The centre $Z(\mathcal{C})$ of a fusion category $\mathcal{C}$, is a unitary modular tensor category.

Question: What about the converse, i.e., can we characterize every unitary 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?

Remark: Everything is supposed over $\mathbb{C}$.

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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 Jul 25 '16 at 14:39

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 Jul 26 '16 at 8:30

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