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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?

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    $\begingroup$ If you're asking whether or not every UMTC is the double of some UFC the answer is no. Fibonacci is a counter example. $\endgroup$
    – Matthew Titsworth
    Commented Jul 25, 2016 at 14:39

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A characterization of Drinfeld centers of fusion categories is given in this paper as braided fusion categories containing a so-called Lagrangian algebra.

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    $\begingroup$ Which is equivalent with $\mathcal C$ being in the the trivial class of the Witt group $\endgroup$
    – Marcel Bischoff
    Commented Jul 26, 2016 at 8:30

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