Dimer Models, Integrable Systems and Quantum Teichmuller Space

Understand

We introduce a correspondence between dimer models (and hence superconformal quivers) and the quantum Teichmuller 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 nodes of valence greater than 3 (equivalently superpotential terms of order greater than 3 in the corresponding quiver gauge theories) must be decomposed by the introduction of 2-valent nodes.