Ordinary modules for vertex algebras of 픬픰픭1|2푛.

We show that the affine vertex superalgebra V k ⁢ (o ⁢ s ⁢ p 1 | 2 ⁢ n ) at generic level 푘 embeds in the equivariant 풲-algebra of s ⁢ p 2 ⁢ n times 4 ⁢ n free fermions. This has two corollaries: (1) it provides a new proof that, for generic 푘, the coset Com ⁡ (V k ⁢ (s ⁢ p 2 ⁢ n ) , V k ⁢ (o ⁢ s ⁢ p 1 | 2 ⁢ n )) is isomorphic to W ℓ ⁢ (s ⁢ p 2 ⁢ n ) for ℓ = − (n + 1) + (k + n + 1) / (2 ⁢ k + 2 ⁢ n + 1) , and (2) we obtain the decomposition of ordinary V k ⁢ (o ⁢ s ⁢ p 1 | 2 ⁢ n ) -modules into V k ⁢ (s ⁢ p 2 ⁢ n ) ⊗ W ℓ ⁢ (s ⁢ p 2 ⁢ n ) -modules. Next, if 푘 is an admissible level and ℓ is a non-degenerate admissible level for s ⁢ p 2 ⁢ n , we show that the simple algebra L k ⁢ (o ⁢ s ⁢ p 1 | 2 ⁢ n ) is an extension of the simple subalgebra L k ⁢ (s ⁢ p 2 ⁢ n ) ⊗ W ℓ ⁢ (s ⁢ p 2 ⁢ n ) . Using the theory of vertex superalgebra extensions, we prove that the category of ordinary L k ⁢ (o ⁢ s ⁢ p 1 | 2 ⁢ n ) -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 ) -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 ) -modules are rigid. [ABSTRACT FROM AUTHOR]