Hodge Theory of the Turaev Cobracket and the Kashiwara--Vergne Problem

TL;DR

This paper demonstrates that the Turaev cobracket, after completion, respects mixed Hodge structures on algebraic curves and uses this to construct solutions to the Kashiwara--Vergne problem across all genera, revealing deep geometric and algebraic connections.

Contribution

It establishes the mixed Hodge structure compatibility of the Turaev cobracket and constructs torsors of solutions to the Kashiwara--Vergne problem for all genera, linking topology, geometry, and algebra.

Findings

Turaev cobracket is a morphism of mixed Hodge structures after completion.

Constructs torsors of solutions to the Kashiwara--Vergne problem for all genera.

Provides a homological description of the Turaev cobracket.

Abstract

In this paper we show that, after completing in the $I$-adic topology, the Turaev cobracket on the vector space freely generated by the closed geodesics on a smooth, complex algebraic curve $X$ with an algebraic framing is a morphism of mixed Hodge structure. We combine this with results of a previous paper (arXiv:1710.06053) on the Goldman bracket to construct torsors of solutions of the Kashiwara--Vergne problem in all genera. The solutions so constructed form a torsor under a prounipotent group that depends only on the topology of the framed surface. We give a partial presentation of these groups. Along the way, we give a homological description of the Turaev cobracket.