Infinitesimal gauge symmetries of closed forms

Previous Article
Euler equations on a semi-direct product of the diffeomorphisms group by itself
JGM Home
This Issue
Next Article

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

Abstract
Related Papers

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.
[2]	R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2^nd 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," 2^nd 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," 2^nd 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," 2^nd 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," 2^nd 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," 2^nd 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.

[1]	Rúben Sousa, Semyon Yakubovich. The spectral expansion approach to index transforms and connections with the theory of diffusion processes. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2351-2378. doi: 10.3934/cpaa.2018112
[2]	Michael Hochman. Lectures on dynamics, fractal geometry, and metric number theory. Journal of Modern Dynamics, 2014, 8 (3&4) : 437-497. doi: 10.3934/jmd.2014.8.437
[3]	Paul Bracken. Connections of zero curvature and applications to nonlinear partial differential equations. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1165-1179. doi: 10.3934/dcdss.2014.7.1165
[4]	Carlos Durán, Diego Otero. The projective symplectic geometry of higher order variational problems: Minimality conditions. Journal of Geometric Mechanics, 2016, 8 (3) : 305-322. doi: 10.3934/jgm.2016009
[5]	Robert J. McCann. A glimpse into the differential topology and geometry of optimal transport. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1605-1621. doi: 10.3934/dcds.2014.34.1605
[6]	Łukasz Rudnicki. Geophysics and Stuart vortices on a sphere meet differential geometry. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022075
[7]	Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127
[8]	Katarzyna Grabowska, Luca Vitagliano. Tulczyjew triples in higher derivative field theory. Journal of Geometric Mechanics, 2015, 7 (1) : 1-33. doi: 10.3934/jgm.2015.7.1
[9]	Penka Georgieva, Aleksey Zinger. Real orientations, real Gromov-Witten theory, and real enumerative geometry. Electronic Research Announcements, 2017, 24: 87-99. doi: 10.3934/era.2017.24.010
[10]	Ulrike Kant, Werner M. Seiler. Singularities in the geometric theory of differential equations. Conference Publications, 2011, 2011 (Special) : 784-793. doi: 10.3934/proc.2011.2011.784
[11]	Christopher Cox, Renato Feres. Differential geometry of rigid bodies collisions and non-standard billiards. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6065-6099. doi: 10.3934/dcds.2016065
[12]	Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703
[13]	Angelo B. Mingarelli. Nonlinear functionals in oscillation theory of matrix differential systems. Communications on Pure and Applied Analysis, 2004, 3 (1) : 75-84. doi: 10.3934/cpaa.2004.3.75
[14]	Santiago Capriotti. Dirac constraints in field theory and exterior differential systems. Journal of Geometric Mechanics, 2010, 2 (1) : 1-50. doi: 10.3934/jgm.2010.2.1
[15]	Leon Petrosyan, David Yeung. Shapley value for differential network games: Theory and application. Journal of Dynamics and Games, 2021, 8 (2) : 151-166. doi: 10.3934/jdg.2020021
[16]	Ioan Bucataru. A setting for higher order differential equation fields and higher order Lagrange and Finsler spaces. Journal of Geometric Mechanics, 2013, 5 (3) : 257-279. doi: 10.3934/jgm.2013.5.257
[17]	Antonio Cañada, Salvador Villegas. Lyapunov inequalities for partial differential equations at radial higher eigenvalues. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 111-122. doi: 10.3934/dcds.2013.33.111
[18]	Baruch Cahlon. Sufficient conditions for oscillations of higher order neutral delay differential equations. Conference Publications, 1998, 1998 (Special) : 124-137. doi: 10.3934/proc.1998.1998.124
[19]	R.S. Dahiya, A. Zafer. Oscillation theorems of higher order neutral type differential equations. Conference Publications, 1998, 1998 (Special) : 203-219. doi: 10.3934/proc.1998.1998.203
[20]	Peiguang Wang, Xiran Wu, Huina Liu. Higher order convergence for a class of set differential equations with initial conditions. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3233-3248. doi: 10.3934/dcdss.2020342

2020 Impact Factor: 0.857

Tools

Metrics

Other articles
by authors

on AIMS
Olivier Brahic
on Google Scholar
Olivier Brahic

[Back to Top]