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 $$\mathfrak {A}_{N,\ell }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ℓ</mml:mi> </mml:mrow> </mml:msub> </mml:math> , which is simultaneously the algebra of quantum observables of the rational spin Ruijsenaars–Schneider model of N particles with $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> spin polarizations. Inside this algebra, we find a loop algebra and Yangian of $$\mathfrak {gl}_\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>gl</mml:mi> <mml:mi>ℓ</mml:mi> </mml:msub> </mml:math> and conjecture that the algebra $$\mathfrak {A}_{N,\ell }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ℓ</mml:mi> </mml:mrow> </mml:msub> </mml:math> can be identified with a truncated Yangian of affine type $$A_{\ell -1}^{(1)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> . Finally, we use the commutation relations inside $$\mathfrak {A}_{N,\ell }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ℓ</mml:mi> </mml:mrow> </mml:msub> </mml:math> to derive a difference equation for eigenstates of the lowest Hamiltonian that reproduces the known quantization of the spinless case when $$\ell =1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> .