Resurgent Structure of the Topological String and the First Painlevé Equation

Resurgent Structure of the Topological String
and the First Painlevé Equation

Kohei IWAKI a and Marcos MARIÑO b

K. Iwaki and M. Mariño

a) Graduate School of Mathematical Sciences, The University of Tokyo,
a) 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan<EMAIL_ADDRESS>
b) Section de Mathématiques et Département de Physique Théorique, Université
de Genève,
b) 1211 Genève 4, Switzerland<EMAIL_ADDRESS>http://www.marcosmarino.net

Received September 13, 2023, in final form March 19, 2024; Published online
April 02, 2024

We present an explicit formula for the Stokes automorphism acting on the
topological string partition function. When written in terms of the dual
partition function, our formula implies that flat coordinates in topological
string theory transform as quantum periods, and according to the
Delabaere–Dillinger–Pham formula. We first show how the formula follows from
the non-linear Stokes phenomenon of the Painlevé I equation, together with the
connection between its $\tau$-function and topological strings on elliptic
curves. Then, we show that this formula is also a consequence of a recent
conjecture on the resurgent structure of the topological string, based on the
holomorphic anomaly equations, and it is in fact valid for arbitrary
Calabi–Yau threefolds.

resurgence; topological string theory; first Painlevé equation; Borel
resummation; Stokes automorphisms

81T45; 14N35; 34M40; 34M55

##  1 Introduction and conclusions

Topological string theory has been at the center of many developments in
modern mathematical physics. For example, through string dualities [11, 26]
and the topological recursion of [32] it is related to matrix models and the
$1/N$ expansion. It has been also shown that certain “dual” partition
functions appearing in topological string theory are closely related to
$\tau$-functions of Painlevé equations [30, 31, 45, 56]. These results have a
close relation with the pioneering works [9, 37].

The basic quantities in topological string theory are given by formal
perturbative series in the string coupling constant $g_{s}$, which are
factorially divergent. One can then ask, in the spirit of the theory of
resurgence [29, 58], what is the resurgent structure of these series, i.e.,
what are the exponentially small trans-series associated to their Borel
singularities, and the associated Stokes constants. This question has been
recently addressed in various works. Based on the trans-series approach of
[19, 20, 21], a conjecture for the resurgent structure of topological string
theory on arbitrary Calabi–Yau (CY) threefolds was put forward in [41, 42].
Although topological strings are closely related to quantum periods (also
known as Voros symbols), their resurgent structure is quite different. For
example, it is expected that the Stokes automorphisms of quantum periods are
given by the Delabaere–Dillinger–Pham (DDP) formula [23, 24, 25] (see [22, 43]
for recent arguments in that direction), but the Stokes automorphisms act on
the topological string partition function with a more complicated structure,
and no closed formula is known for them.

In this paper we provide such a closed formula. It reads

${\mathfrak{S}}Z(\nu;g_{s})=\exp\bigl{\\{}a\operatorname{Li}_{2}\bigl{(}{\rm
e}^{-g_{s}\partial}\bigr{)}-g_{s}a\log\bigl{(}1-{\rm
e}^{-g_{s}\partial}\bigr{)}\partial\bigr{\\}}Z(\nu;g_{s}),$ (1.1)

where $a$ is a Stokes constant, $\nu$ is the flat coordinate (or period)
parametrizing the moduli space, and $\partial$ is the derivative with
reference to $\nu$.111We have written the formula in a somewhat schematic way,
and for a simple case, involving a single modulus; additional details,
generalizations and clarifications can be found below.

Although the partition function transforms in a relatively complicated way
under the Stokes automorphism, the dual partition function (obtained from the
original one after a discrete Fourier transform) transforms in a very simple
way:

${\mathfrak{S}}\tau(\nu,\rho;g_{s})={\rm e}^{a\operatorname{Li}_{2}({\rm
e}^{2\pi{\rm i}\rho/g_{s}})}\tau\big{(}\nu-ag_{s}\log\big{(}1-{\rm
e}^{2\pi{\rm i}\rho/g_{s}}\big{)},\rho;g_{s}\big{)},$ (1.2)

i.e., it picks up a prefactor involving the dilogarithm function, and its
argument transforms indeed as a quantum period, following the DDP formula!

In this paper, we give two lines of argument that lead to (1.1) and (1.2). The
first one is based on the connection found in [45] between $\tau$-functions of
the first Painlevé equation, and a dual partition function in topological
string theory. By using this connection, one can show that the non-linear
Stokes phenomenon of the first Painlevé equation leads to the above
transformation formula for the dual partition function under a Stokes
automorphism. Although this derivation is based on a particular example of the
topological string partition function, the resulting formula turns out to be
universally valid. To show this, we develop a second argument based on the
conjectures for the resurgent structure of the topological string proposed in
[41, 42], and we provide a derivation of (1.1) in the spirit of alien
calculus. Since the conjectures of [41, 42] apply to any topological string
partition function which satisfies the holomorphic anomaly equations (HAE) of
[6], we conclude that (1.1) applies universally to closed topological strings
on arbitrary backgrounds. It can be extended to refined and real topological
strings by using similar ideas and techniques, see [1, 57].

As is mentioned above, the derivation based on [45] naturally relates the DDP
formula to our main formula. The DDP formula has a close relationship with the
BPS invariants in class ${\mathcal{S}}$ theories, which are defined by a
weighted counting of saddle connections in the spectral network [36]. In fact,
the DDP formula for the quantum period $V_{\mu}$ reads

${\rm e}^{V_{\mu}}\mapsto{\rm e}^{V_{\mu}}\bigl{(}1+\sigma(\gamma){\rm
e}^{V_{\gamma}}\bigr{)}^{\Omega(\gamma)\langle\mu,\gamma\rangle},$ (1.3)

where $\gamma$ is the cycle on the spectral curve associated with the saddle
connection, the sign $\sigma(\gamma)\in\\{\pm 1\\}$ is specified by the type
of saddle connection, and $\Omega(\gamma)$ is the BPS invariant; see, e.g.,
[15, 36]. In view of this, we expect the Stokes constant $a$ appearing in
(1.1), (1.2) and in the formulas in Section 3 to be given by a BPS invariant
$\Omega(\gamma)$. The precise relation is

$a=\frac{\Omega(\gamma)}{2\pi{\rm i}}.$ (1.4)

In the more general case considered in this paper, the $\Omega(\gamma)$ are
given by the Donaldson–Thomas invariants of the CY threefold. Here, the cycle
$\gamma$ is dual to the Borel singularity ${\mathcal{A}}$ underlying the
Stokes discontinuity, and as we explain below ${\mathcal{A}}$ is an integral
period of the CY geometry (up to an overall normalization). This
identification between Stokes constants appearing in the resurgent structure
of the topological string and BPS invariants was already suggested in [41,
42], and it is also consistent with the results of [46, 47]. Further evidence
for this identification appears in the recent paper [40].

We should mention that close cousins of (1.2) have appeared before in
different, but related contexts. In [3], the transformation properties of a
certain family of theta series defined on the (twistor space of the)
hypermultiplet moduli space of CY threefolds under wall-crossing were studied.
In the special case $k=1$, the theta series (1.5) in [3] reduces to the dual
topological partition function [2], and their wall-crossing formula (1.9)
agrees with our formula for the Stokes automorphism. In addition, for the
formula to agree, one must have the relation (1.4), giving in this way
additional evidence for the identification between Stokes constants and
Donaldson–Thomas invariants. A detailed comparison between the wall-crossing
formula of [3] and our formula will be made in Section 3. A similar wall-
crossing formula was also found in [18] for the dual partition functions of
some ${\mathcal{N}}=2$ gauge theories. Let us point out that in [2, 3, 18] the
transformation of the flat coordinate according to the DDP formula is
essentially built in, while in our case it follows in a more indirect (and
surprising) way from the resurgent structure of the topological string
perturbative series. It would be very interesting to understand better the
relation between [2, 3, 18] and our approach.

The result obtained in this paper can be understood as relating the Stokes
automorphism acting on the topological string partition function, to the
Stokes automorphism acting on quantum periods. It might have a connection to
the blowup formula which relates in a similar way the topological string free
energy to the Nekrasov–Shatashvili free energy [39, 50]. This idea has also
been developed in [40] and it might lead to a different derivation of our main
formula. It would be also interesting to study the relation between our work
and a series of papers by Bridgeland [12, 13, 14].

After this paper was finished, we were informed by R. Schiappa and M. Schwick
that in forthcoming work they address similar issues and obtain related
results, albeit with different methods.

##  2 From non-linear Stokes phenomenon of the first
Painlevé transcendents to the main formula

In this section, we give a derivation of the formula (1.2) for the topological
recursion partition function through the analysis of the non-linear Stokes
phenomenon of the first Painlevé equation. Our derivation is based on the
exact WKB theoretic approach to the Stokes multipliers of the isomonodromy
system associated with the first Painlevé equation, which was developed in
[45, Section 5]. We note that our result is closely related to [5, 65, 67],
and we will make a comment in the end of this section. We also note that our
derivation is based on several conjectures on the Borel summability of
partition function and wave function etc., which are naturally expected from
the viewpoint of the exact WKB analysis of Schrödinger equations (see [54],
for example). See [29, 58] for materials in Borel summation method and
resurgence.

###  2.1 Topological recursion and Painlevé I

Let us briefly review the relationship between topological recursion and the
first Painlevé equation.

