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

SIGMA 15 (2019), 076, 16 pages      arXiv:1905.02434      https://doi.org/10.3842/SIGMA.2019.076

Momentum Sections in Hamiltonian Mechanics and Sigma Models

Noriaki Ikeda
Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan

Received May 24, 2019, in final form September 29, 2019; Published online October 03, 2019

Abstract
We show a constrained Hamiltonian system and a gauged sigma model have a structure of a momentum section and a Hamiltonian Lie algebroid theory recently introduced by Blohmann and Weinstein. We propose a generalization of a momentum section on a pre-multisymplectic manifold by considering gauged sigma models on higher-dimensional manifolds.

Key words: symplectic geometry; Lie algebroid; Hamiltonian mechanics; nonlinear sigma model.

pdf (405 kb)   tex (23 kb)  

References

  1. Blohmann C., Fernandes M.C.B., Weinstein A., Groupoid symmetry and constraints in general relativity, Commun. Contemp. Math. 15 (2013), 1250061, 25 pages, arXiv:1003.2857.
  2. Blohmann C., Weinstein A., Hamiltonian Lie algebroids, arXiv:1811.11109.
  3. Bouwknegt P., Bugden M., Klimčík C., Wright K., Hidden isometry of ''T-duality without isometry'', J. High Energy Phys. 2017 (2017), no. 8, 116, 19 pages, arXiv:1705.09254.
  4. Bursztyn H., Cavalcanti G.R., Gualtieri M., Reduction of Courant algebroids and generalized complex structures, Adv. Math. 211 (2007), 726-765, arXiv:math.DG/0509640.
  5. Buscher T.H., A symmetry of the string background field equations, Phys. Lett. B 194 (1987), 59-62.
  6. Buscher T.H., Path-integral derivation of quantum duality in nonlinear sigma-models, Phys. Lett. B 201 (1988), 466-472.
  7. Callies M., Frégier Y., Rogers C.L., Zambon M., Homotopy moment maps, Adv. Math. 303 (2016), 954-1043, arXiv:1304.2051.
  8. Cariñena J.F., Crampin M., Ibort L.A., On the multisymplectic formalism for first order field theories, Differential Geom. Appl. 1 (1991), 345-374.
  9. Cavalcanti G.R., Gualtieri M., Generalized complex geometry and T-duality, in A Celebration of the Mathematical Legacy of Raoul Bott, CRM Proc. Lecture Notes, Vol. 50, Amer. Math. Soc., Providence, RI, 2010, 341-365, arXiv:1106.1747.
  10. Chatzistavrakidis A., Deser A., Jonke L., T-duality without isometry via extended gauge symmetries of 2D sigma models, J. High Energy Phys. 2016 (2016), no. 1, 154, 20 pages, arXiv:1509.01829.
  11. Chatzistavrakidis A., Deser A., Jonke L., Strobl T., Beyond the standard gauging: gauge symmetries of Dirac sigma models, J. High Energy Phys. 2016 (2016), no. 8, 172, 28 pages, arXiv:1607.00342.
  12. Chatzistavrakidis A., Deser A., Jonke L., Strobl T., Gauging as constraining: the universal generalised geometry action in two dimensions, PoS Proc. Sci. (2017), PoS(CORFU2016), 087, 20 pages, arXiv:1705.05007.
  13. Chatzistavrakidis A., Deser A., Jonke L., Strobl T., Strings in singular space-times and their universal gauge theory, Ann. Henri Poincaré 18 (2017), 2641-2692, arXiv:1608.03250.
  14. Gotay M.J., Isenberg J., Marsden J.E., Montgomery R., Momentum maps and classical relativistic fields. Part I: Covariant field theory, arXiv:physics/9801019.
  15. Herman J., Weak moment maps in multisymplectic geometry, arXiv:1807.01641.
  16. Higgins P.J., Mackenzie K., Algebraic constructions in the category of Lie algebroids, J. Algebra 129 (1990), 194-230.
  17. Hull C.M., Spence B., The geometry of the gauged sigma-model with Wess-Zumino term, Nuclear Phys. B 353 (1991), 379-426.
  18. Ikeda N., Lectures on AKSZ sigma models for physicists, in Noncommutative Geometry and Physics 4, World Sci. Publ., Hackensack, NJ, 2017, 79-169, arXiv:1204.3714.
  19. Ikeda N., Strobl T., On the relation of Lie algebroids to constrained systems and their BV/BFV formulation, Ann. Henri Poincar'e 20 (2019), 527-541, arXiv:1803.00080.
  20. Ikeda N., Uchino K., QP-structures of degree 3 and 4D topological field theory, Comm. Math. Phys. 303 (2011), 317-330, arXiv:1004.0601.
  21. Kotov A., Strobl T., Lie algebroids, gauge theories, and compatible geometrical structures, Rev. Math. Phys. 31 (2019), 1950015, 31 pages, arXiv:1603.04490.
  22. Liu Z.-J., Weinstein A., Xu P., Manin triples for Lie bialgebroids, J. Differential Geom. 45 (1997), 547-574, arXiv:dg-ga/9508013.
  23. Mackenzie K., Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, Vol. 124, Cambridge University Press, Cambridge, 1987.
  24. Madsen T.B., Swann A., Multi-moment maps, Adv. Math. 229 (2012), 2287-2309, arXiv:1012.2048.
  25. Madsen T.B., Swann A., Closed forms and multi-moment maps, Geom. Dedicata 165 (2013), 25-52, arXiv:1110.6541.
  26. Vaintrob A.Yu., Lie algebroids and homological vector fields, Russian Math. Surveys 52 (1997), 428-429.
  27. Wright K., Lie algebroid gauging of non-linear sigma models, J. Geom. Phys. 146 (2019), 103490, 23 pages, arXiv:1905.00659.

Previous article  Next article  Contents of Volume 15 (2019)