Iterated integrals of superconnections

TL;DR

This paper explores how iterated integrals of flat superconnections on graded bundles produce A-infinity functors, linking superconnection flatness to higher algebraic structures and Reidemeister torsion.

Contribution

It provides a detailed, foundational exposition connecting superconnections, Chen's iterated integrals, and A-infinity structures, clarifying their relationships from basic definitions.

Findings

Iterated integrals of flat superconnections yield A-infinity functors.

Flatness of superconnections is equivalent to the functorial property.

Connections to higher Reidemeister torsion are discussed.

Abstract

Starting with a Z-graded superconnection on a graded vector bundle over a smooth manifold M, we show how Chen's iterated integration of such a superconnection over smooth simplices in M gives an A-infinity functor if and only if the superconnection is flat. If the graded bundle is trivial, this gives a twisting cochain. 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" and a brief discussion of how this is related to higher Reidemeister torsion.