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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
Submission information
Preprint Link: scipost_202305_00007v1  (pdf)
Date accepted: 2023-05-30
Date submitted: 2023-05-05 08:23
Submitted by: Gu, Jie
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Mathematical Physics
Approaches: Theoretical, Computational


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 so-called Nekrasov-Shatashvili limit. We consider the trans-series extension of the holomorphic anomaly equations satisfied by these quantities, and we obtain exact multi-instanton solutions for these trans-series. Building on this result, we propose a unified perspective on the resurgent structure of quantum periods. We show for example that the Delabaere-Pham 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 double-well in quantum mechanics, and for the quantum mirror curve of local $\mathbb{P}^2$.

List of changes

1. Page 2 third paragraph: After "NS free energies and Wilson loops",
"generic results given in (3.79), (3.92) and special results with
boundary conditions relevant for resurgent structure in (3.105),
(3.106)." is added. After "the resurgent structure of the quantum
periods", "given in (3.139)." is added.

2. Eqs. (3.79), (3.92), (3.105), (3.106), (3.118), (3.139) boxed.

3. Page 5 below (2.20): Explanation on $t$ added "where $t$ is the flat
coordinate or the classical A-period".

4. Page 5 last line: "w.r.t." removed.

5. Page 12 below (3.25): $D_S$ changed to $\partial_S$.

6. Page 13 below (3.44): "power series in $y$" changed to "power
series in a formal independent variable $y$"

7. Page 14 below (3.48): "the term of order $y^n$ in the formal power
series $f(y)$" changed to "the coefficient of the term $y^n$ in the
formal power series $f(y)$".

8. Page 19 second line: "From (3.113)" changed to "From (3.82)".

9. Page 20 at the end of third line: "for any integers $n, m$" added.

10. Page 20 below (3.104) after "We will denote the corresponding
solutions by F^(n|m)_l, w^(n|m)_l.": "Note that here and in the
following, whenever we are in the pure instanton sector we will
revert to the notation F^(n), w^(n) to reduce the clutter." added.

11. Page 21 below (3.109): "where we have taken into account (3.27)."
changed to "where we have used (3.42) and also taken into account

12. Page 22 first paragraph in section 3.3: After the first sentence,
"The Borel resummations of quantum periods differ across a Stokes
ray that issue from the origin and pass through some Borel
singularities. The difference between these Borel resummations is
given by the Stokes coefficient times the Borel resummation of the
transseries associated to the singularities." added.

13. Page 23 first line of eq. (3.124): Additional minus sign on the
right hand side added.

14. Page 23 below (3.124): "where we used $F^{(\ell)}_\ell(−\hbar) =
F^{(0|\ell)}_\ell(\hbar), \omega^{(\ell)}_\ell(-\hbar) =
\omega^{(0|\ell)}_\ell(\hbar)$." added.

15. Page 24 above (3.130): Reference 61 is added.

16. Page 35 in *Acknowledgements": After the first sentence "We would
also like to thank one of the SciPost referees for many useful
suggestions and stimulating questions." added.

Published as SciPost Phys. 15, 035 (2023)

Reports on this Submission

Anonymous Report 2 on 2023-5-19 (Invited Report)


I believe that the authors made the required improvements to their work. In particular, the boxed equations now provide a good visual aid to separate main results from derivations, and definitely help the readability of the paper. I would also like to thank the authors for clarifying my queries.

It is my opinion that this work should be published in SciPost in its current form.

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Anonymous Report 1 on 2023-5-5 (Invited Report)


I would like to thank the authors for the improvements they have implemented. I can suggest the paper for publication on SciPost.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

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