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Castrillón López, Marco and Muñoz Masqué, Jaime (2001) Gauge interpretation of characteristic classes. Mathematical Research Letters, 8 (4). pp. 457468. ISSN 10732780

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Official URL: http://www.intlpress.com/site/pub/pages/journals/items/mrl/content/vols/0008/0004/a006/
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Abstract
Let π:P→M be a principal Gbundle. 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 Gbundle, 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 Ginvariant vector fields on P and gau P as the ideal of πvertical Ginvariant 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 qprojection XC∈X(C(P)). A differential form Ω on C(P) is said to be aut Pinvariant [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 Pinvariant 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.
Item Type:  Article 

Uncontrolled Keywords:  Connections on a principal bundle, characteristic classes, gauge invariance, jet bundles. 
Subjects:  Sciences > Mathematics > Differential geometry 
ID Code:  24200 
Deposited On:  15 Jan 2014 14:01 
Last Modified:  18 Feb 2019 12:05 
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