Deformations of Dimer Models

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2022
• ISSN: 1815-0659
• Hauptverfasser: Higashitani, Akihiro, Nakajima, Yusuke
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Інститут математики НАН України 2022
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/211638
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: Deformations of Dimer Models. Akihiro Higashitani and Yusuke Nakajima. SIGMA 18 (2022), 030, 53 pages

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.