Calabi Yau algebras and weighted quiver polyhedra

TL;DR

This paper explores the construction of 3-dimensional Calabi-Yau algebras using dimer models and introduces weighted quiver polyhedra as a generalization to orbifolds, expanding the algebraic framework.

Contribution

It introduces the concept of toric orders and shows how all such orders that are 3-Calabi-Yau can be derived from dimer models on a torus, extending to orbifolds.

Findings

All 3-Calabi-Yau toric orders can be constructed from dimer models on a torus.

Weighted quiver polyhedra generalize dimer models to orbifolds.

The CY-3 condition implies the existence of weighted quiver polyhedra.

Abstract

Dimer models have been used in string theory to construct path algebras with relations that are 3-dimensional Calabi-Yau Algebras. These constructions result in algebras that share some specific properties: they are finitely generated modules over their centers and their representation spaces are toric varieties. In order to describe these algebras we introduce the notion of a toric order and show that all toric orders which are 3-dimensional Calabi-Yau algebras can be constructed from dimer models on a torus. Toric orders are examples of a much broader class of algebras: positively graded cancellation algebras. For these algebras the CY-3 condition implies the existence of a weighted quiver polyhedron, which is an extension of dimer models obtained by replacing the torus with any two-dimensional compact orientable orbifold.