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 KdV equation deduced earlier by two of us (F.D. \& E.I.) 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 given 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 to be bi-hamiltonian. We demonstrate that these 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.

29 pages, LaTeX; a LaTeX error has been corrected

References in corpus (5)

• $N=4$ super KdV equation
• Sigma Models in (4,4) Harmonic Superspace
• THE MINIMAL N=2 SUPEREXTENSION OF THE NLS EQUATION
• N=2 Super Boussinesq Hierarchy: Lax Pairs and Conservation Laws
• On the Super-NLS Equation and its Relation with N=2 Super-KdV within Coset Approach

Cited by in corpus (20)

• N=4 Supersymmetric Mechanics in Harmonic Superspace
• Supersymmetric KP Hierarchy: ``Ghost'' Symmetry Structure, Reductions and Darboux-Backlund Solutions
• New Integrable Extensions of N=2 KdV and Boussinesq Hierarchies
• New super KdV system with the N=4 SCA as the hamiltonian structure
• N=4 Super NLS-mKdV Hierarchies
• N=2 local and N=4 nonlocal reductions of supersymmetric KP hierarchy in N=2 superspace
• The Hamiltonian structure of the N=2 supersymmetric GNLS hierarchy
• Symmetries of Supersymmetric Integrable Hierarchies of KP Type
• Lax pairs for N=2,3 Supersymmetric KdV Equations and their Extensions
• A note on real forms of the complex N=4 supersymmetric Toda chain hierarchy in real N=2 and N=4 superspaces
• Integrable Extensions of N=2 Supersymmetric KdV Hierarchy Associated with the Nonuniqueness of the Roots of the Lax operator
• Integrable System Constructed out of Two Interacting Superconformal Fields
• Division Algebras and Extended N=2,4,8 SuperKdVs
• Extensions of the N=2 Supersymmetric a=-2 Boussinesq Hierarchy
• On gauge-equivalent formulations of N=4 SKdV hierarchy
• Super-Affine Hierarchies and their Poisson Embeddings
• Higher Spin Chern-Simons Theory and the Super Boussinesq hierarchy
• N=4 supersymmetric integrable systems
• Hamiltonian structure of an operator valued extension of Super KdV equations
• Lax pair formulation of the N=4 Toda chain (KdV) hierarchy in N=4 superspace