LAUSR

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 $$ {\mathfrak{gl}}_N $$ 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 $$ {\mathfrak{gl}}_N $$ gl N -chain on M sites with the $$ {\mathfrak{gl}}_M $$ gl M -chain on N sites. For these systems we construct explicitly a subgroup of the cluster mapping class group $$ {\mathcal{G}}_{\mathcal{Q}} $$ G 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 $$ {\mathcal{G}}_{\mathcal{Q}} $$ G Q .