Ribbon graphs and mirror symmetry

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