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- W2946616307 abstract "A bstract 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 $$ {mathfrak{gl}}_N $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>gl</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> 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 $$ {mathfrak{gl}}_N $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>gl</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> -chain on M sites with the $$ {mathfrak{gl}}_M $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>gl</mml:mi> <mml:mi>M</mml:mi> </mml:msub> </mml:math> -chain on N sites. For these systems we construct explicitly a subgroup of the cluster mapping class group $$ {mathcal{G}}_{mathcal{Q}} $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>Q</mml:mi> </mml:msub> </mml:math> 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 $$ {mathcal{G}}_{mathcal{Q}} $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>Q</mml:mi> </mml:msub> </mml:math> ." @default.
- W2946616307 created "2019-05-29" @default.
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- W2946616307 date "2019-10-01" @default.
- W2946616307 modified "2023-09-27" @default.
- W2946616307 title "Cluster integrable systems and spin chains" @default.
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- W2946616307 doi "https://doi.org/10.1007/jhep10(2019)100" @default.
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