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On the resurgent structure of quantum periods
by Jie Gu, Marcos Mariño
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Submission summary
Authors (as registered SciPost users):  Jie Gu · Marcos Mariño 
Submission information  

Preprint Link:  https://arxiv.org/abs/2211.03871v3 (pdf) 
Date submitted:  20230317 04:30 
Submitted by:  Gu, Jie 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
Quantum periods appear in many contexts, from quantum mechanics to local mirror symmetry. They can be described in terms of topological string free energies and Wilson loops, in the socalled NekrasovShatashvili limit. We consider the transseries extension of the holomorphic anomaly equations satisfied by these quantities, and we obtain exact multiinstanton solutions for these transseries. Building on this result, we propose a unified perspective on the resurgent structure of quantum periods. We show for example that the DelabaerePham formula, which was originally obtained in quantum mechanical examples, is a generic feature of quantum periods. We illustrate our general results with explicit calculations for the doublewell in quantum mechanics, and for the quantum mirror curve of local $\mathbb{P}^2$.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2023420 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2211.03871v3, delivered 20230419, doi: 10.21468/SciPost.Report.7077
Strengths
1. General closedform (exact) results for the multiinstanton sectors of the transseries of the NS free energies and Wilson loop vevs. This provides a much more complete picture of their nonperturbative completion, as compared to previous work.
2. The authors present the derivations of their computations in a lot of detail. In general they specify when a particular result is conjectured and not derived.
3. The resurgent analysis of the multiinstanton transseries provides a direct study of the DelabaerePham formula, which is then then thoroughly confirmed in a quantum mechanical and a quantum mirror curve example.
Weaknesses
1. Notation is not consistent throughout, specially after introducing the negative action instantons. The technical nature of the paper makes it difficult to follow if different notation is used for the same object.
2. In parts of the work, specially in section 3, it is difficult to separate the main results from their computations/derivations.
3. It is sometimes difficult to understand when the authors are focusing on particular cases (such as specific boundary conditions) or general cases.
Report
This paper presents a thorough study of the resurgent properties of quantum periods associated to quantum curves of genus one. This is achieved via the analysis of the holomorphic anomaly equations (HAEs), which can be used to obtain the NS free energies and the Wilson loop vevs. These are obtained as asymptotic expansions, then upgraded to transseries, which define implicitly the quantum periods. Section 2 presents a brief introduction to quantum curves and the HAEs, which are the main building blocks to be used in the computations of Section 3. It also presents a brief summary of the work. Section 3 then includes a very detailed analysis of the nonperturbative completion of the NS free energies and Wilson loop vevs, with closed form for general instanton number thoroughly analysed and derived. The resurgent structure of these quantities is then analysed/conjectured. Finally a transseries and its resurgent properties is obtained for the quantum periods. Section 4 is devoted to the study of two examples, the quantum mechanical double well, and the mirror curve of local \mathbb{P}^{2}, which corroborates the results obtained in the previous sections. The paper ends with a summary of the work and some open questions.
The paper is generally well written and clear, well referenced and very detailed, and without any major weaknesses. Although this work comes as a companion paper to another work from the same authors, it is my opinion that this work stand perfectly well on its own, and includes novel analysis of the quantum periods, which will certainly be useful for different quantum problems/different curves.
It is my opinion that this paper meets the expectations and generally all the acceptance criteria, and it should be published in SciPost after some minor revision. In order to be published I believe the authors need to review the consistent use of their notation, and perhaps separate their main results from the corresponding derivation, as at the moment their main results are buried sometimes in the middle of the computations and are difficult to find.
Requested changes
There are somechanges that I would request the authors take into consideration. I also have some questions for the authors, which can be found below.
1. t appears in equation (2.20) but it is not defined until later. There should be some mention of what it is here.
2. there is a typo on the last line of page 5: I believe that "w.r.t" should not be there.
3. Keeping track of the different types of S used (propagator, its holomorphic limit and Stokes coefficients) is sometimes difficult.
4. On page 12, first line after eq (3.25), what is D_S?
5. In the computations of pages 1317, y is treated as an independent variable which is then taken to be Sigma(C). Is this correct? The variable y is not clearly explained in this proof, and the authors do not explicitly state its final value in some of the derivations.
