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(If yes, is there a reference for this statement?)

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  • $\begingroup$ Addendum: I also wanted to know whether there are known examples of unitary modular tensor categories that do not have a Lagrangian algebra but do have zero central charge $\endgroup$
    – Frank
    Commented Feb 9, 2017 at 21:03

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The Drinfeld center of a spherical fusion category has topological central charge $0\pmod 8$ see Remark 5.19 in

Müger, Michael: From subfactors to categories and topology. II. The quantum double of tensor categories and subfactors. J. Pure Appl. Algebra 180 (2003), no. 1-2, 159–219.

There are modular tensor categories with central charge 0, which are not the Drinfeld center of a fusion category.

Take the pointed modular tensor category $\mathcal C(\mathbb Z/{17}\mathbb Z,q)$ with $q(x)=\exp(16\pi i x^2 /17)$. It is easy to check that the metric group $(\mathbb Z/{17}\mathbb Z,q)$ has no Lagrangian subgroup, thus no Lagrangian algebra. The easiest way to see it is that $17$ is no square. This category is $\mathrm{SU}(17)$ at level 1 thus has central charge 16 which means topological central charge $0\pmod 8$. Another way to calculate the central charge is by calculating the Gauss sum.

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