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.