Disorder in AdS$_3$/CFT$_2$

The authors discuss a holographic model corresponding to a 2-dimensional CFT deformed by marginally relevant quenched disorder and in particular they study the effects of disorder on the averaged geometry. Disorder is introduced in a novel way by using a Chern-Simons formulation of AdS3 gravity. The attempt is interesting and the results deserve publication in some form. Nevertheless, I agree with the other anonymous Referee that some technical parts of the paper must be clarified and the analytic results should be supported by some additional numerical confirmations.

My main complaint regards the role/meaning of the unphysical features mentioned in the manuscript and the validity of the Poincare-Lindstedt inspired resummation. More precisely:

(1) In the text, I found different statements regarding the nature of these unphysical features. In the abstract the authors state that these unphysical features are due to the disorder average. In pages 3 and 15, they say that these features are a signal of the breakdown of the perturbative expansion. Also, in page 15 they suggest the possibility that these features are actually a signature of the marginal relevance of the disorder deformation. I believe that a clarification on these lines is necessary and important. Are these features really unphysical? Or are they telling us something about the nature of the introduced disorder and/or the techniques used?

(2) The most important point is probably the question whether the resummation commutes or not with the disorder average. In the References cited where this resummation is employed, the resummation is operatively performed before the average. On the contrary, the authors here introduce the resummation after the average. At page 15, they claim that the two commute because the functions used to regularize the metric (after the average) do not depend on the random phases. This statement is highly not obvious to me. First, after the average, no phases can appear in any case in the regularization functions, so I am not sure how this can be an argument for the commutation of the two operations. Second, if really this was the case, one could take the same phases independent terms and regularize the metric before the average, i.e., commuting the two operations. Then, the average over the regularization terms would be trivial since they do not explicitly depend on the phases. The Authors should provide more solid arguments for this commutation and perhaps confirm it with some numerical computations as suggested by the other Referee.

(3) The resummation not only fixes the ~z^2 unwanted terms in the metric, curvature, etc but it seems also to modify finite physical observables. For example (32) and (56) seem different. How can one understand this? In my opinion, some comments on this point are in order.
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Finally, some minor comments.

(4) In figure 1, the light blue line separating the region could be probably removed.

(5)The introduced disorder clearly breaks the translational symmetry of the dual field theory. From symmetry arguments, one would therefore expect the corresponding gravity theory to display a mass for the graviton, probably modifying the Einstein equations of motion (see for example PhysRevLett.112.071602). This is somehow connected with some brief discussion in the conclusions about topological massive gravity. Could this be linked to the fact that the average metric does not solve anymore the Einstein equations in vacuum? Is this mass appearing somewhere in your setup or do you believe that it would appear only at the level of the fluctuations ?

(6)The Authors introduce the sources of disorder in the gauge fields. Is it possible to show to which components of the metric they correspond to (using Eq.(14) for example)? That would help to gain some physical intuition behind the technical construction.

(7) Some additional references about disorder in holography and CFTs should be added in the introductory part. E.g. : 1509.02547, 1401.7993, 1505.05171, 1802.08650,1409.6875, ...

(8) More importantly, I am quite surprised that 2004.06543, 2110.11978 are not mentioned. Not only they contain important developments following other References mentioned in the introduction but they could be even related to the IR divergences seen by the Authors. I suggest to add them and comment on these.

(9) Finally, when the Authors mention the physics of Anderson localization (which seems to be the long-range task of this exploration somehow) they should provide a few more details about it and in particular about the existing results in the holographic literature. Ref.[21] for example made an important statement about the possibility of (not) having Anderson localization in holography (or at least in simple holographic models) [btw. 1511.05970 could be also mentioned]. Later on, some counterexamples of those statements appear by adding extra bulk couplings (1601.07897,1602.01067 ). Finally, 1711.10953 provided an interesting view on the topic. In my opinion, that part could be a bit enlarged. Also, can Anderson localization appear in a classical theory or is it necessarily tight to quantum effects (interference)? If quantum effects are necessary, then it is at least disputable whether large N holographic models could provide any help in this direction. A comment about this point would help the Reader.

This paper deals with a challenging and interesting topic, that of strongly coupled disordered systems, in a rather new setup. They study disorder via the AdS3/CFT2 duality by means of the Chern-Simons (CS) formulation of AdS gravity. The manuscript is clearly written and the computations and results are nicely described and presented.

Although this work presents an interesting approach to the treatment of disorder via holography; in its present form I am not convinced that the manuscript meets any of the four 'Expectations' acceptance criteria of the journal. I shall go through the article and raise questions related to the points I believe are the weak or unclear steps of this work.

