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Spaces of states of the twodimensional O(n) and Potts models
by Jesper Lykke Jacobsen, Sylvain Ribault, Hubert Saleur
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Submission summary
Authors (as registered SciPost users):  Jesper Lykke Jacobsen · Sylvain Ribault 
Submission information  

Preprint Link:  https://arxiv.org/abs/2208.14298v1 (pdf) 
Code repository:  https://gitlab.com/s.g.ribault/representationtheory/ 
Date submitted:  20221013 08:56 
Submitted by:  Ribault, Sylvain 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We determine the spaces of states of the twodimensional $O(n)$ and $Q$state Potts models with generic parameters $n,Q\in \mathbb{C}$ as representations of their known symmetry algebras. While the relevant representations of the conformal algebra were recently worked out, it remained to determine the action of the global symmetry groups: the orthogonal group for the $O(n)$ model, and the symmetric group $S_Q$ for the $Q$state Potts model. We do this by two independent methods. First we compute the twisted torus partition functions of the models at criticality. The twist in question is the insertion of a group element along one cycle of the torus: this breaks modular invariance, but allows the partition function to have a unique decomposition into characters of irreducible representations of the global symmetry group. Our second method reduces the problem to determining branching rules of certain diagram algebras. For the $O(n)$ model, we decompose representations of the Brauer algebra into representations of its unoriented JonesTemperleyLieb subalgebra. For the $Q$state Potts model, we decompose representations of the partition algebra into representations of the appropriate subalgebra. We find explicit expressions for these decompositions as sums over certain sets of diagrams, and over standard Young tableaux. We check that both methods agree in many cases. Moreover, our spaces of states are consistent with recent bootstrap results on fourpoint functions of the corresponding CFTs.
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Reports on this Submission
Report #2 by Anonymous (Referee 2) on 202318 (Contributed Report)
 Cite as: Anonymous, Report on arXiv:2208.14298v1, delivered 20230108, doi: 10.21468/SciPost.Report.6476
Report
(emailed by a referee to the editor, submitted by the editor)
First, the question that the article addresses.
With the great wealth of results on O(n) and S_Q models and their critical behavior, it is relevant to understand the basis of these models on a more fundamental grouptheoretical and representationtheoretical level.
It already begins with the knowledge of the symmetry groups themselves, which are nontrivial generalisation of the symmetry groups for natural values of n and Q.
Also, while the authors claim that the partition sum is the basis of one of their approaches, this seems at first sight a strange claim, as the partition sum can be written in terms of (twodimensional) diagrams all of which represent objects which themselves are invariant under the action of the symmetry groups. The state space can hardly be understood in terms of invariant elements only.
The key to this riddle appears to be that the authors consider the twisted partition sum, and the fact that the partition sum is written in terms of the trace of a transfer matrix, which acts in a space of onedimensional diagrams. These onedimensional diagrams are not invariant under the symmetry groups. It is not obvious to me that the resulting state space is independent of the approach. But whether or not it is independent, the resulting state space is worth studying. I think that the algebraic approach does not resolve this difficulty, as it is based on the same set of diagrams.
The authors have taken great efforts to make the material accessible, with extensive references to existing material not only to the published (refereed) literature, but also useful wikipedia lemmas. I found it very useful that in their list of references they have included back references to the places in the article that refer to each element in the list. Also they build up the results carefully by first giving a sketch of the results in the introduction, before they jump to their derivations.
In my opinion the article is a very interesting and useful to be published. Admittedly it will appeal to a limited (but growing) audience, but it is of great importance to the community of mathematical statistical physics, and will be very useful outside of the statistical physics community.
I have no doubt that the article should be published, and may even be considered a catch by the journal that does.
Report #1 by Anonymous (Referee 1) on 2022121 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2208.14298v1, delivered 20221201, doi: 10.21468/SciPost.Report.6242
Strengths
The paper uses a pair of rigorous approaches, that leave little space to doubt about the results.
Weaknesses
The paper is very dense and many concepts are introduced, making it hard to access to people who are not already experts in those topic.
The reading of Section 4 is particularly hard, since many things discussed in Section 4 are actually introduced or defined in Appendix A, and the reader has to go back and forth between those two parts of the paper. In my opinion, several parts of Appendix A should be moved to the main text.
Report
The paper confirms a previous conjecture by the authors about the action of $O(n)$ and $S_Q$ groups in the critical twodimensional $O(n)$ and Potts model. I have a few comments after which I recommend publication
Requested changes
1. I do not understand how clusters with cross topology contribute to Eq. (3.29). They are not counted in $S_0$ nor in $S$. For example, if I consider two configuration, one with only trivial clusters and one with only trivial clusters and a cluster with cross topology, do they contribute the same to (3.29)? Maybe a comment about it would be good
2. (3.30): I would avoid the notation $0^{N(C)}$ and use $\delta_{N(C),0}$, since it's clearer and does not make the formulas more cumbersome (it's also the notation used in (3.29))
3. Above (4.17): $u$ is not defined here, but in the appendix. A sentence about what $u$ is, to gain some intuition on it, would be good
4. (4.21): there are 6 more diagrams, related by swapping the two top indices, which are not considered. By reading the paragraph above (4.21), it's not clear why these should not be considered.
5. Below (4.22): the fact that that representation is $W_1^{(4)}$ is not clear given that there has been no explanation what these representation must be in terms of diagrams. There should be some reference to the appendix A, where this is explained
6. Similarly, at the end of the same paragraph, the statement that the defining feature of $W_{L/2}^{(L)}$ is annihilated by $TL_L(n)$ is not something that has been explained here.
7. Typo in eq (A.17b), $N$ instead of $L$