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Solving a family of $T\bar{T}$like theories
by Bruno Le Floch and Márk Mezei
Submission summary
Authors (as registered SciPost users):  Bruno Le Floch · Mark Mezei 
Submission information  

Preprint Link:  scipost_201905_00001v1 (pdf) 
Date submitted:  20190504 02:00 
Submitted by:  Le Floch, Bruno 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We deform twodimensional quantum field theories by antisymmetric combinations of their conserved currents that generalize Smirnov and Zamolodchikov's $T\bar{T}$ deformation. We obtain that energy levels on a circle obey a transport equation analogous to the Burgers equation found in the $T\bar{T}$ case. This equation relates charges at any value of the deformation parameter to charges in the presence of constant background gauge fields. We determine the initial data and solve the transport equations for antisymmetric combinations of flavor symmetry currents and the stress tensor starting from conformal field theories. Among the theories we solve are conformal field theories deformed by $J\bar{T}$ and $T\bar{T}$ simultaneously. We check our answer against results from AdS/CFT.
Current status:
Reports on this Submission
Strengths
1 the subject is interesting
2 novel strategy for solving for the spectrum of $T\bar T$ and $J\bar T$ deformed CFTs, which generalizes the previous approaches
3the authors have put significant effort in explaining their method
Weaknesses
1 the presentation is oftentimes cumbersome and physically nontransparent
2 the physical significance of the calculations in section 5.2 is not clear
3  section 3.4 is inconclusive
Report
This article studies irrelevant deformations of 2d QFTs by composite operators constructed from conserved currents, which have the remarkable property that the finitesize spectrum of the deformed theory is exactly solvable. The current article introduces a new method, based on coupling to background fields, which allows to solve for the spectrum in presence of several simultaneous deformations, thus generalizing the previous results in the literature. This method is in principle also applicable to nonconformal theories.
While most of the paper appears physically sound, it is not at all clear what is the physical motivation for introducing the "spectrumgenerating operators'' of section 5.2. According to their definition on line 578, these operators are simply the original CFT Virasoro/KacMoody generators conjugated by the deformation. Their properties, including the commutation relations (5.10), follow trivially from their definition, and have nothing to do with the symmetries of the deformed theory. The calculation of deformed OPEs using these operators is extremely cumbersome, and it is not clear what are the advantages, if any, of proceeding this way.
In conclusion, this paper introduces a new method to solve a problem of current interest and the main result looks physically robust. At times, the presentation appears unnecessarily complicated and could be improved. I am happy to recommend this article for publication in SciPost, provided the changes below are considered.
Requested changes
1 line 3: since the $T\bar T$ deformation was simultaneously introduced by two different groups, the authors should cite both or neither
2 line 51: it is not true that $T\bar T$type deformations are not plagued by the usual ambiguities, see e.g. discussion in 1205.6805 and 1603.00719
3 line 121: this statement is only true in absence of anomalies
4 eqn (3.3): since the authors are introducing a different method than previously used in the literature (studying the currents, rather than the conserved charges), it would be helpful if they included the details of the derivation here
5 since (3.2) simply amounts to coupling the CFT to a background vielbein, is there a simple way to understand the deformed solution (3.3) from this point of view, including the trace equation on line 180?
6line 282: can the authors spell out here what $O_{Ii}$ stands for?
7 the presentation in section 4.1 seems unnecessarily complicated, as the differential operators $D_1$ and $D_2$ are simply implementing the definitions (4.3) and (4.4). Can the authors perhaps simplify the discussion accordingly?
8  line 382: the results of this table appear to follow from the current commutation relations
9 line 448: this statement is not true in presence of anomalies
10  section 5.2: can the authors explain what is the physical significance of the spectrumgenerating operators and why it was necessary to introduce them?
11  the statement in footnote 5 is incorrect, as the generators in those papers act in the deformed theory; in particular, their zero modes compute the deformed conserved charges, unlike the zero modes $\Lambda_0, \Upsilon_0$
Strengths
1 the article is well written
2 the subject is very interesting
3 the strategy implemented, and carefully described in section 2, is original
Weaknesses
1 section 3.4 is inconclusive
2 the method used is powerful but no explicit use of quantum integrability was made.
Report
This paper deals with irrelevant integrable deformations of conformal field theories. It aims to generalize recent results concerning the perturbation of 2D models with the Zamolodichikov's TTbar operator and, in particular, to find an exact evolution equation for the spectrum of CFTs perturbed by combinations of specific conserved currents. The authors' strategy, carefully described in section 2, is original and relatively robust.
The main result of the paper is the classical evolution equation encoded in Table 3. The authors made checks to support its validity also at the quantum level. However, there must be a more direct way to arrive at the same flow equation, for example, by working directly within the quantum exact Smatrix setup. Maybe the authors should mention this possibility in the "Conclusions and outlook" section.
In conclusion, this is a serious piece of work in a fastdeveloping research field, and I recommend this article for publication.