Nonlinear supersymmetry in the quantum Calogero model

  • Francisco Correa
  • Olaf LechtenfeldEmail author
  • Mikhail Plyushchay
Open Access


It is long known that the rational Calogero model describing n identical particles on a line with inverse-square mutual interaction potential is quantum superintegrable. We review the (nonlinear) algebra of the conserved quantum charges and the intertwiners which relate the Liouville charges at couplings g and g±1. For integer values of g, these intertwiners give rise to additional conserved charges commuting with all Liouville charges and known since the 1990s. We give a direct construction of such a charge, the unique one being totally antisymmetric under particle permutations. It is of order \( \frac{1}{2} \) n(n−1)(2g−1) in the momenta and squares to a polynomial in the Liouville charges. With a natural \( \mathbb{Z} \) 2 grading, this charge extends the algebra of conserved charges to a nonlinear supersymmetric one. We provide explicit expressions for intertwiners, charges and their algebra in the cases of two, three and four particles.


Integrable Equations in Physics Conformal and W Symmetry Extended Supersymmetry 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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© The Author(s) 2014

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Francisco Correa
    • 1
  • Olaf Lechtenfeld
    • 2
    • 3
    Email author
  • Mikhail Plyushchay
    • 4
  1. 1.Centro de Estudios Científicos (CECs)ValdiviaChile
  2. 2.Institut für Theoretische Physik and Riemann Center for Geometry and PhysicsLeibniz Universität HannoverHannoverGermany
  3. 3.Centre for Quantum Engineering and Space-Time ResearchLeibniz Universität HannoverHannoverGermany
  4. 4.Departamento de FísicaUniversidad de Santiago de ChileSantiago 2Chile

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