N=4 super KdV hierarchy in N=4 and N=2 superspaces

Abstract

We present the results of further analysis of the integrability properties of the N=4 supersymmetric Korteweg–de Vries (KdV) equation deduced earlier by two of us [F. Delduc and E. Ivanov, Phys. Lett. B 309, 312 (1993)] as a Hamiltonian flow on N=4 SU(2) superconformal algebra in the harmonic N=4 superspace. To make this equation and the relevant Hamiltonian structures more tractable, we reformulate it in the ordinary N=4 and further in N=2 superspaces. In N=2 superspace it is represented by a coupled system of evolution equations for a general N=2 superfield and two chiral and antichiral superfields, and involves two independent real parameters, a and b. We construct a few first bosonic conserved charges in involution, of dimensions from 1 to 6, and show that they exist only for the following choices of the parameters: (i) a=4, b=0; (ii) a=−2, b=−6; (iii) a=−2, b=6. The same values are needed for the relevant evolution equations, including N=4 KdV itself, to be bi-Hamiltonian. We demonstrate that the above three options are related via SU(2) transformations and actually amount to the SU(2) covariant integrability condition found in the harmonic superspace approach. Our results provide a strong evidence that the unique N=4 SU(2) super KdV hierarchy exists. Upon reduction to N=2 KdV, the above three possibilities cease to be equivalent. They give rise to the a=4 and a=−2 N=2 KdV hierarchies, which thus prove to be different truncations of the single N=4 SU(2) KdV one.