SciPost logo

SciPost Submission Page

A note on the identity module in $c=0$ CFTs

by Yifei He, Hubert Saleur

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Yifei He
Submission information
Preprint Link:  (pdf)
Date accepted: 2022-02-28
Date submitted: 2022-02-15 04:01
Submitted by: He, Yifei
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


It has long been understood that non-trivial Conformal Field Theories (CFTs) with vanishing central charge ($c=0$) are logarithmic. So far however, the structure of the identity module -- the (left and right) Virasoro descendants of the identity field -- had not been elucidated beyond the stress-energy tensor $T$ and its logarithmic partner $t$ (the solution of the "$c\to 0$ catastrophe"). In this paper, we determine this structure together with the associated OPE of primary fields up to level $h=\bar{h}=2$ for polymers and percolation CFTs. This is done by taking the $c\to 0$ limit of $O(n)$ and Potts models and combining recent results from the bootstrap with arguments based on conformal invariance and self-duality. We find that the structure contains a rank-3 Jordan cell involving the field $T\bar{T}$, and is identical for polymers and percolation. It is characterized in part by the common value of a non-chiral logarithmic coupling $a_0=-{25\over 48}$.

Author comments upon resubmission

Dear Editor,

Thank you for the consideration of the publication and the referee for further questions.
Please see below for our answer. We hope this addresses the referee's concern.

Best regards,
Yifei He and Hubert Saleur

List of changes

One of the basic assumption is that for an operator to be well defined in the $c=0$ theory, it must have finite correlation functions. This is the key to the $c\to 0$ analysis. Indeed, one can consider the case as the referee mentioned of an operator defined using the generic $c$ fields $\psi,\phi$ as $\mathcal{O}=#\psi+\frac{1}{c}\phi$, but for it to be a well-defined $c=0$ operator it must have finite correlations at $c=0$. One can analyze its 2-point function in a similar way as $t$. (in this case it will result in a logarithmic operator with certain conditions on $#$) Perhaps the real concern is whether the two-point function analysis guarantees the finiteness of all other correlation functions of $O$ (or $t$) defined this way. To make this generic claim requires further studies. However in the special case of the operator $t$, we know for example that its 3-point function is also well-defined as studied in reference [18].
To summarize, in writing eq. (2.18), we are assuming that $\ldots$ involves only well-defined operators at $c=0$ in the above sense and therefore $\langle t(z,\bar{z})\ldots \rangle$ is finite by this assumption. We have slightly modified the footnote 7 to make this clear.

Published as SciPost Phys. 12, 100 (2022)

Reports on this Submission

Anonymous Report 1 on 2022-2-15 (Invited Report)


I am now happy with the explanation of point 2) and recommend publication

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Login to report or comment