 Those vector fields on "P" which are both right invariant and vertical form an ideal formula_50 of formula_75, which has a relationship to the entire bundle "P" analogous to that of the Lie algebra formula_8 of the gauge group "G" to the individual "G"-torsor fiber "F".The "field theory" of interest is defined in terms of a set of "fields" (smooth maps into various vector spaces) defined on a principal gauge bundle "P".Different fields carry different representations of the gauge group "G", and perhaps of other symmetry groups of the manifold such as the Poincaré group.