On Unitary 2-Group Symmetries

TL;DR

This work develops a higher-categorical generalization of unitary symmetry actions to line operators in quantum field theories with finite 2-group symmetry $\mathcal{G}$. By modeling line operators as forming a 2-Hilbert space and symmetry actions as $\dagger$-2-functors $B\mathcal{G}\to \text{Mat}(\text{Herm})$, the authors classify unitary 2-representations of $\mathcal{G}$, showing irreducibles are induced from sub-2-groups with data $(H,\lambda,u,p)$ and that intertwiners are labeled by double cosets with projective representations on intersections; positivity yields a variant into $\text{Mat}(\text{Hilb})$. The framework recovers the ordinary 2-representation theory when higher data like $q$ or $s$ are forgotten, while highlighting a reflection anomaly encoded by a $q\in \mathrm{Hom}(G,\mathbb{Z}_2)$ that constrains unitary line operator actions. Together, these results extend Wigner-type unitarity to extended observables and connect symmetry, anomaly structure, and higher representation theory in a concrete, computable classification. The work provides a foundation for understanding line operator transformations under 2-group symmetries and offers tools for analyzing twisted sectors and their intertwiners in topological and quantum field-theoretic contexts.

Abstract

Global internal symmetries act unitarily on local observables or states of a quantum system. In this note, we aim to generalise this statement to extended observables by considering unitary actions of finite global 2-group symmetries $\mathcal{G}$ on line operators. We propose that the latter transform in unitary 2-representations of $\mathcal{G}$, which we classify up to unitary equivalence. Our results recover the known classification of ordinary 2-representations of finite 2-groups, but provide additional data interpreted as a type of reflection anomaly for $\mathcal{G}$.