A note on real forms of the complex N = 4 supersymmetric Toda chain hierarchy in real N = 2 and N = 4 superspaces

Three inequivalent real forms of the complex N=4 supersymmetric Toda chain hierarchy (Nucl. Phys. B558 (1999) 545, solv-int/9907004) in the real N = 2 superspace with one even and two odd real coordinates are presented. It is demonstrated that the first of them possesses a global N = 4 supersymmetry, while the other two admit a twisted N = 4 supersymmetry. A new superfield basis in which supersymmetry transformations are local is discussed and a manifest N = 4 supersymmetric representation of the N = 4 Toda chain in terms of a chiral and an anti-chiral N = 4 superfield is constructed. 1) E-Mail: [email protected] 2) E-Mail: [email protected] †) UMR 5672 du CNRS, associéè a l'Ecole Normale Supérieure de Lyon. 1. Introduction. Recently the Lax pair representation of the even and odd flows of the complex N = 4 supersymmetric Toda chain hierarchy in N = 2 superspace were constructed in [1]. The corresponding local and nonlocal Hamiltonians, the finite and infinite discrete symmetries, the first two Hamiltonian structures and the recursion operator connecting all evolution equations and the Hamiltonian structures were also studied. The goal of the present letter is first to analyse the possible real forms of the N = 4 Toda chain hierarchy in N = 2 superspace, second to derive a manifest N = 4 supersymmetric representation of its first nontrivial even flows in the real N = 4 superspace. Let us start with a short summary of the results that we shall need concerning the complex N = 4 supersymmetric Toda chain hierarchy (see [1, 2, 3, 4] for more details). The complex N = 4 supersymmetric Toda chain hierarchy in the complex N = 2 super-space comprises an infinite set of even and odd flows for two complex even N = 2 superfields u(z, θ + , θ −) and v(z, θ + , θ −), where z and θ ± are complex even and odd coordinates, respectively. The flows are generated by complex even and odd evolution derivatives {