On Some Simple Orbifold Affine VOAs at Non-admissible Level Arising from Rank One 4D SCFTs

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 Adamović and Perše 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.