N=1,2 Super-NLS Hierarchies as Super-KP Coset Reductions

TL;DR

This paper develops a supersymmetric coset reduction method for N=1,2 super-KP hierarchies, leading to new super-NLS hierarchies with associated super-$ ext{cw}$ algebras and explicit Lax operators.

Contribution

It introduces a supersymmetric coset approach to derive N=1,2 super-NLS hierarchies and their algebraic structures within a manifestly supersymmetric framework.

Findings

Explicit super-Lax operators for N=2 case

Derivation of super-$ ext{cw}$ algebra structures

Modified hierarchies via free-field mappings

Abstract

We define consistent finite-superfields reductions of the $N=1,2$ super-KP hierarchies via the coset approach we already developped for reducing the bosonic KP-hierarchy (generating e.g. the NLS hierarchy from the $sl(2)/U(1)-{\cal KM}$ coset). We work in a manifestly supersymmetric framework and illustrate our method by treating explicitly the $N=1,2$ super-NLS hierarchies. W.r.t. the bosonic case the ordinary covariant derivative is now replaced by a spinorial one containing a spin ${\textstyle {1\over 2}}$ superfield. Each coset reduction is associated to a rational super-$\cw$ algebra encoding a non-linear super-$\cw_\infty$ algebra structure. In the $N=2$ case two conjugate sets of superLax operators, equations of motion and infinite hamiltonians in involution are derived. Modified hierarchies are obtained from the original ones via free-fields mappings (just as a m-NLS equation arises by representing the $sl(2)-{\cal KM}$ algebra through the classical Wakimoto free-fields).