Deformations of Dimer Models
The combinatorial mutation of polygons, which transforms a given lattice polygon into another one, is an important operation to understand mirror partners for two-dimensional Fano manifolds, and the mutation-equivalent polygons give ℚ-Gorenstein deformation-equivalent toric varieties. On the other h...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Date: | 2022 |
| ISSN: | 1815-0659 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2022
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211638 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Deformations of Dimer Models. Akihiro Higashitani and Yusuke Nakajima. SIGMA 18 (2022), 030, 53 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862646523771224064 |
|---|---|
| author | Higashitani, Akihiro Nakajima, Yusuke |
| author_facet | Higashitani, Akihiro Nakajima, Yusuke |
| citation_txt | Deformations of Dimer Models. Akihiro Higashitani and Yusuke Nakajima. SIGMA 18 (2022), 030, 53 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The combinatorial mutation of polygons, which transforms a given lattice polygon into another one, is an important operation to understand mirror partners for two-dimensional Fano manifolds, and the mutation-equivalent polygons give ℚ-Gorenstein deformation-equivalent toric varieties. On the other hand, for a dimer model, which is a bipartite graph described on the real two-torus, one can assign a lattice polygon called the perfect matching polygon. It is known that for each lattice polygon , there exists a dimer model having as the perfect matching polygon and satisfying certain consistency conditions. Moreover, a dimer model has rich information regarding toric geometry associated with the perfect matching polygon. In this paper, we introduce a set of operations which we call deformations of consistent dimer models, and show that the deformations of consistent dimer models realize the combinatorial mutations of the associated perfect matching polygons.
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| first_indexed | 2026-03-15T11:15:08Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211638 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T11:15:08Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Higashitani, Akihiro Nakajima, Yusuke 2026-01-07T13:43:11Z 2022 Deformations of Dimer Models. Akihiro Higashitani and Yusuke Nakajima. SIGMA 18 (2022), 030, 53 pages 1815-0659 2020 Mathematics Subject Classification: 52B20; 14M25; 14J33 arXiv:1903.01636 https://nasplib.isofts.kiev.ua/handle/123456789/211638 https://doi.org/10.3842/SIGMA.2022.030 The combinatorial mutation of polygons, which transforms a given lattice polygon into another one, is an important operation to understand mirror partners for two-dimensional Fano manifolds, and the mutation-equivalent polygons give ℚ-Gorenstein deformation-equivalent toric varieties. On the other hand, for a dimer model, which is a bipartite graph described on the real two-torus, one can assign a lattice polygon called the perfect matching polygon. It is known that for each lattice polygon , there exists a dimer model having as the perfect matching polygon and satisfying certain consistency conditions. Moreover, a dimer model has rich information regarding toric geometry associated with the perfect matching polygon. In this paper, we introduce a set of operations which we call deformations of consistent dimer models, and show that the deformations of consistent dimer models realize the combinatorial mutations of the associated perfect matching polygons. The authors would like to thank Alexander Kasprzyk for valuable lectures and discussions on the combinatorial mutation of Fano polygons. The authors would also like to thank the anonymous referees for their numerous valuable comments and suggestions. The first author is supported by JSPS Grant-in-Aid for Scientific Research (C) 20K03513. The second author was supported by the World Premier International Research Center Initiative (WPI initiative), MEXT, Japan, and is supported by JSPS Grant-in-Aid for Early-Career Scientists 20K14279. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Deformations of Dimer Models Article published earlier |
| spellingShingle | Deformations of Dimer Models Higashitani, Akihiro Nakajima, Yusuke |
| title | Deformations of Dimer Models |
| title_full | Deformations of Dimer Models |
| title_fullStr | Deformations of Dimer Models |
| title_full_unstemmed | Deformations of Dimer Models |
| title_short | Deformations of Dimer Models |
| title_sort | deformations of dimer models |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211638 |
| work_keys_str_mv | AT higashitaniakihiro deformationsofdimermodels AT nakajimayusuke deformationsofdimermodels |