Reparametrization mode Ward Identities and chaos in higher-pt. correlators in CFT$_2$

The authors further explore the reparametrization mode representation for stress-tensor exchanges in higher point correlation functions in two dimensional CFT and use these results to study the late time behavior of several out-of-time-order configurations in these higher point functions.

While the work has technical merit and interesting conclusions especially regarding the absence of Lyaponov growth in certain out-of-time-order configurations, I believe that further clarification is needed in order to justify the claims presented.

In particular, one of the primary results is equation (3.36), concerning the ${\cal O}(1/c^2)$ contributions to the Virasoro identity block in the comb channel, but it is not clear to me that this is the only contribution to this block at this order. Naively, I would expect that, representing the Virasoro identity block in the comb channel via projectors as $\langle X_3 X_5 \vert 1 \vert \phi_1 \vert \phi \vert \phi_2 \vert 1 \vert Y_4 Y_6 \rangle$, there would be more contributions at ${\cal O}(1/c^2)$ than the $\langle X_3 X_5 \vert T \vert_g \phi_1 \phi_2 \vert T \vert_g Y_4 Y_6 \rangle$ contribution computed from, e.g., ${\cal O}(1/c^2)$ contributions to either of the projectors $\vert 1 \vert$, $\vert \phi \vert$. As it stands, more justification is required to support (3.36) and consequently the resulting behavior in the various out-of-time-order configurations considered later.

Another minor point concerning the result (3.36) concerns the ambiguities present in the auxiliary function ${\cal A}(z)$ presented in equation (3.21). It is claimed that these ambiguities "do not contribute to anything physical", but I think a little elaboration on this point is warranted. Is the claim that such terms drop out upon constructing the correlator with the bilinears ${\cal B}^{(1)}$?

In addition, I find the discussion of the 8- and 9-point blocks confusing, as the contributions considered don't correspond to any standard conformal block decomposition I'm aware of. Due to the fusion of three operators into the stress tensor, it seems like this is a resummation of a particular subset of a full conformal block decomposition (i.e. conventional conformal block diagrams only have trivalent vertices instead of the 4-point vertices in figures 2, 3), rather than an individual block as referred to in the text. So while the calculation of this contribution may be valid, it is unclear to me how to interpret the result in terms of a systematic expansion of the full correlation function.

1- Presentation has been greatly improved compared to the previous version
2- The authors compute the non-maximal OTO braiding and find no evidence of scrambling whatsoever---an interesting result

1- Claims made in the paper are not sufficiently justified
2-There remain a few holes in the presentation of the steps
3-Section on 8- and 9-pt functions requires work

In this paper, the authors engage in the theory of reparametrization modes in 2d CFT--denoted $\epsilon(z,\bar{z})$--which encode (multi-)stress-tensor exchanges in the Virasoro identity block of arbitrary numbers of external operators. The paper attempts to understand how Virasoro ward identities govern the form of multiple $\epsilon$ -insertions.

In particular the authors use this construction to compute the leading-order, 5-pt Virasoro comb channel block, which they then use to compute the scrambling behavior of various OTO braidings of operators. The authors find that, for non-maximally braided combinations of operators, there is no scrambling whatsoever---a peculiar result.

There are a few holes in the argument that I think still need to be cleared up. First and foremost, the most important object computed by the authors is denoted $V^{\rm6 (Vir)}_{T, \text{comb}}$ and is given in equation (3.36). It remains unclear to me why this object is the Virasoro contribution to the block at order $c^{-2}$. For example, in 2005.06440, it was shown that the Virasoro contributions to the 6-pt star block come from higher order terms in the expansion in terms of the reparametrization mode $\epsilon$ (contributions to $\mathcal{B}$ at order $\epsilon^2$ or higher). But equation (3.36) restricts only to the leading piece in this $\epsilon$ expansion, and no argument is given for why no additional contribution appear at the same order in $c^{-2}$, e.g. from objects like $\mathcal{B}^{(2)}$. For example, there is an additional contribution to the star channel Virasoro block that appears at order $c^{-2}$ as computed in 2005.06440. The authors should at least explain why this does not happen in the comb channel.

(I suspect that there actually is an additional contribution, which would explain away the lack of scrambling for the non-maximal OTO braiding analyzed in section 4.2.2. But this remains to be seen. )

It would also be useful to know how (3.36) differs from the comb block given in 1810.03244, part of which is given in (4.18). Does the additional Virasoro piece com from $\mathcal{K}^{(2)}$ in (3.27)? Is there any part that looks like the comb block of 1810.03244? Such a consistency check should not be left up to the reader.

My remaining confusion is about section 3.3.1. The graphs depicted in figure 2, to my eyes, do not form a convergent basis of functions for correlators. This is for the simple reason that fusions of triples of operators doesn't allow for arbitrary fusions of two operators in the subgraph. I would thus not call this a comb channel stress tensor block, as pairwise operators of the same dimension can always fuse into $T$. The section may be relevant since the block related to this type of graph is, in fact, computable, but of limited use since the choice of channel is one few would consider.

1- Explain why (3.36) is the honest Virasoro comb block at order $c^{-2}$ or find the additional contribution from $\epsilon^2$ terms.

2- If extra $\epsilon^2$ terms contribute, find how they affect the calculation of the OTOCs

3-Give an account of how (3.36) differs from (4.18)

4-Additional explanations required in 3.3.1 as to the relevance of these types of graphs to physical correlators.