Bundle-connection pairs and loop group representations

Let $M$ be a connected differentiable manifold. Denote by $\Omega_m(M)$ the space of $H^1$-loops based at a fixed point $m\in M$. Associated to $\Omega_m(M)$ one has $\widetilde\Omega_m(M)$, the group of unparameterized loops. Given a bundle-connection pair $(E,\nabla)$ over $M$ with fiber the finite-dimensional vector space $V$ and structure group $G\subset\operatorname{GL}(V)$ we get (up to equivalence) a smooth representation of $\widetilde\Omega _m(M)$ in $G$ given by the parallel transport operator $P^{\nabla}$. It is possible to find in the literature several versions of the converse theorem, namely: all (smooth) representations of $\widetilde\Omega _m(M)$ arise in the above described way from a bundle-connection pair. It is shown in the present paper that the correct setting for this theorem is the theory of induced representations for groupoids.