• The maximal genus to which perturbative free energies are available in compact models is not sufficient to check the non-trivial multi-instanton amplitudes presented in section 5.6. However, analogous results for local Calabi-Yau manifolds are testable [23], as the topological string amplitudes can be computed to higher genus in these cases.

• Although we will mainly focus on testing solutions of instanton amplitudes in the instanton sectors of action $X^1$ and $P_0$, the results derived in section 5 should apply generally. Once an instanton sector of action $\mathcal{A}$ has been identified (e.g. by studying the Borel singularities of $\mathcal{F}^{(0)}$), the first conjecture in section 5.7 implies that we can find a frame $\Gamma_A$ in which $\mathcal{A}$ is an A-period; the second conjecture in section 5.7 then dictates that the instanton amplitude in frame $\Gamma_A$ is of the simple form (312) and (313). We can then use the results of sections 5.5 and 5.6 to write down the instanton amplitudes in any other frame.

• Both the evaluation of perturbative and instanton free energies depend on the choice of frame. Nevertheless, our numerical studies seem to indicate that the resurgent structure of topological string is independent of the choice of frame. In particular, this means that the location of Borel singularities as well as the associated Stokes constants do not seem to change across different frames, at least as far as the leading Borel singularities, which are accessible by numerics, are concerned.

• When we are not in an $\mathcal{A}$-frame, the instanton amplitude $\mathcal{F}^{(\ell)}$ in the $\ell \mathcal{A}$ instanton sector itself contains a nontrivial power series and we can apply the same resurgent analysis as before: looking for Borel singularities and studying the associated higher instanton amplitudes. On the other hand, as already pointed out in [23] in the case of local Calabi-Yau manifolds, the instanton amplitudes are expressed completely in terms of the perturbative free energy, and consequently the resurgent structure of the former can be deduced from the latter. To discuss this point, it is convenient to use the language of alien derivatives. The formula (343) for the Stokes discontinuity suggests

$$ \dot{\Delta}_{\ell, \mathcal{A}} \mathcal{F}^{(0)}(g_s) = s_{\ell, \mathcal{A}} \mathcal{F}_{\ell}^{(\ell)}(g_s), \qquad (344) $$

where the constants $s_{\ell, \mathcal{A}}$ can be derived from the Stokes constants $S_{\ell, \mathcal{A}}$. The subscript notation $\mathcal{F}_{\ell}^{(n)}$ used on the RHS was introduced below (313) to specify that the instanton amplitude is computed with the boundary condition (313). Then, for the full family of trans-series $\mathcal{F}_{\ell}^{(n|m)}$, we conjecture that

$$ \dot{\Delta}_{\ell, \mathcal{A}} \mathcal{F}_{\ell}^{(\ell|n|\ell m)} = s_{\ell, \mathcal{A}}(n+1) \mathcal{F}_{\ell}^{(\ell(n+1)|\ell m)}(g_s). \qquad (345) $$

This should follow directly from (344) by taking derivatives. The derivation of the case $\ell = 1$ goes through exactly as in [23].

### Another note on normalization conventions:

The numerical calculations performed in this section require choosing a normalization for the periods, as well as a normalization of the topological string amplitudes, determined by the choice of constant $\alpha$ introduced in (143). We will indicate the normalization of the periods by citing the constant term of the holomorphic period at the MUM point. E.g., for the normalization chosen in (79), this constant is $(2\pi i)^3$. The values cited in this section are in terms of a proportionality constant $\kappa$. We performed our computations with a normalization $(2\pi i)^{3/2}$ of the periods (this is the normalization for which the triple intersection number $C_{zzz}$ has no factors of $2\pi i$) and $\alpha^2 = 1$; for this choice, $\kappa$ coincides with the value cited in (177). It is however equally possible to relate our results to a different normalization of the periods, by adjusting both $\alpha^2$ and $\kappa$. Two additional choices are indicated in table 5.
<table><caption>Table 5: The normalization constant $\aleph$ appearing in the numerical results cited in this section as a function of the normalization of the periods and of the topological string amplitudes $F_g$.</caption><thead><tr><th>normalization of periods</th><th>α<sup>2</sup></th><th>κ</th></tr></thead><tbody><tr><td>1</td><td>(2πi)<sup>-3</sup></td><td>i(2πi)<sup>2</sup></td></tr><tr><td>(2πi)<sup>3/2</sup></td><td>1</td><td>i√2πi</td></tr><tr><td>(2πi)<sup>3</sup></td><td>(2πi)<sup>3</sup></td><td>1/(2π)</td></tr></tbody></table>

## 6.1 Mapping out the Borel plane

In this subsection, we investigate the Borel singularities of the perturbative free energy $\mathcal{F}^{(0)}$ evaluated in different frames and at different points of moduli space.

We first consider the vicinity of the MUM point $z = 0$. For the perturbative free energy $\mathcal{F}^{(0)}(X^0, X^1)$ evaluated in the MUM frame, the discussion in section 5.7.1 predicted Borel singularities at<sup>23</sup>

$$ \pm \kappa \ell X^0, \quad \pm \kappa (dX^1 + nX^0), \quad \ell, d \ge 1, n \in \mathbb{Z}. \qquad (346) $$

The first few of these, in the case of the quintic, are nicely visible in the plot reproduced in Figure 4 (left). In particular, we find the branch cuts due to $\pm \kappa X^0$ along the positive and negative imaginary axes, and the two towers of Borel singularities due to $\kappa (\pm X^1 + ZX^0)$, forming a peacock pattern already observed in local Calabi-Yau manifolds [80,84]. The peacock pattern becomes more prominent if we subtract the dominant constant map contributions from the free energies. The result is plotted in Figure 4 (right). On the other hand, the Borel singularities $\kappa \ell X^0$ with higher $\ell$ are hidden behind the branch cuts of $\ell = 1$. But we can uncover the first few by combining the Padé approximant of the Borel transform with a conformal map

$$ \zeta = \frac{2A\xi}{i(1-\xi^2)}. \qquad (347) $$

See for example [95] for a summary of this type of techniques in Borel analysis. The resulting plot, shown in Figure 5, displays clearly the singularities with $\ell = 2, \dots$ in the $\xi$-plane.

We can also evaluate the free energies in other frames, for example in the conifold frame defined by $(X^1, P_0)$, or frames that are not associated to special points in the moduli space, such as e.g. the frame defined by specifying as A-periods $(P_0, P_1)$. The Borel singularities of $\mathcal{F}^{(0)}(X^1, P_0)$ and $\mathcal{F}^{(0)}(P_0, P_1)$ near the MUM point are plotted in Figure 6. We find that the positions of the visible singularities coincide with those of $\mathcal{F}^{(0)}(X^0, X^1)$.

Next, we consider loci in moduli space close to the conifold point $z = \mu$. Figure 7 (left) shows a plot of the Borel singularities of $\mathcal{F}^{(0)}(X^1, P_0)$. The only singular points that are visible are the branch points at

$$ \pm \kappa P_0, \qquad (348) $$

consistent with the asymptotic analysis in section 5.7.2 based on the gapped form (136) of the topological string amplitudes $\mathcal{F}_g$ close to the conifold locus. Figure 7 (right) shows the Borel singularities of $\mathcal{F}^{(0)}(X^1, P_0)$ upon subtracting the leading poles before the gap to obtain the regularized free energies

$$ \mathcal{F}_g^{\text{reg}}(X^1, P_0) = \mathcal{F}_g(X^1, P_0) - \frac{(-1)^{g-1} B_{2g}}{2g(2g-2)} \left( \frac{(2\pi i)^{1/2}}{P_0} \right)^{2g-2}, \quad g > 1. \qquad (349) $$

<footnote n="23"><sup>23</sup>Due to the symmetry of perturbative free energies under $g_s \to -g_s$, Borel singularities $\pm A$ always appear in pairs.</footnote>
<figure><img src="image_7.png" /><figcaption>Figure 4: The location of the poles of the Padé approximant to $\hat{F}^{(0)}(X^0, X^1)$ (the hat indicates the Borel transform), evaluated to order $g = 64$ at $z = 10^{-2}\mu$. On the right, the leading constant map contribution is subtracted. The black dots correspond to the position of the periods $\mathfrak{C}(mX^0 + nX^1)$, $(m, n) = (1, 0)$ (on the imaginary axis), $(0, 1), (1, 1), (2, 1), \dots$.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 5: The location of the poles of the Padé approximant to $\hat{F}^{(0)}(X^0, X^1)$ as a function of $\zeta$, evaluated to order $g = 61$ at $z = 10^{-3}\mu$, mapped to the $\xi$-plane by the conformal map (347). The black dots correspond to the position of the periods $\mathfrak{C}mX^0$, $m = 1, 2, 3$. The red dots inside a circle of radius one are mappings of the singularities from the two towers.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 6: The location of the poles of the Padé approximant to $\hat{\mathcal{F}}^{(0)}(X^1, P_0)$, on the left, and $\hat{\mathcal{F}}^{(0)}(P_0, P_1)$, on the right, evaluated to order $g = 64$ at $z = 10^{-2}\mu$. The black dots correspond to the position of the periods $\Re(mX^0 + nX^1)$, $(m, n) = (1, 0)$ (on the imaginary axis), $(0, 1)$, $(1, 1)$, $(2, 1)$, ...</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 7: The location of the poles of the Padé approximant to $\hat{\mathcal{F}}^{(0)}(X^1, P_0)$, without (left) and with (right) the leading singularity removed, evaluated to order $g = 61$ at $z = (1 - 10^{-6})\mu$. In the left diagram, the black dot is at the position of the period $\Re P_0$. In the right diagram, the black dots are at the position of the periods $\Re X^0$, $\Re X^1$, $\Re(X^0 + X^1)$.</figcaption></figure>

Compared to the figure on the left, additional singularities located at

$$ \pm \Re X^0, \quad \pm \Re X^1, \quad \pm \Re(X^0 \pm X^1), \qquad (350) $$

become visible. Their location cannot be predicted analytically. We will study these additional singular points in section 6.2.

Similarly, we can evaluate the free energies in other frames, for instance in the MUM frame defined by the designation of A-periods $(X^0, X^1)$ or the frame defined by the designation of A-periods $(P_0, P_1)$. The Borel singularities of $\mathcal{F}^{(0)}(X^0, X^1)$ and of $\mathcal{F}^{(0)}(P_0, P_1)$ close to the conifold point are plotted in Figure 8. They coincide with the Borel singularities in the conifold frame close to this point, plotted in Figure 7 (left).

We note that all three frames considered exhibit the same visible Borel singularities both close to the MUM point and close to the conifold point. As far as the leading singularities are concerned, the reason for this universality can perhaps be traced to the fact that the dominant contribution to the topological string amplitudes in both of these regions is due to the
<figure><img src="image_11.png" /><figcaption>Figure 8: The location of the poles of the Padé approximant to $\mathcal{F}^{(0)}(X^0, X^1)$, on the left, and $\mathcal{F}^{(0)}(P_0, P_1)$, on the right, evaluated to order $g = 64$ at $z = \frac{5}{6}\mu$. The black dots correspond to the position of the period $\aleph P_0$.</figcaption></figure>

holomorphic ambiguity $f_g$. In Figure 9, we plot the singularities of the Borel transform of

$$ \mathcal{F}^{\text{hol}}(g_s) = \sum_g f_g g_s^{2g-2}, \qquad (351) $$

close to the conifold and the MUM point. Indeed, the dominant Borel singularities coincide with those in the three frames considered above.

