Affine Toda field theories related to Coxeter groups of noncrystallographic type

Abstract

Abstract:We propose affine Toda field theories related to the non-crystallographic Coxeter groups H 2 , H 3 and H 4 . The classical mass spectrum, the classical three-point couplings and the one-loop corrections to the mass renormalisation are determined. The construction is carried out by means of a reduction procedure from crystallographic to non-crystallographic Coxeter groups. The embedding structure explains for various affine Toda field theories that their particles can be organised in pairs, such that their relative masses differ by the golden ratio.

Non-crystallographic reduction of generalized Calogero–Moser models

“…The purpose of this article has been to extend a recent reduction procedure [28] based on the embedding of non-crystallographic Coxeter groups into crystallographic ones to Calogero-Moser models. This lead us to propose new integrable systems based on H 2,3,4 by extending from the known rational potential to the trigonometric, hyperbolic and elliptic case.…”

“…The first map ω has been previously considered in the literature, see e.g. [33] [28], and is defined as follows…”

“…We will make use of this fact when reducing the crystallographic Calogero-Moser systems to non-crystallographic ones below. Note that the defining relations (2.7), (2.8) also apply to the fundamental weights [28].…”

“…The details of the embeddings (1.4) have been presented previously in the literature [32,33,34] and in particular [28] which we follow in our notation. We therefore omit proofs and only present the necessary formulae for the reduction.…”

“…Employing a reduction procedure recently introduced in the context of affine Toda field theory [28] the models (1.2) are obtained from the Hamiltonians (1.1) of the corresponding crystallographic groups given in (1.4). See also [29] and references therein for similar reduction procedures.…”