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

TL;DR

This paper constructs the $N=2$ super $W_4$ algebra via a reduction of the Gel'fand-Dikii bracket, presenting it in a supersymmetric form and identifying three integrable generalized KdV hierarchies associated with it.

Contribution

It introduces the $N=2$ super $W_4$ algebra as a reduction of the Gel'fand-Dikii bracket and finds three integrable hierarchies with this algebra as Hamiltonian structure.

Findings

Constructed the $N=2$ super $W_4$ algebra in supersymmetric form.

Identified three integrable generalized KdV hierarchies associated with the algebra.

Established the algebra as a reduction of the second Gel'fand-Dikii bracket.

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 $\Phi_i(X)$, with $\Phi_1$ being the $N=2$ energy momentum tensor and $\Phi_2$ and $\Phi_3$ 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.