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

Preprint Link:  https://arxiv.org/abs/2109.05050v2 (pdf) 
Date submitted:  20220208 09:27 
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
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
List of changes
Anonymous Report 2 on 20211220
1, Indeed, as the referee pointed out, the normalization of $\Psi$ is fixed by the normalization of $X$, namely that its twopoint 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 twopoint 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 twopoint functions, then the field is not welldefined at $c=0$ as its twopoint 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 twopoint function and in particular its holomorphicity. This different from directly looking at the antichiral conformal dimension, which might vanish without excluding some logarithmic dependency. An example of this is the logarithmic field $t(z,\bar{z})$: While its antichiral conformal dimension vanishes, it is a nonchiral field and this can be seen from the twopoint 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 twopoint 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.
===================================================================
Anonymous Report 1 on 20211220
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 noninteracting 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 twopoint functions of $(t,T)$ in the introduction. (page 3)
4, The constant in the twopoint 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 nonzero 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 rank2 Jordan block (figure 1) at generic $c$ as recently obtained in references [1,4]. We would like to point out that they are nonchiral 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 twopoint 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 rankthree 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 2point 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 stateoperator 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 2point function eq. (2.41a), and indeed, multiplying $\Psi_2$ by a constant would change the constant appearing in the 2point function. In this sense, the constants appearing in the 2point function are not normalizationindependent quantities. However, for the rank3 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 twopoint 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 fourpoint 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 nonzero 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 selfduality 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 messylooking. 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".
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 3) on 2022211 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2109.05050v2, delivered 20220211, doi: 10.21468/SciPost.Report.4401
Strengths
1. Detailed and precise work
2. Clear presentation
Weaknesses
1. Given the technicality of the paper, the target audience is pretty small
Report
I am happy with the answer and changes the author did for point 1) and 3), but I'm not satisfied with the answer for point 2).
Clearly an object such as $\frac{1}{c} \phi$ by itself is not well defined in the $c \to 0$ limit. However, it could be that there's some operator $\mathcal{O} =\# \psi + \frac{1}{c}\phi$ which is well defined in the $c \to 0$ limit for some appropriate prefactor of some other operator $\psi$. This is what happens for $t$, see eq (2.9); from my understanding it could happen for other operators as well. Now, if I consider a correlation function of the type $\langle t \mathcal{O} \ldots \rangle $, what tells me that I will not have cancellations between powers of $c$ so that it will remain finite as $c \to 0$?
Requested changes
Better explanation of point 2)