On the Completeness of the Set of Classical W-Algebras Obtained from DS Reductions

TL;DR

This paper investigates the structure and completeness of classical W-algebras derived from Drinfeld-Sokolov reductions, revealing new noncanonical reductions and restrictions based on the quasi-primary basis and $sl(2)$ embeddings.

Contribution

It strengthens the connection between $sl(2)$ embeddings and DS reductions, and identifies new noncanonical W-algebras with free field components.

Findings

Canonical DS reductions produce W-algebras with highest weight bases.

Noncanonical reductions can lead to direct products with free fields.

Restrictions on conformal weights of generators are established.

Abstract

We clarify the notion of the DS --- generalized Drinfeld-Sokolov --- reduction approach to classical ${\cal W}$-algebras. We first strengthen an earlier theorem which showed that an $sl(2)$ embedding ${\cal S}\subset {\cal G}$ can be associated to every DS reduction. We then use the fact that a $\W$-algebra must have a quasi-primary basis to derive severe restrictions on the possible reductions corresponding to a given $sl(2)$ embedding. In the known DS reductions found to date, for which the $\W$-algebras are denoted by ${\cal W}_{\cal S}^{\cal G}$-algebras and are called canonical, the quasi-primary basis corresponds to the highest weights of the $sl(2)$. Here we find some examples of noncanonical DS reductions leading to $\W$-algebras which are direct products of ${\cal W}_{\cal S}^{\cal G}$-algebras and `free field' algebras with conformal weights $\Delta \in \{0, {1\over 2}, 1\}$. We also show that if the conformal weights of the generators of a ${\cal W}$-algebra obtained from DS reduction are nonnegative $\Delta \geq 0$ (which is