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

Let W W be a finite dimensional vector space over alttext="double-struck upper C"> C encoding="application/x-tex">\mathbb {C} viewed as purely odd supervector space, and let alttext="s R e p left-parenthesis W right-parenthesis"> s R e p ( stretchy="false">) encoding="application/x-tex">sRep(W) the symmetric tensor category of superrepresentations supergroup . We show that set equivalence classes non-degenerate braided categories alttext="script class="MJX-tex-caligraphic" mathvariant="script">C encoding="application/x-tex">\mathcal containing Lagrangian subcategory is torsor cyclic group Z slash 16 double-struck Z"> mathvariant="double-struck">Z / 16 {Z}/16\mathbb {Z} In particular, we obtain there are alttext="8"> 8 encoding="application/x-tex">8 non-equivalent such which integral non-integral.