A Poincaré lemma for connection forms (Q578603)

Denote by \({\mathcal P}\) the space of piecewise smooth curves in \(R^ n\) beginning at the origin. A path 2-form is a function h on \({\mathcal P}\) such that for each element \(\sigma\) in \({\mathcal P}\), h(\(\sigma)\) is a 2-form at the endpoint of \(\sigma\) with values in a Lie algebra \({\mathcal G}\). For example, if A is a smooth \({\mathcal G}\) valued connection form on \(R^ n\) with curvature F and parallel translation operator P(\(\sigma)\) then the equation \(L^ A(\sigma)=P(\sigma)^{-1}F(\sigma (1))P(\sigma)\) defines \(L^ A\) as a path 2-form. A necessary and sufficient condition is given to characterize those path 2-forms which arise in this way. By way of application it is shown that the Birula-Mandelstam generalization of Maxwell's equations to nonabelian gauge fields is equivalent to the Yang- Mills equation.