Supersymmetry and the KdV equations for Integrable Hierarchies with a Half-integer Gradation

TL;DR

This paper develops a supersymmetry framework for integrable models based on the $sl(2|1)$ loop algebra, deriving N=2 and N=1 supersymmetric equations and revealing their non-local symmetry structures.

Contribution

It introduces a novel supersymmetry formulation for integrable hierarchies with half-integer gradation using the $sl(2|1)$ loop algebra, including new derivations of supersymmetric equations.

Findings

Derived N=2 supersymmetric mKdV and sinh-Gordon equations.

Obtained N=1 equations via algebraic reduction.

Described non-local symmetry structures in supersymmetric models.

Abstract

Supersymmetry is formulated for integrable models based on the $sl(2|1)$ loop algebra endowed with a principal gradation. The symmetry transformations which have half-integer grades generate supersymmetry. The $sl(2|1)$ loop algebra leads to N=2 supersymmetric mKdV and sinh-Gordon equations. The corresponding N=1 mKdV and sinh-Gordon equations are obtained via reduction induced by twisted automorphism. Our method allows for a description of a non-local symmetry structure of supersymmetric integrable models.