Work

Birational transformations on dimer integrable systems

Year: 2025

Type: article

Abstract: We show that when two toric Calabi-Yau 3-folds and their corresponding toric varieties are related by a birational transformation, they are associated with a pair of dimer models on the 2-torus that define dimer integrable systems, which themselves become birationally equivalent. Under this birational equivalence, the Casimirs and the Hamiltonias as well as the spectral curve and the Poisson structure formed by the cluster variables of the dimer integrable systems are identified to each other. These integrable systems defined by dimer models were first introduced by Goncharov and Kenyon. We illustrate this equivalence explicitly using a pair of dimer integrable systems corresponding to the Abelian orbifolds of the form C 3 / Z 4 × Z 2 with orbifold action (1, 0, 3)(0, 1, 1) and C / Z 2 × Z 2 with action (1, 0, 0, 1)(0, 1, 1, 0), whose spectral curves and Hamiltonians are shown to be related by a birational transformation.

Source: Physical review. D/Physical review. D.

DOI: 10.1103/1mll-95cs

Authors: Minsung Kho, Norton Lee, Rak-Kyeong Seong

Institutions: Ulsan National Institute of Science and Technology, Institute for Basic Science

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Funders: BK21 FOUR Future Veterinary Medicine Leading Education and Research Center, College of Veterinary Medicine, Seoul National University, National Research Foundation of Korea, Ministry of Education, Science and Technology, Institute for Basic Science

Awards: G3472888953, G3598130303, G670587537