A Calogero Model with root string representatives of infinite order Coxeter orbits

TL;DR

The paper develops a Calogero-type many-body model endowed with infinite Weyl symmetry by extending to the A3^(1) hyperbolic Kac-Moody framework. It introduces an infinite root system constructed from root-string representatives to capture Coxeter-orbit structure and derives closed-form expressions for the action of Coxeter elements, ensuring affine-Weyl invariance of the potential. The resulting Hamiltonian comprises a Lorentzian kinetic term and an explicitly invariant infinite-sum potential, which reduces to the familiar four-particle A3-Calogero model in an appropriate limit. The work highlights ghost-like features in the kinetic term and outlines future directions for quantisation and extension to other infinite-dimensional algebras.

Abstract

We present a worked example for the new extensions of the multi-particle Calogero model endowed with infinite Weyl group symmetry of affine and hyperbolic type. Building upon the hyperbolic extension of the $A_3$-Kac-Moody algebra, we construct an explicit realisation of the model in terms of infinite root systems generated from Coxeter orbits. To address the challenge of summing over infinitely many roots, we introduce root string representatives that span the invariant root space while preserving invariance under the affine Weyl group. This approach yields closed-form expressions for the potentials, which by construction are invariant under the full affine Weyl symmetry. Moreover, we demonstrate that in an appropriate infinite-coordinate limit the model reduces smoothly to the conventional four particle $A_3$-Calogero system. Our construction constitutes a systematic method for implementing infinite-dimensional symmetries into Calogero-type models, thus broadening their algebraic and physical applicability.