Cluster integrable systems and spin chains

TL;DR

This paper explores the deep connection between cluster integrable systems and spin chains, revealing their equivalence, spectral duality, and symmetries, with implications for understanding 5d supersymmetric gauge theories.

Contribution

It establishes an explicit isomorphism between $rak{gl}_N$ XXZ spin chains and cluster integrable systems with specific Newton polygons, and constructs their symmetry groups.

Findings

$rak{gl}_N$ XXZ spin chain is isomorphic to a cluster integrable system.

Spectral duality relates $rak{gl}_N$-chains on $M$ sites to $rak{gl}_M$-chains on $N$ sites.

Explicit subgroup of the cluster mapping class group acts by permutations of zig-zags.

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 $N \times M$ rectangular Newton polygon and $N \times M$ 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 $\mathcal{G}_\mathcal{Q}$.