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.

i.e.without base-point) can be represented by many paths (choose a base-point and consider the path going from that point around the loop once). All iterated integrals indeed determine the loop, but if you have two different sets of such iterated integrals, they could still describe the same free loop, just with different choice of base-point. – David Vercauteren Aug 6 '13 at 17:45