Elliptic spin Ruijsenaars-Schneider integrable models from 5d $\mathcal{N}=1$ gauge theories

Abstract

We show that the Coulomb branches of 5d $\mathcal{N}=1$ necklace quiver gauge theories on $\mathbb{R}^3 \times T^2$ are identified with the phase spaces of elliptic spin Ruijsenaars-Schneider models with dynamical inhomogeneities. This identification resolves the long-standing problem of determining a Poisson structure and a quantization of these models. We further show that the resulting quantum system is the integrable system governing supersymmetric indices of 4d $\mathcal{N}=1$ theories of class $\mathcal{S}_k$.