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Unifying Constructions of Non-Invertible Symmetries

by Lakshya Bhardwaj, 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: 2023-08-17
Date submitted: 2023-08-07 16:32
Submitted by: Tiwari, Apoorv
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approach: Theoretical


In the past year several constructions of non-invertible symmetries in Quantum Field Theory in $d\geq 3$ have appeared. In this paper we provide a unified perspective on these constructions. Central to this framework are so-called theta defects, which generalize the notion of theta-angles, and allow the construction of universal and non-universal topological symmetry defects. We complement this physical analysis by proposing a mathematical framework (based on higher-fusion categories) that converts the physical construction of non-invertible symmetries into a concrete computational scheme.

Author comments upon resubmission

We thank the referees for their careful reading of our paper and the comments and suggestions. Below is a response to the referees comments and a list of corresponding changes to the paper:

Report 1:

  1. The referee has queried whether the term "theta defects" is necessary or is simply a different terminology for condensation defects. Theta defects are {\it not} the same as condensation. E.g. for 2-group symmetries theta defects are shown to be distinct to condensation [ref 7 in the current version]. Other examples are twisted thetas, which are not condensation defects either. So introducing this terminology is crucial, and it unifies the perspective on non-invertible symmetries substantially: it incorporates condensation, but is much more than these. To stress this point, which is indeed important, we have added a box at the end of section 2 to emphasis this distinction. We hope this clarifies this point.
  2. We illustrated in detail in our companion paper how the mathematical formalism of this paper can be used to perform concrete computations.
  3. We thank the referee for pointing this out. This has now been referred to on top of page 19.
  4. As mentioned in the introduction, it is indeed left to future work, but this does not imply that the program discussed in section 4 is only for non-intrinsic symmetries. The particular aspect of the program that might connect to intrinsic symmetries is the one related to interfaces discussed in section 2.4.

Report 2:

We have corrected the typos pointed out by the referee and provided additional information according to referee's comments. Regarding point 4, what referee points out in the N=4 example is indeed correct, but it is not clear to us that the two types of defects are the same in general, and it is for this reason that we treat the two types separately.

Published as SciPost Phys. 15, 122 (2023)

Reports on this Submission

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


The authors have addressed the referee's comments satisfactorily. The manuscript deserves publication.

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