44 pages, 24 figures, comments welcome. Updated the result Theorem B to require strong consistency throughout, added Lemma 7.12, and made other minor changes
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.
Tata Institute of Fundamental Research - Centre for Applicable Mathematics（应用数学中心，塔塔基础研究所）；Massachusetts Institute of Technology（麻省理工学院）；ETH Zürich（苏黎世联邦理工学院）；Yale University（耶鲁大学）
Twists, Higher Dimer Covers, and Web Duality for Grassmannian Cluster Algebras
University of California, Riverside（加州大学河滨分校）；University of Minnesota, Twin Cities（明尼苏达大学双城分校）；University of California, Berkeley（加州大学伯克利分校）；Johns Hopkins University（约翰斯·霍普金斯大学）
