Anomalies of Coset Non-Invertible Symmetries

TL;DR

The paper develops a unified framework for coset non-invertible symmetries in D spacetime dimensions, parameterized by (G,K,ω_{D+1},α_D) and realized via a sandwich bulk-boundary construction. It shows how twists modify fusion rules, associators, and the Frobenius-Schur data, and it uses bulk G-gauge theory with a K-gauged boundary to analyze anomalies as obstructions to SPT realization. A central result is that finite coset symmetries G/K are anomaly-free and realizable in symmetric gapped phases when G=H⋈K with [ω_{D+1}|_H]=0, while in other cases the coset is anomalous and can enforce gaplessness, with explicit examples including A_5/ Z_2 and A_6/A_5. The work provides lattice models and field-theory illustrations, clarifies obstructions to gauging, and outlines future directions toward fermionic extensions and connections to higher fusion categories.

Abstract

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.

Figures (5)

• Figure 1: Sandwich construction for twisted coset symmetry defects in $D$ spacetime dimensions. The middle region has anomalous $G$ symmetry defect, with anomaly given by cocycle $\omega_{D+1}$. In the outer region on the two sides, the $K\subset G$ symmetry is gauged with topological action $\alpha_D$. The two interfaces separating the region is given by Dirichlet boundary condition of the $K$ gauge field. The subgroup $K$ is finite to ensure the defect is topological. The symmetry defect of the $K$ gauge theory has the form of the sandwich $\tilde{U}_g =\overline{\mathcal{D}}_{\mathrm{Rep}(K)}\times U_g\times \mathcal{D}_{\mathrm{Rep}(K)}$.
• Figure 2: The magnetic defect in $G$ gauge theory with holonomy $g$ in the presence of topological action for $G$ becomes attached to a gauged SPT defect given by twisted compactification of the topological action on a circle with holonomy $g$.
• Figure 3: The topological action in $G$ gauge theory modifies the junction of magnetic or dyonic defects with additional gauged SPT defects.
• Figure 4: Associator of twisted coset symmetry $(G,K,\omega_{D+1},\alpha_D)$ is inherited from the associator $\omega_{D+1}$ of twisted $G$ symmetry using the sandwich construction. The blue lines are interfaces of Dirichlet boundary condition of the $K$ gauge field, and the black lines are the $G$ symmetry defects.
• Figure 5: The edges nearby a vertex $v$ and a plaquette $p$.