On SW-minimal models and N=1 supersymmetric Quantum Toda-field theories

TL;DR

This paper constructs free field representations of SW-algebras related to N=1 supersymmetric Toda-field theories, identifies restrictions on the Cartan matrix for minimal models, and explores their degenerate representations and fusion rules.

Contribution

It introduces a new restriction on the Cartan matrix for super-Lie algebras to form minimal models and constructs their free field representations using Toda-field-theories.

Findings

Only Osp(2n|2n-1) and Osp(2n|2n+1) yield minimal models.

A necessary restriction on the Cartan matrix is identified.

Fusion rules for degenerate representations are analyzed.

Abstract

We construct free field representations of the $SW$-algebras SW(3/2,2) and SW(3/2,3/2,2) by using the corresponding Toda-Field-Theories. In constructing the series of minimal models using covariant vertex operators, we find a necessary restriction on the Cartan matrix of the Super-Lie-Algebra, also for the general case. Within this framework, this restriction claims that there be a minimum of one non-vanishing element on the diagonal of the Cartan matrix, which is without parallel in bosonic conformal field theory. As a consequence only two series of SSLA's yield minimal models, namely Osp(2n|2n-1) and Osp(2n|2n+1). Subsequently some general aspects of degenerate representations of SW-algebras, notably the fusion rules, are investigated. As an application we discuss minimal models of SW(3/2,2), which were constructed with independent methods, in this framework. Covariant formulation is used throughout this paper.