340–361 (1956)
Finkelstein, D., Rubinstein, J.: Connection between spin statisitics and kinks. J. Math. Phys. 9, 1762–1779 (1968)
Giulini, D.: On the possibility of spinorial quantization in the Skyrme model. Mod. Phys. Lett. A8, 1917–1924 (1993)
Gottlieb, D.: Lifting actions in fibrations. In: Geometric applications of homotopy theory I. Lecture Notes in Mathematics 657, Berlin-Heidelberg-New york: Springer, 1978, pp. 217–253
Halzen, F., Martin, A.: Quarks and Leptons: an introductory course in modern particle physics. New York: John Wiley and Sons, 1984
Hattori, A., Yoshida, T.: Lifting group actions in fiber bundles. Japan J. Math. 24, 13–25 (1976)
Helgason, S.: Differential geometry, Lie groups and symmetric spaces. Providence, Rhode Island: Amer. Math. Soc., 2001
Imbo, T.D., Shah Imbo, C., Sudarshan, E.C.G.: Identical particles, exotic statistics and braid groups. Phys. Lett. B 234, 103–107 (1990)
Iwasawa, K.: On some types of topological groups. Ann. of Math. 50:3, 507–558 (1949)
Kirby, R.C.: The topology of 4-manifolds. Lecture Notes in Mathematics 1374, New York: Springer-Verlag, 1985
Krusch, S.: Homotopy of rational maps and quantization of Skyrmions. Annals Phys. 304, 103–127 (2003)
Mimura, M., Toda, H.: Topology of Lie groups, I and II. Transl. Amer. Math. Soc., Providence, Rhode Island: Amer. Math. Soc., 1991
Particle data group: Review of particle physics. http://pdg.lbl.gov
Pontrjagin, L.: A classification of mappings of the 3-dimensional complex into the 2-dimensional sphere. Mat. Sbornik N.S. 9:51, 331–363 (1941)
Ramadas, T.R.: The Wess-Zumino term and fermionic solitons. Commun. Math. Phys. 93, 355–365 (1984)
Schulman, L.: A path integral for spin. Phys. Rev. 176:5, 1558–1569 (1968)
Simms, D.J., Woodhouse, N.M.J.: Geometric quantization. Lecture Notes in Physics 53, Berlin: Springer-Verlag, 1977
Skyrme, T.H.R.: A unified field theory of mesons and baryons. Nucl. Phys. 31, 555–569 (1962)
Sorkin, R.: A general relation between kink-exchange and kink-rotation. Commun. Math. Phys. 115, 421–434 (1988)
Spanier, E.H.: Algebraic topology. New York: Springer, 1966
Steenrod, N.: The topology of fiber bundles. Princeton: Princeton University Press, 1951
Witten, E.: Current algebra, baryons and quark confinement. Nucl. Phys. B233, 433–444 (1983)
Zaccaria, F., Sudarshan, E., Nilsson, J., Mukunda, N., Marmo, G., Balachandran, A.: Universal unfolding of Hamiltonian systems: from symplectic structures to fibre bundles. Phys. Rev. D27, 2327–2340 (1983)
Author information
Authors and Affiliations
Additional information
Communicated by L. Takhtajan
The first author was partially supported by NSF grant DMS-0204651
The second author was partially supported by EPSRC grant GR/R66982/01
Rights and permissions
About this article
Cite this article
Auckly, D., Speight, M. Fermionic Quantization and Configuration Spaces for the Skyrme and Faddeev-Hopf Models. Commun. Math. Phys. 263, 173–216 (2006). https://doi.org/10.1007/s00220-005-1496-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-005-1496-1