Classification of formality morphisms for chains and Drinfel'd associators

Question

In his 1997 preprint q-alg/9709040, M. Kontsevich proved constructively the existence of a $L_\infty$-quasi-isomorphism between the differential graded algebra structure on the deformation complex of the associative algebra of functions on any manifold and its cohomology. The explicit construction of Kontsevich's $L_\infty$-quasi-isomorphism involves a sum on a particular class of graphs as well as a map from these graphs to $\mathbb R$ (the weight function).

He also notes that this $L_\infty$-quasi-isomorphism is not unique. This statement has been straightened in a series of paper due mainly to Willwacher and Dolgushev where it was shown that there is a canonical bijection (of torsors for the Grothendieck-Teichmüller group) between the set of Drinfel'd associators and the set of (homotopy classes of) stable formality morphisms such that the homotopy class of Kontsevich’s morphism with standard propagator is mapped to the Alekseev-Torossian associator. To different Drinfel'd associators correspond therefore different formality morphisms differing in their weight functions.

Secondly, Shoikhet's solution of Tsygan's conjecture on the existence of a formality theorem for chains also involves the construction of a quasi-isomorphism (of $L_\infty$-modules), the latter involving graphs and weights different from Kontsevich's original ones.

Questions:

1) Is there a known classification of the class of formality morphisms for chains?

2) Is there a known relation between such formality morphisms and algebraic objects (similar to Drinfel'd associators) ?

Answer 1

In [The Grothendieck-Teichmüller group action on differential forms and formality morphism of chains] (arxiv:1308.6097), T. Willwacher shows that to each Drinfel'd associator, one can associate both a stable formality morphism $\mathcal U_\Phi$ and a formality morphism of chains $\mathcal V_\Phi$. By choosing $\Phi$ to be the Alekseev-Torossian associator, $\mathcal U_\Phi$ coincides with the Kontsevich formality morphism with standard propagator and $\mathcal V_\Phi$ to the Shoikhet morphism.