Dimer Algebras, Ghor Algebras, and Cyclic Contractions

Abstract

AbstractA 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 $$\Lambda $$ Λ on a torus is a dimer algebra (a quiver with potential) if and only if it is noetherian, and otherwise $$\Lambda $$ Λ is the quotient of a dimer algebra by homotopy relations. Furthermore, we classify the simple $$\Lambda $$ Λ -modules of maximal dimension and give an explicit description of the center of $$\Lambda $$ Λ 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.