September  2011, 3(3): 277-312. doi: 10.3934/jgm.2011.3.277

Infinitesimal gauge symmetries of closed forms

1. 

Instituto Superior Técnico, dep. de Matemática, Av. Rovisco Pais 1049-001 Lisboa, Portugal

Received  April 2011 Revised  October 2011 Published  November 2011

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.
Citation: Olivier Brahic. Infinitesimal gauge symmetries of closed forms. Journal of Geometric Mechanics, 2011, 3 (3) : 277-312. doi: 10.3934/jgm.2011.3.277
References:
[1]

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

[2]

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

[3]

J. C. Baez, Higher Yang-Mills theory,, preprint, (). 

[4]

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

[5]

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

[6]

J. C. Baez and J. Huerta, An invitation to higher Gauge theory,, \arXiv{1003.4485}., (). 

[7]

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.

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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.

[14]

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.

[15]

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.

[16]

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

[17]

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.

[18]

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.

[19]

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

[20]

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.

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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.

[26]

J. F. Martins and R. Picken, A cubical set approach to 2-bundles with connection and Wilson surfaces,, \arXiv{0808.3964}., (). 

[27]

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.

[28]

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.

[29]

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.

[30]

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

[31]

C. L. Rogers, 2-plectic geometry, Courant algebroids and categorified prequantizations,, \arXiv{1001.0040v2}., (). 

[32]

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.

[33]

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.

[34]

U. Schreiber and K. Waldorf, Connections on non-abelian Gerbes and their Holonomy,, \arXiv{0808.1923v1}., (). 

[35]

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.

[36]

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

[37]

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

[38]

B. Uribe, Group actions on dg-manifolds and their relation to equivariant cohomology,, \arXiv{1010.5413v1}., (). 

[39]

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

[40]

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

show all references

References:
[1]

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

[2]

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

[3]

J. C. Baez, Higher Yang-Mills theory,, preprint, (). 

[4]

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

[5]

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

[6]

J. C. Baez and J. Huerta, An invitation to higher Gauge theory,, \arXiv{1003.4485}., (). 

[7]

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.

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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.

[14]

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.

[15]

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.

[16]

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

[17]

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.

[18]

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.

[19]

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

[20]

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.

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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.

[26]

J. F. Martins and R. Picken, A cubical set approach to 2-bundles with connection and Wilson surfaces,, \arXiv{0808.3964}., (). 

[27]

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.

[28]

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.

[29]

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.

[30]

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

[31]

C. L. Rogers, 2-plectic geometry, Courant algebroids and categorified prequantizations,, \arXiv{1001.0040v2}., (). 

[32]

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.

[33]

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.

[34]

U. Schreiber and K. Waldorf, Connections on non-abelian Gerbes and their Holonomy,, \arXiv{0808.1923v1}., (). 

[35]

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.

[36]

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

[37]

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

[38]

B. Uribe, Group actions on dg-manifolds and their relation to equivariant cohomology,, \arXiv{1010.5413v1}., (). 

[39]

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

[40]

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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