Before ending this subsection, we comment on an interesting observation regarding the monodromy action on the Borel singularities. As we have seen, the Borel singularities of the perturbative free energies are given by geometric periods of the Calabi-Yau manifolds. The periods are known to enjoy monodromy actions around singular points of the moduli space. For example, in the case of the quintic, if we start from some locus in the moduli space, circle around the MUM point and come back to the original locus, the periods transform by (we omit here the transformation of the other two periods, which are also non-trivial)

$$ X^1 \to X^1 + X^0. \qquad (352) $$

Therefore, near the MUM point the peacock pattern of Borel singularities arises by consistency: as long as there exists one singular point of the type $\aleph(dX^1 + nX^0)$ with $d \neq 0$, all the other singular points of the type $\aleph(dX^1 + \mathbb{Z}X^0)$ will appear by repeatedly applying the MUM monodromy action, generating a vertical tower of Borel singularities displaced by $dX^1$ from the origin. In other words, the distribution of Borel singularities near the MUM point is invariant under the MUM monodromy action.

The same phenomenon is observed near an orbifold point. Taking again the quintic as an example, which has an orbifold point at $z = \infty$, it is natural to use the orbifold frame defined by the periods $(X_0^0, X_0^1) = (w^{1/5}(1 + O(w)), w^{2/5}(1 + O(w)))$ with $w = 1/z$. A plot of Borel singularities of $\mathcal{F}^{(0)}(X_0^0, X_0^1)$ is given in Figure 10. The visible Borel singularities are located at

$$ \aleph \vec{q} \cdot \vec{\Pi}, \qquad (353) $$

with charge vectors

$$ \vec{q} \in \{\pm(0, 0, 0, 1), \pm(0, 0, -1, 1), \pm(-1, -1, 3, -1), \pm(2, 1, -3, -2), \pm(-1, 0, 1, 1)\}, \qquad (354) $$

defined relative to the period vector defined in (79). This distribution of Borel singularities is also invariant under the monodromy action around the orbifold point given by

$$ \vec{q} \to \vec{q} \cdot M_{\infty}, \qquad (355) $$

where the orbifold monodromy action is

$$ M_{\infty} = \begin{pmatrix} 1 & 1 & -5 & 2 \\ 0 & 1 & -3 & 5 \\ 1 & 1 & -4 & 2 \\ 0 & 0 & -1 & 1 \end{pmatrix}. \qquad (356) $$
<figure><img src="image_12.png" /><figcaption>Figure 9: The location of the poles of the Padé approximant to $\hat{\mathcal{F}}^{\text{hol}}$ evaluated to order $g = 64$ at the point $z = 10^{-2}\mu$ on the left, and $z = \frac{5}{6}\mu$ on the right. The black dots on the left give the positions of the periods $\mathfrak{R}(mX^0 + nX^1)$, $(m, n) = (1, 0), (0, 1), (1, 1), (2, 1)$, while the black dot on the right gives the position of $\mathfrak{R}P_0$.</figcaption></figure>

In fact, the ten charge vectors in (354) with the overall $\pm$ signs form two orbits of length 5 of the monodromy action $M_\infty$.

The situation near the conifold point $z = \mu$ is different. The monodromy action around the conifold point of the quintic is

$$X^0 \to X^0 + P_0. \qquad (357)$$

By the same argument as near the MUM point, near the conifold point we would expect due to the presence of the Borel singularities at $\pm\mathfrak{R}X^0$ two (horizontal) towers of Borel singularities given by $\mathfrak{R}(X^0 + ZP_0)$, which are nevertheless conspicuously absent in Figure 7.

The failure of monodromy invariance is reminiscent of a phenomenon pointed out by Seiberg and Witten [96] regarding the BPS spectrum of $\mathcal{N} = 2$ supersymmetric gauge theory. Stokes constants in topological string theory have been conjectured to be related to BPS invariants [14, 80]. Above, we have found Borel singularities of perturbative free energies to be located at geometric periods, which according to homological mirror symmetry are proportional to central charges of D-branes in type IIA superstring theory. It is tempting to link the Borel singularities to stable D-branes. Then, if a wall of marginal stability ends in the conifold point, it cannot be avoided when circling the conifold point and some stable D-branes will decay. As a consequence, we shall not expect the spectrum of stable D-branes, hence the distribution of Borel singularities, to be invariant under the conifold monodromy action. Conversely, when the distribution of Borel singularities is observed to be invariant under the MUM (resp. orbifold) monodromy action near the MUM point (resp. orbifold point) as above, this may indicate that walls of marginal stability can be avoided upon circling the MUM point (resp. orbifold point).
<figure><img src="image_13.png" /><figcaption>Figure 10: The location of the poles of the Padé approximant to $\mathcal{F}^{(0)}(X_0^0, X_0^1)$, evaluated to order $g = 60$, at $z = 10^8\mu$. The black dots correspond to periods with charge vectors $(0, 0, 0, 1)$, $(0, 0, -1, 1)$, $(-1, -1, 3, -1)$, $(2, 1, -3, -2)$, $(-1, 0, 1, 1)$.</figcaption></figure>

## 6.2 Experimental evidence for boundary conditions

In this subsection, we provide additional numerical evidence for the conjecture that the instanton amplitudes simplify in a frame where the instanton action $\mathcal{A}$ is an A-period and take the form given in (312), (313).

We start by numerically verifying the subleading asymptotics of $\mathcal{F}^{(0)}(X^0, X^1)$ near the MUM point, as worked out based on the Gopakumar-Vafa form of the topological string amplitude in section 5.7.1.

Let us consider the sequence

$$s_g = 2\pi^2 (A_{(1,0)})^{2g-2} \frac{\mathcal{F}_g(X^0, X^1)}{\Gamma(2g-1) + \Gamma(2g-2)}, \qquad (358)$$

which, according to (328) restricted to $d = 1$ and $n \in \{0, \pm 1\}$, should have an oscillatory behavior

$$s_g \sim n_{0,1} \left\{ 1 + 2 \left( 1 + (X^0/X^1)^2 \right)^{1-g} \cos(2(g-1)\phi) \right\}, \qquad (359)$$

where

$$\phi = \tan^{-1}(X^0/X^1). \qquad (360)$$

In Figure 11, on the left, we show the sequence $s_g$ (black dots) as compared to the expected oscillatory behavior on the r.h.s. of (359) (red line), when $z = 10^{-5}$. We can subtract the oscillatory part from $s_g$ to obtain a sequence that converges to $n_{0,1} = 2875$ (without subleading $1/g$ corrections). This is shown in Figure 11, on the right. This gives an approximation to 2875 with a precision of $10^{-10}$.

The discussion regarding the implications of the Gopakumar-Vafa formula for the asymptotics of $\mathcal{F}^{(0)}(X)$ is general. It applies to multi-parameter Calabi-Yau manifolds as well as to local Calabi-Yau manifolds. As the topological string amplitudes are computable to higher genus in the local case, the asymptotics displayed in (328) can be numerically checked beyond degree 1. Consider e.g. local $\mathbb{P}^2$. Having tested the asymptotics (328) for $d = 1$, one subtracts the resolved conifold piece corresponding to $d = 1$, so as to eliminate all the Borel singularities with $d = 1$. One then proceeds with $d = 2$, and so on. Let us give an example. Consider the sequence

$$s_g = \pi^2 \frac{\Gamma(2g)}{2g-2} A_{(3,0)}^{2g-2} \left\{ \mathcal{F}_g(t) - \frac{(-1)^{g-1} B_{2g}}{2g(2g-2)!} \left( 3 \mathrm{Li}_{3-2g}(e^{-t}) - 6 \mathrm{Li}_{3-2g}(e^{-2t}) \right) \right\}. \qquad (361)$$
<figure><img src="image_14.png" /><figcaption>Figure 11: On the left, the sequence $s_g$ (black dots) as compared to the expected oscillatory behavior (359) (red line), when $z = 10^{-5}$. On the right, the sequence $s_g$ minus its oscillatory part (red dots) and the prediction 2875 (black line).</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Figure 12: The sequence $s_g$ for $z = -10^{-8}$ in local $\mathbb{P}^2$ is represented in black dots and compared to the expected oscillatory behavior (362) (red line), where we include only a finite number of terms in the sum. In the first figure, we include only $m = 0$, in the second $m = 0, 1, 2$, and in the third $m = 0, 1, \dots, 8$.</figcaption></figure>

Here, we have subtracted the pieces that correspond to the Gopakumar–Vafa invariants of degree one, $n_{0,1} = 3$, and two, $n_{0,2} = -6$. What remains is the asymptotics governed by the degree 3 genus zero Gopakumar–Vafa invariant $n_{0,3} = 27$. Therefore, according to (328), the sequence should asymptote to

$$s_g \sim 27 \sum_{m \ge 0} \left| \frac{A_{(3,m)}}{A_{(3,0)}} \right|^{2-2g} \cos(2(g-1)\phi_m), \quad (362)$$

where $\phi_m = \arg(A_{(3,m)})$ and only the term with $\ell = 1$ is kept (the terms $\ell \ge 2$ lead to exponential corrections to the asymptotics). In order to check this, we have to go to a region where $A_{(3,0)}$ is smaller than the conifold action. Since the actions $A_{(3,m)}$ are quite close to each other in absolute value, for small values of $m$, one has to add them to correctly reproduce the asymptotics. In Figure 12, we make this comparison, for $z = -10^{-8}$. When enough terms are added, the asymptotic formula reproduces the sequence with high precision. Note that the flat region at the very beginning of the sequence is only reproduced when we included many oscillations, as in Fourier analysis.

We will now move beyond the analytic predictions in Section 5.7, and check the boundary conditions in an instanton sector for which we have no analytic control over the asymptotics of the topological string amplitudes. We will focus on the $X^1$ instanton sector in the conifold frame defined by the A-periods $(X^1, P_0)$. As $X^1$ is an element of the set of A-periods defining the frame, the second conjecture formulated in section 5.7 predicts that the associated instanton amplitude should be of the simple form (312), (313).

