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

Preprint Link:  scipost_202305_00007v1 (pdf) 
Date accepted:  20230530 
Date submitted:  20230505 08:23 
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 $\mathbb{P}^2$.
Published as SciPost Phys. 15, 035 (2023)
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 Aperiod".
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^(nm)_l, w^(nm)_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
(3.27)."
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.
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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.