Generalized Charges, Part I: Invertible Symmetries and Higher Representations

TL;DR

<3-5 sentence high-level summary> The paper develops a unified framework for charges carried by operators in quantum field theories with generalized global symmetries, showing that q-charges for invertible symmetries are naturally described by (q+1)-representations of the corresponding higher groups. It extends the familiar 0-form symmetry story (where 0-charges are representations) to higher-form and higher-group symmetries, and introduces a rich taxonomy that includes non-genuine and twisted charges, as well as symmetry fractionalization phenomena. The analysis uses higher-categorical language (2-representations, 3-representations, Drinfeld centers, and fusion categories) and provides concrete field-theory examples in dimensions 2–4, illustrating how localized and induced symmetries, anomalies, and condensation defects shape the charges. The work lays the groundwork for Part II, which will address non-invertible (categorical) symmetries and their action on charged operators, using tools like SymTFT and the Drinfeld center to generalize the construction. The results offer a principled, unified perspective on the algebraic structure of symmetries in QFT and topological phases with potential broad impact on dualities, defect networks, and higher-category gauge theories.

Abstract

$q$-charges describe the possible actions of a generalized symmetry on $q$-dimensional operators. In Part I of this series of papers, we describe $q$-charges for invertible symmetries; while the discussion of $q$-charges for non-invertible symmetries is the topic of Part II. We argue that $q$-charges of a standard global symmetry, also known as a 0-form symmetry, correspond to the so-called $(q+1)$-representations of the 0-form symmetry group, which are natural higher-categorical generalizations of the standard notion of representations of a group. This generalizes already our understanding of possible charges under a 0-form symmetry! Just like local operators form representations of the 0-form symmetry group, higher-dimensional extended operators form higher-representations. This statement has a straightforward generalization to other invertible symmetries: $q$-charges of higher-form and higher-group symmetries are $(q+1)$-representations of the corresponding higher-groups. There is a natural extension to higher-charges of non-genuine operators (i.e. operators that are attached to higher-dimensional operators), which will be shown to be intertwiners of higher-representations. This brings into play the higher-categorical structure of higher-representations. We also discuss higher-charges of twisted sector operators (i.e. operators that appear at the boundary of topological operators of one dimension higher), including operators that appear at the boundary of condensation defects.

Figures (25)

• Figure 1: The layer structure of genuine (2d $\mathcal{O}_2$) and non-genuine ($\mathcal{O}_1$ and $\mathcal{O}_0$) operators as naturally occurring in theories with $d\geq 3$.
• Figure 2: The layered structure for 2-category, which parallels the layer structure of operators in figure \ref{['fig:LayerCake']}.
• Figure 3: Action of a 0-form symmetry realized by a codimension-1 topological defect $D_{d-1}^{(g)}$, $g \in G^{(0)}$, on a local operator $\mathcal{O}$.
• Figure 4: A 0-form symmetry $g\in G^{(0)}$ may act by changing a line operator $L$ to another line operator $g\cdot L$.
• Figure 5: A topological local operator $D_0^{(h)}$ arising at the junction of a line operator $L$ and the 0-form symmetry generator $D_{d-1}^{(h)},h\in H_L$. Such topological local operators $D_0^{(h)}$ for all $h\in H_L$ generate an induced 0-form symmetry on $L$.