The paper starts with a nice introduction where the authors clearly set the problem and the progress made in past studies of holographic disorder. Then in section 2 they introduce the CS formulation of 3d gravity and how disorder will be introduced through the chemical potentials of the two gauge fields. They show how in CS gravity one can easily obtain the resulting metric for a chosen chemical potential. And here comes my first question:
[from here on the numbering of the questions is related to the corresponding section in the paper]

2.1) If I understood the computations correctly they can in principle obtain an expression of the metric in terms of the disordered chemical potentials (16 and 17) and their derivatives (and possibly ratios of mu, \bar mu and their derivatives). Do I correctly understand that they truncate the solution at 2nd order in the disorder strength only to be able to analytically perform the disorder average?
[If an expression of the metric non-perturbative in disorder is at hand maybe many of the points in the paper could be made stronger by a couple of simulations]

2.2) In the expression for the metric in the appendix (eq. 62) there appear IR divergent terms (~z^2) in g_{phi phi}. Am I right to assume that the expressions in eq 62 are correct in z but truncated at quadratic order in \epsilon?
Then one could worry that unless somehow the z^2 terms vanish upon going to higher orders in epsilon (or beyond perturbative disorder) this is already indicating that disorder is likely to give rise to a divergent IR (something that would not be surprising if the disorder is marginally relevant). Going back to the comment 2.1) above: it would probably be quite easy to get a numerical estimate of the fate of these z^2 terms beyond the perturbative expansion.

3.1) Section 3 is devoted to the study of the state dual to the averaged metric. I would like to ask first about the state resulting from averaging over realisations. The authors mention that the resulting energy-momentum tensor satisfies the QNEC (as opposed to the one resulting from the average metric). I would like to ask about the energy momentum tensor resulting from averaging over realisations and how it compares to the energy momentum tensor they obtain in section 4 after the 'resummation' procedure. Does it also imply the introduction of mass and angular momentum?

3.2) Small question about sections 3.1 and 3.2: given that the average metric does not solve the Einstein equations, is it surprising that the standard results for the trace anomaly and the QNEC do not hold?

In section 4 the authors set on to finding a 'resummed' averaged metric

4.1) They define the procedure as resummation and refer to previous holographic examples. In those, a non-divergent averaged metric was found after first modifying the metric ansatz and then solving the EoMs for the 'resummed' metric ansatz. This seems out of reach within the CS formalism used by the authors. And in my view this 'resummation' is a weakness of the paper since one could argue that the 'resummed' averaged metric they obtain in section 4 is just a new solution of vacuum Einstein's equations whose connection to the initial disorder problem seems tenuous.

4.2) A small question about the 'resummed' averaged metric. Its g_{tt} basically reads g_{tt} = -1/z^2 +1/24 (\epsilon^2 +\bar\epsilon^2), right?
What should I make of the value of z where this g_{tt} vanishes?

4.2) In the 'resummation' process the authors modify the averaged metric in a somewhat minimalistic way such that they arrive to a metric that now solves the vacuum Einstein equations and thus verifies the trace anomaly and QNEC conditions. Ideally it should be made much more clear why one should expect that this modified metric is actually describing the disordered system (for small disorder). Since this modified metric does not follow from solving the initial problem of introducing disorder via the CS chemical potentials I think a stronger case needs to be done about this. (A possible way to go is to just obtain 'numerical' solutions if, as I asked in 2.1, the expression for the non-truncated metric in terms of the disorder distribution is at hand).
I am also worried about the lack of justification for some of the choices in the 'resummation', for instance at some point some coefficients are set to zero 'in order to introduce as little change by the 'resummation' parameters as possible'. Since by this process one is already obtaining a quite different metric from the one we started with it is not obvious to me that this justification holds.

All in all, I think this work presents a nice idea but, up to my understanding of it, it needs to be improved in some key points to meet any of the 'Expectations' criteria of this journal. As I hope is clear from my comments&questions above, I believe that the results presented in the manuscript do not clearly show that the proposed 'resummed metric' is actually describing the effect of the introduction of disorder via the CS chemical potentials as is the stated aim of the work.
Maybe the connection between the 'resummed' metric and the actual effect of the disordered chemical potentials can be supported by further analytical computations (via the Einstein equations?) or a couple of numerical simulations that should be easy to perform (just doing some averages over expressions containing maybe complicated combinations of the disordered distribution) if the authors have an analytic expression for the metric in terms of the disorder distribution. Alternatively, maybe the authors could consider aiming for publication in SciPost Physics core which has less stringent acceptance criteria in terms of exceptionality.