The $N=2$ super $W_4$ algebra and its associated generalized KdV hierarchies

Abstract

We construct the $N=2$ super $W_4$ algebra as a certain reduction of the second Gel'fand-Dikii bracket on the dual of the Lie superalgebra of $N=1$ super pseudo-differential operators. The algebra is put in manifestly $N=2$ supersymmetric form in terms of three $N=2$ superfields $Φ_i(X)$, with $Φ_1$ being the $N=2$ energy momentum tensor and $Φ_2$ and $Φ_3$ being conformal spin $2$ and $3$ superfields respectively. A search for integrable hierarchies of the generalized KdV variety with this algebra as Hamiltonian structure gives three solutions, exactly the same number as for the $W_2$ (super KdV) and $W_3$ (super Boussinesq) cases.