Gapped boundaries of fermionic topological orders and higher central charges

We develop a test for the vanishing of higher central charges of a fermionic topological order, which is a necessary condition for the existence of a gapped boundary, purely in terms of the modular data of the super-modular tensor category. More precisely, we test whether a given super-MTC has $c = 0$ mod $\frac{1}{2}$, and, if so, whether the modular extension with $c =0$ mod $8$ has vanishing higher central charges. The test itself does not require an explicit computation of the modular extensions and is easily carried out. We apply this test to known examples of super-modular tensor categories. Since our test allows us to obtain information about the chiral central charge of a super-modular tensor category in terms of its modular data without direct knowledge of its modular extensions, this can also be thought of as the first step towards a fermionic analogue of the Gauss-Milgram formula.