ITERATED INTEGRALS OF SUPERCONNECTIONS 0902a

Iterated integration of a connection gives the holonomy or parallel transport of the connection. Iterated integration of a superconnection gives something else which we call a “superconnection parallel transport.” We ask under what conditions are the holonomies of a connection on a graded vector bundle chain maps and under what conditions the superconnection parallel transports give homotopies and higher homotopies of these chain maps. This happens if and only if the superconnection is flat. If the graded bundle is trivial, we get a twisting cochain by integrating over simplices. When the bundle is nontrivial we get an A∞-functor. Very similar results were obtained by K.T. Chen using similar methods. This paper is intended to explain this from scratch beginning with the definition and basic properties of a connection and ending with an exposition of Chen’s “formal connections.”