Quantized Quiver Varieties and the Quantum Spin Ruijsenaars-Schneider Model

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 ${\mathfrak{A}}_{N,\mathrm{\ell }}$ 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 $\mathrm{\ell }$ spin polarizations. Inside this algebra, we find a loop algebra and Yangian of ${\mathfrak{g}\mathfrak{l}}_{\mathrm{\ell }}$ and conjecture that in the limit of infinitely many particles, the algebra ${\mathfrak{A}}_{N,\mathrm{\ell }}$ becomes a shifted affine Yangian. We also exhibit a difference equation for eigenstates of the lowest Hamiltonian that reduces to the spinless case when $\mathrm{\ell }=1$.