Conformal embeddings of affine vertex algebras in minimal $W$-algebras I: structural results

Abstract

We find all values of $k\in \mathbb{C}$, for which the embedding of the maximal affine vertex algebra in a simple minimal W-algebra $W_k(\mathfrak{g},\theta )$ is conformal, where $\mathfrak{g}$ is a basic simple Lie superalgebra and $-\theta$ its minimal root. In particular, it turns out that if $W_k(\mathfrak{g},\theta )$ does not collapse to its affine part, then the possible values of these $k$ are either $-\frac{2}{3}{h}^{\vee }$ or $-\frac{h^{\vee }-1}{2}$, where $h^{\vee }$ is the dual Coxeter number of $\mathfrak{g}$ for the normalization $(\theta ,\theta )=2$. As an application of our results, we present a realization of simple affine vertex algebra $V_{-\frac{n+1}{2}}(sl(n+1))$ inside of the tensor product of the vertex algebra $W_{\frac{n-1}{2}}(sl(2|n),\theta )$ (also called the Bershadsky-Knizhnik algebra) with a lattice vertex algebra.

Latex File, 30 pages, minor corrections

References in corpus (6)

• Field Algebras
• Joseph ideals and lisse minimal W-algebras
• Finite vs infinite decompositions in conformal embeddings
• $\mathcal{W}$-algebras with non-admissible levels and the Deligne exceptional series
• A realization of certain modules for the $N=4$ superconformal algebra and the affine Lie algebra $A_2^{(1)}$
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Cited by in corpus (21)

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• The vertex algebras ${\mathcal{R}}^{(p)}$ and ${\mathcal{V}}^{(p)}$
• Tensor categories of affine Lie algebras beyond admissible levels
• A realisation of the Bershadsky--Polyakov algebras and their relaxed modules
• S-duality for the large $N=4$ superconformal algebra
• Trialities of orthosymplectic $\mathcal{W}$-algebras
• An application of collapsing levels to the representation theory of affine vertex algebras
• Realizations of simple affine vertex algebras and their modules: the cases $\widehat{sl(2)}$ and $\widehat{osp(1,2)}$
• Minimal W-superalgebras and modular representations of basic Lie superalgebras
• Singularities of nilpotent Slodowy slices and collapsing levels of W-algebras
• On the semisimplicity of the category $K{L}_k$ for affine Lie superalgebras
• Unitarity of minimal $W$-algebras and their representations I
• On low rank 4d $\mathcal{N}=2$ SCFTs
• On the structure of W-algebras in type A
• On the representation theory of the vertex algebra $L_{-5/2}(sl(4))$
• Rationality of vertex operator superalgebras with rational conformal weights
• Unitarity of minimal $W$-algebras and their representations II: Ramond sector
• Representations of superconformal algebras and mock theta functions
• Associated varieties of simple affine VOAs $L_k(s{l}_3)$ and $W$-algebras $W_k(s{l}_3,f)$
• Conformal nets from minimal W-algebras
• Unitarity of minimal $W$-algebras