We focus on the topological recursion partition function

$Z(t,\nu;g_{s})=\exp\biggl{(}\sum_{g\geq 0}g_{s}^{2g-2}F_{g}(t,\nu)\biggr{)}$
(2.1)

defined from a family of genus 1 spectral curves of the form

$\Sigma\colon\ y^{2}=4x^{3}+2tx+u(t,\nu)$ (2.2)

parametrized by independent complex parameters $t$ and $\nu$. Here, $u(t,\nu)$
is a locally-defined function given by the implicit relation

$\displaystyle\nu=\frac{1}{2\pi{\rm i}}\oint_{A}y\,{\rm d}x$ (2.3)

after fixing a choice of a symplectic basis $A$, $B$ of
$H_{1}(\Sigma,{\mathbb{Z}})$. See Appendix A for the definition of $F_{g}$
(see also [32, 45]). These quantities depend on the choice of the symplectic
basis, and we will choose a specific one to derive our main formula in Section
2.3.

It was shown in [45] that the discrete Fourier transform

$\tau(t,\nu,\rho;g_{s})=\sum_{k\in{\mathbb{Z}}}{\rm e}^{2\pi{\rm
i}k\rho/g_{s}}Z(t,\nu+kg_{s};g_{s})$ (2.4)

gives a $\tau$-function for the Painlevé I equation222This construction is now
generalized to all Painlevé equations; see [30, 31, 56].

$g_{s}^{2}\frac{{\rm d}^{2}q}{{\rm d}t^{2}}=6q^{2}+t.$ (2.5)

That is,

$q(t,\nu,\rho;g_{s})=-g_{s}^{2}\frac{{\rm d}^{2}}{{\rm
d}t^{2}}\log\tau(t,\nu,\rho;g_{s})$ (2.6)

is a formal solution of (2.5). Actually, the formal series (2.4) and (2.6)
(and (2.9) below) are not a usual power series in $\hbar$, but can be regarded
as a two-sided trans-series (which contain both positive and negative
exponential factors). These exponential terms can be summed up to
$\vartheta$-functions. The discrete Fourier transform can be regarded as a
non-perturbative correction to perturbative partition function; see [31, 33,
45] for details. The parameters $\nu$ and $\rho$ are regarded as integration
constants parametrizing the general solution of Painlevé I. The formula (2.4)
is analogous to the formula obtained in [9, 37].

To derive our main formula (1.1), we will also use the isomonodromy system
associated with Painlevé I [51]:

$\displaystyle\begin{split}&\left[g_{s}^{2}\frac{\partial^{2}}{\partial
x^{2}}-\frac{g_{s}}{x-q}\left(g_{s}\frac{\partial}{\partial
x}-p\right)-\left(4x^{3}+2tx+2H\right)\right]Y=0,\\\
&\left[g_{s}\frac{\partial}{\partial
t}-\frac{1}{2(x-q)}\left(g_{s}\frac{\partial}{\partial
x}-p\right)\right]Y=0,\end{split}$ (2.7)

where

$p=g_{s}\frac{{\rm d}q}{{\rm d}t},\qquad H=\frac{1}{2}p^{2}-2q^{3}-tq.$ (2.8)

It is well known that the compatibility condition of the system (2.7) of PDEs
is given by the Painlevé I equation. The compatibility implies that the Stokes
multipliers333This paper discusses two types of Stokes phenomenon. The first
one is related to the limit $g_{s}\to 0$, while the second one is related to
$x\to\infty$. Our main formulas (1.1) and (1.2) are related to the first type
Stokes phenomenon, but we analyze the second type as well to derive the main
formulas. of the first equation of the system (2.7), which is a linear ODE
with an irregular singular point $x=\infty$ of Poincaré rank $5/2$, are
independent of the isomonodromic time $t$. It was also shown in [45] that

$Y_{\pm}(x,t,\nu,\rho;g_{s})=\frac{\sum_{k\in{\mathbb{Z}}}{\rm e}^{2\pi{\rm
i}k\rho/g_{s}}Z(t,\nu+kg_{s};g_{s})\,\chi_{\pm}(x,t,\nu+kg_{s};g_{s})}{\sum_{k\in{\mathbb{Z}}}{\rm
e}^{2\pi{\rm i}k\rho/g_{s}}Z(t,\nu+kg_{s};g_{s})}$ (2.9)

gives a formal solution for the isomonodromy system. Here, $\chi_{\pm}$ is a
WKB-type formal series defined by the “quantum curve formula” as
follows444Precisely speaking, we must regularize the integral in (2.10) for
$(g,n)=(0,1)$ and $(0,2)$; see [45]. (see [10], for example):

$\displaystyle\chi_{\pm}(x,t,\nu;g_{s})$ (2.10)
$\displaystyle=\exp\Biggl{(}\sum_{\begin{subarray}{c}g\geq 0\\\ n\geq
1\end{subarray}}\frac{(\pm
g_{s})^{2g-2+n}}{n!}\int^{z(x)}_{0}\cdots\int^{z(x)}_{0}\left(\omega_{g,n}(z_{1},\dots,z_{n})-\delta_{g,0}\delta_{n,2}\frac{{\rm
d}x(z_{1}){\rm d}x(z_{2})}{(x(z_{1})-x(z_{2}))^{2}}\right)\Biggr{)}.$

Here, $\omega_{g,n}$’s are the topological recursion correlators (see Appendix
A for the definition).

The most important formula to have a formal solution to the isomonodromy
system (2.7) is the following one, which describes the term-wise (in $g_{s}$)
analytic continuation along $A$-cycle and $B$-cycle with respect to
$x$-variable:

$Y_{\pm}(x,t,\nu,\rho;g_{s})\mapsto\begin{cases}{\rm e}^{\pm 2\pi{\rm
i}\nu/g_{s}}\,Y_{\pm}(x,t,\nu,\rho;g_{s})&\text{along $A$-cycle},\\\\[5.0pt]
{\rm e}^{\mp 2\pi{\rm i}\rho/g_{s}}\,Y_{\pm}(x,t,\nu,\rho;g_{s})&\text{along
$B$-cycle}.\end{cases}$ (2.11)

See [45, Theorems 3.9 and 4.8] for the derivation of (2.11). In the spirit of
the exact WKB analysis, and the abelianization in the sense of
Gaiotto–Moore–Neitzke [35], the monodromy and Stokes data of a Schrödinger-
type linear ODE should be written in terms of the quantum periods on the
spectral curve (see [54, Section 3] for more details). Exponentials of those
periods are called Voros symbols, which play an important role in the exact
WKB analysis (see [23, 54, 68], for example). In view of the property (2.11),
it is natural to define the Voros symbols of the isomonodromy system (2.7)
along $A$-cycle and $B$-cycle by ${\rm e}^{2\pi{\rm i}\nu/g_{s}}$ and ${\rm
e}^{2\pi{\rm i}\rho/g_{s}}$, respectively, even though the parameter $\rho$ is
not an actual period integral on the spectral curve. These Voros symbols are
$t$-independent, so it is natural to expect that $Y_{\pm}$ satisfies an
isomonodromy system. This is a philosophical remark on the result of [45].

The discrete Fourier transform is absolutely crucial to obtain the constant
monodromy property (2.11) and the quantum curve (i.e., the first equation in
(2.7)) as an ODE. The property is proved for more general spectral curves in
[31]. It would be interesting to apply the following method to other Painlevé
equations.

###  2.2 Stokes graph of the isomonodromy system

Here we also review the discussion of [45, Section 5] on heuristic computation
of the Stokes multipliers of the isomonodromy system.

Let us take the meromorphic quadratic differential

$\varphi(x)=\big{(}4x^{3}+2tx+u(t,\nu)\big{)}{\rm d}x^{2},$ (2.12)

and define the Stokes graph of (the first equation of) the isomonodromy system
(2.7) as the graph on ${\mathbb{P}}^{1}$ as follows:

  * •

The vertices of the Stokes graph consists of zeros and poles of $\varphi$.

  * •

The edges of the Stokes graph, called Stokes curves, are trajectories of
$\varphi$ emanating from a zero of $\varphi$.

Here, a (horizontal) trajectory of $\varphi$ is any maximal leaf of the
foliation on ${\mathbb{P}}^{1}$ defined by

$\displaystyle{\rm Im}\int^{x}\sqrt{\varphi(x)}={\rm constant}.$ (2.13)

See [63] for more details on trajectories of quadratic differentials.

The Stokes graph is also known as an example of spectral networks of [35].
Note that the Stokes graph depends on the parameters $t$ and $\nu$. Away from
the zero locus of the discriminant, $\varphi$ has three simple zeros, and
three Stokes curves emanate from each simple zero. Each face of the Stokes
graph is called a Stokes region. Some examples are shown in Figure 1. The
neighborhood of the irregular singular point $x=\infty$ is divided into five
sectors by asymptotic directions of Stokes curves. The five directions are
called singular directions.

$t=t_{c}$
$t=t_{c}-{\rm i}\epsilon$ $t=t_{c}+{\rm i}\epsilon$

Figure 1: Stokes graphs for several $t$ (with $\nu=1$). Here we choose
$t_{c}=-5$ and $\epsilon=1/2$.

The main conjectural ansatz for the computation of the Stokes multipliers of
the isomonodromy system (2.7) around $x=\infty$ are the following:

  * (i)

If the Stokes graph does not contain any saddle connection (i.e., a Stokes
curve connecting zeros of $\varphi$), then the partition function (2.1) is
Borel summable. Moreover, the discrete Fourier series

$\displaystyle{\mathscr{T}}(t,\nu,\rho;g_{s})=\sum_{k\in{\mathbb{Z}}}{\rm
e}^{2\pi{\rm i}k\rho/g_{s}}{\mathcal{Z}}(t,\nu+kg_{s};g_{s})$ (2.14)

converges and gives an analytic $\tau$-function for Painlevé I. Here,
${\mathcal{Z}}$ is the Borel sum of the partition function $Z$.

