Ribbon Graphs and Mirror Symmetry I

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 X_Γ with toric components. We prove that our model is equivalent to Perf(X_Γ), the dg category of perfect complexes on X_Γ.