Calabi Yau Algebras and Weighted Quiver Polyhedra

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 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 category algebras with cancellation. For this broader class the CY-3 condition also 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. We discuss which of these quiver polyhedra give rise to Calabi Yau algebras.