[Paper Review] The Ruijsenaars self-duality map as a mapping class symplectomorphism

L. Fehér, C. Klimčı́k|arXiv (Cornell University)|Mar 15, 2012

Nonlinear Waves and Solitons17 references3 citations

TL;DR

This paper establishes the compactified trigonometric Ruijsenaars-Schneider III_b system as a quasi-Hamiltonian reduced phase space of the internally fused double of SU(n), proving its self-duality symplectomorphism arises from the standard mapping class group generator S ∈ SL(2,ℤ). The reduced phase space is symplectomorphic to ℂP(n−1) with the Fubini-Study form, where particle positions and action variables are exchanged under the duality map, rigorously confirming long-standing conjectures by Gorsky et al.

ABSTRACT

This is a brief review of the main results of our paper arXiv:1101.1759 that contains a complete global treatment of the compactified trigonometric Ruijsenaars-Schneider system by quasi-Hamiltonian reduction. Confirming previous conjectures of Gorsky and collaborators, we have rigorously established the interpretation of the system in terms of flat SU(n) connections on the one-holed torus and demonstrated that its self-duality symplectomorphism represents the natural action of the standard mapping class generator S on the phase space. The pertinent quasi-Hamiltonian reduced phase space turned out to be symplectomorphic to the complex projective space equipped with a multiple of the Fubini-Study symplectic form and two toric moment maps playing the roles of particle-positions and action-variables that are exchanged by the duality map. Open problems and possible directions for future work are also discussed.

Motivation & Objective

• To rigorously establish the global structure of the compactified trigonometric Ruijsenaars-Schneider III_b system using quasi-Hamiltonian reduction.
• To confirm the conjecture that the system's self-duality symplectomorphism arises from the mapping class group action on the one-holed torus.
• To identify the reduced phase space as complex projective space equipped with the Fubini-Study symplectic form and two toric moment maps corresponding to particle positions and action variables.
• To clarify the geometric and algebraic origin of Ruijsenaars duality in terms of flat SU(n) connections and the SL(2,ℤ) action on moduli spaces.
• To explore the implications for quantum integrability and potential extensions to spin and hyperbolic variants of the system.

Proposed method

• Utilizes quasi-Hamiltonian reduction on the internally fused double D = G × G for G = SU(n), with moment map constraints encoding flat connections on the one-holed torus.
• Applies the reduction procedure to a diagonal conjugacy class μ₀, yielding a symplectic quotient P(μ₀) diffeomorphic to ℂP(n−1).
• Identifies two toric moment maps on P(μ₀): one for particle positions (J_k), one for action variables (I_k), which are exchanged under the duality map.
• Constructs the self-duality symplectomorphism 𝔖 as the induced map from the standard S-matrix generator S ∈ SL(2,ℤ) acting on the double via automorphisms.
• Demonstrates that the anti-symplectic involution 𝔽 = 𝔠 ∘ 𝔖 arises from the map R_D = ϱ_D ∘ S_D² on the double, descending to the reduced space.
• Introduces a coupling constant Λ via scaling the invariant inner product on su(n), generalizing the symplectic form to ΛΩ_loc, relevant for quantum extensions.

Experimental results

Research questions

• RQ1 How can the compactified trigonometric Ruijsenaars-Schneider III_b system be globally realized via quasi-Hamiltonian reduction?
• RQ2 What is the geometric origin of the self-duality symplectomorphism in terms of mapping class group actions on moduli spaces of flat connections?
• RQ3 How do the particle positions and action variables on the reduced phase space relate to the toric moment maps in the Fubini-Study geometry of ℂP(n−1)?
• RQ4 Can the anti-symplectic involution 𝔽 = 𝔠 ∘ 𝔖 be derived from a canonical automorphism of the double group D = SU(n) × SU(n)?
• RQ5 What is the structure of the reduced phase space P(μ₀) for non-diagonal μ₀, and does it support new integrable systems?

Key findings

• The reduced phase space P(μ₀) is symplectomorphic to ℂP(n−1) equipped with a multiple of the Fubini-Study symplectic form, confirming the classical phase space structure of the III_b system.
• The self-duality symplectomorphism 𝔖 is realized as the induced action of the standard S-generator of SL(2,ℤ) on the reduced phase space, providing a geometric origin for Ruijsenaars duality.
• The two toric moment maps on ℂP(n−1) correspond precisely to the particle positions and action variables of the system, with 𝔖 exchanging them as required by self-duality.
• The anti-symplectic involution 𝔽 = 𝔠 ∘ 𝔖 arises from the map R_D = ϱ_D ∘ S_D² on the double, which descends to an involution on P(μ₀) reversing the Poisson structure.
• The center 𝒻 × 𝒻 of SU(n) × SU(n) acts on the double and descends to a ℤ_n × ℤ_n action on ℂP(n−1), consistent with structures used in earlier works.
• The self-dual hyperbolic Ruijsenaars-Schneider system remains without a known master phase space, representing a key open problem for future reduction-based constructions.