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Journal of High Energy Physics

, 2019:100 | Cite as

Cluster integrable systems and spin chains

  • A. Marshakov
  • M. SemenyakinEmail author
Open Access
Regular Article - Theoretical Physics
  • 34 Downloads

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 \( {\mathcal{G}}_{\mathcal{Q}} \).

Keywords

Quantum Groups Supersymmetric Gauge Theory 

Notes

Open Access

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Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Center for Advanced Studies, SkoltechMoscowRussia
  2. 2.Faculty of Mathematics, NRU HSEMoscowRussia
  3. 3.Institute for Theoretical and Experimental PhysicsMoscowRussia
  4. 4.Theory Department of Lebedev Physics InstituteMoscowRussia

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