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Strebel Differentials and stable Matrix Factorizations

We study the connection between quadratic Strebel differentials on punctured surfaces and the construction of moduli spaces of matrix factorizations for dimer models using GIT-quotients. We show that for each consistent dimer model and each nondegenerate stability condition $θ$ we can find a Strebel differential for which the horizontal trajectories correspond to the $θ$-stable matrix factorizations and the vertical trajectories correspond to the arrows of the dimer quiver. We give explicit expressions for the $θ$-stable matrix factorizations that can be deduced from these horizontal trajectories. Following ideas by Pascaleff and Sybilla we show that each nondegenerate stability condition gives rise to a sheaf of curved algebras coming from consistent dimer models. The corresponding categories of matrix factorizations can be glued together to form the category of matrix factorizations of the original dimer.