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Global Form of Flavor Symmetry Groups in 4d N=2 Theories of 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/2105.08730v2 (pdf) |
| Date submitted: | 2022-01-02 15:37 |
| Submitted by: | Bhardwaj, Lakshya |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We provide a systematic method to deduce the global form of flavor symmetry groups in 4d N=2 theories obtained by compactifying 6d N=(2,0) superconformal field theories (SCFTs) on a Riemann surface carrying regular punctures and possibly outer-automorphism twist lines. Apriori, this method only determines the group associated to the manifest part of the flavor symmetry algebra, but often this information is enough to determine the group associated to the full enhanced flavor symmetry algebra. Such cases include some interesting and well-studied 4d N=2 SCFTs like the Minahan-Nemeschansky theories. The symmetry groups obtained via this method match with the symmetry groups obtained using a Lagrangian description if such a description arises in some duality frame. Moreover, we check that the proposed symmetry groups are consistent with the superconformal indices available in the literature. As another application, our method finds distinct global forms of flavor symmetry group for pairs of interacting 4d N=2 SCFTs (recently pointed out in the literature) whose Coulomb branch dimensions, flavor algebras and levels coincide (along with other invariants), but nonetheless are distinct SCFTs.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2022-3-31 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2105.08730v2, delivered 2022-03-31, doi: 10.21468/SciPost.Report.4834
Strengths
1. The article has useful results.
2. The author discusses several examples including the cases of surfaces with twists.
3. The author discusses how the global form of the full symmetry group can often be obtained from the global form of the manifest symmetry.
4. The author also discusses an application of his results on the topic of distinguishing between distinct SCFTs.
Weaknesses
1. Notation used in the article is heavy. It could be useful to have a glossary of notation in an appendix so that a reader can look up the meaning of each symbol.
Report
It has become clear over the past decade that the information contained in the global form of the global symmetry group is essential to understanding a QFT better. For example, it helps determine the 't Hooft anomalies of the QFT, which are invariant under an RG flow. For Lagrangian theories, determining the global form is easy; however, for non-Lagrangian theories (for example, those in class S), it was not clear how to answer this question. The author has given a neat answer to this question: it is determined by thinking carefully about the punctures and the surface defects of 6d (2,0) theory. This article adds useful information to the knowledge of the community, and I recommend it for publication.
Requested changes
I suggest that the author includes a glossary of notation in the appendix. This would make the article easier to read.
Report #1 by Anonymous (Referee 3) on 2022-1-28 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2105.08730v2, delivered 2022-01-28, doi: 10.21468/SciPost.Report.4254
Strengths
1. The results are interesting and useful.
2. The method discussed in the article can be applied to general theories of class $\mathcal{S}$.
Weaknesses
1. The presentation of the material and style of the article could be significantly improved so that the paper is more readable.
2. The choice of notation renders the article heavy and unreadable.
3. Many computational details need to be to be spelt out explicitly.
Report
This article provides a general method to compute the the global form of flavor symmetry in the theories of class $\mathcal{S}$, with regular punctures and possibly with outer-automorphism twists. The author demonstrated the application of such a method in wide range of examples. Although the results are interesting and useful, the referee would like to ask the following improvement prior to publication.
Requested changes
1. The authors mentioned the lattices of charges $Y'_F$, $\tilde{Y}_F$ and $Y_F$ in Eqs. (3.6), (3.9) and (3.10), respectively. However, none of the detail is spelt out in the subsequent examples. For example, at the end page 23, how did the authors compute $\tilde{Y}_F$? Since these are important for obtaining the main results in the article, I highly recommend the author to spell out the detail explicitly for non-trivial examples, for example in an appendix.
2. Given the lattice $Y_F$, how did the author compute the quotient $\mathcal{Z}$ and its generator? (For example, in the sentence below Eq. (4.6).) Again, the author definitely should elucidate the detail explicitly.
3. Could the author explain why the complex conjugate representation disappears from Eq. (3.2) to Eq. (3.11)?