Quantized Quiver Varieties and the Quantum Spin Ruijsenaars-Schneider Model

TL;DR

This paper constructs a quantum algebraic framework for the rational spin Ruijsenaars--Schneider model using quantum Hamiltonian reduction of quiver varieties, revealing new algebraic structures and conjecturing their limits.

Contribution

It introduces a quantized quiver variety associated with the model and identifies its algebraic structures, including a loop algebra and Yangian, with conjectures on their limits.

Findings

Construction of a quantized quiver variety $rakA_{N, ext{ell}}$ for the model

Identification of a loop algebra and Yangian within the algebra

A difference equation for eigenstates reducing to the spinless case when $ ext{ell}=1$

Abstract

This paper tackles the long-standing problem of quantizing the rational spin Ruijsenaars--Schneider model originating in the work of Krichever and Zabrodin. We make use of the technique of quantum Hamiltonian reduction to construct a quantized quiver variety $\mathfrak{A}_{N,\ell}$ associated to the framed Jordan quiver. This quantized quiver variety is simultaneously the algebra of quantum observables of the rational spin Ruijsenaars--Schneider model of $N$ particles with $\ell$ spin polarizations. Inside this algebra, we find a loop algebra and Yangian of $\mathfrak{gl}_\ell$ and conjecture that in the limit of infinitely many particles, the algebra $\mathfrak{A}_{N,\ell}$ becomes a shifted affine Yangian. We also exhibit a difference equation for eigenstates of the lowest Hamiltonian that reduces to the spinless case when $\ell=1$.