The stable symplectic category and a conjecture of Kontsevich

Abstract

We consider an oriented version of the stable symplectic category defined in [17]. We show that the group of monoidal automorphisms of this category, that fix each object, contains a natural subgroup isomorphic to the solvable quotient (or a graded-abelian quotient) of the Grothendieck-Teichmüller group. This establishes a stable version of a conjecture of Kontsevich which states that groups closely related to the Grothendieck-Teichmüller group act on the moduli space of certain field theories [19]. The above quotient is also shown to be the motivic group of monoidal automorphisms of a canonical representation (or fiber functor) on the stable symplectic category.

Toward a Galois theory of the integers over the sphere spectrum

“…This suggests possible connections with the study of automorphisms of the braid groups (i.e. understanding the work of Grothendieck and Drinfel'd on the tower of mapping class groups [14,19]); we won't discuss this further here, but it is one of the motivations for our interest in this subject [25,40].…”