Dimer algebras, ghor algebras, and cyclic contractions

TL;DR

This paper characterizes ghor algebras from dimer quivers on surfaces, showing when they are dimer algebras and classifying their simple modules, with implications for physics and algebraic structures.

Contribution

It provides a complete classification of ghor algebras on tori, linking noetherian property to being a dimer algebra and describing their centers explicitly.

Findings

A ghor algebra on a torus is a dimer algebra if and only if it is noetherian.

Non-noetherian ghor algebras are quotients of dimer algebras by homotopy relations.

Explicit description of the center of ghor algebras using perfect matchings.

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.