Letters in Mathematical Physics

, Volume 106, Issue 10, pp 1429–1449 | Cite as

Trigonometric and Elliptic Ruijsenaars–Schneider Systems on the Complex Projective Space



We present a direct construction of compact real forms of the trigonometric and elliptic \({n}\)-particle Ruijsenaars–Schneider systems whose completed center-of-mass phase space is the complex projective space \({{\mathbb{CP}}^{n-1}}\) with the Fubini–Study symplectic structure. These systems are labeled by an integer \({p\in\{1,\ldots,n-1\}}\) relative prime to \({n}\) and a coupling parameter \({y}\) varying in a certain punctured interval around \({p\pi/n}\). Our work extends Ruijsenaars’s pioneering study of compactifications that imposed the restriction \({0 < y < \pi/n}\), and also builds on an earlier derivation of more general compact trigonometric systems by Hamiltonian reduction.


integrable systems Ruijsenaars–Schneider models compact phase space 

Mathematics Subject Classification

37J35 70H06 37J15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alekseev, A., Malkin, A., Meinrenken, E.: Lie group valued moment maps. J. Differ. Geom. 48, 445–495 (1998). arXiv:dg-ga/9707021
  2. 2.
    Blondeau-Fournier, O., Desrosiers, P., Mathieu, P.: The supersymmetric Ruijsenaars–Schneider model. Phys. Rev. Lett. 114, 121602 (2015). arXiv:1403.4667 [hep-th]
  3. 3.
    Bogomolny, E., Giraud, O., Schmit, C.: Random matrix ensembles associated with Lax matrices. Phys. Rev. Lett. 103, 054103 (2009). arXiv:0904.4898 [nlin.CD]
  4. 4.
    Etingof, P.: Calogero–Moser Systems and Representation Theory. European Mathematical Society, Zurich (2007)Google Scholar
  5. 5.
    Fehér, L., Ayadi, V.: Trigonometric Sutherland systems and their Ruijsenaars duals from symplectic reduction. J. Math. Phys. 51, 103511 (2010). arXiv:1005.4531 [math-ph]
  6. 6.
    Fehér, L., Klimčík, C.: Self-duality of the compactified Ruijsenaars–Schneider system from quasi-Hamiltonian reduction. Nucl. Phys. B 860, 464–515 (2012). arXiv:1101.1759 [math-ph]
  7. 7.
    Fehér, L., Klimčík, C.: On the spectra of the quantized action-variables of the compactified Ruijsenaars–Schneider system. Theor. Math. Phys. 171, 704–714 (2012). arXiv:1203.2864 [math-ph]
  8. 8.
    Fehér, L., Kluck, T.J.: New compact forms of the trigonometric Ruijsenaars–Schneider system. Nucl. Phys. B 882, 97–127 (2014). arXiv:1312.0400 [math-ph]
  9. 9.
    Gorsky, A., Nekrasov, N.: Relativistic Calogero–Moser model as gauged WZW theory. Nucl. Phys. B 436, 582–608 (1995). arXiv:hep-th/9401017
  10. 10.
    Nekrasov, N.: Infinite-dimensional algebras, many-body systems and gauge theories. In: Morozov, A.Yu., Olshanetsky, M.A. (eds.) Moscow Seminar in Mathematical Physics. AMS Transl. Ser. 2, pp. 263–299. American Mathematical Society, Providence (1999)Google Scholar
  11. 11.
    Olshanetsky M.A., Perelomov A.M.: Classical integrable finite-dimensional systems related to Lie algebras. Phys. Rep. 71, 313–400 (1981)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Pusztai, B.G., Görbe, T.F.: Lax representation of the hyperbolic van Diejen dynamics with two coupling parameters (preprint) (2016). arXiv:1603.06710 [math-ph]
  13. 13.
    Ruijsenaars S.N.M., Schneider H.: A new class of integrable systems and its relation to solitons. Ann. Phys. 170, 370–405 (1986)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ruijsenaars S.N.M.: Complete integrability of relativistic Calogero–Moser systems and elliptic function identities. Commun. Math. Phys. 110, 191–213 (1987)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ruijsenaars, S.N.M.: Finite-dimensional soliton systems. In: Kupershmidt, B. (ed.) Integrable and Superintegrable Systems, pp. 165–206. World Scientific, Singapore (1990)Google Scholar
  16. 16.
    Ruijsenaars S.N.M.: Action-angle maps and scattering theory for some finite-dimensional integrable systems. III. Sutherland type systems and their duals. Publ. RIMS Kyoto Univ. 31, 247–353 (1995)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ruijsenaars, S.N.M.: Systems of Calogero–Moser type. In: Semenoff, G.W., Vinet, L. (eds.) Proceedings of the 1994 CRM-Banff Summer School ‘Particles and Fields’, pp. 251–352. Springer (1999)Google Scholar
  18. 18.
    van Diejen, J.F.: Deformations of Calogero–Moser systems. Theor. Math. Phys. 99, 549–554 (1994). arXiv:solv-int/9310001
  19. 19.
    van Diejen, J.F., Vinet, L.: The quantum dynamics of the compactified trigonometric Ruijsenaars–Schneider model. Commun. Math. Phys. 197, 33–74 (1998). arXiv:math/9709221 [math-ph]
  20. 20.
    van Diejen, J.F., Vinet, L. (eds.): Calogero–Moser–Sutherland Models. Springer, New York (2000)Google Scholar
  21. 21.
    van Diejen, J.F., Emsiz, E.: Orthogonality of Macdonald polynomials with unitary parameters. Math. Z. 276, 517–542 (2014). arXiv:1301.1276 [math.RT]

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Theoretical PhysicsUniversity of SzegedSzegedHungary
  2. 2.Department of Theoretical PhysicsWIGNER RCP, RMKIBudapestHungary

Personalised recommendations