We first consider the case of the quintic. Unfortunately, $X^1$ is never the dominant instanton sector in the range between the MUM point and the conifold point. This is because, as we have seen from Figure 6 (left) and Figure 7 (right), both $X^0$ and $X^1$ appear as Borel singularities in this region, and the $X^0$ instanton sector always dominates the $X^1$ instanton sector, as $X^0$ is
<table><caption>Table 6: For the quintic in the conifold frame defined by the A-periods ($X^1, P_0$) at $z = (1 - 10^{-6})\mu$, we compare the Stokes discontinuity (underlined are stabilized digits) across the positive real axis where the Borel singular point $\mathfrak{R}X^1$ is located, and the value of the simple one-instanton amplitude (312) with $\tau_k = \delta_{k,1}/(2\pi)$ of instanton action $\mathcal{A} = \mathfrak{R}X^1$, evaluated with $\mathcal{F}^{(0)}$ up to order $g = 61$. The ratio of the two results is constant for a suitable range of $g_s$ and it is close to $n_{0,1} = 2875$.</caption><thead><tr><th>g<sub>s</sub></th><th>disc<sub>0</sub>F<sup>(0)</sup></th><th>F<sup>(1)</sup><sub>A</sub></th><th>disc<sub>0</sub>F<sup>(0)</sup>/F<sup>(1)</sup><sub>A</sub></th></tr></thead><tbody><tr><td>1/2</td><td>2.0727336107126 × 10<sup>-40</sup> i</td><td>7.2095082301714 × 10<sup>-43</sup> i</td><td>2874.99999</td></tr><tr><td>1</td><td>1.5789470403404 × 10<sup>-18</sup> i</td><td>5.4919897055452 × 10<sup>-22</sup> i</td><td>2874.99999</td></tr><tr><td>3/2</td><td>2.6261206071978 × 10<sup>-11</sup> i</td><td>9.1343325745150 × 10<sup>-15</sup> i</td><td>2874.99999</td></tr><tr><td>2</td><td>9.8839645377145 × 10<sup>-8</sup> i</td><td>3.4379000230359 × 10<sup>-11</sup> i</td><td>2875.00057</td></tr><tr><td>5/2</td><td>1.3181344321030 × 10<sup>-5</sup> i</td><td>4.5848197859366 × 10<sup>-9</sup> i</td><td>2874.99725</td></tr></tbody></table>

smaller than $X^1$ in absolute values here, cf. Figure 3. As the $X^0$ instanton amplitude is non-trivial in the conifold frame, its effect on the asymptotics of perturbative free energies cannot be easily subtracted. It is therefore difficult to extract information on the $X^1$ instanton sector from the large order asymptotics of the perturbative free energies of the quintic.

Instead, we compute the Stokes discontinuity across the positive real axis, on which $X^1$ is located. By (343), the result will be proportional to the sum over all Borel resummed instanton amplitudes associated to actions lying on the real axis. In addition to $X^1$, $P^0 \in \mathbb{R}$ close to the conifold point. However, unlike the contribution of the $X^0$ instanton sector, the contribution of the $P^0$ sector (and the associated multi-instanton sectors) can readily be subtracted from the asymptotics. To compute the Stokes discontinuity associated to the $X^1$ instanton sector, we therefore consider the Borel transforms of the sum of the regularized free energies $\mathcal{F}_g^{\text{reg}}(X^1, P_0)$ defined in (349). As we only have access to finitely many orders in the perturbation series, the Borel transforms we work with numerically do not exhibit logarithmic singularities. It turns out however that the poles of the Padé approximant suffice to obtain the discontinuity of the Borel resummation to high numerical precision. The Laplace transform integral (154) can either be performed numerically along a path slightly above/below the positive real axis, or the discontinuity (156) can be computed directly as a sum over the residues of the integrand (the difference of the upper/lower semi-circle contributions from the (+), (-) integration path respectively yield the full residue at each pole). Either way, the result should be proportional to the Borel resummed $X^1$ instanton amplitude, with proportionality constant the associated Stokes constant. We give the result of the ratio of the discontinuity and the instanton amplitude $\mathcal{F}_A^{(1)}$ given by (312) associated to the instanton action $\mathcal{A} = \mathfrak{R}X^1$, evaluated at the special boundary condition $\tau_k = \delta_{1,k}/(2\pi)$, in Table 6. We find that this ratio is constant for a suitable range of $g_s$ (dependent on the order to which the perturbative series is available), and it agrees with

$$n_{0,1} = 2875. \qquad (363)$$

Note that this is the same Stokes constant associated to the $X^1$ instanton sector in the MUM frame, cf. (330).

Another Calabi-Yau manifold which is of particular interest is $X_{2,2,2,2}(1^8)$, which has the property that near the conifold point, $\mathfrak{R}X^1$ is smaller than $\mathfrak{R}X^0$ in magnitude, so that $\pm \mathfrak{R}X^1$ are the dominant Borel singularities for the regularized conifold free energy $\mathcal{F}^{(0)\text{reg}}(X^1, P_0)$ defined in (349). For instance at $z = (1 - 10^{-6})\mu$,

$$\mathfrak{R}X^0 = 44.1619\dots i, \quad \mathfrak{R}X^1 = 35.6383\dots \qquad (364)$$
We can then apply the technique of large order asymptotics. We find that

$$ \mathcal{F}_{g}^{(0)\text{reg}}(X^1, P_0) \sim \frac{S_1}{\pi} \frac{\Gamma(2g-1)}{\mathcal{A}_1^{2g-1}} \left( \mathcal{F}_0^{(1)}(X^1, P_0) + \mathcal{F}_1^{(1)}(X^1, P_0) \frac{\mathcal{A}_1}{2g-2} \right), \quad (365) $$

where

$$ S_1 = n_{0,1} = 512, \quad \mathcal{A}_1 = \aleph X^1, \quad \mathcal{F}_0^{(1)}(X^1, P_0) = \frac{\mathcal{A}_1}{2\pi}, \quad \mathcal{F}_1^{(1)}(X^1, P_0) = \frac{1}{2\pi}. \quad (366) $$

This implies that the one-instanton amplitude associated to the instanton action $\mathcal{A}_1$ is indeed of the type (312), (313). For instance, to find $S_1 \mathcal{F}_0^{(1)}(X^1, P_0)/\pi$, we can study the sequence

$$ s_g^{(1)} = \mathcal{F}_g^{(0)\text{reg}}(X^1, P_0) \frac{\mathcal{A}_1^{2g-1}}{\Gamma(2g-1)}, \quad (367) $$

which asymptotes to $S_1 \mathcal{F}_0^{(1)}(X^1, P_0)/\pi$ in large $g$ and the convergence can be made faster using the Richardson transformation. Similarly, once $S_1 \mathcal{F}_0^{(1)}(X^1, P_0)/\pi$ is found, we can study the sequence

$$ s_g^{(2)} = \left( \mathcal{F}_g^{(0)\text{reg}}(X^1, P_0) \frac{\mathcal{A}_1^{2g-1}}{\Gamma(2g-1)} - \frac{S_1 \mathcal{F}_0^{(1)}(X^1, P_0)}{\pi} \right) \frac{2g-2}{\mathcal{A}_1}, \quad (368) $$

which asymptotes to $S_1 \mathcal{F}_1^{(1)}(X^1, P_0)/\pi$ at large $g$. Finally, we can use the sequence

$$ s_g^{(3)} = \left( \mathcal{F}_g^{(0)\text{reg}}(X^1, P_0) \frac{\mathcal{A}_1^{2g-1}}{\Gamma(2g-1)} - \frac{S_1 \mathcal{F}_0^{(1)}(X^1, P_0)}{\pi} - \frac{S_1 \mathcal{F}_1^{(1)}(X^1, P_0)}{\pi} \frac{\mathcal{A}_1}{2g-2} \right) \frac{(2g-2)(2g-3)}{\mathcal{A}_1^2}, \quad (369) $$

to explore a possible third term $S_1 \mathcal{F}_2^{(1)}(X^1, P_0)/(\pi)$. Here we always use

$$ S_1 = n_{0,1} = 512. \quad (370) $$

The sequences of $s_g^{(1)}, s_g^{(2)}, s_g^{(3)}$ are illustrated in Figs. 13. It is clear that $\mathcal{F}_{0,1}^{(1)}(X^1, P_0)$ fit well and $\mathcal{F}_2^{(1)}(X^1, P_0)$ should be zero.

As in the case of the quintic, another test we can perform is to compute the Stokes discontinuity. The Stokes discontinuity of the perturbative free energy across the positive real axis should be

$$ \mathrm{disc}_0 \mathcal{F}^{(0)\text{reg}}(X^1, P_0) = \frac{S_1}{2\pi} \left( \frac{\mathcal{A}_1}{g_s} + 1 \right) e^{-\mathcal{A}_1/g_s}. \quad (371) $$

This is also verified by numerical calculation as shown in Table 7.

## 6.3 Checking one-instanton amplitudes against asymptotics and Stokes discontinuities

In this section, we provide strong evidence for the solution of the nontrivial one-instanton amplitude (278) from the holomorphic anomaly equations. We will focus on two instanton sectors: the $P_0$ instanton sector in the MUM frame defined by the A-periods $(X^0, X^1)$, and the $X^0$ instanton sector in the conifold frame defined by the A-periods $(X^1, P_0)$.

The asymptotic behavior of $\mathcal{F}_g = \mathcal{F}_g(X_*^0, X_*^1)(z)$ at large $g$ is governed by instanton amplitudes via the relation

$$ \mathcal{F}_g \sim \sum_{\omega} \frac{S_{\omega}}{2\pi} \frac{\Gamma(2g-b_{\omega})}{\mathcal{A}_{\omega}^{2g-b_{\omega}}} \left( \mathcal{F}_0^{(\omega)} + \frac{\mathcal{F}_1^{(\omega)} \mathcal{A}_{\omega}}{2g-b_{\omega}-1} + \frac{\mathcal{F}_1^{(\omega)} \mathcal{A}_{\omega}^2}{(2g-b_{\omega}-1)(2g-b_{\omega}-2)} + \dots \right), \quad (372) $$
<figure><img src="image_16.png" /><figcaption>Figure 13: Comparison between auxiliary sequences (367), (368), (369) (red dots) which asymptote to $S_1 \mathcal{F}_{0,1,2}^{(1)}(X^1, P_0)/\pi$ with $S_1 = n_{0,1}$, their Richardson transforms (1st-order, blue dots), and the prediction (366) (black lines) for the example of the $X_{2,2,2,2}(1^8)$ model at $z = (1 - 10^{-6})\mu$.</figcaption></figure>

according to (320). The RHS of this relation depends on the Stokes constants $S_\omega$, the constants $b_\omega$, the instanton actions $A_\omega$, and the associated instanton amplitudes $\mathcal{F}_g^{(\omega)}$. As noted above, we will also label, when convenient, an instanton sector $\omega$ by the associated action $A_\omega$.

The instanton sectors which dominate the asymptotics of $\mathcal{F}_g$ can be read off the Borel plane plots studied in section 6.1. When the asymptotics is dominated by a single instanton sector,<fnref n="24" /> the corresponding instanton action can be extracted from the asymptotics by considering the auxiliary sequence

$$s_g^{(A)} = 2g \sqrt{\frac{\mathcal{F}_g}{\mathcal{F}_{g+1}}} \xrightarrow{g \gg 1} A. \qquad (373)$$

We can accelerate the convergence of the sequence by using Richardson transformations.

