Moduli Spaces of Gauge Theories from Dimer Models: Proof of the   Correspondence
  Recently, a new way of deriving the moduli space of quiver gauge theories that arise on the world-volume of D3-branes probing singular toric Calabi-Yau cones was conjectured. According to the proposal, the gauge group, matter content and tree-level superpotential of the gauge theory is encoded in a periodic tiling, the dimer graph. The conjecture provides a simple procedure for determining the moduli space of the gauge theory in terms of perfect matchings.   For gauge theories described by periodic quivers that can be embedded on a two-dimensional torus, we prove the equivalence between the determination of the toric moduli space with a gauged linear sigma model and the computation of the Newton polygon of the characteristic polynomial of the dimer model. We show that perfect matchings are in one-to-one correspondence with fields in the linear sigma model. Furthermore, we prove that the position in the toric diagram of every sigma model field is given by the slope of the height function of the corresponding perfect matching. 
Sebastian Franco and David Vegh