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Nonlinear supersymmetry in the quantum Calogero model

Abstract

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.

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Correspondence to Olaf Lechtenfeld.

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ArXiv ePrint: 1312.5749

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Correa, F., Lechtenfeld, O. & Plyushchay, M. Nonlinear supersymmetry in the quantum Calogero model. J. High Energ. Phys. 2014, 151 (2014). https://doi.org/10.1007/JHEP04(2014)151

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Keywords

  • Integrable Equations in Physics
  • Conformal and W Symmetry
  • Extended Supersymmetry