Gapped Boundaries of Fermionic Topological Orders and Higher Central Charges

The existence of gapped boundaries of bosonic topological orders can be tested in terms of the vanishing of higher central charges, which can be easily computed in terms of the modular data. For fermionic topological orders, even the chiral central charge admits no simple expression in terms of the modular data. Using the congruence property of representations formed by the modular data, we develop a method that tests whether the higher central charges of a fermionic topological order, including the chiral central charge, vanish. The test can be carried out entirely in terms of the modular data of the super-modular tensor category describing the fermionic topological order, and does not require explicit computation of modular extensions. This leads to a stringent set of easily computable necessary conditions for a fermionic topological order to admit a gapped boundary. We apply our test to known examples of fermionic topological orders to determine which of them potentially admit a gapped boundary.