Ascending the attractor flow in the D1-D5 system

In this work, the authors study a rather large generalization (namely, a 21-parameter family) of a class of spacetimes that have attracted considerable recent interest due to their connections to non-$\mathrm{AdS}$ holography and the single-trace $T \overline{T}$ deformation. The canonical example within this class is obtained from a solution of type IIB supergravity which is obtained from taking a "partial decoupling" limit of a collection of F1-strings and NS5-branes, which corresponds to going to the near-horizon region of the five-branes but not of the strings, and interpolates from a deep-bulk $\mathrm{AdS}$ (or BTZ) solution to an asymptotically linear dilaton region; this solution is believed to be related to a $2d$ vacuum of little string theory (LST). This article extends the above construction to a larger collection of decoupling limits, all of which -- the authors argue -- correspond to LST-like theories (including related models such as NCOS) and may be identified with controlled irrelevant deformations of a dual $\mathrm{CFT}_2$.

One should note that this work involves only a pure gravity analysis, and quantities such as spacetime masses are computed using the covariant phase space formalism in classical general relativity. These results, therefore, do not directly address the identity of the corresponding controlled irrelevant deformations in the holographic duals, which would be analogues of the single-trace $T \overline{T}$ operator. Despite this, I still find the results to be interesting and worthy of publication; to be fair, even in the standard case, the single-trace $T \overline{T}$ operator cannot be unambiguously defined in the field theory except at the symmetric product point.

There is one point which I might ask the authors to clarify since, even though they appear to explain it in multiple places, the explanations seem somewhat different to me and I fear that I have not fully understood the claim. It has been believed since some fairly old work, e.g. hep-th/0006023 which is reference [14] of this paper, that applying a TsT transformation correctly generates the *extremal* versions of the brane bound states considered in this article, but fails for the *non-extremal* versions. The authors of this manuscript seem to partly agree with this conclusion since they state, e.g. in footnote 13, that TsT "does not provide a useful parametrisation at finite temperature" (although they do not say that it *fails*, only that it is not useful). But then the authors seem to go ahead and do the TsT anyway.

The question of whether TsT generates the correct solutions is distinct from the question of how to take the appropriate decoupling limit, which the authors sidestep by going to the so-called "spatially non commutative SYM frame" as explained in Appendix C. Instead I am asking the authors to clarify why TsT works at all. It seems to me that the authors are claiming that TsT does generate the correct non-extremal solutions *if* one carefully works with the correct choices of coordinates and parameters, but I feel that this should be stated explicitly, since the authors appear to use a similar parameterization as [14].

Another small comment is that the authors should cite other papers where $T \overline{T}$-like flow equations for the masses of gravity solutions have been observed in special cases of their construction, such as 1911.12359, 2111.02243, 2302.03041, and 2303.12422.

On a more trivial note, I might suggest fixing the following typos before proceeding to publication:

(1) Change "symmetric product orbifold if $T \overline{T}$-deformed CFTs" to "symmetric product orbifold of $T \overline{T}$-deformed CFTs" at the bottom of page 4.

(2) Change "proprerties" to "properties" below (2.19).

(3) Change "tensor multiples" to "tensor multiplets" below (2.28).

(4) Change "noting but" to "nothing but" on page 16.

(5) Change "For de OD3" to "For the OD3" below (3.49).

(6) Change "illustation" to "illustration" below (C.26).

(7) Change "impliying" to "implying" below (C.40).

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The paper discusses D1-D5-like solutions of type IIB supergravity, which interpolate between flat space and $AdS_3\times S^3 \times K3$, focusing on decoupling limits of the asymptotic space-time where it degenerates to linear dilaton or similar space-times. The paper notes that from the point of view of the IR CFT, at leading order 22 irrelevant deformations appear in these flows (corresponding to scalars fixed by the ``attractor mechanism''), and that the flows corresponding to decoupled theories in the UV satisfy one condition on the coefficients of these deformations. The analysis of these flow solutions and their decoupling limits is clear and interesting, and they are analyzed in detail both for the extremal solutions and for non-extremal solutions.

