Ivancevic_Applied-Diff-Geom

918 Applied Differential Geometry: A Modern IntroductionIn order to study the gauge invariance of one or another object ingauge theory, it suffices to examine its invariance under an arbitrary 1–parameter subgroup [Φ P ] of the gauge group. Its infinitesimal generator isa G−invariant vertical vector–field ξ on a principal bundle P or, equivalently,a sectionξ = ξ p (x)e p (5.351)of the gauge algebra bundle V G P → Q (4.35). We will call it a gaugevector–field. One can think of its components ξ p (q) as being gauge parameters.Gauge vector–fields (5.351) are transformed under the infinitesimalgenerators of gauge transformations (i.e., other gauge vector–fields) ξ ′ bythe adjoint representationL ξ ′ξ = [ξ ′ , ξ] = c p rqξ ′ r ξ q e p ,(ξ, ξ ′ ∈ V G P (Q)).Therefore, gauge parameters are subject to the coadjoint representationξ ′ : ξ p ↦→ − c p rqξ ′ r ξ q . (5.352)Given a gauge vector–field ξ (5.351) seen as the infinitesimal generatorof a 1–parameter gauge group [Φ P ], let us get the gauge vector–fields on aP −associated bundle Y and the connection bundle C.The corresponding gauge vector–field on the P −associated vector bundleY → Q issues from the relation (5.346), and readsξ Y = ξ p I i p∂ i ,where I p are generators of the group G, acting on the typical fibre V of Y .The gauge vector–field ξ (5.351) acts on elements a (5.46) of the connectionbundle by the lawL ξ a = [ξ, a] F N = (−∂ α ξ r + c r pqξ p a q α)dq α ⊗ e r .In view of the vertical splitting (5.339), this quantity can be regarded asthe vertical vector–fieldξ C = (−∂ α ξ r + c r pqξ p a q α)∂ α r (5.353)on the connection bundle C, and is the infinitesimal generator of the 1–parameter group [Φ C ] of vertical automorphisms (5.347) of C, i.e., a desiredgauge vector–field on C.