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 chai n hierarchy (Nucl, Phys, B 558 (1999) 545) in the real N = 2 superspace wit h one even and two odd real coordinates are presented, It is demonstrated t hat the first of them possesses a global N = 4 supersymmetry, while the oth er 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, Its relation to the com plex N = 4 supersymmetric KdV hierarchy is established. Darboux-Backlund sy mmetries and a new real form of this last hierarchy possessing a twisted N = 4 supersymmetry are derived. (C) 2000 Elsevier Science B.V. All rights re served.