Anomalous Actions of Groups on Tensor Categories
Noah Lanier
TL;DR
This work categorifies the notion of anomalous group actions on algebras to tensor categories by introducing π-anomalous actions as 3-functors from 3-Gr(Q,π) to 2-Aut_⊗(C). Given a surjection ρ: G → Q with kernel K, a 4-cocycle π and a 3-cochain ω lifting π (dω = ρ^*(π)), together with a G-action on C, the authors construct an explicit π-anomalous action of Q on the twisted crossed product C ⋉_ω K, thereby realizing extensions of fusion-like categories in the presence of anomalies. The main theorem is proved via a careful higher-categorical construction: endofunctors q_*, monoidal data ψ^q, pseudo-natural isomorphisms χ, and coherence modifications Ω, ensuring the necessary trihomomorphism axioms hold. They illustrate the method with two broad classes of examples—free groups and finite groups—showing how to produce explicit π-anomalous actions, including concrete realizations for Q = S_4 and constructions from group extensions G = Z_n ⋊_c Q. The results provide a practical framework for generating π-anomalous actions and have potential applications to gauging, topological boundaries in (3+1)-D orders, and the extension theory of fusion categories.
Abstract
For a group $G$ and a 4-cocycle $π\in Z^{4}(G,\mathbb{k}^{\times})$, a $π$-anomalous action of $G$ on a linear monoidal category $C$ is a linear monoidal 2-functor between 3-groups $3\text{-Gr}(G,π)\rightarrow \text{2-Aut}_\otimes(C)$ where the latter denotes the 3-group of autoequivalences of $C$. Given $G$ and $π$, we provide a method of constructing anomalous actions of $3\text{-Gr}(G,π)$ on a tensor categories.