TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.17.5.131
TI  - Anomalies of non-invertible symmetries in (3+1)d
PY  - 2024/11/12
UR  - https://scipost.org/SciPostPhys.17.5.131
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 17
IS  - 5
SP  - 131
A1  - Córdova, Clay
AU  - Hsin, Po-Shen
AU  - Zhang, Carolyn
AB  - Anomalies of global symmetries are important tools for understanding the dynamics of quantum systems. We investigate anomalies of non-invertible symmetries in 3+1d using 4+1d bulk topological quantum field theories given by Abelian two-form gauge theories, with a 0-form permutation symmetry. Gauging the 0-form symmetry gives the 4+1d "inflow" symmetry topological field theory for the non-invertible symmetry. We find a two levels of anomalies: (1) the bulk may fail to have an appropriate set of loop excitations which can condense to trivialize the boundary dynamics, and (2) the "Frobenius-Schur indicator" of the non-invertible symmetry (generalizing the Frobenius-Schur indicator of 1+1d fusion categories) may be incompatible with trivial boundary dynamics. As a consequence we derive conditions for non-invertible symmetries in 3+1d to be compatible with symmetric gapped phases, and invertible gapped phases. Along the way, we see that the defects characterizing $\mathbb{Z}_{4}$ ordinary symmetry host worldvolume theories with time-reversal symmetry $\mathsf{T}$ obeying the algebra $\mathsf{T}^{2}=C$ or $\mathsf{T}^{2}=(-1)^{F}C,$ with $C$ a unitary charge conjugation symmetry. We classify the anomalies of this symmetry algebra in 2+1d and further use these ideas to construct 2+1d topological orders with non-invertible time-reversal symmetry that permutes anyons. As a concrete realization of our general discussion, we construct new lattice Hamiltonian models in 3+1d with non-invertible symmetry, and constrain their dynamics.
ER  -

@Article{10.21468/SciPostPhys.17.5.131,
	title={{Anomalies of non-invertible symmetries in (3+1)d}},
	author={Clay Córdova and Po-Shen Hsin and Carolyn Zhang},
	journal={SciPost Phys.},
	volume={17},
	pages={131},
	year={2024},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.17.5.131},
	url={https://scipost.org/10.21468/SciPostPhys.17.5.131},
}