Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spin Ruijsenaars-Schneider models are Coulomb branches

Gleb Arutyunov, Lukas Hardi

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.

Spin Ruijsenaars-Schneider models are Coulomb branches

Abstract

In this paper, we show that the Poisson algebras of cohomological and -theoretic Coulomb branches of 3d 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 -theory, quantum toroidal) superintegrability structure manifest. We conjecture that the Poisson algebras of elliptic Coulomb branches similarly reproduce the elliptic spin Ruijsenaars-Schneider model.
Paper Structure (17 sections, 21 theorems, 85 equations)

This paper contains 17 sections, 21 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Cohomological and $K$-theoretic Coulomb branches
  3. Rational spin RS models are cohomological Coulomb branches
  4. GKLO representation of the cohomological Coulomb branch
  5. Affine Yangian of $\mathfrak{gl}_\ell$
  6. $L$-operator algebra
  7. Superintegrability
  8. Equations of motion
  9. Poisson bracket of the spin variables
  10. Hyperbolic spin RS models are $K$-theoretic Coulomb branches
  11. GKLO representation of the $K$-theoretic Coulomb branch
  12. Quantum toroidal algebra of $\mathfrak{gl}_\ell$
  13. $L$-operator algebra
  14. Equations of motion
  15. Poisson bracket of the spin variables
  16. ...and 2 more sections

Key Result

Proposition 3.2

There is an injective homomorphism $\psi\colon \mathfrak{C}_{N,\ell} \to \mathfrak{A}_{N,\ell}/\gamma \mathfrak{A}_{N,\ell}$ of Poisson algebras that sends

Theorems & Definitions (45)

  • Definition 2.1
  • Remark
  • Definition 2.2
  • Definition 3.1
  • Proposition 3.2
  • Remark
  • proof
  • Proposition 3.3
  • Remark
  • Corollary 3.4
  • ...and 35 more