The Calogero Model - Anyonic Representation, Fermionic Extension and Supersymmetry

TL;DR

This paper explores the Calogero model's connections to anyons, extends it with fermionic and supersymmetric structures, and provides explicit solutions and algebraic frameworks for these extended systems.

Contribution

It introduces a complex representation linking the Calogero model to anyons, extends the model with supersymmetry and fermions, and constructs explicit solutions and algebraic structures.

Findings

Established equivalence between Calogero system and anyons in the lowest Landau level.

Extended the operator solution to include fermions using supersymmetry.

Constructed a supersymmetric extension of the deformed Heisenberg algebra.

Abstract

We discuss several applications and extensions of our previous operator solution of the $N$-body Calogero problem, \ie N particles in 1 dimension subject to a two-body interaction of the form $\half \sum_{i,j}[ (x_i - x_j)^2 + g/ {(x_i - x_j)^2}]$. Using a complex representation of the deformed Heisenberg algebra underlying the Calogero model, we explicitly establish the equivalence between this system and anyons in the lowest Landau level. A construction based on supersymmetry is used to extend our operator method to include fermions, and we obtain an explicit solution of the supersymmetric Calogero model constructed by Freedman and Mende. We also show how the dynamical $OSp(2;2)$ supersymmetry is realized by bilinears of modified creation and annihilation operators, and how to construct a supersymmetic extension of the deformed Heisenberg algebra.