The N=2 supersymmetric Calogero-Sutherland model and its eigenfunctions

TL;DR

This paper develops the theory of N=2 Jack superpolynomials, showing they are eigenfunctions of a supersymmetric Calogero-Sutherland model and establishing their orthogonality and integrability.

Contribution

It introduces a novel construction of N=2 Jack superpolynomials from non-symmetric Jack polynomials and characterizes them as eigenfunctions of a supersymmetric integrable model.

Findings

N=2 Jack superpolynomials are eigenfunctions of the supersymmetric Calogero-Sutherland model.

The model's complete integrability is demonstrated with explicit conserved quantities.

The superpolynomials are orthogonal under both analytical and combinatorial scalar products.

Abstract

In a recent work, we have initiated the theory of N=2 symmetric superpolynomials. As far as the classical bases are concerned, this is a rather straightforward generalization of the N=1 case. However this construction could not be generalized to the formulation of Jack superpolynomials. The origin of this obstruction is unraveled here, opening the path for building the desired Jack extension. Those are shown to be obtained from the non-symmetric Jack polynomials by a suitable symmetrization procedure and an appropriate dressing by the anticommuting variables. This construction is substantiated by the characterization of the N=2 Jack superpolynomials as the eigenfunctions of the N=2 supersymmetric version of the Calogero-Sutherland model, for which, as a side result, we demonstrate the complete integrability by displaying the explicit form of four towers of mutually commuting (bosonic) conserved quantities. The N=2 Jack superpolynomials are orthogonal with respect to the analytical scalar product (induced by the quantum-mechanical formulation) as well as a new combinatorial scalar product defined on a suitable deformation of the power-sum basis.