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

Abstract: We develop a systematic framework for constructing (3+1)-dimensional topological quantum field theories (TQFTs) that realize specified anomalies of finite symmetries, as encountered in gauge theories with fermions or fermionic lattice systems. Our approach generalizes the Wang-Wen-Witten symmetry-extension construction to the fermionic setting, building on two recent advances in the study of fermionic TQFTs and related homotopy theory. The first is the categorical classification of anomalous TQFTs in (3+1)d. The second, which we develop further in a planned sequel to this paper, is a hastened Adams spectral sequence for computing supercohomology groups, closely paralleling techniques from cobordism theory. By integrating supercohomology and cobordism methods within the recently developed categorical framework of fusion 2-categories, we provide a concrete and systematic route to constructing fermionic TQFTs with specified anomalies, thereby establishing a conceptual bridge between anomaly realization, cobordism, and higher-categorical structures.