Abstract
We discuss relation between the cluster integrable systems and spin chains in the context of their correspondence with 5d supersymmetric gauge theories. It is shown that \( {\mathfrak{gl}}_N \) XXZ-type spin chain on M sites is isomorphic to a cluster integrable system with N × M rectangular Newton polygon and N × M fundamental domain of a ‘fence net’ bipartite graph. The Casimir functions of the Poisson bracket, labeled by the zig-zag paths on the graph, correspond to the inhomogeneities, on-site Casimirs and twists of the chain, supplemented by total spin. The symmetricity of cluster formulation implies natural spectral duality, relating \( {\mathfrak{gl}}_N \) -chain on M sites with the \( {\mathfrak{gl}}_M \) -chain on N sites. For these systems we construct explicitly a subgroup of the cluster mapping class group \( {\mathcal{G}}_{\mathcal{Q}} \) and show that it acts by permutations of zig-zags and, as a consequence, by permutations of twists and inhomogeneities. Finally, we derive Hirota bilinear equations, describing dynamics of the tau-functions or A-cluster variables under the action of some generators of \( {\mathcal{G}}_{\mathcal{Q}} \).
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Marshakov, A., Semenyakin, M. Cluster integrable systems and spin chains. J. High Energ. Phys. 2019, 100 (2019). https://doi.org/10.1007/JHEP10(2019)100
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DOI: https://doi.org/10.1007/JHEP10(2019)100