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On affine Toda field theories related to ${\bf D}_r$ algebras and their real Hamiltonian forms

Vladimir S. Gerdjikov, Georgi G. Grahovski

Abstract

The paper deals with affine 2-dimensional Toda field theories related to simple Lie algebras of the classical series ${\bf D}_r$. We demonstrate that the complexification procedure followed by a restriction to a specified real Hamiltonian form commutes with the external automorphisms of $\mathfrak{g}$. This is illustrated on the examples ${\bf D}_{r+1}^{(1)} \to {\bf B}_r^{(1)}$ and ${\bf D}_4^{(1)} \to {\bf G}_2^{(1)}$ using external automorphisms of the corresponding extended Dynkin diagrams.

On affine Toda field theories related to ${\bf D}_r$ algebras and their real Hamiltonian forms

Abstract

The paper deals with affine 2-dimensional Toda field theories related to simple Lie algebras of the classical series . We demonstrate that the complexification procedure followed by a restriction to a specified real Hamiltonian form commutes with the external automorphisms of . This is illustrated on the examples and using external automorphisms of the corresponding extended Dynkin diagrams.
Paper Structure (8 sections, 1 theorem, 42 equations, 2 figures)

This paper contains 8 sections, 1 theorem, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Complexification of ATFT
  4. RHF of ATFT from external automorphisms
  5. From ${\bf D}_{r+1}^{(1)}$ to ${\bf B}_{r}^{(1)}$
  6. From ${\bf D}_{4}^{(1)}$ to ${\bf B}_{3}^{(1)}$
  7. From ${\bf D}_4^{(1)}$ to ${\bf G}_2^{(1)}$
  8. Conclusions

Key Result

Proposition 1

Let us consider the RHF of ATFT using the external automorphisms of the algebras ${\bf D}^{(1)}_{r+1}$. If we use the second order external automorphisms, then If we use the third order external automorphism of $D_4$, then we obtain

Figures (2)

  • Figure 1: Extended Dynkin diagrams of the complex untwisted affine Kac-Moody algebras ${\bf D}_{r+1}^{(1)}$ and ${\bf B}_r^{(1)}$ (upper and lower panels respectively).
  • Figure 2: Reductions of ${\bf D}_4^{(1)}$ affine Lie algebra: (a)${\bf D}_4^{(1)}\rightarrow {\bf B}_3^{(1)}$; (b)${\bf D}_4^{(1)}\rightarrow {\bf G}_2^{(1)}$.

Theorems & Definitions (2)

  • Remark 1
  • Proposition 1