As argued for in section 5, consistency of our ansatz requires $b_\omega = 1$. This condition can also be verified numerically: if the asymptotics is dominated by the instanton action $A$, then the associated constant $b$ will be the limit of the sequence

$$s_g^{(b)} = -\frac{1}{2} \left( \frac{A^2 \mathcal{F}_{g+1}}{2g \mathcal{F}_g} - 2g - 1 \right) \xrightarrow{g \gg 1} b. \qquad (374)$$

Most significantly, the one-instanton amplitudes can be extracted from the asymptotics and compared to our theoretical prediction. Again assuming that the asymptotics is dominated by

<footnote n="24">²⁴Recall that due to the occurrence of only even powers of $g_s$ in the asymptotic series (1), all instanton sectors occur in pairs $\pm A$. In the following, we will refer to such pairs as a “single” instanton sector.</footnote>
<table><caption>Table 7: For the model $X_{2,2,2,2}(1^8)$ in the conifold frame defined by the A-periods $(X^1, P_0)$ at $z = (1 - 10^{-6})\mu$, we compare the Stokes discontinuity (underlined are stablised digits) across the positive real axis where the Borel singular point $\mathfrak{R}X^1$ is located, with $\mathcal{F}^{(0)}$ evaluated up to order $g = 32$, and the value of the simple one-instanton amplitude (312) with $\tau_k = \delta_{k,1}/(2\pi)$ of instanton action $\mathcal{A} = \mathfrak{R}X^1$. The ratio of the two results is constant for a suitable range of $g_s$ and it is close to $n_{0,1} = 512$.</caption><thead><tr><th>g<sub>s</sub></th><th>disc<sub>0</sub>F<sup>(0)</sup></th><th>F<sub>A</sub><sup>(1)</sup></th><th>disc<sub>0</sub>F<sup>(0)</sup>/F<sub>A</sub><sup>(1)</sup></th></tr></thead><tbody><tr><td>1/2</td><td>6.532093616 &times; 10<sup>-28</sup> i</td><td>1.275799523 &times; 10<sup>-30</sup> i</td><td>512.00000</td></tr><tr><td>1</td><td>9.942777405 &times; 10<sup>-13</sup> i</td><td>1.941948685 &times; 10<sup>-15</sup> i</td><td>512.00000</td></tr><tr><td>3/2</td><td>9.693393658 &times; 10<sup>-8</sup> i</td><td>1.893240419 &times; 10<sup>-10</sup> i</td><td>512.00014</td></tr><tr><td>2</td><td>2.798526539 &times; 10<sup>-5</sup> i</td><td>5.465895145 &times; 10<sup>-8</sup> i</td><td>511.99784</td></tr><tr><td>5/2</td><td>8.007846050 &times; 10<sup>-4</sup> i</td><td>1.563991489 &times; 10<sup>-6</sup> i</td><td>512.01340</td></tr></tbody></table>

the instanton action $\mathcal{A}$, we have

$$ \mathcal{F}_g \sim \frac{S_A}{2\pi} \frac{\Gamma(2g-1)}{\mathcal{A}^{2g-1}} \mathcal{F}_0^{(A)} + \frac{S_{-A}}{2\pi} \frac{\Gamma(2g-1)}{(-\mathcal{A})^{2g-1}} \mathcal{F}_0^{(-A)}. \quad (375) $$

Assuming symmetry under $\mathcal{A} \to -\mathcal{A}$, such that $S_A = S_{-A}$ and $\mathcal{F}_0^{(A)} = -\mathcal{F}_0^{(-A)}$, this implies

$$ s_{A,g}^{0} = \frac{\mathcal{A}^{2g-1}}{\Gamma(2g-1)} \mathcal{F}_{g} \xrightarrow{g \gg 1} \frac{S_{A}}{\pi} \mathcal{F}_{0}^{(A)}. \quad (376) $$

Given a theoretical prediction for $\mathcal{F}_0^{(A)}$, we can similarly obtain a numerical prediction for the one-loop contribution to the one-instanton amplitude via

$$ s_{A,g}^{1} = \left( \frac{\mathcal{A}^{2g-1}}{\Gamma(2g-1)} \mathcal{F}_{g} - \frac{S_{A}}{\pi} \mathcal{F}_{0}^{(A)} \right) \frac{2g-2}{\mathcal{A}} \xrightarrow{g \gg 1} \frac{S_{A}}{\pi} \mathcal{F}_{1}^{(A)}. \quad (377) $$

These asymptotic estimates for the loop coefficients $\mathcal{F}_g^{(A)}$ of the one-instanton amplitude can be compared to our theoretical predictions, extracted from the exact formula (278). The first few loop orders are extracted from the general formula in (284). These formulae depend on the two constants $c^0$ and $c^1$ defined in (229). In the one-modulus case, we can evaluate all derivatives in terms of the free energies $\mathcal{F}_0(t)$ and $\mathcal{F}_1(t)$ as

$$ \begin{aligned} \tau_{00} &= 2\mathcal{F}_0(t) - 2t\partial_t\mathcal{F}_0(t) + t^2\partial_t^2\mathcal{F}_0(t), \\ \tau_{01} &= \partial_t\mathcal{F}_0(t) - t\partial_t^2\mathcal{F}_0(t), \\ \tau_{11} &= \partial_t^2\mathcal{F}_0(t), \end{aligned} \quad (378) $$

and

$$ \begin{aligned} C_{000} &= -\frac{t^3}{X^0} \partial_t^3 \mathcal{F}_0(t), & C_{001} &= \frac{t^2}{X^0} \partial_t^3 \mathcal{F}_0(t), \\ C_{011} &= -\frac{t}{X^0} \partial_t^3 \mathcal{F}_0(t), & C_{111} &= \frac{1}{X^0} \partial_t^3 \mathcal{F}_0(t), \end{aligned} \quad (379) $$

as well as

$$ \frac{\partial \mathcal{F}_1(t)}{\partial X^0} = -\frac{t}{X^0} \partial_t \mathcal{F}_1(t), \qquad \frac{\mathcal{F}_1(t)}{X^1} = \frac{1}{X^0} \partial_t \mathcal{F}_1(t). \quad (380) $$

When the contributions from two instanton sectors are comparable, we can still extract numerical predictions under the condition that the contribution of one of the instanton sectors is known completely. We will see instances of this in examples, to which we now turn.
<table><caption>Table 8: The asymptotic estimate of the instanton action obtained via the sequence $s_g^{(A)}$ compared to the periods $P_0$ and $X^0$. The computation is based on $\mathcal{F}_g(X^0, X^1)$ evaluated up to genus 64 for the example of the quintic.</caption><thead><tr><th>z</th><th>Asymptotic estimate of action</th><th>\mathcal{R}P<sub>0</sub></th><th>\mathcal{R}X<sup>0</sup></th></tr></thead><tbody><tr><td>10<sup>-6</sup>μ</td><td>39.4784191203291289117078i</td><td>-1138.03925609468142282947</td><td>-39.4784191203291289125828i</td></tr><tr><td>10<sup>-4</sup>μ</td><td>39.47856920606554978139i</td><td>-494.0326608439227865458</td><td>-39.47856920606554978519i</td></tr><tr><td>10<sup>-2</sup>μ</td><td>39.4936233785076876i</td><td>-141.223891994711932</td><td>-39.4936233785077847i</td></tr><tr><td>1/8μ</td><td>39.67553248i</td><td>-42.99972954</td><td>-39.67553551i</td></tr><tr><td>1/7μ</td><td>39.704120i</td><td>-39.321466</td><td>-39.705033i</td></tr><tr><td>1/6μ</td><td>44.i</td><td>-35.</td><td>-40.i</td></tr><tr><td>1/5μ</td><td>30.73</td><td>-30.65</td><td>-39.80i</td></tr><tr><td>1/3μ</td><td>19.0714633228</td><td>-19.0714633440</td><td>-40.0447429295i</td></tr><tr><td>1/2μ</td><td>11.1503736933130532</td><td>-11.150373693313239</td><td>-40.390151835645641i</td></tr><tr><td>5/6μ</td><td>2.65608862910239270</td><td>-2.65608862910239633</td><td>-41.3365927932450749i</td></tr><tr><td>23/24μ</td><td>0.6030509245517648306</td><td>-0.6030509245517648990</td><td>-41.92408741381918292i</td></tr></tbody></table>

We will begin by studying the asymptotics of the topological string amplitudes $\mathcal{F}_g(X^0, X^1)$ in the MUM frame. We will perform this analysis for the quintic.

We numerically evaluate the $\mathcal{F}_g(X^0, X^1)$, at real values of $z$ in between the MUM point and the conifold point,

$$0 < z < \mu = 5^{-5}, \qquad (381)$$

and obtain a sequence of values of $\mathcal{F}_g(X^0, X^1)$. We then substitute these values into the sequence $s_g^{(A)}$ defined in (373) and perform the number of Richardson transformations which best stabilizes the sequence (i.e. which leads to the largest number of coincident digits in the last two entries of the Richardson transformed sequence). These values are given in Table 8 for a range of values for $z$. From our study of the Borel plane, see Figure 4, we expect the dominant instanton action to transition from $\mathcal{R}X^0$ close to the MUM point to $\mathcal{R}P_0$ close to the conifold point. These two periods are also listed in Table 8 for each value of $z$. We find that the respective actions are reproduced to high precision by the asymptotics close to the associated singular point. The method fails when the two actions are of comparable size.

We next determine asymptotic estimates for $b$ at points in moduli space at which the asymptotics is dominated by one instanton action. The results are collected in Table 9, and not surprisingly confirm the theoretical value $b = 1$.

Finally, we turn to the asymptotic predictions for $\mathcal{F}_0^{(1)}$ and $\mathcal{F}_1^{(1)}$. Close to the MUM point,

<table><caption>Table 9: The asymptotic estimate of the constant $b$ via the sequence $s_g^{(b)}$ evaluated for $\mathcal{A} = \mathcal{R}P_0$, $\mathcal{A} = \mathcal{R}X^0$ as appropriate. The computation is based on $\mathcal{F}_g(X^0, X^1)$ evaluated up to genus 64 for the example of the quintic.</caption><thead><tr><th>z</th><th>Asymptotic estimate of b</th></tr></thead><tbody><tr><td>10<sup>-6</sup>μ</td><td>1.00000000000000000376</td></tr><tr><td>10<sup>-4</sup>μ</td><td>1.00000000000000000376</td></tr><tr><td>10<sup>-2</sup>μ</td><td>1.00000000000000157</td></tr><tr><td>1/8μ</td><td>1.0000042</td></tr><tr><td>1/3μ</td><td>1.00000000537</td></tr><tr><td>1/2μ</td><td>1.0000000000001046</td></tr><tr><td>5/6μ</td><td>1.00000000000001197</td></tr><tr><td>23/24μ</td><td>1.000000000000001512</td></tr></tbody></table>
<table><caption>Table 10: Comparison, in the frame $(X^0, X^1)$, between the asymptotic estimate for the normalized genus 0 one-instanton amplitude $\frac{S_\mu}{\pi} \mathcal{F}_0^{(\mu)}$ and the prediction, using $S_\mu = 1$, for the example of the quintic.</caption><thead><tr><th>z</th><th>Asymptotic estimate I</th><th>Asymptotic estimate II</th><th>Prediction</th></tr></thead><tbody><tr><td>μ/8</td><td>1. × 10<sup>9</sup></td><td>-2.0685 × 10<sup>-8</sup></td><td>-2.0618 × 10<sup>-8</sup></td></tr><tr><td>μ/7</td><td>10000.</td><td>-4.659137 × 10<sup>-8</sup></td><td>-4.658992 × 10<sup>-8</sup></td></tr><tr><td>μ/6</td><td>0.01</td><td>-1.163074023 × 10<sup>-7</sup></td><td>-1.163074007 × 10<sup>-7</sup></td></tr><tr><td>μ/5</td><td>-3.335 × 10<sup>-7</sup></td><td>-3.313309985143 × 10<sup>-7</sup></td><td>-3.313309985104 × 10<sup>-7</sup></td></tr><tr><td>μ/3</td><td>-5.1510310321251 × 10<sup>-6</sup></td><td>-5.151031032069825626 × 10<sup>-6</sup></td><td>-5.151031032069825187 × 10<sup>-6</sup></td></tr><tr><td>μ/2</td><td>-0.000038081205262381317350</td><td>-0.000038081205262381317350</td><td>-0.000038081205262381316984</td></tr><tr><td>5μ/6</td><td>-0.000374000001694825755160</td><td>-0.000374000001694825755160</td><td>-0.000374000001694825754743</td></tr><tr><td>23μ/24</td><td>-0.0004997585182539551567396</td><td>-0.0004997585182539551567396</td><td>-0.0004997585182539551566954</td></tr></tbody></table>

