How Complete is the Classification of W-Symmetries ?

TL;DR

This paper investigates the classification completeness of W-symmetries in 2D conformal quantum field theory, revealing new models and highlighting the existence of W-algebras beyond known Casimir algebra patterns.

Contribution

It introduces new rational W-algebra models with two generators and identifies W(2,4,6) features that extend beyond Casimir algebra classifications.

Findings

Most rational models fit known Casimir algebra series

New W-algebras with two generators belong to known series

Existence of W(2,4,6) features outside Casimir algebra patterns

Abstract

In 2D conformal quantum field theory, we continue a systematic study of W-algebras with two and three generators and their highest weight representations focussing mainly on rational models. We review the known facts about rational models of W(2,\delta)-algebras. Our new rational models of W-algebras with two generators all belong to one of the known series. The majority of W-algebras with three generators -including the new ones constructed in this letter- can be explained as subalgebras or truncations of Casimir algebras. Nonetheless, for one solution of W(2,4,6) we reveal some features that do not fit into the pattern of Casimir algebras or orbifolds thereof. This shows that there are more W-algebras than those predicted from Casimir algebras (or Toda field theories). However, most of the known rational conformal field theories belong to the minimal series of some Casimir algebra.