  * (ii)

Under the same saddle-free condition, the WKB-type series $\chi_{\pm}$ defined
in (2.10) is Borel summable on each Stokes region. Moreover,

$\displaystyle{\mathscr{Y}}_{\pm}(x,t,\nu,\rho;g_{s})=\frac{\sum_{k\in{\mathbb{Z}}}{\rm
e}^{2\pi{\rm
i}k\rho/g_{s}}{\mathcal{Z}}(t,\nu+kg_{s};g_{s})\,{\mathcal{X}}_{\pm}(x,t,\nu+kg_{s};g_{s})}{\sum_{k\in{\mathbb{Z}}}{\rm
e}^{2\pi{\rm i}k\rho/g_{s}}{\mathcal{Z}}(t,\nu+kg_{s};g_{s})}$ (2.15)

converges and give an analytic solution of the isomonodromy system (2.7) on
the Stokes region. Here, ${\mathcal{X}}_{\pm}$ is the Borel sum of
$\chi_{\pm}$.

  * (iii)

Under the same saddle-free condition, the Borel sums ${\mathcal{X}}_{\pm}$ of
the WKB-type series (2.10) defined on adjacent faces of the Stokes graph are
related by the Voros connection formula [54, 62, 68] (or the path-lifting rule
in the sense of Gaiotto–Moore–Neitzke [35]).

The assumption (i) is consistent with the conjecture in [27, 41, 42] which
claims the Borel singularities appear on the lattice generated by the integral
periods of $y{\rm d}x$. For a class of genus $0$ spectral curves, the
conjecture is proved rigorously in [46, 47]. Note that the saddle-free
condition in (i) is satisfied if all the integral periods of $y{\rm d}x$ on
the spectral curve have a non-zero imaginary part.

The assumption (ii) is inferred from the known result on the Borel summability
of WKB solution of Schrödinger equations (cf. [28, 59, 60] and an unpublished
work by Koike–Schäfke). These known rigorous results are not applicable to our
case since $\chi_{\pm}$ defined in (2.10) does not satisfy an ODE but a PDE,
but we expect that the same statement is true since the PDE also has $\Sigma$
as the classical limit (see [45, Theorem 3.7]). Under the assumption (ii), we
have five canonical solutions ${\mathscr{Y}}^{(j)}_{\pm}$ defined in the
Stokes region $D_{j}$ in Figure 1 ($j=0,\pm 1,\pm 2$ mod $5$). Then, we define
the Stokes multiplier $s_{j}$ attached to the $j$-th singular direction $\arg
x=2\pi j/5$ as the non-trivial entry of the Stokes matrix ${\mathcal{S}}_{j}$
defined by

$\displaystyle\big{(}{\mathscr{Y}}^{(j+1)}_{+},{\mathscr{Y}}^{(j+1)}_{-}\big{)}=\big{(}{\mathscr{Y}}^{(j)}_{+},{\mathscr{Y}}^{(j)}_{-}\big{)}\cdot{\mathcal{S}}_{j},$
(2.16)

where

${\mathcal{S}}_{j}=\begin{cases}\begin{pmatrix}1&s_{j}\\\
0&1\end{pmatrix},&j=0,\pm 2,\\\\[10.00002pt] \begin{pmatrix}1&0\\\
s_{j}&1\end{pmatrix},&j=\pm 1.\end{cases}$ (2.17)

The Voros connection formula in (iii) is also well established in the case of
Schrodinger equation in [54, 68]. It enables us to describe these Stokes
multipliers explicitly via the Voros symbols ${\rm e}^{2\pi{\rm i}\nu/g_{s}}$
and ${\rm e}^{2\pi{\rm i}\rho/g_{s}}$. A rough explanation is the following:
The Voros connection formula allows us to describe the analytic continuation
of the Borel resummed WKB solution by the term-wise analytic continuations on
the spectral curve, and the formula (2.11) gives an explicit description of
those analytic continuations in terms of the Voros symbols (see [45, Section
5] for details.). We will present the resulting Stokes multipliers in the next
subsection.

###  2.3 Delabaere–Dillinger–Pham formula and non-linear Stokes phenomenon

Figure 2: $A$-cycle and $B$-cycle.

Now, let us discuss the non-linear Stokes phenomenon for Painlevé I. We borrow
the idea of Takei used in [64, 65] where he relates the mutation of Stokes
graphs (which we will explain shortly) to non-linear Stokes phenomenon of
Painlevé transcendents, through the invariance of the Stokes multipliers of
(2.7). This idea has been recently further developed in [5, 67]. We also note
that our result is closely related to [34, 53] based on the Riemann–Hilbert
method.

Figure 1 depicts Stokes graphs for $\nu=1$ and several $t$’s which are close
to $t_{c}=-5$. Let us label the three zeros of $\varphi(x)$ at $t=t_{c}$ as

$e_{1}\fallingdotseq 1.59075,\qquad e_{2}\fallingdotseq-0.01288,\qquad
e_{3}\fallingdotseq-1.57157,$ (2.18)

and take the $A$ cycle (resp., the $B$ cycle) as the cycle encircling the
zeros $e_{1}$ and $e_{2}$ (resp., $e_{2}$ and $e_{3}$). See Figure 2. Here the
orientation of these cycle is given so that

$\oint_{A}y{\rm d}x=2\int^{e_{1}}_{e_{2}}y{\rm d}x,\qquad\oint_{B}y{\rm
d}x=-2\int^{e_{2}}_{e_{3}}y{\rm d}x,$ (2.19)

where the branch of $y$ is chosen so that it has a positive imaginary part
(resp., a negative real part) on the segment $[e_{2},e_{1}]$ (resp.,
$[e_{3},e_{2}]$). We may observe that a saddle connection appears at
$t=t_{c}$, where the $B$-period

$\oint_{B}y{\rm d}x\fallingdotseq 5.87065$ (2.20)

of $y{\rm d}x$ has zero imaginary part. As we have mentioned in Section 2.2,
we expect that this induces singularities on the positive real axis on the
Borel-plane, and we will discuss the action of the Stokes automorphism below.

The saddle connection is resolved and we have saddle-free Stokes graphs under
a small variation of $t$ since the $B$-periods of $ydx$ are no longer real:

$\oint_{B}y{\rm d}x\fallingdotseq\begin{cases}5.83465+1.47447{\rm i}&\text{at
$t=t_{c}-{\rm i}\epsilon$},\\\ 5.83465-1.47447{\rm i}&\text{at $t=t_{c}+{\rm
i}\epsilon$}.\end{cases}$ (2.21)

Here we take $\epsilon=1/2$. We can observe that the topology of the Stokes
graphs changes discontinuously before and after the appearance of the saddle
connection. This is what we call the mutation of Stokes graph.

Since the Stokes graphs at $t=t_{c}\pm{\rm i}\epsilon$ are saddle-free, we can
use the recipe of [45, Section 5] to compute the Stokes multipliers around
$x=\infty$. The resulting Stokes multipliers are

$\displaystyle\text{At $t=t_{c}-{\rm i}\epsilon$}\colon\
\begin{cases}s_{-2}={\rm i}{\rm e}^{2\pi{\rm i}\nu/g_{s}},\\\ s_{-1}={\rm
i}\big{(}{\rm e}^{-2\pi{\rm i}\nu/g_{s}}-{\rm e}^{-2\pi{\rm
i}(\nu+\rho)/g_{s}}+{\rm e}^{-2\pi{\rm i}\rho/g_{s}}\big{)},\\\ s_{0}={\rm
i}{\rm e}^{2\pi{\rm i}\rho/g_{s}},\\\ s_{1}={\rm i}{\rm e}^{-2\pi{\rm
i}\rho/g_{s}}\big{(}1-{\rm e}^{2\pi{\rm i}\nu/g_{s}}\big{)},\\\ s_{2}={\rm
i}{\rm e}^{-2\pi{\rm i}\nu/g_{s}}\big{(}1-{\rm e}^{2\pi{\rm
i}\rho/g_{s}}\big{)},\end{cases}$ (2.22) $\displaystyle\text{At $t=t_{c}+{\rm
i}\epsilon$}\colon\ \begin{cases}{s}_{-2}={\rm i}{\rm e}^{2\pi{\rm
i}\nu/g_{s}}\big{(}1-{\rm e}^{2\pi{\rm i}\rho/g_{s}}\big{)},\\\ {s}_{-1}={\rm
i}{\rm e}^{-2\pi{\rm i}\rho/g_{s}}\big{(}1-{\rm e}^{-2\pi{\rm
i}\nu/g_{s}}\big{)},\\\ {s}_{0}={\rm i}{\rm e}^{2\pi{\rm i}\rho/g_{s}},\\\
{s}_{1}={\rm i}\big{(}{\rm e}^{-2\pi{\rm i}\rho/g_{s}}-{\rm e}^{2\pi{\rm
i}(\nu-\rho)/g_{s}}+{\rm e}^{2\pi{\rm i}\nu/g_{s}}\big{)},\\\ {s}_{2}={\rm
i}{\rm e}^{-2\pi{\rm i}\nu/g_{s}}.\end{cases}$ (2.23)

Although Figure 1 is drawn for $\epsilon=1/2$, the formulas (2.22) and (2.23)
are valid for any sufficiently small $\epsilon>0$ since the topological
configuration of the Stokes graphs are same for those $\epsilon$. We can check
that the consistency conditions (cf. [66])

$1+s_{j-1}s_{j}+{\rm i}s_{j}+2=0,\qquad s_{j}=s_{j+5}$ (2.24)

of the Stokes multipliers are satisfied for both cases. We note that the
result agrees with the computation of [5, Section 7.4]. This is also
consistent with [65, 67] through the identification of [5, equations
(7.71)–(7.72)].

