On some simple orbifold affine VOAs at non-admissible level arising from rank one 4D SCFTs

Abstract

We study the representations of some simple affine vertex algebras at non-admissible level arising from rank one 4D SCFTs. In particular, we classify the irreducible highest weight modules of $L_{-2}(G_2)$ and $L_{-2}(B_3)$. It is known by the works of Adamovi\'{c} and Per\v{s}e that these vertex algebras can be conformally embedded into $L_{-2}(D_4)$. We also compute the associated variety of $L_{-2}(G_2)$, and show that it is the orbifold of the associated variety of $L_{-2}(D_4)$ by the symmetric group of degree 3 which is the Dynkin diagram automorphism group of $D_4$. This provides a new interesting example of associated variety satisfying a number of conjectures in the context of orbifold vertex algebras.