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Infinitesimal gauge symmetries of closed forms

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  • Motivated by the relationship between symplectic fibrations and classical Yang-Mills theories, we study the closedness of a $n$-form ($n$=2,3) defined on the total space of a fibration as a simple model for an abstract field theory. We introduce $2$-plectic fibrations and interpret geometrically the corresponding equations for coupling in terms of higher analogues of connections.
    Mathematics Subject Classification: Primary: 53C8, 70S15; Secondary: 55R20.


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  • [1]

    A. Alekseev, A. Malkin and E. Meinreken, Lie group valued moment maps, J. of Differential Geom., 48 (1998), 445-495.


    R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2nd edition, Addison-Wesley, 1987.


    J. C. BaezHigher Yang-Mills theory, preprint, arXiv:hep-th/0206130.


    J. C. Baez and A. S. Crans, Higher-dimensional algebra. VI. Lie 2-algebras, Theory and Applications of Categories, 12 (2004), 492-538.


    J. C. Baez and A. D. Lauda, Higher-dimensional algebra. V. 2-groups, Theory and Applications of Categories, 12 (2004), 423-491.


    J. C. Baez and J. HuertaAn invitation to higher Gauge theory, arXiv:1003.4485.


    J. C. Baez, A. E. Hoffnung and C. L. Rogers, Categorified symplectic geometry and the classical string, Comm. Math. Phys., 293 (2010), 701-725.doi: 10.1007/s00220-009-0951-9.


    J. C. Baez and U. Schreier, Higher gauge theory: 2-connections on 2-bundles, preprint, arXiv:hep-th/0412325, 2004.


    O. Brahic, Extensions of Lie brackets, Journal of Geometry and Physics, 60 (2010), 352-374.doi: 10.1016/j.geomphys.2009.10.006.


    O. Brahic and C. Zhu, Lie algebroid fibrations, Adv. Math., 226 (2011), 3105-3135.doi: 10.1016/j.aim.2010.10.006.


    L. Breena nd W. Messing, Differential geometry of gerbes, Adv. Math., 198 (2005), 732-846.doi: 10.1016/j.aim.2005.06.014.


    J.-L. Brylinski, "Loop Spaces, Characteristic Classes and Geometric Quantization," Progr. Math., 107, Birkhäuser Boston, Inc., Boston, MA, 1993.


    H. Bursztyn and M. Crainic, Dirac structures, momentum maps, and quasi-Poisson manifolds, in "The Breadth of Symplectic and Poisson Geometry," 1-40, Progr. Math., 232, Birkhäuser Boston, Boston, MA, 2005.


    H. Bursztyn, Gil R. Cavalcanti and M. Gualtieri, Reduction of Courant algebroids and generalized complex structures, Advances in Mathematics, 211 (2007), 726-765.doi: 10.1016/j.aim.2006.09.008.


    D. Conduché, Modules croisés généralisés de longueur 2, in "Proceedings of the Luminy Conference on Algebraic K-Theory" (Luminy, 1983), J. Pure Appl. Algebra, 34 (1984), 155-178.


    M. Crainic, Prequantization and Lie brackets, J. Symplectic Geom., 2 (2004), 579-602.


    M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math. (2), 157 (2003), 575-620.doi: 10.4007/annals.2003.157.575.


    C. Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, in "Séminaire Bourbaki," Vol. 1, Exp. No. 24, 153-168, Soc. Math. France, Paris, 1995.


    V. Guillemin, E. Lerman and S. Sternberg, "Symplectic Fibrations and Multiplicity Diagrams," Cambridge University Press, Cambridge, 1996.doi: 10.1017/CBO9780511574788.


    M. Gotay, R. Lashof, J. Śniatycki and A. Weinstein, Closed forms on symplectic fibre bundles, Comm. Math. Helv., 58 (1983), 617-621.doi: 10.1007/BF02564656.


    V. Guillemin and S. Sternberg, "Symplectic Techniques in Physics," 2nd edition, Cambridge University Press, Cambridge, 1990.


    F. Lalonde and D. McDuff, Symplectic structures on fibre bundles, Topology, 42 (2003), 309-347.doi: 10.1016/S0040-9383(01)00020-9.


    D. McDuff and D. Salamon, "Introduction to Symplectic Topology," 2nd edition, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998.


    D. McDuff, Enlarging the Hamiltonian group, Conference on Symplectic Topology, J. Symplectic Geom., 3 (2005), 481-530.


    J. F. Martins and R. Picken, On two-dimensional holonomy, Trans. Amer. Math. Soc., 362 (2010), 5657-5695.doi: 10.1090/S0002-9947-2010-04857-3.


    J. F. Martins and R. PickenA cubical set approach to 2-bundles with connection and Wilson surfaces, arXiv:0808.3964.


    J. F. Martins and R. Picken, The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module, Differential Geom. Appl., 29 (2011), 179-206.doi: 10.1016/j.difgeo.2010.10.002.


    J. Pradines, Théorie de Lie pour les groupoïdes différentiables. Calcul différentiel dans la catégorie des groupoïdes infinitésimaux, C.R. Acad. Sci. Paris Sér. A-B, 264 (1967), A245-A248.


    D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, in "Quantization, Poisson Brackets and Beyond" (Manchester, 2001) (ed. Theodore Voronov), Contemp. Math., Vol. 315, Amer. Math. Soc., Providence, RI, 2002.


    D. Roytenberg and A. Weinstein, Courant algebroids and strongly homotopy Lie algebras, Lett. Math. Phys., 46 (1998), 81-93.doi: 10.1023/A:1007452512084.


    C. L. Rogers2-plectic geometry, Courant algebroids and categorified prequantizations, arXiv:1001.0040v2.


    P. Ševera and A. Weinstein, Poisson geometry with a 3-form background. Noncommutative geometry and string theory, (Yokohama, 2001), Progr. Theoret. Phys. Suppl. No., 144 (2001), 145-154.


    U. Schreiber and K. Waldorf, Smooth functors vs. differential forms, Homology, Homotopy Appl., 13 (2011), 143-203.doi: 10.4310/HHA.2011.v13.n1.a6.


    U. Schreiber and K. WaldorfConnections on non-abelian Gerbes and their Holonomy, arXiv:0808.1923v1.


    S. Sternberg, Minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field, Proc. Nat. Acad. Sci. USA, 74 (1977), 5253-5254.doi: 10.1073/pnas.74.12.5253.


    F. Toppan, On anomalies in classical dynamics, Journal of Nonlinear Mathematical Physics, 8 (2001), 518-533.doi: 10.2991/jnmp.2001.8.4.6.


    A. Tsemo, Gerbes, 2-gerbes and symplectic fibrations, Rocky Mountain J. Math., 38 (2008), 727-777.


    B. UribeGroup actions on dg-manifolds and their relation to equivariant cohomology, arXiv:1010.5413v1.


    A. WeinsteinA universal phase space for particles in a Yang-Mills field, Letters in Mathematical Physycs, 2 (1977/78), 417-420. doi: 10.1007/BF00400169.


    A. Weinstein, Fat bundles and symplectic manifolds, Adv. in Math., 37 (1980), 239-250.doi: 10.1016/0001-8708(80)90035-3.

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