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oai:docta.ucm.es:20.500.14352/589012023-08-25T20:00:07Zcom_20.500.14352_14col_20.500.14352_15

Gauge interpretation of characteristic classes Castrillón López, Marco Muñoz Masqué, Jaime 514.7 Connections on a principal bundle characteristic classes gauge invariance jet bundles. Geometría diferencial 1204.04 Geometría Diferencial Let π:P→M be a principal G-bundle. Then one can consider the following diagram of fibre bundles: \CD J^{1}(P) @>\pi_{10}>> P\\ @VqVV @VV\pi V\\ C(P) @>p>> M\endCD where p is the bundle of connections of π. As is well known, q is also a principal G-bundle, and the canonical contact form θ on J1(P) can be considered as a connection form on q, with curvature form Θ. One defines aut P as the Lie algebra of G-invariant vector fields on P and gau P as the ideal of π-vertical G-invariant vector fields on P. If X∈autP⊂X(P), then one defines the infinitesimal contact transformation associated to X, X1∈X(J1(P)), and its q-projection XC∈X(C(P)). A differential form Ω on C(P) is said to be aut P-invariant [resp. gauge invariant] if LXCΩ=0 for every X∈autP [resp. X∈gauP]. On the other hand, let us denote by g the Lie algebra of G. An element of the symmetric algebra of g∗ will be called a Weil polynomial. The main result of the paper is the following theorem: If G is connected, for every gauge invariant form Ω on C(P) there exist differential forms ω1,…,ωk on M and Weil polynomials f1,…,fk such that Ω=p∗(ω1)∧f1(Θ)+⋯+p∗(ωk)∧fk(Θ). As a consequence, the authors prove that a differential form Ω on C(P) is aut P-invariant iff Ω=f(Θ), where f is a Weil polynomial, and then Ω is closed. Explicit examples are shown and the link between the above theorem and the geometric formulation of Utiyama's theorem is explained. DGESIC (Spain) Depto. de Álgebra, Geometría y Topología Fac. de Ciencias Matemáticas TRUE pub 2023-06-20T18:54:25Z 2023-06-20T18:54:25Z 2001 journal article https://hdl.handle.net/20.500.14352/58901 1073-2780 10.4310/MRL.2001.v8.n4.a6 eng PB98–0533. open access application/pdf International Press