Anomalies of fermionic CFTs via cobordism and bootstrap

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 $\mathbb{Z}^f_2$ 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 $\Delta>2$ 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 $\mathrm{Spin}\text{-}\mathbb{Z}^f_{2^{l+1}}$ structures.