Symmetry TFTs for Non-Invertible Defects

TL;DR

This work develops a comprehensive Symmetry TFT framework for non-invertible duality defects across dimensions, explicitly building SymTFTs for (1+1)d TY$(Z_N)$-like settings and their (3+1)d analogs. By gauging electromagnetic exchanges in bulk DW theories and analyzing the Drinfeld centers, it reproduces the non-invertible fusion rules and F-symbols of duality interfaces and twist defects, while clarifying how higher duality interfaces arise as boundaries of bulk topological data. The authors extend the construction to (4+1)d, deriving the twist defects and their fusion in odd/even $N$, and show how duality defects in lower dimensions emerge upon shrinking the bulk, establishing detailed correspondences among lines, surfaces, and higher-manifold operators. A key application is distinguishing intrinsically non-invertible from non-intrinsic non-invertible duality defects in 1+1d, with a clear criterion based on the invariance of the SymTFT under topological manipulations. Overall, the paper provides a unifying, gauge-theoretic method to compute fusion, F-symbols, and higher-categorical data for non-invertible symmetries across multiple spacetime dimensions and emphasizes the role of SymTFTs as a topological invariant under topological manipulations.

Abstract

Given any symmetry acting on a $d$-dimensional quantum field theory, there is an associated $(d+1)$-dimensional topological field theory known as the Symmetry TFT (SymTFT). The SymTFT is useful for decoupling the universal quantities of quantum field theories, such as their generalized global symmetries and 't Hooft anomalies, from their dynamics. In this work, we explore the SymTFT for theories with Kramers-Wannier-like duality symmetry in both $(1+1)$d and $(3+1)$d quantum field theories. After constructing the SymTFT, we use it to reproduce the non-invertible fusion rules of duality defects, and along the way we generalize the concept of duality defects to \textit{higher} duality defects. We also apply the SymTFT to the problem of distinguishing intrinsically versus non-intrinsically non-invertible duality defects in $(1+1)$d.

Figures (45)

• Figure 1: Schematic picture of the SymTFT. By imposing Dirichlet (resp. Neumann) boundary conditions on the left and the non-topological boundary condition (\ref{['eq:dynamboundary']}) on the right, we may compactify to obtain ${\mathcal{X}}$ (resp. ${\mathcal{X}}/G$). In general, there exist other topological boundary conditions as well.
• Figure 2: A theory with anomaly $\alpha$ can be realized on the boundary of an invertible phase Inv$^\alpha(A)$. This is possible if and only if the Symmetry TFT is a (generalized) DW theory.
• Figure 3: Decomposition of $X_2$ along a neck. The interface is located at $x=0$.
• Figure 4: The duality defect from gauging $\mathbb{Z}_N$ over half of the spacetime $X_2^{\geq 0}$ with Dirichlet boundary conditions.
• Figure 5: Fusion of two duality interfaces. The partition function on $X^{\geq 0}$ is given by \ref{['eq:NN']}.