Extended supersymmetric Calogero model

Abstract

We present a surprising redefinition of matrix fermions which brings the supercharges of the N -extended supersymmetric An−1 Calogero model introduced in [1] to the standard form maximally cubic in the fermions. The complexity of the model is transferred to a non-canonical and nonlinear conjugation property of the fermions. Employing the new cubic supercharges, we apply a supersymmetric generalization of a "folding" procedure for A2n−1 ⊕ A1 to explicitly construct the supercharges and Hamiltonian for arbitrary even-N supersymmetric extensions of the Bn, Cn and Dn rational Calogero models. We demonstrate that all considered models possess a dynamical osp(N |2) superconformal symmetry.

N=4 supersymmetric Calogero-Sutherland models

“…In contrast with the rational Calogero-Moser and/or Euler-Calogero-Moser system admitting an arbitrary number of supersymmetries [4,5,6], their trigonometric/hyperbolic versions can so far be supersymmetrized up to the N = 4 cases only. If we try to construct the additional supercharges, they will span a soft variant of the super Poincaré algebra with purely fermionic conserved R-charges in the commutator of Q a with Q b (2.22).…”

“…Using these ingredients in [6] the supercharges and Hamiltonian have been constructed for arbitrary even-N supersymmetric extensions of the A n , B n , C n and D n rational Calogero models. In what follows we will use the same ingredients to construct N = 2, 4 trigonometric/hyperbolic Calogero-Sutherland models with the supercharges and the Hamiltonian obeying the N = 2, 4-extended super-Poincaré algebra.…”

“…There is a lot of confirmation that N = 4 supersymmetric extensions of Calogero-Moser systems must include a large number of fermions -far more than the 4n fermions expected within the standard (but not very successful) approach [1,2,3,4,5,6]. The source of these fermions is the supersymmetrization of the matrix models from which, in the purely bosonic cases, the Calogero-Moser systems can be obtained by a reduction (see e.g.…”

“…However, for N = 4 their bosonic part does not reproduce the ordinary Calogero systems but only spin-Calogero ones. 1 In a series of papers [4,5,6] we developed a different approach. Mainly working in the Hamiltonian formulation, we worked out an ansatz for the supercharges which accommodates all Calogero models associated with the classical A n , B n , C n and D n Lie algebras.…”

Supersymmetric Calogero and Calogero-Sutherland models from gauging

“…There exists other types of superextensions of the n-particle Calogero model with su(n) or so(n) spin variables [24,25]. Here, the su(n) spin variables can be removed by a Hamiltonian reduction, keeping only the N n 2 fermions for any number N of supersymmetries.…”