The Stable Symplectic category and a conjecture of Kontsevich

TL;DR

This paper establishes a stable version of Kontsevich's conjecture by linking the automorphisms of a symplectic category to a quotient of the Grothendieck--Teichmüller group, revealing deep symmetries in moduli spaces.

Contribution

It demonstrates that the automorphism group of an oriented stable symplectic category contains a subgroup isomorphic to a quotient of the Grothendieck--Teichmüller group, confirming a conjecture of Kontsevich.

Findings

Automorphism group contains a solvable quotient of the Grothendieck--Teichmüller group.

Establishes a connection between symplectic categories and the motivic Galois group.

Provides a stable version of Kontsevich's conjecture relating to moduli spaces.

Abstract

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--Teichm\"uller group. This establishes a stable version of a conjecture of Kontsevich which states that groups closely related to the Grothendieck--Teichm\"uller group act on the moduli space of certain field theories \cite{KO}. The above quotient of the Grothendieck--Teichm\"uller group is also shown to be the motivic group of monoidal automorphisms of a canonical representation (or fiber functor) on the stable symplectic category.