Abstract
Given a ribbon graph $\Gamma$ with some extra structure, we define, using
constructible sheaves, a dg category $CPM(\Gamma)$ meant to model the Fukaya
category of a Riemann surface in the cell of Teichm"uller 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 $X_\Gamma$ with toric components. We prove that our
model is equivalent to $Perf(X_\Gamma)$, the dg category of perfect
complexes on $X_\Gamma$.
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