Higher genus Kashiwara-Vergne problems and the Goldman-Turaev Lie bialgebra

TL;DR

This paper introduces a family of generalized Kashiwara-Vergne problems for higher genus surfaces, proves their solvability, and explores their implications for the Goldman-Turaev Lie bialgebra's formality.

Contribution

It extends the classical Kashiwara-Vergne problem to higher genus surfaces and establishes solutions using elliptic associators, linking to Lie bialgebra formality.

Findings

Solutions exist for all higher genus cases

Every solution induces a Lie bialgebra isomorphism

Connections to the Goldman-Turaev Lie bialgebra formality

Abstract

We define a family ${\rm KV}^{(g,n)}$ of Kashiwara-Vergne problems associated with compact connected oriented 2-manifolds of genus $g$ with $n+1$ boundary components. The problem ${\rm KV}^{(0,3)}$ is the classical Kashiwara-Vergne problem from Lie theory. We show the existence of solutions of ${\rm KV}^{(g,n)}$ for arbitrary $g$ and $n$. The key point is the solution of ${\rm KV}^{(1,1)}$ based on the results by B. Enriquez on elliptic associators. Our construction is motivated by applications to the formality problem for the Goldman-Turaev Lie bialgebra $\mathfrak{g}^{(g, n+1)}$. In more detail, we show that every solution of ${\rm KV}^{(g,n)}$ induces a Lie bialgebra isomorphism between $\mathfrak{g}^{(g, n+1)}$ and its associated graded ${\rm gr} \, \mathfrak{g}^{(g, n+1)}$. For $g=0$, a similar result was obtained by G. Massuyeau using the Kontsevich integral. This paper is a summary of our results. Details and proofs will appear elsewhere.