Birational transformations on dimer integrable systems

We show that when two toric Calabi-Yau 3-folds and their corresponding toric varieties are related by a birational transformation, they are associated with a pair of dimer models on the 2-torus that define dimer integrable systems, which themselves become birationally equivalent. Under this birational equivalence, the Casimirs and the Hamiltonias as well as the spectral curve and the Poisson structure formed by the cluster variables of the dimer integrable systems are identified to each other. These integrable systems defined by dimer models were first introduced by Goncharov and Kenyon. We illustrate this equivalence explicitly using a pair of dimer integrable systems corresponding to the Abelian orbifolds of the form <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi mathvariant="double-struck">C</a:mi><a:mn>3</a:mn></a:msup><a:mo>/</a:mo><a:msub><a:mi mathvariant="double-struck">Z</a:mi><a:mn>4</a:mn></a:msub><a:mo>×</a:mo><a:msub><a:mi mathvariant="double-struck">Z</a:mi><a:mn>2</a:mn></a:msub></a:math> with orbifold action (1, 0, 3)(0, 1, 1) and <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mi mathvariant="script">C</f:mi><f:mo>/</f:mo><f:msub><f:mi mathvariant="double-struck">Z</f:mi><f:mn>2</f:mn></f:msub><f:mo>×</f:mo><f:msub><f:mi mathvariant="double-struck">Z</f:mi><f:mn>2</f:mn></f:msub></f:math> with action (1, 0, 0, 1)(0, 1, 1, 0), whose spectral curves and Hamiltonians are shown to be related by a birational transformation.