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

We would like to thank the referees for their thorough reading of our manuscript and for their comments. Here are the answers to their questions.

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 field-dependent 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 finite-size 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 (modular-invariant) 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.