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We introduce a correspondence between dimer models (and hence superconformal quivers) and the quantum Teichmüller space of the Riemann surfaces associated to them by mirror symmetry. Via the untwisting map, every brane tiling gives rise to a tiling of the Riemann surface with faces surrounding punctures. We explain how to obtain an ideal triangulation by dualizing this tiling. In order to do so, tiling...
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