Gauge forms on SU(2)-bundles

Abstract

Let π:P→M be a principal SU(2)-bundle, let autP [resp. gauP⊂autP] be the Lie algebra [resp. the ideal] of all G-invariant [resp. G-invariant π-vertical] vector fields in X(P), and let p:C(P)→M be the bundle of connections of P. A differential form ωr on C(P) of arbitrary degree 0≤r≤4n, n=dimM, is said to be autP-invariant [resp. gauP-invariant] if it is invariant under the natural representation of autP [resp. gauP] on X(C(P)). The Z-graded algebra over Ω∙(M) of autP-invariant [resp. gauP-invariant] differential forms is denoted by IautP [resp. IgauP]. The basic results of this paper are the following: (1) The algebra of gauge invariant differential forms on p:C(P)→M is generated over the algebra of differential forms on M by a 4-form η4, i.e., IgauP(C(P))=(p∗Ω∙(M))[η4], where the form η4 is globally defined on C(P) by using the canonical su(s)-valued 1-form of the bundle T∗(M)⊗su(2) and the determinant function det:su(2)→R; its local expression is η4=14S123(dA1i∧dxi∧dA1j∧dxj+2A2jA3kdxj∧dxk∧dA1i∧dxi), (Aij,xj), 1≤i≤3, 1≤j≤n, being the coordinate system induced from (xj) and the standard basis (B1,B2,B3) of su(2) on C(P). (2) Assume M is connected. Then, IautP(C(P))=R[η4]. (3) The cohomology class of η4 in H4(C(P);R) coincides with −4π2p∗(c2(P)), where c2(P) stands for the second Chern class of P. Remark that p:C(P)→M is an affine bundle and hence one has a natural isomorphism p∗:H∙(M;R)→H∙(C(P);R). Another important remark is the following. If dimM≤3, then every principal SU(2)-bundle π:P→M is trivial and hence its Chern class vanishes, but the form η4 is not zero although its pull-back along every section of C(P) does vanish