Lax pair formulation of the N = 4 Toda chain ( KdV ) hierarchy in N = 4 superspace

Lax pair and Hamiltonian formulations of the N = 4 supersymmetric Toda chain (KdV) hierarchy in N = 4 superspace are proposed. The general formulae for the infinite tower of its bosonic flows in terms of the Lax operator in N = 4 superspace are derived. A new N = 4 superfield basis in which the flows are local is constructed and its embedding in the N = 4 O(4) superconformal supercurrent is established. A proof that the flows possess five complex conjugations and an infinite-dimensional group of discrete symmetries in N = 4 superspace is presented. A relation between the two descriptions of the hierarchy in N = 4 superspace used in the literature is established. All known N = 2 superfield representations of the N = 4 KdV hierarchy are shown to derive from our N = 4 superspace Lax representation. 1) E-Mail: [email protected] 2) E-Mail: [email protected] †) UMR 5672 du CNRS, associéè a l'Ecole Normale Supérieure de Lyon.