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Disorder in AdS$_3$/CFT$_2$

by Moritz Dorband, Daniel Grumiller, René Meyer, Suting Zhao

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Authors (as registered SciPost users): Moritz Dorband · Daniel Grumiller · Rene Meyer
Submission information
Preprint Link:  (pdf)
Date submitted: 2023-09-01 10:58
Submitted by: Dorband, Moritz
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Gravitation, Cosmology and Astroparticle Physics
  • High-Energy Physics - Theory
Approach: Theoretical


We perturbatively study marginally relevant quenched disorder in AdS$_3$/CFT$_2$ to second order in the disorder strength. Using the Chern-Simons formulation of AdS$_3$ gravity for the Poincar\'e patch, we introduce disorder via the chemical potentials. We discuss the bulk and boundary properties resulting from the disorder averaged metric. The disorder generates a small mass and angular momentum. In the bulk and the boundary, we find unphysical features due to the disorder average. Motivated by these features, we propose a Poincar\'e-Lindstedt-inspired resummation method. We discuss how this method enables us to remove all of the unphysical features.

Author comments upon resubmission

Dear Editor,
first of all we apologise for the delay in our response. We thank the referee for carefully reading our manuscript and for the detailed comments that we address below. A list of all changes of our manuscript is given after our answers to the referees comments.
Below we address the comments of the report dated 13.02.2023.

We confirm the statements by the referee. The disorder is introduced into the system by the chemical potentials of the CS formulation of gravity. By the relation between the CS gauge fields $A$, $\bar{A}$ and the metric, the chemical potentials appear in the metric components, see e.g. app. A, eq. (66). As we commented on p. 8 before starting sec. 2.2, the IR divergent behaviour of the averaged metric is, due to the marginal relevance of the disorder in the sense of the Harris criterion, not unexpected.

The CS gauge fields $A$ and $\bar{A}$, as given eqs. (5-8) with the solutions for ${\cal L}$ and $\bar{\cal L}$ in eqs. (12-13) are solutions to the flatness conditions, eq. (9). This is true for arbitrary $\mu(\phi)$, $\bar{\mu}(\phi)$, so it holds in particular also for our choice of the disorder sources in eqs. (16-17), even after expanding to second order in the disorder strengths. The flatness conditions are equivalent to the vacuum Einstein equations. Since we have a vacuum solution before averaging, we aim to also find vacuum solutions after the average has been implemented.

In the gravity description of QCD, singularities in the IR region are indeed necessary. However, these singular spacetimes still satisfy certain conditions, as discussed e.g. in [1005.4690] (ref.~[41] in our manuscript). As an example, these spacetimes can be used to define the semiclassical approximation for a string. As we discuss around eq. (29), this is not valid for our averaged metric. Moreover, our metric contains closed timelike curves in the deep IR region. Due to these properties, we do not expect that our metric has any ``reasonable'' field theory physics as a dual description. We will further address this below in the answer to point 4 a).

Note also that in the work on disordered IR fixed points, it is the averaged function $\langle A^{(2)}(x,z)\rangle_R$, contained in the metric component $g_{tt}$, which diverges in the IR. However, the Ricci scalar of this metric is finite everywhere. This is not true in our case: the Ricci scalar (eq. (26) in our manuscript) of the averaged metric is divergent in the IR.

a) We will give two arguments for why a modification of our metric is necessary. First, in the naively averaged disordered metric with the IR singularity, the conformal anomaly equation (34) is not fulfilled. Like all quantum anomalies, the conformal anomaly is a UV effect, and switching on an IR relevant (i.e. UV irrelevant) disorder potential should not affect it. On the other hand, our resummed metrics both in the metric and Chern-Simons formulation fulfill the anomaly equation (34), which is a clear-cut criterion for pinning down the free parameters in the resummation. Second, while in AdS/CFT naked singularities can be physically acceptable (c.f. e.g. [hep-th/0002160]), certain admissibility criteria need to be fulfilled. In [1005.4690], it was put forward that a naked singularity is admissible in particular if a string world sheet is repelled by it, i.e. cannot reach the singularity. If it could, the string world sheet fluctuations would grow large, and the string would exit the semiclassical regime. We have shown in equation (29) of the draft that this criterion is violated for the naked singularity in the naively averaged disordered metric. On the other hand, the resummed metrics are completely regular, and this problem ceases to exist. We believe that these two arguments clearly show the unphysical nature of the naked singularity in the naively disordered space-time, which is cured by our resummation.

