Gauge interpretation of characteristic classes

Abstract

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.