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The non-rational limit of D-series minimal models

by Sylvain Ribault

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Submission summary

Authors (as registered SciPost users): Sylvain Ribault
Submission information
Preprint Link:  (pdf)
Date submitted: 2020-03-10 01:00
Submitted by: Ribault, Sylvain
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


We study the limit of D-series minimal models when the central charge tends to a generic irrational value $c\in (-\infty, 1)$. We find that the limit theory's diagonal three-point structure constant differs from that of Liouville theory by a distribution factor, which is given by a divergent Verlinde formula. Nevertheless, correlation functions that involve both non-diagonal and diagonal fields are smooth functions of the diagonal fields' conformal dimensions. The limit theory is a non-trivial example of a non-diagonal, non-rational, solved two-dimensional conformal field theory.

Author comments upon resubmission

According to the reviewer's report, some important subtleties in the article are not well explained. I have tried to clarify them.

List of changes

In response to the reviewer's suggestions:

1. I have written further explanations on the non-diagonal sector, and related it to the model's $\mathbb{Z}_2$ symmetry as discussed in Ref. [10]. See page 5 after Eq. (2.8), and page 6 the beginning of the last paragraph.

2. Writing that the limit of D-series minimal models was expected to be a diagonal extension of Liouville theory was actually an understatement of the major surprise that occurred. I have tried to explain this in more detail at the beginning of the Conclusion.

3. The role of the degenerate fields is now elaborated in more detail at the beginning of Section 2.2 (second paragraph).

4. The lack of symmetry under $\beta \to \frac{1}{\beta}$ is now demonstrated after Eq. (2.13).

Additional changes:

5. In the Conclusion (Outlook part), I have added a paragraph on possible generalizations.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 1 on 2020-4-28 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1909.10784v2, delivered 2020-04-28, doi: 10.21468/SciPost.Report.1644


As the anonymous referee, I must apologise for the late report.

This is certainly a very stimulating paper. I have spent some time thinking about it and have found it hard to come to a conclusion for reasons below.

I finds it hard to accept one of the main conclusions, which is "We will therefore have to conclude that the diagonal and mixed four-point functions cannot belong to the same CFT". If the limiting procedure results in a CFT then it is a CFT; if it does not, then what is it?

The main problem appears to be the non-analyticity in $\beta^2$ arising from the shift $\sigma(P)$ in equation (2.25). I was for some time misled by this formula which appears to suggest that $\sigma(P)$ is defined independently of $\{r_2,s_2,r_3,s_3\}$. However, since the sign has to change to accommodate changes in sign of three point functions of fields with spin, this cannot be true - for some pairs of $(r_2,s_2)$ and $(r_3,s_3)$, the three point function has to change sign and so $\sigma(P)$ does depend on $\{r_2,s_2,r_3,s_3\}$.

Unfortunately, I cannot find a formula for $\sigma(P)$, or for $C_{P_1,{r_2,s_2},{r_3,_3}}$ itself. It says in [5] that the shift equations are solved (for a sign $f_{2,3}(P)$) but I could not find that solution in that paper, which may very well just be my fault. As it is, the absence of a clear formula for $C_{P_1,{r_2,s_2},{r_3,_3}}$ makes it hard to check whether there are other ways to fix the problem of non-analyticity.

I could imagine that the sign $\sigma(P)$ might have a component (like a cocyle) that depends on $\{r_2,s_2,r_3,s_3\}$ and part that depends on $P$ in such a way that one could split the fields into two sets, one for two possible signs, and for which new fields then have structure constants that are analytic, up to some signs required because of the presence of fields with spin - but it is hard to start thinking about this in the absence of a formula for $C_{P_1,{r_2,s_2},{r_3,_3}}$.

So, to summarise, I have found the ideas very stimulating but have been unable to decide what I think of the results presented here. They seem to me outside what I regard as "likely" and I have not been able to do the sorts of tests of the ideas I would like to do because I have not been able to find the required formulae. This could just be my incompetence, for which I apologise.

If the author can provide a formula for $\sigma(P)$, either explicitly, or a clear indication where it is to be found, then I think that would be a substantial improvement and at that stage I would be very likely to recommend publication. Even if the eventual conclusions are wrong, the paper is still very interesting and stimulating.

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Author:  Sylvain Ribault  on 2020-05-04  [id 814]

(in reply to Report 1 on 2020-04-28)
reply to objection

I am grateful for the reviewer's work and interest. On the function $\sigma(P)$ and on the "hard to accept" conclusion, I realize that the article is not clear and explicit enough. I would be happy to submit a revised version where I would attempt to clarify these points.

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Sylvain Ribault  on 2020-05-11  [id 820]

PS: here are the changes in the arxiv version 3:

1.I have written more details on the sign factor $\sigma(P)$. To begin with, I have explained how the shift equations (2.26) are deduced from ref. [5], and why they do not depend on $r_i,s_i$ in our case. Then, I have written explicit expressions (2.29) and (2.30)
for the solutions of the shift equations. Moreover, in order to better show how the signs appear in four-point functions, I have added the diagrams (2.32).

2.In order to clarify the statement that "the diagonal and mixed four-point functions cannot belong to the same CFT", I have added a few lines of explanation after this statement, at the very end of Section 4. I have also added explanations at the end of Section 5.1.

Sylvain Ribault  on 2020-05-11  [id 819]

In order to clarify the points raised by the reviewer, I have sent a revised version to arXiv: .