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Comments on a stateoperator correspondence for the torus
by Alexandre Belin, Jan de Boer, Jorrit Kruthoff
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Submission summary
Authors (as registered SciPost users):  Alexandre Belin · Jorrit Kruthoff · Jan de Boer 
Submission information  

Preprint Link:  https://arxiv.org/abs/1802.00006v2 (pdf) 
Date accepted:  20181129 
Date submitted:  20181016 02:00 
Submitted by:  Kruthoff, Jorrit 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We investigate the existence of a stateoperator 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 HorowitzMyers conjecture for the AdS soliton.
Author comments upon resubmission
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 stresstensors 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 ReehSchlieder 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 ReehSchlieder theorem has not been proved in curved space.
List of changes
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 stateoperator correspondence in the discussion section on page 27.
2. Point 2 of referee 1: we extended the section on the HorowitzMyers 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 stateoperator 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 twotorus. 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 stateoperator 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.
Published as SciPost Phys. 5, 060 (2018)