Holonomy and Ran Spaces

Let $M$ be a flat (i.e., zero curvature) compact manifold, and let $p_x$ be a basepoint. Define a curve $\gamma: p_x \to p_x$ such that $\gamma(0) = p_x$ and $\gamma(1) = p_x$ but $\gamma(c) =p_y \; \forall c \in ]0,1[ \; \forall y \neq x$. Define the \emph{holonomy representation} by:
\begin{equation}
    \mathcal{H}_\gamma: \pi_1(M,p) \to Aut(E_p)
\end{equation}
where $E$ is the fiber over $p$. This immediately necessitates the notion of a \emph{holonomy bundle} of the fiber, which we will set to be:
\begin{equation}
    \mathcal{H}_E = \bigcup_i \; E_{p_i}
\end{equation}