Towards a classification of graded unitary ${\mathcal W}_3$ algebras

TL;DR

This paper classifies four-dimensional unitary ${ m W}_3$ vertex algebras, showing they correspond to specific minimal models and are linked to certain Argyres--Douglas theories.

Contribution

It demonstrates that only minimal model ${ m W}_3$ algebras are compatible with four-dimensional unitarity under certain filtrations, connecting them to boundary-admissible affine algebras.

Findings

All compatible ${ m W}_3$ algebras are minimal models with specific central charges.

These algebras are realized via Drinfel'd--Sokolov reduction of boundary-admissible affine algebras.

The compatible algebras are associated with $(A_2,A_q)$ Argyres--Douglas theories.

Abstract

We study constraints imposed by four-dimensional unitarity (formalised as graded unitarity in recent work by the first author) on possible ${\mathcal W}_3$ vertex algebras arising from four-dimensions via the SCFT/VOA correspondence. Under the assumption that the $\mathfrak{R}$-filtration is a weight-based filtration with respect to the usual strong generators of the vertex algebra, we demonstrate that all values of the central charge other than those of the $(3,q+4)$ minimal models are incompatible with four-dimensional unitarity. These algebras are precisely the ones that are realised by performing principal Drinfel'd--Sokolov reduction to boundary-admissible $\mathfrak{sl}_3$ affine current algebras; those affine algebras were singled out by a similar graded unitarity analysis in \cite{ArabiArdehali:2025fad}. Furthermore, these particular vertex algebras are known to be associated with the $(A_2,A_q)$ Argyres--Douglas theories.