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Anomalies of Non-Invertible Symmetries in (3+1)d
by Clay Cordova, Po-Shen Hsin, Carolyn Zhang
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Submission summary
| Authors (as registered SciPost users): | Po-Shen Hsin |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2308.11706v2 (pdf) |
| Date submitted: | 2024-04-23 04:39 |
| 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.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2024-8-25 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2308.11706v2, delivered 2024-08-25, doi: 10.21468/SciPost.Report.9649
Report
This work provides a new way to determine the anomaly of non-invertible symmetry in higher dimensions. The key idea is to consider the abelian symmetry TQFT associated with the abelian part of the symmetry, and examine its topological boundary conditions.
The first level of anomaly comes from the non-existence of duality invariant topological boundary condition of the abelian symmetry TQFT.
The second level of anomaly comes from the FS indicator, which, by decorated domain wall construction, reduces to the anomaly of time reversal symmetry of the non-invertible defect worldvolume theory. Then (1) the non-invertible symmetry being anomaly free (in the sense of admitting a trivially gapped phase) is argued to be equivalent to (2) the existence of a 3d world volume TQFT that matches the time reversal anomaly from the FS indicator. This equivalence between (1) and (2) are scattered in different parts of the paper, and I would suggest the authors to establish the equivalence better by, e.g. having a dedicated subsection, and explain the proof of (1) -> (2) and (2) -> (1) directions explicitly, therefore showing their equivalence. I believe this would make the paper, especially Section 3, more readable.
Apart from the above suggestion, there is a small gap for the argument around eq 2.2. One needs to argue that simple surface induces only 1 copy of identity line. Otherwise eq 2.2 can be modified to $n \times n’ = \sum_i n_i$, which does not lead to contradiction.
Recommendation
Ask for minor revision
Report #1 by Anonymous (Referee 2) on 2024-5-14 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2308.11706v2, delivered 2024-05-14, doi: 10.21468/SciPost.Report.9055
Strengths
1 - The paper studies anomalies for non-invertible symmetries in 3+1d. The authors identify two levels of anomalies, generalizing analogous results for anomalies of Tambara-Yamagami non-invertible symmetries in 1+1d
2 - Along the way, the authors study anomalies in the time reversal symmetry of an interesting class of 2+1d TQFTs, and construct an infinite family of 2+1d TQFTs enjoying a non-invertible version of time reversal symmetry
3 - The authors present and study explicit lattice models for 3+1d theories that enjoy a ZN 1-form symmetry and are invariant under its gauging
Weaknesses
1 - At times, the discussion is rather technical, especially in section 3
Report
Uncovering dynamical consequences of non-invertible symmetries in 3+1d dimensions is an important endeavor. This paper makes substantial progress in this direction by performing a systematic analysis of 't Hooft anomalies for Kramers-Wannier-like symmetries.
This paper meets this Journals' acceptance criteria and is recommended for publication, after addressing some minor points listed below.
Requested changes
1 - The notation GL(n,Z) is often used to indicate n x n matrices with integer entries that are invertible and whose inverses are also matrices with integer entries. Thus, these matrices have determinant equal to +1, -1. The authors use the notation GL(2r,Z) around (2.8), but remark that U needs not have determinant +1 or -1. The authors could comment briefly on this for clarification purposes.
2 - Some minor typos: "that the the Lagrangian" beginning of sec 2.1.2; "an quadratic" above (2.12); "an symmetric" above (2.24); "affect out discussion" below (3.19); "odd N cause" 4th line page 29; "symmetry symmetry" between (3.42) and (3.43); "can can produce" above Thm 2; "J0 and and J1" bottom page 36; "the anomaly field theory that [...] form symmetries with is given by" 2nd bullet page 40
Recommendation
Ask for minor revision