Anomalies of Non-Invertible Symmetries in (3+1)d Article Swipe

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Clay Córdova , Po-Shen Hsin , Carolyn Zhang · YOU? · · 2023 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2308.11706

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.

Type

preprint

Language

en

Landing Page

http://arxiv.org/abs/2308.11706

PDF

https://arxiv.org/pdf/2308.11706

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green

Cited By

11

Related Works

10

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https://openalex.org/W4386150599

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https://openalex.org/W4386150599

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DOI

https://doi.org/10.48550/arxiv.2308.11706

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Title

Anomalies of Non-Invertible Symmetries in (3+1)d

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Type

preprint

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Language

en

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Publication year

2023

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Publication date

2023-08-22

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Authors

Clay Córdova, Po-Shen Hsin, Carolyn Zhang

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https://arxiv.org/abs/2308.11706

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https://arxiv.org/pdf/2308.11706

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Yes

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OA status

green

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https://arxiv.org/pdf/2308.11706

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Concepts

Homogeneous space, Physics, Invertible matrix, Symmetry (geometry), Hamiltonian (control theory), Theoretical physics, Pure mathematics, Mathematics, Quantum mechanics, Geometry, Mathematical optimization

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Cited by

11

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2025: 5, 2024: 6

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Related works (count)

10

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