A Note on Dimer Models and D-brane Gauge Theories

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.