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Real Hamiltonian forms of affine Toda field theories: spectral aspects

Vladimir S. Gerdjikov, Georgi G. Grahovski, Alexander A. Stefanov

Abstract

The paper is devoted to real Hamiltonian forms of 2-dimensional Toda field theories related to exceptional simple Lie algebras, and to the spectral theory of the associated Lax operators. Real Hamiltonian forms are a special type of "reductions" of Hamiltonian systems, similar to real forms of semi-simple Lie algebras. Examples of real Hamiltonian forms of affine Toda field theories related to exceptional complex untwisted affine Kac-Moody algebras are studied. Along with the associated Lax representations, we also formulate the relevant Riemann-Hilbert problems and derive the minimal sets of scattering data that determine uniquely the scattering matrices and the potentials of the Lax operators.

Real Hamiltonian forms of affine Toda field theories: spectral aspects

Abstract

The paper is devoted to real Hamiltonian forms of 2-dimensional Toda field theories related to exceptional simple Lie algebras, and to the spectral theory of the associated Lax operators. Real Hamiltonian forms are a special type of "reductions" of Hamiltonian systems, similar to real forms of semi-simple Lie algebras. Examples of real Hamiltonian forms of affine Toda field theories related to exceptional complex untwisted affine Kac-Moody algebras are studied. Along with the associated Lax representations, we also formulate the relevant Riemann-Hilbert problems and derive the minimal sets of scattering data that determine uniquely the scattering matrices and the potentials of the Lax operators.
Paper Structure (22 sections, 1 theorem, 92 equations, 6 figures)

This paper contains 22 sections, 1 theorem, 92 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Affine Toda field theories: preliminaries
  3. Real Hamiltonian forms and affine Toda field theories
  4. Real Hamiltonian forms
  5. Affine Toda field theories related to $E_6^{(1)}$
  6. Affine Toda field theories related to $E_7^{(1)}$
  7. Affine Toda field theories related to $E_8^{(1)}$
  8. Affine Toda field theories related to $F_4^{(1)}$
  9. Affine Toda field theories related to $G_2^{(1)}$
  10. On the spectral properties of the Lax operators with $\mathbb{Z}_h$-reduction
  11. General theory
  12. The FAS of the Lax operators $L$
  13. The inverse scattering problem and the Riemann-Hilbert problem
  14. The minimal sets of scattering data
  15. The minimal sets of scattering data from the RHP
  16. ...and 7 more sections

Key Result

Theorem 1

Let us assume that the Lax operator $\mathcal{L}$ (eq:calL) is constructed along the ideas in Section 4 above, i.e. it is related to a simple Lie algebra $\mathfrak{g}$ and possesses $\mathbb{Z}_h$ symmetry where $h$ is the Coxeter number of $\mathfrak{g}$. Let us also assume that its potential $Q_x Let also Then: A) $\mathcal{T}_1$ (resp. $\mathcal{T}_2$) provide a minimal set of scattering data

Figures (6)

  • Figure 1: The extended Dynkin diagram of the complex untwisted affine Kac-Moody algebra $E_6^{(1)}$. The white roots are invariant with respect to the automorphism $S_2$ in \ref{['eq:DCox-E6']}.
  • Figure 2: The extended Dynkin diagram of the complex untwisted affine Kac-Moody algebra $E_7^{(1)}$. The white roots are invariant with respect to the automorphism $S_1$ in \ref{['eq:DCox-E7']}.
  • Figure 3: The extended Dynkin diagram of the complex untwisted affine Kac-Moody algebra $E_8^{(1)}$. The white roots are invariant with respect to the automorphism $S_2$ in \ref{['eq:DCox-E8']}.
  • Figure 4: The extended Dynkin diagram of the complex untwisted affine Kac-Moody algebra $F_4^{(1)}$. The white roots are invariant with respect to the automorphism $S_2$ in \ref{['eq:DCox-F4']}.
  • Figure 5: The extended Dynkin diagram of the complex untwisted affine Kac-Moody algebra $G_2^{(1)}$.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof