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 K\"ahler 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.
Categories: math.SG math-ph math.AG math.MP