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Anomalies of Non-Invertible Symmetries in (3+1)d
by Clay Cordova, Po-Shen Hsin, Carolyn Zhang
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Po-Shen Hsin |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202409_00035v1 (pdf) |
| Date accepted: | 2024-10-29 |
| Date submitted: | 2024-09-29 21:46 |
| Submitted by: | Hsin, Po-Shen |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
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.
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Author comments upon resubmission
List of changes
p8 expanding footnote 6 to account for the second report.
p10 above (2.8) changing the sentence to 2r x 2r integer invertible matrix U to account for the request change 1 of the first report.
p12 above (2.16) similarly changing the sentence to 2x2 integer matrix to account for the request change 1 of the first report.
p16 new figure 1 and two new paragraphs "At first glance,.." and "More precisely.." to clarify the logic flow along the suggestion in the second report.
In addition, the typos pointed out in request change 2 of the first report are corrected.
Published as SciPost Phys. 17, 131 (2024)