The center $Z(\mathcal{C})$ of a spherical fusion category $\mathcal{C}$ (over $\mathbb{C}$) is a modular tensor category.


*Question:* What about the converse, i.e., can we characterize every modular tensor category $\mathcal{M}$ such that the equation $Z(\mathcal{C}) \simeq \mathcal{M}$ admits a solution $\mathcal{C}$ which is a fusion category?



On the existence of a square root for a modular tensor category
A characterization of Drinfeld centers of fusion categories is given in [this paper](http://arxiv.org/abs/1009.2117) as braided fusion categories containing a so-called Lagrangian algebra.


