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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
Submission summary
Authors (as registered SciPost users):  Ayan Patra · Rohan R. Poojary 
Submission information  

Preprint Link:  https://arxiv.org/abs/2103.00824v5 (pdf) 
Date submitted:  20230116 13:03 
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: $\langle\epsilon\epsilon\phi\phi\rangle$. 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
The channel we end up computing is the stresstensor comb channel which can be obtained by demanding that 2 of the pairs (of identical operators) in the comb channel fuse to produce the stresstensor. This uniquely fixes the operator propagating in the remaining internal leg (c.f. Fig1). We term this the stresstensor contribution to the 6pt comb channel (defined in eq(3.37)) which we compute to $\mathcal{O}(1/c^2)$.
The changes to the manuscript with relevant comments are as follows:
1) The value of $\mathcal{A}$ can be shifted by a function of the form in eq(3.22) and this would not effect the final form of $\mathcal{K}^{(2)}$ computed in eq(3.27). $c.f.$ footnote16 on pg10. This has been checked explicitly.
2) We check for additional contributions to the above stresstensor Virasoro comb channel at $\mathcal{O}(1/c^2)$ by inserting the projector onto the Virasoro descendants of the vacuum (eq(3.38) and AppendixD) between the pair of identical operators. We find that the additional contribution simply symmetrizes the answer w.r.t. the 3 pairs of operators. The final answer is eq(3.41). This contribution can be interpreted as that coming from $\epsilon^2$ as the additional contributions are similar in form to eq(3.36).
3) We take into the additional contributions to the Virasoro stresstensor comb channel in computing the OTOC of the kind where only the X and Y pairs are out of time ordered (section 4.2) and still find no exponential growth.
4) We cannot see the global stresstensor comb channel eq(4.18) in any obvious form within the final answer eq(3.41) or in a part of it like eq(3.36). We state this below eq(4.19). It could be due to the fact that the method used to compute the full Virasoro contribution (Fig1b) makes use of the 2pt function of $\phi$ while the global contribution (Fig1a) computed for example in 2005.06440 (c.f. eq (5.1)of 2205.06440) explicitly makes use of the projector onto the global descendants of $\phi$.
5) We mention that the computation of simpler higher pt functions are done since the method allows them (beginning of section 3.3.1). These higher pt. functions have 4pt vertices which we show (pg17, Fig4) can be decomposed into a diagram consisting of 2 3pt vertices and an internal exchange of an additional stresstensor. We also argue that if this susbtitution is made into the 4pt vertices of the higher pt comb channel we get stresstensor comb channels where the operators propagating in the internal legs are fixed by demanding that the like operator (pairs) on either ends fuse to give the stresstensor.
6) We also appropriately correct the answers for the Virasoro stresstensor comb channel for these higher pt. functions considered in subsection3.3.1 by accounting for all $\mathcal{O}(1/c^2)$ corrections in a manner similar to the 6pt case in eq3.57 & eq3.58 and AppendixE.
List of changes
1) Footnote16 on pg10.
2) Footnote19 on pg11 explaining the relation between $\mathcal{K}^{(2)}$ and $\mathcal{C}^{(2)}$ better in eq3.27
3) New equations for checking all relevant $1/c^{2}$ contributions from eq3.27 to eq3.41 along with AppendixD.
4) New expressions accounting corrections to simpler higherpt stresstensor comb channels eq3.57 & eq3.58 and relevant correction in AppendixE.
5) Comments and a new Fig4 below eq3.58 explaining a possible way to split 4pt vertex into one consisting 2 3pt vertex.
6) Comments below eq4.13.
7) Comments below eq4.19 explaining how expression in eq4.18 (Global T combchannel) is not obvious in eq3.41 (Virasoro T comb channel).
8) Eq4.20, eq4.23 & eq4.24 and relevant comments accounting for the corrections due to eq3.41.
Current status:
Reports on this Submission
Strengths
1 A potentially interesting result regarding the lack of exponential growth for nonmaximally braided OTO configurations in a 6 pt function
Weaknesses
1 Despite multiple refereeing rounds and a year since submission, the authors have not addressed the clarity issues in the manuscript
2Their result for the 6pt block does not match the literature to leading order
3There are incorrect claims regarding 8 and 9pt functions
4Given the lack of clarity and clear mistakes in the leadup, it is difficult to take the results seriously
Report
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 hepth: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 notationalthough 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 answeralthough 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 operatorsthus 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 subexponential 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 2as it does not represent a complete basis of operators in an OPE expansionthe 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.
Requested changes
None
Author: Rohan R. Poojary on 20230207 [id 3327]
(in reply to Report 2 on 20230201)
We thank the referees for pointing out several lacunae in our presentation. We make the following points regarding our analysis.
