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

1. Detailed and precise work
2. Clear presentation

1. Given the technicality of the paper, the target audience is pretty small

The paper studies in detail the structure under Virasoro of fields appearing in the identity blocks for some logarithmic theories with vanishing central charge. The precise structure is worked out using the c->0 limit, which is unique and well defined in the theories considered.

The paper is technical but clear. I recommend publication in SciPost, after few very minor changes.

1. Eq (2.5): how are the normalizations fixed? For $b_{12}$ in eq (2.6) to have meaning that should be specified. I assume it has to do with the two point function of $X$ but I wasn't able to find this.

2. Is there an argument why no operators would have factors such as $1/c$? Eq (2.18) assumes that, for any n-point function, terms such as $c t$ always vanish, meaning that other operators cannot involve inverse factors of $c$ in general. An explanation would be great.

3. Why does (2.21) follow from (2.20)? It could be that $\bar \partial X $ has vanishing 2pf but not higher npf, given that the theory is not unitary. If there is some other reason for (2.21) it would be good to have it written down.

This paper addresses the structure of the indecomposable Virasoro-times-Virasoro module containing the identity field in the CFTs believed to describe the appropriate limits of the $O(n)$ and Potts models. This is an old problem that has not received as much attention as it should. The present paper argues convincingly that the states in this module of conformal weights $(h,\bar{h})$, with $h,\bar{h} \le 2$, include Jordan blocks for the Virasoro mode $L_0$ of ranks $2$ and $3$. The deeper structure is not investigated.

The methodology is a straightforward, though technically involved, generalisation of the well known $c \to 0$ catastrophe argument. This combines some recent bootstrapping results with basic physical principles. The results are shown to be self-consistent and extend a recent proposal of Nivesvivat and Ribault. (They also extend several other earlier proposals, though this is not discussed in detail.)

There is a lot to learn from this paper, including technical asides that one may not have thought of prior to commencing calculation. The results are interesting and warrant publication and I recommend it warmly with some minor rewrites. I also hope that this paper stimulates further investigation of further (non-chiral) logarithmic structures.

