A Topological Invariant for Modular Fusion Categories

The modular data of a modular category $\mathcal{C}$, consisting of the $S$-matrix and the $T$-matrix, is known to be an incomplete invariant of $\mathcal{C}$. More generally, the invariants of framed links and knots defined by a modular category as part of a topological quantum field theory can be viewed as numerical invariants of the category. Among these invariants, we study the invariant defined by the Borromean link colored by three objects. Thus we obtain a tensor that we call $B$. We derive a formula for the Borromean tensor for the twisted Drinfeld doubles of finite groups. Along with $T$, it distinguishes the $p$ non-equivalent modular categories of the form $\mathcal{Z}({\rm Vec}_G^\omega)$ for $G$ the non-abelian group $\mathbb{Z}/q\mathbb{Z} \rtimes \mathbb{Z}/p\mathbb{Z}$, which are not distinguished by the modular data.