Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 915the vector space V . The curvature (5.34) of this connection readsR = − 1 2 F p λµ I p i jy j dq α ∧dq µ ⊗ ∂ i , (5.333)where F p λµare coefficients (5.326) of the strength of a principal connectionA.In particular, any principal connection A induces the associated linearconnection on the gauge algebra bundle V G P → Q. The correspondingcovariant differential ∇ A ξ (5.30) of its sections ξ = ξ p e p reads∇ A ξ : Q → T ∗ Q ⊗ V G P, ∇ A ξ = (∂ α ξ r + c r pqA p αξ q )dq α ⊗ e r .It coincides with the Nijenhuis differentiald A ξ = [A, ξ] F N = ∇ A ξ (5.334)of ξ seen as a V G P −valued 0–form, and is given by the local expressiongiven by the local expressionwhere A is the local connection form (5.43).∇ A ξ = dξ + [A, ξ], (5.335)5.11.3 Classical Gauge FieldsSince gauge potentials are represented by global sections of the connectionbundle C → Q (5.45), its 1–jet space J 1 (Q, C) plays the role of a configurationspace of classical gauge theory. The key point is that the jet spaceJ 1 (Q, C) admits the canonical splitting over C which leads to a uniquecanonical Yang–Mills Lagrangian density of gauge theory on J 1 (Q, C).Let us describe this splitting. One can show that the principalG−bundleJ 1 (Q, P ) → J 1 (Q, P )/G = C (5.336)is canonically isomorphic to the trivial pull–back bundleP C = C × P → C, (5.337)and that the latter admits the canonical principal connection [García(1977); Giachetta et. al. (1997)]A = dq α ⊗ (∂ α + a p αe p ) + da r α ⊗ ∂ α r ∈ O 1 (C) ⊗ T G (P C )(C). (5.338)