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On Higher-dimensional Carrollian and Galilean Conformal Field Theories
by Bin Chen, Reiko Liu, Yu-fan Zheng
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Submission summary
| Authors (as registered SciPost users): | Yu-fan Zheng |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202210_00048v1 (pdf) |
| Date submitted: | 2022-10-06 09:26 |
| Submitted by: | Zheng, Yu-fan |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
In this paper, we study the Carrollian and Galilean conformal field theories (CCFT and GCFT) in $d>2$ dimensions. We construct the highest weight representations (HWR) of Carrollian and Galilean conformal algebra (CCA and GCA). Even though the two algebras have different structures, their HWRs share similar structure, because their rotation subalgebras are isomorphic. In both cases, we find that the finite dimensional representations are generally reducible but indecomposable, and can be organized into the multiplets. Moreover, it turns out that the multiplet representations in $d>2$ CCA and GCA carry not only the simple chain structure appeared in logCFT or $2d$ GCFT, but also more generally the net structures. We manage to classify all the allowed chain representations. Furthermore we discuss the two-point and three-point correlators by using the Ward identities. We mainly focus on the two-point correlators of the operators in chain representations. Even in this relative simple case, we find some novel features: multiple-level structure, shortage of the selection rule on the representations, undetermined 2-pt coefficients, etc.. We find that the non-trivial correlators could only appear for the representations of certain structure, and the correlators are generally polynomials of time coordinates for CCFT (spacial coordinates for GCFT), whose orders depend on the levels of the correlators.
List of changes
In this of the preprint, we have added following comments comparing to 2112.10514v2:
1. Added comments on the representation in the introduction. Stressed that the highest weight representation is not of the form of Wigner classification.
2. Added comments on Carrollian-BMS symmetry in the introduction, as well as added citations of related papers.
3. Corrected a statement in section 2.2. The Casimir operators of Galilean conformal algebra can be deduced by taking limit.
4. Added a foot note on page 20 section 3.2 to explain the convenience using $SU$ rather than $SO$ Young tableau.
5. Added a comment in section 3.5 about dual relation of Carrollian/Galilean representations is a result of the geometric structure.
6. Changed the equation (4.5) to be the same with (B.1).
7. Added on page 33 section 4.1 explicit theories with different correlation functions.
8. Corrected some typos.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 4) on 2022-11-3 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202210_00048v1, delivered 2022-11-03, doi: 10.21468/SciPost.Report.6040
Report
In the new version of the paper, the authors attempt to answer several questions that were raised in my earlier report. Some of the errors have been removed and some explanations have been provided. However there are some further comments that need to be made.
• Point 4: This is still the main remaining issue. There is still no proper explanation why SU(n) Young Tableaux appeared. This seems ad hoc and I am not sure one can trust the constructions that follow from this assumption. The authors need to clarify this further.
• Point 7: The statement that the magnetic scalar action gives spatial delta function correlations is wrong. See e.g. 2207.03468, where it is shown that the theory can effectively be reduced to a lower dimensional Euclidean CFT. Since the authors don’t write anything else regarding the magnetic theory, this statement should be removed.
Once these points are addressed, the paper can be considered further.
Report #1 by Anonymous (Referee 3) on 2022-10-12 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202210_00048v1, delivered 2022-10-12, doi: 10.21468/SciPost.Report.5877
Report
The authors have replied to my comments in a satisfactory way. There is only one point I would like to add:
Regarding point 4 in the attached pdf, the terms electric and magnetic Galilean action that you use are swapped with respect to the definitions in the reference that I mentioned in my report, 2206.12177. Their magnetic Galilean action in equation 7.4 of 2206.12177 corresponds to what you call electric in your file. Moreover, I am not sure if their electric Galilean action in equation 7.9 is not the same as your magnetic one coming from the Hamiltonian action. There is also an extra field in the action playing the role of a Lagrange multiplier.
In the electric case it is clear that the magnetic action involving a Lagrange multiplier, 7.15 in 2206.12177, is the same as 5.3 in 2109.06708. The Lagrange multiplier is simply what was the momentum in the Hamiltonian action before taking the limit. At first glance I don't see this equivalence in the Galilean case. This could be worth to check.
Since the authors will treat electric and magnetic scalar actions in a further publication, I think this new version of their manuscript can be published in SciPost.
Author: Yu-fan Zheng on 2022-12-04 [id 3101]
(in reply to Report 2 on 2022-11-03)Dear referee,
Thank you very much for your patience and valuable comments and suggestions. Please see the attachment for our detailed reply, and see the resubmission page for new version of the priprint.
link to resubmission: https://scipost.org/submissions/scipost_202210_00048v2/
link to new priprint: https://scipost.org/preprints/scipost_202210_00048v2/
Best regards,
Bin Chen, Reiko Liu and Yu-fan Zheng
Attachment:
Reply_to_report_1_2_on__2112_10514_.pdf