Dimer models and toric diagrams

TL;DR

This work establishes a duality between quiver gauge theories and dimer models by tying toric diagram multiplicities to the coefficients in the Kasteleyn determinant $P(z,w)=\det K(z,w)$. It provides general, analytic formulas for multiplicities in orbifold cases, employs block-determinant techniques to handle high-rank orbifolds, and presents an algorithmic Higgsing framework to enumerate toric phases via partial resolutions. The approach yields both recovery of known results and new predictions for toric phases, offering a computationally efficient route to explore the phase structure of quiver theories and their string-theoretic embeddings. The methods illuminate the combinatorial origin of multiplicities and connect dimer models to toric geometry in a concrete, actionable way, with potential extensions to broader classes of CY manifolds and dualities.

Abstract

We propose a duality between quiver gauge theories and the combinatorics of dimer models. The connection is via toric diagrams together with multiplicities associated to points in the diagram (which count multiplicities of fields in the linear sigma model construction of the toric space). These multiplicities may be computed from both sides and are found to agree in all known examples. The dimer models provide new insights into the quiver gauge theories: for example they provide a closed formula for the multiplicities of arbitrary orbifolds of a toric space, and allow a new algorithmic method for exploring the phase structure of the quiver gauge theory.