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  • Letter
  • Open Access

Birational transformations on dimer integrable systems

Minsung Kho*

Norton Lee†

Rak-Kyeong Seong‡

  • *Contact author: minsung@unist.ac.kr
  • †Contact author: norton.lee@ibs.re.kr
  • ‡Contact author: seong@unist.ac.kr
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Phys. Rev. D 112, L041901 – Published 4 August, 2025

DOI: https://doi.org/10.1103/1mll-95cs

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 C3/Z4×Z2 with orbifold action (1, 0, 3)(0, 1, 1) and C/Z2×Z2 with action (1, 0, 0, 1)(0, 1, 1, 0), whose spectral curves and Hamiltonians are shown to be related by a birational transformation.

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Outline

Information

Phys. Rev. D 112, L041901– Published 4 August, 2025
  • Received 30 April 2025
  • Accepted 16 July 2025
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