Generalized parafermions of orthogonal type

Abstract

There is an embedding of affine vertex algebras $V^k({\mathfrak{gl}}_n)\hookrightarrow V^k({\mathfrak{sl}}_{n+1})$, and the coset ${\mathcal{C}}^k(n)=\text{Com}(V^k({\mathfrak{gl}}_n),V^k({\mathfrak{sl}}_{n+1}))$ is a natural generalization of the parafermion algebra of ${\mathfrak{sl}}_2$. It was called the algebra of generalized parafermions by the third author and was shown to arise as a one-parameter quotient of the universal two-parameter ${\mathcal{W}}_{\infty }$-algebra of type $\mathcal{W}(2,3,\dots )$. In this paper, we consider an analogous structure of orthogonal type, namely ${\mathcal{D}}^k(n)=\text{Com}(V^k({\mathfrak{so}}_{2n}),V^k({\mathfrak{so}}_{2n+1}){)}^{{\mathbb{Z}}_2}$. We realize this algebra as a one-parameter quotient of the two-parameter even spin ${\mathcal{W}}_{\infty }$-algebra of type $\mathcal{W}(2,4,\dots )$, and we classify all coincidences between its simple quotient ${\mathcal{D}}_k(n)$ and the algebras ${\mathcal{W}}_{\ell }({\mathfrak{so}}_{2m+1})$ and ${\mathcal{W}}_{\ell }({\mathfrak{so}}_{2m}{)}^{{\mathbb{Z}}_2}$. As a corollary, we show that for the admissible levels $k=-(2n-2)+\frac{1}{2}(2n+2m-1)$ for ${\widehat{\mathfrak{so}}}_{2n}$ the simple affine algebra $L_k({\mathfrak{so}}_{2n})$ embeds in $L_k({\mathfrak{so}}_{2n+1})$, and the coset is strongly rational. As a consequence, the category of ordinary modules of $L_k({\mathfrak{so}}_{2n+1})$ at such a level is a braided fusion category.

arXiv commentMinor corrections, final version to appear in J. Algebra

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Cited by in corpus (3)

• Trialities of orthosymplectic $\mathcal{W}$-algebras
• A general mirror equivalence theorem for coset vertex operator algebras
• Ordinary modules for vertex algebras of ${\mathfrak{osp}}_{1|2n}$