Anomalies of Coset Non-Invertible Symmetries

Anomalies of global symmetries provide important information on the quantum dynamics. We show the dynamical constraints can be organized into three classes: genuine anomalies, fractional topological responses, and integer responses that can be realized in symmetry-protected topological (SPT) phases. Coset symmetry can be present in many physical systems including quantum spin liquids, and the coset symmetry can be a non-invertible symmetry. We introduce twists in coset symmetries, which modify the fusion rules and the generalized Frobenius-Schur indicators. We call such coset symmetries twisted coset symmetries, and they are labeled by the quadruple $(G,K,ω_{D+1},α_D)$ in $D$ spacetime dimensions where $G$ is a group and $K\subset G$ is a discrete subgroup, $ω_{D+1}$ is a $(D+1)$-cocycle for group $G$, and $α_{D}$ is a $D$-cochain for group $K$. We present several examples with twisted coset symmetries using lattice models and field theory, including both gapped and gapless systems (such as gapless symmetry-protected topological phases). We investigate the anomalies of general twisted coset symmetry, which presents obstructions to realizing the coset symmetry in (gapped) symmetry-protected topological phases. We show that finite coset symmetry $G/K$ becomes anomalous when $G$ cannot be expressed as the bicrossed product $G=H\Join K$, and such anomalous coset symmetry leads to symmetry-enforced gaplessness in generic spacetime dimensions. We illustrate examples of anomalous coset symmetries with $A_5/\mathbb{Z}_2$ symmetry, with realizations in lattice models.

💡 Research Summary

The paper develops a comprehensive framework for “coset non‑invertible symmetries,” which arise when a subgroup (K) of a larger symmetry group (G) is gauged. The authors introduce a “twisted” description of such symmetries by a quadruple ((G,K,\omega_{D+1},\alpha_D)). Here (\omega_{D+1}) is a ((D+1))‑cocycle on (G) encoding a possible ’t Hooft anomaly of the original (G) symmetry, while (\alpha_D) is a (D)‑cochain on (K) that specifies a topological action for the gauged subgroup. Both twists can modify the fusion rules and associators of the resulting defect category, turning the ordinary coset (G/K) into a genuinely non‑invertible symmetry.

A central theme is the classification of anomalies associated with these twisted coset symmetries. The authors separate anomalies into three mutually exclusive classes: (1) genuine ’t Hooft anomalies ((