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

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Copyright information

© The Author(s) 2014

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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