Thus, we have seen that the mutation of Stokes graphs induces a discontinuous
change of the expressions of the Stokes multipliers. It is easy to observe
that the Stokes data (2.23) are obtained from (2.22) by the transformation

$\big{(}{\rm e}^{2\pi{\rm i}\nu/g_{s}},{\rm e}^{2\pi{\rm
i}\rho/g_{s}}\big{)}\mapsto\big{(}{\rm e}^{2\pi{\rm i}\nu/g_{s}}\big{(}1-{\rm
e}^{2\pi i\rho/g_{s}}\big{)},{\rm e}^{2\pi{\rm i}\rho/g_{s}}\big{)}.$ (2.25)

In fact, this is an example of a cluster transformation (or a
Kontsevich–Soibelman transformation). Our observation is consistent with the
results [23, 25, 36, 48, 49] where the mutation of Stokes graphs induces a
cluster transformation for the Voros symbols (the quantum period, the
Fock–Goncharov coordinates). (2.25) is an example of the
Delabaere–Dillinger–Pham (DDP) formula.

Now we get to the crucial point. Since $t$ is the isomonodromic time, these
Stokes multipliers must be preserved under the variation of $t$. Therefore,
the above formulas suggest the following: Let $(\nu^{+},\rho^{+})$ and
$(\nu^{-},\rho^{-})$ be possibly different pairs of parameters, and let
${\mathscr{T}}^{\pm}(t,\nu^{\pm},\rho^{\pm},g_{s})$ be the Borel sum of
$\tau(t,\nu^{\pm},\rho^{\pm},g_{s})$ defined at $t=t_{c}\pm{\rm i}\epsilon$.
If these $\tau$-functions correspond to a common solution of Painlevé I, then
they must be glued at $t=t_{c}$ so that the corresponding Stokes multipliers
are identical. Namely, we have

${\mathscr{T}}_{P_{\rm I}}^{-}(t,\nu^{-},\rho^{-};g_{s})={\rm
e}^{\frac{1}{2\pi{\rm i}}\operatorname{Li}_{2}({\rm e}^{2\pi{\rm
i}\rho^{+}/g_{s}})}{\mathscr{T}}_{P_{\rm I}}^{+}(t,\nu^{+},\rho^{+};g_{s})$
(2.26)

with

$(\nu^{+},\rho^{+})=\left(\nu^{-}-\frac{g_{s}}{2\pi{\rm i}}\log\big{(}1-{\rm
e}^{2\pi{\rm i}\rho^{-}/g_{s}}\big{)},\rho^{-}\right).$ (2.27)

Here, the prefactor $\exp\big{(}\frac{1}{2\pi{\rm
i}}\operatorname{Li}_{2}\big{(}e^{2\pi{\rm i}\rho^{+}/g_{s}}\big{)}\big{)}$ on
the right hand side is related to the generating function of the monodromy
symplectomorphism (which is the cluster transformation in this case). The
2-form $2\pi{\rm i}g_{s}^{-2}{\rm d}\nu\wedge{\rm d}\rho$ is a natural
symplectic form on the space of initial conditions for Painlevé I, and (2.27)
is a symplectic transform on the space. It was proposed in [55, equation
(3.17)] that the (extended) $\tau$-function for Painelvé I should be defined
by taking the exponential of the primitive of $-2\pi{\rm i}g_{s}^{-2}\nu{\rm
d}\rho$ as a normalization factor (see also [7, 8, 18]). The relation (2.27)
implies

$-\frac{2\pi{\rm i}}{g_{s}^{2}}\left(\nu^{-}{\rm d}\rho^{-}-\nu^{+}{\rm
d}\rho^{+}\right)={\rm d}\log{\rm e}^{\frac{1}{2\pi{\rm
i}}\operatorname{Li}_{2}({\rm e}^{2\pi{\rm i}\rho^{+}/g_{s}})},$ (2.28)

and hence we have taken the prefactor in (2.26).555We would like to thank I.
Coman and F. Del Monte for helpful discussion on the prefactor.

The formula (2.26) can be regarded as the connection formula which describes
the non-linear Stokes phenomenon for the Painlevé transcendents on the
negative real axis in $t$-plane. We may write the formula in terms of the
Stokes automorphism as

${\mathfrak{S}}\tau(t,\nu,\rho;g_{s})={\rm e}^{\frac{1}{2\pi{\rm
i}}\operatorname{Li}_{2}(e^{2\pi{\rm
i}\rho/g_{s}})}\tau\left(t,\nu-\frac{g_{s}}{2\pi{\rm
i}}\log\big{(}1-e^{2\pi{\rm i}\rho/g_{s}}\big{)},\rho;g_{s}\right).$ (2.29)

If we look at the zero Fourier mode (i.e., coefficient of $e^{2\pi{\rm
i}k\rho/g_{s}}$ with $k=0$), we have the all order instanton corrections of
the partition function:

${\mathfrak{S}}Z(t,\nu;g_{s})=\sum_{n=0}^{\infty}Z^{(n)}(t,\nu;g_{s})$ (2.30)

with the terms of the form

$Z^{(n)}(t,\nu;g_{s})=\mathfrak{Z}^{(n)}(t,\nu-ng_{s};g_{s})\,Z(t,\nu-
ng_{s};g_{s}).$ (2.31)

Here, $\mathfrak{Z}^{(n)}(t,\nu;g_{s})$ is a differential polynomial of the
free energy $F(t,\nu;g_{s})$ with respect to $\nu$. The Seiberg–Witten
relation $\partial_{\nu}F_{0}=\oint_{B}y{\rm d}x$ implies that $Z^{(n)}$ is a
formal power series in $g_{s}$ with an exponential factor ${\rm
e}^{-n\oint_{B}y{\rm d}x/g_{s}}$. Namely, $Z^{(n)}$ is an $n$-instanton
amplitude. It turns out that the first few terms

$\displaystyle Z^{(0)}(t,\nu;g_{s})$ $\displaystyle=Z(t,\nu;g_{s}),$ (2.32)
$\displaystyle Z^{(1)}(t,\nu;g_{s})$
$\displaystyle=\left(1+\frac{g_{s}}{2\pi{\rm i}}\frac{\partial
F}{\partial\nu}(t,\nu-g_{s};g_{s})\right)Z(t,\nu-g_{s};g_{s})$ (2.33)

of (2.30) precisely agree (up to a normalization factor) with the multi-
instanton results for the topological string obtained in [41, 42]. In the next
section, we will show that the agreement occurs for all $n$ as well. We may
also observe that the first few terms of the 1-instanton part (2.33) are
consistent with a known connection formula for Painlevé I (see, e.g., [52,
53]). This supports our heuristic derivation of (2.29).

Before ending the section, let us make a remark on a relation with the results
of [5, 65, 67]. These works also derive a connection formula for solutions of
Painlevé I through the isomonodromy property. Our result (2.29) describes the
connection formula at the level of $\tau$-functions, and the main difference
is the appearance of the prefactor given by the dilogarithm function. The
factor disappears in the solution of Painlevé I due to the logarithmic
derivative (2.6). As we will see in the next section, we can relate the
connection formula of Painlevé I with the results of [41, 42] thanks to the
prefactor. This is our new observation.

##  3 Resurgent structure and Stokes automorphisms
in topological string theory

In this section, we show that the main result from the previous section,
(2.29), is a consequence of the conjectures of [41, 42] on the resurgent
structure of the topological string.

###  3.1 Resurgent structure of the topological string

In [41, 42], a general conjecture on the resurgent structure of the
topological string on arbitrary Calabi–Yau manifolds was put forward. This
conjecture is based on a trans-series solution of the HAE of [6], as proposed
in [19, 20]. For this reason, it applies to the free energies obtained by
doing topological recursion on curves of genus $g\geq 1$, but it also applies
to the free energies of compact Calabi–Yau threefolds, since both are
perturbative solutions to the HAE. For simplicity, we will first present the
results in the one-modulus, local case originally studied in [42]. The
generalization to the multi-modulus, general case is straightforward and will
be presented below.

