Nonlocality and the central geometry of dimer algebras

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.

This paper was reorganized and expanded, with new results, into the four papers: arXiv:1711.09771; arXiv:1805.08047; arXiv:1902.11299; and arXiv:1903.10460

References in corpus (2)

• Dimer models and cluster categories of Grassmannians
• Cancellativization of dimer models