Consistent dimer models on surfaces with boundary
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 dime...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425000180/type/journal_article |
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| author | Jonah Berggren Khrystyna Serhiyenko |
| author_facet | Jonah Berggren Khrystyna Serhiyenko |
| author_sort | Jonah Berggren |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-1cb7dbbbefd9485f88aff45200121cdf |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-1cb7dbbbefd9485f88aff45200121cdf2025-08-20T03:28:51ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2025.18Consistent dimer models on surfaces with boundaryJonah Berggren0Khrystyna Serhiyenko1University of Kentucky; E-mail:University of KentuckyA 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.https://www.cambridge.org/core/product/identifier/S2050509425000180/type/journal_article16G2013F6018N2514D21 |
| spellingShingle | Jonah Berggren Khrystyna Serhiyenko Consistent dimer models on surfaces with boundary Forum of Mathematics, Sigma 16G20 13F60 18N25 14D21 |
| title | Consistent dimer models on surfaces with boundary |
| title_full | Consistent dimer models on surfaces with boundary |
| title_fullStr | Consistent dimer models on surfaces with boundary |
| title_full_unstemmed | Consistent dimer models on surfaces with boundary |
| title_short | Consistent dimer models on surfaces with boundary |
| title_sort | consistent dimer models on surfaces with boundary |
| topic | 16G20 13F60 18N25 14D21 |
| url | https://www.cambridge.org/core/product/identifier/S2050509425000180/type/journal_article |
| work_keys_str_mv | AT jonahberggren consistentdimermodelsonsurfaceswithboundary AT khrystynaserhiyenko consistentdimermodelsonsurfaceswithboundary |