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As Contributors: | Yuji Tachikawa |

Arxiv Link: | https://arxiv.org/abs/1712.09542v2 |

Date submitted: | 2018-06-07 |

Submitted by: | Tachikawa, Yuji |

Submitted to: | SciPost Physics |

Domain(s): | Theoretical |

Subject area: | High-Energy Physics - Theory |

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.

Editor-in-charge assigned

Submission 1712.09542v2 (7 June 2018)

1. clearly written

2. gives nice overview of important subject of generalized symmetry

3. finds good balance between general theory and concrete examples

none

The paper deals with the subject of generalized symmetries in the context of (topological) quantum field theories with defects. More concretely, it studies the gauging of finite subgroups of symmetry subgroups, bringing together ideas and results from the perspective of domain walls and algebraic topology, and a clear discussion of how anomalies relate to this description. I strongly recommend publication of the paper, and I suggest to address the points raised below as the author sees fit.

1. Footnote 1: Why does the doubled coset appear as a label set for domain walls?

2. Footnote 1: Does "consistent set of topological defects" mean that it is closed with respect to fusion?

3. Page 3, Notations and conventions: Please give a brief reminder on "n-form symmetry". How does it relate to n-groupoids? (This is connected to item 14 below.)

4. Page 3, Notations and conventions: At first I was confused by the dimensions of the spaces X and Y. Please briefly explain why the dimension D+1 appears even though spacetime is D-dimensional.

5. Page 4, last paragraph: Its seems that this discussion is also a partial TFT interpretation of Pachner's theorem on D-dimensional triangulations. Has the author considered this in more detail?

6. Page 6, boxed result: What exactly is Y here, and how does the result depend on it? Why was it not mentioned more prominently earlier in Section 2?

7. Page 7: Why does gh in G serve as a source (and not as target) of the domain wall?

8. Page 8: Please give more details for the last paragraph before Section 2.3.

9. Footnote 5: I would have thought that the non-commutative algebra "defines" the space M indirectly, not directly.

10. Page 9, last item: One could argue that the extension to a 3-functor from BG to the 3-category of certain fusion categories and their bimodule categories (as studied by Etingof, Nikshych and Ostrik) is even more relevant in TQFT; in this case there is also an H^4-obstruction.

11. Page 14, second paragraph of Section 2.7: Please give a reference for the well-known result.

12. Page 17, first paragraph of Section 3.3: In my copy of the paper Section 3.1 treats the case Z_n, not only Z_2.

13. Page 18, last paragraph: What do the \otimes-symbols mean?

14. Page 23: Is G_{[n]} really equal to B^nG?

15. Typos:

- "Gaitto" on Page 2

- "two analysis" on Page 3

- "3)-chain" on Page 4

- "as also as" on Page 7

- "we know have" on Page 8

- "described blow" on Page 11