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Noninvertible Symmetries and Higher Representation Theory II
by Thomas Bartsch, Mathew Bullimore, Andrea E. V. Ferrari, Jamie Pearson
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Submission summary
Authors (as registered SciPost users):  Thomas Bartsch · Andrea Ferrari 
Submission information  

Preprint Link:  https://arxiv.org/abs/2212.07393v2 (pdf) 
Date submitted:  20231003 17:20 
Submitted by:  Bartsch, Thomas 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
In this paper we continue our investigation of the global categorical symmetries that arise when gauging finite higher groups and their higher subgroups with discrete torsion. The motivation is to provide a common perspective on the construction of noninvertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. We propose that the symmetry categories obtained by gauging higher subgroups may be defined as higher grouptheoretical fusion categories, which are built from the projective higher representations of higher groups. As concrete applications we provide a unified description of the symmetry categories of gauge theories in three and four dimensions based on the Lie algebra $\mathfrak{so}(N)$, and a fully categorical description of noninvertible symmetries obtained by gauging a 1form symmetry with a mixed 't Hooft anomaly. We also discuss the effect of discrete torsion on symmetry categories, based a series of obstructions determined by spectral sequence arguments.
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Strengths
1 Very clear presentation of technical results.
2 Systematic analysis of higher dimensional cases.
3 Plenty of examples worked out in great detail.
Weaknesses
1 The physical motivation for the author's studies is not stated clearly.
2 The paper becomes rather technical towards the end, especially in the last Section regarding 3Categories.
Report
The article describes the symmetry categories which are obtained by gauging (a subgroup of) an invertible symmetry in d=2,3,4.
This is prompted by the surge of interest during the last two years in studying noninvertible symmetries in higher dimensions.
Overall the paper is very well written and the presentation is clear and rather concise.
In Section 2 the authors review the known theory of bimodule categories after gauging a group H. The review is very crisp and clear.
In Section 3 the authors move on to the case of 2categories, which is also described in great detail.
In Section 4 the authors describe gauging of H 0form symmetry inside the category 3Vec(G). I feel this section is less conclusive and clear than the rest of the paper, also owing to the fact that the theory of 3Categories is much less developed. In particular I feel that the discussion on the stacking of TFTs and the inclusion of ReshetikinTuraev type theories could be improved.
I would suggest publication, but I will give some (optional) minor revisions which in my opinion would improve the readability of the work.
 The main results (the classification of symmetry defects after gauging) are quite compact, it would be nice if they were either included in the introduction or otherwise highlighted (maybe by a colored box) to improve reading.
 Since this is supposed to be a physics paper, I feel that not enough physical justification is given for the author's study (e.g. what are we going to learn? what is the physical relevance of the examples discussed? How to the symmetry structures impact physical observables?)
Regarding the body of the paper I also have some notes:
In Section 3 around eqn. (3.1) the authors discuss stacking with decoupled 2d TFTs. I would notice that such TFTs are also determined by a series of Euler counterterms, which fix the disk partition function in the presence of a boundary condition (that is, a state on S^1). I believe these would also show up as TFT coefficients as in the 4d case.
 In Section 4 the authors introduce the category of 3Representations of a group and various generalizations thereof. It is my understanding that the authors claim that such category has an infinite number of simple objects. Could there be some equivalence relation that has to be imposed to reduce it to a finite category? Is the number of connected components pi_0(C) finite?
There are also some minor typos to be fixed which I list below.
Requested changes
1 Above eqn. 1.1 it is stated that Psi is an element of C^{D1}, but should be C^D. In the rest of the paper the counting is done correctly.
2 On page 9 (and in other occurrences throughout the paper) the notation H_g is introduced. I understand this is a kind of bistabilizer, as it is clear from the Figure. However the mathematical notation is quite unclear at least to me. The authors should explain their notation better.
3 On page 24 there is a missing reference. This is also a rather important reference for people wishing to learn about higher reps.
4 On page 25 it is stated that a projective 2Rep is given by a trivialization dc = psi1  psi2 of the difference between discrete torsions. Is it obvious that psi1psi2 is trivial in cohomology?