Ordinary modules for vertex algebras of ${\mathfrak{osp}}_{1|2n}$

Abstract

We show that the affine vertex superalgebra $V^k({\mathfrak{osp}}_{1|2n})$ at generic level $k$ embeds in the equivariant $\mathcal{W}$-algebra of ${\mathfrak{sp}}_{2n}$ times $4n$ free fermions. This has two corollaries: (1) it provides a new proof that for generic $k$, the coset $\text{Com}(V^k({\mathfrak{sp}}_{2n}),V^k({\mathfrak{osp}}_{1|2n}))$ is isomorphic to ${\mathcal{W}}^{\ell }({\mathfrak{sp}}_{2n})$ for $\ell =-(n+1)+\frac{k+n+1}{2k+2n+1}$, and (2) we obtain the decomposition of ordinary $V^k({\mathfrak{osp}}_{1|2n})$-modules into $V^k({\mathfrak{sp}}_{2n})\otimes {\mathcal{W}}^{\ell }({\mathfrak{sp}}_{2n})$-modules. Next, if $k$ is an admissible level and $\ell$ is a non-degenerate admissible level for ${\mathfrak{sp}}_{2n}$, we show that the simple algebra $L_k({\mathfrak{osp}}_{1|2n})$ is an extension of the simple subalgebra $L_k({\mathfrak{sp}}_{2n})\otimes {\mathcal{W}}_{\ell }({\mathfrak{sp}}_{2n})$. Using the theory of vertex superalgebra extensions, we prove that the category of ordinary $L_k({\mathfrak{osp}}_{1|2n})$-modules is a semisimple, rigid vertex tensor supercategory with only finitely many inequivalent simple objects. It is equivalent to a certain subcategory of ${\mathcal{W}}_{\ell }({\mathfrak{sp}}_{2n})$-modules. A similar result also holds for the category of Ramond twisted modules. Due to a recent theorem of Robert McRae, we get as a corollary that categories of ordinary $L_k({\mathfrak{sp}}_{2n})$-modules are rigid.

23 pages

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Cited by in corpus (2)

• Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras
• Duality via convolution of W-algebras