Ribbon graphs and mirror symmetry

Given a ribbon graph Γ with some extra structure, we define, using constructible sheaves, a dg category CPM(Γ) meant to model the Fukaya category of a Riemann surface in the cell of Teichmüller space described by Γ. When Γ is appropriately decorated and admits a combinatorial “torus fibration with section,” we construct from Γ a one-dimensional algebraic stack XΓ with toric components. We prove that our model is equivalent to Perf(XΓ), the dg category of perfect complexes on XΓ.