Quantized Quiver Varieties and the Quantum Spin Ruijsenaars–Schneider Model

Abstract This paper tackles the long-standing problem of quantizing the rational spin Ruijsenaars–Schneider model originating in the work of Krichever and Zabrodin (Russ Math Surv 50:1101, 1995. arXiv:hep-th/9505039 ). We make use of the technique of quantum Hamiltonian reduction to construct a quantized quiver variety $$\mathfrak {A}_{N,\ell }$$ A N , ℓ , which is simultaneously the algebra of quantum observables of the rational spin Ruijsenaars–Schneider model of N particles with $$\ell $$ ℓ spin polarizations. Inside this algebra, we find a loop algebra and Yangian of $$\mathfrak {gl}_\ell $$ gl ℓ and conjecture that the algebra $$\mathfrak {A}_{N,\ell }$$ A N , ℓ can be identified with a truncated Yangian of affine type $$A_{\ell -1}^{(1)}$$ A ℓ - 1 ( 1 ) . Finally, we use the commutation relations inside $$\mathfrak {A}_{N,\ell }$$ A N , ℓ to derive a difference equation for eigenstates of the lowest Hamiltonian that reproduces the known quantization of the spinless case when $$\ell =1$$ ℓ = 1 .