Hypergeometric Systems in two Variables, Quivers, Dimers and Dessins d'Enfants

This paper presents some parallel developments in Quiver/Dimer Models, Hypergeometric Systems and Dessins d'Enfants. The setting in which Gelfand, Kapranov and Zelevinsky have formulated the theory of hypergeometric systems, provides also a natural setting for dimer models. The Fast Inverse Algorithm, the untwisting procedure and the Kasteleyn matrix for dimer models are recasted in this setting. There is a relation between triangulations in GKZ theory and some perfect matchings in the dimer models, so that the secondary polygon coincides with the Newton polygon of the Kasteleyn determinant. Finally it is observed in examples and conjectured to hold in general, that the determinant of the Kasteleyn matrix with suitable weights becomes after a simple transformation equal to the principal A-determinant in GKZ theory.