The Stable Symplectic category and a conjecture of Kontsevich

We consider an oriented version of the stable symplectic category defined in \cite{N}. 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 \cite{KO}. The above quotient of the Grothendieck--Teichmüller group is also shown to be the motivic group of monoidal automorphisms of a canonical representation (or fiber functor) on the stable symplectic category.

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