Ordinary modules for vertex algebras of 𝔬𝔰𝔭1|2𝑛

Abstract We show that the affine vertex superalgebra V k ⁢ (o ⁢ s ⁢ p 1 | 2 ⁢ n) V^k (osp₁|₂₍) at generic level 𝑘 embeds in the equivariant 𝒲-algebra of s ⁢ p 2 ⁢ n sp₂₍ 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) ) Com (V^k (sp₂₍), V^k (osp₁|₂₍) ) is isomorphic to W ℓ ⁢ (s ⁢ p 2 ⁢ n) W^ (sp₂₍) for ℓ = − (n + 1) + (k + n + 1) / (2 ⁢ k + 2 ⁢ n + 1) =- (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 (osp₁|₂₍) -modules into V k ⁢ (s ⁢ p 2 ⁢ n) ⊗ W ℓ ⁢ (s ⁢ p 2 ⁢ n) V^k (sp₂₍) ^ (sp₂₍) -modules. Next, if 𝑘 is an admissible level and ℓ is a non-degenerate admissible level for s ⁢ p 2 ⁢ n sp₂₍, we show that the simple algebra L k ⁢ (o ⁢ s ⁢ p 1 | 2 ⁢ n) L₊ (osp₁|₂₍) is an extension of the simple subalgebra L k ⁢ (s ⁢ p 2 ⁢ n) ⊗ W ℓ ⁢ (s ⁢ p 2 ⁢ n) L₊ (sp₂₍) W_ (sp₂₍). Using the theory of vertex superalgebra extensions, we prove that the category of ordinary L k ⁢ (o ⁢ s ⁢ p 1 | 2 ⁢ n) L₊ (osp₁|₂₍) -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) W_ (sp₂₍) -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₊ (sp₂₍) -modules are rigid.