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Reparametrization mode Ward Identities and chaos in higherpt. correlators in CFT$_2$
by Arnab Kundu, Ayan K. Patra, Rohan R. Poojary
This is not the latest submitted version.
Submission summary
Authors (as registered SciPost users):  Ayan Patra · Rohan R. Poojary 
Submission information  

Preprint Link:  https://arxiv.org/abs/2103.00824v4 (pdf) 
Date submitted:  20220930 14:26 
Submitted by:  Poojary, Rohan R. 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Recently introduced reparametrization mode operators in CFTs have been shown to govern stress tensor interactions $via$ the shadow operator formalism and seem to govern the effective dynamics of chaotic systems. We initiate a study of Ward identities of reparametrization mode operators $i.e.$ how two dimensional CFT Ward identities govern the behaviour of insertions of reparametrization modes $\epsilon$ in correlation functions: $\bra\epsilon\epsilon\phi\phi\ket$. We find that in the semiclassical limit of large $c$ they dictate the leading $\mathcal{O}(c^{1})$ behaviour. While for the $4$pt function this reproduces the same computation as done by Heahl, Reeves \& Rozali in \cite{Haehl:2019eae}, in the case of 6pt function of pairwise equal operators this provides an alternative way of computing the Virasoro block in stresstensor comb channel. We compute a maximally out of time ordered correlation function in a thermal background and find the expected behaviour of an exponential growth governed by Lyapunov index $\lambda_L=2\pi/\beta$ lasting for twice the scrambling time of the system $t^*=\frac{\beta}{2\pi}\log\,c$ for the maximally braided type of $out$$of$$time$$ordering$. However when only the internal operators of the comb channel are \emph{outoftimeordered}, the correlator sees no exponential behaviour despite the inclusion of the Virasoro contribution. From a bulk perspective for the \emph{outoftimeordered} $4$pt function we find that the Casimir equation for the stress tensor block reproduces the linearised back reaction in the bulk.
Author comments upon resubmission
We verify that the Virasoro contribution to the 6pt maximally OTOC does exhibit the chaotic behaviour found for the global contribution to the same channel in \ref[36]. We also investigate the 4pt OTO in the 6pt comb channel when the X and Y pairs in Fig1 are OTOed only w.r.t. each other.
Contrary to our previous reply we find that both the global and Virasoro contributions to the 6pt function do not show an exponential behaviour for such an OTOing. This we find in contrast with the computation for the 6pt star channel in \ref[36] where taking into account the Virasoro contribution gave rise to the exponentially chaotic behaviour.
The understanding arrived at in \ref[36] was that one needs to take into account the gravitational(Virasoro) interaction at the desired order in 1/c which corresponds to taking into account the Virasoro descendants in the internal legs of (star) channel.
In the 6pt comb channel depicted in Fig1 this amounts to considering the Virasoro descendants of the operator $\phi$ propagating in the internal leg of the channel. However, the global and the Virasoro contributions do not show an exponentially chaotic behaviour. We find this result pertaining to the Virasoro stresstensor 6pt comb channel intriguing.
Thus, we are able to show that the Ward identity related to $\langle\epsilon\epsilon\phi\phi\rangle$ allows us to compute the Virasoro contribution to the stresstensor comb channel of 6pt correlator. This Virasoro contribution has a peculiarity that when only the X & Y pairs as depicted in Fig1 are OTOed w.r.t. each other, the correlator does not show any exponential behaviour.
We thank the referees for insisting on the check of the former claims made in the previous versions of the document with regards to the results obtained in \ref[36]. This though cumbersome at first, has allowed us to improve the manuscript to a great extent and a better understanding of the physics.
We hope the present version also is more clearer in its explanation and we eagerly await any comments. We apologize for the delay in bringing out the current version of the manuscript.
List of changes
Below is a short list of changes to sections and subsections. The changes are enumerated according to the (sub,subsub)sections they occur.
