I recently found out about Chen's iterated integrals for paths in a differentiable manifold, and I was wondering if an analogous construction exists for free loops, i.e. a set of variables one computes from a free loop and which uniquely determine this loop up to reparametrizations and insertions/deletions of trees.
Another thing I was wondering is whether it is on some way possible to write down a Haar measure for integration over path/loop space using these iterated integrals. If I discretize the problem I'm working on by replacing the manifold $\mathbb R^n$ by a cubic lattice, I find I have to sum over all loops not containing tree parts on this lattice, each loop having the same weight. How does this translate to a continuous manifold using these iterated integrals (if it translates at all)?
Thanks for any help people have to offer.