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The non-rational limit of D-series minimal models

by Sylvain Ribault

This is not the current version.

Submission summary

As Contributors: Sylvain Ribault
Arxiv Link: (pdf)
Date submitted: 2019-10-22
Submitted by: Ribault, Sylvain
Submitted to: SciPost Physics Core
Discipline: Physics
Subject area: High-Energy Physics - Theory
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.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 1 on 2019-12-16 Invited Report


1- The paper makes a serious attempt to understand the limit(s) of the D-series minimal models. Not all the results are rigorous, but I did not find any mistakes.


1- The analysis depends strongly on the fields being clearly divided into diagonal and non-diagonal, but the "non-diagonal" sector includes diagonal fields of the sort $(r,s),(-r,s) = (0,s),(0,s)$. This means that further analysis is required to justify equations (2.16), (2.18), (2.19) etc. I think this could be done, for example it is noted before equation (2.27) that these can be distinguished by their 3 point functions, but at the very least the words "non-diagonal sector" are misleading. This needs to be explained clearly.

2- I though the discussion in in the conclusion was a little disingenuous. The D-series diagonal fields have an identical set of structure constants and correlation functions with a subset of the A-series models, but the A-series models have extra diagonal fields and so it is not especially surprising if the structure constants of the limits are different.

3- I do not know in what sense the degenerate fields "exist" in the theory if they are excluded from the spectrum. They are outside the theory and I would like some justification why they should be able to be included consistently and that deductions from their correlation functions are still valid.


This is an interesting paper adding to the study of limit CFTs and CFTs with continuous spectrum. There are some clear issues involved with defining the limit CFT(s) of the D-models which are discussed and tackled.

I would have liked a clearer demonstrations that there are in fact two different CFTs in the limit, one from $\beta$ and one from $1/\beta$. It is asserted a couple of times but I did not find a clear explanation: apologies if I just missed it.

Requested changes

1- please clarify that the "non-diagonal sector" also includes diagonal fields and that the restrictions on fusion rules etc follow from other considerations than "diagonality".

  • validity: high
  • significance: high
  • originality: top
  • clarity: ok
  • formatting: good
  • grammar: acceptable

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