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 n( n-1)(2 g-1) in the momenta and squares to a polynomial in the Liouville charges. With a natural 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.
- Publication:
-
Journal of High Energy Physics
- Pub Date:
- April 2014
- DOI:
- 10.1007/JHEP04(2014)151
- arXiv:
- arXiv:1312.5749
- Bibcode:
- 2014JHEP...04..151C
- Keywords:
-
- Integrable Equations in Physics;
- Conformal and W Symmetry;
- Extended Supersymmetry;
- High Energy Physics - Theory;
- Mathematical Physics
- E-Print:
- 1+21 pages