1) The global Tcomb channel is computed as $\langle X_{3,5}T_g\phi_1 \phi_g \phi_2T_g\psi_{4,6}\rangle$ where $T_g$ and $\phi_g$ are projectors onto global states descended from $T$ and $\phi$ (as in eq(5.1) of 2005.06440). There are therefore 2 reasons we say we compute the Virasoro contribution to this up to order $1/c^2$:
a) $\phi_g$ can get $1\c$ (and higher powers of $1/c$) corrections as pointed by one of the referees. However if we compute $\langle X_{3,5}T\phi_1 \phi_2T\psi_{4,6}\rangle$ we allow for all (orders in $1/c$) descendants associated with $\phi$ to propagate between $\phi_1$ and $\phi_2$. Therefore all(orders of) the $1/c$ corrections to $\phi_g$ are accounted for as states descended from $\phi$ alone can propagate. This is the only justification we can offer in this regard.
b) One can use the traditional projectors onto the Virasoro descendants constructed out of $L_i$s to write $T=I_1 + I_2+\dots$ (or $ \mathbb{I}=1+I_1 +I_2+\dots $) as done in the works by Fitzpatrick, Kaplan, Walters (1403.6829) and Fitzpatrick & Kaplan(1512.03052) as a power series in $1/c$ with $I_1\simT_g$. We show show using these that the connected contribution from $I_2$ ends up giving the same answer as one computed in eq(3.36) at $\mathcal{O}(1/c^2)$, but with operator pairs exchanged. This is explained in appendix A and we can definitely explain this more thoroughly with better notations so as to not to cause any confusion. The method followed here is similar to that in 1512.03052 and the same reasoning enables us to show that higher terms like $I_3$ in $T$ do not contribute at order $1/c^2$.
2) The 'simpler' higher point generalizations of the comb channel are computed because the technique used to compute the 6pt Tcomb channel allows for such a computation. We do not(and can not) make any statements if these can be used to decompose an arbitrary higherpt correlator and if therefore are universal. However such channels can only exist if the 3pt correlator coefficients $C_{XXX},C_{\phi,\phi\phi},C_{YYY}$ are nonzero.
The rhs of of Fig4 indeed vanishes and the correct decomposition into 3pt vertices of the lhs implies that the internal line in the rhs is that of the primary $X$ as pointed out. Here again (as in (1a) above) the use of 3pt Ward Identity $\langle XXXT \rangle$ in the computation implies that all descendants of $X$ propagate between the states $\langle XX$ and $XT\rangle$. Such diagramatic decomposition would only make the higherpt functions computed appear more like the comb channels drawn in Fig2.
3) The expression in eq(4.18) is indeed not the complete expression of the global Tcomb channel block as in eq(5.6) of 2005.06440 as pointed out. We wished to illustrate how the nonanalytic structure doesn't give rise to any exponential growth for the OTO of 2 pairs of operators studied in subsection4.2 and this result stands even if we included the remaining term of eq(5.6) in 2005.06440, as it has the similar analytic behaviour (of the log terms) under $z\rightarrow1/z,u\rightarrow u/z,v\rightarrow v/z$.
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. In particular, the authors present results for the 6,8, and 9point Virasoro identity block in the comb channel to order ${\cal O}(1/c^2)$ and consider the outoftimeorder 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 6point 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 9point blocks remains unclear. First, there are the same concerns regarding organization of the $1/c$ expansion as in the 6point case. More conceptually, while the added discussion at the end of section 3.3.1 (regarding the decomposition of 4pt vertices into 3pt 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 "stresstensor 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).
Author: Rohan R. Poojary on 20230207 [id 3326]
(in reply to Report 1 on 20230131)
We thank the referees for pointing out several lacunae in our presentation. We make the following points regarding our analysis.
1) The global Tcomb channel is computed as $\langle X_{3,5}T_g\phi_1 \phi_g \phi_2T_g\psi_{4,6}\rangle$ where $T_g$ and $\phi_g$ are projectors onto global states descended from $T$ and $\phi$ (as in eq(5.1) of 2005.06440). There are therefore 2 reasons we say we compute the Virasoro contribution to this up to order $1/c^2$:
a) $\phi_g$ can get $1\c$ (and higher powers of $1/c$) corrections as pointed by one of the referees. However if we compute $\langle X_{3,5}T\phi_1 \phi_2T\psi_{4,6}\rangle$ we allow for all (orders in $1/c$) descendants associated with $\phi$ to propagate between $\phi_1$ and $\phi_2$. Therefore all(orders of) the $1/c$ corrections to $\phi_g$ are accounted for as states descended from $\phi$ alone can propagate. This is the only justification we can offer in this regard.
b) One can use the traditional projectors onto the Virasoro descendants constructed out of $L_i$s to write $T=I_1 + I_2+\dots$ (or $ \mathbb{I}=1+I_1 +I_2+\dots $) as done in the works by Fitzpatrick, Kaplan, Walters (1403.6829) and Fitzpatrick & Kaplan(1512.03052) as a power series in $1/c$ with $I_1\simT_g$. We show show using these that the connected contribution from $I_2$ ends up giving the same answer as one computed in eq(3.36) at $\mathcal{O}(1/c^2)$, but with operator pairs exchanged. This is explained in appendix A and we can definitely explain this more thoroughly with better notations so as to not to cause any confusion. The method followed here is similar to that in 1512.03052 and the same reasoning enables us to show that higher terms like $I_3$ in $T$ do not contribute at order $1/c^2$.
