The Grothendieck-Teichm\"uller group action on differential forms and formality morphism of chains

TL;DR

This paper explores how the Grothendieck-Teichmüller group acts on differential forms and extends formality morphisms to chains and cochains, enriching the mathematical framework linking associators and formality.

Contribution

It introduces an extension of Kontsevich-type formality morphisms to chains and cochains via the Grothendieck-Teichmüller group action, up to homotopy.

Findings

Extended formality morphisms to chains and cochains.

Described the Grothendieck-Teichmüller group action up to homotopy.

Connected associators with formality morphisms in a new way.

Abstract

It is known that one can associate a Kontsevich-type formality morphism to every Drinfeld associator. We show that this morphism may be extended to a Kontsevich-Shoikhet formality morphism of cochains and chains, by describing the action of the Grothendieck-Teichm\"uller group on such objects (up to homotopy).