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Fermionic Higherform Symmetries
by YiNan Wang, Yi Zhang
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Submission summary
Authors (as registered SciPost users):  Yinan Wang · Yi Zhang 
Submission information  

Preprint Link:  scipost_202304_00003v2 (pdf) 
Date submitted:  20230629 11:32 
Submitted by:  Wang, Yinan 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
In this paper, we explore a new type of global symmetriesthe fermionic higherform symmetries. They are generated by topological operators with fermionic parameter, which act on fermionic extended objects. We present a set of field theory examples with fermionic higherform symmetries, which are constructed from fermionic tensor fields. They include the free fermionic tensor theories, a new type of fermionic topological quantum field theories, as well as the exotic 6d (4,0) theory. We also discuss the gauging and breaking of such global symmetries and the relation to the no global symmetry swampland conjecture.
Author comments upon resubmission
We have made significant changes to the draft, addressing many points by the referees. In particular, we expanded on the following physics points:
(1) The gauging of fermionic symmetries and 't Hooft anomalies;
(2) The presence of magnetic fermionic symmetries;
(3) The physical details of the new fermionic TQFTs proposed in the paper, including the gapped properties and edge modes.
We have also added a number of physical explanations regarding the questions of the referees. For the other followup questions, such as a full exploration of such symmetries on curved spacetime or a lattice, we will investigate them in future works.
Sincerely,
The authors
List of changes
We have made the following changes to the manuscript according to each referee's request.
Referee 1:
(1) We have expanded Section 4 on the gauging of fermionic symmetries for a free fermionic pform gauge field. We showed explicitly around (4.3) that the ddimensional action is not gauge invariant, and there's a 't Hooft anomaly. We also added the I_{d+2} 't Hooft anomaly polynomial in (4.7), which is different from the bosonic cases. Note that it cannot be written in terms of a characteristic class of the background gauge field itself.
(2) On the minor point 2, we added a sentence in the first paragraph of page 4, explaining that the fermionic symmetry group is taken as R^s, where s is the number of spinor components.
(3) On the minor point 3, we added explanations below (2.11) about the parameter \eta, which is an unquantized "charge" of the fermionic Wilson loop.
(4) On the minor point 5, we changed the statement to that there exists a topological generator for the magnetic symmetry, but the existence of EM duality in fermionic theories is unknown, and we do not find a charged object.
(5) We also revised the draft according to the other minor points.
Referee 2:
(1) On page 12 above (3.24), we comment on the spectrum of such fermionic TQFTs. They are gapped analogous to the bosonic counter parts, because the e. o. m. gives rise to pure gauge.
Referee 3:
(1) We commented on the interpretation of fermionic Wilson loop as order parameters at the end of Section 5.1. Nonetheless, the full quantization of such a physical model is still unknown.
(2) We commented on page 12 above (3.24) that such TQFTs are gapped.
(3) On page 12 below (3.22), we discussed the edge modes of the fermionic TQFT in a 3d example.
(4) We expanded the discussion of gauging and 't Hooft anomaly on page 16.
We have also added some references.
Current status:
Reports on this Submission
Anonymous Report 2 on 2023722 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202304_00003v2, delivered 20230722, doi: 10.21468/SciPost.Report.7553
Report
I would like to thank the authors for their efforts in answering all the questions. As far as I can understand, the authors have partially addressed the questions I raised in the previous report, but some important points remain unclear. In particular, the behavior of the vacuum expectation value of the fermionic Wilson is not analyzed. As the authors themselves state in the manuscript, this is left for future work. In principle, this analysis should be possible at least in the case of free fermionic theories. In addition, the added discussion about 't Hooft anomalies around Eq. (4.7) does not address the question about the type of constraints that follow from the existence of such anomalies in the IR sector. In this way, it is seems to be hard to extract much physical information from fermionic higherform symmetries. I think it would be enlightening to discuss more about these points.
Anonymous Report 1 on 202374 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202304_00003v2, delivered 20230704, doi: 10.21468/SciPost.Report.7443
Report
The authors have exhaustively addressed all the minor points raised in the previous report. The major point about ´t Hooft anomalies has been clarified in one example, but not in full generality.
The main problem that remains is that the general construction and analysis assumes to work in flat space, while the analysis of ´t Hooft anomalies requires placing the theories on a nontrivial topology $M^{(d)}$. While the expanded discussion seems to justify the procedure for a free $p$form spinor, it is still unclear why this step is justified in general.
The issue is the following: the authors observe an "obstruction to gauging", and (understandably) refer to it as a ´t Hooft anomaly. However, while there exists a mathematical definition and classification of anomalies based on cohomology, it does not seem to apply to the present case in general, because the entire construction of the authors is in flat space and cohomology is trivial.
I acknowledge that a comprehensive characterization of this "fermionic" version of ´t Hooft anomalies is beyond the scope of the work, but the authors should more carefully distinguish throughout Section 4 between what is derived and what is a proposal justified heuristically by analogy with known anomalies.
Requested changes
Explicitly clarify the usage of the phrasing "´t Hooft anomaly" and the tension with the fact of working in flat space in Section 4.
Author: Yinan Wang on 20230728 [id 3849]
(in reply to Report 1 on 20230704)
Dear referee,
We would like to clarify the notion of 't Hooft anomaly we used in the paper. In general, there are two types of anomalies: (1) local anomalies, which arise from infinitesimal gauge transformations, and is mathematically described by anomaly polynomials and descent formalism; (2) global anomalies, which come from large gauge transformations, and is mathematically described by group cohomology.
The 't Hooft anomalies discussed in the paper belong to local anomalies, hence it does not need to be assigned with a nontrivial group cohomology class.
Sincerely,
The authors
Author: Yinan Wang on 20230728 [id 3848]
(in reply to Report 2 on 20230722)Dear referee,
Thanks for the comment on the vev of fermionic Wilson loops. We will add an appendix on the computation of fermionic Wilson loop vevs for the examples of free RaritaSchwinger fields in the next version.
For the 't Hooft anomaly polynomial, it is quite unconventional as it involves both the gauge field and matter field. We do not have more detailed comments on the UV/IR matching at this stage, and we would like to clarify it in the future.
Sincerely,
The authors