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

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

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.