Trigonometric Sutherland systems and their Ruijsenaars duals from symplectic reduction

Abstract

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.

34 pages, minor additions and corrections of typos in v2

References in corpus (1)

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

Cited by in corpus (14)

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• BPS States in Omega Background and Integrability
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• Spin versions of the complex trigonometric Ruijsenaars-Schneider model from cyclic quivers
• 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
• Trigonometric and elliptic Ruijsenaars-Schneider systems on the complex projective space
• Global description of action-angle duality for a Poisson-Lie deformation of the trigonometric ${\mathrm{BC}}_n$ Sutherland system
• Self-duality and scattering map for the hyperbolic van Diejen systems with two coupling parameters (with an appendix by S. Ruijsenaars)
• Poisson reductions of master integrable systems on doubles of compact Lie groups
• An integrable BC(n) Sutherland model with two types of particles
• On the derivation of Darboux form for the action-angle dual of trigonometric BC(n) Sutherland system