How to Build Anomalous (3+1)d Topological Quantum Field Theories

We develop a systematic framework for constructing (3+1)-dimensional topological orders or topological quantum field theories (TQFTs) that realize specified anomalies of finite symmetries, as encountered in gauge theories with fermions or in fermionic lattice systems. Our approach generalizes the symmetry-extension construction to the fermionic setting, and is grounded in recent advances in the categorical classification of anomalous TQFTs in (3+1)d. In this framework, symmetry-extension data of a supercohomology theory are translated into a fusion 2-category, on which the anomalous TQFT is built. Building on this machinery, we demonstrate explicit calculations for various symmetry groups and their associated anomalies, with the help of a hastened Adams spectral sequence for computing supercohomology groups which we will detail in a planned sequel. Finally, we prove that all supercohomology anomalies can be realized by fermionic topological orders, whereas beyond-supercohomology anomalies cannot, resolving a question of C\'ordova--Ohmori for fermionic (3+1)d systems with finite symmetries.