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

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Let W W be a finite dimensional vector space over C \mathbb {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 \mathcal {C} containing s R e p ( W ) sRep(W) as a Lagrangian subcategory is a torsor over the cyclic group Z / 16 Z \mathbb {Z}/16\mathbb {Z} . In particular, we obtain that there are 8 8 non-equivalent such braided tensor categories C \mathcal {C} which are integral and 8 8 which are non-integral.