VOAs labelled by complex reflection groups and 4d SCFTs

We define and study a class of N = 2 vertex operator algebras W_G labelled by complex reflection groups. They are extensions of the N = 2 super Virasoro algebra obtained by introducing additional generators, in correspondence with the invariants of the complex reflection group G. If G is a Coxeter group, the N = 2 super Virasoro algebra enhances to the (small) N = 4 superconformal algebra. With the exception of G = ℤ₂, which corresponds to just the N = 4 algebra, these are non-deformable VOAs that exist only for a specific negative value of the central charge. We describe a free-field realization of W_G in terms of rank(G) βγbc ghost systems, generalizing a construction of Adamovic for the N = 4 algebra at c = −9. If G is a Weyl group, W_G is believed to coincide with the N = 4 VOA that arises from the four-dimensional super Yang-Mills theory whose gauge algebra has Weyl group G. More generally, if G is a crystallographic complex reflection group, W_G is conjecturally associated to an N = 3 4d superconformal field theory. The free-field realization allows to determine the elusive “R-filtration” of W_G, and thus to recover the full Macdonald index of the parent 4d theory.