the asymptotics will be governed by the analytically derived instanton amplitudes (322) associated to the instanton action $\mathcal{A}_{\text{MUM}} = \mathfrak{R}X^0$. Recall that these were imposed as boundary conditions in our computations. Close to the conifold point, we predict that the instanton amplitudes (284) associated to the instanton action $\mathcal{A}_\mu = \mathfrak{R}P_0$, i.e. evaluated with

$$c^0 = \mathfrak{R}, \quad c^1 = 0, \qquad (382)$$

should govern the asymptotics. In the second column of Table 10 and of Table 11, under the heading “Asymptotic estimate I”, we have listed the asymptotic estimates for $\frac{S_\mu}{\pi} \mathcal{F}_0^{(\mu)}$ and $\frac{S_\mu}{\pi} \mathcal{F}_1^{(\mu)}$ respectively, based on the sequences $s_{A_{\mu,g}}^0$ and $s_{A_{\mu,g}}^1$ introduced above.

The estimates match the prediction based on (382), given in the fourth column of Table 10 and of Table 11, convincingly close to the conifold point, but break down around $z \sim \frac{\mu}{6}$. At this value of $z$, $|A_\mu| < |A_{\text{MUM}}|$, i.e. the conifold action is still dominant, see Table 8. Given that the action enters in the asymptotics to the power of $1-2g$, the breakdown of our estimate already at this value of $z$ implies that the instanton amplitudes associated to $\mathcal{A}_{\text{MUM}}$ must be appreciably larger than the amplitude associated to $A_\mu$. And indeed, at genus $g=64$, we compute

$$\left| \frac{\Gamma(2 \times 64 - 1)}{A_{\mu}^{2 \times 64 - 1}} \right| \sim 2 \times 10^{15}, \qquad \left| \frac{\Gamma(2 \times 64 - 1)}{A_{\text{MUM}}^{2 \times 64 - 1}} \right| \sim 8 \times 10^{7}, \qquad (383)$$

while

$$|\mathcal{F}_0^{(\mu)}| \sim 10^{-7}, \qquad |\mathcal{F}_0^{(\text{MUM})}| \sim 40, \qquad (384)$$

and

$$|\mathcal{F}_1^{(\mu)}| \sim 10^{-7}, \qquad |\mathcal{F}_1^{(\text{MUM})}| \sim 1. \qquad (385)$$

<table><caption>Table 11: Comparison, in the frame $(X^0, X^1)$, between the asymptotic estimate for the normalized genus 1 one-instanton amplitude $\frac{S_\mu}{\pi} \mathcal{F}_1^{(\mu)}$ and the prediction, using $S_\mu = 1$, for the example of the quintic.</caption><thead><tr><th>z</th><th>Asymptotic estimate I</th><th>Asymptotic estimate II</th><th>Prediction</th></tr></thead><tbody><tr><td>μ/8</td><td>-4. × 10<sup>9</sup></td><td>-1.6724 × 10<sup>-8</sup></td><td>-1.6650 × 10<sup>-8</sup></td></tr><tr><td>μ/7</td><td>-50000.</td><td>-3.206806 × 10<sup>-8</sup></td><td>-3.206579 × 10<sup>-8</sup></td></tr><tr><td>μ/6</td><td>-0.05</td><td>-6.405055587 × 10<sup>-8</sup></td><td>-6.405055488 × 10<sup>-8</sup></td></tr><tr><td>μ/5</td><td>-1.3 × 10<sup>-7</sup></td><td>-1.280693556310 × 10<sup>-7</sup></td><td>-1.280693556226 × 10<sup>-7</sup></td></tr><tr><td>μ/3</td><td>5.88102454425421 × 10<sup>-7</sup></td><td>5.88102454418872040 × 10<sup>-7</sup></td><td>5.88102454418871349 × 10<sup>-7</sup></td></tr><tr><td>μ/2</td><td>0.000025534824045449402553</td><td>0.000025534824045449402553</td><td>0.000025534824045449402293</td></tr><tr><td>5μ/6</td><td>0.001214358107695072071777</td><td>0.001214358107695072071777</td><td>0.001214358107695072070550</td></tr><tr><td>23μ/24</td><td>0.00609369882423593493007</td><td>0.00609369882423593493007</td><td>0.00609369882423593492297</td></tr></tbody></table>
<figure><img src="image_17.png" /><figcaption>Figure 14: The significant digits of the asymptotic estimates for 1 instanton genus 1, 2, 3 contributions without (left) and with (right) constant map subtraction for the quintic $X_5(1^5)$ to $g = 64$.</figcaption></figure>

Hence, beyond $z \sim \frac{\mu}{5}$, the contribution of the two instanton sectors associated to $\mathcal{A}_{\mu}$ and $\mathcal{A}_{\text{MUM}}$ to the asymptotics become first comparable and then dominated by the latter. However, as we know the leading asymptotics of $\mathcal{F}_g(X^0, X^1)$ close to the MUM point exactly – it is given by (322) – we can subtract it from $\mathcal{F}_g(X^0, X^1)$ and test the prediction based on (382) also in this region. The results of this computation, $s_{\mathcal{A}_{\mu,g}}^0, s_{\mathcal{A}_{\mu,g}}^1$, evaluated on $\mathcal{F}_g(X^0, X^1)$ with the leading asymptotics subtracted, are given in the third column, under the heading “Asymptotics II” of Table 10 and Table 11 respectively and show convincing agreement with the predictions.

Very close to the MUM point (e.g. at the values $10^{-2}\mu, 10^{-4}\mu, 10^{-6}\mu$ considered in Table 8 and Table 9), our numerical precision is no longer sufficient to resolve the subleading contribution of the $\mathcal{A}_{\mu}$ instanton sector to the asymptotics.

The numerical study of asymptotic estimates is illustrated graphically in Figure 14, which also compares the asymptotic estimates for $\mathcal{F}_2^{(\mu)}$ and $\mathcal{F}_3^{(\mu)}$ to the theoretical predictions. Indeed, as we have predictions for the instanton amplitudes $\mathcal{F}_g^{(\mu)}$ at arbitrary loop order, we can obtain asymptotic predictions for $\mathcal{F}_n^{(\mu)}$ from a sequence $s_{\mathcal{A}_{\mu,g}}^n$ in which the contributions from the lower loop coefficients to the asymptotics of $\mathcal{F}_g(X^0, X^1)$ have been subtracted. With increasing $n$, we must compute the topological string amplitudes to increasingly high genus and evaluate them with ever increasing numerical precision. Below, we will compute asymptotic estimates of the one-instanton amplitudes up to loop order 8 for $\mathcal{F}_g(X^1, P_0)$.

We now change frames and perform the analogous asymptotic analysis for the topological string amplitudes $\mathcal{F}_g(X^1, P_0)$ in the conifold frame. Here, the leading asymptotics close to the conifold point is known exactly – it is given by (337). We can hence test the prediction of (284) for the instanton sector associated to the instanton action $\mathcal{A} = \mathcal{R}X^0$, i.e. for

$$c_0 = 0, \quad c_1 = \mathcal{R}. \qquad (386)$$
<table><caption>Table 12: Comparison, in the frame $(X^1, P_0)$, between the asymptotic estimate for $\frac{S_{MUM}}{\pi} \mathcal{F}_0^{(A_{MUM})}$ and the prediction, using $S_{MUM} = -\chi$, for the example of the quintic.</caption><thead><tr><th>z</th><th>Asymptotic estimate I</th><th>Asymptotic estimate II</th><th>Prediction</th></tr></thead><tbody><tr><td>10<sup>-6</sup> μ</td><td>-382.257657577866488i</td><td>-382.257657577866488i</td><td>-382.257657577867968i</td></tr><tr><td>10<sup>-4</sup> μ</td><td>-364.260537318513919i</td><td>-364.260537318513919i</td><td>-364.260537318518819i</td></tr><tr><td>10<sup>-2</sup> μ</td><td>-312.00085776255i</td><td>-312.00085776255i</td><td>-312.00085777409i</td></tr><tr><td>μ/8</td><td>-401.997458i</td><td>-402.2i</td><td>-401.997221i</td></tr><tr><td>μ/7</td><td>300i</td><td>-229.4055067i</td><td>-229.4055313i</td></tr><tr><td>μ/6</td><td>2 × 10<sup>7</sup> i</td><td>-221.78747i</td><td>-221.78865i</td></tr><tr><td>μ/5</td><td>2 × 10<sup>13</sup> i</td><td>-212.105904i</td><td>-212.106060i</td></tr></tbody></table>

<table><caption>Table 13: Comparison, in the frame $(X^1, P_0)$, between the asymptotic estimate for $\frac{S_{MUM}}{\pi} \mathcal{F}_1^{(A_{MUM})}$ and the prediction, using $S_{MUM} = -\chi$, for the example of the quintic.</caption><thead><tr><th>z</th><th>Asymptotic estimate I</th><th>Asymptotic estimate II</th><th>Prediction</th></tr></thead><tbody><tr><td>10<sup>-6</sup> μ</td><td>-19.96043925387713</td><td>-19.96043925387713</td><td>-19.96043925387822</td></tr><tr><td>10<sup>-4</sup> μ</td><td>-32.52506815749229</td><td>-32.52506815749229</td><td>-32.52506815749648</td></tr><tr><td>10<sup>-2</sup> μ</td><td>-49.74119660140</td><td>-49.74119660140</td><td>-49.74119660508</td></tr><tr><td>μ/8</td><td>10.132210</td><td>10.33</td><td>10.132118</td></tr><tr><td>μ/7</td><td>-1000</td><td>-53.5985829</td><td>-53.5985968</td></tr><tr><td>μ/6</td><td>-6 × 10<sup>7</sup></td><td>-53.1246552</td><td>-53.1246694</td></tr><tr><td>μ/5</td><td>-5 × 10<sup>13</sup></td><td>-52.3718764</td><td>-52.3718908</td></tr></tbody></table>

According to the Borel plane plots on the left in Figure 6 and Figure 8, this should capture the leading asymptotics near the MUM point and the subleading asymptotics near the conifold point. In the second column of Table 12 and Table 13, under the heading “Asymptotics I”, we have listed the asymptotic estimates for $\frac{S_{MUM}}{\pi} \mathcal{F}_0^{(A_{MUM})}$ and $\frac{S_{MUM}}{\pi} \mathcal{F}_1^{(A_{MUM})}$ respectively, based on the sequences $s_0^{A_{MUM},g}$ and $s_1^{A_{MUM},g}$. The estimates match the prediction based on (386), given in the fourth column of the tables, all the way up to $z = \mu/8$. By $z = \mu/7$, the instanton action $A_\mu$ has inched closer to the origin of the Borel plane than $A_{MUM}$. The prediction (386) now applies to the subleading asymptotics. Subtracting the leading asymptotics yields the asymptotic estimates listed in the third column of Table 12 and Table 13, under the heading “Asymptotics II” – these show good agreement with the predictions for all points $z$ in the moduli space studied. Note that unlike the case for $\mathcal{F}_g(X^0, X^1)$, the breakdown of the asymptotic estimate for the instanton amplitudes (“Asymptotics I” in the tables) coincides with the point in moduli space at which $A_\mu$ moves past $A_{MUM}$ to become the closest action to the origin of the Borel plane. The difference between the two cases is that for $\mathcal{F}_g(X^1, P_0)$, the size of the instanton actions in the two sectors is comparable, and cannot compensate for the difference in the contribution from the actions to the asymptotics (cf. discussion around (383)).

