  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.
