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

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