The conjecture of [41, 42] is as follows. First, it asserts that Borel
singularities of the topological string free energy are integral periods of
the Calabi–Yau manifold, up to some overall normalization (in the local case,
this was already conjectured in [27]). In the one-modulus, local case, this
means that the singularity ${\mathcal{A}}$ can be written as

${\mathcal{A}}=c\partial_{\nu}F_{0}+d\nu+d_{0},$ (3.1)

where $c$, $d$, $d_{0}$ are integer numbers, times a normalization factor
which depends on the normalization of $g_{s}$ (see [41] for a detailed
discussion of normalizations). We will assume that ${\mathcal{A}}$ is a
primitive vector of the period lattice. Then, $\ell{\mathcal{A}}$, with
$\ell\in{\mathbb{Z}}_{>0}$ is also a Borel singularity, and we are interested
in the structure of these “multi-instanton” singularities. There are two
different situations. When $c=0$, the resurgent structure is of the
Pasquetti–Schiappa form [61]. This means the following. Let us define

$F^{(\ell)}_{\mathcal{A}}=\left(\frac{1}{\ell}\frac{{\mathcal{A}}}{g_{s}}+\frac{1}{\ell^{2}}\right){\rm
e}^{-\ell{\mathcal{A}}/g_{s}}.$ (3.2)

Then, the alien derivatives of the free energy are given by

$\dot{\Delta}_{\ell{\mathcal{A}}}F=aF^{(\ell)}_{\mathcal{A}},\qquad\ell\in{\mathbb{Z}}_{>0},$
(3.3)

where $a$ is a Stokes constant. When $c\not=0$, one defines a modified genus
zero free energy $\tilde{F}_{0}$ by the equation
${\mathcal{A}}=c\partial_{\nu}\tilde{F}_{0}$. We note that $\tilde{F}_{0}$
differs from the original $F_{0}$ in a quadratic polynomial in $\nu$. The
total free energy appearing in the formulas for the trans-series involves
$\tilde{F}_{0}$, i.e., it is given by

$F(\nu;g_{s})=g_{s}^{-2}\tilde{F}_{0}(\nu)+\sum_{g\geq
1}g_{s}^{2g-2}F_{g}(\nu).$

Let us now define

$F^{(\ell)}(\nu;g_{s})=\left(\frac{\hbar
F^{\prime}(\nu-\ell\hbar;g_{s})}{\ell}+\frac{1}{\ell^{2}}\right){\rm
e}^{F(\nu-\ell\hbar;g_{s})-F(\nu;g_{s})},$ (3.4)

where we have introduced the rescaled coupling constant $\hbar=cg_{s}$. The
prime in (3.4) and other equations in this section denotes the derivative with
reference to $\nu$. Then, the alien derivatives of $F$ are given by

$\dot{\Delta}_{\ell{\mathcal{A}}}F=aF^{(\ell)},\qquad\ell\in{\mathbb{Z}}_{>0}.$
(3.5)

These are the main conjectures of [41, 42]. They recover and extend partial
results along this direction in [19, 20, 21]. We note that both in (3.5) and
(3.3) it is assumed that the Stokes constant is independent of $\ell$.

From the formulas for the alien derivatives, one can compute the action of the
Stokes automorphism, through Écalle’s formula (see, e.g., [58])

$\displaystyle\mathfrak{S}_{\mathcal{C}}=\exp\left(\sum_{\ell=1}^{\infty}{\mathcal{C}}^{\ell}\dot{\Delta}_{\ell{\mathcal{A}}}\right).$
(3.6)

We have introduced an additional formal parameter ${\mathcal{C}}$ to keep
track of $\ell$. In view of the results of the previous section, we should
consider the action of the Stokes automorphism on the partition function, $Z$,
and we have
$\mathfrak{S}_{{\mathcal{C}}}(Z)=\exp\left(\mathfrak{S}_{{\mathcal{C}}}(F)\right)$.
We have again two cases to consider. The simplest one is when $c=0$. In that
case, the action of more than one alien derivative vanishes, and we simply
have

$\displaystyle\mathfrak{S}_{{\mathcal{C}}}(Z)=\exp\left(a\sum_{\ell=1}^{\infty}{\mathcal{C}}^{\ell}F^{(\ell)}_{\mathcal{A}}\right)Z=\exp\left(a\sum_{\ell=1}^{\infty}{\mathcal{C}}^{\ell}\left(\frac{1}{\ell}\frac{{\mathcal{A}}}{g_{s}}+\frac{1}{\ell^{2}}\right){\rm
e}^{-\ell{\mathcal{A}}/g_{s}}\right)Z,$ (3.7)

which we can write as

$\mathfrak{S}_{{\mathcal{C}}}(Z)=\exp\left(a\operatorname{Li}_{2}\big{(}{\mathcal{C}}{\rm
e}^{-{\mathcal{A}}/g_{s}}\big{)}-a\frac{{\mathcal{A}}}{g_{s}}\log\big{(}1-{\mathcal{C}}{\rm
e}^{-{\mathcal{A}}/g_{s}}\big{)}\right)Z.$ (3.8)

This is the result obtained for the resolved conifold after using the
Pasquetti–Schiappa form (see, e.g., [4]). We note that the ingredients for our
main formula (the dilogarithm and the logarithm) are already here, and they
follow from the Pasquetti–Schiappa form of the multi-instanton amplitudes,
which is in turn ultimately due to the universal behavior of the topological
string at the conifold point [38].

The non-trivial case for the calculation of the Stokes automorphism occurs
when $c\not=0$, since one has to act with multiple alien derivatives. Explicit
calculations show that the result has the form

$\mathfrak{S}_{{\mathcal{C}}}(Z)=\sum_{\ell\geq
0}{\mathcal{C}}^{\ell}\mathfrak{Z}^{(\ell)}(\nu-\ell\hbar)Z(\nu-\ell\hbar;g_{s}),$
(3.9)

where $\mathfrak{Z}^{(\ell)}(\nu)$ can be computed explicitly and
$\mathfrak{Z}^{(0)}=1$. One finds, for the very first values,

$\displaystyle\begin{split}&\mathfrak{Z}^{(1)}=a\left(1+\hbar
F^{\prime}\right),\\\ &\mathfrak{Z}^{(2)}=\frac{a^{2}}{2}\left(1+2\hbar
F^{\prime}+(\hbar
F^{\prime})^{2}+\hbar^{2}F^{\prime\prime}\right)+a\left(\frac{1}{4}+\frac{\hbar
F^{\prime}}{2}\right).\end{split}$ (3.10)

This is in accord with (2.30)–(2.33). We will show in the next subsection that
the structure (3.9) follows from the results of [42]. The arguments in Section
2 suggest in addition the following explicit generating functional for the
functions $\mathfrak{Z}^{(\ell)}(\nu)$:

$Z(\nu;g_{s})\sum_{\ell\geq
0}{\mathcal{C}}^{\ell}\mathfrak{Z}^{(\ell)}(\nu)={\rm
e}^{a\operatorname{Li}_{2}({\mathcal{C}})}Z\left(\nu-\hbar
a\log(1-{\mathcal{C}});g_{s}\right).$ (3.11)

Indeed, it follows from (3.9) and (3.11) that

$\mathfrak{S}_{{\mathcal{C}}}(Z)=\exp\bigl{\\{}a\operatorname{Li}_{2}\big{(}{\mathcal{C}}{\rm
e}^{-\hbar\partial}\big{)}-\hbar a\log\big{(}1-{\mathcal{C}}{\rm
e}^{-\hbar\partial}\big{)}\partial\bigr{\\}}Z,$ (3.12)

where $\partial$ is the derivative with reference to $\nu$. If we introduce
now the discrete Fourier transform as in (2.4),

$\displaystyle\tau(\nu,\rho;g_{s})=\sum_{k\in{\mathbb{Z}}}{\rm e}^{2\pi{\rm
i}k\rho/g_{s}}Z(\nu+kg_{s};g_{s}),$ (3.13)

one easily finds

$\mathfrak{S}_{{\mathcal{C}}}\tau(\nu,\rho;g_{s})={\rm
e}^{a\operatorname{Li}_{2}({\mathcal{C}}{\rm e}^{2\pi{\rm
i}c\rho/g_{s}})}\tau\big{(}\nu-\hbar a\log\big{(}1-{\mathcal{C}}{\rm
e}^{2\pi{\rm i}c\rho/g_{s}}\big{)},\rho;g_{s}\big{)},$ (3.14)

where we have made a choice of normalizations in such a way that the
coefficient $c$ is an integer. The formula above has precisely the structure
anticipated in (2.29) (the two formulas agree after setting
${\mathcal{C}}=c=1$, $a=1/(2\pi{\rm i})$.) In Section 3.3, we will consider
more general cases for the transformation of the dual partition function and
make contact with the results of [3].

###  3.2 A derivation of the formula for Stokes automorphisms

We will now show that the formulas (3.11) and (3.12) follow from the
conjecture on the alien derivatives of the free energy, (3.5). To do this, we
will rely on various results of [41, 42], which we summarize very briefly
here. We refer to those papers for more details.

In the framework of the HAE, the perturbative free energies $F_{g}$ are non-
holomorphic but global functions on the moduli space. More precisely, they are
polynomials in a non-holomorphic propagator $S$, whose coefficients are
functions of a complex coordinate $z$ on the moduli space (not necessarily
flat). The conventional free energies are holomorphic but can be defined in
different frames, which are determined by a choice of $A$ and $B$ periods. The
holomorphic free energies in different frames are obtained by considering the
non-holomorphic free energies, and taking different holomorphic limits of the
propagator. We define the ${\mathcal{A}}$-frame as the frame in which
${\mathcal{A}}$ is the $A$-period, and the holomorphic propagator appropriate
for the ${\mathcal{A}}$-frame will be denoted by $S_{\mathcal{A}}$. The
boundary conditions to solve the HAE are obtained by evaluating the
holomorphic free energies in different frames and imposing particular
behaviors at special points in moduli space, in particular the universal
behaviour at the conifold point [38] and the resulting gap condition [44].

It was pointed out in [19, 20, 21] that the resurgent structure of the
topological string free energy can be obtained by considering trans-series
solutions to the HAE of [6]. The solutions corresponding to the $\ell$-th
instanton sector are formal power series in $g_{s}$, whose coefficients are
also polynomials in $S$ with $z$-dependent coefficients, and they also involve
${\mathcal{A}}$ and their derivatives (the second derivative of
${\mathcal{A}}$ can be however re-expressed in terms of $S_{\mathcal{A}}$).
The $\ell$-instanton amplitude involves of course an exponential prefactor of
the form ${\rm e}^{-\ell{\mathcal{A}}/g_{s}}$. Explicit trans-series solutions
were obtained in [41, 42] by using an operator formalism first suggested in
[16, 17]. The main operator in this formalism is666For the factors of $g_{s}$,
we follow the conventions in [41].

