The inverse spectral map for dimers (Q6072344)

This paper considers the spectral transform of dimer clusters. Based on the paper [\textit{V. V. Fock}, ``Inverse spectral problem for GK integrable systems'', Preprint, \url{arXiv:1503.2015}] it is known that a spectral transform that associates an element in a dimer cluster integrable system to its spectral data is birational. V. V. Fock does this by constructing an inverse map with theta functions on Jacobians of spectral curves. Here the authors provide an alternative version of the inverse map using only rational functions of the spectral data. Planar dimer modeling arises in classical statistical mechanics, and uses models that consider dimer covers of planar edge-weighted graphs. Associated with a dimer model on a bipartite graph on a torus is a Poisson variety with an integrable Hamiltonian system. Also associated with this system is an algebraic curve \(\mathcal{C}\) called the spectral curve and a divisor on this curve, which is a set of distinct points \((p_1, q_1), (p_2, q_2), \dots, (p_g, q_g)\) on \(\mathcal{C}\). An interested reader is advised to consult additional background material in [\textit{A. B. Goncharov} and \textit{R. Kenyon}, Ann. Sci. Éc. Norm. Supér. (4) 46, No. 5, 747--813 (2013; Zbl 1288.37025)]. The main result in this paper is a proof that the inverse map is given by an explicit rational expression depending on the divisor points of a certain open spectral curve \(\mathcal{C}^0\).