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Abstract We show that the affine vertex superalgebra V k β’ (o β’ s β’ p 1 | 2 β’ n) V^k (ospβ|ββ) at generic level π embeds in the equivariant π²-algebra of s β’ p 2 β’ n spββ times 4 β’ n 4n free fermions. This has two corollaries: (1) it provides a new proof that, for generic π, the coset Com β‘ (V k β’ (s β’ p 2 β’ n), V k β’ (o β’ s β’ p 1 | 2 β’ n) ) Com (V^k (spββ), V^k (ospβ|ββ) ) is isomorphic to W β β’ (s β’ p 2 β’ n) W^ (spββ) for β = β (n + 1) + (k + n + 1) / (2 β’ k + 2 β’ n + 1) =- (n+1) + (k+n+1) / (2k+2n+1), and (2) we obtain the decomposition of ordinary V k β’ (o β’ s β’ p 1 | 2 β’ n) V^k (ospβ|ββ) -modules into V k β’ (s β’ p 2 β’ n) β W β β’ (s β’ p 2 β’ n) V^k (spββ) ^ (spββ) -modules. Next, if π is an admissible level and β is a non-degenerate admissible level for s β’ p 2 β’ n spββ, we show that the simple algebra L k β’ (o β’ s β’ p 1 | 2 β’ n) Lβ (ospβ|ββ) is an extension of the simple subalgebra L k β’ (s β’ p 2 β’ n) β W β β’ (s β’ p 2 β’ n) Lβ (spββ) W_ (spββ). Using the theory of vertex superalgebra extensions, we prove that the category of ordinary L k β’ (o β’ s β’ p 1 | 2 β’ n) Lβ (ospβ|ββ) -modules is a semisimple, rigid vertex tensor supercategory with only finitely many inequivalent simple objects. It is equivalent to a certain subcategory of W β β’ (s β’ p 2 β’ n) W_ (spββ) -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 β’ (s β’ p 2 β’ n) Lβ (spββ) -modules are rigid.
Creutzig et al. (Fri,) studied this question.