As pointed out above, we can push the comparison between asymptotic estimates for the one-instanton amplitudes and the theoretical prediction based on (278) to higher loop order. In Figure 15, we perform this comparison for $\mathcal{F}^{(0)}(X^1, P_0)$ at $z = 10^{-2}\mu$ up to loop order 8, and find good agreement.

Finally, we check the entire one-instanton amplitude formula using the Stokes discontinuity
<table><caption>Table 14: Comparison, in the frame $(P_0, P_1)$, between the asymptotic estimate for $\frac{S_{\text{MUM}}}{\pi} \mathcal{F}_0^{(A_{\text{MUM}})}$ and the prediction using $S_{\text{MUM}} = -\chi$ for the first three values of $z$, and between the asymptotic estimate for $\frac{S_{\mu}}{\pi} \mathcal{F}_0^{(\mu)}$ and the prediction using $S_{\mu} = 1$ for the last four values of $z$, for the example of the quintic.</caption><thead><tr><th>z</th><th>Asymptotic estimate</th><th>Prediction</th></tr></thead><tbody><tr><td>10<sup>-6</sup>μ</td><td>2.578293673861729631 – 350.320544587085132123 i</td><td>2.578293673861729616 – 350.320544587085132654 i</td></tr><tr><td>10<sup>-4</sup>μ</td><td>4.283968180390488729 – 314.909621199366985163 i</td><td>4.283968180390488721 – 314.909621199366985961 i</td></tr><tr><td>10<sup>-2</sup>μ</td><td>6.26156388754039 – 241.46605914584608 i</td><td>6.26156388754057 – 241.46605914585064 i</td></tr><tr><td>μ/3</td><td>-0.966171619903455385</td><td>-0.966171619903454465</td></tr><tr><td>μ/2</td><td>-0.5648845303304413622559</td><td>-0.5648845303304413621804</td></tr><tr><td>5μ/6</td><td>-0.13455902187980453773105</td><td>-0.13455902187980453771306</td></tr><tr><td>23μ/24</td><td>-0.03055091673609547562773</td><td>-0.03055091673609547562365</td></tr></tbody></table>

<table><caption>Table 15: Comparison, in the frame $(P_0, P_1)$, between the asymptotic estimate for $\frac{S_{\text{MUM}}}{\pi} \mathcal{F}_1^{(A_{\text{MUM}})}$ and the prediction using $S_{\text{MUM}} = -\chi$ for the first three values of $z$, and between the asymptotic estimate for $\frac{S_{\mu}}{\pi} \mathcal{F}_1^{(\mu)}$ and the prediction using $S_{\mu} = 1$ for the last four values of $z$, for the example of the quintic.</caption><thead><tr><th>z</th><th>Asymptotic estimate</th><th>Prediction</th></tr></thead><tbody><tr><td>10<sup>-6</sup>μ</td><td>-22.118606937252992852 + 0.154244032879358745 i</td><td>-22.118606937252992805 + 0.154244032879358748 i</td></tr><tr><td>10<sup>-4</sup>μ</td><td>-25.715437369360251839 - 0.575488280479718081 i</td><td>-25.715437369360252154 - 0.575488280479718072 i</td></tr><tr><td>10<sup>-2</sup>μ</td><td>-23.58594545231497 - 2.10986661318608 i</td><td>-23.58594545232135 - 2.10986661318582 i</td></tr><tr><td>μ/3</td><td>0.050660591821169677</td><td>0.050660591821168886</td></tr><tr><td>μ/2</td><td>0.05066059182116888572605</td><td>0.05066059182116888572194</td></tr><tr><td>5μ/6</td><td>0.05066059182116888572605</td><td>0.05066059182116888572194</td></tr><tr><td>23μ/24</td><td>0.05066059182116888572605</td><td>0.05066059182116888572194</td></tr></tbody></table>

relation (343). The Stokes discontinuity of $\mathcal{F}^{(0)}(X^1, P_0)$ across the negative imaginary axis should be proportional to the Borel resummation of the one-instanton amplitude with action $\mathcal{O}X^0$. At $z = 1/100\mu$ and with $g_s = -i$, the Stokes discontinuity, with $\mathcal{F}^{(0)}(X^1, P_0)$ expanded up to order $g = 61$, evaluates to

$$ \text{disc}_{\pi/2} \mathcal{F}^{(0)}(X^1, P_0)(z = 1/100, g_s = -i) = \underline{1.85787242132104924598} \times 10^{-16}, \quad (387) $$

where stable digits are underlined. The Borel resummation of $\mathcal{F}^{(1)}(X^1, P_0)$ is

$$ s_{\mathcal{F}}^{(1)}(X^1, P_0)(z = 1/100, g_s = -i) = \underline{1.8578724213210491439127620868310874} \times 10^{-16} \\ -i \underline{2.144138518276324130} \times 10^{-31}, \quad (388) $$

where stable digits are underlined. Both the real and the imaginary part of these two results agree up to $10^{-31}$, which is the order of magnitude of the second instanton amplitude with action $2\mathcal{O}X^0$.

As a final example, we consider the asymptotics of the topological string amplitudes $\mathcal{F}_g(P_0, P_1)$. Unlike the two previous cases, we do not have access to exact asymptotics in this frame anywhere on moduli space. Hence, we are restricted to studying leading asymptotics. The predictions in Table 14 and Table 15 are for $\frac{S_{\text{MUM}}}{\pi} \mathcal{F}_g^{(A_{\text{MUM}})}$, $g = 0, 1$ for the first three values of $z$ close to the MUM point, and for $\frac{S_{\mu}}{\pi} \mathcal{F}_g^{(\mu)}$, $g = 0, 1$, for the remaining four values of $z$. We find convincing agreement at all loop orders considered.

All of the analysis in this section so far has been for the quintic. We can of course equally well study other one-parameter models. A comparison of asymptotic estimates and theoretical prediction for one-instanton amplitudes of $\mathcal{F}_g(X^0, X^1)$ is performed for the dectic $X_{10}(1^3, 2, 5)$ in Figure 16.
<figure><img src="image_18.png" /><figcaption>Figure 15: Comparison between auxiliary sequences (similar to (376), red dots) that asymptote to $\frac{S_{MUM}}{\pi} \mathcal{F}^{(MUM)}$ ($g = 0, 1, \dots, 7$) constructed from evaluation of $\mathcal{F}_g(X^1, P_0)$ up to $g = 61$ with $S_{MUM} = -\chi$, their Richardson transformations (8-th order, blue dots), and the prediction from the 1-instanton formula (278) (black line) for the $X^0$ instanton sector in the case of the quintic.</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>Figure 16: The significant digits of the asymptotic estimates for the one-instanton genus 1, 2, 3 contributions without (left) and with (right) constant map subtraction for the dectic $X_{10}(1^3, 2, 5)$ to $g = 50$.</figcaption></figure>

## 7 Conclusions

The asymptotic nature of the perturbative series in topological string theory indicates that there are additional non-perturbative sectors which can be decoded from perturbation theory. These sectors are characterized by their Borel singularities or instanton actions, by asymptotic series or instanton amplitudes, and by Stokes constants. In this paper, we have found general results for the instanton amplitudes for arbitrary compact Calabi-Yau manifolds, as well as particular results on the structure of Borel singularities, providing a far-reaching generalization of previous results in [20, 21, 23, 94], which focused on the one-modulus, local case. More concretely, we have found that the Borel singularities of perturbative topological string free energies in a generic compact Calabi-Yau threefold are integral periods of the form

$$ \mathcal{A} = \ell(c^J P_J + d_J X^J), \qquad (389) $$

similar to the findings made in the local cases [20, 21, 26]. Here $X^J, P_J$ are integral A- and B-periods respectively whose choice determines a symplectic frame, $\{c^J, d_J\}$ are coprime integers, and $\ell = 1, 2, \dots$. We checked this result with the examples of hypergeometric one parameter Calabi-Yau threefolds listed in Table 1. We have found elegant closed-form expressions for the instanton amplitudes, i.e. the non-perturbative corrections to free energies, in the form of exponentially small trans-series, by solving the trans-series extension of the holomorphic anomaly equations. They are encoded in terms of non-perturbative partition functions given in (309), related to the free energies through the usual exponential-logarithm relation. The non-perturbative partition functions can be evaluated in the holomorphic limit associated to a chosen symplectic frame. The results depend on the type of the Borel singularity the instanton amplitude is evaluated at. If the Borel singularity is an A-period, in the sense that all
the coefficients $c^J$ vanish, the holomorphic limit reduces simply to (312), (313); otherwise, the holomorphic limit is obtained with the rules (310), (311). In the case of one-instanton sectors with $\ell = 1$, the two types of holomorphic limit of the instanton amplitude are respectively (179) and (278).

Our results for the amplitudes show that, in the holomorphic limit, they are simple functionals of the perturbative free energies, as showcased by (278) in the case of one-instanton amplitude, and by the replacement rules (310), (311) in the multi-instanton amplitudes. They also have experimental implications, since they determine the large genus asymptotics of the topological string free energies. We have verified that this is the case in many one-parameter compact Calabi-Yau 3-folds, including the famous quintic Calabi-Yau manifold – see Section 6.

The structure of the instanton amplitudes is intriguing from the physics point of view. These amplitudes involve shifts of the background – specified by the coordinates $X^I$ of the big moduli space – by integer multiples of the string coupling constant. This phenomenon was observed in [23] in the local case, but as we have mentioned in this paper, its appearance in the compact case is somewhat unexpected. It suggests a quantization of the big moduli space coordinates in units of the string coupling constant, as in large $N$ dualities.

Another surprising aspect of the non-perturbative sectors that we have described in this paper is that the corresponding instanton actions are closed string periods, i.e. masses of even/odd-dimensional D-branes in the A/B model, respectively. It has been sometimes suggested (see e.g. [97]) that non-perturbative corrections in the A/B topological string should be due to Lagrangian/holomorphic D-branes, respectively. These corrections do not seem to appear in the resurgent structure uncovered in this paper and in the previous works [20, 21, 23]. We should however emphasize that the resurgent structure associated to the perturbative series does not necessarily cover all the relevant non-perturbative sectors in a physical theory. Consider for example the non-linear $O(3)$ sigma model. There are instanton amplitudes in this model that capture the dependence of observables on the topological $\theta$ angle, and that cannot be detected from perturbation theory alone (this has been made completely explicit in recent studies [98, 99]). Therefore, it is conceivable that the non-perturbative sectors that we have described in this paper should be thought of as “renormalon” sectors of topological string theory, controlling the factorial divergence of the perturbative series due to integration over moduli space. There could be in addition purely “instanton” sectors, perhaps due to Lagrangian/holomorphic branes as suggested in [97], and undetectable in the topological string perturbative series. This possibility remains for the moment rather uncertain in the compact case, since in the absence of a concrete non-perturbative definition of the theory, it is difficult to talk about non-perturbative sectors, other than the ones obtained from the perturbative series.

