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Monday, November 17, 2014

**Abstract:** If $E\rightarrow X$ is a flat vector bundle, and $\pi: LX \rightarrow X$ is the map given by evaluation at $1 \in S^1$, then the pull back bundle $\pi^*E$ is a flat bundle equipped with a canonical automorphism given by the holonomy. I will explain that this construction naturally generalizes to the case of flat $\mathbb{Z}$-graded connections on $X$. Moreover, the restriction of the holonomy automorphism to the based loop space provides a representation of the Pontryagin algebra $C_*(\Omega^{\mathsf{M}} X)$. I will describe how this construction fits into the general story of higher dimensional local systems. The talk is based on joint work in progress with Florian Sch\"atz.