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$T\bar T$ and the mirage of a bulk cutoff
by Monica Guica, Ruben Monten
This Submission thread is now published as
Submission summary
As Contributors:  Ruben Monten 
Arxiv Link:  https://arxiv.org/abs/1906.11251v2 (pdf) 
Date accepted:  20201218 
Date submitted:  20201210 03:11 
Submitted by:  Monten, Ruben 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We use the variational principle approach to derive the large $N$ holographic dictionary for twodimensional $T\bar T$deformed CFTs, for both signs of the deformation parameter. The resulting dual gravitational theory has mixed boundary conditions for the nondynamical 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 onshell 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 BrownYork 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 statedependent Virasoro algebra, with the same central extension as in the original CFT.
Published as SciPost Phys. 10, 024 (2021)
Author comments upon resubmission
The equation (1.7) in the article 1809.09593 by Conti, Negro and Tateo is indeed the same the first line of equation (2.12) in our paper, with the identification $\gamma^{[\mu]}_{\alpha \beta} = \delta_{\alpha \beta}$, $\gamma^{[0]}_{\alpha \beta} = g^{\prime}_{\alpha\beta}$, $\mu = \tau$ and rewritten so as to express $\gamma^{[0]}$ in terms of $\gamma^{[\mu]}$ and $T^{[0]}$. This relation amounts to a fielddependent coordinate transformation if the metric $\gamma^{[\mu]}$ is flat, a property that we exploit at length in section 3. However, more generally the two metrics are not diffeomorphic to each other (as one can see from the fact that it is $\sqrt{\gamma} R$, and not $R$, that stays constant along the flow). We have added a footnote on page 6 to address this relation.
Concerning the interpretation of the imaginary energy states, the referee is correct that we do not provide a very compelling interpretation for them. We do not, however, find this to be a vital problem, because we do not believe that the theory ultimately makes sense in this range of parameter space. Concretely, we adhere to the point of view of 1312.2021, according to whom it is inconsistent to place the deformed theory on a circle for $\mu < 0$, because the superluminal modes in compact space give rise to the same contradictions one would encounter if spacetime had closed timelike curves\footnote{For similar reasons, the $\mu > 0$ theory cannot be placed on a circle of size smaller than $\sqrt{\mu c}$. The fact that observables such as the finitesize energy still match precisely between field theory and gravity for $\mu < 0$ simply appears to indicate that our methods for evaluating this observable in the bulk theory have the same range of validity as the Burger's equation used for evaluating them in the boundary theory, both of which are unable to capture the physics of these modes. It would be very interesting to find the physics that these imaginary energy states are a signal of, perhaps an instability. It is possible that they are resolved if one considers the full $AdS_3 \times X$ background of string theory, rather than its truncation to $3d$ gravity.
We would like to point out in passing that the proposal of McGough et al.\ does not resolve the problem with the imaginary energy states, either. The reason is that, as we show in this paper, their proposed dictionary fails to reproduce the correct energy of the state for any bulk configuration that contains matter fields in addition to gravity. In particular, due to the fact that a black hole is the most massive object one can fit in a given size, their sharp radial cutoff proposal would in general be excluding states whose energy is not yet complex. One may try to salvage this proposal by claiming it should only apply to a CFT that is dual to pure gravity in the bulk, but it is not clear whether such a (modularinvariant) CFT exists and, even if it does, this does not, in our opinion, solve the general problem of imaginary energy states, which is present for any CFT.
It is indeed interesting to note that energy levels with $E < 0$ at $\mu = 0$ can become imaginary for positive values of the deformation parameter. A prominent family of examples are empty AdS and the conical deficit spacetimes. By equating the conical deficit (or smoothness) of the deformed geometry to the conical deficit in the original metric, we can derive the relation between $\mathcal{L}_\mu$ and $\mathcal{L}_0$. It takes the same form as equation (3.16) found before. We have added this new calculation on page 12. The boundary metric $\gamma^{[0]}$ features closed timelike curves for certain negative values of $\rho_c$, in exact agreement with the expectation from the $T \bar T$deformed energy spectrum.
Unlike the negative $\mu$ theory, the positive $\mu$ one leads to time delays and is expected to make sense as a physical theory. The complex value of the vacuum energy is simply identified with the tachyonic instability in the choice of vacuum in string theory. It would be very interesting indeed to find a resolution for this instability in the gravity picture.
List of changes
We have made the following changes with respect to version 1 of the manuscript:
 added references to relevant work in the introduction;
 added a comparison to the article 1809.09593 by Conti, Negro and Tateo on page 6;
 added a discussion of empty AdS and conical defect spacetimes, for which $\mathcal{L} = \bar{\mathcal{L}} < 0$ on page 12;
 corrected a typo in equations 3.34 and 3.35;
 added a note on page 18 addressing a subtlety that was recently addressed in 2011.05445;
 clarified the discussion of the Virasoro algebra on page 18.