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Dimer algebras, ghor algebras, and cyclic contractions

Charlie Beil

TL;DR

This work studies ghor algebras $\Lambda$ arising from dimer quivers on a torus, clarifying when $\Lambda$ is itself a dimer algebra and how it relates to cancellative dimer algebras via cyclic contractions. It develops a framework of impressions $\tau_\psi$ to classify simple $\Lambda$-modules of maximal dimension and describes the center in terms of vertex corner rings, linking the nonnoetherian setting to well-understood cancellative cases. The main result identifies $\Lambda$ with the quotient of a dimer algebra $A$ by non-cancellative relations $\langle p-q\rangle$, and shows how Higgsing-inspired cyclic contractions reduce complex, non-cancellative structures to tractable cancellative models while preserving cyclic data. The work bridges noncancellative quiver gauge theories with their abelian (toric) counterparts, providing explicit descriptions of centers and representation theory and connecting to the mesonic chiral ring via the cycle algebra.

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

Dimer algebras, ghor algebras, and cyclic contractions

TL;DR

This work studies ghor algebras arising from dimer quivers on a torus, clarifying when is itself a dimer algebra and how it relates to cancellative dimer algebras via cyclic contractions. It develops a framework of impressions to classify simple -modules of maximal dimension and describes the center in terms of vertex corner rings, linking the nonnoetherian setting to well-understood cancellative cases. The main result identifies with the quotient of a dimer algebra by non-cancellative relations , and shows how Higgsing-inspired cyclic contractions reduce complex, non-cancellative structures to tractable cancellative models while preserving cyclic data. The work bridges noncancellative quiver gauge theories with their abelian (toric) counterparts, providing explicit descriptions of centers and representation theory and connecting to the mesonic chiral ring via the cycle algebra.

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

Paper Structure

This paper contains 8 sections, 39 theorems, 159 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algebra homomorphisms from perfect matchings
  4. Impressions
  5. Cyclic contractions
  6. Cycle structure
  7. Proof of main theorem
  8. A brief account of Higgsing with quivers

Key Result

Theorem 1.1

(Theorems first main and impression prop.) Let $Q$ be a nondegenerate dimer quiver on a torus, and fix a cyclic contraction $\psi: A \to A'$. Let $\mathcal{P}_0 \subseteq \mathcal{P}$ be the perfect matchings $x$ of $Q$ for which $\psi(x)$ is a simple matching of $Q'$. Set $B := k[\mathcal{P}_0]$.

Figures (15)

  • Figure 1: Some cyclic contractions. Each quiver is drawn on a torus, and $\psi$ contracts the green arrows.
  • Figure 2: The contraction of a unit cycle (drawn in green) to a vertex. Such a contraction cannot induce a contraction of dimer algebras $\psi: A \to A'$ if $A'$ has a perfect matching, by Lemma \ref{['cannot contract']}.
  • Figure 3: Setup for Lemma \ref{['here2']}.1. The paths $p = p_m \cdots p_1$, $q = q_n \cdots q_1$, $p' = p' _1\cdots p'_m$, and $q' = q'_1 \cdots q'_n$ are drawn in red, blue, purple, and violet respectively. Each product $p'_{\alpha}p_{\alpha}$ and $q'_{\beta}q_{\beta}$ is a unit cycle. Note that the region $\mathcal{R}_{p',q'}$ is properly contained in the region $\mathcal{R}_{p,q}$.
  • Figure 4: Cases for Lemma \ref{['permanent 2-cycles']}. In each case, $a$ and $b$ are arrows, and $p$ and $q$ are paths. In case (i) $ap$, $bq$, $ab$ are unit cycles; in cases (ii) and (iii) $qbpa$, $ab$ are unit cycles, and $p$, $q$ are cycles. (In case (ii), $q$ may be a vertex, and $q$ need not be a unit cycle, that is, $q^+$ may contain arrows in its interior.) In case (i) $ab$ is a removable 2-cycle, and in cases (ii) and (iii) $ab$ is a permanent 2-cycle.
  • Figure 5: Setup for Remark \ref{['weird']}. Let $m \geq 1$. Then the path $p := baq^{m+1}dc$ satisfies $p \sigma_i^m = \sigma_i$.
  • ...and 10 more figures

Theorems & Definitions (92)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Remark 2.5
  • Definition 2.6
  • Definition 3.1
  • Definition 3.2
  • ...and 82 more