Cluster integrable systems and spin chains

A. Marshakov, M. Semenyakin

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5 Citations (Scopus)


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 glN 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 glN -chain on M sites with the glM -chain on N sites. For these systems we construct explicitly a subgroup of the cluster mapping class group GQ 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 GQ.

Original languageEnglish
Article number100
JournalJournal of High Energy Physics
Issue number10
Publication statusPublished - 1 Oct 2019


  • Quantum Groups
  • Supersymmetric Gauge Theory


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