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2Group Symmetries in Class S
by Lakshya Bhardwaj
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Submission summary
Authors (as registered SciPost users):  Lakshya Bhardwaj 
Submission information  

Preprint Link:  https://arxiv.org/abs/2107.06816v1 (pdf) 
Date accepted:  20220425 
Date submitted:  20220102 15:39 
Submitted by:  Bhardwaj, Lakshya 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Abstract
2group symmetries are generalized symmetries that arise when 1form and 0form symmetries mix with each other. We uncover the existence of a class of 2group symmetries in general 4d N=2 theories of Class S that can be constructed by compactifying 6d N=(2,0) SCFTs on Riemann surfaces carrying arbitrary regular punctures and outerautomorphism twist lines. The 2group structure can be captured in terms of equivalence classes of line defects plus flavor Wilson lines, which can be thought of as accounting for screening of line defects while keeping track of flavor charges. We describe a method for computing these equivalence classes for a general Class S theory using the data on the Riemman surface used for compactifying its parent 6d N=(2,0) theory.
Published as SciPost Phys. 12, 152 (2022)
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2022413 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2107.06816v1, delivered 20220413, doi: 10.21468/SciPost.Report.4919
Strengths
1) Very clear discussion of generalities of 2group symmetries
2) A nice algorithm to determine 2group structure of class ${\cal S }$ theories
Weaknesses
No weaknesses I could find.
Report
In this paper the author studies the 2group structures in class ${\cal S}$ theories. The discussion does not rely on a Lagrangian description of such theories, which almost always is absent, but rather on geometric understanding of various line operators. The author starts in section 2 discussing and developing a general language and set of tools to understand 2group structures from properties of line operators of a given theory. In section 3 the methods are applied to gauge theories while in section 4 the machinery developed in previous sections is applied to class S theories. Finally, in section 5 an author discusses an example of a class S theory with a non trivial 2group which can be also analyzed using a Lagrangian.
This paper is very well written. It is on a very topical subject of understanding higher group and higher form symmetries of general SCFTs. In particular it makes interesting use of geometric constructions of such theories. I think the result of this paper will be interesting to many researchers working in the field. I thus recommend it for publication in SciPost.
Report #1 by Anonymous (Referee 1) on 202225 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2107.06816v1, delivered 20220205, doi: 10.21468/SciPost.Report.4324
Report
In this paper the author describes a method for identifying 2group symmetries of 4d $\mathcal{N}=2$ theories of Class S. The method involves starting with a punctured Riemann surface (twists are allowed, but all punctures are required to be regular) and wrapping surface defects of the 6d $(2,0)$ theory on 1 and 2cycles/chains, giving rise to line defects and local operators respectively in the 4d theory. The local operators can serve as junctions for the line defects, with any two line defects related by such a junction having the same 1form symmetry charge. In other words, the 1form symmetry group is given by the Pontryagin dual of the Abelian group whose elements are equivalence class of lines $\{L_i\}/\sim$, with $L_1 \sim L_2$ if there exists a local junction operator connecting $L_1$ and $L_2$.
The key observation is that if the local junction operators are charged under a 0form flavor symmetry, the 0 and 1form symmetries can have nontrivial interplay, giving rise to a 2group. The goal then becomes to identify the flavor charges of the local operators. This is again possible from examination of the Riemann surface—schematically, when the 2chains giving the local junction operator pass over a puncture, the corresponding 4d local operator will be charged under the flavor symmetry associated with that puncture.
Altogether, this paper is insightful and very clearly written. In addition, the rather technical discussion in Section 4 is complimented with a beautifully explicit example in Section 5. For these reasons, I recommend this paper for publication in SciPost.
Requested changes
One very minor correction: above (2.19), I believe "2form symmetry” should be “2group symmetry”. Also, I think that the equality in (2.19) should actually be $\neq$?