Dimer algebras, ghor algebras, and cyclic contractions

Abstract

A ghor algebra is the path algebra of a dimer quiver on a surface, modulo relations that come from the perfect matchings of its quiver. Such algebras arise from abelian quiver gauge theories in physics. We show that a ghor algebra $\Lambda$ on a torus is a dimer algebra (a quiver with potential) if and only if it is noetherian, and otherwise $\Lambda$ is the quotient of a dimer algebra by homotopy relations. Furthermore, we classify the simple $\Lambda$-modules of maximal dimension and give an explicit description of the center of $\Lambda$ using a special subset of perfect matchings. In our proofs we introduce formalized notions of Higgsing and the mesonic chiral ring from quiver gauge theory.

41 pages. Replaces part of arXiv:1412.1750, with new results

References in corpus (5)

• Dimer models and toric diagrams
• M2-branes on Orbifolds of the Cone over Q^{1,1,1}
• Towards M2-brane Theories for Generic Toric Singularities
• Calabi-Yau Varieties: from Quiver Representations to Dessins d'Enfants
• Noetherian criteria for dimer algebras

Cited by in corpus (1)

• A generalization of cancellative dimer algebras to hyperbolic surfaces