Dimers and Beauville integrable systems

Abstract

Associated to a convex integral polygon $N$ in the plane are two integrable systems: the cluster integrable system of Goncharov and Kenyon, constructed from the dimer model on bipartite torus graphs, and the Beauville integrable system associated with the toric surface of $N$. These two systems are related by a birational map called the spectral transform. In this paper we study the case when $N$ is the standard triangle of side length $d$, equivalently when the toric surface is $¶^2$, and prove that the spectral transform is a birational isomorphism of integrable systems. Since the Hamiltonians are identified by construction, the essential content is that the spectral transform intertwines the two Poisson structures. In particular, this shows that Beauville integrable systems admit cluster algebra structures.

Figures (8)

• Figure 1: An illustration of the Beauville integrable system on ${\mathbb P}^2$ associated with the standard triangle $N$ of side length $d=4$, for which $g=3$ (cf. OkounkovTalk). The three black lines are the three coordinate axes of ${\mathbb P}^2$ forming the toric boundary $D$, the blue curve is the spectral curve $C$, the red points are the divisor points $(p_i,q_i)_{i=1}^3$, and the green points are the boundary points $C\cap D$.
• Figure 2: The $3\times 3$ fundamental domain of the hexagonal lattice and a zig-zag path in $\bm \alpha_2$.
• Figure 3: The ribbon structure induced by the embedding of $\Gamma$ in ${\mathbb T}$ (left) and the conjugate surface $\widehat{S}$, obtained by reversing the cyclic order at each black vertex (right). The red paths illustrate the bijection between zig-zag paths of $\Gamma$ and boundary components of the conjugate surface (equivalently, faces of $\Gamma$ in $\widehat{S}$).
• Figure 4: A ${\mathbb C}^\times$-local system $\operatorname{wt}$ (red) on a $2 \times 2$ fundamental domain for the honeycomb lattice along with the $z$ and $w$ factors (green).
• Figure 5: The regularized Kasteleyn matrix $\widetilde{\mathsf K}$ on the dense torus (left) and its homogeneous form on $\widetilde{{\mathbb P}}^2$ (right).

Theorems & Definitions (60)

• Theorem 1.1
• Theorem 2.1: Kast
• Example 2.2
• Theorem 2.3: GK13
• Remark 3.1
• Remark 3.2
• Definition 3.3
• Theorem 3.4: Beauville
• Theorem 4.1
• Remark 4.2