I find this paper interesting and think that it deserves to be published. However, I think some of the speculations in the paper are presented too strongly, and their status should be clarified further before the paper can be published. In particular, the authors speculate (and state this as a result, say, in the paragraph before last of the introduction) that the irrelevant flows from the low-energy CFTs to the decoupled theories in the UV are intrinsically well-defined from the IR point of view, even though no evidence is given for this. As the authors review nicely, there is only a tiny class of irrelevant deformations that are known to be well-defined (including the $T{\bar T}$ deformation and similar ones), and their deformation parameters satisfy special properties, that are not believed to arise for the different deformations that are described in this paper. As described in the paper, the irrelevant deformations discussed in this paper are the lowest single-trace operators preserving all the symmetries, but if one would try in the CFT to extend them to higher energies, one would encounter both multi-trace operators and stringy modes that also preserve all the symmetries, and one would need to perform an infinite number of fine-tunings to end up at any specific UV theory (including either the decoupled limits, or flat space string theory). This can be avoided (as in similar examples like the maximally SUSY irrelevant deformation of ${\cal N}=4$ SYM) by going to a limit where all the multi-trace operators and stringy modes decouple, but such a limit requires taking the number of branes (in particular the number of 5-branes) to infinity, and it is not clear what the attractor flow solution means then (from the CFT point of view, in such a limit the CFT becomes a collection of generalized free fields, so the deformation is not very interesting). I do not see any argument in the paper that the flows described in the paper are well-defined from the IR point of view (though of course the full flows starting from the UV are well-defined), so I think the statements in the paper about this (in several places in the introduction, and in section 2.4) should be weakened accordingly (the authors can speculate that perhaps there is a way to define these flows directly from the IR point of view, but they should emphasize that there is no evidence for this).

There is one exception to the discussion above, noted by the authors -- in the specific limit of the parameters of the IR CFT where it is a free symmetric orbifold, one can perform a $T {\bar T}$ deformation of the seed of the symmetric orbifold, and its symmetric orbifold is still well-defined. However, I do not know any evidence that this argument extends to the same ``single-trace $T{\bar T}$'' deformation away from the free orbifold subspace in the parameters of the IR CFT, and certainly not that it extends to the much larger deformation space described in the paper (those deformations do not seem to be well-defined in the CFT even in the free orbifold subspace).

So, I recommend that the authors weaken their statements about the existence of the flow from the IR point of view, in sections 1, 2 and 5, and after this the paper should be suitable for publication. Statements like the one made in the abstract, that generalizations of the known construction ``could provide possibly tractable effective two-dimensional descriptions of...'' are OK, but in other places in the paper it is implied that such two-dimensional descriptions exist and make sense, and I do not see the evidence for this (beyond the similarity of some thermodynamic quantities to the ones appearing in the well-defined flows, but since this similarity is expected from the UV point of view, I do not see why it provides evidence for an intrinsically two-dimensional description).

In addition to this main issue, let me mention a few more minor issues that the authors may consider modifying or clarifying before the publication of the paper :

1) In the last paragraph of section 4.1, the authors mention the similarities and differences between their results and the ones for global $AdS_5$. It is perhaps worth mentioning in this context that for global $AdS_3$ the results are very different than global $AdS_5$, and the ``small black hole'' solutions of the latter do not exist; only the ``large black hole'' solutions do, and they are the ones that then match to the dominant ``small black hole'' solutions in the background of this paper.

2) In sections 4.3 and 4.4 the asymptotic symmetries are described for asymptotically flat solutions, while truncating just to zero modes on the $S^3$. Such a truncation is sensible in asymptotically linear dilaton backgrounds, where the $S^3$ remains at finite size asymptotically with a gap to its Kaluza-Klein modes. However, it is not clear why it is sensible in asymptotically flat space-times, where the $S^3$ grows and there is no gap to its excitations. Presumably, this means that the asymptotic symmetries analyzed in the paper are a subgroup of the full asymptotic symmetries of six dimensional supergravity in flat space, and it would be nice to explain more clearly where the symmetries discussed in this paper fit into the (known) asymptotic symmetries of six dimensional gravity.

3) There are spelling mistakes in the last paragraph of section 3.1 and below (3.49).

After all these issues are considered, and the main point above is clarified, I will be happy to recommend the publication of this paper in SciPost.

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