Unifying Constructions of Non-Invertible Symmetries

TL;DR

This work presents a unified framework showing that a broad class of non-invertible symmetries in $d\geq 3$ QFTs arise from gauging invertible higher-group symmetries via theta defects. It introduces a comprehensive mathematical program based on higher-fusion categories to translate these physical constructions into computable data, including higher vector spaces, higher representations, and module/bimodule structures. The paper distinguishes universal theta symmetries from theory-specific twisted theta defects and condensation defects, and analyzes their fusion, obstruction, and interface behavior, including ABJ anomalies and duality defects. The combined physical and mathematical program aims to classify non-invertible symmetries systematically and to provide a concrete computational toolkit for constructing and manipulating them in diverse dimensions and theories.

Abstract

In the past year several constructions of non-invertible symmetries in Quantum Field Theory in $d\geq 3$ have appeared. In this paper we provide a unified perspective on these constructions. Central to this framework are so-called theta defects, which generalize the notion of theta-angles, and allow the construction of universal and non-universal topological symmetry defects. We complement this physical analysis by proposing a mathematical framework (based on higher-fusion categories) that converts the physical construction of non-invertible symmetries into a concrete computational scheme.

Figures (17)

• Figure 1: Stacking a $\Gamma$-symmetric $d$-dimensional QFT $\mathfrak{T}_\mathcal{S}$ with a $d$-dimensional SPT phase $\mathfrak{I}_{\mathcal{S}"}$ protected by $\Gamma$ higher-group symmetry leads to a new $\Gamma$-symmetric $d$-dimensional QFT $\mathfrak{T}_\mathcal{S}\otimes_\Gamma\mathfrak{I}_{\mathcal{S}"}$ which can be identified with the $d$-dimensional QFT $\mathfrak{T}_{\mathcal{S}'}$ obtained by choosing a different coupling $\mathcal{S}'$ of $\mathfrak{T}$ to $\Gamma$ backgrounds. Gauging further the $\Gamma$ symmetry leads to the $d$-dimensional QFT $(\mathfrak{T}_\mathcal{S}\otimes_\Gamma\mathfrak{I}_{\mathcal{S}"})/\Gamma=\mathfrak{T}_{\mathcal{S}'}/\Gamma$ which is also referred to as the $d$-dimensional QFT obtained by adding theta angle $\mathfrak{I}_{\mathcal{S}"}$ to the $d$-dimensional QFT $\mathfrak{T}_\mathcal{S}/\Gamma$ obtained by gauging the $\Gamma$ symmetry of $\mathfrak{T}_\mathcal{S}$.
• Figure 2: $\mathfrak{T}$ is a $d$-dimensional QFT and $\mathsf{T}$ is a $p$-dimensional QFT. Stacking $\mathsf{T}$ inside the spacetime occupied by $\mathfrak{T}$ produces a $p$-dimensional defect $D_p^{(\mathsf{T})}$ of $\mathfrak{T}$. If $\mathsf{T}$ is a TQFT, then $D_p^{(\mathsf{T})}$ is a topological defect of $\mathfrak{T}$.
• Figure 3: $\mathfrak{T}$ is a $d$-dimensional QFT, $\mathsf{T}$ is a $p$-dimensional QFT and $D_p$ is a $p$-dimensional defect of $\mathfrak{T}$. Stacking $\mathsf{T}$ inside the worldvolume occupied by $D_p$ produces a $p$-dimensional defect $D_p^{(\mathsf{T})}\otimes D_p$ of $\mathfrak{T}$. If $\mathsf{T}$ is a TQFT and $D_p$ is a topological defect, then $D_p^{(\mathsf{T})}\otimes D_p$ is a topological defect of $\mathfrak{T}$.
• Figure 4: $\mathfrak{T}$ is a $d$-dimensional QFT, $D_p$ is a $p$-dimensional topological defect of $\mathfrak{T}$, $D'_p$ is a general, possibly non-topological, defect of $\mathfrak{T}$. Stacking $D_p$ inside the worldvolume occupied by $D'_p$ produces a general $p$-dimensional defect $D_p\otimes D'_p$ of $\mathfrak{T}$. If $D'_p$ is also topological, then $D_p\otimes D'_p$ is a topological defect of $\mathfrak{T}$.
• Figure 5: $\mathfrak{T}$ is a $d$-dimensional QFT with a non-anomalous $\Gamma^{(p)}$$p$-form symmetry, and $\mathfrak{I}_{\widehat{\gamma}}$ is a $p+1$-dimensional SPT phase protected by $\Gamma^{(p)}$. Stacking $\mathfrak{I}_{\widehat{\gamma}}$ inside the spacetime occupied by $\mathfrak{T}$ produces a $\Gamma^{(p)}$-symmetric topological defect $D_{p+1}^{(\mathfrak{I}_{\widehat{\gamma}})}$ of $\mathfrak{T}$. Upon gauging $\Gamma^{(p)}$, we land on a $d$-dimensional QFT $\mathfrak{T}/\Gamma^{(p)}$ with a topological defect $D^{(\widehat{\gamma})}_{p+1}$. These $(p+1)$-dimensional topological defects generate a $\widehat{\Gamma}^{(p)}$$(d-p-2)$-form symmetry of $\mathfrak{T}/\Gamma^{(p)}$.