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Non-Invertible Symmetry Webs

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

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Submission summary

Authors (as registered SciPost users): Lakshya Bhardwaj · Apoorv Tiwari
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-09-08
Date submitted: 2023-08-07 16:25
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.

Author comments upon resubmission

We thank the referees for their careful reading of our paper and the comments and suggestions. We have implemented the following changes to the manuscript following the referees suggestions:

1) Referee 1 and 2 point out that we have not included any applications at this point. This is true, however the goal of this paper is to first describe the structure of the symmetry and their gauging, and how these are manifest in the context of fusion 2-categories. Applications will appear in subsequent works (e.g. they also have appeared already in the context of lattice versions of these precise constructions in [see for e.g., arXiv:2301.01259 and arXiv:2307.01266] and we currently have some on-going work in this direction).

2) Referee 1's comments about a) adding discrete theta and citation to is addressed in the introduction.

3) Regarding both referees comments related to connections to the condensed matter literature---we have now added a comment as well as several references in the introduction. While the phenomenon of symmetry fractionalization has been extensively studied in the condensed matter literature, its fusion 2-categorical aspects as well as the general mechanism of how it arises via gauging invertible symmetries has been less appreciated.

4) We have added a footnote in the introduction stating that we plan to develop the gauging of non-invertible co-dimension 1 defects, which typically arise in the categorical symmetry webs as condensation defects or from the gauging of non-normal subgroups, in future works.

5) It is true that gauging non group extensions of 0-form symmetries would yield new examples. For instance gauging 1-form symmetries in 4d or gauging a 1-form symmetry that participates in a 2-group with a non-trivial Postnikov class would deliver symmetry fractionalization patterns controlled by certain H^3 elements. Both these kinds of examples do not appear in the categorical symmetry webs we used to exemplify the approach in this work. However general computations carried out in this work can be used to study such symmetry fractionalizations.

6) We thank the referees for pointing out the typos. These have now been fixed.

Published as SciPost Phys. 15, 160 (2023)

Reports on this Submission

Anonymous Report 2 on 2023-8-28 (Invited Report)


The authors have addressed the comments.

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Anonymous Report 1 on 2023-8-15 (Invited Report)


The authors have addressed the comments appropriately. I'd like to recommend for publication as it is.

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