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Calabi–Yau algebras and weighted quiver polyhedra

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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.

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  1. School of Mathematics and Statistics, Herschel Building, Newcastle University, Newcastle Upon Tyne, NE1 7RU, UK

    Raf Bocklandt

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  1. Raf Bocklandt
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Bocklandt, R. Calabi–Yau algebras and weighted quiver polyhedra. Math. Z. 273, 311–329 (2013). https://doi.org/10.1007/s00209-012-1006-z

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