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,\omega_{D+1},\alpha_D) ( G , K , ω D + 1 , α D ) in D D spacetime dimensions where G G is a group and K\subset G K ⊂ G is a discrete subgroup, \omega_{D+1} ω D + 1 is a (D+1) ( D + 1 ) -cocycle for group G G , and \alpha_{D} α D is a D D -cochain for group K 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 G / K becomes anomalous when G G cannot be expressed as the bicrossed product G=H\Join K G = H ⋈ 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/\Z_2 symmetry, with realizations in lattice models.