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On Higherdimensional Carrollian and Galilean Conformal Field Theories
by Bin Chen, Reiko Liu, Yufan Zheng
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Authors (as registered SciPost users):  Yufan Zheng 
Submission information  

Preprint Link:  https://arxiv.org/abs/2112.10514v2 (pdf) 
Date submitted:  20220609 04:54 
Submitted by:  Zheng, Yufan 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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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 twopoint and threepoint correlators by using the Ward identities. We mainly focus on the twopoint correlators of the operators in chain representations. Even in this relative simple case, we find some novel features: multiplelevel structure, shortage of the selection rule on the representations, undetermined 2pt coefficients, etc.. We find that the nontrivial 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.
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Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2022930 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2112.10514v2, delivered 20220930, doi: 10.21468/SciPost.Report.5697
Report
The authors construct highest weight representations of the Carrollian and Galilean limits of the conformal algebra in dimension greater that two. Subsequently they compute twopoint and threepoint correlation functions in fourdimensional Carrollian CFT and Galilean CFT with emphasis in the Carrollian case.
The paper is somehow hard to read due to its extension and technicality, but this is usually the case in standard conformal bootstrap papers. The construction of the representation and correlators is given in detail in each case considered and the diagrams presented are useful to follow the paper.
In my opinion, the paper presents an important number of new and original results, which can definitely be of use in further study of nonLorentzian field theories, and thus it is suitable for publication in SciPost.
I have only a couple of comments that I recommend the author to consider prior publication, which are given below:
 In the introduction and later in equation 2.7 the authors comment on the infinite dimensional extension of the conformal Carroll algebra, which is isomorphic to the BMS algebra in one dimension higher. I think it is also worth mentioning that infinitedimensional enhancements of the Carroll algebra have been considered in 2110.15834 and 2207.11359, which I presume could be extended to the conformal case.
 At the end of page 9 the authors claim that the Casimir operators of the Galilean conformal algebra cannot be obtained taking the non relativistic limit of the conformal symmetry. However, later ( in equation 2.16) they show that using such limit to obtain the Casimir operators indeed works. I think this point should be clarified.
 Equation 4.11 corresponds to the electric Carrollian limit of the KleinGordon action. There is also a magnetic Carrollian scalar field action, which has been considered for example in 2109.06708 (Reference [40] of your manuscript) and 2206.12177. Is it possible to associate this magnetic model with one of the correlation functions developed in section 4.1?
 Following the previous question, could the authors comment on the relation of the scalar Galilean correlator given in equation 5.4 and the electric and magnetic Galilean limits of the KleinGordon action given, for example, in 2206.12177?
While reading the paper I spotted some typos :
 Second paragraph of page 13: for an example > as an example.
Paragraph below equation 4.17: mix >mixes
Beginning of page 46: As has discussed > As it has been discussed (or) As discussed
 Beginning of section 5: is exactly the same with the ones > are exactly the same as the ones
Report #1 by Anonymous (Referee 1) on 2022818 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2112.10514v2, delivered 20220818, doi: 10.21468/SciPost.Report.5552
Strengths
1. Contains detailed analysis of generically nondiagonal representations of Galilean and Carrollian CFTs in dimensions greater than 2.
2. The authors provide a novel analysis of "chain" and "net" representations which generalise similar structures found in log CFTs.
3. Given new emergent connections of nonLorentzian CFTs in various contexts, the analysis could be useful for several fields.
Weaknesses
1. Not a very reader friendly paper. One struggles to understand the larger goals of the authors over and beyond the results presented.
2. A general lack of connection to physics throughout the paper, which means that potential applications to important physical questions are glossed over.
3. Some serious issues which are noted below.
Report
In the paper under review, the authors conduct a detailed study on non Lorentzian quantum field theories, in particular, Carrollian and Galilean conformal field theories in dimensions greater than two. The paper deals primarily with algebraic aspects and representations and correlation functions of these theories. The main achievement of the authors is to understand the “chain” and “net” representations that respectively resemble and generalize the logarithmic representations of relativistic CFTs in this context. There are several points that need to be clarified by the authors before the article can be considered further. Below I provide a list of my questions and concerns.
