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

Dear Editor,

We would like to thank you for considering the publication and thank the referees for reading the manuscript so carefully, posing many interesting questions/comments and making valuable suggestions for improvement of the paper. Please see below for the replies to the referees and changes made under the referees' suggestions.

Best regards,
Yifei He and Hubert Saleur

Anonymous Report 2 on 2021-12-20

1, Indeed, as the referee pointed out, the normalization of $\Psi$ is fixed by the normalization of $X$, namely that its two-point function is normalized to 1. This can be seen straightforwardly by considering the relation $A^{\dagger}\Psi=b_{12}\bar{X}$ and write:
\begin{equation}
\langle A\bar{X}|\Psi\rangle=\langle\bar{X}|A^{\dagger}\Psi\rangle=b_{12}\langle\bar{X}|\bar{X}\rangle=b_{12}
\end{equation}
and use the relation between the inner product of the states with the CFT two-point function. Regarding this point, we have added footnote 2 with comments.

2, The standard argument that no operators can have factors such as $\frac{1}{c}$ is as follows: Suppose there is an operator which takes the form $\frac{1}{c}\phi$ where $\phi$ has order 1 two-point functions, then the field is not well-defined at $c=0$ as its two-point function diverges, and therefore it does not make sense to consider operator insertion of $\frac{1}{c}\phi$ into correlation functions. We have added a brief comment on this in footnote 7.

3, A way to see that a field is chiral is to look at its two-point function and in particular its holomorphicity. This different from directly looking at the anti-chiral conformal dimension, which might vanish without excluding some logarithmic dependency. An example of this is the logarithmic field $t(z,\bar{z})$: While its anti-chiral conformal dimension vanishes, it is a non-chiral field and this can be seen from the two-point function of $\bar{\partial}t$ as we have written in eq. (2.22). It is by this standard that we claim $-\sqrt{-\frac{c}{2}}\bar{\partial}X$ a chiral field: from (2.20), the two-point function of this field goes as
\begin{equation}
\langle\Big(-\sqrt{-\frac{c}{2}}\bar{\partial}X\Big)(z,\bar{z})\Big(-\sqrt{-\frac{c}{2}}\bar{\partial}X\Big)(0,0))\rangle=\mathcal{O}(c^2)
\end{equation}
vanishing at $c=0$. This agrees with our identification with $T(z)$ which is further expected from lattice considerations. It would be however interesting to study further to find a more rigorous argument (including also the physical interpretation of this identification) which we leave for future work.

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Anonymous Report 1 on 2021-12-20

