Charting the Conformal Manifold of Holographic CFT$_2$'s

1. The authors employ new techniques to construct a large multi-parameter family of novel AdS_3 supergravity solutions with NS-NS fluxes that can be viewed as consistent backgrounds of type II or heterotic string theory.

2. The authors present explicitly some examples of these supergravity solutions and calculate the spectrum of KK excitations in supergravity by employing ExFT techniques and discuss the corresponding 2d CFT interpretation.

1. The identification of the continuous supergravity parameters with exactly marginal couplings in the holographically dual 2d CFT is discussed only briefly.

2. The discussion of the Zamolodchikov metric on the conformal manifold is also very brief and incomplete.

The authors construct large families of new AdS_3 supergravity solutions by employing techniques from 3d gauged supergravity and Exceptional Field Theory. The paper should be of interest to researchers working on string theory, supergravity, and holography. I think this work merits publication.

I recommend that the authors address the following points before publication

1. The difference between marginal and exactly marginal should be explained and emphasized more strongly in the Introduction, especially given that the discussion in the paper is mostly valid to leading order in the large N limit of the dual 2d CFT and thus exact marginality is not always guaranteed.

2. The meaning of parameter \alpha introduced in Section 2 from the 10d perspective should be explained more clearly. Is it the same as the parameter \alpha used in Section 4.3? If yes, this too should be emphasized.

3. The notion/definition of a TsT transformation should be explained better somewhere in the paper, perhaps at the beginning of Section 5 or in the Introduction.

4. The authors should provide more details on how the metric in (5.26) is computed. The authors should also consider calculating and presenting the Zamolodchikov metric for the examples studied in Section 5.2, 5.3 and 5.4.

5. It is unclear to me why below (6.27) the authors say that one needs to calculate 3pt functions of KK modes in order to check exact marginality. Presumably this should be done by doing a worldsheet calculation? I think that this point needs further clarification.

Ask for minor revision

1 - The authors construct 17(15)-dimensional deformations of known $AdS_3$ backgrounds in type IIB and Heterotic supergravity. They generalize the results of [40,8,18] to include more parameters.

2 - The authors present a unified framework to study the two cases in gauged supergravity.

3 - They discuss the uplift of some of the solutions to 10D and discuss the spectrum of fluctuations around the background.

4 - They then discuss the holographic intrerpretation of the flat directions as $J\bar{J}$ deformation of the original 2D SCFT.

1 - In general the text (notably in the introduction) is not always clear or accurate, it could benefit from some revising.

2 - The paper heavily relies on exceptional field theory and gauged supergravity. However it seems that many of the results can be obtained directly from 10 dimensions using known results in the literature. This is already presented in section 5, but could be presented more prominently earlier on in the paper.

3 - Sometimes it is not clear which parameters are new and which are found in previous work. Also the range of parameters is not stated clearly when it is relevant. I think this should be organized better and maybe stated clearly when (2.51) is first written down.

4 - It is slightly confusing that the authors use half-maximal supergravity instead of maximal given their own earlier works. This is not explained in the paper.

5- the goal of section 6 is not completely clear.

This paper discussed deformations of the well-known $AdS_3\times S^3\times T^4$ and $AdS_3\times S^3\times S^3\times S^1$ solutions of type IIB/Heterotic supergravity.

The paper builds on earlier works by the authors, notably in [40,8,18] and it deserverves publication. However, before recommending publication in Scipost I would like to ask the authors a few questions and request clarifications.

1 - In eq. (2.40) a parameter $\alpha$ is introduced in the embedding tensor. As far as I can tell $\alpha=0$ corresponds to type IIB on $S^3\times T^4$ and $\alpha\ne0$ corresponds to type IIB on $S^3\times S^3\times S^1$. Is this the correct understanding. If so this can be anticipated already in sec. 2.3 to clarify the discussion.

2 - Also, is $\alpha$ a real parameter? or is it a discrete parameter in the sense that for any $\alpha\ne0$ one can perform a field redefinition that sets $\alpha=1$. If this is the case then this must be highlighted for clarity. On the other hand if $\alpha$ is a true parameter, then what does it correspond to in 10D?

