A Note on Dimer Models and D-brane Gauge Theories

TL;DR

This paper explores the relationship between dimer models and quiver gauge theories, proposing a simplified method to extract charge matrices and analyzing the construction of superpotentials for D-brane gauge theories on toric singularities.

Contribution

It introduces a conjecture that a subset of perfect matchings suffices to determine the charge matrix and performs explicit computations on partial resolutions using graph theory techniques.

Findings

A subset of perfect matchings can determine the charge matrix.

Explicit computations on orbifold resolutions using dimer models.

Analysis of symmetries in dimer models via string techniques.

Abstract

The connection between quiver gauge theories and dimer models has been well studied. It is known that the matter fields of the quiver gauge theories can be represented using the perfect matchings of the corresponding dimer model.We conjecture that a subset of perfect matchings associated with an internal point in the toric diagram is sufficient to give information about the charge matrix of the quiver gauge theory. Further, we perform explicit computations on some aspects of partial resolutions of toric singularities using dimer models. We analyse these with graph theory techniques, using the perfect matchings of orbifolds of the form $\BC^3/\Gamma$, where the orbifolding group $\Gamma$ may be noncyclic. Using these, we study the construction of the superpotential of gauge theories living on D-branes which probe these singularities, including the case where one or more adjoint fields are present upon partial resolution. Applying a combination of open and closed string techniques to dimer models, we also study some aspects of their symmetries.