Ivancevic_Applied-Diff-Geom

960 Applied Differential Geometry: A Modern IntroductionRecall that we have a composite bundleY → Σ → X (5.478)of a bundle Y → X denoted by Y Σ and a bundle Σ → X. It is coordinatedby (x α , σ m , y i ) where (x µ , σ m ) are coordinates of Σ and y i are the fibrecoordinates of Y Σ . We further assume that Σ has a global section.The application of composite bundles to field theory is founded on thefollowing [Sardanashvily (1992)]. Given a global section h of Σ, the restrictionY h of Y Σ to h(X) is a subbundle of Y → X. There is the 1–1correspondence between the global sections s h of Y h and the global sectionsof the composite bundle (5.478) which cover h. Therefore, one canthink of sections s h of Y h as describing fermion fields in the presence of abackground parameter field h, whereas sections of the composite bundle Ydescribe all the pairs (s h , h). The configuration space of these pairs is the1–jet space J 1 (X, Y ) of the composite bundle Y .Every connectionA Σ = dx α ⊗ (∂ α + Ãi α∂ i ) + dσ m ⊗ (∂ m + A i m∂ i )on the bundle Y Σ induces the horizontal splittingV Y = V Y Σ ⊕ (Y × V Σ),locally given byẏ i ∂ i + ˙σ m ∂ m = (ẏ i − A i m ˙σ m )∂ i + ˙σ m (∂ m + A i m∂ i ).Using this splitting, one can construct the first–order differential operator(5.41) on the composite bundle Y , namely˜D : J 1 (X, Y ) → T ∗ X ⊗V Y Σ , ˜D = dx α ⊗(y i α−Ãi α−A i mσ m α )∂ i . (5.479)This operator possess the following property. Given a global section h ofΣ, let Γ be a connection on Σ whose integral section is h, that is, Γ ◦ h =j 1 h. Note that the differential (5.479) restricted to J 1 (X, Y ) h ⊂ J 1 (X, Y )becomes the familiar covariant differential relative to the connection on Y h ,A h = dx α ⊗ [∂ α + (A i m∂ α h m + Ãi α)∂ i ].Thus, it is ˜D that we may use in order to construct a Lagrangian densityL : J 1 (X, Y )eD−→ T ∗ X ⊗ V Y Σ → ∧ n T ∗ Xfor sections of the composite bundle Y .In particular, in gravitation theory, we have the composite bundleLX → Σ → X, where Σ is the quotient bundle (5.473) and LX Σ =