Classification of Fermionic Topological Orders from Congruence Representations

The fusion rules and braiding statistics of anyons in (2+1)D fermionic topological orders are characterized by the modular data of a super-modular category. On the other hand, the modular data of a super-modular category form a congruence representation of the Gamma_theta subgroup of the modular group SL_2(Z). We provide a method to classify the modular data of super-modular categories by first obtaining the congruence representations of Gamma_theta and then building candidate modular data out of those representations. We carry out this classification up to rank 10. We obtain both unitary and non-unitary modular data, including all previously known unitary modular data, and also discover new classes of modular data of rank 10. We also determine the central charges of all these modular data, without explicitly computing their modular extensions.