N=4 Super NLS-mKdV Hierarchies

TL;DR

This paper constructs new N=4 supersymmetric integrable hierarchies from N=2 affine algebras, revealing hidden supersymmetry structures and providing scalar Lax formulations, thus advancing the classification of supersymmetric integrable systems.

Contribution

It introduces two new N=4 supersymmetric hierarchies based on N=2 affine algebras and establishes their relation to known hierarchies, along with scalar Lax formulations.

Findings

Identified N=4 supersymmetry in N=2 affine algebra extensions.

Constructed scalar Lax operators for the new hierarchies.

Proposed a classification scheme for N=4 supersymmetric hierarchies.

Abstract

N=2 extension of affine algebra $\hat{sl(2)\oplus u(1)}$ possesses a hidden global N=4 supersymmetry and provides a second hamiltonian structure for a new N=4 supersymmetric integrable hierarchy defined on N=2 affine supercurrents. This system is an N=4 extension of at once two hierarchies, N=2 NLS and N=2 mKdV ones. It is related to N=4 KdV hierarchy via a generalized Sugawara-Feigin-Fuks construction which relates N=2 $\hat{sl(2)\oplus u(1)}$ algebra to ``small'' N=4 SCA. We also find the underlying affine hierarchy for another integrable system with the N=4 SCA second hamiltonian structure, ``quasi'' N=4 KdV hierarchy. It respects only N=2 supersymmetry. For both new hierarchies we construct scalar Lax formulations. We speculate that any N=2 affine algebra admitting a quaternionic structure possesses N=4 supersymmetry and so can be used to produce N=4 supersymmetric hierarchies. This suggests a way of classifying all such hierarchies.