# Dimer models on cylinders over Dynkin diagrams and cluster algebras

## TL;DR

This paper extends dimer model theory to cylinders over Dynkin diagrams, linking them to cluster algebras and proving rigidity of superpotentials for certain quivers in Kac--Moody algebras.

## Contribution

It introduces a general framework for dimer models on cylinders over Dynkin diagrams and connects them to Berenstein--Fomin--Zelevinsky quivers in Kac--Moody algebra theory.

## Key findings

- Dimer models on cylinders over Dynkin diagrams generalize known disc models.
- All Berenstein--Fomin--Zelevinsky quivers for Schubert cells produce such dimer models.
- Superpotentials associated with these quivers are proven to be rigid.

## Abstract

In this paper, we describe a general setting for dimer models on cylinders over Dynkin diagrams which in type A reduces to the well studied case of dimer models on a disc. We prove that all Berenstein--Fomin--Zelevinsky quivers for Schubert cells in a symmetric Kac--Moody algebra give rise to dimer models on the cylinder over the corresponding Dynkin diagram. We also give an independent proof of a result of Buan, Iyama, Reiten and Smith that the corresponding superpotentials are rigid using the dimer model structure of the quivers.