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|As Contributors:||Jorrit Kruthoff · Jan de Boer|
|Submitted by:||Kruthoff, Jorrit|
|Submitted to:||SciPost Physics|
|Subject area:||High-Energy Physics - Theory|
We investigate the existence of a state-operator correspondence on the torus. This correspondence would relate states of the CFT Hilbert space living on a spatial torus to the path integral over compact Euclidean manifolds with operator insertions. Unlike the states on the sphere that are associated to local operators, we argue that those on the torus would more naturally be associated to line operators. We find evidence that such a correspondence cannot exist and in particular, we argue that no compact Euclidean path integral can produce the vacuum on the torus. Our arguments come solely from field theory and formulate a CFT version of the Horowitz-Myers conjecture for the AdS soliton.
We thank the referees for the interesting comments. We agree with the referees that not all aspects of our discussion are completely rigorous, which is precisely why we were not able to produce a proof (or theorem) that a state-operator correspondence cannot exist for the torus. Formulating actual theorems in quantum field theory is notoriously difficult so it is perhaps not so surprising. Nevertheless, we believe we have made completely explicit what we were able to prove and what only constitutes evidence towards a possible proof.
We have added two paragraphs before section 2.1 to further discuss why we consider compact Euclidean manifolds as well as a discussion of the assumption of considering manifolds without operator insertions. It is true that we were not able to prove that such configurations are even local minima of the energy functional. In fact, we believe that they are not global minima of the energy functional and they cannot prepare the vacuum on the torus. The vacuum is most likely only prepared by the half infinite line times the torus, but one can presumably get very close to the vacuum by considering a hemisphere with many stress-tensors inserted near the south pole, effectively creating a long Euclidean throat.
In connection to this, we added a paragraph in the discussion connecting our approach to the Reeh-Schlieder theorem, as an attempt to see how one can compare the approach in this paper to one of the few proven theorems in field theory. It is interesting to note that even the Reeh-Schlieder theorem has not been proved in curved space.
We have modified the specific points addressed by the referees in the following way:
1. Point 1 of referee 1: we added a discussion of the implication of the absence of a state-operator correspondence in the discussion section on page 27.
2. Point 2 of referee 1: we extended the section on the Horowitz-Myers conjecture.
3. Point 1 of referee 2: we added two paragraphs above section 2.1 on page 5 discussing these assumptions. We would like to emphasize again that a compact manifold is a necessary ingredient for a state-operator correspondence. The boundary condition at t = −∞ must be replaced by a lower codimension boundary condition. On the other hand, the assumption about not adding any operators to the path integral was a starting assumption. We do not believe that such configurations are global (or even local) minima of the energy functional.
4. Point 2 of referee 2: we did not make any assumptions in this section, we simply computed the expectation value of the energy in the hemisphere state for two theories: the free boson and a holographic CFT. In both cases, we found that the energy is always greater than the known vacuum energy on the two-torus. This shows that the hemisphere state is not the right candidate to produce the vacuum in all CFTs.
5. Point 3 of referee 2: this is a hard question. We failed to provide a necessary set of conditions such that there does not exist a state-operator correspondence. Providing sufficient conditions is much easier, and we gave such an example in section 4.4. If we assume sufficiently nice analytic properties of the correlation functions, then one naturally obtains a conformal killing vector on the Euclidean section. Nice analytic properties of the Lorentzian correlator thus provides a sufficient condition to prove the other direction of the theorem. However, we would like to emphasize that we do not know whether it is natural to impose such a nice analytic property of the correlator, since their analytic properties can be quite subtle in quantum field theory. It would be very interesting to try to understand this question in more detail.
The authors have improved their presentation as requested.
I think that the revised manuscript can be published in Scipost.