Boundary Description of Microstates of the Two-Dimensional Black Hole

1- the paper contains interesting and important new results on 2d black holes and their microscopic dual.
2-most of the technical calculations are clearly presented and stepwise can be followed (reasonably) well also by non-expert readers.

1-some of the conceptual steps are made in an intuitive fashion and appear a bit ad hoc.
2-the mechanism by which the gap in the eigenvalue distribution is created is not fully made clear.
3-the stringy justification presented in the discussion is incomplete

The authors make a significant step forward in the study of 2d black holes by identifying (or proposing) an explicit microscopic dual of a 2 black hole in the c=1 matrix model. The presented results are original and interesting and will stimulate further research in this direction. The paper is reasonably self-contained and contains sufficient background material to be readable also for non-experts. The remaining open questions may lead to interesting follow up work. The paper could be improved, however, by explaining in a better way why the assumptions that go into the calculation are justified.

The technical calculation in section 4.2.3 is hard to follow, which makes it difficult to understand why the proposed procedure of creating a large gap in the distribution of eigenvalue is justified. In my opinion the authors should improve this part of the paper by giving a physical reason why one can expect that the remaining terms in (4.55) are of this orde (for instance, by identifying a physical mechanism that ensures this).

1- A Novel technique and perspective for analyzing non-singlets in Matrix Quantum Mechanics, directly from the Hamiltonian and the Schrodinger equation of the system
2-The technique applies to a microscopic model that could possibly describe a non-supersymmetric black hole (which is very rare)
3-The novel (non-singlet) coarse grained states found might have a more general applicability in other systems related to black holes
4-The authors provide a concise review on this difficult subject

1- Sometimes the analysis is too qualitative
2-Difficult to comprehend (for non experts) and the discussion sometimes becomes too dense and lacks clarity
3-It is not clear whether one can compute interesting physical observables with the authors' techniques and make more precise statements beyond the calculation of the free energy/thermodynamics

This is an interesting paper, related to the difficult subject of understanding the physics of a non supersymmetric black hole in string theory, through the lense of a dual matrix quantum mechanics (MQM) model.

The main novelty of the paper is a very interesting direct analysis of the non-singlet sectors of the said MQM model in the double scaling limit (albeit quite qualitative at some points) and a definition of various (coarse grained) states, one of which has the property of describing a black hole-like object.

Of course there is still an ongoing debate on whether this object is actually a black hole with an actual horizon in the Lorentzian signature or should be better thought as a long string condensate. It would be interesting if the authors would have been able to make a comparison and distinction of the two objects (especially if they can somehow encode the presence of a horizon in their coarse grained state), but I understand that this is a tall order.

In general I think that the paper deserves to be published, should the authors attend to some thorny points and make some appropriate modifications.

1- Since there is still discussion in the literature on whether the background should be thought of as either a black hole or a long-string (winding mode) condensate the authors should be very clear about these two possibilities and not commit to just one interpretation, unless they have solid proof for it (they should also try to clarify when they expect either description to be better suited depending on the parameter space and size etc of the object)

2-I am not sure I understand the statement on the last paragraph on page 10. Where is the black hole entropy coming from according to the authors?

3- It is not clear how the adjoints can be condensed near the tip of the inverted oscillator and their wavefunction to have support only in this region from a target space perspective. In particular using the long string picture of ref. [16], the long strings have endpoints (fzzt branes) with support in the asymptotic weakly coupled region. Is the state considered corresponding to the tips of the long strings? Did the authors have to take a decoupling limit where the extend of the long-strings does not matter?

4- In Fig. 8: Should $\lambda_{\nu+1}$ be considered as the effective string coupling for the singlet states on this background? Can singlet excitations tunnel inside the hole? Otherwise how can the black hole absorb singlets?

5- In similar spin-Calogero models the wavefunction also carries spin-degrees of freedom. Where did they go in the coarse-grained description of the authors? Did the condensation somehow effectively removed them?

6- Fig. 9: What about the other process of moving the support of the adjoints towards the support of the singlets (making the "solid wider"
and the gap smaller). Is the object stable under this perturbative deformation?


Minor edits:
-Page 5: The mass of the black hole is related to the value of the dilaton and the string coupling at the tip of the cigar, is this not right?
-Page 5: 2.6 seems to be wrong since in the kinetic term one should have $X_L + X_R$, while in the sine-Liouville term $X_L - X_R$, right?
-Below eqn. 2.9. Is this clear that this constitutes the ER=EPR duality, or one should also discuss the entangled condensate of long strings on the ER side?
- 2.16 should hold for $R<2$.
- In 3.2 it would be useful to compare and contrast the authors notation with the usual notation of boxes and anti-boxes such as the one in reference [16].
- It would have been useful to separate into a new paragraph between eqns. 4.16 and 4.17, since before the authors discuss kinematics and then dynamics using the kinematic notation until 4.16.
- It might be worthwhile to mention some literature on spin-Calogero
models