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 ${\bf D}r$. 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.