P. Etingof's conjecture about Drinfeld associators

Abstract

We construct a family of Drinfeld associators interpolating between the Knizhnik-Zamolodchikov associator, the Alekseev-Torossian associator and the anti-Knizhnik-Zamolodchikov associator. We give explicit integral formul\ae\ for the family of elements of the Grothendieck-Teichm\"uller Lie algebra tangent to the family of associators. As an application, we settle a conjecture of Pavel Etingof about the Alekseev-Torossian associator. Furthermore, we give explicit integral formul\ae\ for the family of stable formality morphisms corresponding (in a precise way) to the above family of associators, and for the family of graph cohomology classes corresponding to the above family of elements of the Grothendieck-Teichm\"uller Lie algebra. It follows in particular that the ``logarithmic'' Kontsevich formality morphism corresponds to the Knizhnik-Zamolodchikov associator.