Surface Drinfeld Torsors I : Higher Genus Associators

Abstract

We develop a higher genus version of Drinfeld associators by means of operad theory. We start by introducing a framed version of rational associators and Grothendieck-Teichm\"uller groups and show that their definition is independent of the framing data. Next, we define a framed version of the universal KZ connection and we use it to show that over the complex numbers, the rational framed Drinfeld torsor is not empty. Next, we concentrate on the higher genus version of this story. We define an operad module of framed parenthesized higher genus braidings in prounipotent groupoids and we define its chord diagram counterpart. We then use these operadic modules to operadicly define higher genus associators and Grothendieck-Teichm\"uller groups, which again do not depend on the framing data. Finally, we compare our results in the genus $1$ case with those appearing in the litterature.