Defect Charges, Gapped Boundary Conditions, and the Symmetry TFT

TL;DR

<3-5 sentence high-level summary> The paper develops a unified, computational framework for higher representations (defect charges) of generalized symmetries using the Symmetry TFT $\mathcal{Z}(\mathcal{C})$. By dimensionally reducing gapped boundaries on spheres $S^{p-1}$, defect charges are identified with boundary data and their interactions are encoded via module-category structures, junctions, and defect operator multiplets. The approach is illustrated through detailed examples across 2d–4d theories, including Dijkgraaf-Witten theories, anomalous 1-form symmetries, KOZ non-invertible defects, and (3+1)d duality symmetry, with particular attention to 't Hooft anomalies and obstructions to symmetric defects. The framework provides a practical TFT-based toolkit for analyzing defect RG flows, GW operators, and symmetry actions on defects, with potential applications to a broad class of quantum field theories exhibiting higher-form and non-invertible symmetries.

Abstract

We offer a streamlined and computationally powerful characterization of higher representations (higher charges) for defect operators under generalized symmetries, employing the powerful framework of Symmetry TFT $\mathcal{Z}(\mathcal{C})$. For a defect $\mathscr{D}$ of codimension p, these representations (charges) are in one-to-one correspondence with gapped boundary conditions for the SymTFT $\mathcal{Z}(\mathcal{C})$ on a manifold $Y = Σ_{d-p+1} \times S^{p-1}$, and can be efficiently described through dimensional reduction. We explore numerous applications of our construction, including scenarios where an anomalous bulk theory can host a symmetric defect. This generalizes the connection between 't Hooft anomalies and the absence of symmetric boundary conditions to defects of any codimension. Finally we describe some properties of surface charges for (3 + 1)d duality symmetries, which should be relevant to the study of Gukov-Witten operators in gauge theories.

Figures (10)

• Figure 1: SymTFT setup. Left the sandwich construction for the theory $X$, right the identification of charged multiplets.
• Figure 2: Correspondence between defects and boundary conditions. First we excise a neighbourhood bounded by $\widehat{\Sigma}$ from spacetime to obtain a boundary condition $B\mathscr{D}$. Finally, reducing on the sphere $S^{p-1}$, we study a related boundary condition in the dimensionally reduced bulk theory.
• Figure 3: Sym TFT setup for a boundary condition (Left) an for a defect (Right).
• Figure 4: SymTFT setup for a boundary multiplet (Right) and for a defect multiplet (Left).
• Figure 5: Wedge compactification allows to describe a boundary condition as a transparant interface between $X$ and a gapped theory $\mathcal{T}_B$.