On the flow of states under $T\overline{T}$

1- Clearly written.
2- The Hamiltonian point of view is less commonly applied to TTbar than the Lagrangian point of view.
3- Several detailed discussions of relations with other work (on the S-matrix, energy levels, 3d gravity) and an especially interesting comparison with tensor networks.

1- The text does not make it entirely clear which of the results are new compared to [17].
2- Many expressions are formal and may be ill-defined due to UV divergences.
3- On the plane, dressed operators seem to just be operators of the undeformed theory.

My apologies to the authors for the delay in reviewing their interesting paper.

1- The authors find that the TTbar deformation of 2d field theories on the plane formally amounts to a unitary transformation on the Hilbert space, while it is more elaborate on the cylinder. From this very interesting point of view they rederive various results on energy levels, S-matrix, and correlators. It is not clearly stated which results are new, except for the flow of states themselves.

2- Around (3.26) it is stated that deforming by the dressed $\tilde{T}\tilde{\bar{T}}$ will be equivalent to a one-shot shift by $\lambda T\bar{T}$. This seems very close to the classical claims, where the classical TTbar-deformed action is given in worldsheet coordinates by the one-shot-deformed action: see eq (2.10) of [Coleman, Aguilera-Damia, Freedman, Soni arXiv:1906.05439], building upon work by Tateo et al.

3- More generally, the authors should make an effort to relate their Cauchy string variable X to previous literature, such as Cardy's random metric reformation of TTbar (also using Hubbard-Stratanovich fields), or the JT-gravity approach (or massive gravity, following Tolley).

4- On the plane, it seems dressed operators are simply operators of the original undeformed theory. It is not clear that their correlators teach us anything about the TTbar-deformed theory. If one sees the TTbar theory as viewing the original theory from a worldsheet (à la dynamical change of coordinates of Tateo et al), then the dressed operators are simply forgetting the worldsheet and directly working with the original theory.

5- I share the first reviewer's worries about UV divergences.

6- I must correct the first referee in that treating the TTbar deformation in the Hamiltonian formalism is not novel, see for instance [Le Floch, Mezei arXiv:1907.02516] which heavily relies on it. However, it is true that tracking the state quite so precisely as the present submitted paper is novel.

7- Section 4.2 ends without really giving a holographic interpretation: the authors make the Einstein-Hilbert action appear, but there remains a rather under-explained path-ordered exponential. Besides, (I'm probably just being confused here), isn't the Einstein-Hilbert action only the first three integrals in (4.8)? I don't understand how to interpret the $R^{(\gamma)}$ term geometrically.

8- The authors provide a very suggestive comparison with tensor networks that seems promising, but I am not knowledgeable about that aspect.

1- Compare their X with the literature describing the classical TTbar deformation as a dynamical change of coordinates.

2- Relatedly, clarify whether their dressed operators are non-trivial (on the plane, say) or are simply the original operators in the original theory.

3- Discuss UV divergences.

4- Correct the following minor points

4a- Using sgn(x) in eq (2.6) is weird: in the interval (0,L) mentioned below the equation the sign is just +1, and sgn(x) is not enough to deal with arbitrary real x. One should either just write -x/L for x in (0,L), or write something with the fractional part of x to make it valid for all real x and L-periodic.

4b- Second line of (2.8) should have the opposite sign.

4c- Below (2.22), replace reference to eq (2.22) by (2.21). Inside the integrals in (2.24) and (2.33), replace lambda by lambda' (twice in each). Missing exp(epsilon s) next to O TTbar in (2.27). Between (2.27) and (2.28), replace reference to eq (2.26) by (2.11).

4d- In (2.29) the first $\xi$ and $Q$ should either both or neither have the superscript $a$; the first $S$ is missing $\phi$ as an argument. In (2.30) change $(Q^1)_k$ to $(Q^1_k)$. The $(Q^1)^2$ and $Q^1P$ terms seem to give $P^2$ without factor of~$2$ once integrated out. Below (2.30) it should be stated explicitly that $\xi$ has vanishing spatial integral.

4e- In (3.9), $x-y$ should be $x_1-y_1$.

4f- In section 4.2 missing parentheses around 2.25, 2.3.

4g- Typos: degenracy, cirlce, ``Let the seed theory by'', detials

1. A significant contribution to the subject
2. Well written and explained

1. somewhat formal in its approach
2. fails to discuss UV divergences
3. one result possibly empty of content

1. This paper is an interesting addition to the literature on TTbar. It adopts a novel approach in using the operator-state description rather than the euclidean path integral, and in focussing on the deformed states rather than the spectrum, although the method does rederive previous results on the latter.

2. The paper relies heavily on a method introduced in [17] and not surprisingly yields results which are similar, although expressed in the language of states rather than operators of the path integral approach. However eq. (1.2), which is one way of expressing the solvability of the deformation, is very elegant, especially in the way it shows the difference between the full line and the compactified circle, a difference which was not directly addressed in [17].

3. However I do find it strange that nowhere is there a discussion of potential UV divergences in the somewhat formal expressions which appear. Indeed it was argued extensively in [17] that these do appear when local operators are dressed by a string, and that they may be regulated either by point-splitting or by regulating the Green's function. But as they are written, the formal expressions for the deformed operators simply do not exist.

4. i also do not understand the second way of defining dressed operators by conjugation with U. This leads to the somewhat paradoxical conclusion that the conformal symmetry of the seed theory is preserved, if hidden. This is reminiscent of the (false) statement that an interacting QFT can be thought of as a dressed version of the free theory and therefore retains all of its algebraic features, which is clearly wrong, largely because of UV divergences which mean that the Hilbert spaces are totally different. It is true that the cylinder spectrum does retain memory of the Virasoro algebra, but that is UV finite unlike local operators. I suspect that the conclusions in this part of the paper are merely formal and possibly erroneous as stated.

5. The last part of the paper contains interesting ideas about the tensor network interpretation.

1. The authors should explicitly discuss the matter of UV divergences at appropriate points and how they might modify their results.
2. They should address point 4 above, which might lead to significant changes in this section.