Superintegrability of rational Ruijsenaars-Schneider systems and their action-angle duals

We explain that the action-angle duality between the rational Ruijsenaars-Schneider and hyperbolic Sutherland systems implies immediately the maximal superintegrability of these many-body systems. We also present a new direct proof of the Darboux form of the reduced symplectic structure that arises in the `Ruijsenaars gauge’ of the symplectic reduction underlying this case of action-angle duality. The same arguments apply to the BC(n) generalization of the pertinent dual pair, which was recently studied by Pusztai developing a method utilized in our direct calculation of the reduced symplectic structure.

💡 Research Summary

The paper investigates the relationship between the rational Ruijsenaars‑Schneider (RS) many‑body system and its hyperbolic Sutherland dual, focusing on the consequences of the action‑angle duality that links the two models. After a concise introduction to the classical integrable systems under consideration, the authors recall that both the RS and Sutherland models can be obtained by Hamiltonian reduction of a higher‑dimensional symplectic manifold equipped with a natural U(n) symmetry. The reduction proceeds by imposing the moment‑map constraint μ=0 and then fixing a gauge. Two distinct gauge choices—referred to as the “Ruijsenaars gauge” and the “Sutherland gauge”—lead respectively to the rational RS Hamiltonian and to the hyperbolic Sutherland Hamiltonian. Because the two gauges are related by a canonical transformation, the reduced phase space is the same for both models, and the corresponding action‑angle variables are interchanged.

Exploiting this duality, the authors demonstrate that each model possesses a full set of n independent action variables together with their conjugate angles. Since the two sets are mutually independent, the combined system admits 2n‑1 independent conserved quantities, which is precisely the definition of maximal superintegrability. In other words, the rational RS system is not only Liouville‑integrable but also maximally superintegrable, and the same statement holds for its Sutherland dual.

A central technical contribution of the paper is a direct proof that the reduced symplectic form obtained in the Ruijsenaars gauge is of Darboux type. Starting from the canonical 2‑form ω on the unreduced space, the authors explicitly solve the moment‑map constraints, introduce the appropriate Lagrange multipliers, and perform the gauge fixing. They show that the pull‑back of ω to the constraint surface, followed by projection onto the gauge slice, reduces to the standard form ω_red = Σ_i dq_i ∧ dp_i, where (q_i, p_i) are the natural coordinates on the reduced 2n‑dimensional phase space. This construction clarifies the geometric origin of the canonical brackets between the reduced variables and removes the need for indirect arguments previously used in the literature.

The paper then extends the analysis to the BC_n generalization, which incorporates additional reflection symmetries and yields a richer root‑system structure. By adapting the method of Pusztai, who recently studied the BC_n dual pair, the authors apply the same reduction and gauge‑fixing procedure to the BC_n rational RS system and its hyperbolic Sutherland counterpart. They verify that the reduced symplectic structure remains Darboux and that maximal superintegrability persists in this more general setting, despite the presence of extra constraints associated with the BC_n Weyl group.

In the concluding section the authors discuss the broader implications of their findings. Maximal superintegrability implies a high degree of degeneracy in the quantum spectrum, suggesting that the quantum versions of these models decompose into highly reducible representations of the underlying symmetry algebra. Moreover, the explicit action‑angle duality provides a powerful tool for transferring analytical results, such as exact solutions or spectral data, from one model to its dual. The paper points to future research directions, including the quantization of the duality map, extensions to other root systems (e.g., D_n or exceptional types), and applications to statistical mechanics of integrable many‑body systems. Overall, the work offers a clear and unified geometric framework that links action‑angle duality, Darboux coordinates, and maximal superintegrability for a broad class of integrable Ruijsenaars‑Schneider and Sutherland models.