$\displaystyle\mathsf{D}=g_{s}\partial_{z}{\mathcal{A}}(S-S_{\mathcal{A}})\partial_{z}.$
(3.15)

When evaluated in the holomorphic limit, this operator becomes
$\hbar\partial_{\nu}$, i.e., a derivative with reference to the flat
coordinate $\nu$.

We are now ready to prove the formula (3.11). The Stokes automorphism, acting
on $Z$, produces a formal sum of multi-instanton sectors which has to solve
the HAE for the partition function, and in addition it has to satisfy the
following boundary condition: when evaluated at the ${\mathcal{A}}$-frame, it
is equal to (3.8). This determines its form uniquely. A general $n$-th
instanton solution to the HAE for the partition function was determined in
[42] and has the structure:

$Z^{(n)}=\mathfrak{A}_{n}\,{\rm e}^{-\Phi_{n}}Z,$ (3.16)

where

$\displaystyle\Phi_{n}=\frac{1}{\mathsf{D}}\big{(}1-{\rm
e}^{-n\mathsf{D}}\big{)}G,\qquad
G=\frac{1}{g_{s}}{\mathcal{A}}+\mathsf{D}\big{(}F-g_{s}^{-2}F_{0}\big{)}.$
(3.17)

Let us note that, in the holomorphic limit, $\Phi_{n}\rightarrow
F(\nu;g_{s})-F(\nu-n\hbar;g_{s})$. The prefactor $\mathfrak{A}_{n}$ is
determined as follows. Let $X_{n}={\rm e}^{-n\mathsf{D}}G$. Then, the
$\mathfrak{A}_{n}$ are arbitrary linear combinations of the objects
$\mathfrak{w}_{\ell}$, defined by

$\displaystyle\mathfrak{w}_{\ell}=\sum_{\boldsymbol{k},\,d(\boldsymbol{k})=\ell}C_{\boldsymbol{k}}\mathfrak{X}_{\boldsymbol{k}}.$
(3.18)

In this formula, $\boldsymbol{k}=(k_{1},k_{2},\ldots)$ is a vector of non-
negative entries, $d(\boldsymbol{k})$ is given by
$d(\boldsymbol{k})=\sum_{j}jk_{j}$, the coefficients $C_{\boldsymbol{k}}$ are
of the form,

$\displaystyle C_{\boldsymbol{k}}=\frac{\ell!}{\prod_{j\geq
1}k_{j}!(j!)^{k_{j}}},$ (3.19)

and
$\mathfrak{X}_{\boldsymbol{k}}=X_{n}^{k_{1}}(\mathsf{D}X_{n})^{k_{2}}\big{(}\mathsf{D}^{2}X_{n}\big{)}^{k_{3}}\cdots$.
We would like to emphasize that this structure is determined by requiring that
(3.16) is a solution to the HAE (we recall that the HAE for the partition
function is linear, so we can solve it for each instanton sector separately).
Let us also note that all the $X$’s appearing in $\mathfrak{A}_{n}$ are
shifted, i.e., they are acted upon by the automorphism ${\rm
e}^{-n\mathsf{D}}$, so it is convenient to introduce the “unshifted” prefactor
$\mathfrak{B}_{n}$ defined by

$\mathfrak{A}_{n}={\rm e}^{-n\mathsf{D}}\mathfrak{B}_{n}.$ (3.20)

In $\mathfrak{B}_{n}$, $X_{n}$ is replaced by $X=G$. The precise linear
combination of $\mathfrak{w}_{\ell}$ appearing in $\mathfrak{A}_{n}$ is
uniquely determined by the boundary condition, i.e., by its form in the
${\mathcal{A}}$-frame. As we will see in a moment, when evaluated in the
${\mathcal{A}}$-frame, the holomorphic limit of $Z^{(n)}$ is of the form

$Z^{(n)}_{\mathcal{A}}=\left\\{\sum_{k}c_{n,k}\left(\frac{{\mathcal{A}}}{g_{s}}\right)^{k}\right\\}{\rm
e}^{-n{\mathcal{A}}/g_{s}}Z_{\mathcal{A}},$ (3.21)

where the prefactor is an arbitrary polynomial in ${\mathcal{A}}/g_{s}$. Then,
$\mathfrak{B}_{n}=\sum_{k}c_{n,k}\mathfrak{w}_{k}$.

Let us now apply these results to the calculation of the Stokes automorphism.
As we explained before, the Stokes automorphism produces the holomorphic limit
of a formal linear combination of solutions of the form (3.16). Therefore, we
must have

$\displaystyle\mathfrak{S}_{{\mathcal{C}}}(Z)=\sum_{\ell\geq
0}{\mathcal{C}}^{\ell}\mathfrak{A}_{\ell}Z(\nu-\hbar\ell;g_{s}).$ (3.22)

(To lighten the notation, we are using the same symbols for the non-
holomorphic quantities appearing in the HAE, and for their holomorphic limits.
Hopefully, which one is being used at a given moment is clear from the
context.) This is precisely the structure of (3.9), which follows from the
general results for multi-instantons. We deduce that
$\mathfrak{A}_{\ell}=\mathfrak{Z}^{(\ell)}(\nu-\hbar\ell)$. Both sides involve
functions whose argument is shifted by $-\hbar\ell$. In terms of the unshifted
prefactors introduced in (3.20), we have

$\mathfrak{B}_{\ell}=\mathfrak{Z}^{(\ell)}(\nu).$ (3.23)

The boundary condition obtained from (3.8) is

$\displaystyle\sum_{n\geq
0}{\mathcal{C}}^{n}\mathfrak{B}_{n,{\mathcal{A}}}={}$
$\displaystyle\exp\left(a\operatorname{Li}_{2}({\mathcal{C}})-a\frac{{\mathcal{A}}}{g_{s}}\log(1-{\mathcal{C}})\right)$
$\displaystyle={}$ $\displaystyle{\rm
e}^{a\operatorname{Li}_{2}({\mathcal{C}})}\sum_{k\geq
0}\frac{1}{k!}\left(-a\log(1-{\mathcal{C}})\right)^{k}\left(\frac{{\mathcal{A}}}{g_{s}}\right)^{k},$
(3.24)

which is indeed of the form (3.21). According to what we explained above, we
can already write the general solution to the HAE, by simply replacing
$({\mathcal{A}}/g_{s})^{k}$ by $\mathfrak{w}_{k}$:

$\displaystyle\sum_{n\geq 0}{\mathcal{C}}^{n}\mathfrak{Z}^{(n)}={\rm
e}^{a\operatorname{Li}_{2}({\mathcal{C}})}\sum_{k\geq
0}\frac{1}{k!}\left(-a\log(1-{\mathcal{C}})\right)^{k}\mathfrak{w}_{k},$
(3.25)

where we have already used (3.23). It was proven in [42] that

$\displaystyle\Xi(\xi)=\sum_{\ell\geq
0}\frac{\xi^{\ell}}{\ell!}\mathfrak{w}_{\ell}=\exp\left(\sum_{j=1}^{\infty}\frac{\xi^{j}}{j!}\mathsf{D}^{j-1}X\right),$
(3.26)

where $\xi$ is an arbitrary complex parameter. We conclude that

$\displaystyle\sum_{n\geq 0}{\mathcal{C}}^{n}\mathfrak{Z}^{(n)}={\rm
e}^{a\operatorname{Li}_{2}({\mathcal{C}})}\exp\left(\frac{1}{\mathsf{D}}\big{(}{\rm
e}^{-a\log(1-{\mathcal{C}})\mathsf{D}}-1\big{)}X\right).$ (3.27)

In the holomorphic limit, we have that $X\rightarrow\hbar\partial_{\nu}F$,
where $F$ is the total free energy, and we get in the end

$\displaystyle\sum_{n\geq
0}{\mathcal{C}}^{n}\mathfrak{Z}^{(n)}=\exp\left(a\operatorname{Li}_{2}({\mathcal{C}})+F(\nu-a\hbar\log(1-{\mathcal{C}});g_{s})-F(\nu;g_{s})\right).$
(3.28)

This is precisely (3.11).

###  3.3 Generalizations

The above results concern the one-modulus, local case. However, the
generalization to arbitrary CY threefolds is straightforward, by using the
results of [41] (to which we refer for further details). We will now write in
some detail the more general formula for the Stokes automorphism. In the case
of an arbitrary CY, the genus $g$ free energies depend on the “big moduli
space” flat coordinates $X^{I}$ of the CY, where $I=0,1,\dots,n$. The Borel
singularities or instanton actions are again integral periods, given by linear
combinations,

$\kappa^{-1}{\mathcal{A}}=c^{I}\frac{\partial F_{0}}{\partial
X^{I}}+d_{I}X^{I}.$ (3.29)

We note that, in the local case, $X^{0}=1$, and $\partial F_{0}/\partial
X^{0}$ does not appear, so in the one-modulus case this expression reduces to
(3.1). In (3.29), we have introduced explicitly the normalization factor
$\kappa$ relating the action to the integral periods. If all $c^{I}=0$, the
multi-instanton amplitudes are again of the Pasquetti–Schiappa form (3.2) and
the Stokes automorphism is given by (3.8). When not all $c^{I}$ vanish, one
defines a new genus zero free energy by ${\mathcal{A}}=\kappa
c^{I}\frac{\partial\tilde{F}_{0}}{\partial X^{I}}$, as in the local case. It
can be written as

