Symmetry TFTs and Anomalies of Non-Invertible Symmetries

TL;DR

This paper develops a practical, higher-dimensional framework to diagnose 't Hooft anomalies for non-invertible symmetries via the Symmetry TFT (SymTFT). It shows that linking invariants in DW-based SymTFTs provide a sufficient diagnostic: anomaly-free non-invertible symmetries admit choices of defect representatives with trivial bulk linkings, while nontrivial linkings signal anomalies, with explicit computations in 2d and 4d. By gauging finite invertible symmetries, the authors construct non-invertible defects and analyze their anomalies through bulk linkings, applying the method to the Abelian Higgs Model in 2d, adjoint QCD in 4d, and N=4 SYM. The work clarifies how anomalies of non-invertible symmetries constrain IR dynamics (e.g., forbidding trivially gapped phases) and provides a concrete, computable toolkit in higher dimensions. Overall, the study links higher-category symmetry data to tangible dynamical consequences in both two and four dimensions, enhancing our understanding of generalized symmetries and their anomalies.

Abstract

It is known that the 't Hooft anomalies of invertible global symmetries can be characterized by an invertible TQFT in one higher dimension. The analogous statement remains to be understood for non-invertible symmetries. In this note we discuss how the linking invariants in a non-invertible TQFT known as the Symmetry TFT (SymTFT) can be used as a diagnostic for 't Hooft anomalies of non-invertible symmetries. When the non-invertible symmetry is non-intrinsically non-invertible, and hence the SymTFT is a Dijkgraaf-Witten model, the linking invariants can be computed explicitly. We illustrate this proposal through the examples of the abelian Higgs model in 2d, as well as adjoint QCD and $\mathcal{N}=4$ super Yang-Mills in 4d. We also comment on how the 't Hooft anomalies of non-invertible symmetries impose new constraints on the dynamics.

Figures (8)

• Figure 1: Definition of the $F$-symbols, which generalize 't Hooft anomalies for non-invertible symmetries in $(1+1)$d. The sum is over all $L_y$ appearing in the fusion of $L_h$ with $L_k$. When the defects are invertible, the sum on the right hand side only contains one term.
• Figure 2: Any $d$-dimensional QFT can be expanded into a $(d+1)$-dimensional slab with the SymTFT living inside of it, with a topological boundary condition on the left boundary and a dynamical boundary condition on the right boundary.
• Figure 3: Bulk topological operators terminating on the topological boundary. We refer to $a$ as "terminable" and $b$ as "non-terminable."
• Figure 4: Two topological operators that are terminable on the topological boundary should have trivial linking invariant in the bulk.
• Figure 5: Decomposition of $X_2$ along a neck. The interface is located at $x=0$.