Algebra of the observables in the Calogero model and in the Chern-Simons matrix model

TL;DR

This paper explores the algebraic structure of observables in the Calogero and Chern-Simons matrix models, revealing their similarities and differences through multiple realizations of their symmetry algebra.

Contribution

It introduces four realizations of the dynamical symmetry algebra for the Calogero model and extends the analysis to the Chern-Simons matrix model based on shared algebraic properties.

Findings

Identified four different realizations of the Calogero model's symmetry algebra.

Extended the algebraic analysis to the Chern-Simons matrix model.

Highlighted algebraic similarities and differences between the models.

Abstract

The algebra of observables of an N-body Calogero model is represented on the S_N-symmetric subspace of the positive definite Fock space. We discuss some general properties of the algebra and construct four different realizations of the dynamical symmetry algebra of the Calogero model. Using the fact that the minimal algebra of observables is common to the Calogero model and the finite Chern-Simons (CS) matrix model, we extend our analysis to the CS matrix model. We point out the algebraic similarities and distinctions of these models.