Perhaps the main challenge is to find methods, other than numerical, to determine the remaining ingredients of the resurgent structure, namely the location of Borel singularities and the values of the Stokes constants. We believe that these ingredients are deeply related to stability structures and BPS invariants, and we have indeed shown in this paper that genus zero Gopakumar-Vafa invariants are realized as Stokes constants. A complete and more precise picture however is still lacking. Perhaps an extension of our non-perturbative analysis to degeneration points other than the MUM and conifold points studied above (those that occur at $z = \infty$ in hypergeometric models) could shed further light on these matters.

## Acknowledgments

We would like to thank Michael Borinsky, David Sauzin, Ricardo Schiappa and Samson Shatashvili for useful comments and discussions.
**Funding information** A.K. and M.M. would like to thank the IHES and the Physics Department of the École Normale Supérieure (Paris) respectively for hospitality during the inception of this paper. The work of M.M. has been supported in part by the ERC-SyG project "Recurrent and Exact New Quantum Theory" (ReNewQuantum), which received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program, grant agreement No. 810573. J.G. is supported by the Startup Funding no. 4007022316 and no. 4060692201/011 of the Southeast University. A.K.K.P. acknowledges support under ANR grant ANR-21-CE31-0021.

## References

[1] D. J. Gross and V. Periwal, *String perturbation theory diverges*, Phys. Rev. Lett. **60**, 2105 (1988), doi:**10.1103/PhysRevLett.60.2105**.

[2] S. H. Shenker, *The strength of nonperturbative effects in string theory*, in *Cargese study institute: Random surfaces, quantum gravity and strings* Cargese, France (1990).

[3] J. Polchinski, *Combinatorics of boundaries in string theory*, Phys. Rev. D **50**, R6041 (1994), doi:**10.1103/PhysRevD.50.R6041**.

[4] S. Y. Alexandrov, V. A. Kazakov and D. Kutasov, *Non-perturbative effects in matrix models and D-branes*, J. High Energy Phys. **09**, 057 (2003), doi:**10.1088/1126-6708/2003/09/057**.

[5] F. J. Dyson, *Divergence of perturbation theory in quantum electrodynamics*, Phys. Rev. **85**, 631 (1952), doi:**10.1103/PhysRev.85.631**.

[6] C. M. Bender and T. T. Wu, *Large-order behavior of perturbation theory*, Phys. Rev. Lett. **27**, 461 (1971), doi:**10.1103/PhysRevLett.27.461**.

[7] T. M. Seara and D. Sauzin, *Resumació de Borel i teoria de la resurgencia*, Butl. Soc. Catalana Mat. **18**, 131 (2003).

[8] M. Mariño, *Lectures on non-perturbative effects in large N gauge theories, matrix models and strings*, Fortsch. Phys. **62**, 455 (2014), doi:**10.1002/prop.201400005**.

[9] I. Aniceto, G. Başar and R. Schiappa, *A primer on resurgent transseries and their asymptotics*, Phys. Rep. **809**, 1 (2019), doi:**10.1016/j.physrep.2019.02.003**.

[10] C. Mitschi and D. Sauzin, *Divergent series, summability and resurgence I*, Springer, Cham, Switzerland, ISBN 9783319287355 (2016), doi:**10.1007/978-3-319-28736-2**.

[11] M. Kontsevich and Y. Soibelman, *Stability structures, motivic Donaldson-Thomas invariants and cluster transformations*, (arXiv preprint) doi:**10.48550/arXiv.0811.2435**.

[12] D. Gaiotto, G. W. Moore and A. Neitzke, *Wall-crossing, Hitchin systems, and the WKB approximation*, Adv. Math. **234**, 239 (2013), doi:**10.1016/j.aim.2012.09.027**.

[13] A. Grassi, J. Gu and M. Mariño, *Non-perturbative approaches to the quantum Seiberg-Witten curve*, J. High Energy Phys. **07**, 106 (2020), doi:**10.1007/JHEP07(2020)106**.

[14] M. Mariño, *From resurgence to BPS states*, in *Strings 2019*, Brussels, Belgium (2019), https://member.ipmu.jp/yuji.tachikawa/stringsmirrors/2019/2_M_Marino.pdf.
[15] S. Garoufalidis, J. Gu and M. Mariño, *The resurgent structure of quantum knot invariants*, Commun. Math. Phys. **386**, 469 (2021), doi:10.1007/s00220-021-04076-0.

[16] S. Garoufalidis, J. Gu and M. Mariño, *Peacock patterns and resurgence in complex Chern-Simons theory*, doi:10.1007/s40687-023-00391-1.

[17] M. Mariño, *Open string amplitudes and large order behavior in topological string theory*, J. High Energy Phys. **03**, 060 (2008), doi:10.1088/1126-6708/2008/03/060.

[18] M. Mariño, R. Schiappa and M. Weiss, *Nonperturbative effects and the large-order behavior of matrix models and topological strings*, Commun. Num. Theor. Phys. **2**, 349 (2008), doi:10.4310/CNTP2008.v2.n2.a3.

[19] M. Mariño, *Nonperturbative effects and nonperturbative definitions in matrix models and topological strings*, J. High Energy Phys. **12**, 114 (2008), doi:10.1088/1126-6708/2008/12/114.

[20] R. Couso-Santamaría, J. D. Edelstein, R. Schiappa and M. Vonk, *Resurgent transseries and the holomorphic anomaly*, Ann. Henri Poincaré **17**, 331 (2015), doi:10.1007/s00023-015-0407-z.

[21] R. Couso-Santamaría, J. D. Edelstein, R. Schiappa and M. Vonk, *Resurgent transseries and the holomorphic anomaly: Nonperturbative closed strings in local $CP^2$*, Commun. Math. Phys. **338**, 285 (2015), doi:10.1007/s00220-015-2358-0.

[22] M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, *Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes*, Commun. Math. Phys. **165**, 311 (1994), doi:10.1007/BF02099774.

[23] J. Gu and M. Mariño, *Exact multi-instantons in topological string theory*, SciPost Phys. **15**, 179 (2023), doi:10.21468/SciPostPhys.15.4.179.

[24] M.-x. Huang, A. Klemm and S. Quackenbush, *Topological string theory on compact Calabi-Yau: Modularity and boundary conditions*, in *Homological mirror symmetry*, Springer, Berlin, Heidelberg, Germany, ISBN 9783540680291 (2008), doi:10.1007/978-3-540-68030-7_3.

[25] S. Alexandrov, S. Feyzbakhsh, A. Klemm, B. Pioline and T. Schimannek, *Quantum geometry, stability and modularity*, (arXiv preprint) doi:10.48550/arXiv.2301.08066.

[26] N. Drukker, M. Mariño and P. Putrov, *Nonperturbative aspects of ABJM theory*, J. High Energy Phys. **11**, 141 (2011), doi:10.1007/JHEP11(2011)141.

[27] J. M. Henn, *Multiloop integrals in dimensional regularization made simple*, Phys. Rev. Lett. **110**, 251601 (2013), doi:10.1103/PhysRevLett.110.251601.

[28] G. Tian, *Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric*, in *Mathematical aspects of string theory*, World Scientific, Singapore, ISBN 9789971502744 (1987), doi:10.1142/9789812798411_0029.

[29] A. N. Todorov, *The Weil-Petersson geometry of the moduli space of SU(n ≥ 3) (Calabi-Yau) manifolds I*, Commun. Math. Phys. **126**, 325 (1989), doi:10.1007/BF02125128.

[30] A. Strominger, *Special geometry*, Commun. Math. Phys. **133**, 163 (1990), doi:10.1007/BF02096559.
[31] P. Candelas and X. C. de la Ossa, *Moduli space of Calabi-Yau manifolds*, Nucl. Phys. B **355**, 455 (1991), doi:10.1016/0550-3213(91)90122-E.

[32] T. W. Grimm, A. Klemm, M. Mariño and M. Weiss, *Direct integration of the topological string*, J. High Energy Phys. **08**, 058 (2007), doi:10.1088/1126-6708/2007/08/058.

[33] M. Alim and J. D. Länge, *Polynomial structure of the (open) topological string partition function*, J. High Energy Phys. **10**, 045 (2007), doi:10.1088/1126-6708/2007/10/045.

[34] S. Hosono, *BCOV ring and holomorphic anomaly equation*, Adv. Stud. Pure Math. **59**, 79 (2010), doi:10.2969/aspm/05910079.

[35] S. Katz, A. Klemm, T. Schimannek and E. Sharpe, *Topological strings on non-commutative resolutions*, (arXiv preprint) doi:10.48550/arXiv.2212.08655.

[36] A. Klemm and E. Zaslow, *Local mirror symmetry at higher genus*, (arXiv preprint) doi:10.48550/arXiv.hep-th/9906046.

[37] K. Bönisch, A. Klemm, E. Scheidegger and D. Zagier, *D-brane masses at special fibres of hypergeometric families of Calabi-Yau threefolds, modular forms, and periods*, (arXiv preprint) doi:10.48550/arXiv.2203.09426.

[38] P. Candelas, X. C. De La Ossa, P. S. Green and L. Parkes, *A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory*, Nucl. Phys. B **359**, 21 (1991), doi:10.1016/0550-3213(91)90292-6.

[39] M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, *Holomorphic anomalies in topological field theories*, Nucl. Phys. B **405**, 279 (1993), doi:10.1016/0550-3213(93)90548-4.

[40] D. R. Morrison, *Picard-Fuchs equations and mirror maps for hypersurfaces*, in *Essays on mirror manifolds*, International Press, Hong Kong, 9789627670018 (1992).

[41] A. Klemm and S. Theisen, *Considerations of one-modulus Calabi-Yau compactifications: Picard-Fuchs equations, Kähler potentials and mirror maps*, Nucl. Phys. B **389**, 153 (1993), doi:10.1016/0550-3213(93)90289-2.

[42] A. Font, *Periods and duality symmetries in Calabi-Yau compactifications*, Nucl. Phys. B **391**, 358 (1993), doi:10.1016/0550-3213(93)90152-F.

[43] A. Libgober and J. Teitelbaum, *Lines on Calabi-Yau complete intersections, mirror symmetry, and Picard-Fuchs equations*, Internat. Math. Res. Notices 29 (1993), doi:10.1155/S1073792893000030.

[44] A. Klemm and S. Theisen, *Mirror maps and instanton sums for complete intersections in weighted projective space*, Mod. Phys. Lett. A **9**, 1807 (1994), doi:10.1142/S0217732394001660.