$\displaystyle\tilde{F}_{0}\big{(}X^{I}\big{)}=F_{0}\big{(}X^{I}\big{)}+\frac{1}{2}a_{IJ}X^{I}X^{J},\qquad
a_{IJ}c^{I}=d_{J}.$ (3.30)

Of course, the final formulas will not depend on $a_{IJ}$, but only on
$c^{I}$, $d_{J}$. As shown in [41], one has to define a new genus one free
energy

$\displaystyle\tilde{F}_{1}=F_{1}-\left(\frac{\chi}{24}-1\right)\log X^{0}.$
(3.31)

Such a redefinition has appeared before, e.g., in[2, equation (2.77)]. The
total free energy relevant for the multi-instanton amplitudes will be denoted
by $\tilde{F}\big{(}X^{I};g_{s}\big{)}$, and is given by

$\displaystyle\widetilde{F}\big{(}X^{I};g_{s}\big{)}$
$\displaystyle=g_{s}^{-2}\tilde{F}_{0}\big{(}X^{I}\big{)}+\tilde{F}_{1}\big{(}X^{I}\big{)}+\sum_{g\geq
2}g_{s}^{2g-2}F_{g}\big{(}X^{I}\big{)}$
$\displaystyle=\frac{1}{2g_{s}^{2}}a_{IJ}X^{I}X^{J}+F\big{(}X^{I};g_{s}\big{)}.$
(3.32)

Then, one has the following generalization of (3.12),

$\mathfrak{S}_{{\mathcal{C}}}(\tilde{Z})=\exp\bigl{\\{}a\operatorname{Li}_{2}\bigl{(}{\mathcal{C}}{\rm
e}^{-\kappa g_{s}c^{I}\partial_{I}}\bigr{)}-a\kappa
g_{s}\log\bigl{(}1-{\mathcal{C}}{\rm e}^{-\kappa
g_{s}c^{I}\partial_{I}}\bigr{)}c^{I}\partial_{I}\bigr{\\}}\tilde{Z},$ (3.33)

where we have denoted $\tilde{Z}={\rm e}^{\tilde{F}}$, and
$\partial_{I}=\frac{\partial}{\partial X^{I}}$.

As we have explained before, the action of the Stokes automorphism has a
simpler form when it acts on an appropriate dual partition function. We could
obtain a direct generalization of (3.14) involving the redefined partition
function $\tilde{Z}$. However, in order to make contact with the results of
[3], it is convenient to consider the dual partition function to the original
$Z$. This means that in (3.33) we have to treat separately the quadratic term
in $X^{I}$ appearing in the second line of (3.32). If we denote

$\displaystyle Y^{I}=X^{I}-a\kappa g_{s}c^{I}\log(1-{\mathcal{C}}),$ (3.34)

we find

$\displaystyle\tilde{Z}\big{(}Y^{I};g_{s}\big{)}={}$
$\displaystyle\exp\left\\{\frac{\kappa^{2}}{2}a^{2}d^{I}c_{I}\log^{2}(1-{\mathcal{C}})-g_{s}^{-1}a\kappa\log(1-{\mathcal{C}})d_{I}X^{I}+\frac{1}{2g_{s}^{2}}a_{IJ}X^{I}X^{J}\right\\}$
$\displaystyle\times Z\big{(}Y^{I};g_{s}\big{)}.$ (3.35)

We also note that, for $n\in{\mathbb{Z}}$,

$\displaystyle{\rm e}^{-n\kappa
g_{s}c^{I}\partial_{I}}\exp\left\\{\frac{1}{2g_{s}^{2}}a_{IJ}X^{I}X^{J}\right\\}=\exp\left\\{-g_{s}^{-1}\kappa
d_{I}X^{I}n+\frac{n^{2}}{2}\kappa^{2}c^{I}d_{I}\right\\}$
$\displaystyle\hphantom{{\rm e}^{-n\kappa
g_{s}c^{I}\partial_{I}}\exp\left\\{\frac{1}{2g_{s}^{2}}a_{IJ}X^{I}X^{J}\right\\}=}{}\times\exp\left\\{\frac{1}{2g_{s}^{2}}a_{IJ}X^{I}X^{J}\right\\}.$
(3.36)

We have to be more concrete about the normalization factor $\kappa$. It was
found in [41] that, with the canonical normalization of $g_{s}$, one has
$\kappa^{2}=-2\pi{\rm i}$, and this means that

$\displaystyle{\rm e}^{\frac{n^{2}}{2}\kappa^{2}c^{I}d_{I}}={\rm e}^{-\pi{\rm
i}nc^{I}d_{I}},$ (3.37)

since $n,d_{I},c^{I}\in{\mathbb{Z}}$. We will now put together $c^{I}$,
$d_{I}$ in a symplectic vector $\gamma=\big{(}c^{I},d_{I}\big{)}$. Let us
introduce

$\displaystyle\boldsymbol{X}_{\gamma}=\sigma(\gamma)\exp\big{[}-\kappa
g_{s}^{-1}\big{(}d_{I}X^{I}-\rho_{I}c^{I}\big{)}\big{]},$ (3.38)

where $\rho_{I}$, $I=0,1,\dots,n$, are additional variables, and
$\sigma(\gamma)=(-1)^{d_{I}c^{I}}$. Then, one finds that (3.33) is equivalent
to

$\displaystyle\mathfrak{S}_{{\mathcal{C}}}\left(\sum_{\boldsymbol{\ell}\in{\mathbb{Z}}^{n}}{\rm
e}^{\kappa\rho_{I}\ell^{I}/g_{s}}Z\big{(}X^{I}+\kappa\ell^{I}g_{s};g_{s}\big{)}\right)$
$\displaystyle\qquad={\rm
e}^{a\operatorname{Li}_{2}({\mathcal{C}}\boldsymbol{X}_{\gamma})-\pi{\rm
i}a^{2}d^{I}c_{I}\log^{2}(1-{\mathcal{C}}\boldsymbol{X}_{\gamma})}\sum_{\boldsymbol{\ell}\in{\mathbb{Z}}^{n}}{\rm
e}^{-ag_{s}^{-1}\kappa\log(1-{\mathcal{C}}\boldsymbol{X}_{\gamma})d_{I}(X^{I}+\kappa
g_{s}\ell^{I})}$ $\displaystyle\phantom{\qquad=}{}\times Z\left(X^{I}+\kappa
g_{s}\ell^{I}-ag_{s}\kappa
c^{I}\log\left(1-{\mathcal{C}}\boldsymbol{X}_{\gamma}\right);g_{s}\right){\rm
e}^{\kappa\rho_{I}\ell^{I}/g_{s}}.$ (3.39)

The appropriate definition of the dual partition function in this general case
is [3]:

$\displaystyle\tau\big{(}X^{I},\rho_{I};g_{s}\big{)}={\rm
e}^{\frac{1}{2g_{s}^{2}}X^{I}\rho_{I}}\sum_{\boldsymbol{\ell}\in{\mathbb{Z}}^{n}}{\rm
e}^{\kappa\rho_{I}\ell^{I}/g_{s}}Z\big{(}X^{I}+\kappa\ell^{I}g_{s};g_{s}\big{)},$
(3.40)

and the action of the Stokes automorphism is

$\displaystyle\mathfrak{S}\tau\big{(}X^{I},\rho_{I};g_{s}\big{)}={}$
$\displaystyle\exp(aL_{\sigma(\gamma)}(\boldsymbol{X}_{\gamma}))$
$\displaystyle\times\tau\big{(}X^{I}-ag_{s}\kappa
c^{I}\log(1-\boldsymbol{X}_{\gamma}),\rho_{I}-ag_{s}\kappa
d_{I}\log(1-\boldsymbol{X}_{\gamma});g_{s}\big{)},$ (3.41)

where we have put ${\mathcal{C}}=1$, and $L_{\epsilon}(z)$ is the twisted
Rogers dilogarithm, as in [3]:

$\displaystyle
L_{\epsilon}(z)=\operatorname{Li}_{2}(z)+\frac{1}{2}\log\big{(}\epsilon^{-1}z\big{)}\log(1-z).$
(3.42)

It is easy to see that (3.41) agrees precisely with the wall-crossing formula
(1.9) in [3], where their variables $\xi^{I}$, $\tilde{\xi}_{I}$ are related
to ours by $\big{(}X^{I},\rho_{I}\big{)}=-\kappa
g_{s}\big{(}\xi^{I},\tilde{\xi}_{I}\big{)}$.777[3] also give a wall-crossing
formula for $Z$, in terms of an integral transform, which is equivalent to
(3.33). We would like to thank Boris Pioline for many discussions on the
relation between the approach of [2, 3] and the one presented in this paper.
In addition, the agreement between the formulas requires the identification
(1.4). One advantage of (3.41) is that, when ${c^{I}=0}$, one recovers as well
the transformation formula (3.8) (this is easily seen by looking, e.g., at the
mode with $\ell^{I}=0$).

