Modular extension of topological orders from congruence representations

We present an efficient method to compute the modular extension of both fermionic topological orders and Z₂-symmetric bosonic topological orders in two spatial dimensions, from congruence representations of SL₂(Z) and its subgroups. To demonstrate the validity of our approach, we provide explicit calculations for topological orders with ranks up to 10 for the fermionic cases and up to 6 for the bosonic cases. Along the way, we clarify the relation between fermionic rational conformal field theories, which live on the boundary of the corresponding fermionic topological orders, and modular extensions. We show that the SL₂(Z) representation of the Ramond-Ramond sector can be determined from the Nuveu-Schwarz-Nuveu-Schwarz sector using the modular extensions.