When Does a Three-Dimensional Chern–Simons–Witten Theory Have a Time Reversal Symmetry?

Geiko, R.V.; Moore, Gregory W.

doi:10.1007/s00023-023-01303-3

Ann. Henri Poincaré

2023

DOI: 10.1007/s00023-023-01303-3

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When Does a Three-Dimensional Chern–Simons–Witten Theory Have a Time Reversal Symmetry?

R.V. Geiko

Gregory W. Moore

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2024

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

Córdova,

Hsin,

Zhang

2024

SciPost Phys.

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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}ℤ4 ordinary symmetry host worldvolume theories with time-reversal symmetry \mathsf{T}𝖳 obeying the algebra \mathsf{T}^{2}=C𝖳2=C or \mathsf{T}^{2}=(-1)^{F}C,𝖳2=(−1)FC, with CC 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.

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

Córdova,

Hsin,

Zhang

2024

SciPost Phys.

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When Does a Three-Dimensional Chern–Simons–Witten Theory Have a Time Reversal Symmetry?

Cited by 1 publication

References 63 publications

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

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

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