Geometry and analysis in nonlinear sigma models

D. Auckly ; L. Kapitanski ; J. M. Speight

Algebra i analiz, Tome 18 (2006) no. 1, pp. 3-33.

Résumé

The configuration space of a nonlinear sigma model is the space of maps from one manifold to another. This paper reviews the authors' work on nonlinear sigma models with target a homogeneous space. It begins with a description of the components, fundamental group, and cohomology of such configuration spaces, together with the physical interpretations of these results. The topological arguments given generalize to Sobolev maps. The advantages of representing homogeneous-space-valued maps by flat connections are described, with applications to the homotopy theory of Sobolev maps, and minimization problems for the Skyrme and Faddeev functionals. The paper concludes with some speculation about the possibility of using these techniques to define new invariants of manifolds.

Bibliographie

[1] Auckly D., Kapitanski L., “Holonomy and Skyrme's model”, Comm. Math. Phys., 240 (2003), 97–122 | DOI | MR | Zbl

[2] Auckly D., Kapitanski L., “Analysis of $S^2$-valued maps and Faddeev's model”, Comm. Math. Phys., 256 (2005), 611–620 | DOI | MR | Zbl

[3] Auckly D., Kapitanski L., The Pontrjagin–Hopf invariants for Sobolev maps, submitted

[4] Auckly D., Kapitanski L., Integrality of homotopy invariants in the Skyrme model, in preparation

[5] Auckly D., Speight J. M., Fermionic quantization and configuration spaces for the Skyrme and Faddeev–Hopf models, preprint, http://arxiv.org/abs/hep-th/0411010 | MR

[6] Balachandran A., Marmo G., Skagerstam B., Stern A., Classical topology and quantum states, World Sci. Publ. Co., Inc., River Edge, NJ, 1991 | MR

[7] Battye R. A., Sutcliffe P. M., “Skyrmions, fullerenes and rational maps”, Rev. Math. Phys., 14 (2002), 29–85 | DOI | MR | Zbl

[8] Bopp F., Haag Z., “Über die Möglichkeit von Spinmodellen”, Z. Naturforschung, 5a (1950), 644–653 | MR

[9] Bott R., Tu L. W., Differential forms in algebraic topology, Grad. Texts is Math., 82, Springer-Verlag, New York–Berlin, 1982 | MR | Zbl

[10] Brezis H., “The interplay between analysis and topology in some nonlinear PDE problems”, Bull. Amer. Math. Soc. (N.S.), 40 (2003), 179–201 | DOI | MR | Zbl

[11] Dirac P., “The theory of magnetic poles”, Phys. Rev. (2), 74 (1948), 817–830 | DOI | MR | Zbl

[12] Faddeev L. D., Quantization of solitons, Preprint IAS print-75-QS70, 1975 | MR

[13] Faddeev L. D., “Knotted solitons and their physical applications”, Topological Methods in the Physical Sciences (London, 2000), R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 359, 2001, 1399–1403 | DOI | MR | Zbl

[14] Faddeev L. D., Niemi A. J., “Stable knot-like structures in classical field theory”, Nature, 387 (1997), 58–61 | DOI

[15] Federer H., “A study of function spaces by spectral sequences”, Trans. Amer. Math. Soc., 82 (1956), 340–361 | DOI | MR | Zbl

[16] Finkelstein D., Rubinstein J., “Connection between spin, statisitics, and kinks”, J. Math. Phys., 9 (1968), 1762–1779 | DOI | MR | Zbl

[17] Gisiger T., Paranjape M. B., “Recent mathematical developments in the Skyrme model”, Phys. Rep., 306:3 (1998), 109–211 | DOI | MR

[18] Giulini D., “On the possibility of spinorial quantization in the Skyrme model”, Modern Phys. Lett. A, 8 (1993), 1917–1924 | DOI | MR | Zbl

[19] Hansen V. L., http://www.mat.dtu.dk/people/V.L.Hansen/string.html

[20] Hietarinta J., Salo P., “Ground state in the Faddeev–Skyrme model”, Phys. Rev. D, 62 (2000), 081701(R) | DOI

[21] Kapitanski L., “On Skyrme's model”, Nonlinear Problems in Mathematical Physics and Related Topics, II, Int. Math. Ser. (N.Y.), 2, Kluwer/Plenum, New York, 2002, 229–241 | MR | Zbl

[22] Kirby R. C., The topology of 4-manifolds, Lecture Notes in Math., 1374, Springer-Verlag, Berlin, 1989 | MR | Zbl

[23] Koshkin S., Homogeneous spaces and Faddeev–Skyrme models, Dissertation, Kansas State Univ., 2006

[24] Mandl F., Shaw G., Quantum field theory, Wiley, 1991

[25] Pontrjagin L., “A classification of mappings of the three-dimensional complex into the two-dimensional sphere”, Mat. sb., 9(51):2 (1941), 331–363 | MR | Zbl

[26] Rajaraman R., Solitons and instantons, North-Holland, Amsterdam, 1982 | MR | Zbl

[27] Schulman L., “A path integral for spin”, Phys. Rev. (2), 176 (1968), 1558–1569 | DOI | MR

[28] Simms D. J., Woodhouse N. M. J., Lectures in geometric quantization, Lecture Notes in Phys., 53, Springer-Verlag, Berlin–New York, 1976 | MR | Zbl

[29] Skyrme T. H. R., “A unified field theory of mesons and baryons”, Nuclear Phys., 31 (1962), 556–569 | DOI | MR

[30] Sorkin R., “A general relation between kink-exchange and kink-rotation”, Comm. Math. Phys., 115 (1988), 421–434 | DOI | MR | Zbl

[31] Vakulenko A. F., Kapitanskii L. V., “Ustoichivost solitonov v $S^2$-nelineinoi $\sigma$-modeli”, Dokl. AN SSSR, 246:4 (1979), 840–842 | MR | Zbl

[32] Weinberg S., The quantum theory of fields, vol. 1, Cambridge Univ. Press, Cambridge, 1996

[33] White B., “Homotopy classes in Sobolev spaces and the existence of energy minimizing maps”, Acta Math., 160 (1988), 1–17 | DOI | MR | Zbl

[34] Witten E., “Current algebra, baryons, and quark confinement”, Nuclear Phys. B, 233 (1983), 433–444 | DOI | MR

[35] Zaccaria F., Sudarshan E., Nilsson J., Mukunda N., Marmo G., Balachandran A., “Universal unfolding of Hamiltonian systems: from symplectic structure to fiber bundles”, Phys. Rev. D (3), 27 (1983), 2327–2340 | DOI | MR