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

1- A potentially interesting result regarding the lack of exponential growth for non-maximally braided OTO configurations in a 6 pt function

1- Despite multiple refereeing rounds and a year since submission, the authors have not addressed the clarity issues in the manuscript
2-Their result for the 6pt block does not match the literature to leading order
3-There are incorrect claims regarding 8- and 9-pt functions
4-Given the lack of clarity and clear mistakes in the leadup, it is difficult to take the results seriously

With each refereeing round the authors have had to pare back their claimed results. At this stage, as far as I can tell, the authors have applied the algorithm and graviton basis of appendix B of hep-th:1403.6829 by Fitzpatrick, Kaplan and Walters (FKW), in addition to some tricks from the reparametrization mode formalism, in order to compute the 6pt comb "Virasoro'' block. This calculation, on its own, is not novel, and the application of computing a certain type of OTO configuration, is mired with confusions (more on that below).

At referees request, the authors have tried to clarify their algorithm, leading to equations (3.37) and (3.38) . However, the authors do not define their normalization factors in equation (3.38), leaving the reader to guess whether they are applying the same formalism as FKW, since they are using the same notation---although without attribution. If so then, their claim each normalization factor is of a certain order in $c$ is incorrect, but it is impossible to tell what the authors actually mean without a proper definition. If they do not mean to follow FKW, then how is one supposed to check their calculation?

Let us give the authors the benefit of the doubt. The authors then restrict the expansion to order $1/c^2$, coming from insertions of $\mathcal{B}^{(1)}$ which implies that they are only considering the global part of the comb block, despite their claim that this is somehow a Virasoro block. This result for the global block was already known and satisfies a simple Casimir equation (for reference, one can find the answer in 1810.03244 by Rosenhaus). However, their result perhaps disagrees with this known answer---although this may be due to a trivial mistake by the authors, it is difficult to tell.

To see this, one needs to go to section 4, where the authors claim that equation (4.18), copied from section 5 of 2005.06440 by Anous and Haehl, is the global comb channel block, but is crucially not symmetric under exchanges of pairs of operators---thus differing from their answer. If one traces back, it appears that they have miscopied the result from Anous and Haehl (c.f. equation (5.6) rather than (5.7)), which *is* symmetric (and, incidentally, is the same function as the one found by Rosenhaus, which also is symmetric).

The authors then go on to compute some OTO braidings and make claims about sub-exponential growth in these configurations.

I am inclined to reject this paper at this stage, as, after multiple refereeing rounds, the authors are still making casual mistakes (ones that can not simply be typos) and it does not seem like there is a concerted effort to find and rid their paper of them. Moreover, and despite requests in previous rounds to improve the clarity of exposition, is still impossible as a referee to check all their claims, even in principle without extra effort.

As a simple example of one of these casual mistakes, after refereeing round 2, in response to a request by referees to elaborate on Figure 2---as it does not represent a complete basis of operators in an OPE expansion---the authors added figure 4 to clarify the meaning of their expansion. But this OPE expansion trivially gives zero by the Virasoro Ward identity and translation invariance (said in another way $C_{TT\phi}=0$). In my humble opinion, such casual errors at the 3rd refereeing round of a paper should be grounds for rejection.

None

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. In particular, the authors present results for the 6,8, and 9-point Virasoro identity block in the comb channel to order ${\cal O}(1/c^2)$ and consider the out-of-time-order continuations of their results.

While the presentation has been improved, there are still some confusions and readability issues that remain and that should be addressed. Most substantially, the justification of equation (3.41) for the 6-point Virasoro block is still rather opaque. For instance, the standard presentation for the Virasoro block considered in (3.37) should have a projector $|\phi|$ for the projection onto the conformal family of $\phi$, corresponding to a diagram with only trivalent vertices. This additional projector could in turn be organized into a $1/c$ expansion. The authors should include some discussion on if (and how, if so) this would affect their claim for the ${\cal O}(1/c^2)$ result.

Furthermore, in order to increase readability, the authors should include definitions of the normalization factors ${\cal N}_{i,j}$ and their notation (e.g. presumably $L_{-(m,n)}$ stands for $L_{-m} L_{-n}$). For example, the authors claim (both in the main text below eq. (3.38) and in appendix D) that all of the ${\cal N}^{-1}$ factors in $I_2$ are ${\cal O(1/c^2)}$, while the last term in $I_2$ seems to contradict this statement (assuming ${\cal N}_{(m+n), (m+n)}^{-1}$ is the same object as ${\cal N}_{i,i}^{-1} \sim {\cal O}(1/c)$ in $I_1$). Without such definitions, it is an unreasonable burden for the reader to verify the claims (such as that in eq. (D.8) or (D.9)).

Regarding the discussion in appendix D, it is not obvious that all contributions to ${\cal O}(1/c^2)$ have been included (beyond any potential contributions from the $\phi$ projector discussed above). In particular, eq. (D.3) includes terms that are naively order $1/c^3$ which nevertheless contribute at order $1/c^2$ due to the various contractions between Virasoro generators (namely those coming from the partial contraction of $L_{-i}$ with $L_{(m,n)}$). It would be helpful if the authors could comment on why analogous terms, e.g. $L_{-(i,j)}$ partially contracting with $L_{(l,m,n)}$, couldn't also contribute at this order.

Finally, the relation of the computations in this 3.3.1 and genuine Virasoro 8- and 9-point blocks remains unclear. First, there are the same concerns regarding organization of the $1/c$ expansion as in the 6-point case. More conceptually, while the added discussion at the end of section 3.3.1 (regarding the decomposition of 4-pt vertices into 3-pt vertices) does help clarify the computation, the argument as presented isn't quite correct. In particular, the expansion depicted in Figure 4 can't be correct, since the right hand diagram in Figure 4 vanishes for any 2d CFT (as $C_{TTX}= 0$ for any Virasoro primary $X\neq 1$). At first glance, it does seem like the left hand side of Figure 4 would be uniquely expanded into a block where the internal $T$ line is instead an $X$ (corresponding to the OPEs $X_5 \times X_7 \to X$ and then $X\times X_3 \to 1$). However, even in this case, the authors should then comment on whether this OPE channel, which is of a different character than the "stress-tensor comb" blocks considered in the rest of the paper and requires that $X$ appears in the OPE $X\times X$, is generic or universal in any sense (cf. there are operators in minimal models where fusion of this type does not occur).