Cluster-Enriched Yang-Baxter Equation from SUSY Gauge Theories

TL;DR

This paper introduces a generalized Yang-Baxter equation incorporating cluster variables, constructed via supersymmetric gauge theories, linking R-matrices to 2D gauge theory partition functions and cluster algebra structures.

Contribution

It presents a novel generalization of the Yang-Baxter equation involving cluster variables, derived from the correspondence with supersymmetric gauge theories.

Findings

Constructed solutions to the generalized Yang-Baxter equation.

Linked R-matrices to 2D supersymmetric gauge theory partition functions.

Identified FI parameters with cluster y-variables.

Abstract

We propose a new generalization of the Yang-Baxter equation, where the R-matrix depends on cluster $y$-variables in addition to the spectral parameters. We point out that we can construct solutions to this new equation from the recently-found correspondence between Yang-Baxter equations and supersymmetric gauge theories. The $S^2$ partition function of a certain 2d $\mathcal{N}=(2,2)$ quiver gauge theory gives an R-matrix, whereas its FI parameters can be identified with the cluster $y$-variables.