A promenade through singular symplectic geometry

• Autores: Pablo Nicolás Martínez
• Localización: Reports@SCM: an electronic journal of the Societat Catalana de Matemàtiques, ISSN-e 2385-4227, Vol. 9, Nº. 1, 2024, págs. 65-76
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen
  • català

    En aquest article, presentem la geometria simplèctica i de Poisson des de la mecànica hamiltoniana. Després introduïm els algebroides de Lie simplèctics, objectes al mig de la geometria simplèctica i de Poisson. Posteriorment, recordem la noció de reducció simplèctica en presència d’una aplicació moment. Com a aplicació d’aquesta construcció, descrivim els espais de fase de partícules carregades sota la presència de camps de Yang–Mills. Finalment, introduïm un anàleg singular d’aquesta construcció i donem exemples físics.

  • English

    In this article, we present symplectic and Poisson geometry from the perspective of Hamiltonian mechanics. We then introduce symplectic Lie algebroids, objects which lie between symplectic and Poisson manifolds. Afterwards, we recall the notion of symplectic reduction under the existence of a moment map. As an application of this construction, we describe the phase space of a charged particle interacting with a Yang–Mills field. Finally, we introduce a singular analogue of this construction and provide physical examples.

• Referencias bibliográficas
  • M. de León, J.C. Marrero, E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A 38(24) (2005), R241–R308.
  • V. Guillemin, E. Miranda, A.R. Pires, Symplectic and Poisson geometry on b-manifolds, Adv. Math. 264 (2014), 864–896.
  • J.C. Marrero, E. Padr´on, M. Rodr´ıguez-Olmos, Reduction of a symplectic-like Lie algebroid with momentum map and its application to fiberwise...
  • J. Marsden, A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys. 5(1) (1974), 121–130.
  • R.B. Melrose, The Atiyah–Patodi–Singer Index Theorem, Res. Notes Math. 4, A K Peters, Ltd., Wellesley, MA, 1993.
  • P. Mir, E. Miranda, P. Nicol´as, Hamiltonian facets of classical gauge theories on E-manifolds, J. Phys. A 56(23) (2023), Paper no. 235201,...
  • E. Miranda, C. Oms, The singular Weinstein conjecture, Adv. Math. 389 (2021), Paper no. 107925, 41 pp.
  • R. Montgomery, Canonical formulations of a classical particle in a Yang–Mills field and Wong’s equations, Lett. Math. Phys. 8(1) (1984), 59–67.
  • R.W. Montgomery, The bundle picture in mechanics (symplectic geometry, Yang–Mills, Poisson manifolds, Wong’s equations), Thesis (Ph.D.)-University...
  • G. Scott, The geometry of b k manifolds, J. Symplectic Geom. 14(1) (2016), 71–95.
  • J.-P. Serre, Faisceaux alg´ebriques coh´erents, Ann. of Math. (2) 61 (1955), 197–278.
  • S. Sternberg, Minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang–Mills field, Proc. Nat. Acad....
  • R.G. Swan, Vector bundles and projective modules, Trans. Amer. Math. Soc. 105 (1962), 264–277.
  • A. Weinstein, A universal phase space for particles in Yang–Mills fields, Lett. Math. Phys. 2(5) (1977/78), 417–420.
  • A. Weinstein, The local structure of Poisson manifolds, J. Differential Geom. 18(3) (1983), 523–557.