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Global symmetry and conformal bootstrap in the two-dimensional $Q$-state Potts model

by Rongvoram Nivesvivat

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Submission summary

Authors (as registered SciPost users): Rongvoram Nivesvivat
Submission information
Preprint Link: https://arxiv.org/abs/2205.09349v1  (pdf)
Code repository: https://gitlab.com/s.g.ribault/Bootstrap_Virasoro/-/tree/Stests
Date submitted: 2022-07-08 10:28
Submitted by: Nivesvivat, Rongvoram
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical

Abstract

Four-point functions of the Potts conformal field theory are dictated by two constraints: the crossing-symmetry equation and $S_Q$ symmetry. We numerically solve the crossing-symmetry equation for the simplest 28 four-point functions of the Potts conformal field theory for $Q\in\mathbb{C}$. In all examples, we find crossing-symmetry solutions that are consistent with $S_Q$ symmetry of the Potts conformal field theory. In particular, we have determined their numbers of crossing-symmetry solutions, their exact spectra, and a few corresponding fusion rules. Furthermore, in contrast to our results for the $O(n)$ model, in 17 out of 28 cases, there are extra crossing-symmetry solutions whose interpretations are still unknown.

Current status:
Has been resubmitted

Reports on this Submission

Report #3 by Anonymous (Referee 4) on 2023-2-4 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2205.09349v1, delivered 2023-02-04, doi: 10.21468/SciPost.Report.6673

Strengths

Very clear presentation of the results and organization of the paper

Weaknesses

A physical interpretation of the technical results is absent

Report

The author discusses the existence of crossing symmetric four-point functions in a CFT characterized by two hypotheses:

1-The spectrum of the CFT coincides with the proposal of [2] for the $Q$-state Potts model in its geometrical representation in terms of FK clusters.

2-The fields belong to particular $S_{Q}$ representations with complex $Q$. It is known in the mathematical literature how tensor products of such $S_Q$ representations decompose into irreps. Thus further constraints, based on $S_{Q}$ symmetry, can be put on the (inter-chiral) conformal block expansion of the four-point functions.

The author shows that in certain cases there exist more solutions to the bootstrap equations than the number of linearly independent functions that can be obtained by counting the possible irreps in the tensor product of four $S_Q$ representations. The interpretation of those additional functions is not clarified. The paper is written logically and clearly, including in my opinion the introduction.

The material presented is sufficiently interesting (although not of strong impact) for a specialist in the problem. In particular, it shows once again and in a non-trivial way the consistency of the spectrum proposed in [2] and the usefulness of the symmetry considerations formulated in [7]. Altogether, I recommend publication.

Here are some minor points/remarks to which the author is invited to ask but not necessarily in the paper:

1-The linearly independent four-point connectivities were first considered in arXiv:1104.4323. The author could add a reference to this paper.

2-Can the author explain how the number of crossing symmetric solutions to the bootstrap equations without the $S_Q$ constraints is obtained?
Is it possible to obtain the dimension of this vector space without solving the bootstrap equations?

3-Are the structure constants obtained analytic functions of $c$? In principle, I think, in its geometrical representation the model considered makes sense only for $c\leq 1$.

4-The additional not $S_Q $invariant solutions are present also for integer $0\leq Q\leq4$?

A final curiosity: the papers, arXiv:1008.1216 and arXiv:1111.4033 presented a construction of spin clusters in the $Q$-state Potts model at arbitrary $Q$. Are some of the fields that appear there, also present in the spectrum of [2]?

Requested changes

The author is asked to answer the questions in the report and to add a reference.

  • validity: high
  • significance: ok
  • originality: ok
  • clarity: high
  • formatting: perfect
  • grammar: perfect

Author:  Rongvoram Nivesvivat  on 2023-02-11  [id 3342]

(in reply to Report 3 on 2023-02-04)
Category:
remark
answer to question

I have updated the revised version on arXiv.
https://arxiv.org/abs/2205.09349

Answers to Report 3:

%1-The linearly independent four-point connectivities were first considered in arXiv:1104.4323. The author could add a reference to this paper.

Answer to %1:

I have referred to this paper at the introduction of the four-point connectivities in (1.4).

%2-Can the author explain how the number of crossing symmetric solutions to the bootstrap equations without the $S_Q$ constraints is obtained? Is it possible to obtain the dimension of this vector space without solving the bootstrap equations?

Answer to %2:

For each four-point function, solving the crossing-symmetry equation with the spectrum of the Potts CFT gives us a linear combination of solutions. Then we look into each of these solutions whether their spectra satisfy the $S_Q$ constraint (3.12). In general, we find that not all of them satisfies (3.12). I have attempted to summarize this concept in "Main results". Examples of how to extract these extra solutions are also presented in Section 5.

