Logarithmic operators in $c=0$ bulk CFTs

This paper presents a thorough study of conformal field theories in the c -> 0 limit, corresponding to models such as percolation and the self-avoiding walk (SAW).

One of the most intriguing results is the non-trivial four-point function of an energy-type operator discussed in Section 5.3.

While the paper is quite technical and not always easy to follow, it already covers a large amount of material in considerable depth. It would be difficult to improve the clarity significantly without substantially increasing the length of the paper.

• It would be helpful to precisely define what is meant by 'Kac operators.' In particular, the term 'Kac operator' already appears in the mathematical physics literature, referring to an operator introduced by M. Kac in 1966, which corresponds to the transfer operator for a lattice model in statistical mechanics. As I understand it, in the present paper, the author intends to define 'Kac operators' as all operators with integer Kac indices.

• I found it somewhat confusing that the author initially motivates the study using the bulk CFTs of percolation and self-avoiding walks, linking these models to cluster and loop model CFTs. However, by the end of the introduction, the terminology shifts to 'the cluster, dilute, and dense loop models,' corresponding respectively to 'percolation, SAW, and percolation hulls.' This change in vocabulary may confuse readers unfamiliar with these distinctions. I suggest that the introduction include a more detailed explanation of the differences between these three cases—particularly between the dilute and dense loop models—and clarify their connections to the physical models mentioned.

• Reference [21] appears to be incomplete; I believe it corresponds to arXiv:cond-mat/0111031.

Publish (meets expectations and criteria for this Journal)

1-Interesting observations on open problems that are central to different research communities.
2-Well-structured and articulate presentation of ideas

1-These results are a natural extension of previous research and not wholly unexpected

Dear Editor,

iIn this paper, the authors highlight the emergence of a logarithmic structure at the c=0c=0 dense and dilute loop critical point. This is achieved primarily by interpreting new bootstrap solutions of c≤1c≤1 theories, both diagonal and non-diagonal. Particular attention is given to the energy operator and its logarithmic properties.

On one hand, this approach is not entirely new. In arXiv:1311.2055, where the occurrence of logarithmic structures for arbitrary c was first identified, similar results were obtained through the analysis of Coulomb gas correlation functions. These earlier results relied on the interplay between the so-called imaginary DOZZ formula and conformal blocks, which are ultimately central to the bootstrap solutions analyzed in the present work.

On the other hand, the authors here apply conformal field theories that have been explicitly connected to statistical models, which adds further significance to their findings. I therefore recommend this paper for publication.

The paper is very well written and clear. The results arXiv:1311.2055 should be cited.

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The paper deals with the c->0 limit of a few families of CFTs, and studies in great detail the logarithmic operators that one finds in this limit. Both rank-2 and higher ranks logarithmic multiplets are considered. The paper has a great deals of computations and details, that could help someone already familiar with these topics. On the other hand, the amount of detailed computations, often jumping from one CFT to another, also make it harder to understand what the main points of the paper are.

1. Below (2.2), diagonal is a property of the theory rather than the operator: a non-diagonal theory can still contain scalar operators. Their OPE contains opertors with non zero spin.

2. Below (2.14), it is unclear why <TT>=0 should mean the operator T being at risk of being removed, given that nothing is said about other correlation functions of T with other operators of the theory. Indeed the operator is not removed.

3. Typo in (2.39), bb instead of b

4. I cannot understand the sentence below (5.5)

5. What is exactly the definition of $\hat{\Phi}_{2,1}$ appearing in (5.48)?

6. The notation of $w \to 0$ in(5.53) is very confusing, given that we already have an operator living at $z=0$. I reccommend sending $w \to \infty$ with the proper rescaling (what is done in (5.50)) rather than defining $w = \frac 1z$ and sending $w \to 0$.

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