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Calabi–Yau algebras and weighted quiver polyhedra

Lookup NU author(s): Dr Rafael Bocklandt

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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.

Publication metadata

Author(s): Bocklandt R

Publication type: Article

Publication status: Published

Journal: Mathematische Zeitschrift

Year: 2013

Volume: 273

Issue: 1-2

Pages: 311-329

Print publication date: 22/03/2012

ISSN (print): 0025-5874

ISSN (electronic): 1432-1823

Publisher: Springer

URL: http://dx.doi.org/10.1007/s00209-012-1006-z

DOI: 10.1007/s00209-012-1006-z

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