  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.
