| Summary: | Abstract We define and study a class of N $$ \mathcal{N} $$ = 2 vertex operator algebras W G $$ {\mathcal{W}}_{\mathrm{G}} $$ labelled by complex reflection groups. They are extensions of the N $$ \mathcal{N} $$ = 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 $$ \mathcal{N} $$ = 2 super Virasoro algebra enhances to the (small) N $$ \mathcal{N} $$ = 4 superconformal algebra. With the exception of G = ℤ2, which corresponds to just the N $$ \mathcal{N} $$ = 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 $$ {\mathcal{W}}_{\mathrm{G}} $$ in terms of rank(G) βγbc ghost systems, generalizing a construction of Adamovic for the N $$ \mathcal{N} $$ = 4 algebra at c = −9. If G is a Weyl group, W G $$ {\mathcal{W}}_{\mathrm{G}} $$ is believed to coincide with the N $$ \mathcal{N} $$ = 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 $$ {\mathcal{W}}_{\mathrm{G}} $$ is conjecturally associated to an N $$ \mathcal{N} $$ = 3 4d superconformal field theory. The free-field realization allows to determine the elusive “R-filtration” of W G $$ {\mathcal{W}}_{\mathrm{G}} $$ , and thus to recover the full Macdonald index of the parent 4d theory.
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