Synthetic Differential Geometry of Chen's Iterated Integrals

TL;DR

This paper explores Chen's iterated integrals using synthetic differential geometry, demonstrating that they form a subcomplex of the de Rham complex on various path spaces, providing a new geometric perspective.

Contribution

It introduces a synthetic differential geometric framework for Chen's iterated integrals and shows their subcomplex structure in the de Rham complex on path spaces.

Findings

Iterated integrals form a subcomplex of the de Rham complex

Application to free and based path spaces

Provides a new geometric perspective on Chen's integrals

Abstract

Chen's iterated integrals are treated within synthetic differential geometry. The main result is that iterated integrals produce a subcomplex of the de Rham complex on the free path space as well as based path spaces.