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

Dear editor, 

Thank you for providing us with feedback from the referees. We proceed to addressing their points in order.

Regarding the comments in Report 1,

• In our conventions, the second factor in (6.5) should indeed appear with a minus sign. We thank the referee for spotting this typo. It has been fixed in the current submission together with a sign in $j_1^L + i j_2^L$ in eq. (6.8).

• We have expanded the discussion in section 6.1 to clarify our method, and expanded on the results. These currents that participate in the deformations are defined in terms of the Killing vectors in eq. (6.14) via (6.13). We have checked explicitly that the currents thus defined are (anti-)holomorphic upon imposing the equations of motion, and details of this computation are included in the current manuscript in app. C. The notation to refer to the currents has also been updated for the sake of clarity.

• As mentioned in point 2 above, we have now proven that all currents involved in the deformations are (anti-)holomorphic. Exact marginality can thus be claimed following the results of [7, 68, 70].

• The deformations in (6.17) (previously (6.14)) and below need to be described in terms of tensor products, and not wedge products in general. For the TsT deformation in (6.17), which is parameterised by $\beta$ as discussed in section 5.1 of ref. [69], the deformation is indeed proportional \mbox{$\delta\beta_{1}(j_{\varphi_{1}}\otimes \bar{j}{y^{7}}-j}\otimes \bar{j}{\varphi})$}. Nevertheless, it should be noted that associating TsT deformations to antisymmetric products is a gauge-dependent statement. An instance of this fact is the deformation in (6.28) (previous (6.23)) with $\delta\beta_1=\delta\tilde{\beta}_1=\delta\tilde{\beta}_2=0$, which is symmetric in the currents but still a pure TsT transformation in the frame described below eq. (5.40).

Concerning the weaknesses that the second report notes,

• Some extra details have been introduced that hopefully make the discussion clearer.

• The fact that the solutions can be obtained by other means was only apparent after the $3d$ vacua were noted to be inside SO(7,7), as discussed in Section 5. The gauged supergravity techniques are more general and could lead to other solutions, as our set has not been proven to be exhaustive, and are a necessary ingredient for the spectral analysis which is used to assess perturbative stability.

• Some further comments have been added regarding this point below (2.51), although given the different parameterisation with respect to the other relevant references in the literature, we defer the detailed comparison to section 5. Regarding the range of the parameters, it remains in general an open question.

• Half-maximal supergravity both in $3d$ and $10d$ is used for the uplifts into heterotic supergravity. For type IIB, the solutions we present can be found in the half-maximal truncation, but their spectral analysis requires the complete theory. We have added some extra remarks on this in the introduction and section 3.

• A brief motivation has been included at the beginning of the section.

As for the requested changes in that report,

• The understanding of the parameter $\alpha$ is correct, and we have added clarifications around eq. (2.40), (4.2), (4.12) and (4.30).

• The parameter is real and takes values in the unit interval, as noted in footnote 6. Its value affects the geometry and the masses, as can be seen in e.g. eq. (4.33), and therefore different values correspond to physically different solutions. In $10d$, $\alpha$ corresponds to the ratio of the sphere radii. This has been added around eq. (2.40), (4.12) and (4.30).

• In this case, this understanding is not correct. As remarked before, the solutions were obtained directly in $3d$ by a careful analysis of the scalar potential, and only later uplifted via ExFT methods; and this $3d$ origin allows an otherwise unattainable study of the solutions. We have highlighted this before (2.51).

• The results in app. A of ref. [19] would inform consistent truncations only if the SL$(3,\mathbb{R})$ transformation acted exclusively on the $T^3$ angles. In this case, we do agree it would be unsurprising. However, even in the $S^3\times T^4$ case, the TsT transformations discussed in the text do involve angles on the $S^3$, where the link to the consistent truncation is much less obvious due to the absence of a global symmetry. We have clarified this on page 46 in the discussion, with the sentence ``Among the directions in the conformal manifold, the possibility of describing TsT deformations acting on spheres in a consistent truncation is a three-dimensional peculiarity, as in higher-dimensions the moduli triggering those transformations sit within higher Kaluza-Klein levels''.

• To the best of the authors' knowledge, no QFT techniques are available for non-supersymmetric conformal manifolds, except for the case of current-current deformations.

Regarding the requested changes in the third report,

• We have added new comments about the supergravity approximation and the distinction between classically marginal and exactly marginal in the second paragraph of the introduction.

• We have added clarifications around eq. (2.40), (4.2), (4.12) and (4.30) about the r\^ole of the parameter $\alpha$.

• In section 5, the notion of TsT transformation is now explained before eq. (5.3).

• Also in section 5, the general strategy to obtain the Zamolodchikov metric is introduced (after eq. (5.26)) as well as the line elements corresponding to the different examples in sections 5.2--5.4.

• The relation between three-point functions and marginal deformations is advanced in the introduction and better explained at the beginning of sec. 6.2. Following ref. [71], the corrections to the beta-functions in CFT perturbation theory are captured by the three-point functions of the supergravity fields, and one can extend this computation to higher orders in perturbation theory. A worldsheet calculation would indeed be necessary to claim exact marginality, as we do for the current-current deformations. We acknowledge that the holographic description of the marginal deformations identified in the text is brief, and plan to expand it future work.

Best regards,

The authors