Ordinary modules for vertex algebras of osp1|2n

We show that the affine vertex superalgebra V-k(osp(1|2n)) at generic level embeds in the equivariant W-algebra of sp2nsp2n times 4n4n free fermions. This has two corollaries: (1) it provides a new proof that, for generic , the coset Com(V-k(sp(2n)),V-k(os(p1|2n))) is isomorphic to W-l(sp(2n))W-l(sp(2n)) for l=-(n+1)+(k+n+1)/(2k+2n+1) , and (2) we obtain the decomposition of ordinaryV(k)(os(p1|2n)) -modules intoVk(sp2n)circle times W-l(sp2n)-modules. Next, if is an admissible level and l is a non-degenerate admissible level for sp(2n) , we show that the simple algebra L-k(osp(1|2n))L-k(os(p1|2n)) is an extension of the simple subalgebra L-k(sp(2n))circle times W-l(sp(2n)) . Using the theory of vertex superalgebra extensions, we prove that the category of ordinaryL(k)(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(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(sp(2n)) -modules are rigid