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$T\bar T$ and the mirage of a bulk cutoff

by Monica Guica, Ruben Monten

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Authors (as registered SciPost users): Ruben Monten
Submission information
Preprint Link:  (pdf)
Date submitted: 2019-07-31 02:00
Submitted by: Monten, Ruben
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


We use the variational principle approach to derive the large $N$ holographic dictionary for two-dimensional $T\bar T$-deformed CFTs, for both signs of the deformation parameter. The resulting dual gravitational theory has mixed boundary conditions for the non-dynamical graviton; the boundary conditions for matter fields are undeformed. When the matter fields are turned off and the deformation parameter is negative, the mixed boundary conditions for the metric at infinity can be reinterpreted on-shell as Dirichlet boundary conditions at finite bulk radius, in agreement with a previous proposal by McGough, Mezei and Verlinde. The holographic stress tensor of the deformed CFT is fixed by the variational principle, and in pure gravity it coincides with the Brown-York stress tensor on the radial bulk slice with a particular cosmological constant counterterm contribution. In presence of matter fields, the connection between the mixed boundary conditions and the radial "bulk cutoff" is lost. Only the former correctly reproduce the energy of the bulk configuration, as expected from the fact that a universal formula for the deformed energy can only depend on the universal asymptotics of the bulk solution, rather than the details of its interior. The asymptotic symmetry group associated with the mixed boundary conditions consists of two commuting copies of a state-dependent Virasoro algebra, with the same central extension as in the original CFT.

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Anonymous Report 2 on 2019-9-22 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1906.11251v1, delivered 2019-09-22, doi: 10.21468/SciPost.Report.1193


1. The paper makes a significant step towards giving a holographic interpretation of TTbar deformations, building on ideas of McGough et al.
2. They convincingly show that there should be a remnant of the left- and right-Virasoro symmetry after TTbar-deforming a CFT.


1. Partly because the authors are mostly concerned with the holographic dictionary, they do not really comment on how the states that acquire an imaginary energy should be interpreted (and in a sense, whether a QFT featuring such states makes sense at all). That makes some of their conclusions on the holographic interpretation of such theories less strong that one would ideally like.


The authors study the holographic interpretation of the deformation of CFT2s built out of Zamolodchikov "TTbar" operator. This was first considered by McGough et al., who considered one distinguished sign of the deformation parameter (that the authors call the negative sign). In that case, all but a finite number of states acquire imaginary energy, and should be expunged from the spectrum; because of that McGough et al. propose that this should define a holographic cutoff.

The authors expand on these ideas and propose a holographic dictionary for TTbar deformed theories. In the case studied by McGough et al., this leads to compatible results: the authors set boundary conditions for the fields that can be reinterpreted as Dirichlet boundary
conditions at finite bulk radius. However, the authors propose that the region outside the would-be cutoff surface should not be removed from the geometry, in contrast from McGough et al.. The authors also extend their holographic dictionary to more general set-ups, involving the other sign of the deformation as well as matter (in which cases, the cut-off interpretation is lost). Moreover, very interestingly the authors study the asymptotic symmetries of these set-ups, finding that the Virasoro symmetries of the original CFT2 remain, though the generators take a state dependent form.

I find the paper interesting and well-written and I recommend it for publication. I would however like the authors to address in more detail how imaginary-energy states should be interpreted in these deformed theories. Almost all states acquire imaginary energy for negative values of the deformation parameter, casting doubt on whether such a deformation is meaningful at all. While at the technical level the authors explain that such singularities are related to the singularity of a certain coordinate, I do not find it clear why one can simply live with such an unphysical behavior. Moreover, I would like the authors to comment on the imaginary energy states that appear for positive values of the deformation parameter. Finally, as noticed by another referee, the map in eq. (2.12) is very similar to the one recently found by Conti et al., and the authors should comment on this similarity.

Requested changes

1. Comment, in broad terms, on the physical interpretation and soundness of deformed theories with imaginary-energy states.
2. Comment, also at the technical level, on how imaginary-energy states should be understood holographically for positive sign of the deformations.
3. Comment on the similarity of their eq. (2.12) with earlier work of Conti et al.

  • validity: high
  • significance: good
  • originality: high
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  • formatting: excellent
  • grammar: excellent

Anonymous Report 1 on 2019-8-30 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1906.11251v1, delivered 2019-08-30, doi: 10.21468/SciPost.Report.1141


In this paper, the authors derived the holographic dictionary for $T\bar{T}$-deformed CFTs using variational principle approach. The results in this paper are quite interesting and important for the holographic understanding of $T\bar{T}$ deformation.

The holographic dictionary for $T\bar{T}$ deformed CFTs was first proposed by McGough, Mezei and Verlinde (MMV) to be a cut-off geometry. Although this proposal is intriguing, it has several limitations and raises a number of puzzles. The proposal of MMV works only for one of the signs for the deformation parameter in the pure gravity sector. The proposal given in the current paper works for both signs and the cases with matter fields in the bulk. In the special case considered by MMV it is possible to reinterpret the result as a cut-off geometry. The holographic dictionary given in this paper amounts to modifying boundary conditions for the metric at infinity, which is more natural and expected from standard AdS/CFT since $T\bar{T}$ deformation is a double trace deformation. In a sense, the authors demystified the MMV proposal and makes the holographic dictionary more complete.

In addition to this, the authors also derived several other interesting results. Most notably, they analyzed the asymptotic symmetry associated with the new boundary conditions and found that it is still two copies of Virasoro algebra, with the same central extension as the original CFT. The only difference is that now the generators are state dependent. This finding, to some extent, gives another explanation to the solvability of the $T\bar{T}$ deformed theory.

With the above summary and comments, I am glad to recommend the paper to be published. There's only one point which worth clarifying a bit further. Eq(2.12) of the paper is basically the same as Eq(1.7) of the paper by Conti, Negro and Tateo (arXiv:1809.09593) proposed from a different consideration. The quantities on the right hand side are functions of the \emph{field dependent coordinate}. However, this point is not clear from the derivation by variational principle approach. Are the two formulae the same ? Since both papers claim that the corresponding formulae are consistent with the work of Dubovsky et al, they should be the same. In this case, the authors need to comment on this point. If the two formulae are different, it would also be nice to comment on the difference and the relation.

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