Flat Z-graded connections and loop spaces

Abstract

The pull back of a flat bundle $E\to X$ 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 ${\Omega }_{\ast }X$ 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 $C_{\bullet }({\Omega }_{\ast }X)$, the dg algebra of singular chains on ${\Omega }_{\ast }X$.

Comments are welcome

References in corpus (2)

• Morita cohomology
• String topology and the based loop space