Towards a classification of graded unitary ${\mathcal W}_3$ algebras
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910130807767040 |
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| author | Beem, Christopher Kulkarni, Harshal |
| author_facet | Beem, Christopher Kulkarni, Harshal |
| contents | We study constraints imposed by four-dimensional unitarity (formalised as graded unitarity in recent work by the first author) on possible ${\mathcal W}_3$ vertex algebras arising from four-dimensions via the SCFT/VOA correspondence. Under the assumption that the $\mathfrak{R}$-filtration is a weight-based filtration with respect to the usual strong generators of the vertex algebra, we demonstrate that all values of the central charge other than those of the $(3,q+4)$ minimal models are incompatible with four-dimensional unitarity. These algebras are precisely the ones that are realised by performing principal Drinfel'd--Sokolov reduction to boundary-admissible $\mathfrak{sl}_3$ affine current algebras; those affine algebras were singled out by a similar graded unitarity analysis in \cite{ArabiArdehali:2025fad}. Furthermore, these particular vertex algebras are known to be associated with the $(A_2,A_q)$ Argyres--Douglas theories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15944 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Towards a classification of graded unitary ${\mathcal W}_3$ algebras Beem, Christopher Kulkarni, Harshal High Energy Physics - Theory Quantum Algebra Representation Theory We study constraints imposed by four-dimensional unitarity (formalised as graded unitarity in recent work by the first author) on possible ${\mathcal W}_3$ vertex algebras arising from four-dimensions via the SCFT/VOA correspondence. Under the assumption that the $\mathfrak{R}$-filtration is a weight-based filtration with respect to the usual strong generators of the vertex algebra, we demonstrate that all values of the central charge other than those of the $(3,q+4)$ minimal models are incompatible with four-dimensional unitarity. These algebras are precisely the ones that are realised by performing principal Drinfel'd--Sokolov reduction to boundary-admissible $\mathfrak{sl}_3$ affine current algebras; those affine algebras were singled out by a similar graded unitarity analysis in \cite{ArabiArdehali:2025fad}. Furthermore, these particular vertex algebras are known to be associated with the $(A_2,A_q)$ Argyres--Douglas theories. |
| title | Towards a classification of graded unitary ${\mathcal W}_3$ algebras |
| topic | High Energy Physics - Theory Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2602.15944 |