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Codimension-2 defects and higher symmetries in (3+1)D topological phases
by Maissam Barkeshli, Yu-An Chen, Sheng-Jie Huang, Ryohei Kobayashi, Nathanan Tantivasadakarn, Guanyu Zhu
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Submission summary
| Authors (as registered SciPost users): | Yu-An Chen · Ryohei Kobayashi · Nathanan Tantivasadakarn |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2208.07367v1 (pdf) |
| Date submitted: | 2022-08-29 09:02 |
| Submitted by: | Chen, Yu-An |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
(3+1)D topological phases of matter can host a broad class of non-trivial topological defects of codimension-1, 2, and 3, of which the well-known point charges and flux loops are special cases. The complete algebraic structure of these defects defines a higher category, and can be viewed as an emergent higher symmetry. This plays a crucial role both in the classification of phases of matter and the possible fault-tolerant logical operations in topological quantum error correcting codes. In this paper, we study several examples of such higher codimension defects from distinct perspectives. We mainly study a class of invertible codimension-2 topological defects, which we refer to as twist strings. We provide a number of general constructions for twist strings, in terms of gauging lower dimensional invertible phases, layer constructions, and condensation defects. We study some special examples in the context of $\mathbb{Z}_2$ gauge theory with fermionic charges, in $\mathbb{Z}_2 \times \mathbb{Z}_2$ gauge theory with bosonic charges, and also in non-Abelian discrete gauge theories based on dihedral ($D_n$) and alternating ($A_6$) groups. The intersection between twist strings and Abelian flux loops sources Abelian point charges, which defines an $H^4$ cohomology class that characterizes part of an underlying 3-group symmetry of the topological order. The equations involving background gauge fields for the 3-group symmetry have been explicitly written down for various cases. We also study examples of twist strings interacting with non-Abelian flux loops (defining part of a non-invertible higher symmetry), examples of non-invertible codimension-2 defects, and examples of interplay of codimension-2 defects with codimension-1 defects. We also find an example of geometric, not fully topological, twist strings in (3+1)D $A_6$ gauge theory.
Current status:
Reports on this Submission
Report #2 by Lakshya Bhardwaj (Referee 1) on 2022-11-6 (Invited Report)
- Cite as: Lakshya Bhardwaj, Report on arXiv:2208.07367v1, delivered 2022-11-06, doi: 10.21468/SciPost.Report.6064
Report
This paper has excellent scientific content and is written clearly. I recommend the publication of this paper wholeheartedly.
This paper studies a novel construction of 1+1 dimensional invertible topological defects in 2+1 and 3+1 dimensional gauge theories with discrete gauge group G. These defects are called twist strings and they arise from 1+1 dimensional SPT phases protected by G symmetry. The paper also describes the action of the twist strings on flux operators, the realization of twist strings as condensation defects, and 3-group symmetries that twist strings participate in. This is part of a wider effort to understand spectrum and properties of topological defects of topological phases, with an eye towards applications in topological quantum computation.
I have a complaint regarding referencing though: This exact same construction of topological defects was utilized in the paper (co-authored by me) https://arxiv.org/pdf/2208.05973.pdf to construct non-invertible symmetries of quantum field theories. The first version of this paper appeared a few days before the submitted paper. In this paper, we construct not only invertible twist strings arising from 1+1 dimensional SPT phases with G symmetry, but also non-invertible twist strings arising from 1+1 dimensional non-invertible TQFTs with G symmetry. We also describe the fusion 2-category formed by twist strings, and show that all such twist strings (in the case when G is a 0-form symmetry) are condensation defects.
Report #1 by Anonymous (Referee 2) on 2022-10-1 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2208.07367v1, delivered 2022-10-01, doi: 10.21468/SciPost.Report.5812
Strengths
1. Although the paper is written primarily in CMP language, it also contains some hep-th terminology and is accessible to a broader community.
2. Contains many concrete examples.
Report
The paper discusses the construction/classification of line and surface defects in various 3+1d, 2+1d, and 1+1d topological phases / TQFTs. In particular the paper introduces a novel construction of such defects in discrete G-gauge theories. They start with a lower dimensional TQFT/SPT phases with symmetry G and upon gauging G in the whole spacetime they construct various topological defects of G-gauge theories.
The paper contains new result with interesting examples. I strongly recommend the draft for publication.
I just want to point out a relevant paper to the authors. The paper [arXiv:2208.05973] appeared about the same time as this draft on arixv and contains some similar results from a hep-th perspective.
Author: Yu-An Chen on 2022-12-12 [id 3125]
(in reply to Report 1 on 2022-10-01)We thank the referee for the accurate summary of our manuscript and for pointing out the reference we missed. We have included the new reference in the updated version.
Author: Yu-An Chen on 2022-12-12 [id 3126]
(in reply to Report 2 by Lakshya Bhardwaj on 2022-11-06)We appreciate the referee for the precise summary of our manuscript and for pointing out the reference we missed. The reference is relevant to our manuscript since it used similar concepts (from a hep-th perspective). We have added this reference to our manuscript in the updated version.