# 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.

## Key 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 $\mathcal{G}_\mathcal{Q}$.

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Source: https://tomesphere.com/paper/1905.09921