Spin Ruijsenaars-Schneider models are Coulomb branches

TL;DR

This paper establishes a connection between Coulomb branch algebras in 3d gauge theories and integrable spin Ruijsenaars-Schneider models, revealing new algebraic structures underlying these integrable systems.

Contribution

It demonstrates that Coulomb branch Poisson algebras encode the dynamics of spin Ruijsenaars-Schneider models, linking gauge theory and integrable systems in a novel way.

Findings

Poisson algebras reproduce equations of motion for rational and hyperbolic models

Monopole operators in GKLO representation elucidate superintegrability

Conjecture extends the framework to elliptic models

Abstract

In this paper, we show that the Poisson algebras of cohomological and $K$-theoretic Coulomb branches of 3d $\mathcal{N}=4$ necklace quiver gauge theories provide Poisson structures and Hamiltonians that reproduce the equations of motion of the rational and hyperbolic spin Ruijsenaars-Schneider models, respectively. The construction is carried out in terms of monopole operators in the GKLO representation, also making the affine Yangian (and, in $K$-theory, quantum toroidal) superintegrability structure manifest. We conjecture that the Poisson algebras of elliptic Coulomb branches similarly reproduce the elliptic spin Ruijsenaars-Schneider model.