Conformal mechanical treatment of Calogero-Moser model and infinite dimensional Lie algebra of conformal Galilei type

TL;DR

This paper explores the connection between Calogero-Moser particles in harmonic potentials and an infinite-dimensional Lie algebra, revealing a correspondence with algebraic structures via free field realization.

Contribution

It introduces a novel link between Calogero-Moser models and the representation theory of a semi-direct sum of Virasoro algebra and its modules, using free field realization.

Findings

Establishes a correspondence between excited states and singular vectors.

Provides explicit examples of singular vectors in Verma modules.

Develops a free field realization of the time evolution operator.

Abstract

We present a relationship between the Calogero-Moser particles confined in harmonic oscillator potentials and a representation theory of the infinite dimensional Lie algebra which is a semi-direct sum of Virasoro algebra and its module. More precisely, it is a correspondence of excited states of the model and singular vectors in Verma modules over the algebra. This is found by a free field realization of the time evolution operator of the model. We investigate the Verma modules and some explicit example of singular vectors are given.