We do not know yet how to count the extra solutions without solving the crossing-symmetry equation.

However, in the critical $O(n)$ model, we now understand how to count solutions without solving the crossing-symmetry equation by counting two-dimensional graphs, known as the combinatorial maps. This new approach of counting solutions will appear in our upcoming paper. It should be interesting to find a similar approach which also works with the Potts model. If succeed, it will allow us to count the extra solutions.


%3-Are the structure constants obtained analytic functions of $c$? In principle, I think, in its geometrical representation the model considered makes sense only for $c\leq 1$

Answer to %3

Yes, structure constants are analytic function of the central charge, and we expect that the structure constants themselves are valid for generic value of the central charge in (1.1).

For instance, in arxiv:2007.04190, we have tested that the formula for the structure constants $D_{(0, 1/2)}$ in the four-point connectivities from arxiv:1009.1314 coincide with the numerical bootstrap at very high precision for complex value of the central charge.

This also agrees with the fact that the Fortuin-Kasteleyn description of the Potts model makes sense for generic $Q$.


%4-The additional not $S_Q$ invariant solutions are present also for integer $0\leq Q\leq 4$?

Answer to %4:

For integer $Q$, we expect that the extra solutions would still appear. For instance, results for these special values of $Q$ could be obtained by taking limits of our results for generic $Q$, which are solutions to the linear system (3.2). In general, a limit should not change the number of solutions to linear system. This situation should indeed be studied in more details by considering the limits $Q$ approaching integers of these extra solutions.


%5-A final curiosity: the papers, arXiv:1008.1216 and arXiv:1111.4033 presented a construction of spin clusters in the $Q$-state Potts model at arbitrary $Q$. Are some of the fields that appear there, also present in the spectrum of [2]?

Answer to %5:

The critical exponents in (3) of 1008.1218 is related to conformal dimension (2.1) as follows

$h_{l_1 - l_2, 2*l_1} = \Delta_{2*l_1, l_1 - l_2} $

where $\kappa$ in (4) of 1008.1218 is our $4\beta^2$.

Since $\Delta_{r, s}$ with $r \in 2\mathbb{N}$ and $s \in \mathbb{{Z}$, so the critical exponents of 1008.1218 appear in the spectrum (2.2) for integer $l_1$ and $l_2$. It might be interesting to investigate further if some of our bootstrap solutions are observables of the spin-cluster Potts model in 1008.1218

Report #2 by Anonymous (Referee 5) on 2022-10-11 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2205.09349v1, delivered 2022-10-11, doi: 10.21468/SciPost.Report.5872

Strengths

1. interesting topic
2. proper methodology

Weaknesses

1. not clearly written
2. several places with no clear assumptions made

Report

The paper deals with the study of four-point functions in Potts model. The topic is interesting on its own and also the applied methodology is implemented in a novel way. However, I have several comments and questions that need to be addressed to make the paper in a publishable format.

Requested changes

1. The introduction is extremely technical. I would recommend a rewriting to make it clearer what the author would like to show/prove and to contextualize the study. For instance in the last paragraph, it is written: "more general four point functions...", it would be good to make more precise statements and put them into context.

2. The section on main results, while could be quite interesting due to the technical level of the paper, is not very illuminating since it is mostly a table of content. Also here, "the simplest 28 four point functions" is a bit arbitrary, either the author explains which are the simplest before or it should be specified which class has been considered. I would recommend either to remove the section and put it in the concluding section as a summary or to rewrite it completely to make more clear what are the main results.

3. In eq (3.1) how can one compute the coefficients D? Later on it is written that they are all computable and it would be nice to give an (even short) account of how to do it.

4. After eq. (3.2), two comments:
-why the intermediate states need to be the same in all channels?
-the comment regarding having three relations to avoid infinitely many solutions sounds very interesting but it is not clear what it means and how to interpret it. I would like the author to clarify it.

5. In the examples sections it seems that the author is inputting the exact spectrum into the crossing relations to compute the three point function. Am I right? If so, how the procedure is carried over needs to be discussed before eq. (3.8), meaning in that subsection.

  • validity: high
  • significance: good
  • originality: ok
  • clarity: low
  • formatting: good
  • grammar: good

Author:  Rongvoram Nivesvivat  on 2023-02-11  [id 3343]

(in reply to Report 2 on 2022-10-11)

I have updated the revised version on arXiv. https://arxiv.org/abs/2205.09349


Answers to Report 2:

%1. The introduction is extremely technical. I would recommend a rewriting to make it clearer what the author would like to show/prove and to contextualize the study. For instance in the last paragraph, it is written: "more general four point functions...", it would be good to make more precise statements and put them into context.

