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 is an element of M. Associated to Omega(m)(M) one has <(Omega)over tilde>(m)(M), the group of unparameterized loops. Given a bundle-connection pair (E, del) over M with fiber the finite-dimensional vector space V and structure group G subset of GL(V) we get (up to equivalence) a smooth representation of <(Omega)over tilde>(m)(M) in G given by the parallel transport operator P-del. It is possible to find in the literature several versions of the converse theorem, namely: all (smooth) representations of <(Omega)over tilde>(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.