Dimer Models, Integrable Systems and Quantum Teichmuller Space

TL;DR

This paper establishes a correspondence between dimer models, superconformal quivers, and the quantum Teichmuller space, revealing how tilings and triangulations relate to integrable systems and quantum geometry.

Contribution

It introduces a novel framework connecting dimer models with quantum Teichmuller space, including explicit examples and a generalization of Fock coordinates for complex dimer models.

Findings

Correspondence between dimer models and quantum Teichmuller space.

Explicit examples illustrating the construction process.

Generalization of Fock coordinates for non-3-valent nodes.

Abstract

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. From a quiver gauge theory perspective, this operation corresponds to integrating-in massive fields. Fock coordinates in Teichmuller space are in one-to-one correspondence with chiral fields in the quiver. We present multiple explicit examples, including infinite families of theories, illustrating how the right number of Fock coordinates is generated by this procedure. Finally, we explain how Chekhov and Fock commutation relations between coordinates give rise to the commutators associated to dimer models by Goncharov and Kenyon in the context of quantum integrable systems. For generic dimer models (i.e. those containing nodes that are not 3-valent), this matching requires the introduction of a natural generalization of Chekhov and Fock rules. We also explain how urban renewal in the original brane tiling (Seiberg duality for the quivers) is mapped to flips of the ideal triangulation.