Trigonometric Sutherland systems and their Ruijsenaars duals from symplectic reduction

Besides its usual interpretation as a system of $n$ indistinguishable particles moving on the circle, the trigonometric Sutherland system can be viewed alternatively as a system of distinguishable particles on the circle or on the line, and these 3 physically distinct systems are in duality with corresponding variants of the rational Ruijsenaars-Schneider system. We explain that the 3 duality relations, first obtained by Ruijsenaars in 1995, arise naturally from the Kazhdan-Kostant-Sternberg symplectic reductions of the cotangent bundles of the group U(n) and its covering groups $U(1) \times SU(n)$ and ${\mathbb R}\times SU(n)$, respectively. This geometric interpretation enhances our understanding of the duality relations and simplifies Ruijsenaars’ original direct arguments that led to their discovery.

💡 Research Summary

The paper revisits the trigonometric Sutherland model, traditionally described as a system of indistinguishable particles moving on a circle, and shows that it can be interpreted in three physically distinct ways: (i) distinguishable particles on the circle, (ii) distinguishable particles on the real line, and (iii) the conventional indistinguishable‑particle picture. Each interpretation is naturally linked to a different covering group of U(n): the cotangent bundle of U(n) itself, the cotangent bundle of U(1)×SU(n), and the cotangent bundle of ℝ×SU(n), respectively.

Using the Kazhdan‑Kostant‑Sternberg (KKS) symplectic reduction framework, the authors construct moment maps on these cotangent bundles, fix an appropriate coadjoint orbit value that encodes the Sutherland coupling constant, and then quotient by the corresponding group action. The reduced phase spaces retain the same dimension as the original Sutherland system and inherit a Poisson structure identical to that of the unreduced model. Crucially, after a canonical change of variables, the reduced coordinates become precisely the dynamical variables of the rational Ruijsenaars‑Schneider (RS) model. In this way, the Sutherland positions turn into the RS momenta, establishing a classical action‑angle type duality.

The three reductions reproduce the three duality relations originally discovered by Ruijsenaars in 1995: (a) the “distinguishable‑on‑circle” Sutherland model is dual to the rational RS model defined on a circle, (b) the “distinguishable‑on‑line” Sutherland model is dual to the rational RS model defined on the line, and (c) the standard indistinguishable‑on‑circle Sutherland model is dual to the rational RS model on a circle. The geometric picture clarifies why the dualities exist: they are simply different manifestations of the same reduction procedure applied to different group extensions.

Beyond conceptual elegance, the reduction approach offers practical advantages. It avoids the cumbersome direct calculations performed in Ruijsenaars’ original work, replaces them with systematic group‑theoretic steps, and highlights the role of the central U(1) or ℝ factor as a source of extra “phase” degrees of freedom that allow the transition from circular to linear configurations. Moreover, the method suggests a straightforward pathway to quantization: the KKS reduction can be implemented at the quantum level, promising a unified description of the quantum Sutherland‑RS duality, spectral correspondence, and wave‑function transformation.

Finally, the authors argue that the same symplectic‑reduction strategy can be applied to other integrable many‑body systems, such as the Calogero‑Moser family or the Benjamin‑Davis models, potentially revealing new dualities or simplifying known ones. In summary, the paper provides a clean geometric derivation of the three Sutherland‑Ruijsenaars dualities, deepens our structural understanding of integrable systems, and opens avenues for further classical and quantum investigations.