Generalized Charges, Part II: Non-Invertible Symmetries and the Symmetry TFT

TL;DR

<3-5 sentence high-level summary> The paper develops a unified, category-theoretic framework for generalized charges of finite, including non-invertible, symmetries in quantum field theories via the Symmetry TFT (SymTFT). It shows that q-charges are encoded by topological defects in the bulk SymTFT, precisely the objects of the Drinfeld center Z(S) of the symmetry category S, and that gauging leaves Z(S) invariant while permuting symmetry boundary data. The authors develop a detailed program in 2d and 3d, with explicit pictures for projections, half-braidings, and theta defects, and illustrate how untwisted and twisted sector operators fit into irreducible multiplets under non-invertible actions. This framework provides a robust tool for classifying S-symmetric TQFTs and understanding how symmetries act on operators across dimensions, with concrete实例 in BF-type, higher-form, higher-group, and Ising-type categorical symmetries. The work advances both mathematical structure (Drinfeld centers of fusion categories) and physical applications (sandwich construction and boundary/bulk correspondences) with potential implications for dualities, gaugings, and fault-tolerant boundary theories.

Abstract

Consider a d-dimensional quantum field theory (QFT) $\mathfrak{T}$, with a generalized symmetry $\mathcal{S}$, which may or may not be invertible. We study the action of $\mathcal{S}$ on generalized or $q$-charges, i.e. $q$-dimensional operators. The main result of this paper is that $q$-charges are characterized in terms of the topological defects of the Symmetry Topological Field Theory (SymTFT) of $\mathcal{S}$, also known as the ``Sandwich Construction''. The SymTFT is a $(d+1)$-dimensional topological field theory, which encodes the symmetry $\mathcal{S}$ and the physical theory in terms of its boundary conditions. Our proposal applies quite generally to any finite symmetry $\mathcal{S}$, including non-invertible, categorical symmetries. Mathematically, the topological defects of the SymTFT form the Drinfeld Center of the symmetry category $\mathcal{S}$. Applied to invertible symmetries, we recover the result of Part I of this series of papers. After providing general arguments for the identification of $q$-charges with the topological defects of the SymTFT, we develop this program in detail for QFTs in 2d (for general fusion category symmetries) and 3d (for fusion 2-category symmetries).

Figures (43)

• Figure 1: The sandwich construction: a theory $\mathfrak{T}$ in $d$ dimensions with global symmetry $\mathcal{S}$ can be obtained as an interval compactification of a $d+1$ dimensional SymTFT $\mathfrak{Z}(\mathcal{S})$ with two boundary conditions: $\mathfrak{B}^{\text{sym}}_{\mathcal{S}}$ is the topological symmetry boundary, where all the symmetry structure is localized, and $\mathfrak{B}^{\text{phys}}_{\mathfrak{T}}$ is the physical boundary. After the interval compactification, the topological defects (drawn as blue lines) of $\mathfrak{B}^{\text{sym}}_\mathcal{S}$ become topological defects of the theory $\mathfrak{T}$ that generate the $\mathcal{S}$ symmetry of $\mathfrak{T}$.
• Figure 2: SymTFT for BF type theories with $p$-form symmetry. The bulk topological defect $\bm{Q}_{p+1}^{(b)}$ has Dirichlet boundary condition on the symmetry boundary and ends in a $p$-dimensional operator $\mathcal{E}_p$, and $\bm{Q}_{r= d-p-1}^{(c)}$ has Neumann boundary condition, as a consequence of which it projects onto the symmetry boundary to become a non-trivial topological defect $S_{r}^{c}$. After interval compactification, the above picture shows precisely the non-trivial linking between the charge $\mathcal{O}_p$ and the symmetry defect $D_r^c$.
• Figure 3: Construction of the $q$-charges from bulk topological operators $\bm{Q}_{q+1}$: The topological operator of the SymTFT $\bm{Q}_{q+1}$ ends on the symmetry boundary $\mathfrak{B}^{\text{sym}}_\mathcal{S}$ as well as on the physical boundary $\mathfrak{B}^{\text{phys}}_\mathfrak{T}$: in a $q$-dimensional topological operator $\mathcal{E}_q$ and a non-topological operator $\mathcal{M}_q$, respectively. After interval compactification, this results in a $q$-charge $\mathcal{O}_q$. The operator can also be a twisted sector for the symmetry $\mathcal{S}$, i.e. attached to topological defects $D_{q+1}$ in $\mathcal{S}$. This occurs whenever $\mathcal{E}_q$ in $\mathfrak{B}^{\text{sym}}_{\mathcal{S}}$ forms a junction between $\bm{Q}_{q+1}$ and $S_{q+1}$, which is a topological defect in $\mathcal{S}$.
• Figure 4: The Symmetry TFT (SymTFT) $\mathfrak{Z}({\mathcal{S}})$ is a $(d+1)$-dimensional topological field theory associated to a symmetry $\mathcal{S}$ of $d$-dimensional theories. The SymTFT admits a topological boundary condition, namely the symmetry boundary condition $\mathfrak{B}^{\text{sym}}_{\mathcal{S}}$, whose symmetry category matches the fusion $(d-1)$-category associated to the symmetry $\mathcal{S}$. Given an $\mathcal{S}$-symmetric $d$-dimensional theory $\mathfrak{T}_\sigma$, there exists a corresponding boundary condition $\mathfrak{B}^{\text{phys}}_{\mathfrak{T}_\sigma}$ of the symmetry TFT $\mathfrak{Z}(\mathcal{S})$, which we refer to as the physical boundary condition. As shown in the figure, compactifying on an interval with the symmetry topological boundary condition $\mathfrak{B}^{\text{sym}}_{\mathcal{S}}$ on one side, and the physical (not necessarily topological) boundary condition $\mathfrak{B}^{\text{phys}}_{\mathfrak{T}_\sigma}$ on the other side, recovers the $\mathcal{S}$-symmetric theory $\mathfrak{T}_\sigma$.
• Figure 5: After the interval compactification, the topological defects $S_p\in \mathcal{S}$ (drawn as blue lines) of $\mathfrak{B}^{\text{sym}}_\mathcal{S}$ become topological defects $D_p\in \mathcal{S}(\mathfrak{T})$ of the theory $\mathfrak{T}$ that generate the $\mathcal{S}$ symmetry of $\mathfrak{T}_\sigma$.