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

Abstract

We define a family ${\mathrm{KV}}^{(g,n)}$ of Kashiwara-Vergne problems associated with compact connected oriented 2-manifolds of genus $g$ with $n+1$ boundary components. The problem ${\mathrm{KV}}^{(0,3)}$ is the classical Kashiwara-Vergne problem from Lie theory. We show the existence of solutions of ${\mathrm{KV}}^{(g,n)}$ for arbitrary $g$ and $n$. The key point is the solution of ${\mathrm{KV}}^{(1,1)}$ 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 ${\mathrm{KV}}^{(g,n)}$ induces a Lie bialgebra isomorphism between ${\mathfrak{g}}^{(g,n+1)}$ and its associated graded $\mathrm{gr}{\mathfrak{g}}^{(g,n+1)}$. 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.

6 pages, 2 figures