b) We have already addressed this point above in the second paragraph. The metric defined by the connections of the Chern--Simons formulation, where we introduce the disorder, is a solution to the vacuum Einstein equations. This is true for any choice of $\mu(\phi)$, $\bar{\mu}(\phi)$. This can be understood by the fact that the flatness conditions of the Chern--Simons formalism, which the connections satisfy for any choice of the chemical potentials, are equivalent to the vacuum Einstein equations.

c) In choosing our ansatz for the resummation we have proceeded analogously to the choices made in earlier studies of resummation in disordered systems, such as [1402.0872] and [1504.03288]. These conditions are given in the list in the lower half of p.~12. Subject to these conditions, our ansatz of a general polynomial in $z$ given in eqs. (42-43) is the most generic possible choice.

In our ansatz, we did not allow for non-perturbative dependencies on $z$, such as $\ln z$. Including terms such as $z^n\ln z$ with open coefficients $a_n,b_n$ to the ansatz for the resummation functions $\alpha(z),\beta(z)$ and calculating the Ricci scalar of the resulting metric however shows that these terms do only lead to further divergences. In particular, the open coefficients cannot be fixed such that the divergences already present before resummation get cancelled. Therefore, such terms are excluded.

Regarding the BPS bound, the resummed metric can always be written as a Ba$\tilde{\text{n}}$ados geometry, i.e. a vacuum solution. Therefore, the BPS bound has to hold for the resummed metric.

d) Motivated by the comments of the referee, we took a closer look at implementing the resummation procedure in the CS formulation. In fact, we found a way to perform a resummation in the same spirit as for the metric, in that certain regularity conditions should be satisfied. Due to the disorder, the averaged connections do not satisfy the gauge flatness conditions of the CS theory. By a minimal modification of the connection, in particular of the lowest weight components of the $t$-component, we uniquely determine the modification such that the flatness conditions are satisfied. The metric resulting from the resummed connections does not have curvature singularities; its dual EM tensor does not have a conformal anomaly and the relation between the trace of the boundary EM tensor and the boundary Ricci scalar is satisfied. In particular, up to an overall sign in the $t-\phi$-component, the resulting metric (eq.~(78) in our manuscript) matches what we found by the resummation procedure in the metric formulation (e.g.~eq.~(60) with $c_1=0$ in our manuscript). The sign is due to the fact that averaging and computing the metric from the connections does not commute and can be removed by a time reversal $t\to-t$. These results are discussed in the new section 5 of our manuscript.

Indeed, from the point of view of the energy-momentum tensor averaged only once its components are calculated, nothing particular worrisome appears. Then, no resummation would be necessary. However, as we point out e.g. on page 8 before eq. (25), this is not the approach of our paper. We in particular study the disordered system with the average implemented once the metric components are calculated, following the procedure of [1402.0872] (ref. [14] in our paper). The energy-momentum tensor following from this averaged metric, as we discuss in our paper, does not satisfy the QNEC and the conformal anomaly equation.

