 One may define the space "Pl" of local polynomials in these fields and their derivatives.The fundamental Lagrangian density of one's theory is presumed to lie in the subspace "Pl"0 of polynomials which are real-valued and invariant under any unbroken non-gauge symmetry groups.It is also presumed to be invariant not only under the left action (passive coordinate transformations) and the global right action of the gauge group but also under local gauge transformations—pullback along the infinitesimal diffeomorphism associated with an arbitrary choice of right invariant vertical vector field formula_79.