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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 accepted: 2020-07-31
Date submitted: 2020-06-23 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

I have corrected the overall sign in Eq. (2.25), so that the three-point structure constant has the right behaviour under permutations of the non-diagonal fields. The values of $\sigma(P)$ are still given by Eqs. (2.29) and (2.30), where the constant prefactor can now be set to one, i.e. it is not just $P$-independent.

Let me emphasize that only the $P$-dependence of $\sigma(P)$ matters for taking the limit of a given four-point function. And $\sigma(P)$ does not appear in the numerical tests of crossing symmetry in Section 4.2, as taking the continuum limit transforms $\sigma(P)$ into the distribution (4.11).

Published as SciPost Phys. Core 3, 002 (2020)

Reports on this Submission

Anonymous Report 1 on 2020-7-19 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1909.10784v4, delivered 2020-07-19, doi: 10.21468/SciPost.Report.1839


I would like yet again to thank the author for clarifying the results in this paper. I still find the ultimate analysis unconvincing, but the calculations presented are still very interesting and are now sufficiently detailed to allow a reader to repeat the derivations and decide for themselves whether there is an alternative explanation, and I am happy to recommend publication.

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

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Sylvain Ribault  on 2020-06-24  [id 865]


To be more accurate: when testing crossing symmetry in Section 4.2, the sign prefactor $(-1)^{r_2s_3}$ of the three-point structure constant does appear. But this sign is not included in the definition of $\sigma(P)$ in version 4 of the submitted article. The factor $\sigma(P)$ itself is taken to be one in the $t$- and $u$-channel calculations, where it plays no role as we do not integrate over continuous momentums. In the $s$-channel calculation that factor is transformed into the distribution (4.11) by taking the limit.