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

Infinitesimal gauge symmetries of closed forms

Olivier Brahic 1,

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.
Keywords: Differential geometry, connections, higher gauge theory..
Mathematics Subject Classification: Primary: 53C8, 70S15; Secondary: 55R2.
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.  Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[38]

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

[39]

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

[40]

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

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.  Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[38]

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

[39]

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

[40]

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

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