Consistent Dimer Models on Surfaces with Boundary

TL;DR

This paper introduces new notions of consistency for dimer models on surfaces with boundary, linking combinatorial properties to algebraic structures and categorification of cluster algebras.

Contribution

It defines weak and strong consistency conditions for dimer models on general surfaces and explores their algebraic and categorical implications, extending known results from disk and torus cases.

Findings

Weak consistency is equivalent to absence of bad configurations.

Strong consistency implies bimodule 3-Calabi-Yau property of associated algebras.

Consistency conditions facilitate reduction and submodel analysis.

Abstract

A dimer model is a quiver with faces embedded in a surface. We define and investigate notions of consistency for dimer models on general surfaces with boundary which restrict to well-studied consistency conditions in the disk and torus case. We define weak consistency in terms of the associated dimer algebra and show that it is equivalent to the absence of bad configurations on the strand diagram. In the disk and torus case, weakly consistent models are nondegenerate, meaning that every arrow is contained in a perfect matching; this is not true for general surfaces. Strong consistency is defined to require weak consistency as well as nondegeneracy. We prove that the completed as well as the noncompleted dimer algebra of a strongly consistent dimer model are bimodule internally 3-Calabi-Yau with respect to their boundary idempotents. As a consequence, the Gorenstein-projective module category of the completed boundary algebra of suitable dimer models categorifies the cluster algebra given by their underlying quiver. We provide additional consequences of weak and strong consistency, including that one may reduce a strongly consistent dimer model by removing digons and that consistency behaves well under taking dimer submodels.