Ribbon Graphs and Mirror Symmetry I

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