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

A bstract We study constraints imposed by four-dimensional unitarity (formalised as graded unitarity in recent work by the first author) on possible $$ {\mathcal{W}}_3 $$ W 3 vertex algebras arising from four-dimensions via the SCFT/VOA correspondence. Under the assumption that the $$ \mathbf{\mathfrak{R}} $$ 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 $$ {\mathbf{\mathfrak{sl}}}_3 $$ sl 3 affine current algebras; those affine algebras were singled out by a similar graded unitarity analysis in [1]. Furthermore, these particular vertex algebras are known to be associated with the ( A 2 , A q ) Argyres-Douglas theories.