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

TL;DR

This paper explores the deep connections between hypergeometric systems, quiver/dimer models, and dessins d'enfants, revealing how these mathematical structures interrelate through algorithms, triangulations, and determinants.

Contribution

It introduces a unified framework linking hypergeometric systems with dimer models and dessins d'enfants, including new insights into the Kasteleyn matrix and principal A-determinant relations.

Findings

Kasteleyn matrix determinant relates to the principal A-determinant in GKZ theory.

Triangulations in GKZ theory correspond to perfect matchings in dimer models.

The secondary polygon matches the Newton polygon of the Kasteleyn determinant.

Abstract

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.