Classification and Birational Equivalence of Dimer Integrable Systems for Reflexive Polygons

TL;DR

This work delivers a complete classification of dimer integrable systems tied to the 30 brane tilings whose toric Calabi–Yau 3-folds correspond to the 16 reflexive polygons in two dimensions. For each reflexive-polygon–associated dimer integrable system, the authors provide the Casimirs, the single Hamiltonian built from 1-loops, the spectral curve, and the Poisson commutation relations, and they map all birational equivalences across the classification by explicitly presenting the transformations that align Casimirs, Hamiltonians, spectral curves, and Poisson structures. In total, 16 birationally equivalent pairs are identified, which together with Seiberg duality yield 5 buckets (equivalence classes) of dimer integrable systems. The study further shows that deformations of brane tilings, including mass deformations, correspond to these birational transformations, leaving invariant the mesonic moduli space’s generator count and the U(1)$_R$-refined Hilbert series, thus mirroring phenomena observed for higher-dimensional brane brick models. Together, these results illuminate the intricate web linking toric geometry, dimer integrable systems, and gauge theory dualities, providing a concrete, computable atlas of birational relations within this reflexive-polygon sector.

Abstract

Brane tilings are bipartite periodic graphs on the 2-torus and realize a large family of 4d N=1 supersymmetric gauge theories corresponding to toric Calabi-Yau 3-folds. We present a complete classification of dimer integrable systems corresponding to the 30 brane tilings whose toric Calabi-Yau 3-folds are given by the 16 reflexive polygons in 2 dimensions. For each dimer integrable system associated to a reflexive polygon, we present the Casimirs, the single Hamiltonian built from 1-loops, the spectral curve, and the Poisson commutation relations. We also identify all birational equivalences between dimer integrable systems in this classification by presenting the birational transformations that match the Casimirs and the Hamiltonians as well as the spectral curves and Poisson structures between equivalent dimer integrable systems. In total, we identify 16 pairs of birationally equivalent dimer integrable systems which combined with Seiberg duality between the corresponding brane tilings form 5 distinct equivalence classes. Echoing phenomena observed for brane brick models realizing a family of 2d (0,2) supersymmetric gauge theories corresponding to toric Calabi-Yau 4-folds, we illustrate that deformations of brane tilings, including mass deformations, correspond to the birational transformations we discover in this work, and leave invariant the number of generators of the mesonic moduli space as well as the corresponding U(1)R-refined Hilbert series.

Figures (43)

• Figure 1: The 16 reflexive polygons in 2 dimensions with labels corresponding to the associated 30 brane tilings classified in Hanany:2012hi. Birational transformations between toric Calabi-Yau 3-folds correspond to birational equivalence between the associated dimer integrable systems. Combined with Seiberg duality, we identify 5 equivalence classes called buckets amongst the 30 brane tilings and dimer integrable systems classified in this work.
• Figure 2: The 30 brane tilings corresponding to the 16 reflexive polygons in dimension 2 are related by Seiberg duality (red), specular duality (blue), and birational transformations (yellow). Under Seiberg duality and under birational transformations, the associated dimer integrable systems are equivalent and form equivalence classes, which we call buckets. The labels correspond to the 30 brane tilings classified in Hanany:2012hi with the corresponding 16 reflexive toric diagrams given in Figure \ref{['fig_bucketsummary']}.
• Figure 3: (a) Seiberg duality on brane tilings Feng:2000miFeng:2001xrFeng:2002zwFeng:2001bnBeasley:2001zp is a local deformation of the bipartite graph on the 2-torus acting on square faces, which is also known as urban renewal or spider moves goncharov2012dimersclusterintegrablesystemsCIUCU1998341999math......3025K. (b) Specular duality on brane tilings Hanany:2012vc swaps directed paths along edges corresponding to zig-zag paths with directed paths around faces and vice versa while preserving intersections between these paths.
• Figure 4: (a) A product of perfect matchings weights $\overline{p}_a$ and $(\overline{p}_b)^{-1}$ in terms of directed edge variables, and (b) a product of closed directed paths given by permutations in $S_{2|E|}$ with a cancellation between a pair of directed edges.
• Figure 5: The brane tiling for the suspended pinch point (SPP) with chiral fields $X_{ij}$, node labels $w_j$ and $b_k$, and edge labels $e_{jk}$. The superpotential $W$ and the corresponding permutation tuples $\sigma_W^{-1}$ and $\sigma_B$ in terms of edge labels are also shown.