Geometry and analysis in nonlinear sigma models

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

doi:10.1090/S1061-0022-06-00940-X

Geometry and analysis in nonlinear sigma models
HTML articles powered by AMS MathViewer

by D. Auckly, L. Kapitanski and J. M. Speight
Translated by: the authors

St. Petersburg Math. J. 18 (2007), 1-19

DOI: https://doi.org/10.1090/S1061-0022-06-00940-X

Published electronically: November 27, 2006

PDF | Request permission

Abstract:

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.

References

Dave Auckly and Lev Kapitanski, Holonomy and Skyrme’s model, Comm. Math. Phys. 240 (2003), no. 1-2, 97–122. MR 2004981, DOI 10.1007/s00220-003-0901-x
Dave Auckly and Lev Kapitanski, Analysis of $S^2$-valued maps and Faddeev’s model, Comm. Math. Phys. 256 (2005), no. 3, 611–620. MR 2161273, DOI 10.1007/s00220-005-1289-6
—, The Pontryagin–Hopf invariants for Sobolev maps (submitted).
—, Integrality of homotopy invariants in the Skyrme model (in preparation).
Dave Auckly and Martin Speight, Fermionic quantization and configuration spaces for the Skyrme and Faddeev-Hopf models, Comm. Math. Phys. 263 (2006), no. 1, 173–216. MR 2207327, DOI 10.1007/s00220-005-1496-1
A. P. Balachandran, G. Marmo, B.-S. Skagerstam, and A. Stern, Classical topology and quantum states, World Scientific Publishing Co., Inc., River Edge, NJ, 1991. MR 1254876, DOI 10.1142/9789814271912
Richard A. Battye and Paul M. Sutcliffe, Skyrmions, fullerenes and rational maps, Rev. Math. Phys. 14 (2002), no. 1, 29–85. MR 1877915, DOI 10.1142/S0129055X02001065
Fritz Bopp and Rudolf Haag, Über die Möglichkeit von Spinmodellen, Z. Naturforschung 5a (1950), 644–653 (German). MR 0043309
Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. MR 658304
Haim Brezis, The interplay between analysis and topology in some nonlinear PDE problems, Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 2, 179–201. MR 1962295, DOI 10.1090/S0273-0979-03-00976-5
P. A. M. Dirac, The theory of magnetic poles, Phys. Rev. (2) 74 (1948), 817–830. MR 27713
L. D. Faddeev, Quantization of solitons, Preprint IAS print-75-QS70 (1975).
Ludwig Faddeev, Knotted solitons and their physical applications, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 359 (2001), no. 1784, 1399–1403. Topological methods in the physical sciences (London, 2000). MR 1853628, DOI 10.1098/rsta.2001.0842
L. D. Faddeev and A. J. Niemi, Stable knot-like structures in classical field theory, Nature 387 (1997), 58–61.
Herbert Federer, A study of function spaces by spectral sequences, Trans. Amer. Math. Soc. 82 (1956), 340–361. MR 79265, DOI 10.1090/S0002-9947-1956-0079265-2
David Finkelstein and Julio Rubinstein, Connection between spin, statistics, and kinks, J. Mathematical Phys. 9 (1968), 1762–1779. MR 234695, DOI 10.1063/1.1664510
T. Gisiger and M. B. Paranjape, Recent mathematical developments in the Skyrme model, Phys. Rep. 306 (1998), no. 3, 109–211. MR 1659576, DOI 10.1016/S0370-1573(98)00037-4
Domenico Giulini, On the possibility of spinorial quantization in the Skyrme model, Modern Phys. Lett. A 8 (1993), no. 20, 1917–1924. MR 1228374, DOI 10.1142/S0217732393001641
V. L. Hansen, http://www.mat.dtu.dk/people/V.L.Hansen/string.html
J. Hietarinta and P. Salo, Ground state in the Faddeev-Skyrme model, Phys. Rev. D 62 (2000), 081701(R).
Lev Kapitanski, On Skyrme’s model, Nonlinear problems in mathematical physics and related topics, II, Int. Math. Ser. (N. Y.), vol. 2, Kluwer/Plenum, New York, 2002, pp. 229–241. MR 1971999, DOI 10.1007/978-1-4615-0701-7_{1}3
Robion C. Kirby, The topology of $4$-manifolds, Lecture Notes in Mathematics, vol. 1374, Springer-Verlag, Berlin, 1989. MR 1001966, DOI 10.1007/BFb0089031
S. Koshkin, Homogeneous spaces and Faddeev-Skyrme models, Dissertation, Kansas State Univ., 2006.
F. Mandl and G. Shaw, Quantum field theory, Wiley, 1991.
L. Pontrjagin, A classification of mappings of the three-dimensional complex into the two-dimensional sphere, Rec. Math. [Mat. Sbornik] N.S. 9 (51) (1941), 331–363 (English, with Russian summary). MR 0004780
R. Rajaraman, Solitons and instantons, North-Holland Publishing Co., Amsterdam, 1982. An introduction to solitons and instantons in quantum field theory. MR 719693
Lawrence Schulman, A path integral for spin, Phys. Rev. (2) 176 (1968), 1558–1569. MR 237149
David J. Simms and Nicholas M. J. Woodhouse, Lectures in geometric quantization, Lecture Notes in Physics, vol. 53, Springer-Verlag, Berlin-New York, 1976. MR 0672639
T. H. R. Skyrme, A unified field theory of mesons and baryons, Nuclear Phys. 31 (1962), 556–569. MR 0138394
Rafael D. Sorkin, A general relation between kink-exchange and kink-rotation, Comm. Math. Phys. 115 (1988), no. 3, 421–434. MR 931669
A. F. Vakulenko and L. V. Kapitanskiĭ, Stability of solitons in $S^{2}$ of a nonlinear $\sigma$-model, Dokl. Akad. Nauk SSSR 246 (1979), no. 4, 840–842 (Russian). MR 543541
Steven Weinberg, The quantum theory of fields. Vol. I, Cambridge University Press, Cambridge, 1996. Foundations; Corrected reprint of the 1995 original. MR 1410064, DOI 10.1017/CBO9781139644174
Brian White, Homotopy classes in Sobolev spaces and the existence of energy minimizing maps, Acta Math. 160 (1988), no. 1-2, 1–17. MR 926523, DOI 10.1007/BF02392271
Edward Witten, Current algebra, baryons, and quark confinement, Nuclear Phys. B 223 (1983), no. 2, 433–444. MR 717916, DOI 10.1016/0550-3213(83)90064-0
F. Zaccaria, E. C. G. Sudarshan, J. S. Nilsson, N. Mukunda, G. Marmo, and A. P. Balachandran, Universal unfolding of Hamiltonian systems: from symplectic structure to fiber bundles, Phys. Rev. D (3) 27 (1983), no. 10, 2327–2340. MR 704119, DOI 10.1103/PhysRevD.27.2327