SciPost Submission Page
Anomalies of Non-Invertible Symmetries in (3+1)d
by Clay Cordova, Po-Shen Hsin, Carolyn Zhang
Submission summary
| Authors (as registered SciPost users): | Po-Shen Hsin |
| Submission information | |
|---|---|
| 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: |
|
| 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
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