Non-Invertible Symmetry Webs

TL;DR

This work develops a comprehensive framework for gauging 0-form symmetries in the presence of non-invertible symmetry structures by embedding the problem into fusion 2-categories. It introduces and analyzes sequential and partial gaugings, theta and condensation defects, and symmetry fractionalization within a 2-categorical language, producing detailed symmetry webs for several groups G (notably Z_2 × Z_2, Z_4, S_3, and D_8) and their applications to 3d orthogonal gauge theories. A key achievement is the explicit construction of complete categorical symmetry webs, including non-invertible sectors, and the demonstration of how line and surface fractionalization is controlled by 4-cocycles and 3-cocycles in associated 2-groups. The results generalize to higher dimensions via maximal symmetry categories tied to higher-group symmetries, offering a principled route to understanding categorical symmetries in a broad class of gauge theories. The framework provides computational tools for gauging in the presence of non-invertibles and paves the way for higher-dimensional and higher-form generalizations of symmetry webs.

Abstract

Non-invertible symmetries have by now seen numerous constructions in higher dimensional Quantum Field Theories (QFT). In this paper we provide an in depth study of gauging 0-form symmetries in the presence of non-invertible symmetries. The starting point of our analysis is a theory with $G$ 0-form symmetry, and we propose a description of sequential partial gaugings of sub-symmetries. The gauging implements the theta-symmetry defects of the companion paper [1]. The resulting network of symmetry structures related by this gauging will be called a non-invertible symmetry web. Our formulation makes direct contact with fusion 2-categories, and we uncover numerous interesting structures such as symmetry fractionalization in this categorical setting. The complete symmetry web is derived for several groups $G$, and we propose extensions to higher dimensions. The highlight of this analysis is the complete categorical symmetry web, including non-invertible symmetries, for 3d pure gauge theories with orthogonal gauge groups and its extension to arbitrary dimensions.

Figures (26)

• Figure 1: Categorical symmetry web for 3d gauge theories with gauge algebra $\mathfrak{so}(4N)$. We label each theory by its global gauge group, and the 2-category which is its symmetry category. The arrows denote gaugings of 0-form symmetries and each arrow is labelled by the subgroup of $D_8$ that is gauged along that arrow. We discuss each of these steps in the text. The notation for the groups $\mathbb{Z}_2^{[g]}$ is explained in the text around (\ref{['leftcoset']}). The section labels indicate where the particular gauging is discussed.
• Figure 2: Categorical symmetry web for 3d gauge theories starting with the theory $\mathfrak{T}_{1}$ with 0-form symmetry ${\mathbb Z}_2\times {\mathbb Z}_2$ (top) and $\mathbb{Z}_4$ (bottom), and gauging subsequently $\mathbb{Z}_2$. The category in the middle for theory $\mathfrak{T}_2$ has a trivial (top) and non-trivial (bottom) cocycle $\omega$, which is encoded in the group extension class $\epsilon$. This cocycle is key in order to obtain $\mathsf{2}\text{-}\mathsf{Rep} (\mathbb{Z}_4)$ after gauging the remaining $\mathbb{Z}_2$ to reach theory $\mathfrak{T}_3$. For $G= \mathbb{Z}_2 \times \mathbb{Z}_2$ the main difference is in the absence of the non-trivial cocycle, i.e. $\epsilon$ is trivial.
• Figure 3: In order to make $nD_2^{(\text{id})}$$H$-symmetric, we need to choose line operators (shown in blue) living at the junction of $D_2^{(h)}$ for all $h\in H$ and $nD_2^{(\text{id})}$.
• Figure 4: A 1-form symmetry generating line $D_1^{(\widehat{h})}$ (shown in blue) changes to another 1-form symmetry generating line $D_1^{(k\cdot \widehat{h})}$ (shown in teal) upon sliding it across a 0-form symmetry generating surface $D_2^{(k)}$ (shown in red). See equation (\ref{['FUS']}).
• Figure 5: Symmetry Fractionalization: The junctions of topological surface operators $D_2^{(k_i)}$ generating the $K$ 0-form symmetry with a fixed line $D_1^{(\widehat{h})}$ generating the 1-form symmetry do not obey $K$ multiplication laws, but instead acquire a projectivity $\widehat{h}(\epsilon(k_1,k_2))\in U(1)$.