We consider general fermionic quantum field theories with a global finite group symmetry G, focusing on the case of 2-dimensions and torus spacetime. The modular transformation properties of the family of partition functions with different backgrounds is determined by the 't Hooft anomaly of G and fermion parity. For a general possibly non-abelian G we provide a method to determine the modular transformations directly from the bulk 3d invertible topological quantum field theory (iTQFT) corresponding to the anomaly by inflow. We also describe a method of evaluating the character map from the real representation ring of G to the group which classifies anomalies. Physically the value of the map is given by the anomaly of free fermions in a given representation. We assume classification of the anomalies/iTQFTs by spin-cobordisms. As a byproduct, for all abelian symmetry groups G, we provide explicit combinatorial expressions for corresponding spin-bordism invariants in terms of surgery representation of arbitrary closed spin 3-manifolds. We work out the case of G = Z 2 in detail, and, as an application, we consider the constraints that 't Hooft anomaly puts on the spectrum of the infrared conformal field theory.

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We study constraints on the space of d = 2 fermionic CFTs as a function of non-perturbative anomalies exhibited under a fermionic discrete symmetry group G f , focusing our attention also on cases where G f is non-abelian or presents a non-trivial twist of the Z f 2 subgroup. For the cases we selected, among our results we find that modular bootstrap consistency bounds predict the presence of relevant/marginal operators only for some groups and anomalies. From this point of view, the appearance in the analysis of several kinks around irrelevant operators with ∆ > 2 means that for fermionic systems with increasingly larger symmetry groups modular bootstrap is able to give less constraining bounds than its bosonic counterpart. Within our analysis we show how the anomaly constraints on fermionic CFTs can be effectively recovered from the structure of the abelian subgroups of G f . Finally, we extend the previous surgery description of bordism invariants that describe 3d abelian spin-TQFTs, in order to include the case of theories with Spin-Z f 2 l+1 structures.