Cluster-Enriched Yang-Baxter Equation from SUSY Gauge Theories

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.