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Generalized Symmetries in Ftheory and the Topology of Elliptic Fibrations
by Max Hubner, David R. Morrison, Sakura SchäferNameki, YiNan Wang
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Submission summary
Authors (as registered SciPost users):  Max Hubner · Yinan Wang 
Submission information  

Preprint Link:  https://arxiv.org/abs/2203.10022v2 (pdf) 
Date submitted:  20220427 00:55 
Submitted by:  Hubner, Max 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We realize higherform symmetries in Ftheory compactifications on noncompact elliptically fibered CalabiYau manifolds. Central to this endeavour is the topology of the boundary of the noncompact elliptic fibration, as well as the explicit construction of relative 2cycles in terms of Lefschetz thimbles. We apply the analysis to a variety of elliptic fibrations, including geometries where the discriminant of the elliptic fibration intersects the boundary. We provide a concrete realization of the 1form symmetry group by constructing the associated charged line operator from the elliptic fibration. As an application we compute the symmetry topological field theories in the case of elliptic threefolds, which correspond to mixed anomalies in 5d and 6d theories.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2022529 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2203.10022v2, delivered 20220529, doi: 10.21468/SciPost.Report.5148
Strengths
1 Mathematically clear exposition and general statements of new geometrical methods to work out higher form symmetries in FTheory compactifications
2 Detailed discussion of a number of examples
Weaknesses
1 No real introduction to any of the concepts used, so only understandable by people already well versed in the field. Some aspects are hard to follow and further details might be useful. E.g. statement of the long exact sequence and definition of maps used in 2.1 or a reference; definition of \mathbb{E};
Report
The literature on higherform symmetries (HFS) from Mtheory so far only covered QFTs engineered using noncompact geometries with localized singularities, i.e. those not intersecting the boundary. When engineering QFTs in Ftheory this is not the typical situation (e.g. in cases with flavour symmetry) and the above methods need to be extended to singular geometries. This generalization is what this works aims to address. This is particularly relevant as the topology of the boundary is crucial in determining the HFS. Equally relevant, this work explains how to work out HFS by using the natural data in terms of which FTheory compactifications are typically presented. The crucial tool here are Lefschetz thimbles, i.e. onecycles of the elliptic fibration which are collapsed somewhere on the discriminant locus. The authors give a range of instrutive examples and even discuss how to work out couplings in the associated symmetry TFT.
This work is a great exploration of how to work out HFS from the topology of Lefschetz thimbles. This new approach opens the door for indepth studies of a great variety of QFTs that can be engineered via FTheory.
Requested changes
none
Report #1 by Anonymous (Referee 1) on 2022520 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2203.10022v2, delivered 20220520, doi: 10.21468/SciPost.Report.5107
Strengths
The authors developed some new and interesting Ftheory techniques that can be applied to the study of higherform symmetries of 6 dimensional field theories.
Weaknesses
The style of writing in its current form is not very physical, not very accessible, and not very "readable" to quantum field theorists who are not an experts on Ftheory. The authors could have made more effort to improve the "readability" of the article.
Report
The authors of this article developed certain techniques in Ftheory to study higher form symmetries and their mixed anomalies in some 6d and the corresponding 5d KaluzaKlein theories. In particular, the correspondence between the Kodaira thimbles and the group of line operators (defect group) are pointed out in this article. The authors then applied this correspondence to compute the defect group of several nonHiggsable clusters for 6 dimensional SCFTs. The mixed anomalies of higherform symmetries were then determined from the triple intersection numbers. Since this article has some interesting material, I recommend this article for publication in the SciPost after addressing the comments on the "Requested Changes" section.
Requested changes
1. In the introduction of the article, I would recommend the authors to spell out further the physical meaning of the following phrase: "the discriminant of the elliptic fibration intersects the boundary". It would be useful if the authors could give some explicit physical examples of theories that have this property and those that do not have this property. This would serve as a nice motivation for the study of the material in this article.
2. The 6 dimensional conformal matter theories discussed in this article contain one gauge group. Can this be generalised to those with many gauge groups?
3. Can the technique developed in this article be applied to detect the mixed anomaly that arises from gauging the 1form symmetry that participates in the 2group structure, for example, those discussed in Section 3.4 of [arXiv:2110.14647]? If so, could the authors explain how? And if not, could the authors explain what the problem is?
Author: Max Hubner on 20220720 [id 2671]
(in reply to Report 1 on 20220520)We thank both referees for their valuable feedback and suggestions.
Regarding the first point raised in the this report, we have now made modifications in the introduction highlighting that the noncompact discriminant components intersecting the boundary have the interpretation of flavor branes and that their effect on the boundary topology geometrizes the screening effects of matter fields.
The second remark in the report concerns the question whether our methods apply in the context of semisimple gauge groups. Here we point out that the general discussion makes no assumption on whether gauge groups are simple or semisimple. Indeed, the examples discussed in section 4 of the paper which cover cases such as SU(n)xSU(m) gauge theories, also demonstrate our formalism in such cases.
Regarding the final comment on studying further anomalies involving flavor symmetries we point out that such questions lie outside the scope of the presented paper. Such computations would require geometrizing the global form of the flavor symmetry (as discussed in later papers such as 2203.10097 and 2203.10102) and involve methods not developed in this paper  purposefully as these two papers were already in progress, and build on the current paper.
We hope that the revisions have now made the paper suitable for publication.