American Institute of Mathematical Sciences

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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.   Google Scholar

[2]

R. Abraham and J. E. Marsden, "Foundations of Mechanics,", 2nd edition, (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.   Google Scholar

[5]

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

[9]

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

[10]

O. Brahic and C. Zhu, Lie algebroid fibrations,, Adv. Math., 226 (2011), 3105.  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.  doi: 10.1016/j.aim.2005.06.014.  Google Scholar

[12]

J.-L. Brylinski, "Loop Spaces, Characteristic Classes and Geometric Quantization,", Progr. Math., 107 (1993).   Google Scholar

[13]

H. Bursztyn and M. Crainic, Dirac structures, momentum maps, and quasi-Poisson manifolds,, in, 232 (2005), 1.   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.  doi: 10.1016/j.aim.2006.09.008.  Google Scholar

[15]

D. Conduché, Modules croisés généralisés de longueur 2,, in, 34 (1984), 155.   Google Scholar

[16]

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

[17]

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

[18]

C. Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable,, in, (1995), 153.   Google Scholar

[19]

V. Guillemin, E. Lerman and S. Sternberg, "Symplectic Fibrations and Multiplicity Diagrams,", Cambridge University Press, (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.  doi: 10.1007/BF02564656.  Google Scholar

[21]

V. Guillemin and S. Sternberg, "Symplectic Techniques in Physics,", 2nd edition, (1990).   Google Scholar

[22]

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

[23]

D. McDuff and D. Salamon, "Introduction to Symplectic Topology,", 2nd edition, (1998).   Google Scholar

[24]

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

[25]

J. F. Martins and R. Picken, On two-dimensional holonomy,, Trans. Amer. Math. Soc., 362 (2010), 5657.  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.  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).   Google Scholar

[29]

D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids,, in, (2001).   Google Scholar

[30]

D. Roytenberg and A. Weinstein, Courant algebroids and strongly homotopy Lie algebras,, Lett. Math. Phys., 46 (1998), 81.  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.   Google Scholar

[33]

U. Schreiber and K. Waldorf, Smooth functors vs. differential forms,, Homology, 13 (2011), 143.  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.  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.  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.   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.  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.   Google Scholar

[2]

R. Abraham and J. E. Marsden, "Foundations of Mechanics,", 2nd edition, (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.   Google Scholar

[5]

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

[9]

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

[10]

O. Brahic and C. Zhu, Lie algebroid fibrations,, Adv. Math., 226 (2011), 3105.  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.  doi: 10.1016/j.aim.2005.06.014.  Google Scholar

[12]

J.-L. Brylinski, "Loop Spaces, Characteristic Classes and Geometric Quantization,", Progr. Math., 107 (1993).   Google Scholar

[13]

H. Bursztyn and M. Crainic, Dirac structures, momentum maps, and quasi-Poisson manifolds,, in, 232 (2005), 1.   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.  doi: 10.1016/j.aim.2006.09.008.  Google Scholar

[15]

D. Conduché, Modules croisés généralisés de longueur 2,, in, 34 (1984), 155.   Google Scholar

[16]

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

[17]

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

[18]

C. Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable,, in, (1995), 153.   Google Scholar

[19]

V. Guillemin, E. Lerman and S. Sternberg, "Symplectic Fibrations and Multiplicity Diagrams,", Cambridge University Press, (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.  doi: 10.1007/BF02564656.  Google Scholar

[21]

V. Guillemin and S. Sternberg, "Symplectic Techniques in Physics,", 2nd edition, (1990).   Google Scholar

[22]

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

[23]

D. McDuff and D. Salamon, "Introduction to Symplectic Topology,", 2nd edition, (1998).   Google Scholar

[24]

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

[25]

J. F. Martins and R. Picken, On two-dimensional holonomy,, Trans. Amer. Math. Soc., 362 (2010), 5657.  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.  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).   Google Scholar

[29]

D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids,, in, (2001).   Google Scholar

[30]

D. Roytenberg and A. Weinstein, Courant algebroids and strongly homotopy Lie algebras,, Lett. Math. Phys., 46 (1998), 81.  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.   Google Scholar

[33]

U. Schreiber and K. Waldorf, Smooth functors vs. differential forms,, Homology, 13 (2011), 143.  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.  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.  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.   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.  doi: 10.1016/0001-8708(80)90035-3.  Google Scholar

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