There is another generalization of the formula (3.11) that one could consider.
So far we have only included forward alien derivatives, and correspondingly
purely instanton sectors. We can also consider alien derivatives in both the
negative and the positive directions, which lead to amplitudes with both
instantons and “negative instantons”. Let us define

$F^{(0|\ell)}(\nu;g_{s})=-F^{(\ell)}(\nu;-g_{s}),\qquad
F^{(0|\ell)}_{\mathcal{A}}(\nu;g_{s})=-F^{(\ell)}_{\mathcal{A}}(\nu;-g_{s}).$
(3.43)

The basic alien derivative in the negative direction is simply,

$\dot{\Delta}_{-\ell{\mathcal{A}}}F=aF^{(0|\ell)}\quad\text{or}\quad
aF^{(0|\ell)}_{\mathcal{A}},$ (3.44)

depending on whether $c\not=0$ or $c=0$. We can then consider the “mixed”
Stokes automorphism:

$\displaystyle\mathfrak{S}_{{\mathcal{C}}_{1},{\mathcal{C}}_{2}}=\exp\Bigg{(}\sum_{\ell\geq
1}{\mathcal{C}}_{1}^{\ell}\dot{\Delta}_{\ell{\mathcal{A}}}+{\mathcal{C}}_{2}^{\ell}\dot{\Delta}_{-\ell{\mathcal{A}}}\Bigg{)}.$
(3.45)

Acting on $Z$, it has the structure

$\displaystyle\mathfrak{S}_{{\mathcal{C}}_{1},{\mathcal{C}}_{2}}(Z)=\sum_{\ell_{1},\ell_{2}\geq
0}{\mathcal{C}}_{1}^{\ell_{1}}{\mathcal{C}}_{2}^{\ell_{2}}\mathfrak{Z}^{(\ell_{1}|\ell_{2})}(\nu-(\ell_{1}-\ell_{2})\hbar)Z(\nu-(\ell_{1}-\ell_{2})\hbar),$
(3.46)

where $\mathfrak{Z}^{(0|0)}=1$. In this case, the boundary condition follows
from (3.44), (3.43) and (3.2). It is given by

$\displaystyle\exp\left\\{a\sum_{\ell=1}^{\infty}\big{(}{\mathcal{C}}_{1}^{\ell}F^{(\ell)}_{\mathcal{A}}+{\mathcal{C}}_{2}^{\ell}F^{(0|\ell)}_{\mathcal{A}}\big{)}\right\\}$
$\displaystyle\qquad=\exp\left\\{a\left(\operatorname{Li}_{2}({\mathcal{C}}_{1})-\operatorname{Li}_{2}({\mathcal{C}}_{2})\right)-a\frac{{\mathcal{A}}}{g_{s}}\left(\log(1-{\mathcal{C}}_{1})+\log(1-{\mathcal{C}}_{2})\right)\right\\}.$
(3.47)

By using the results in [42], we can generalize (3.11) to

$\displaystyle Z(\nu;g_{s})\sum_{\ell_{1},\ell_{2}\geq
0}{\mathcal{C}}_{1}^{\ell_{1}}{\mathcal{C}}_{2}^{\ell_{2}}\mathfrak{Z}^{(\ell_{1}|\ell_{2})}(\nu)$
$\displaystyle\qquad={\rm
e}^{a\left(\operatorname{Li}_{2}({\mathcal{C}}_{1})-\operatorname{Li}_{2}({\mathcal{C}}_{2})\right)}Z\left(\nu-a\hbar\log(1-{\mathcal{C}}_{1})-a\hbar\log(1-{\mathcal{C}}_{2});g_{s}\right).$
(3.48)

It is easy to write this formula in the form (3.33) or (3.41).

##  Appendix A Definition of correlators and free energy
in topological recursion

To apply the topological recursion, we regard (2.2) as a family of spectral
curves in the sense of [32] (i.e., a data consisting of a compact Riemann
surface $C$ and a pair $(x,y)$ of meromorphic functions on it), through the
Weierstrass parametrization:

$\displaystyle C={\mathbb{C}}/\Lambda,\qquad x(z)=\wp(z),\qquad
y(z)=\wp^{\prime}(z).$ (A.1)

Here $\wp(z)=\wp(z;g_{2},g_{3})$ is the Weierstrass $\wp$-function with
$g_{2}=-2t$ and $g_{3}=-u$, which is doubly-periodic with periods $\omega_{A}$
and $\omega_{B}$ (we omit the $t$ and $\nu$ dependence for simplicity).
$\Lambda={\mathbb{Z}}\omega_{A}+{\mathbb{Z}}\omega_{B}$ is the lattice
generated by the periods of the elliptic curve (2.2).

Let $z_{o}\in{\mathbb{C}}$ be a generic point, and $\Omega$ be the
quadrilateral with $z_{o}$, $z_{o}+\omega_{A}$, $z_{o}+\omega_{B}$ and
${z_{o}+\omega_{A}+\omega_{B}}$ on its vertices; that is, a fundamental domain
of ${\mathbb{C}}/\Lambda$. The ramification points (i.e., zeros of ${\rm d}x$)
on $\Omega$ are given by the half-periods $r_{1}\equiv\omega_{A}/2$,
$r_{2}\equiv\omega_{B}/2$ and $r_{3}\equiv(\omega_{A}+\omega_{B})/2$ modulo
$\Lambda$. These points correspond to the branch points $e_{i}=x(r_{i})$
($i=1,2,3$) of the elliptic curve which are defined by
$4x^{3}+2tx+u=4(x-e_{1})(x-e_{2})(x-e_{3})$. The covering involution
${y\mapsto-y}$ is realized by $z\mapsto\sigma(z)\equiv-z$ mod $\Lambda$.

To run the topological recursion, we also need the Bergman bidifferential
normalized along the chosen A-cycle. For our spectral curve, it is given by

$\displaystyle
B(z_{1},z_{2})=\left(\wp(z_{1}-z_{2})+\frac{\eta_{A}}{\omega_{A}}\right){\rm
d}z_{1}{\rm d}z_{2},$ (A.2)

where

$\eta_{A}=-\oint_{A}\frac{x{\rm d}x}{y}.$

Then, the topological recursion recursively constructs a doubly-indexed
sequence of meromorphic multi-differentials $\omega_{g,n}(z_{1},\dots,z_{n})$
($g\geq 0$, $n\geq 1$) on the spectral curve, called correlators, as follows:

$\displaystyle\omega_{0,1}(z_{1})=y(z_{1}){\rm
d}x(z_{1}),\qquad\omega_{0,2}(z_{1},z_{2})=B(z_{1},z_{2}),$ (A.3)

and for $2g-2+n\geq 1$, we define

$\displaystyle\omega_{g,n}(z_{1},\dots,z_{n})=\sum_{j=1}^{3}\mathop{\rm
Res}_{z=r_{j}}K(z_{1},z)R_{g,n}(z,z_{2},\dots,z_{n}),$ (A.4)

where

$\displaystyle R_{g,n}(z,z_{2},\dots,z_{n})={}$
$\displaystyle\omega_{g-1,n+1}(z,\sigma(z),z_{2},\dots,z_{n})$
$\displaystyle+\sum^{\prime}_{\begin{subarray}{c}g_{1}+g_{2}=g\\\ I\sqcup
J=\\{2,\dots,n\\}\end{subarray}}\omega_{g_{1},|I|+1}(z,z_{I})\omega_{g_{2},|J|+1}(\sigma(z),z_{J}).$
(A.5)

Here, the recursion kernel $K(z_{1},z)$ is given by

$\displaystyle K(z_{1},z)=\frac{1}{(y(z)-y(\sigma(z))){\rm
d}x(z)}\int^{w=z}_{w=0}\omega_{0,2}(z_{1},w).$ (A.6)

We use the convention for a tuple of variables as
$z_{I}=(z_{i_{1}},\dots,z_{i_{k}})$ if $I=\\{i_{1},\dots,i_{k}\\}$, and the
prime in the right-hand side of (A.5) means that only indices satisfying
$(g_{i},I_{i})\neq(0,\varnothing)$ are taken (i.e., $\omega_{0,1}$ does not
appear) in the summation.

Here we also recall the definition of the genus $g$ free energy
$F_{g}=F_{g}(t,\nu)$ introduced in [32]. The genus $0$ free energy $F_{0}$ is
defined in [32, Section 4.2.2]. In our case, it is given by

$\displaystyle F_{0}=\frac{tu}{5}+\frac{\nu}{2}\oint_{B}y{\rm d}x.$ (A.7)

The genus $1$ free energy $F_{1}$ is also defined in [32, Section 4.2.3] up to
a multiplicative constant. We employ

$\displaystyle
F_{1}=-\frac{1}{12}\log\big{(}\omega_{A}^{6}{\mathcal{D}}\big{)}$ (A.8)

as the definition. Here

$\displaystyle{\mathcal{D}}=-8t^{3}-27u(t,\nu)^{2}$ (A.9)

is the discriminant of (2.2). Finally, we define the genus $g$ free energy
$F_{g}$ for $g\geq 2$ by

$\displaystyle F_{g}=\frac{1}{2-2g}\sum_{j=1}^{3}\mathop{\rm
Res}_{z=r_{j}}\Phi(z)\omega_{g,1}(z),$ (A.10)

where $\Phi$ is any primitive of $\omega_{0,1}$. See [32] for properties of
$\omega_{g,n}$ and $F_{g}$.

### Acknowledgements

We would like to thank Ioana Coman, Fabrizio del Monte, Alba Grassi, Paolo
Gregori, Jie Gu, Shinobu Hosono, Omar Kidwai, Oleg Lisovyy, Pietro Longhi,
Kento Osuga, Boris Pioline, Ricardo Schiappa, Masa-Hiko Saito, Maximilian
Schwick, Atsushi Takahashi, Yoshitugu Takei and Joerg Teschner for useful
conversations. We would also like to thank to Maxim Kontsevich and Yan
Soibelman for organizing the IHES school “Wall-crossing structures,
analyticity and resurgence”, which made possible this collaboration. The work
of K.I has been supported by the JSPS KAKENHI Grand Numbers 20K14323,
21K18576, 21H04994, 22H00094, 23K17654. The work of M.M. has been supported in
part by the ERC-SyG project “Recursive 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. Finally, we would like to thank the anonymous
referees for their helpful comments that improved the quality of the paper.

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