Cartan proved that for a connected compact Lie group $G$ the left invariant differential forms yield the correct cohomology of $G$. The same argument works for a connected compact $G$-manifold: the idea is to "average" left invariant forms on $G$ using a Haar measure.


Can we extend this result for non compact infinite dimensional manifolds? In particular, consider the free loop space $LM$ of a manifold $M$; this is an infinite dimensional $S^1$-manifold. Is there a way to compute the cohomology of $LM$ using a model of "invariant forms" and the idea of averaging?


By a result of Chen, we know that iterated integrals of differential forms in $M$ yield the correct cohomology of $LM$. Is this model related to Cartan's story of invariant forms?


These questions are a bit vague, but I guess how to make them precise is part of my question.



free loop space and invariant forms
Iterated integrals define a map
$$
\sigma: C(\Omega(M)) \to \Omega(LM)
$$
where $C(\Omega(M))$ is the cyclic bar complex of $\Omega(M)$. It has various nice properties; for instance, it induces an isomorphism in cohomology, when $M$ is simply-connected. 


Unfortunately, the forms in the image of $\sigma$ are *not* invariant with respect to the $S^1$-action on $LM$, which probably means that there is no relation of the kind you expect.


However, the forms in the image of $\sigma$ are basic (in particular, invariant) with respect to the action of $Diff([0,1])$ by reparameterization, but the image of $\sigma$ is not even dense in the space of all basic forms. (The image is characterized by a stronger invariance that cannot be formulated in terms of a group action: a kind of *thin homotopy invariance*).


Just in order to add a layer of confusion, I remark that if $G$ is a compact group acting on $M$, there exists a $G$-equivariant extension of the iterated integral map $\sigma$, introduced by Getzler-Jones-Petrack. It uses the Cartan model for $G$-equivariant forms on $LM$, and an equivariant version of the cyclic bar complex of equivariant forms on $M$.


