# The Space of Integrable Systems from Generalised $T\bar{T}$-Deformations

### Submission summary

 As Contributors: Takato Yoshimura Arxiv Link: https://arxiv.org/abs/2105.03326v3 (pdf) Date submitted: 2021-09-17 23:46 Submitted by: Yoshimura, Takato Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Approach: Theoretical

### Abstract

We introduce an extension of the generalised $T\bar{T}$-deformation described by Smirnov-Zamolodchikov, to include the complete set of extensive charges. We show that this gives deformations of S-matrices beyond CDD factors, generating arbitrary functional dependence on momenta. We further derive from basic principles of statistical mechanics the flow equations for the free energy and all free energy fluxes. From this follows, without invoking the microscopic Bethe ansatz or other methods from integrability, that the thermodynamics of the deformed models are described by the integral equations of the thermodynamic Bethe-Ansatz, and that the exact average currents take the form expected from generalised hydrodynamics, both in the classical and quantum realms.

###### Current status:
Editor-in-charge assigned

We thank the reviewer for his/her positive assessment. We have implemented several changes in response to the referee's each comment.

### List of changes

1. We rearranged the way we cite these references so that it's more accurate. We also cited the paper mentioned by the referee as well as another article on non-relativistic $T\Bar{T}$-deformations.
2. We added the definition of $T\Sigma^\mathrm{Int}$.
3. We corrected the typo.
4. We meant supplement material" by SM. We rephrased it as the appendix.
5. The reference to eq (5) is corrected.
6. We added the reference to a particular section in the appendices.
7. The definition of the indicator function $\chi$ is added.
8. The definition of $\rho(\theta)$ is added.

### Submission & Refereeing History

Resubmission 2105.03326v3 on 17 September 2021
Submission 2105.03326v2 on 25 June 2021

## Reports on this Submission

### Strengths

- proposes an interesting generalisation of TTbar deformations
- shows the implications on the infinite-volume S-matrix
- derives flow equations for the deformed theory

### Weaknesses

- imprecise on the properties of these generalised transformations
- does not compare with previously determined flow equations
- no examples or physical discussion provided

### Report

Dear Editor,
this article proposes a generalisation of the current-current deformations discussed by Smirnov and Zamolodchikov (that already generalise the celebrated TTbar deformation). The authors derive the effects of these deformations on the S-matrix of the theory (in infinite volume) and write down flow equations for the charges.

The topic is interesting and some of the authors' result seem correct. However, this work needs revision in several points, which I discuss below. It is my recommendation that the paper should not be accepted for publication until such a major revision has been made.

The referee

### Requested changes

1. In the introduction, the authors state that "A physical insight into TTbar was gained [...]" by relating them to changes of particle width. To put it mildly, this is a very partial statement. Physical insights in TTbar include their description as quasi-local deformations, their relation to two-dimensional gravity to string theories on the worldsheet and in target space, and to holography. The authors should mention all that.
2. In section two the authors talk about a "more judiciously chosen" set of charges. However, it becomes clear later that these charges do not necessarily satisfy physical unitarity and crossing (not to mention real / Hermitian analyticity). The authors should make it clear "what are this charges good for", see also my points 3. and 6. below.
3. Related to point 2., it would be good if the authors discussed in some detail the deformation of one simple theory (such a Sinh-Gordon), for some example of deformations that they propose that were not previously in the literature. In particular, the authors should present and discuss the finite-volume spectrum for such deformations (for instance , the kappa, eta and lambda deformations that they introduce), also as a way to put their newly-developed formalism to the test.
4. The names eta and lambda deformations, and to a lesser extent kappa, are commonly used in the literature of integrable deformations of sigma models (they are types of quantum deformations). The authors should probably pick new names.
5. In section 4 and 5 the authors discuss the flow equations and thermodynamic Bethe ansatz for their deformations. Throughout the discussion it is unclear to me whether the theory is in finite volume or at finite temperature. If I recall correctly, this made quite a difference in the authors ref. [16]. The authors should clarify this, and explain in detail whether or not their results match the one of [16] in the case where they are both applicable. This is far from immediately clear.
6. Related to point 2., I find the discussion of section 6 imprecise. The requirement of crossing symmetry and of physical unitarity (for real momenta) seem sufficient to rule out these newly-constructed deformations. These requirement are considerably weaker than imposing that the S-matrix is analytic in the whole physical strip. More generally, for two dimensional integrable QFTs there is a well define list of properties that may be demanded of the S-matrix, related to a well-defined list of physical principles: Poincare' invariance, locality, causality, unitarity, parity, time-reversal, particle-to-antiparticle symmetry, existence of bound-states. The authors should clarify which of these properties are broken by their new deformations with respect to the "usual TTbar" ones, and if possible provide example of known theories of such a type.

• validity: good
• significance: good
• originality: ok
• clarity: ok
• formatting: excellent
• grammar: excellent

### Report

This work extends the notion of generalised $T\bar{T}$ deformations, including the complete set of extensive charges. They show that the deformation leads to a general deformation of the S-matrix. The authors derive flow equations to the free energy and its fluxes. Moreover, they show that the substitution of the deformed S-matrix in the TBA equation leads to the same results.

The article meets the publication criteria of SciPost Physics, and I do recommend the publication on SciPost Physics after some clarification (see Requested changes).

### Requested changes

1- In the first paragraph of Section 5 the authors write the following confusing sentence: "Here we show that the generalised $T\bar{T}$-deformation provides a novel derivation of TBA", but in the Conclusion they write "We showed [...] that the thermodynamics of the deformed theories coincides with that obtained by TBA". The former sentence should be rephrased since the derivation in Appendix F starts with stating the deformed TBA equations and results in the flow equation. I also suppose that in the last paragraph of Section 5 the authors intended to refer to Appendix F instead of C.

• validity: high
• significance: high
• originality: high
• clarity: high
• formatting: perfect
• grammar: excellent