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Non-Invertible Symmetry Webs

by Lakshya Bhardwaj, Lea E. Bottini, Sakura Schäfer-Nameki, Apoorv Tiwari

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Submission summary

Authors (as registered SciPost users): Lakshya Bhardwaj · Apoorv Tiwari
Submission information
Preprint Link:  (pdf)
Date submitted: 2023-05-17 21:36
Submitted by: Tiwari, Apoorv
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


Non-invertible symmetries have by now seen numerous constructions in higher dimensional Quantum Field Theories (QFT). In this paper we provide an in depth study of gauging 0-form symmetries in the presence of non-invertible symmetries. The starting point of our analysis is a theory with $G$ 0-form symmetry, and we propose a description of sequential partial gaugings of sub-symmetries. The gauging implements the theta-symmetry defects of the companion paper [1]. The resulting network of symmetry structures related by this gauging will be called a non-invertible symmetry web. Our formulation makes direct contact with fusion 2-categories, and we uncover numerous interesting structures such as symmetry fractionalization in this categorical setting. The complete symmetry web is derived for several groups $G$, and we propose extensions to higher dimensions. The highlight of this analysis is the complete categorical symmetry web, including non-invertible symmetries, for 3d pure gauge theories with orthogonal gauge groups and its extension to arbitrary dimensions.

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Reports on this Submission

Anonymous Report 2 on 2023-7-15 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2212.06842v1, delivered 2023-07-14, doi: 10.21468/SciPost.Report.7505


The manuscript discussed many examples of gauging symmetry 2Vec G, where various non-invertible symmetries/ higher group can occur in various cases.

I recommend publication if the following comments are taken into account (as well as the comments in the first referee report):

- As mentioned in the referee report 1, it would be helpful to have applications such as any observable physical consequence that would be hard to obtain without using the method in the manuscript.

- The manuscript discussed symmetry fractionalizations, which has been extensively studied in the condensed matter literature, e.g. the earliest ones include the ENO paper and
What is the relation between the discussion in the manuscript and the literature?

- In the manuscript, the fractionalization classes are related to group extension and described H^2. However, there are more general fractionalization classes when the theories have two form or higher symmetries such as O(N) theory in 3+1d, as described by H^* of higher degree. Do they give more examples?

- a 't Hooft anomaly -> an 't Hooft anomaly

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Anonymous Report 1 on 2023-7-14 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2212.06842v1, delivered 2023-07-14, doi: 10.21468/SciPost.Report.7501


This paper studies webs of theories with non-invertible symmetries mainly in 3d. There is also a section explaining how the method can be generalized to higher dimensions.

The main idea was to start with a non-anomalous, invertible, symmetry given by the fusion 2-category 2-Vec(G), and perform sequential partial gauging of all possible sub-symmetries. The authors found that the sequential gauging is consistent with gauging 2-Vec(G) in one go. Along the way, many interesting features in the intermediate steps are uncovered, most notably symmetry fractionalization on lines and condensation defects.

One strength of this work is that it contains plenty of examples, which the referee found quite helpful. However, the style of the presentation makes the paper not very readable. There are way too many notations, superscripts, subscripts, with or without parentheses, etc.

Although the paper discussed a very systematic construction of non-invertible symmetries, it seems that the authors still miss some important possibilities:

1. The authors did not discuss the possibility of including discrete theta terms when they gauge 0-form symmetries. For example, including them into the D8-web produces the Pin^- theories, as discussed in (fig 3 of) (By the way, please also consider citing this work as it already contains many results regarding the \mathfrak{so} theories revisited by the current paper.)

2. The authors did not discuss the possibility of gauging non-invertible co-dimension 1 defects (even determining the whether they are gaugable is unclear to the referee).

It would be useful to make these points clear: either by making some comments on these points, or by saying that they are beyond the scope of this work and will be left for future work.

Moreover, the paper spent over 100 pages exploring the huge webs of symmetries, and it will be a pity that the paper does not contain possible applications on these symmetries. Do these new symmetries tell us anything about the dynamics of Yang-Mills theories in the webs unknown before? The referee thinks it would be very interesting (and necessary) to at least include a subsection on the physical implications of these symmetries.

Finally, two extremely minor typos:
1. In Table 1: PSU should be \mathrm{PSU}.
2. The notations in eq (2.21) and eq (2.27) conflict. The latter have superscript V while the former doesn’t.

Overall, I would recommend for publication, once the comments are addressed.

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