I have a short list of comments and suggestions:
0. The paper is generally well written, but there are local occurrences where this is not the case. I suggest a thorough proof reading to catch certain paragraphs in which the grammar obscures the arguments being made.
0'. It may not be my place to say, but the usage of ``Jordan cell'' instead of the much more standard ``Jordan block'' throughout is known to be a mistranslation on the part of Gurarie that has propagated through the community. I suggest changing it.
1. The abstract seems to claim that non-trivial CFTs with $c=0$ are logarithmic. I wonder in what sense the product $M(4,5) \otimes M(5,8) \otimes M(5,8)$ of Virasoro minimal models is trivial.
2. There is occasional mention of ``Loewy diagrams'', eg in the first paragraph of Sec 2.1. I think it is worth mentioning that a Loewy diagram does not fix the action of the algebra completely (as suggested in that paragraph) and also that it is not the same as the pictures we meet in (3.11) and Figure 2. In fact, I cannot find any Loewy diagrams in the paper, though I think adding one to summarise the paper's main result would be very welcome!
3. In the introduction and Sec 2.1, fields $t$ and $\bar{t}$ appear without much information. They end up being defined in Sec 2.2. However, non-experts are likely to be very confused until they read the later section --- the notation even suggests that $t$ is holomorphic and $\bar{t}$ is antiholomorphic, neither of which is true. I suggest that more information about these fields is included in Sec 2.1 to avoid such potential confusion.
4. Before (2.6), a parenthetical reminds us that certain constants appearing in correlators depend on the choice of logarithmic partner field. However, this is only the case if the logarithmic coupling is non-zero. [This is the case for almost all $c$ here, but we don't actually know that at this point.]
5. (2.6) itself consists of some non-trivial statements about chiral logarithmic couplings. I'd like to see a derivation or a citation here, please.
6. Footnote 3 is curious. It appears to say that linear combinations of fields of different dimensions are problematic (which is of course untrue). Maybe this could be clarified.
7. (2.13) is noted to be an assumption required for the analysis, at least for the $O(n)$ models. I think it would be appropriate to explicitly note this assumption, along with others that come later, when describing the results in the introduction and conclusions.
8. One thing I found confusing was the authors' use of $=$ to mean not only ``$=$'' but also ``$=$ once a $c\to0$ limit is taken''. The reader is warned about this explicitly, but I think it would significantly improve the exposition if ``$\overset{c\to0}{=}$'' was used for the latter (as it is in (2.20)). Also, one comes across big-$\mathcal{O}$ notation, eg (2.29). Maybe just one precise way, $c\to0$ or $\mathcal{O}$, of explaining such equations would suffice?
9. Sec 2.3 is also, in my opinion, unnecessarily complicated by not explaining the appearance of new symbols until well after they appear. Here, I refer to (2.27) in which $\Phi_2$ has been previous introduced, but $\Phi_1$ and $\Phi_0$ are not until (2.32). This destroys the reader's flow as they search the previous text in vain for the latter's definitions.
10. (2.29) and (2.31) are, like (2.13), assumptions that should be mentioned in both the introduction and conclusions when asserting the fine results of this paper.
11. The way it is written, there seems to be no reason to give (2.32). I suggest rewriting this part of the text and adding a forward reference to the later results that depend on these expressions.
12. (2.42) is one of the key results it seems, the rank $3$ Jordan block logarithmic coupling. But it is stated without derivation. Possibly (2.44) is actually the derivation, but I could not be sure if that was the intention. I also note that the "easy to verify" claim leading to (2.43) doesn't appear to be possible to verify without such a derivation. Please rewrite these paragraphs.
13. In a few places, the authors refer to a Hilbert space. It isn't clear which Hilbert space they are referring to, but perhaps it should be mentioned that any space of states including non-trivial Jordan blocks for $L_0$ cannot form a Hilbert space as the $L_0$-eigenvector has zero norm.
14. The first two paragraphs of Sec 3.1 were a little difficult to understand and I suggest a rewrite. In particular, did the authors forget the Jordan block for the barred fields?
15. After (3.9), it may be wise to spell out what is meant by ``change of basis'' and what it implies for other calculations.
16. (3.11) suddenly uses kets, without comment, whereas all previous equations do not. Actually, kets are introduced in the following section, Eq. (3.19), so perhaps this is a typo?
17. As (3.11) is a first approximation to one of the final results, it may be useful to indicate precisely what is being pictured here. A casual glance leads one to question what it means if an arrow is missing --- eg there is no arrow representing the action of $A$ on $\Psi_2$: does that mean that the result is $0$? I think not, because there are many $L_0$ arrows missing which definitely don't give $0$.
18. After (3.11), the phrase ``independent of the normalization'' is used, referring to $a_0$. Could the authors explain what they mean? I would assume that ``normalization'' refers to some measure of the size of some field, eg $\Psi_2$ in (2.41a). However, multiplying $\Psi_2$ by $7$ clearly changes $a_0$, so the authors must mean something different.
19. Before (3.22), the authors mention the logarithmic conformal block of the identity module. Could they please explain what they mean, precisely? I suspect that they mean the right-hand side of (3.22), which is unfortunately not equal to the 4-pt function on the left-hand side but rather a projection of it.
20. (3.33) is curious: why would one expect both calculations to give $\Psi_1$ on the nose? Perhaps one thinks to tune $\alpha$ to make it so, but then why not different $\alpha$ for the two calculations? It wasn't clear to me if the authors weren't making an implicit assumption here. Could they clarify?
21. In (3.39) and below, there seems to be a systematic typo in which $|A\bar{t}>$ is written instead of $A|\bar{t}>$, etc.
22. Figure 2 is ``the final structure'' of the identity module, up to fields with $h,\bar{h} \le 2$. However, it is missing many arrows describing the action of $L_0$. The state $\tilde{\Psi}_1$ in the middle has no arrows coming in and out, so the final structure appears to be indicating that it is a primary field that doesn't belong to the non-chiral vacuum module, but another direct summand. Please add some discussion to clarify what is being asserted here. [Some of this is clarified in Sec 3.3.1, but this is another instance in which the fact that there is clarification to come needs to be stated clearly at the right place. Otherwise, it cannot really be ``the final structure''.]
25. (3.47) is preceded by a reference to Sec 2.3, which should really be a reference to an equation number.
26. Sec 4 suggests that the structure of the non-chiral vacuum module has been fixed for fields with $h+\bar{h}\le4$, which is stronger than what was asserted in previous sections.