Gauge algebras, curvature and symplectic structure

García, Pedro Luis Rodríguez

doi:10.4310/jdg/1214433983

Cited by 57 publications

(24 citation statements)

References 4 publications

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“…The formulation of (classical) gauge theories in the language of modern differential geometry is an extensive subject that, since its beginnings in the 1970's, has been addressed by many authors; some references in this direction which have been useful in the course of our work are [8][9][10][11][12][13]. It should also be mentioned that Lie group bundles can be regarded as a special class of Lie groupoids, namely, locally trivial Lie groupoids for which the source projection and the target projection coincide [14], and the use of Lie groupoids in gauge theories has been advocated before by some authors [15,16].…”

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confidence: 99%

Local symmetries in gauge theories in a finite-dimensional setting

Forger¹,

Soares²

2012

Journal of Geometry and Physics

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It is shown that the correct mathematical implementation of symmetry in the geometric formulation of classical field theory leads naturally beyond the concept of Lie groups and their actions on manifolds, out into the realm of Lie group bundles and, more generally, of Lie groupoids and their actions on fiber bundles. This applies not only to local symmetries, which lie at the heart of gauge theories, but is already true even for global symmetries when one allows for fields that are sections of bundles with (possibly) non-trivial topology or, even when these are topologically trivial, in the absence of a preferred trivialization.

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confidence: 99%

Local symmetries in gauge theories in a finite-dimensional setting

Forger¹,

Soares²

2012

Journal of Geometry and Physics

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“…The following proposition is the basic link between this approach to multisymplectic forms and the classical Poincaré-Cartan formalism in the calculus of variations, as in, for example, [11,14]. …”

Section: The Multisymplectic Formalismmentioning

confidence: 99%

“…This is a particular case of the Utiyama theorem (see, for example [11]). Following the guide from the infinite dimensional space of fields point of view, one should consider variations of sections of the bundle Y that are needed to form the Euler-Lagrange equations.…”

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confidence: 95%

Some remarks on Lagrangian and Poisson reduction for field theories

López

Marsden

2003

Journal of Geometry and Physics

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Given a Hamiltonian system on a fiber bundle, the Poisson covariant formulation of the Hamilton equations is described. When the fiber bundle is a G-principal bundle and the Hamiltonian density is G-invariant, the reduction of this formulation is studied thus obtaining the analog of the Lie-Poisson reduction for field theories. The relation of this reduction with the Lagrangian reduction and the Lagrangian and Poisson reduction for electromagnetism are also analyzed.

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“…M is by de¢nition the sub-bundle of HomðTM; ðTPÞ=GÞ de¢ned by those elements G x : T x M ! ððTPÞ=GÞ x that satisfy p Ã G x ¼ Id T x M (see, for example, [4,5]). As is well known, C is an af¢ne bundle modeled over the vector bundle T Ã M adP !…”

Section: Preliminariesmentioning

confidence: 99%