6. On the second line of page 19 I think the wrong equation is being referenced.
7. On the third line of page 20, for which n,m is the condition valid?
8. The superscript in (3.105) and (3.106) seems to have reverted back to the notation prior to introducing the negative action instantons, but this is not clear. I suggest the authors review the notation they use throughout to be consistent, and if they do need to revert back to previously used notation, they should state it explicitly.
9. Is the condition on eq (3.108) still true if one includes both types of instantons (and not only the (\ell0) ones)?
10. Below equation (3.109) the authors mention using (3.27), but haven't they used either (3.92) or (3.42)?
11. When generally calling an equation that is on a previous page, to improve readability quite a lot it would be very useful to state in very few words what that equation is.
12. Is the resurgent analysis in section 3.3 for the perturbative series alone (meaning that it related the perturbative series with nonperturbative sectors, but the relation between nonperturbative sectors is not given)?
13. What happens at alpha=0, does the perturbative series truncate or is it still asymptotic?
14. Would it be possible to include more information on where (3.124) comes from?
15. The results (3.122) and (3.124) are different from the alien derivatives expected for a two parameter transseries, why is this the case?
16. The resurgent analysis in pages 23 and 24 seems to be related to the resurgence of an implicit function (see Delabaere's paper on this subject), is this the case? Can these results be written with that in mind?
17. It is often not clear when the choice (3.104) was taken and when then results are more general. I suggest the authors review this and explicitly state the generality of the results.
Report #1 by Anonymous (Referee 1) on 2023412 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2211.03871v3, delivered 20230412, doi: 10.21468/SciPost.Report.7041
Strengths
1. The computations are thorough.
2. The paper is clearly written.
Weaknesses
1. Very technical
2. Some features of the results are obscured.
Report
Nonperturbative completions to physical observables are notoriously both hard to compute and of key importance in understanding global, e.g. monodromy, properties for physical observables.
In this work the authors consider the resurgence structure of quantum periods, which can be thought of as the quantization of classical periods associated with integrals of usual differentials on curves. Two specific cases are considered here: quantum periods associated with WKB quantum mechanics and the quantum mirror curve associated with $\mathbb{P}^2$.
This is a companion paper to 2211.01403, by the same authors, where more emphasis is put on the topological string side of the story and more details are provided.
A unifying picture is obtained from studying the holomorphic anomaly equation (HAE) satisfied by the NS free energy and the Wilson loop vevs (sort of ''generating functions'' for quantum periods).
Once this system of equations is set into place, the authors are able to construct a multiinstanton transseries solution for both the NS free energy and the Wilson loop.
By computing the socalled Alien derivative on their conjectured transseries coefficients, the authors are able to reproduce the DelabaerePham formula showing how general this result is, following essentially from the resurgence structure of multiinstantons.
Finally, the authors provide many convincing evidences that their conjectured transseries expansion is indeed correct by numerically studying the cases of the double well potential in QM and the local $\mathbb{P}^2$.
The paper is clearly written and most certainly deserves publication although I wonder whether it would be more suitable for a different journal, for example Annales Henri Poincar\'e.
Requested changes
1. I found it rather difficult to pin down the main key equations/results of this paper. It would be perhaps helpful for the reader to present already at the level of the introduction what the key results are, e.g. the multiinstanton transseries expansion for the NS free energy and Wilson loop and the general DelabaerePham formula.
2. Secondly, I am afraid section 3 was rather hard to follow. It is a very technical yet necessary section, since it contains all the meat for the calculation, however, I wonder whether it would be best to relegate some of the most technical points to an appendix.
3. Finally, it might be useful to remind the reader unfamiliar with the Stokes automorphism, what this object actually does, i.e. it connects across two different Stokes wedges.
 We will box the important formulas and refer to these formulas in the introduction.
 After boxing the results we hope that this is clearer and more readable.
 We will add some explanation in the beginning of section 3.3.
Author: Jie Gu on 20230425 [id 3613]
(in reply to Report 2 on 20230420)