The PT Symmeter

Andreas Fring, Monique Smith

We provide a general construction procedure for antilinearly invariant complex root spaces. The proposed method is generic and may be applied to any Weyl group allowing to take any element of the group as a starting point for the construction. Worked out examples for several specific Weyl groups are presented, focusing especially on those cases for which no solutions were found previously. When applied in the defining relations of models based on root systems this usually leads to non-Hermitian models, which are nonetheless physically viable in a self-consistent sense as they are antilinearly invariant by construction. We discuss new types of Calogero models based on these complex roots. In addition we propose an alternative construction leading to q-deformed roots. We employ the latter type of roots to formulate a new version of affine Toda field theories based on non-simply laced roots systems. These models exhibit on the classical level a strong-weak duality in the coupling constant equivalent to a Lie algebraic duality, which is known for the quantum version of the undeformed case.

http://arxiv.org/abs/1108.1719
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Andreas Fring, Monique Smith

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.

http://arxiv.org/abs/1010.2218
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)