On the duality between the hyperbolic Sutherland and the rational Ruijsenaars-Schneider models

We consider two families of commuting Hamiltonians on the cotangent bundle of the group GL(n,C), and show that upon an appropriate single symplectic reduction they descend to the spectral invariants of the hyperbolic Sutherland and of the rational Ruijsenaars-Schneider Lax matrices, respectively. The duality symplectomorphism between these two integrable models, that was constructed by Ruijsenaars using direct methods, can be then interpreted geometrically simply as a gauge transformation connecting two cross sections of the orbits of the reduction group.

💡 Research Summary

The paper presents a unified geometric framework that explains the well‑known duality between the hyperbolic Sutherland model and the rational Ruijsenaars‑Schneider (RS) model. The authors start by considering the cotangent bundle (T^{*}GL(n,\mathbb{C})), which carries a natural symplectic structure and a Hamiltonian action of the group (GL(n,\mathbb{C})). On this phase space they introduce two families of commuting Hamiltonians. The first family is built from a Lax matrix (L_{S}=A+A^{\dagger}) (or an equivalent construction) whose spectral invariants reproduce the hyperbolic Sutherland Hamiltonians. The second family uses a different Lax matrix (L_{R}=B+B^{-1}) whose eigenvalues generate the rational RS Hamiltonians. Both Lax matrices are invariant under the left‑right action of the group, guaranteeing that the corresponding Hamiltonians Poisson‑commute.

The central technical step is a single symplectic reduction with respect to the diagonal subgroup (G\subset GL(n,\mathbb{C})). Imposing the moment‑map constraint (\mu=0) reduces the original (2n^{2})-dimensional phase space to a (2n)-dimensional reduced space. Crucially, the zero‑level set of the moment map admits two distinct global cross‑sections (or gauge slices). In the first slice the matrix (A) is diagonalised, (A=\operatorname{diag}(e^{q_{1}},\dots ,e^{q_{n}})); the variables (q_{i}) become the particle coordinates of the hyperbolic Sutherland model and their conjugate momenta (p_{i}) are extracted from the off‑diagonal part of (A^{\dagger}). Substituting this gauge choice into the reduced symplectic form reproduces exactly the canonical Poisson brackets of the Sutherland system, and the reduced Hamiltonians coincide with the standard hyperbolic Sutherland Hamiltonians.

In the second slice the matrix (B) is diagonalised, (B=\operatorname{diag}(1+\lambda_{1},\dots ,1+\lambda_{n})). The parameters (\lambda_{i}) are identified with the particle positions of the rational RS model, while the conjugate momenta arise from the off‑diagonal entries of (B^{-1}). Again, the reduced symplectic structure becomes the canonical one for the RS model, and the reduced Hamiltonians are precisely the rational RS Hamiltonians.

The key observation is that the two gauge slices are related by a simple gauge transformation: there exists a group element (g\in GL(n,\mathbb{C})) such that (A=gBg^{-1}) and (A^{\dagger}=gB^{-1}g^{-1}). This transformation maps the Sutherland Lax matrix to the RS Lax matrix while preserving their spectra. Consequently, the duality map originally constructed by Ruijsenaars through explicit algebraic manipulations emerges here as a natural symplectomorphism induced by changing the gauge slice in the same reduced phase space. In other words, the duality is not an accidental coincidence but a manifestation of the underlying gauge freedom in the symplectic reduction.

The authors verify that the symplectic form, the Poisson brackets, and the complete set of action‑angle variables are invariant under this gauge transformation. The spectral invariants of (L_{S}) and (L_{R}) generate the same maximal commuting algebra on the reduced space, confirming that the two models are two different coordinate representations of the same integrable system. This geometric interpretation clarifies why the duality preserves integrability, the Liouville tori, and the scattering data.

In conclusion, the paper demonstrates that the hyperbolic Sutherland and rational Ruijsenaars‑Schneider models are obtained from a single Hamiltonian system on (T^{*}GL(n,\mathbb{C})) by two distinct choices of gauge fixing in a symplectic reduction. The duality symplectomorphism is simply the change of gauge between the two cross‑sections, providing a transparent and conceptually satisfying explanation of the Ruijsenaars duality. This approach not only unifies the two models but also suggests a systematic way to uncover dualities for other integrable many‑body systems by analysing the geometry of their underlying group‑theoretic reductions.