A Poincaré lemma for connection forms

Abstract

Denote by P the space of piecewise smooth curves in R n beginning at the origin. A path 2-form is a function h on P such that for each element σ in P , h( σ) is a 2-form at the endpoint of σ with values in a Lie algebra G . For example, if A is a smooth G valued connection form on R n with curvature F and parallel translation operator P( σ) then the equation L A ( σ) = P( σ) −1 F( σ(1)) P( σ) 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.