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 ${\mathcal{S}}_k$.

9 pages, 2 figures, clarified quantum vs classical treatment, fixed typos