Abstract
We define and study a class of \( \mathcal{N} \) = 2 vertex operator algebras \( {\mathcal{W}}_{\mathrm{G}} \) labelled by complex reflection groups. They are extensions of the \( \mathcal{N} \) = 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 \( \mathcal{N} \) = 2 super Virasoro algebra enhances to the (small) \( \mathcal{N} \) = 4 superconformal algebra. With the exception of G = ℤ2, which corresponds to just the \( \mathcal{N} \) = 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 \( {\mathcal{W}}_{\mathrm{G}} \) in terms of rank(G) βγbc ghost systems, generalizing a construction of Adamovic for the \( \mathcal{N} \) = 4 algebra at c = −9. If G is a Weyl group, \( {\mathcal{W}}_{\mathrm{G}} \) is believed to coincide with the \( \mathcal{N} \) = 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, \( {\mathcal{W}}_{\mathrm{G}} \) is conjecturally associated to an \( \mathcal{N} \) = 3 4d superconformal field theory. The free-field realization allows to determine the elusive “R-filtration” of \( {\mathcal{W}}_{\mathrm{G}} \), and thus to recover the full Macdonald index of the parent 4d theory.
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Bonetti, F., Meneghelli, C. & Rastelli, L. VOAs labelled by complex reflection groups and 4d SCFTs. J. High Energ. Phys. 2019, 155 (2019). https://doi.org/10.1007/JHEP05(2019)155
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DOI: https://doi.org/10.1007/JHEP05(2019)155