Field-Theoretical Construction of Conserved Currents, Non-Invertible Symmetries, and Mixed Anomalies in Three-Dimensional Non-Abelian Topological Order

TL;DR

This work develops a continuum field-theoretic framework to identify and characterize generalized symmetries in a (3+1)D Borromean-Rings topological order described by a twisted BF theory with gauge group $G=igotimes_i \mathbb{Z}_{N_i}$ and a twisted term $S_{aab}$. By extracting conserved currents from the equations of motion, the authors separate currents into Type-I (invertible) and Type-II (non-invertible) classes; the latter rely on projector constraints that prevent inverses, yielding non-invertible higher-form symmetries. They construct explicit symmetry operators, analyze their fusion rules, and classify mixed anomalies by gauging currents with background fields, identifying both bulk-cancellable and intrinsic obstructions. The results illuminate how invertible and non-invertible higher-form symmetries organize in 3D topological orders and how anomaly inflow can resolve certain, but not all, gauging inconsistencies. This framework provides a concrete field-theoretic handle on symmetry structures in 3D topological order and lays groundwork for lattice realizations, boundary analyses, and broader diagrammatic consistency in higher-dimensional topological phases.

Abstract

..In this work, we investigate generalized symmetries, with particular emphasis on non-invertible ones, in three-dimensional non-Abelian topological orders hosting both particle- and loop-like excitations. We adopt a continuum topological field theory description, focusing on twisted $BF$ theories with gauge group $G=\prod_i \mathbb{Z}_{N_i}$ and an $a \wedge a \wedge b$ twisted term. This field theory supports Borromean-Rings braiding and realizes non-Abelian topological order, which for $G=(\mathbb{Z}_2)^3$ admits a microscopic realization via the $\mathbb{D}_4$ Kitaev quantum double lattice model. We systematically identify all generalized symmetry operators by extracting conserved currents from the equations of motion. Two distinct classes of currents emerge: type-I currents, which generate invertible higher-form symmetries, and type-II currents, which give rise to non-invertible higher-form symmetries. The non-invertibility originates from projectors accompanying the symmetry operators, which restrict admissible gauge-field configurations. We further analyze the fusion rules of these symmetries, showing that invertible symmetries admit inverses, while non-invertible symmetries fuse through multiple channels. Finally, we study mixed anomalies among these generalized symmetries by simultaneously coupling multiple currents to proper types of background gauge fields and examining their gaugeability. We identify two types of mixed anomalies: one cancellable by topological field theories in one higher dimension, and another representing an intrinsic gauging obstruction encoded in the $(3+1)$D continuum topological field theory...