[45] C. F. Doran and J. W. Morgan, *Mirror symmetry and integral variations of Hodge structure underlying one-parameter families of Calabi-Yau threefolds*, in *BIRS workshop on Calabi-Yau varieties and mirror symmetry*, Banff, Canada (2006).

[46] A. Landman, *On the Picard-Lefschetz transformation for algebraic manifolds acquiring general singularities*, Trans. Amer. Math. Soc. **181**, 89 (1973), doi:10.2307/1996622.

[47] D. van Straten, *Calabi-Yau operators*, in *Uniformization, Riemann-Hilbert correspondence, Calabi-Yau manifolds & Picard-Fuchs equations*, International Press of Boston, Somerville, USA, ISBN 9781571463630 (2018).
[48] A. Joshi and A. Klemm, *Swampland distance conjecture for one-parameter Calabi-Yau three-folds*, J. High Energy Phys. **08**, 086 (2019), doi:10.1007/JHEP08(2019)086.

[49] E. L. Ince, *Ordinary differential equations*, Dover Publications, Mineola, USA, ISBN 9780486603490 (1944).

[50] J. Hofmann and D. van Straten, *Some monodromy groups of finite index in $Sp_4(Z$), J. Aust. Math. Soc. **99**, 48 (2015), doi:10.1017/S1446788715000014.

[51] K. Bönisch, M. Elmi, A.-K. Kashani-Poor and A. Klemm, *Time reversal and CP invariance in Calabi-Yau compactifications*, J. High Energy Phys. **09**, 019 (2022), doi:10.1007/JHEP09(2022)019.

[52] *One parameter Calabi-Yau operators*, Universität Mainz, Mainz, Germany, https://cydb.mathematik.uni-mainz.de.

[53] J. Walcher, *Opening mirror symmetry on the quintic*, Commun. Math. Phys. **276**, 671 (2007), doi:10.1007/s00220-007-0354-8.

[54] K. Bönisch, C. Duhr, F. Fischbach, A. Klemm and C. Nega, *Feynman integrals in dimensional regularization and extensions of Calabi-Yau motives*, J. High Energy Phys. **09**, 156 (2022), doi:10.1007/JHEP09(2022)156.

[55] S. Yamaguchi and S.-T. Yau, *Topological string partition functions as polynomials*, J. High Energy Phys. **07**, 047 (2004), doi:10.1088/1126-6708/2004/07/047.

[56] E. Witten, *Quantum background independence in string theory*, (arXiv preprint) doi:10.48550/arXiv.hep-th/9306122.

[57] S. Codesido, M. Mariño and R. Schiappa, *Non-perturbative quantum mechanics from non-perturbative strings*, Ann. Henri Poincaré **20**, 543 (2018), doi:10.1007/s00023-018-0751-x.

[58] M.-x. Huang, S. Katz and A. Klemm, *Topological string on elliptic CY 3-folds and the ring of Jacobi forms*, J. High Energy Phys. **10**, 125 (2015), doi:10.1007/JHEP10(2015)125.

[59] M. Borinsky, *Renormalized asymptotic enumeration of Feynman diagrams*, Ann. Phys. **385**, 95 (2017), doi:10.1016/j.aop.2017.07.009.

[60] C. Duhr, A. Klemm, F. Loebbert, C. Nega and F. Porkert, *Yangian-invariant fishnet integrals in two dimensions as volumes of Calabi-Yau varieties*, Phys. Rev. Lett. **130**, 041602 (2023), doi:10.1103/PhysRevLett.130.041602.

[61] B. Haghighat, A. Klemm and M. Rauch, *Integrability of the holomorphic anomaly equations*, J. High Energy Phys. **10**, 097 (2008), doi:10.1088/1126-6708/2008/10/097.

[62] R. Gopakumar and C. Vafa, *Branes and fundamental groups*, Adv. Theor. Math. Phys. **2**, 399 (1998), doi:10.4310/ATMP.1998.v2.n2.a7.

[63] P. Candelas, X. de la Ossa, M. Elmi and D. van Straten, *A one parameter family of Calabi-Yau manifolds with attractor points of rank two*, J. High Energy Phys. **10**, 202 (2020), doi:10.1007/JHEP10(2020)202.

[64] R. Gopakumar and C. Vafa, *M-theory and topological strings–II*, (arXiv preprint) doi:10.48550/arXiv.hep-th/9812127.
[65] M.-x. Huang, A. Klemm, M. Mariño and A. Tavanfar, *Black holes and large order quantum geometry*, Phys. Rev. D **79**, 066001 (2009), doi:10.1103/PhysRevD.79.066001.

[66] P. S. Aspinwall and D. R. Morrison, *Topological field theory and rational curves*, Commun. Math. Phys. **151**, 245 (1993), doi:10.1007/BF02096768.

[67] C. Faber and R. Pandharipande, *Hodge integrals and Gromov-Witten theory*, Invent. math. **139**, 173 (2000), doi:10.1007/s002229900028.

[68] M. Mariño and G. Moore, *Counting higher genus curves in a Calabi-Yau manifold*, Nucl. Phys. B **543**, 592 (1999), doi:10.1016/S0550-3213(98)00847-5.

[69] I. Antoniadis, E. Gava, K. S. Narain and T. R. Taylor, *N = 2 type II-heterotic duality and higher-derivative F-terms*, Nucl. Phys. B **455**, 109 (1995), doi:10.1016/0550-3213(95)00467-7.

[70] Z. Liu and Y. Ruan, *Castelnuovo bound and higher genus Gromov-Witten invariants of quintic 3-folds*, (arXiv preprint) doi:10.48550/arXiv:2210.13411.

[71] S. Katz, A. Klemm and C. Vafa, *M-theory, topological strings and spinning black holes*, Adv. Theor. Math. Phys. **3**, 1445 (1999), doi:10.4310/ATMP.1999.v3.n5.a6.

[72] *One parameter Calabi-Yau higher genus data*, Universität Bonn, Bonn, Germany, http://www.th.physik.uni-bonn.de/Groups/Klemm/data.php.

[73] D. Ghoshal and C. Vafa, *c = 1 string as the topological theory of the conifold*, Nucl. Phys. B **453**, 121 (1995), doi:10.1016/0550-3213(95)00408-K.

[74] M. Aganagic, A. Klemm, M. Mariño and C. Vafa, *Matrix model as a mirror of Chern-Simons theory*, J. High Energy Phys. **02**, 010 (2004), doi:10.1088/1126-6708/2004/02/010.

[75] V. Bouchard and R. Cavalieri, *On the mathematics and physics of high genus invariants of $[C^3/Z_3]$*, Adv. Theor. Math. Phys. **13**, 695 (2009), doi:10.4310/ATMP.2009.v13.n3.a4.

[76] G. Moore, *Arithmetic and attractors*, (arXiv preprint) doi:10.48550/arXiv.hep-th/9807087.

[77] T. Schimannek, *Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges*, J. High Energy Phys. **02**, 007 (2022), doi:10.1007/JHEP02(2022)007.

[78] J. C. Le Guillou and J. Zinn-Justin (eds.), *Large order behavior of perturbation theory*, Elsevier, Amsterdam, Netherlands, ISBN 9780444596208 (1990).

[79] J. Écalle, *Les fonctions résurgentes*, Publications Mathématiques d'Orsay, Paris, France (1981).

[80] J. Gu and M. Mariño, *Peacock patterns and new integer invariants in topological string theory*, SciPost Phys. **12**, 058 (2022), doi:10.21468/SciPostPhys.12.2.058.

[81] T. Reis, *On the resurgence of renormalons in integrable theories*, PhD thesis, University of Geneva, Geneva, Switzerland (2022), (arXiv preprint) doi:10.48550/arXiv.2209.15386.

[82] N. Drukker, M. Mariño and P. Putrov, *From weak to strong coupling in ABJM theory*, Commun. Math. Phys. **306**, 511 (2011), doi:10.1007/s00220-011-1253-6.
[83] R. Couso-Santamaría, R. Schiappa and R. Vaz, *On asymptotics and resurgent structures of enumerative Gromov-Witten invariants*, Commun. Num. Theor. Phys. **11**, 707 (2017), doi:10.4310/CNTP2017.v11.n4.a1.

[84] S. Pasquetti and R. Schiappa, *Borel and Stokes nonperturbative phenomena in topological string theory and $c=1$ matrix models*, Ann. Henri Poincaré **11**, 351 (2010), doi:10.1007/s00023-010-0044-5.

[85] M. Alim, A. Saha, J. Teschner and I. Tulli, *Mathematical structures of non-perturbative topological string theory: From GW to DT invariants*, Commun. Math. Phys. **399**, 1039 (2022), doi:10.1007/s00220-022-04571-y.

[86] M. Alim, L. Hollands and I. Tulli, *Quantum curves, resurgence and exact WKB*, SIGMA **19**, 009 (2023), doi:10.3842/SIGMA.2023.009.

[87] A. Grassi, Q. Hao and A. Neitzke, *Exponential networks, WKB and topological string*, SIGMA **19**, 064 (2023), doi:10.3842/SIGMA.2023.064.

[88] B. Candelpergher, J.-C. Nosmas and F. Pham, *Approche de la résurgence*, Editions Hermann, Paris, France, ISBN 9782705662011 (1993).

[89] O. Costin, *Asymptotics and Borel summability*, CRC press, New York, USA, ISBN 9780429141485 (2008), doi:10.1201/9781420070323.

[90] S. Codesido, *A geometric approach to non-perturbative quantum mechanics*, PhD thesis, University of Geneva, Geneva, Switzerland (2018).

[91] E. Witten, *Some computations in background-independent off-shell string theory*, Phys. Rev. D **47**, 3405 (1993), doi:10.1103/PhysRevD.47.3405.

[92] S. L. Shatashvili, *Comment on the background independent open string theory*, Phys. Lett. B **311**, 83 (1993), doi:10.1016/0370-2693(93)90537-R.

[93] M. Mariño, R. Schiappa and M. Weiss, *Multi-instantons and multicuts*, J. Math. Phys. **50**, 052301 (2009), doi:10.1063/1.3097755.

[94] R. Couso-Santamaría, M. Mariño and R. Schiappa, *Resurgence matches quantization*, J. Phys. A: Math. Theor. **50**, 145402 (2017), doi:10.1088/1751-8121/aa5e01.

[95] O. Costin and G. V. Dunne, *Conformal and uniformizing maps in Borel analysis*, Eur. Phys. J. Spec. Top. **230**, 2679 (2021), doi:10.1140/epjs/s11734-021-00267-x.

[96] N. Seiberg and E. Witten, *Electric-magnetic duality, monopole condensation, and confinement in $N=2$ supersymmetric Yang-Mills theory*, Nucl. Phys. B **426**, 19 (1994), doi:10.1016/0550-3213(94)90124-4.

[97] N. Nekrasov, *Z theory: Chasing m/f theory*, (arXiv preprint) doi:10.48550/arXiv.hep-th/0412021.

[98] M. Mariño, R. Miravitllas and T. Reis, *Instantons, renormalons and the theta angle in integrable sigma models*, SciPost Phys. **15**, 184 (2023), doi:10.21468/SciPostPhys.15.5.184.

[99] Z. Bajnok, J. Balog and I. Vona, *The full analytic trans-series in integrable field theories*, Phys. Lett. B **844**, 138075 (2023), doi:10.1016/j.physletb.2023.138075.