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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: 2020-06-02 02:00
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

This resubmitted version is on arXiv since May 8th, I am now formally submitting it at the Editor's request.

I am grateful for the reviewer's work and interest. On the function σ(P) and on the "hard to accept" conclusion, I realize that the article is not clear and explicit enough. The resubmitted version addresses these shortcomings.

List of changes

1. I have written more details on the sign factor σ(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.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 1 on 2020-6-14 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1909.10784v3, delivered 2020-06-14, doi: 10.21468/SciPost.Report.1762


Again, I would like to thank the author for their changes. The comments on the limit process require more thought, but they present a point of view that I think is definitely acceptable.

On the issue of the sign $\sigma(P)$, I am sorry that I was not more explicit as I think that the author has not understand my concerns. I was not worried by the shift equations themselves, whether they depend on $(r_2,s_2)$ and $(r_3,s_3)$, only in the solutions and the actual values of $\sigma(P_1)$.

The sign $\sigma(P)$ has to depend on $(r_2,s_2)$ and $(r_3,s_3)$ since, for example, $C_{P,(2,1/2),(4,1/2)} = - C_{P,(4,1/2),(2,1/2)}$ from the standard result for the three point structure constant of fields with integer spin,
$C_{abc} = (-1)^{S_c+S_a+S_b} C_{acb}$ where $S_a$ is the spin of the field $a$, [see e.g. eqn (2.2.48) in arxiv:1406.4290] and the facts that the spin of the field $V_P$ is $0$, and of $V_{(r,s)}$ is $rs$.
Hence, if we replace $\sigma(P)$ in eqn (2.25) by the more general notation
$\sigma_{P_1,(r_2,s_2),(r_3,s_3)}$, it has to be the case that
$\sigma_{P_1,(2,1/2),(4,1/2)} = - \sigma_{P_1,(4,1/2),(2,1/2)}$.
I would like to know what values the author has given for $\sigma(P_1)=\sigma_{P_1,(r_2,s_2),(r_3,s_3)}$ to understand if there is any way to undo the non-analyticity by field redefinitions, splitting the fields into two sets, etc, or (as is obviously suggested) there is none.
I really think that the author should provide the values of $\sigma_{P_1,(r_2,s_2),(r_3,s_3)}$ so that one could check the numerical calculations.

I am sorry that such a seemingly small point should hold up publication, but it seems essential to me to allow readers to reproduce the calculations and to decide for themselves on the possibility of an analytic solution or not.

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Author:  Sylvain Ribault  on 2020-06-15  [id 854]

(in reply to Report 1 on 2020-06-14)
answer to question

In Eq. (2.25) for the three-point structure constant, in addition to $\sigma(P)$, there should be a prefactor $(-1)^{r_2s_3}$. This prefactor obeys $(-1)^{r_2s_3} = (-1)^{r_2s_2+r_3s_3} (-1)^{r_3s_2}$, because $r_1+r_2\in 2\mathbb{Z}$ and $s_2+s_3\in\mathbb{Z}$. So it leads to the expected behaviour of the three-point structure constant when exchanging the fields $2$ and $3$.

Does this answer the question? Please object if it does not. If I receive no objection within a few days, I plan to submit a revised version with the additional prefactor.

Author:  Sylvain Ribault  on 2020-06-24  [id 864]

(in reply to Sylvain Ribault on 2020-06-15 [id 854])
answer to question

Yes, this sign factor is described in the notebook Correlators.ipynb in the second text cell, and implemented by the line "product = (-1)**sum(fields[i].indices[0]*fields[i+1].indices[1] for i in range(3))" in the method three_shift() of the class FourPoint.

Anonymous on 2020-06-23  [id 863]

(in reply to Sylvain Ribault on 2020-06-15 [id 854])

Just to confirm, is this the value of the structure constants used in the numerical checks?

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