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NonInvertible HigherCategorical Symmetries
by Lakshya Bhardwaj, Lea E. Bottini, Sakura SchaferNameki, Apoorv Tiwari
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Submission summary
Authors (as registered SciPost users):  Lakshya Bhardwaj · Lea Bottini · Apoorv Tiwari 
Submission information  

Preprint Link:  https://arxiv.org/abs/2204.06564v3 (pdf) 
Date accepted:  20220920 
Date submitted:  20220908 18:38 
Submitted by:  Bottini, Lea 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We sketch a procedure to capture general noninvertible symmetries of a ddimensional quantum field theory in the data of a highercategory, which captures the local properties of topological defects associated to the symmetries. We also discuss fusions of topological defects, which involve condensations/gaugings of highercategorical symmetries localized on the worldvolumes of topological defects. Recently some fusions of topological defects were discussed in the literature where the dimension of topological defects seems to jump under fusion. This is not possible in the standard description of highercategories. We explain that the dimensionchanging fusions are understood as highermorphisms of the highercategory describing the symmetry. We also discuss how a 0form subsymmetry of a highercategorical symmetry can be gauged and describe the highercategorical symmetry of the theory obtained after gauging. This provides a procedure for constructing noninvertible highercategorical symmetries starting from invertible higherform or highergroup symmetries and gauging a 0form symmetry. We illustrate this procedure by constructing noninvertible 2categorical symmetries in 4d gauge theories and noninvertible 3categorical symmetries in 5d and 6d theories. We check some of the results obtained using our approach against the results obtained using a recently proposed approach based on 't Hooft anomalies.
Author comments upon resubmission
We thank the referee for their very careful reading and insightful comments and suggestions.
We have implemented changes according to the referee's suggestion, as detailed below.
1.) We very much appreciate the comment by the referee that the paper is long. In the new version we have tried to streamline the presentation and moved some of the examples from section 5 and 6 to the appendices. The reason for providing such a large number of examples is, however, that each illustrates a somewhat different point, which  in subsequent works following this paper  were used by various other authors.
The outer automorphism gauging of Spin$(4N+2)$ and Spin$(4N)$ are indeed similar and we retained only one in the main text.
However, the nonabelian gauging, as well as $O(2)$ gauge theory examples reveal some distinct features, which we would like to highlight and thus retain in the main text.
2.) The decision to relegate most of the details of section 8 to an appendix is largely due to the fact that the conceptual point of this construction is due to the paper by Kaidi, Ohmori, Zheng (KOZ), and thus not original to this paper. We do extend some of their analysis to other theories, including the use of other minimal TQFTs that are used to stack the invertible defects with, but conceptually the main points were made in KOZ. However we do think that presenting the results is very insightful as they provide an independent crosscheck to some of our results in section 6.
3.) The new version includes a half page summary of the method based on mixed anomalies.
4.) This is a very nice observation and we very much agree with this, however the motivation of the paper is to study the symmetry category of a single theory, rather than the space of all theories. We in fact hope to return to the latter question in a future paper.
5.) We are unsure what the referee was trying to say here, clearly there is some typo in the report... We are guessing that the question is to define what the number of vacua for a defect is. This is number of topological local operators.
6.) 11.) 13.)14.) These are all implemented in the new version.
7.) That is good point and we added the definition of D/A already in section 3.2.
8.) Regarding the fusion of D/A:
This in fact requires further technology, which was developed by a subset of us (LB, SSN) with Wu, and is not included in this paper. We have added a forward reference to three papers, which subsequently have elucidated this point, see right after (4.26).
The full symmetry category will of course close on itself, however since in this paper we do not discuss (or had the knowledge to compute) the fusion of condensation defects, the analysis is limited to fusion of simple objects modulo condensation.
9.) We added comments in sections 5 and 6 to clarify the interpretation of the operators. Also we stressed the flatness of the backgrounds.
10.) We clarified that the notation used here  we define $\mathcal{C}^{\text{ob}}$ as simple objects modulo condensation. In the full symmetry category the simple objects would include the condensation defects as well, however as we explained under 8.) we discuss only a subset of fusions here, which excludes condensation defects.
We appreciate that points 8./10.) are confusing, and have thus added a  hopefully  clarifying remark in the introduction as well.
12.) We added some explanations for this.
Again, thank you very much for the detailed reading and very useful suggestions for improvement!
REFEREE 2:
We thank Prof. Chang for his report and questions regarding our paper. Let us respond to the three queries that he made:
1.) The key point to distinguish here is composition versus fusion. What the referee has in mind is composition, which is indeed defined also for other dimensions. Fusion on the other hand, as we define it, requires the existence of 2morphisms that live at the junction of 1morphisms. In the case of 3d, these would be local operators (2morphisms).
2.) In the general section 4 we restrict our analysis to abelian groups, in which case this point does not arise. Note however that in section 5.3 we do provide a nonabelian example, and indeed there is a deviation from this formula in (4.19), which is applicable for abelian groups. For example the fusion in (5.51). We added another clarification about this in the paper, stating the restriction to abelian groups in section 4.
3.) We have added some comments to explain this.
Published as SciPost Phys. 14, 007 (2023)
Reports on this Submission
Report 2 by ChiMing Chang on 202299 (Invited Report)
Report
The authors have addressed all the comments in my previous report. I recommend this paper for publication on SciPost.
Report
The improvements made by the authors were appropriate and satisfactory. (I apologize for the missing words and typos in the point 5. The authors correctly guessed what I meant and made the necessary modifications.) The paper can now be recommended to be published.