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The Toda-Weyl mass spectrum

Martin T. Luu

TL;DR

This paper extends the affine Toda mass-spectrum paradigm by formulating Toda-Weyl theories where the Lagrangian is built from eigenvectors of general Weyl-group elements. The authors develop a method to compute the classical mass spectrum using pairings between a Weyl-eigenvector and orbit representatives of root data, illustrating the approach with two detailed examples: E6 and F4. They show how Carter matrices generalize the link between root geometry and mass spectra, recovering familiar trigonometric expressions in the Coxeter-like cases and revealing new mass-geometry relations beyond the Cartan-matrix framework. The work lays groundwork for further exploration of three-point couplings and integrability in these generalized theories, with full mathematical details reserved for future publication. Overall, it provides a conceptual and computational bridge between Weyl-group linear algebra and the relative geometry of special root sets in determining classical masses.

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.

The Toda-Weyl mass spectrum

TL;DR

This paper extends the affine Toda mass-spectrum paradigm by formulating Toda-Weyl theories where the Lagrangian is built from eigenvectors of general Weyl-group elements. The authors develop a method to compute the classical mass spectrum using pairings between a Weyl-eigenvector and orbit representatives of root data, illustrating the approach with two detailed examples: E6 and F4. They show how Carter matrices generalize the link between root geometry and mass spectra, recovering familiar trigonometric expressions in the Coxeter-like cases and revealing new mass-geometry relations beyond the Cartan-matrix framework. The work lays groundwork for further exploration of three-point couplings and integrability in these generalized theories, with full mathematical details reserved for future publication. Overall, it provides a conceptual and computational bridge between Weyl-group linear algebra and the relative geometry of special root sets in determining classical masses.

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.

Paper Structure

This paper contains 7 sections, 1 theorem, 67 equations, 2 tables.

Table of Contents

  1. Introduction
  2. The Toda-Weyl Lagrangians
  3. Mass calculations
  4. Example I
  5. Example II
  6. A general theory
  7. Conclusions

Key Result

Theorem 1

Let $\mathfrak g$ be a simple finite-dimensional complex Lie algebra with a Cartan subalgebra $\mathfrak h^{\textrm{Weyl}}$. Let $\sigma$ be an element in the Weyl group of $\mathfrak g$ such that: Let $\Lambda_{+} \in \mathfrak h^{\textrm{Weyl}}$ be an eigenvector of $\sigma$ and let $\Lambda_{-} = *(\Lambda_{+})$. Let $\gamma_1,\cdots, \gamma_{s}$ denote orbit representatives for the action of

Theorems & Definitions (4)

  • Definition 1
  • Theorem 1
  • proof
  • Remark 1