# On gauging finite subgroups

### Submission summary

 As Contributors: Yuji Tachikawa Arxiv Link: https://arxiv.org/abs/1712.09542v3 (pdf) Date accepted: 2020-01-09 Date submitted: 2019-12-24 01:00 Submitted by: Tachikawa, Yuji Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory High-Energy Physics - Theory Approach: Theoretical

### Abstract

We study in general spacetime dimension the symmetry of the theory obtained by gauging a non-anomalous finite normal Abelian subgroup $A$ of a $\Gamma$-symmetric theory. Depending on how anomalous $\Gamma$ is, we find that the symmetry of the gauged theory can be i) a direct product of $G=\Gamma/A$ and a higher-form symmetry $\hat A$ with a mixed anomaly, where $\hat A$ is the Pontryagin dual of $A$; ii) an extension of the ordinary symmetry group $G$ by the higher-form symmetry $\hat A$; iii) or even more esoteric types of symmetries which are no longer groups. We also discuss the relations to the effect called the $H^3(G,\hat A)$ symmetry localization obstruction in the condensed-matter theory and to some of the constructions in the works of Kapustin-Thorngren and Wang-Wen-Witten.

### Ontology / Topics

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Published as SciPost Phys. 8, 015 (2020)

I am very sorry that my extremely slow reply to the very detailed referee report.
I summarized below the changes made according to the suggestions by the referee.

### List of changes

3: I added the information that an $n$-form symmetry comes with a map to $K(G,n+1)$.

4: This is due to the fact that the general yoga says that the anomaly of a $D$-dimensional theory is captured by a $(D+1)$-dimensional invertible topological phase. I added a few references on this point. A short paragraph was added just before "organization of the note" to mention this important fact.

5: This was studied in the past; two references were added. I also explicitly mentioned which of the spacetimes $X_D$ and $Y_{D+1}$ were considered in various steps involved.

6: Indeed I should have mentioned the role of $Y$ more prominently. This I think is covered by the added paragraph mentioned in the point 4 above, and the clarification mentioned in the point 5 above.

7: The word "source" was used in the sense of the "source of (electric and other) charges" used in physics, and not in the sense of the source and the target. The word is changed to the "boundary", which more accurately captures the situation and should cause less confusion.

8: I tried to clarify the explanation by adding a few sentences describing an additional intermediate step, so that the original Figures 3,4,5 were replaced by the new Figures 3,4,5,6.

9: I agree with the referee's comment. I tried to clarify the footnote.

10: I agree with the referee's comment. A paragraph and two references were added.

11: The original article by Eilenberg and Mac Lane was added.

12: It was a carry-over from an earlier version of the note; I thank the referee for the careful reading. They are corrected.

13: They are tensor product as $\mathbb{Z}$-modules. This fact was made explicit.

14: Essentially, yes, but I would like to keep the notation $G_{[n]}$ as a shorthand for the (ordinary) group $G$ regarded as an $n$-form symmetry.

15: I'd like to thank again the referee for the careful reading. They are all corrected.