1. The authors write in the introduction that although the finite version of the Carrollian conformal algebra is isomorphic to Poincare algebra, one cannot use the usual Wigner classification of representations. I understand that the authors wish to concentrate on highest weight representations throughout the paper. But I am unsure why it is said that one cannot use the Wigner classification. This would clearly give another class of representations that would be useful for some purposes.
2. Sec 2: The authors are considering iso(4,1). This has to be a contraction of so(4,2) and not so(4,1) as is written. Below Fig 1, the authors say that the method of taking limits cannot work for GCFTs and then go on to show that the limit indeed does work. This contradiction needs to be addressed.
3. The highlighted statement on top of page 13 about the npoint invariants seems to be an assumption for which some justification has been given. As the authors go on to show later, there is another branch of the correlation functions which give temporal dependence and spatial delta functions. It seems that this statement would not be true for this branch. The authors should clarify this.
4. I am very confused about the description of the higher rank tensor
representations and Young tableaux that is presented in Sec 3.2.2. The vector representation is well worked out and clearly presented. Why do we need to take recourse to SU(4) Young tableau? While composing two SO(3)’s the appearance of SO(4) is of course understandable. But how and why do we need to consider SU(4) in the general representation?
5. In Sec 3.5, there is a dual map constructed. It would be good to link this up to the more intuitively obvious geometric duality of the Carroll and Galilean manifolds where two fibre bundle structures are related by an interchange of the base and the fibre.
6. The equation (4.5) is an important set of equations, but it is written in a rather cumbersome way. I think the equations in the appendix B, where the authors work out the twopoint function looks much neater. Perhaps one can give equations B.1 in the main text to give the reader a better understanding of what is happening here.
7. Below Eq (4.11), the authors write that the vacuum in the time dependent branch is a factorized one. The deltafunction does not have anything to do with a factorized vacuum. Even if one adds a potential, like was done in 2110.02319, the twopoint function remains the same. The authors should modify their statements accordingly.
8. This is a general question. It seems that the singlet sector is perhaps the relevant sector for at least some physical applications. Is it possible to restrict to this subsector? Or does one necessarily need to look at these intricate chain and net structures? For the class of Carrollian and Galilean theories that are derived as a limit of relativistic theories, it seems that one should be able to restrict to the singlet sector. This also seems to be the case for the explicit examples that have been considered in the literature. It would be good if the authors could comment about this.
9. Relatedly, it seems in most known examples, the correlations of Carrollian theories give the time dependent branch. Do the authors have some comments on which theories give the branch that they have concentrated on? Is there also a simple example of a Galilean conformal field theory which gives the structures that the authors describe?
Author: Yufan Zheng on 20220926 [id 2849]
(in reply to Report 1 on 20220818)
Dear referee,
Thank you very much for your patience and valuable comments and suggestions. Please see the attachment for our detailed reply.
Best regards,
Bin Chen, Reiko Liu and Yufan Zheng
Attachment:
Reply_to_report_on__2112_10514.pdf
Author: Yufan Zheng on 20221007 [id 2895]
(in reply to Yufan Zheng on 20220926 [id 2849])
Dear referee,
Please see the resubmission page for new version of the priprint.
link to resubmission: https://scipost.org/submissions/scipost_202210_00048v1/
link to new priprint: https://scipost.org/preprints/scipost_202210_00048v1/
Best regards,
Bin Chen, Reiko Liu and Yufan Zheng
Author: Yufan Zheng on 20221006 [id 2882]
(in reply to Report 2 on 20220930)Dear referee,
Thank you very much for your patience and valuable comments and suggestions. Please see the attachment for our detailed reply.
Best regards,
Bin Chen, Reiko Liu and Yufan Zheng
Attachment:
Reply_to_report_2_on__2112_10514_.pdf
Author: Yufan Zheng on 20221007 [id 2894]
(in reply to Yufan Zheng on 20221006 [id 2882])Dear referee,
Please see the resubmission page for new version of the priprint.
link to resubmission: https://scipost.org/submissions/scipost_202210_00048v1/
link to new priprint: https://scipost.org/preprints/scipost_202210_00048v1/
Best regards,
Bin Chen, Reiko Liu and Yufan Zheng
Anonymous on 20221006 [id 2886]
(in reply to Yufan Zheng on 20221006 [id 2882])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 after the modifications listed in their response to my first report, their manuscript can be published in SciPost.