Anomalies of non-invertible symmetries in (3+1)d

• Published in: SciPost physics Vol. 17; no. 5; p. 131
• Main Authors: Córdova, Clay, Hsin, Po-Shen, Zhang, Carolyn
• Format: Journal Article
• Language: English
• Published: Netherlands Stichting SciPost 01.11.2024; SciPost
• ISSN: 2542-4653; 2542-4653
• DOI: 10.21468/SciPostPhys.17.5.131

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} ℤ 4 ordinary symmetry host worldvolume theories with time-reversal symmetry \mathsf{T} obeying the algebra \mathsf{T}^{2}=C 2 = C or \mathsf{T}^{2}=(-1)^{F}C, 2 = ( − 1 ) F C , with C 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.