Disorder in AdS$_3$/CFT$_2$

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
\begin{align}
\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.
\end{align}
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

- 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