Integrability vs. Supersymmetry

TL;DR

This paper examines (1,0)-superconformal Toda theories, revealing that classical integrability and certain conserved currents do not extend to their supersymmetric versions, and clarifies misconceptions about their supersymmetry properties.

Contribution

It demonstrates that classical integrability and W-algebra generators do not generalize to supersymmetric Toda theories based on simple Lie algebras, and refutes claims of (1,1) supersymmetry in these models.

Findings

Classical integrability does not survive in supersymmetric Toda theories.

W-algebra generators cannot be generalized to supersymmetric cases.

Superconformal models do not admit (1,1) supersymmetry.

Abstract

We investigate (1,0)-superconformal Toda theories based on simple Lie algebras and find that the classical integrability properties of the underlying bosonic theories do not survive. For several models based on algebras of low rank, we show explicitly that none of the conserved W-algebra generators can be generalized to the supersymmetric case. Using these results we deduce that at least one W-algebra generator fails to generalize in any model based on a classical Lie algebra. This argument involves a method for relating the bosonic Toda theories and their conserved currents within each classical series. We also scrutinize claims that the (1,0)-superconformal models actually admit (1,1) supersymmetry and find that they do not. Our results are consistent with the belief that all integrable Toda models with fermions arise from Lie superalgebras.