PT invariant complex E(8) root spaces

TL;DR

This paper introduces a method to construct complex root spaces invariant under antilinear transformations for any Coxeter group, exemplified on E(8), leading to new models in integrable systems and field theories.

Contribution

It presents a novel construction procedure for invariant complex root spaces applicable to all Coxeter groups, including E(8), with applications to integrable models and field theories.

Findings

Constructed complex root spaces invariant under antilinear transformations for Coxeter groups.

Applied the method to E(8), a previously unsolved case.

Proposed new generalizations of Calogero-Moser and affine Toda models.

Abstract

We provide a construction procedure for complex root spaces invariant under antilinear transformations, which may be applied to any Coxeter group. The procedure is based on the factorisation of a chosen element of the Coxeter group into two factors. Each of the factors constitutes an involution and may therefore be deformed in an antilinear fashion. Having the importance of the E(8)-Coxeter group in mind, such as underlying a particular perturbation of the Ising model and the fact that for it no solution could be found previously, we exemplify the procedure for this particular case. As a concrete application of this construction we propose new generalisations of Calogero-Moser Sutherland models and affine Toda field theories based on the invariant complex root spaces and deformed complex simple roots, respectively.