0', Following the referee's comments, we have changed all the names "Jordan cells" to "Jordan blocks". Before Gurarie, we note that some old french books used "cellule de Jordan" quite commonly.
1, We are not considering the case of $c=0$ CFTs that are constructed by tensor products. Indeed, one can resolve the $c\to 0$ catastrophe in tensor product CFTs, for example by taking two non-interacting CFTs with equal and opposite central charges, and introducing a chiral field $t(z)$. However in this case the algebra of $T$ and $t$ form are trivial. A nice argument can be found in section II A of reference [20] which we refrain from repeat here.
2, To avoid confusing the readers between the diagrams we draw in the paper and Loewy diagrams, we have suppressed mentioning ``Loewy diagrams" throughout the paper.
3, To avoid the potential confusion about $t,\bar{t}$ as the referee mentioned, we have added the two-point functions of $(t,T)$ in the introduction. (page 3)
4, The constant in the two-point function of the logarithmic field (for example $\Psi$ in eq. (2.5a)) depends on the choice of basis in the Jordan block. As we have commented in the manuscript, this constant can be modified by a change of basis. Of course, as the referee pointed out, this is only the case for non-zero logarithmic coupling. (With zero logarithmic coupling, there is no logarithmic operator and thus no Jordan block.)
5, Eq. (2.6) are the logarithmic couplings of the rank-2 Jordan block (figure 1) at generic $c$ as recently obtained in references [1,4]. We would like to point out that they are non-chiral logarithmic couplings, contrary to what the referee said. By the request of the referee, we have added footnote 3 for a clear reference to both references [1,4] where the derivations can be found.
6, The linear combination of fields with different dimensions do not correspond to scaling fields (for example, their correlation functions do not have a scaling behavior with certain exponent). It is in this sense that we are referring to this as ``dimensionally problematic'' in footnote 5.
7, The singularity cancellation conditions (2.13), (2.28) and (2.30) are obtained by requiring the finite behavior of the $c=0$ theory which is a key to our analysis. (This dates back to Gurarie's original analysis on the $t$ field although we have studied beyond that.) To make this clear, as the referee suggested, we have added the following sentence to the introduction (second to last paragraph on page 2): “By requiring the finiteness of two-point functions at $c=0$, we obtain singularity cancellation conditions (2.13), (2.28) and (2.30), which allow us to establish the existence of a rank-three Jordan block of fields of weight $h=\bar{h}=2$ when $c=0$ with the bottom field being $T\bar{T}$, and determine the corresponding universal logarithmic coupling.”
8, We have followed the referee's suggestion to use the notation $\overset{c\to 0}{=}$ at various places when we mean ``in the $c\to 0$" limit (in addition to the comments we had made before). This appears in eqs. (2.17), (2.18), (2.19), (2.21), (2.37) and (2.38). In eq. (2.20) we have kepted the big-$\mathcal{O}$ notation but removed $\overset{c\to 0}{=}$ as the referee suggested.
9, We have rewritten the beginning of section 2.3 to make the flow of the reading more smooth. In this part, we would like to first focus on the field $\Phi_2$ and obtain the conditions (2.28) and (2.30) as required by the finiteness of its 2-point function. This is why we postpone the definition of $\Phi_1,\Phi_2$ to avoid interruption. Now we have postponed writing down the logarithmic OPE (2.32) after the definition of the fields $\Phi_1,\Phi_0$ to avoid the confusion as the referee mentioned. We believe this is a more logical order for writing this part.
10, This question is related to question 7 and we have answered both above.
11, This is related to the answer 9 above. We believe it is good to keep the definition (2.31) (previously (2.32)) in order for the readers to check straightforwardly the logarithmic OPE (2.32). Therefore we have kept these definitions.
12, Based on the referee’s suggestion, we have slightly rewritten the part between eq. (2.40) and (2.44) to make it more clear. The expression of $a_0$ is by direct calculation using the definitions of the fields $\Psi_{0,1,2}$. Using the current expression of (2.42), it should be straightforward to verify (2.43) which makes it clear that the logarithmic coupling $a_0$ is identical for Potts and $O(n)$ models. Eq. (2.44) is not a derivation, but rather an interesting observation which makes it more manifest of the claim above. It remains an interesting open question whether there is a structural derivation of (2.44).
13, By Hilbert space, we meant the CFT state space. Indeed as the referee pointed out, there are zero norm states, and therefore we have changed all the places we used ``Hilbert space" to ``CFT state space" to avoid any confusion.
14, We have rewritten the beginning of section 3.1 to make it clear the procedure of applying conformal invariance to obtain the actions of $L_n,\bar{L}_n$. On the other hand, the Jordan block of the barred fields $(\bar{t},\bar{T})$ is the same as the ones for $(t,T)$ with the replacement $L_0\to \bar{L}_0$. We have added a brief comment on this in footnote 9.
15, As the referee suggested, we have added comment on this in footnote 10.
16, Before depicting (3.11), we have mentioned in the last paragraph on page 11 of ``using state-operator correspondence” and then switched to the ket notation. This allowed us to directly depict the structures on the CFT states under Virasoro algebra based on the actions we obtained earlier in this section on the operators.
17, The structure depicted in (3.11) is a result of imposing simply conformal invariance on the logarithmic OPE (2.45) for all fields up to $h,\bar{h}=2$. To make this clear, we have added comments after the figure as the referee suggested.
18, As the referee mentioned, one can consider the normalization of the field eg. $\Psi_2$ as measured by its 2-point function eq. (2.41a), and indeed, multiplying $\Psi_2$ by a constant would change the constant appearing in the 2-point function. In this sense, the constants appearing in the 2-point function are not normalization-independent quantities. However, for the rank-3 Jordan cell, if one modifies the normalization of $\Psi_2$, one has to do the same for $\Psi_1$ and $\Psi_0$ (with the same normalization factor) in order to preserve the form of their two-point functions as in eq. (2.41) which is required by conformal invariance. In this case. the $a_0$ appeared in the algebraic relation (3.13) remains unchanged, and it is in this sense that the $a_0$ here is normalization independent. (We have chosen a normalization for the fields $\Psi_{0,1,2}$ such that the $a_0$ appearing in (3.13) agrees with the constant in (2.41) but the constants in (2.41) can be modified by a different normalization while the one in (3.13) cannot.) We have added footnote 11 to comment on this.
19, Indeed, as the referee pointed out, the rhs of eq. (3.22) needs to be reinterpreted. By logarithmic conformal block of the identity module, we mean the four-point function projected onto the subspace of the CFT state space made of the states constructed from the identity module by Virasoro algebra. We have rewritten the part between eq. (3.22) and (3.24) to make this clear. In the meantime, we interpret the rhs of (3.22) as summing over all the states in the CFT state space which in particular includes the identity module.
20, The logic of (3.34) (originally (3.33)) is as following: from the computation of Gram matrix in (3.21), we see that $\Psi_1$ has non-zero inner product with $\Psi_1,\Psi_2$ which leads to (3.32) from (3.31). In (3.33), we then see that the option of $\Psi_2$ violate self-duality and therefore choose $\Psi_1$. The normalization in (3.31) and comparison with (3.21) allows to conclude that this is precisely $\Psi_1$ (no rescaling by some constant) In this derivation, one does not tune $\alpha$, just leave it as an unknown number which is fixed only later in (3.36).
21, We have corrected the typos of $A|\bar{t}\rangle$ as suggested by the referee.
22, Indeed as the referee pointed out, we have refrained from drawing the actions of $L_0,\bar{L}_0$ in figure 2. These actions are explicitly stated in eqs. (3.4) and (3.5) and one reason to not draw them is to avoid making the figure 2 too messy-looking. We have added comments after the figure, also commenting and referencing to the next subsection on more details of the state/operator $\tilde{\Psi}_1$. We hope this is enough to make things clear for the readers.
25, We have added the reference to the equations that defines the field $\Psi_1$ as suggested by the referee.
26, We have modified the phrase ``What happens for $h+\bar{h}>4$ remains to be explored" to ``What happens for $h,\bar{h}>2$ remains to be explored".