2) The 'simpler' higher point generalizations of the comb channel are computed because the technique used to compute the 6pt Tcomb channel allows for such a computation. We do not(and can not) make any statements if these can be used to decompose an arbitrary higherpt correlator and if therefore are universal. However such channels can only exist if the 3pt correlator coefficients $C_{XXX},C_{\phi,\phi\phi},C_{YYY}$ are nonzero.
The rhs of of Fig4 indeed vanishes and the correct decomposition into 3pt vertices of the lhs implies that the internal line in the rhs is that of the primary $X$ as pointed out. Here again (as in (1a) above) the use of 3pt Ward Identity $\langle XXXT \rangle$ in the computation implies that all descendants of $X$ propagate between the states $\langle XX$ and $XT\rangle$. Such diagramatic decomposition would only make the higherpt functions computed appear more like the comb channels drawn in Fig2.
3) The expression in eq(4.18) is indeed not the complete expression of the global Tcomb channel block as in eq(5.6) of 2005.06440 as pointed out. We wished to illustrate how the nonanalytic structure doesn't give rise to any exponential growth for the OTO of 2 pairs of operators studied in subsection4.2 and this result stands even if we included the remaining term of eq(5.6) in 2005.06440, as it has the similar analytic behaviour (of the log terms) under $z\rightarrow1/z,u\rightarrow u/z,v\rightarrow v/z$.
Anonymous on 20230207 [id 3325]
We thank the referees for pointing out several lacunae in our presentation. We make the following points regarding our analysis.
1) The global Tcomb channel is computed as $\langle X_{3,5}T_g\phi_1 \phi_g \phi_2T_g\psi_{4,6}\rangle$ where $T_g$ and $\phi_g$ are projectors onto global states descended from $T$ and $\phi$ (as in eq(5.1) of 2005.06440). There are therefore 2 reasons we say we compute the Virasoro contribution to this up to order $1/c^2$:
a) $\phi_g$ can get $1\c$ (and higher powers of $1/c$) corrections as pointed by one of the referees. However if we compute $\langle X_{3,5}T\phi_1 \phi_2T\psi_{4,6}\rangle$ we allow for all (orders in $1/c$) descendants associated with $\phi$ to propagate between $\phi_1$ and $\phi_2$. Therefore all(orders of) the $1/c$ corrections to $\phi_g$ are accounted for as states descended from $\phi$ alone can propagate. This is the only justification we can offer in this regard.
b) One can use the traditional projectors onto the Virasoro descendants constructed out of $L_i$s to write $T=I_1 + I_2+\dots$ (or $ \mathbb{I}=1+I_1 +I_2+\dots $) as done in the works by Fitzpatrick, Kaplan, Walters (1403.6829) and Fitzpatrick & Kaplan(1512.03052) as a power series in $1/c$ with $I_1\simT_g$. We show show using these that the connected contribution from $I_2$ ends up giving the same answer as one computed in eq(3.36) at $\mathcal{O}(1/c^2)$, but with operator pairs exchanged. This is explained in appendix A and we can definitely explain this more thoroughly with better notations so as to not to cause any confusion. The method followed here is similar to that in 1512.03052 and the same reasoning enables us to show that higher terms like $I_3$ in $T$ do not contribute at order $1/c^2$.
2) The 'simpler' higher point generalizations of the comb channel are computed because the technique used to compute the 6pt Tcomb channel allows for such a computation. We do not(and can not) make any statements if these can be used to decompose an arbitrary higherpt correlator and if therefore are universal. However such channels can only exist if the 3pt correlator coefficients $C_{XXX},C_{\phi,\phi\phi},C_{YYY}$ are nonzero.
The rhs of of Fig4 indeed vanishes and the correct decomposition into 3pt vertices of the lhs implies that the internal line in the rhs is that of the primary $X$ as pointed out. Here again (as in (1a) above) the use of 3pt Ward Identity $\langle XXXT \rangle$ in the computation implies that all descendents of $X$ propagate between the states $\langle XX$ and $XT\rangle$. Such diagramatic decomposition would only make the higherpt functions computed appear more like the comb channels drawn in Fig2.
3) The expression in eq(4.18) is indeed not the complete expression of the global Tcomb channel block as in eq(5.6) of 2005.06440 as pointed out. We wished to illustrate how the nonanalytic structure doesn't give rise to any exponential growth for the OTO of 2 pairs of operators studied in subsection4.2 and this result stands even if we included the remaining term of eq(5.6) in 2005.06440, as it has the similar analytic behaviour (of the log terms) under $z\rightarrow1/z,u\rightarrow u/z,v\rightarrow v/z$.