Nonlocality and the central geometry of dimer algebras

TL;DR

This paper explores the properties of dimer algebras, revealing that non-cancellative cases have nonnoetherian centers with Krull dimension 3 and unique geometric features, connecting algebraic and geometric perspectives.

Contribution

It establishes the nonnoetherian nature of centers in non-cancellative dimer algebras and introduces new geometric insights and formalized concepts from gauge theory.

Findings

Non-cancellative dimer algebras have nonnoetherian centers.

The center has Krull dimension 3 and is generically noetherian.

The reduced center corresponds to a Gorenstein algebraic variety with a unique 'smeared-out' point.

Abstract

Let $A$ be a dimer algebra and $Z$ its center. It is well known that if $A$ is cancellative, then $A$ and $Z$ are noetherian and $A$ is a finitely generated $Z$-module. Here we show the converse: if $A$ is non-cancellative (as almost all dimer algebras are), then $A$ and $Z$ are nonnoetherian and $A$ is an infinitely generated $Z$-module. Although $Z$ is nonnoetherian, we show that it nonetheless has Krull dimension 3 and is generically noetherian. Furthermore, we show that the reduced center is the coordinate ring for a Gorenstein algebraic variety with the strange property that it contains precisely one 'smeared-out' point of positive geometric dimension. In our proofs we introduce formalized notions of Higgsing and the mesonic chiral ring from quiver gauge theory.