Non-invertible symmetries and mixed anomalies from conserved current construction in (3+1)D twisted $BF$ topological quantum field theories

TL;DR

This paper develops a systematic method to construct and analyze non-invertible higher-form symmetries and their anomalies in (3+1)D twisted $BF$ topological quantum field theories, with applications to non-Abelian topological orders.

Contribution

It introduces a current-based construction of generalized symmetries, including non-invertible types, and provides fusion rules and anomaly diagnostics directly from the continuum action.

Findings

Constructed topological symmetry operators from conserved currents.

Identified two classes of symmetries: invertible and non-invertible.

Diagnosed mixed anomalies and their cancellation mechanisms.

Abstract

We develop a current-based construction of generalized symmetries in $(3+1)$D twisted $BF$ topological quantum field theories (TQFTs), focusing on intrinsically non-invertible higher-form symmetries and their mixed anomalies. Starting from the equations of motion, we extract conserved currents and exponentiate the corresponding charges to obtain topological symmetry operators. This gives a step-by-step procedure for constructing symmetry operators, fusion, and anomaly diagnostics directly from the continuum action. We focus on twisted $BF$ theories with gauge group $G=\prod_i \mathbb{Z}_{N_i}$ and an $a\wedge a\wedge b$ twist, where $a$'s and $b$ are 1-form and 2-form gauge fields, respectively. These theories realize non-Abelian $(3+1)$D TQFTs supporting Borromean-rings braiding and describe three-dimensional non-Abelian topological orders in condensed matter. For $G=(\mathbb{Z}_2)^3$, a microscopic realization is given by the $\mathbb{D}_4$ Kitaev quantum double model. Two distinct classes of conserved currents emerge: Type-I currents generate invertible higher-form symmetries with group-like fusion, while Type-II currents require additional consistency conditions on gauge-field configurations, leading to intrinsically non-invertible symmetries dressed by projectors. We compute the fusion algebra: invertible operators admit inverses, while non-invertible ones exhibit multi-channel fusion governed by projector fusion. We diagnose mixed anomalies by coupling multiple conserved currents to background gauge fields, revealing two outcomes: anomalies canceled by anomaly inflow from a higher-dimensional theory, and intrinsic gauging obstructions encoded in the $(3+1)$D continuum theory. Overall, our results provide a unified and practical approach for constructing and characterizing higher-form symmetries, which can be extended to more general TQFTs and topological orders.