Gauge invariance on principal SU(2)-bundles

Abstract

Let π:P→M be a principal G-bundle. One denotes by J1P the 1-jet bundle of local sections of π, by autP the Lie algebra of G-invariant vector fields of P and by gauP the ideal of π-vertical vector fields in autP. A differential form ω on J1P is said to be autP-invariant [resp. gauP-invariant] if LX(1)ω=0 for every X∈autP [resp. X∈gauP], where X(1) is the natural lift of X∈X(P) to J1P, i.e., X(1) is the infinitesimal contact transformation associated to X. The authors of the present paper study the structure of autP- and gau P-invariant forms, when the structure group is G=SU(2). They prove that the algebra of autP-invariant [resp. gauP-invariant] forms is differentiably generated over the real numbers [resp. over the graded algebra of differential forms on M] by the standard structure forms. These are the 1-forms ϑa obtained when one decomposes the standard su(2)-valued 1-form ϑ on J1P as ϑ=ϑa⊗Ba, a∈{1,2,3}, where Ba is the standard basis of the Lie algebra su(2). On the other hand, by means of the identification between the affine bundle C(P)→M of connections on P and the quotient bundle (J1P)/G→M, they show that the representation autP→X(C(P)) can be obtained by infinitesimal contact transformations