%2. The section on main results, while could be quite interesting due to the technical level of the paper, is not very illuminating since it is mostly a table of content. Also here, "the simplest 28 four point functions" is a bit arbitrary, either the author explains which are the simplest before or it should be specified which class has been considered. I would recommend either to remove the section and put it in the concluding section as a summary or to rewrite it completely to make more clear what are the main results.


Answer to %1 and %2:

Regarding these two points, I have attempted to write a less technical introduction by referring to some physical models on the table (1.2). I have also rewritten "Main results" and added more specific details.


%3. In eq (3.1) how can one compute the coefficients D? Later on it is written that they are all computable and it would be nice to give an (even short) account of how to do it.

Answer to %3:


Coefficients D in the inter-chiral block expansion (3.1) can be considered as products of three-point structure constants, which can be completely determined by using the BPZ equation, the crossing-symmetry equation, and the single-valuedness of four-point functions of the types: $<V_{1,2}^D V_1 V_2 V_3 >$ and $<V_{1,2}^D V_1 V_{1,2}^D V_1 >$ where $V_{1,2}^D$ is the level-2 degenerate field and $V_1, V_2 V_3$ are generic primary fields.

I have added this explanation below (3.1), as well as a reference for examples of detailed calculations of $D$ in (3.1).


%4. After eq. (3.2), two comments:
%4.1-why the intermediate states need to be the same in all channels?
%4.2-the comment regarding having three relations to avoid infinitely many solutions sounds very interesting but it is not clear what it means and how to interpret it. I would like the author to clarify it.

Answer to %4.1:

For generic four-point functions, the intermediate states are not the same in all channels.

In particular, the spectra $S^{(s)}$, $S^{(t)}$, $S^{(u)}$ are the full spectrum of the Potts model allowed by the degenerate fusion rules (3.2). So $S^{(s)}$, $S^{(t)}$, $S^{(u)}$ are identical for four-point functions of identical fields but not for generic four-point functions. This ensures us the resulting solutions are at least all crossing-symmetry solutions in the Potts model, as will be demonstrated for several examples in Sections 4 and 5.


I have added this explanation above (3.3), as well as an example of four-point functions whose the intermediate states are not the same below (3.7).



Answer to %4.2:

We do not know yet the interpretation of having infinitely many solutions, and I have clarified that this issue is still an open problem below (3.2).



%5. In the examples sections it seems that the author is inputting the exact spectrum into the crossing relations to compute the three point function. Am I right? If so, how the procedure is carried over needs to be discussed before eq. (3.8), meaning in that subsection.

Answer to %5

Not quite, we compute the four-point structure constants $D$ by inputting the exact spectrum into the crossing-symmetry equation.

However, we could deduce three-point functions from our method. For instance, we wrote down the fusion rules (4.14) and (4.15) by checking that (4.14) and (4.15) always hold for many examples of four-point functions of the type $<V_{2,1/2}V_{0, 1/2}V_1V_2$ and $<V_{2,0}V_{0, 1/2}V_1V_2>$ of the Potts model.

This clarification has been added below (4.14) and (4.15).

Report #1 by Anonymous (Referee 6) on 2022-10-8 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2205.09349v1, delivered 2022-10-08, doi: 10.21468/SciPost.Report.5850

Strengths

1)-Results are new, conistent with previous results and the numerical evidences are very convincing.
2)-These results support the existence of a consistent CFT describing the geometrical critical phases of a very important class of models
3)-The paper is well written, easy to consult for researcher working in the fields

Weaknesses

1)The paper requires the study of the previous work on the subject. This of course makes this paper difficult to read for people not working on this problem. I believe, on the other side, thiat this allows the people working on this subject to extract the new results quite easily

2i) As far as I know, the new crossing symmetry correlation functions studied in this paper have not a clear interpretation in terms of the lattice model observables.

Report

Dear editor,
these last years have seen very important progress in the definition of a CFT describing the Potts and the O(n) critical points. The importance of these progresses go beyond the description of the statistical models, as they point out, for the first time after decades of efforts from different perspectives, how to construct explicitly a local, non-unitary and logarithmic CFT. So this line of research is among the most active and fruitful in the general Bootstrap program.

This paper contributes greatly to the comprehension of the Potts CFT, exploring many other different four point functions of this theory. From one side provides new consistency checks of the framework elaborated in previous work, including in particular the way to implement the global permutation S_q symmetry for general q. From the other side, is hints to some new solutions with respect to the predictions of the global symmetry that deserve future investigations.

The paper is written in a "list of results" way, but in a very clear way. I found this choice particularly adapted for the people working in this area.

I recommend publication of the manuscript in the present form

  • validity: top
  • significance: top
  • originality: high
  • clarity: top
  • formatting: perfect
  • grammar: excellent

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