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Comments on a stateoperator correspondence for the torus
by Alexandre Belin, Jan de Boer, Jorrit Kruthoff
This is not the latest submitted version.
This Submission thread is now published as SciPost Phys. 5, 060 (2018)
Submission summary
As Contributors:  Alexandre Belin · Jorrit Kruthoff · Jan de Boer 
Arxiv Link:  http://arxiv.org/abs/1802.00006v1 (pdf) 
Date submitted:  20180321 01:00 
Submitted by:  Kruthoff, Jorrit 
Submitted to:  SciPost Physics 
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.
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Submission & Refereeing History
Published as SciPost Phys. 5, 060 (2018)
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Reports on this Submission
Anonymous Report 2 on 201865 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1802.00006v1, delivered 20180605, doi: 10.21468/SciPost.Report.489
Strengths
1 The author investigate an interesting question, addressing a very relevant and modern subject.
2 The approach is original and can lead to further progresses in the field.
3 The discussion is organised in increasing complexity, first explaining a simple example and then proceeding with the general argument.
Weaknesses
1 The rigour of the discussion is a bit low and not up to the subject.
2 The authors could not produce a general proof of their argument and leave the main point of the paper as a conjecture.
Report
The manuscript investigates the existence of a statesoperators correspondence for Conformal Field Theories defined on a more general manifold than the flat one, namely $S_d\times \mathds R$. In particular they focus on manifolds whose spacial slices are two dimensional tori.
As preliminary step, the author explore the possibility to create the vacuum state. They assume that the vacuum state is created by a Path Integral over a compact manifold with no insertion of operators. Despite the assumption is natural, it is not justified.
Next, an example is considered: the half twosphere times a circle. In this case the authors manage to obtain (under a few assumptions) a relation between the thermal energy and the Casimir energy of the ground state. They then show in two realizations, the free boson and holographic CFTs that this two quantities differ. They conclude that this particular manifold doesn't admit a statesoperators correspondence.
Despite the examples are quite instructive, they lack of rigour. In particular the authors do not show that all the assumptions made in the previous section are fulfilled.
The subsequent part of the paper is an attempt to disprove the existence of the state operator correspondence for a generic manifold with boundary $T_2$. The authors manage to prove that if a manifold satisfies certain properties (existence of a conformal killing vector normal to the torus) then it can create the vacuum state. They also point out that that the only manifold with this properties is noncompact and therefore is not a good candidate according to their original assumptions.
The other direction of the theorem is instead left as a conjecture. This is unfortunate since it is the most important part.
In conclusion, the work represents an original attempt to solve a very important and actual issue. It clarifies the problem and reduces it to a geometrical one.
The methods presented can spur further developments in the field.
After a minor revision, I think the work will qualify for a publication on SciPost.
Requested changes
Before publication I would like to see addressed the following points:
1 Comment on the assumptions of compactness of the manifold and insertion of operators in the Path Integral: why these are valid assumptions to create a the vacuum state and what happens if one relaxes them.
2 Show explicitly that the two examples presented satisfy all the assumptions made in Section 3 and comment on why these examples do not work.
3 Find a minimal set of assumption that allow to prove the second part of the theorem in Section 4.
Anonymous Report 1 on 201852 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1802.00006v1, delivered 20180502, doi: 10.21468/SciPost.Report.437
Strengths
1) Original analysis on a not so explored important subject
Weaknesses
1) Methodology not rigorous
2) Inconclusive results
Report
Great progress has been achieved in the last years in the study of CFTs using the conformal bootstrap.
This method uses crossing symmetry to constrain the local data of a CFT, the spectrum of local primary operators and their OPE coefficients.
However, other constraints are expected to arise by demanding consistency of the CFT when defined on other manifolds.
A notable example of this sort is modular invariance in 2d CFTs on tori. In contrast to the twodimensional case, not much is known about the nature
of such constraints for CFTs in $d>2$.
Motivated by these considerations, the authors investigate the existence of a stateoperator correspondence in CFTs defined on manifolds of the form
$T^{d1}\times R$, generalizing the known correspondence on the cylinder $S^{d1}\times R$. The authors did not reach a definite answer, but provide
some evidence for a negative one, through examples and heuristic arguments. They also point out a connection between their arguments and a conjecture
by Horowitz and Myers, ref.[12].
The paper analyzes important and not so much explored ``global" aspects of CFTs. The paper is well written, though it is sometimes a bit sloppy.
The final results are inconclusive, but the arguments provided might be interesting enough for publication in Scipost.
Before publication, however, the authors should improve their presentation in two aspects:
1) The implications of the alleged nonexistence of the stateoperator correspondence for tori to the possible global constraints in CFTs  the starting motivation of the paper  should be better and more extensively explained.
2) The explanation of the connection of the paper with the HorowitzMyers conjecture, limited to a few lines at the end of page 23, should be expanded and improved.
Requested changes
See report
Jorrit Kruthoff on 20180407 [id 237]
Please note the following typo in the text: In the second line below eq. 3.10 we say 'In what follows, it will be convenient to make the smaller circle contractible...', but this should be 'the larger circle contractible...'.