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Exact multi-instantons in topological string theory

by Jie Gu, Marcos Mariño

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Submission summary

Authors (as registered SciPost users): Jie Gu · Marcos Mariño
Submission information
Preprint Link: scipost_202309_00009v1  (pdf)
Date accepted: 2023-09-20
Date submitted: 2023-09-11 03:24
Submitted by: Gu, Jie
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Mathematical Physics
Approach: Theoretical


Topological string theory has multi-instanton sectors which lead to non-perturbative effects in the string coupling constant and control the large order behavior of the perturbative genus expansion. As proposed by Couso, Edelstein, Schiappa and Vonk, these sectors can be described by a trans-series extension of the BCOV holomorphic anomaly equations. In this paper we find exact, closed form solutions for these multi-instanton trans-series in the case of local Calabi-Yau manifolds. The resulting multi-instanton amplitudes turn out to be very similar to the eigenvalue tunneling amplitudes of matrix models. Their form suggests that the flat coordinates of the Calabi-Yau manifold are naturally quantized in units of the string coupling constant, as postulated in large $N$ dualities. Based on our results we obtain a general picture for the resurgent structure of the topological string in the local case, which we illustrate with explicit calculations in the case of local $\mathbb{P}^2$.

Published as SciPost Phys. 15, 179 (2023)

List of changes

- Page 6, line 4: "the F_gs" => "the free energies F_g".
- Page 7, line 1 below (3.7): "trans-series part" => "instanton amplitudes".
- Page 9, immediately above (3.24): "by using (2.14)" => "by using the relation between propagators (2.14)".
- Page 11, below (3.40): the original equation (3.72) is moved up and is now given in (3.41), as suggested in Point 5 of Report 2.
- Page 12: eqs. (3.50) and (3.51) are added to clarify the vanishing of the right hand side of (3.49) (originally (3.48)), in response to Point 6 of Report 2.
- Page 15, immediately below (3.83): "satisfy (2.22)" => "satisfy the master equation (2.22)".
- Page 16, in (3.90): The bound of summation in $k$ is added.
- Page 17, in (3.98): The bound of summation in $k$ is added.
- Page 22, in (3.141): A factor of $i$ is added on the right hand side.
- Page 23, immediately above (3.150): "trans-series" => "trans-series sectors".
- Page 26, in Figure 1: The branch point in the positive imaginary axis in the right plot is highlighted by a black dot. In the last line of the caption, "\ell = 2,3,4" => "\ell=1,2,3,4". This is in response to Point 11 of Report 2.
- Page 27, line 2 in the caption of Figure 2: "the F_gs" => "the free energies F_g".

Reports on this Submission

Anonymous Report 2 on 2023-9-12 (Invited Report)


The authors took into consideration my (minor) suggestions, and changed the draft accordingly. My opinion is that the work in its current form is published.

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Anonymous Report 1 on 2023-9-11 (Invited Report)


The proposed minor changes have been implemented. I recommend publication.

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