Sequential Flows by Irrelevant Operators

1 - The paper is well written
2 - The studied setup is original

1 - The holographic interpretation is fairly speculative
2 - In various points, the authors have to resort to numerical analysis

This manuscript deals with theories obtained by applying sequential $T\bar{T}$ deformations.

In Section 2, the authors consider the case where a single CFT is subject to two consecutive $T\bar{T}$ deformations with parameters $λ_1$ and $λ_2$. They observe that in the zero-momentum sector, the spectrum is the same as if obtained through a single deformation with parameter $λ_1+λ_2$.

In Section 3, the authors study the deformation of a theory obtained by tensoring two CFTs that were previously separately deformed. The resulting theory is made non-trivial thanks to the non-linear nature of the $T\bar{T}$ operator, introducing an interaction between the two sectors.
The analysis is, in various points, based on numerical evidence.

In Section 4, the same setup of Section 3 is considered from a Lagrangian standpoint. Specifically, the authors consider as a starting point a theory of two free scalars or, alternatively, two free fermions.

By the authors' own admission, this work is somewhat explorative in nature, but the results presented seem to me correct. However, one of the main points studied in the manuscript concerns what happens when one combines deformations with opposite signs. A $T\bar{T}$ deformation with negative $λ$ acting on a CFT is known to induce a complexification of the energy eigenvalues for some upper portion of the spectrum, leaving only a finite number of real energy levels.
To understand to which extent a second deformation with positive $λ$ can, at least partially, restore the real spectrum one should first understand how to properly interpret a deformation with a negative parameter in the first place. Do the complex energies signal a non-unitarity of the theory? Should they be removed from the spectrum? Is the deformed finite-volume theory even well-defined?
Given that the effect of a single deformation is still not fully understood from a physical standpoint, it is difficult to draw any conclusion regarding the outcome of sequential deformations.

Furthermore, the results of Section 2 seem to me just an obvious consequence of the very definition of the $T\bar{T}$ deformation, where at any point $λ$ along the flow, the infinitesimal deformation is induced by the $T\bar{T}$ operator computed at $λ$. Finite deformations $g_\lambda$, thus, act as elements of a one-parameter group. Schematically: $g_{λ_1} \cdot g_{λ_2} = g_{λ_1+λ_2}$.
In fact, this property should hold for general two-dimensional theories (i.e. not just conformal field theories), and it is not at all restricted to the zero-momentum sector.
While the authors seem to briefly hint at this fact in the Introduction, they then proceed to derive all the results by studying certain solutions of the flow equation. Yet, it seems to me that, being familiar with how a single $T\bar{T}$ deformation acts on CFTs, one could anticipate the results of Section 2 without performing any explicit computation.

I recommend publication, provided the authors address the points raised below:

1 - It is not clear to me what Eq. (1.6) represents. It is supposed to be the first order term in $λ_i$ of the deforming operator, which in principle should be obtained by expanding $$\lambda_3 (T_1(\lambda_1)+T_2(\lambda_2)) (\bar{T}_1(\lambda_1)+\bar{T}_2(\lambda_2))$$ at the first order in $λ_1$ and $λ_2$. In Eq. (1.7), the case $λ_3=-λ_1=-λ_2$ is considered. However, the flows induced by $λ_1$ and $λ_2$ do not commute with the flow induced by $λ_3$. Therefore, as indicated in Figure 1, when $λ_3$ is small, $λ_1$ and $λ_2$ should be finite.
The authors should clarify this paragraph.
Similar considerations apply to Eq. (3.23).

2 - The authors should stress that the results of Section 2 are, as one would expect, consistent with a single deformation and that this is a general fact.

3 - There is a typo in Eq. (1.6), where parentheses do not match.

4 - In Eq. (3.5), $\mathcal{E}_n$ should read $\mathcal{E}_{m,n}$.

1- The authors propose a creative and interesting generalization of the known $T \overline T$ deformations.
2- They perform an extensive analysis, both analytical and numerical, of many of the qualitatively different cases their deformation allows.
3- This analysis sheds some light on the complex energy levels, one of the big open questions related to the $T \overline T$ deformation.

1- It is not clear whether some of the bounds are optimal, see full report.
2- Despite mentioning the holographic interpretation in the abstract, the authors are unfortunately rather brief and noncommittal in their conclusion on this aspect.

The authors study an analog of the $T \overline T$ deformation obtained by deforming a pair of, already deformed, field theories by the operator $(T_1 + T_2) (\overline T_1 + \overline T_2)$. They analyze extensively the energy spectrum of these theories for several values of each of the individual deformations. Notably, it turns out that one can "recover" states whose energy had become complex after the first $T \overline T$ deformation by applying a judicially chosen "joint" $T \overline T$ deformation. The authors furthermore give numerical evidence that these theories remain invariant under S-duality.
They conclude by studying the classical Lagrangian for some examples and comment briefly on possible holographic interpretations.

This paper presents an interesting perspective on an important unanswered question for the $T \overline T$ deformation, namely the faith of the complex energy states. The analysis suggests that these energies become real again by a reverse $T \overline T$ deformation, or even, more surprisingly, by different deformations.

That said, I would like to add some remarks and questions. First off, I think that section 2 would benefit from a precise clarification of what the nontrivial statement is. It does not seem surprising that, within a certain range of parameters, a $T \overline T$ deformation can be undone by a reverse deformation because the spectrum is given by the Burger's flow, which can be solved by the method of characteristics and is reversible as long as a unique solution exists, i.e. as long as the characteristics do not intersect. It seems therefore that the nontrivial claim in section 2 is that the flow is reversible even beyond this point, if one specifies a finite amount of data such as the specific branch of the square root that is chosen. Is this a fair summary, and could the authors provide more comments?

Second, it is surprising that the bound $r < 1/2$ from the diagonal spectrum (3.22) is much stronger than the bound $r < 1$ from the off-diagonal spectrum (3.24). This seems to suggest that (3.8), which was derived by first making a large energy approximation and then flowing with the joint $T \overline T$ deformation, is not an accurate estimate. One can for example compare (3.8) with the exact result when $\alpha = \beta$, $\lambda_1 = \lambda_2 = \lambda$ and $\lambda_3 = - r \lambda$ for $r \approx 1/2$, these do not seem to match.

Finally, it would be very interesting if the authors could provide some more precise comments on the holographic correspondence they have in mind.