Birational Transformations on Dimer Integrable Systems

Abstract

Dimers, also known as brane tilings, are bipartite periodic graphs on a 2-torus, that represent a Type IIB brane configuration in string theory, which realizes a family of 4-dimensional supersymmetric quiver gauge theories corresponding to toric Calabi-Yau 3-folds. By Goncharov and Kenyon, these dimer models have been shown to define also integrable systems. In this talk, we illustrate a recent discovery that when two toric Calabi-Yau 3-folds and their corresponding toric varieties are related by a birational transformation, the corresponding dimer models define two integrable systems, which are also birational equivalent. I illustrate this discovery with an explicit example and give also a brief overview on how this discovery can lead us to new results in the future.