Ordinary modules for vertex algebras of 𝔬𝔰𝔭1|2𝑛

Abstract

Abstract We show that the affine vertex superalgebra V k ⁢ ( o ⁢ s ⁢ p 1 | 2 ⁢ n ) V^{k}(\mathfrak{osp}_{1|2n}) at generic level 𝑘 embeds in the equivariant 𝒲-algebra of s ⁢ p 2 ⁢ n \mathfrak{sp}_{2n} times 4 ⁢ n 4n free fermions. This has two corollaries: (1) it provides a new proof that, for generic 𝑘, the coset Com ⁡ ( V k ⁢ ( s ⁢ p 2 ⁢ n ) , V k ⁢ ( o ⁢ s ⁢ p 1 | 2 ⁢ n ) ) \operatorname{Com}(V^{k}(\mathfrak{sp}_{2n}),V^{k}(\mathfrak{osp}_{1|2n})) is isomorphic to W ℓ ⁢ ( s ⁢ p 2 ⁢ n ) \mathcal{W}^{\ell}(\mathfrak{sp}_{2n}) for ℓ = − ( n + 1 ) + ( k + n + 1 ) / ( 2 ⁢ k + 2 ⁢ n + 1 ) \ell=-(n+1)+(k+n+1)/(2k+2n+1) , and (2) we obtain the decomposition of ordinary V k ⁢ ( o ⁢ s ⁢ p 1 | 2 ⁢ n ) V^{k}(\mathfrak{osp}_{1|2n}) -modules into V k ⁢ ( s ⁢ p 2 ⁢ n ) ⊗ W ℓ ⁢ ( s ⁢ p 2 ⁢ n ) V^{k}(\mathfrak{sp}_{2n})\otimes\mathcal{W}^{\ell}(\mathfrak{sp}_{2n}) -modules. Next, if 𝑘 is an admissible level and ℓ is a non-degenerate admissible level for s ⁢ p 2 ⁢ n \mathfrak{sp}_{2n} , we show that the simple algebra L k ⁢ ( o ⁢ s ⁢ p 1 | 2 ⁢ n ) L_{k}(\mathfrak{osp}_{1|2n}) is an extension of the simple subalgebra L k ⁢ ( s ⁢ p 2 ⁢ n ) ⊗ W ℓ ⁢ ( s ⁢ p 2 ⁢ n ) 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 ⁢ ( o ⁢ s ⁢ p 1 | 2 ⁢ n ) 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 W ℓ ⁢ ( s ⁢ p 2 ⁢ n ) \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 ⁢ ( s ⁢ p 2 ⁢ n ) L_{k}(\mathfrak{sp}_{2n}) -modules are rigid.