Loops, Holonomy and Signature

TL;DR

This paper establishes a topological framework for loop groups in Euclidean space, linking them to a Fréchet-Lie group and principal bundle holonomy, and provides a new proof and generalization of the Chen signature theorem.

Contribution

It introduces a topology on loop groups that embeds them into a Fréchet-Lie group and offers a novel geometric proof and broader generalization of the Chen signature theorem.

Findings

Loop groups are embedded in a Fréchet-Lie group with a compatible topology.

A new geometric proof of the Chen signature theorem is provided.

The theorem is generalized to broader classes of loops.

Abstract

We show that there is a topology on certain groups of loops in Euclidean space such that these groups are embedded in a Fr\'echet-Lie group which is the structural group of a principal bundle with connection whose holonomy coincides with the Chen signature map. We also give an alternative geometric new proof of the Chen signature theorem and a generalization of this theorem in classes strictly containing the one originally considered by Chen.