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

Abstract

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.

16 pages, v2: comments and references added at the end of the text

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Cited by in corpus (27)

• On Three Dimensional Quiver Gauge Theories and Integrability
• Poisson-Lie interpretation of trigonometric Ruijsenaars duality
• Self-duality of the compactified Ruijsenaars-Schneider system from quasi-Hamiltonian reduction
• Multiplicative quiver varieties and generalised Ruijsenaars-Schneider models
• Supersymmetric hyperbolic Calogero-Sutherland models by gauging
• Three-particle Integrable Systems with Elliptic Dependence on Momenta and Theta Function Identities
• Brief lectures on duality, integrability and deformations
• Trigonometric Sutherland systems and their Ruijsenaars duals from symplectic reduction
• Action-angle duality between the C(n)-type hyperbolic Sutherland and the rational Ruijsenaars-Schneider-van Diejen models
• Spin versions of the complex trigonometric Ruijsenaars-Schneider model from cyclic quivers
• Difference equation for the Heckman-Opdam hypergeometric function and its confluent Whittaker limit
• Duality between the trigonometric BC(n) Sutherland system and a completed rational Ruijsenaars-Schneider-van Diejen system
• 3d Mirror Symmetry for Instanton Moduli Spaces
• Lax representation of the hyperbolic van Diejen dynamics with two coupling parameters
• On the superintegrability of the rational Ruijsenaars-Schneider model
• Bi-Hamiltonian structure of a dynamical system introduced by Braden and Hone
• $\mathcal{N}=2$ supersymmetric hyperbolic Calogero-Sutherland model
• On the scattering theory of the classical hyperbolic C(n) Sutherland model
• Poisson Structures of Calogero-Moser and Ruijsenaars-Schneider Models
• Global description of action-angle duality for a Poisson-Lie deformation of the trigonometric ${\mathrm{BC}}_n$ Sutherland system
• Quantum spectral curve for (q,t)-matrix model
• Self-duality and scattering map for the hyperbolic van Diejen systems with two coupling parameters (with an appendix by S. Ruijsenaars)
• Integrable many-body systems of Calogero-Ruijsenaars type
• Poisson-Nijenhuis structures on quiver path algebras
• Bi-Hamiltonian structure of Sutherland models coupled to two $\mathfrak{u}(n{)}^{\ast }$-valued spins from Poisson reduction
• Wave functions for quantum integrable particle systems via partial confluences of multivariate hypergeometric functions
• On the derivation of Darboux form for the action-angle dual of trigonometric BC(n) Sutherland system