Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 030, 53 pages      arXiv:1903.01636      https://doi.org/10.3842/SIGMA.2022.030

Deformations of Dimer Models

Akihiro Higashitani a and Yusuke Nakajima b
a) Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Osaka 565-0871, Japan
b) Department of Mathematics, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-Ku, Kyoto, 603-8555, Japan

Received August 06, 2021, in final form April 10, 2022; Published online April 16, 2022

Abstract
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 ${\mathbb Q}$-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 $P$ there exists a dimer model having $P$ 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.

Key words: dimer models; combinatorial mutation of polygons; mirror symmetry.

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