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A note on the identity module in $c=0$ CFTs
by Yifei He, Hubert Saleur
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Submission summary
Authors (as registered SciPost users):  Yifei He 
Submission information  

Preprint Link:  https://arxiv.org/abs/2109.05050v3 (pdf) 
Date accepted:  20220228 
Date submitted:  20220215 04:01 
Submitted by:  He, Yifei 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
It has long been understood that nontrivial 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 stressenergy 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 selfduality. We find that the structure contains a rank3 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 nonchiral logarithmic coupling $a_0={25\over 48}$.
Author comments upon resubmission
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 welldefined $c=0$ operator it must have finite correlations at $c=0$. One can analyze its 2point 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 twopoint 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 3point function is also welldefined as studied in reference [18].
To summarize, in writing eq. (2.18), we are assuming that $\ldots$ involves only welldefined 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)