On identically closed forms locally constructed from a field

Let M be an n-dimensional manifold with derivative operator ∇a and let B(M) be an arbitrary vector bundle over M, equipped with a connection. A cross section of B defines a field φ on M. Let α be a p-form on M (with p<n) which is locally constructed from φ and finitely many of its derivatives (as well as, possibly, some ‘‘background fields’’ ψ and their derivatives) such that dα=0 for all cross sections φ. Suppose further that α=0 for the zero cross section, φ=0. It is proven here that there exists a (p−1)-form β that also is a local function of φ,ψ and finitely many of their derivatives, such that α=dβ. A number of applications of this result are described. In particular, gauge invariance is established for the charges and the total fluxes derived from gauge-dependent conserved currents, and severe limitations are established on the the possibilities for gravitational analogs of magnetic charges.