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.
Highlights
Introduction and summaryAny four-dimensional N = 2 superconformal field theory (SCFT) contains a subsector isomorphic to a vertex operator algebra (VOA) [1]
As we review in detail below, any VOA that descends from a 4d N = 2 SCFT inherits a filtration associated to the 4d R-symmetry quantum number — the details of the cohomological construction of [1] imply that while R is in general not preserved by the OPE, it can at most decrease
In this work we study a class of N = 2 VOAs labelled by complex reflection groups
Summary
There is a heuristic understanding of these free fields as corresponding to the low-energy degrees of freedom of the 4d theory at a generic point on its Higgs branch of vacua This physical picture will be discussed elsewhere [25]: it appears to be much more general, possibly valid for all VOAs that arise from N = 2 SCFTs. Our proposal generalizes to all g a construction of Adamovic [26], who exhibited a free-field realization of VirN =4 with c = −9 (in our framework, the g = sl(2) case) in terms of a single βγbc system. As we explain below, the moduli space of vacua of the putative parent 4d theory would be the orbifold R6n/Γ, where n is the rank of Γ, but general consistency conditions on the low-energy effective theory restrict Γ to be crystallographic [23, 29].4 This whole circle of ideas admits a natural extension to a class of VOAs WG ⊃ WΓ, labelled by a general complex reflection group G. We preview some our findings in the outlook subsection of this introduction
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