3 - Equation (2.51) constitutes one of the main new results in the paper, but it is unclear to me how it is obtained. From the text above it, I understand that this solution is obtained by first uplifting to 10 dimensions where squashings and fibrations can be obviously generalized. Then this general deformation is reduced back to 3D leading to (2.51). Is this a correct understanding? if so what is the benefit of performing this analysis in 3D rather than just directly in 10D? Indeed, a more careful reading of section 5 seems to confirm that all deformations given in (2.51) correspond to rather straight-forward deformations of the original AdS background that are more clearly understood in 10D (I am referring to the discussion on page 29 of the manuscript). If this is true then the paper may benefit from this explanation earlier in the manuscript as for many readers this will improve readability. Currently the manuscripts demands quite extensive knowledge on gauged supergravity and exceptional field theory to follow the text.

4 - In the paragraph below (2.51) it is pointed out that some of the parameters in the solution corresponds to TsT transfromations of the original undeformed vacuum. The authors then comment that it is remarkable that TsT parameters are contained in gauged supergravity. However, I do not think this is surprising for the $S^3\times T^4$ reduction of type IIB as the TsT transformation can be formulated as a transformation in type IIB (or 11D) on T^3 (see appendix A in ref [19]). It is perhaps a bit more surprising for the $S^3\times S^3\times S^1$ case.

5 - In section 6 it is argued that the flat directions in the supergravity solution correspond to $J\bar{J}$ deformation of the 2D SCFT. Is the claim that for all 17 parameters this is the case or only a subset of them? If all 17 parameters can be thought of as $J\bar{J}$ deformtaions, can you then predict how large the moduli space of deformations should be using QFT perspective? I ask since in the conclusion it is stated that a search for flat directions in 3D sugra has not been carried out to a conclusive extent, and it would be nice to know how far of the 17-dimensional space is.

Ask for minor revision

The discussion on the worldsheet description of the deformations and the interpretation as current-current deformations should be clarified and possibly corrected. See the requested changes below.

1 - The parameterisation of the group element of $SU(2)$ in (6.5) does not seem right. In fact, both a left and a right transformation generated by $\sigma_3$ would affect just the combination $\varphi_1 + \varphi_2$. Another way to put it is that it seems that there is no $SU(2)$ transformation capable of affecting $\varphi_1 - \varphi_2$. On one of the two sides of $g$, then, the relative sign of the angles should change.

2 - The definition of the currents in (6.12) is not clear. On the one hand, because they are defined in terms of the Killing vectors, they should probably be interpreted as the Noether currents. It is unclear, however, why the authors use a notation in which it appears that these currents are chiral/antichiral. In general, the Noether currents will not have a definite chirality, and in particular they will be non-vanishing for both values of the worldsheet indices. See for example eq. (5.7) of ref. [69]. For the understanding of (6.12), it does not help that the Killing vectors $k^i$ and $\bar k^i$ have not been defined, for example to obtain the currents in (6.13). I suspect (I hope to be corrected if I'm wrong) that the authors are taking $k^i=\bar k^i$, and that $j^i$ and $\bar j^i$ in (6.12) are not two different currents, but just the two worldsheet components of the same Noether current. If that is the case, I would ask the authors to stress in the text that, despite the notation, those $j,\bar{j}$ are not necessarily chiral/antichiral, and that they are just the two components of the same Noether current. I would also ask them not to use $k$ and $\bar k$ but just $k$, for example, because otherwise the definition becomes very confusing.

3 - The marginal current-current deformations constructed in ref. [7, 68,70] are of the form $J^1\bar{J}^2$ where $J^1$ is chiral and $\bar J^2$ is antichiral. If my interpretation in point 2 above is correct, the authors are not describing current-current deformations of that type, so they cannot really make any statement about the relation to the marginal current-current deformations of [7, 68,70]. Following for example the discussion in section 5.2.2 of ref. [69], it is possible to rewrite the Noether currents as chiral/antichiral currents plus "improvement terms", and to expose the form of the current-current deformations in terms of the chiral/antichiral currents. I would ask the authors to either do this explicitly in one of their example, or at least to comment on this point in the paper. The reason why this is important, for example, is that (6.14) does not identify the marginal operator driving the deformation.

4 - I would like to ask the authors to confirm if in (6.14) and in the following equations the tensor product $\otimes$ is precisely the operation that they mean to write (and not $\wedge$). The reason I ask is that TsT deformations are current-current deformations of the form $J^1\wedge J^2$, where the currents may or may not have chirality properties, and the wedge product is important. The reason why the authors have $\otimes$ and not $\wedge$ should be related to the fact that they don't discuss "pure" TsT transformations, since they mix them with target-space coordinate transformations and B-field gauge transformations. I would like to ask them to comment on this point.

Ask for major revision