Abstract
We describe the correspondence of the Matsuo–Cherednik type between the quantum n-body Ruijsenaars–Schneider model and the quantum Knizhnik–Zamolodchikov equations related to supergroup GL(N|M). The spectrum of the Ruijsenaars–Schneider Hamiltonians is shown to be independent of the Z2-grading for a fixed value of N+M, so that N+M+1 different qKZ systems of equations lead to the same n-body quantum problem. The obtained results can be viewed as a quantization of the previously described quantum-classical correspondence between the classical n-body Ruijsenaars–Schneider model and the supersymmetric GL(N|M) quantum spin chains on n sites.
Highlights
The KZ-Calogero and qKZ-Ruijsenaars correspondences are the Matsuo-Cherednik type constructions [12, 10, 18, 19] for solutions of the Calogero-Moser-Sutherland [4] and RuijsenaarsSchneider [14] quantum problems by means of solutions of the Knizhnik-Zamolodchikov (KZ) [8] and quantum Knizhnik-Zamolodchikov equations [11] respectively
For example, the qKZ equations4 related to the Lie group GL(K): eη ∂xi Φ = Ki( ) Φ, i = 1, . . . , n, (1.1)
Using the asymptotics of solutions to the (q)KZ equations [15] it was argued in [18, 19] that the qKZ-Ruijsenaars correspondence can be viewed as a quantization of the quantum-classical duality [1, 7, 2], which relates the generalized inhomogeneous quantum spin chains and the classical Ruijsenaars-Schneider model
Summary
The KZ-Calogero and qKZ-Ruijsenaars correspondences are the Matsuo-Cherednik type constructions [12, 10, 18, 19] for solutions of the Calogero-Moser-Sutherland [4] and RuijsenaarsSchneider [14] quantum problems by means of solutions of the Knizhnik-Zamolodchikov (KZ) [8] and quantum Knizhnik-Zamolodchikov (qKZ) equations [11] respectively. Using the asymptotics of solutions to the (q)KZ equations [15] it was argued in [18, 19] that the qKZ-Ruijsenaars correspondence can be viewed as a quantization of the quantum-classical duality [1, 7, 2] (see [13, 5]), which relates the generalized inhomogeneous quantum spin chains and the classical Ruijsenaars-Schneider model. We construct generalizations of the vector Ω (1.8) and show that the quantum K-body Ruijsenaars-Schneider model follows from all K + 1 qKZ systems of equations related to the supergroups GL(N|M) with N + M = K (1.17). The skew-symmetric vectors Ω− with the property Ω− Pij = − Ω− (instead of symmetric vector (1.9)) are described as well They lead to the Ruijsenaars-Schneider model with different sign of the coupling constant η and. We briefly describe the notations and definitions related to the graded Lie algebras (groups) in the Appendix
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