Toda field theories and Calogero models associated to infinite Weyl groups

TL;DR

This paper explores the extension of Toda field theories and Calogero models to systems based on hyperbolic and Lorentzian Kac-Moody algebras, focusing on their integrability and symmetry properties related to infinite Weyl groups.

Contribution

It introduces and analyzes new integrable models associated with infinite-dimensional Kac-Moody algebras and their Weyl groups, expanding the scope of algebraic integrable systems.

Findings

Models exhibit integrability properties similar to finite cases.

Invariance under infinite Weyl groups is established.

Extensions to hyperbolic and Lorentzian algebras are consistent with known structures.

Abstract

Many integrable theories can be formulated universally in terms of Lie algebraic root systems. Well-studied are conformally invariant scalar field theories of Toda type and their massive versions, which can be expressed in terms of simple roots of finite Lie and affine Kac-Moody algebras, respectively. Also, multi-particle systems of Calogero-Moser-Sutherland type, which require the entire root system in their formulation, are extensively studied. Here, we discuss recently proposed extensions of these models to similar systems based on hyperbolic and Lorentzian Kac-Moody algebras. We explore various properties of these models, including their integrability and their invariance with respect to infinite Weyl groups of affine, hyperbolic, and Lorentzian types.