Finally, before comparing the $T_{ij}$ of 3.1) of our last reply to the resummed one, we apologise for the typo in the last reply. The r.h.s. of the equation was supposed to display as a matrix with components
\langle\langle T_{tt}^{\text{ren}}\rangle\rangle&=\frac{c}{288\pi}\left(-\frac{3}{2}(\bar{\epsilon}-\epsilon)^2\right),\\
\langle\langle T_{\phi\phi}^{\text{ren}}\rangle\rangle&=\frac{c}{288\pi}\left(\frac{1}{2}(\epsilon^2+10\epsilon\bar{\epsilon}+3\bar{\epsilon}^2\right),\\
\langle\langle T_{t\phi}^{\text{ren}}\rangle\rangle&=\frac{c}{288\pi}(\bar{\epsilon}-\epsilon)^2=\langle\langle T_{\phi t}^{\text{ren}}\rangle\rangle.
Comparing this result to the resummed energy-momentum tensor, we find that in both cases the trace anomaly equation for curved backgrounds holds, although in different ways. While both $R[\gamma^{(0)}]$ and $\text{tr}(T_{ij}^{\text{ren}})$ vanish after resummation, these quantities are generally non-trivial when calculated before averaging, as shown in our previous reply. There does however not seem to be a deeper relation between the energy-momentum tensor given above and the one obtained after resummation. In particular, the components of the respective energy-momentum tensors behave very differently, without any similarity even for equal disorder strengths. In the resummed energy-momentum tensor, the disorder strengths appear as $\epsilon^2$ and $\bar{\epsilon}^2$ only, but never in a combination $(\epsilon-\bar{\epsilon})^2$ as in the above result. Moreover, in the resummed energy-momentum tensor the $tt$- and the $\phi\phi$-components are equal even for generic disorder strengths, while in the above result, they differ.

With best regards,
M. Dorband, D. Grumiller, R. Meyer and S. Zhao

List of changes

- in the introduction, added the last paragraph on the bottom of page 4, ''Having discussed ...'' until ''... after averaging.'' on the top of page 5.
- in the final paragraph of the introduction, added two sentences referring to sec. 5, ''In section 5 ... in the Chern-Simons formulation.''
- added new section 5 on pages 16-18
- added two paragraphs in the conclusion on page 19, addressing sec .5, ''Apart from the metric formalism ... by a sign after averaging.''
- On the top of page 20, removed the comment about resummation in the Chern-Simons formulation, ''While it is clear how our resum-
mation method works on the metric level, it is currently unclear how to implement it in the Chern–Simons formulation. This may require to go beyond the radial gauge (3)-(4).''

All page numbers refer to the resubmitted version of the manuscript

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 1 on 2023-11-1 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2204.00596v6, delivered 2023-11-01, doi: 10.21468/SciPost.Report.8036


Let me first apologise for my late reply and thank the authors for their patience and their effort in addressing my questions.

I thank the authors for their clarifications on why the averaged metric (before resummation) cannot be considered as dual to a disordered CFT.
And I also like the new results they add on the implementation of averaging and resummation in the Chern-Simons formalism.

However, the last part of the authors reply brings the focus to a question that still nags me and which I believe needs be clearly addressed in the paper.

The energy momentum tensor (EMT) resulting from averaging the EMT resulting from each disorder realization is healthy (conformal anomaly respected), but very different from that resulting from the resummed metric.

I would then say that this work should clarify the following question:

One can introduce disorder in this Chern-Simons holographic model and (beautifully) solve it obtaining an inhomogeneous metric (for any value of the disorder strength) which results in an EMT that can be computed analytically.
Upon averaging over disorder realizations (this can be nicely done analytically at second order in disorder strength as the authors show or numerically for any disorder) one obtains an EMT that passes the usual checks for a holographic system. Let me call this EMT, EMT_a
However, if one averages the metric over disorder, then the averaged metric is sick, and, unsurprisingly, unphysical issues (violation of QNEC, etc) occur.
Then the authors find a resummation-inspired averaged metric that is smooth and results in a ‘healthy’ energy momentum tensor that I will call EMT_b

What should be the EMT of the dual disordered field theory? EMT_a or EMT_b? And why?

(Related to this, let me ask again a question that might help: have the authors looked at the inhomogeneous metric before averaging? Is it as sick as the resulting averaged metric?)

[This apparent existence of two different answers is, as the authors are aware, quite different from what happens in their ref [14]. There, a physical result like the value of the IR scaling exponent agrees when computed via
a) the resummed metric
b) from averaging the numerical solution that needs not be resummed]

I think that if this question can be discussed and the need for relying in the resummed metric made clear in the paper, this whole saga can be put to rest and the manuscript be finally published in SciPost.

P.S. I could not find any reference in the main text to the results in Appendix D where the behavior of the system beyond perturbative disorder is discussed. (I think it is a nice advantage of this setup the fact that one can quite easily compute the disordered metric at any value of disorder strength without having to solve any differential equation)

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