The Toda-Weyl mass spectrum

TL;DR

This paper generalizes the mass spectrum calculation of affine Toda theories by using eigenvectors of arbitrary Weyl group elements, extending the known Cartan matrix-based approach to a broader class of theories.

Contribution

It introduces a new framework for deriving Lagrangians and mass spectra using eigenvectors of Weyl group elements beyond the affine case.

Findings

Classical mass spectrum calculated for generalized Weyl group elements

Relation established between mass spectrum and geometry of special roots

Extension of affine Toda mass spectrum description to more general settings

Abstract

The masses of affine Toda theories are known to correspond to the entries of a Perron-Frobenius eigenvector of the relevant Cartan matrix. The Lagrangian of the theory can be expressed in terms of a suitable eigenvector of a Coxeter element in the Weyl group. We generalize this set-up by formulating Lagrangians based on eigenvectors of arbitrary elements in the Weyl group. Under some technical conditions (that hold for many Weyl group elements), we calculate the classical mass spectrum. In particular, we indicate the relation to the relative geometry of special roots, generalizing the affine Toda mass spectrum description in terms of the Cartan matrix. Related questions of three point coupling and integrability are left to be addressed on a future occasion.