Floer theory of higher rank quiver 3-folds

We study threefolds $Y$ fibred by $A_m$-surfaces over a curve $S$ of positive genus. An ideal triangulation of $S$ defines, for each rank $m$, a quiver $Q(Delta_m)$, hence a $CY_3$-category $(C,W)$ for any potential $W$ on $Q(Delta_m)$. We show that for $omega$ in an open subset of the Kahler cone, a subcategory of a sign-twisted Fukaya category of $(Y,omega)$ is quasi-isomorphic to $(C,W_{[omega]})$ for a certain generic potential $W_{[omega]}$. This partially establishes a conjecture of Goncharov concerning `categorifications of cluster varieties of framed $PGL_{m+1}$-local systems on $S$, and gives a symplectic geometric viewpoint on results of Gaiotto, Moore and Neitzke.