Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 123, 11 pages      arXiv:0712.3385      https://doi.org/10.3842/SIGMA.2007.123
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Integrability and Diffeomorphisms on Target Space

Christoph Adam a, Joaquin Sanchez-Guillen a and Andrzej Wereszczynski b
a) Department of Particle Physics, University of Santiago de Compostela, Spain
b) Institute of Physics, Jagellonian University, Reymonta 4, 30-059 Krakow, Poland

Received October 18, 2007, in final form December 10, 2007; Published online December 20, 2007

Abstract
We briefly review the concepts of generalized zero curvature conditions and integrability in higher dimensions, where integrability in this context is related to the existence of infinitely many conservation laws. Under certain assumptions, it turns out that these conservation laws are, in fact, generated by a class of geometric target space transformations, namely the volume-preserving diffeomorphisms. We classify the possible conservation laws of field theories for the case of a three-dimensional target space. Further, we discuss some explicit examples.

Key words: integrability; zero curvature; conservation laws; nonlinear field theories.

pdf (236 kb)   ps (167 kb)   tex (15 kb)

References

  1. Alvarez O., Ferreira L.A., Sanchez-Guillen J., A new approach to integrable theories in any dimension, Nuclear Phys. B 529 (1998), 689-736.
  2. Aratyn H., Ferreira L.A., Zimerman A., Toroidal solitons in (3+1)-dimensional integrable theories, Phys. Lett. B 456 (1999), 162-170.
  3. Aratyn H., Ferreira L.A., Zimerman A., Exact static soliton solutions of (3+1)-dimensional integrable theory with nonzero Hopf numbers, Phys. Rev. Lett. 83 (1999), 1723-1726.
  4. De Carli E., Ferreira L.A., A model for Hopfions on the space-time S3 × R, J. Math. Phys. 46 (2005), 012703, 10 pages, hep-th/0406244.
  5. Wereszczynski A., Integrability and Hopf solitons in models with explicitly broken O(3) symmetry, Eur. Phys. J. C Part. Fields 38 (2004), 261-265, hep-th/0405155.
  6. Ferreira L.A., Exact time dependent Hopf solitons in 3+1 dimensions, J. High Energy Phys. 2006 (2006), no. 3, 075, 9 pages, hep-th/0601235.
  7. Riserio do Bonfim A.C., Ferreira L.A., Spinning Hopf solitons on S3 × R, J. High Energy Phys. 2006 (2006), no. 3, 097, 12 pages, hep-th/0602234.
  8. Nicole D.A., Solitons with nonvanishing Hopf index, J. Phys. G 4 (1978), 1363-1369.
  9. Wereszczynski A., Toroidal solitons in Nicole-type models, Eur. Phys. J. C Part. Fields 41 (2005), 265-268, math-ph/0504008.
  10. Wereszczynski A., Generalized eikonal knots and new integrable dynamical systems, Phys. Lett. B 621 (2005), 201-207, hep-th/0508121.
  11. Adam C., Sanchez-Guillen J., Vazquez R.A., Wereszczynski A., Investigation of the Nicole model, J. Math. Phys. 47 (2006), 052302, 22 pages, hep-th/0602152.
  12. Ward R.S., Hopf solitons on S3 and R3, Nonlinearity 12 (1999), 241-246.
  13. Adam C., Sanchez-Guillen J., Wereszczynski A., Hopf solitons and Hopf Q-balls on S3, Eur. Phys. J. C Part. Fields 47 (2006), 513-524, hep-th/0602008.
  14. Skyrme T.H.R., A nonlinear field theory, Proc. R. Soc. Lond. A 260 (1961), 127-138.
  15. Skyrme T.H.R., A unified field theory of mesons and baryons, Nuclear Phys. 31 (1962), 556-569.
  16. Ferreira L.A., Sanchez-Guillen J., Infinite symmetries in the Skyrme model, Phys. Lett. B 504 (2001), 195-200, hep-th/0010168.
  17. Babelon O., Ferreira L.A., Integrability and conformal symmetry in higher dimensions: a model with exact Hopfion solutions, J. High Energy Phys. 2002 (2002), no. 11, 020, 26 pages.
  18. Adam C., Sanchez-Guillen J., Generalized integrability conditions and target space geometry, Phys. Lett. B 626 (2005), 235-242, hep-th/0508011.
  19. Adam C., Sanchez-Guillen J., Wereszczynski A., Integrability from an Abelian subgroup of the diffeomorphism group, J. Math. Phys. 47 (2006), 022303, 8 pages, hep-th/0511277.
  20. Adam C., Sanchez-Guillen J., Wereszczynski A., Conservation laws in Skyrme-type models, J. Math. Phys. 48 (2007), 032302, 16 pages, hep-th/0610227.
  21. Adam C., Sanchez-Guillen J., Wereszczynski A., Integrable subsystem of Yang-Mills dilaton theory, hep-th/0703224.
  22. Schlesinger L., Parallelverschiebung und Krümmungstensor, Math. Ann. 99 (1927), 413-434.
  23. Adam C., Sanchez-Guillen J., Symmetries of generalized soliton models and submodels on target space S2, J. High Energy Phys. 2005 (2005), no. 1, 004, 15 pages, hep-th/0412028.

Previous article   Next article   Contents of Volume 3 (2007)