W Algebras and Superalgebras from Constrained WZW Models: A Group Theoretical Classification

TL;DR

This paper classifies W algebras and superalgebras derived from constrained WZW models, linking them to specific subalgebras of Lie algebras and superalgebras, and analyzes their conformal spin structures.

Contribution

It provides a systematic group-theoretical classification of W algebras and superalgebras from Toda theories, including the role of U(1) factors and conformal spins.

Findings

Classification of W and super W algebras from Toda models.

Identification of U(1)_Y factors for algebra characterization.

Derivation of superconformal algebra structures with spins ≤ 2.

Abstract

We present a classification of $W$ algebras and superalgebras arising in Abelian as well as non Abelian Toda theories. Each model, obtained from a constrained WZW action, is related with an $Sl(2)$ subalgebra (resp. $OSp(1|2)$ superalgebra) of a simple Lie algebra (resp. superalgebra) $\cg$. However, the determination of an $U(1)_Y$ factor, commuting with $Sl(2)$ (resp. $OSp(1|2)$), appears, when it exists, particularly useful to characterize the corresponding $W$ algebra. The (super) conformal spin contents of each $W$ (super)algebra is performed. The class of all the superconformal algebras (i.e. with conformal spins $s\leq2$) is easily obtained as a byproduct of our general results.