Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 9135.11 Application: Gauge Fields on Principal Connections5.11.1 Connection StrengthGiven a principal G−bundle P → Q, the Frölicher–Nijenhuis bracket(4.140) on the space ∧ ∗ (P ) ⊗ V 1 (P ) of tangent–valued forms on P is compatiblewith the canonical action R G (4.31) of G on P , and induces theFrölicher–Nijenhuis bracket on the space ∧ ∗ (Q) ⊗ T G P (Q) of T G P −valuedforms on Q. Its coordinate form issues from the Lie bracket (4.36).Then any principal connection A ∈ ∧ 1 (Q) ⊗ T G P (Q) (5.42) sets theNijenhuis differentiald A : ∧ r (Q) ⊗ T G P (Q) → ∧ r+1 (Q) ⊗ V G P (Q),d A φ = [A, φ] F N , φ ∈ ∧ r (Q) ⊗ T G P (Q), (5.323)on the space ∧ ∗ (Q) ⊗ T G P (Q).The curvature R (5.34) can be equivalently defined as the NijenhuisdifferentialR : Y → ∧ 2 T ∗ Q ⊗ V Y, given by R = 1 2 d ΓΓ = 1 2 [Γ, Γ] F N . (5.324)Let us define the strength of a principal connection A, asF A = 1 2 d AA = 1 2 [A, A] F N ∈ ∧ 2 (Q)⊗V G P (Q), (5.325)F A = 1 2 F r λµdq α ∧ dq µ ⊗e r , F r λµ = ∂ α A r µ − ∂ µ A r α + c r pqA p αA q µ.(5.326)It is locally given by the expressionF A = dA + 1 [A, A] = dA + A ∧ A, (5.327)2where A is the local connection form (5.43). By definition, the strength(5.325) of a principal connection obeys the second Bianchi identityd A F A = [A, F A ] F N = 0. (5.328)It should be emphasized that the strength F A (5.325) is not the standardcurvature (5.34) of a connection on P , but there are the local relationsψ P ζ F A = zζ ∗Θ, where Θ = dà + 1 [Ã, Ã] (5.329)2