Iterated integrals over higher dimensional loops

TL;DR

This paper introduces a new definition of higher dimensional iterated integrals over membranes, explores their fundamental properties, and proposes a conjecture extending Chen's de Rham Theorem to this context.

Contribution

It provides a novel framework for higher dimensional iterated integrals and extends classical theorems to the membrane case, advancing the mathematical understanding of these integrals.

Findings

Definition of higher dimensional iterated integrals over membranes

Basic properties of these integrals established

Conjecture extending Chen's de Rham Theorem to membranes

Abstract

We give a definition of higher dimensional iterated integrals based on integration over membranes. We prove basic properties of this definition and formulate a conjecture which extends Chen's de Rham Theorem for iterated integrals to the membrane case.