Section2) has two subsections as compared to the older version: subsec
2.1) Deals with the shadow formalism in d dim.
2.2) Deals with the same in the 2d case.
Section3) has had substantial changes. We claim that the Ward identity with 2 insertions of the reparametrization mode computes the comb channel and not the star channel. The claim of a representation of the projector onto the the Virasoro block using the reparametrization modes has been dropped in its entirety along with the subsequent statements about heavy operators in a correlator.
3.1) Reviews known results for Ward identity associated with single insertion of the reparametrization mode.
3.2) Analyses the Ward identity for 2 insertions of the reparametrization mode into a 2pt function of scalars. The analysis upto eq 3.27 (3.28 in old ver) is unchanged.
3.2.1) The last paragraph from this subsubsection (3.0.1 old ver) has been removed.
3.2)This subsection titled Channel Diagramatics replaces subsec3.1,,3.2 & 3.3 of the old ver. This section now contains the comb channel computation for 6pt function.
3.2.1) Generalizes the 6pt comb channel to simpler 8pt and 9pt comb channel computations by replacing identical pairs of operators with identical triplets.
Section4) This section too has substantial changes:
4.1) Contains with few changes and includes the contents up till eq 4.13 of the old version as it still computes the same OTOC but now interpreted for the 6pt comb channel with Virasoro contribution. This OTOC sees the same chaotic behaviour as the global 6pt comb channel unsurprisingly.
4.2) Computes the specified 4pt OTO in the 6pt comb channel
4.2.1) Finds that such a 4pt OTO in the 6pt global comb channel does not show any exponential chaotic behaviour.
4.2.2) Finds that even the Virasoro contribution to the 6pt comb channel for the 4pt OTO does not show any exponential chaotic behaviour. This is in stark contrast with the computation for the 6pt star channel in ref[36] where the inclusion of the Virasoro contribution endows a chaotic behaviour which the global 6pt star channel lacked.
4.3) Finds the chaotic behaviour for a 6pt function by generalizing a 4pt function and finds that such simpler generalizations do not change the chaotic behaviour. Thus suggesting the simpler generalizations to higher pt functions in subsubsec3.2.1 do not give rise to any different chaotic beahviour than that of the parent 6pt function.
Section5) Stays unchanged.
Appendices: A, B & C remain unchanged (the contents of the old D & E appendices have been removed)
D) Summarizes the Virasoro comb channel contribution to the 8pt and 9pt functions obtained in subsubsec3.2.1 in as succinct a form as possible. Any further exposition of the results would lead cumbersome expressions.
Current status:
Reports on this Submission
Report
The authors further explore the reparametrization mode representation for stresstensor exchanges in higher point correlation functions in two dimensional CFT and use these results to study the late time behavior of several outoftimeorder configurations in these higher point functions.
While the work has technical merit and interesting conclusions especially regarding the absence of Lyaponov growth in certain outoftimeorder 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 outoftimeorder 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 9point 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 4point 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.
Strengths
1 Presentation has been greatly improved compared to the previous version
2 The authors compute the nonmaximal OTO braiding and find no evidence of scrambling whatsoeveran interesting result
Weaknesses
1 Claims made in the paper are not sufficiently justified
2There remain a few holes in the presentation of the steps
3Section on 8 and 9pt functions requires work
Report
In this paper, the authors engage in the theory of reparametrization modes in 2d CFTdenoted $\epsilon(z,\bar{z})$which encode (multi)stresstensor 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 leadingorder, 5pt Virasoro comb channel block, which they then use to compute the scrambling behavior of various OTO braidings of operators. The authors find that, for nonmaximally braided combinations of operators, there is no scrambling whatsoevera 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 6pt 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 nonmaximal 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.
Requested changes
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
3Give an account of how (3.36) differs from (4.18)
4Additional explanations required in 3.3.1 as to the relevance of these types of graphs to physical correlators.