Flat Z-graded connections and loop spaces

TL;DR

This paper establishes a deep connection between flat Z-graded connections on a space and representations of the chain algebra of its based loop space, generalizing holonomy concepts.

Contribution

It proves an $A_$ quasi-equivalence between dg categories of flat Z-graded connections and chain algebra representations of the based loop space.

Findings

Holonomy automorphism extends to flat Z-graded connections.

Provides an $A_$ quasi-equivalence between categories.

Connects geometric connections with algebraic loop space representations.

Abstract

The pull back of a flat bundle $E\rightarrow X$ along the evaluation map $\pi: \mathcal{L} X \to X$ from the free loop space $\mathcal{L} X$ to $X$ comes equipped with a canonical automorphism given by the holonomies of $E$. This construction naturally generalizes to flat $\mathbb{Z}$-graded connections on $X$. Our main result is that the restriction of this holonomy automorphism to the based loop space $\Omega_* X$ of $X$ provides an $A_\infty$ quasi-equivalence between the dg category of flat $\mathbb{Z}$-graded connections on $X$ and the dg category of representations of $C_\bullet(\Omega_* X)$, the dg algebra of singular chains on $\Omega_* X$.