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

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