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Curiosities above c = 24

by A. Ramesh Chandra, Sunil Mukhi

This is not the latest submitted version.

This Submission thread is now published as SciPost Phys. 6, 053 (2019)

Submission summary

As Contributors: A. Ramesh Chandra · Sunil Mukhi
Arxiv Link: (pdf)
Date submitted: 2019-01-31 01:00
Submitted by: Mukhi, Sunil
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


Two-dimensional rational CFT are characterised by an integer $\ell$, the number of zeroes of the Wronskian of the characters. For $\ell<6$ there is a finite number of theories and most of these are classified. Recently it has been shown that for $\ell\ge 6$ there are infinitely many admissible characters that could potentially describe CFT's. In this note we examine the $\ell=6$ case, whose central charges lie between 24 and 32, and propose a classification method based on cosets of meromorphic CFT's. We illustrate the method using theories on Kervaire lattices with complete root systems. In the process we construct the first known two-character RCFT's beyond $\ell=2$.

Ontology / Topics

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Conformal field theory (CFT)
Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2019-3-25 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1812.05109v2, delivered 2019-03-25, doi: 10.21468/SciPost.Report.887


1 - Useful and original approach to the classification of RCFT
2 - The authors' methods might lead to potentially new interesting RCFT with c>24 could be found.


1 - The article is not completely self-contained: a review of the previous results by the authors (especially the most recent ones) would make it much clearer


The article's goal is to classify CFTs whose chiral algebra has two characters and whose Wronskian has a certain number of zeroes (l/6=1 in the authors' notation). The starting point is a classification (that was done in previous articles by the authors' and by others) of pairs of admissible characters, i.e. such that their Fourier coefficients are non-negative integers. It is proposed that all (or most) pairs of such admissible characters might correspond to RCFTs obtained by taking cosets of some meromorphic CFT of central charge c=32.

The approach is original and innovative. The methods described here might potentially lead to the discovery of several new RCFT at central charge c>24.

Regarding the clarity of the manuscript, my main concern is that the article is heavily based on results obtained in previous recent articles, and that are not really reviewed, explained or motivated in the present manuscript. This makes the article very difficult to understand even for a reader with a good background in CFT. In particular, for this reason, I was not able to verify that all the authors' statements are correct.

I suggest the authors to add a couple of pages (either in an appendix, or even better as an introductory section), where they review the main results in the references (in particular, reference [15]) that they use in this paper. Once this is done (and provided that the derivations in the article are correct), I think that the manuscript deserves to be published.

Requested changes

As explained in my report, the main change that I would suggest is a (brief) introductory section where the main results that are used by the authors are reviewed and explained. Once this is properly done, some of the following comments might be superfluous:

1 - end of page 1: "The case c=24 ... 71 RCFTs" strictly speaking, the fact that there are 71 meromorphic CFTs at c=24 is based on the unproved conjecture that the Monster VOA is the only one without currents.
2- page 2, line 10: " the number of zeroes of the leading Wronskian, denoted l/6..." I guess it is fractional because it is a function defined on an orbifold (which I guess is the upper half-plane divided by SL(2,Z) ). I would suggest the authors to explain explicitly where the Wronskian is defined and their conventions in counting the multiplicities of zeroes. (In the abstract, the authors claim that l is the number of zeroes of the Wronskian, which means that they are using a different convention for the order of a zero. I understand that it is not a good idea to insert too many complications in the abstract, but at least in the main body this point should be clarified).
3- page 2, lines 18 - 21: "Note that ... given c and l" I would help if the authors write these relations explicitly (and maybe explain in more detail where they come from), at least in the case of two characters.
4- page 3, point (iii) and the following lines: is one supposed add (or take linear combinations of) quasi-characters with the same l? Otherwise, the statement that l is augmented by "multiples of 6" would not make much sense. If this is the case, please say this explicitly. In any case, it is not explained why adding quasi-characters should change the central charge and the dimension of the primary in the way that is described here. I suggest the authors to include some more details and some reference (but the reference alone is not enough).
5- page 7, after point (iii): "Thus all even unimodular lattices ... with dimension less than 24" less or equal to 24.

  • validity: good
  • significance: good
  • originality: good
  • clarity: ok
  • formatting: good
  • grammar: excellent

Author:  Sunil Mukhi  on 2019-04-02  [id 481]

(in reply to Report 2 on 2019-03-25)

Thank you for the helpful report. In a revised version, we have added a couple of pages of introductory material in Sections 1 and 2, as requested. We hope this will make the paper more self-contained. We have also addressed all the points listed in your "requested changes".

Anonymous Report 1 on 2019-3-4 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1812.05109v2, delivered 2019-03-04, doi: 10.21468/SciPost.Report.855


Clear exposition of results and assumptions
Several interesting examples


The ultimate goal of this line of research is not clear.


This paper makes the first steps on a new avenue towards an understanding of CFTs beyond c=24. Although it is not clear where this path will lead, for example given the fact that the c=32 theories are probably not enumerable in practice, there are enough interesting results in this work to warrant publication, so that future work can build on these results.

I only have a minor comment: The abbreviation LY (presumably Lee-Yang) in table 1 should be explained. One should not leave the reader guessing.

Requested changes

Define LY

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

Author:  Sunil Mukhi  on 2019-04-02  [id 480]

(in reply to Report 1 on 2019-03-04)
reply to objection

Sorry, we are including an explanation of LY in a revised version.

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