On finite non-degenerate braided tensor categories with a Lagrangian subcategory

Let W W be a finite dimensional vector space over C C viewed as a purely odd supervector space, and let s R e p ( W ) sRep(W) be the finite symmetric tensor category of finite dimensional superrepresentations of the finite supergroup W W . We show that the set of equivalence classes of finite non-degenerate braided tensor categories C C containing s R e p ( W ) sRep(W) as a Lagrangian subcategory is a torsor over the cyclic group Z / 16 Z Z/16Z . In particular, we obtain that there are 8 8 non-equivalent such braided tensor categories C C which are integral and 8 8 which are non-integral.