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Loops in 4+1d Topological Phases
by Xie Chen, Arpit Dua, PoShen Hsin, ChaoMing Jian, Wilbur Shirley, Cenke Xu
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Submission summary
Authors (as registered SciPost users):  Xie Chen · Arpit Dua · PoShen Hsin · ChaoMing Jian · Cenke Xu 
Submission information  

Preprint Link:  https://arxiv.org/abs/2112.02137v1 (pdf) 
Date submitted:  20211221 19:11 
Submitted by:  Dua, Arpit 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
2+1d topological phases are well characterized by the fusion rules and braiding/exchange statistics of fractional point excitations. In 4+1d, some topological phases contain only fractional loop excitations. What kind of loop statistics exist? We study the 4+1d gauge theory with 2form $\mathbb{Z}_2$ gauge field (the loop only toric code) and find that while braiding statistics between two different types of loops can be nontrivial, the self `exchange' statistics are all trivial. In particular, we show that the electric, magnetic, and dyonic loop excitations in the 4+1d toric code are not distinguished by their selfstatistics. They tunnel into each other across 3+1d invertible domain walls which in turn give explicit unitary circuits that map the loop excitations into each other. The $SL(2,\mathbb{Z}_2)$ symmetry that permutes the loops, however, cannot be consistently gauged and we discuss the associated obstruction in the process. Moreover, we discuss a gapless boundary condition dubbed the `fractional Maxwell theory' and show how it can be Higgsed into gapped boundary conditions. We also discuss the generalization of these results from the $\mathbb{Z}_2$ gauge group to $\mathbb{Z}_N$.
Current status:
Reports on this Submission
Anonymous Report 1 on 202227 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2112.02137v1, delivered 20220206, doi: 10.21468/SciPost.Report.4329
Strengths
1. The results are mathematically elegant.
Weaknesses
1. The authors do not provide an adequate review of where this work fits into the literature.
2. The writing is of uneven quality, especially in the latter parts of the paper (e.g., sec VI).
Report
Overall, the parts that I understand in this paper appear correct. The paper is unfortunately not clearly written. The explanations presented are often rushed, especially so for the most technically difficult arguments. Not enough trouble has been taken to put the work in context within the field.
I'll try to summarise my understanding, and list questions as they arise. This paper examines a family of 4+1 D TQFTs with $\mathbb{Z_N}$ gauge group and two 2 form fields $\mathcal{L}=\frac{N}{2\pi} b^{(1)}db^{(2)}$.
The model thus described has looplike excitations of two types: magnetic and electric. The authors first focus on the $N=2$ case, and consider bound states of electric and magnetic excitations. They show that, whereas such em bound states are fermionic in 2+1D, they have no universal nontrivial exchange statistic in 4+1D. The key difference is that the bilinear form $\int \alpha d \beta $ is symmetric for two 1forms in $2+1$ D, but antisymmetric for two 2forms in $4+1$ D.
The authors note that the model has a number of different possible gapped boundary conditions corresponding to different $3+1$D WalkerWang models. You can condensed e loops, m loops, and dyonic psi loops. In the former two cases, one obtains 3+1 D toric code (or fermionic toric code models), in the last case one obtains a 3DSem model.
Already at this point, I have some serious confusions. Below Eq 23 the authors say:
"The boundary is Z2 2form gauge theory .... describes a Z2 topological order with confined charges"
I'm confused: The 3+1 D toric code has **deconfined** charges! It's possible I've misunderstood the construction, but in any case the text is very confusing without further elaboration. Relatedly, just above eq 36 the authors say:
"Similar to the e condensed and m condensed boundaries, there are two variants of the ψ condensed boundary, corresponding to whether the particle in the boundary Z2 topological
order is a boson or fermion"
Again I'm confused: The 3D semion model has no deconfined particles, so it's meaningless to talk about exchange statistics of the particle. Again, the text is confusing.
Continuing through the paper. Related to these different boundary conditions, the paper notes the apparent $SL(2,Z_2)$ symmetry of the model, which can also be associated with permutations of the e,m, and psi loop excitations. The authors provide explicit unitary maps on an explicit lattice model implementing said automorphisms of the excitations. I don't have any problems with this section (IV).
Sec V and VI are very difficult to follow, and appear rather rushed.
To begin:
"As discussed for the Z2 gauge theory, this symmetry is nonanomalous on orientable spacetime, where the integral of total derivative vanishes"
I've searched the text, and cannot find the discussion of anomalies in the Z_2 case prior to the quoted sentence.
Continuing, I'm confused as to exactly what role time reversal symmetry is playing in the remainder of this paper. Does the H^4 obstruction only exist only when time reversal is imposed? I strongly suspect not, but the text above (63) suggests otherwise (or is a distracting aside whose relevance should be clarified).
Focussing on Sec VI: My understanding of the obstruction is that it captures the fact you cannot proliferate D^N defects in the N even case. Correct? I'm less clear as to why you cannot proliferate D^N defects/gauge $m\leftrightarrow\psi$ symmetry. Some questions:
Where does the final equality in (76) come from? Citation needed.
"Thus we find that the domain wall D^N is completely equivalent to insertion of the electric surface operator at a suitable
locus in any correlation function."
Why is this important? Does this prove there's an obstruction? Why?
I think 7779 form the proof that you cannot gauge $m\leftrightarrow\psi$ symmetry. But I do not understand the construction in (77). Where does this come from? The authors need to spend more time explaining where (77) comes from, and why it corresponds to an attempt to gauge $m\leftrightarrow\psi$ symmetry.
Requested changes
1. The text should be modified to address each of the questions listed above/headoff the confusion described. Particularly sec VI, which will likely require a major rewriting for clarity.
2. The authors cite no other works in the introduction. This must be fixed, and the relation of the manuscript to existing work clarified.
3. Below (69) "am" > "an"
4. Repeated reference [20] and [36].
5. Below (73) aciton> action
Author: Arpit Dua on 20221101 [id 2971]
(in reply to Report 1 on 20220207)Dear Editor and Referee,
Please have a look at the attached pdf in which we reply to the referee's comments and summarize the changes made in the manuscript.
Thanks,
Authors
Attachment:
reply.pdf