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

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 ${\mathfrak{s}\mathfrak{l}}_3$ 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.