Comments on a state-operator correspondence for the torus

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.

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.

The manuscript investigates the existence of a states-operators 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 two-sphere 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 states-operators 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 non-compact 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.

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.

1) Original analysis on a not so explored important subject

1) Methodology not rigorous
2) Inconclusive results

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 two-dimensional 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 state-operator correspondence in CFTs defined on manifolds of the form
$T^{d-1}\times R$, generalizing the known correspondence on the cylinder $S^{d-1}\times R$. 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 non-existence of the state-operator 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 Horowitz-Myers conjecture, limited to a few lines at the end of page 23, should be expanded and improved.

See report