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#  Resurgence and Riemann-Hilbert problems for elliptic Calabi-Yau threefolds

Tom Bridgeland and Iván Tulli

###### Abstract.

Let $X$ be a Calabi-Yau threefold with an elliptic fibration. We investigate
the non-linear Riemann-Hilbert problems associated to the Donaldson-Thomas
theory of fibre classes, and relate them to the Borel sum of the $A$-model
topological string free energy for such classes.

##  1\. Introduction

There has been a great deal of interest recently in applying techniques from
resurgence analysis to study non-perturbative effects in topological string
theory. A central object in the theory is the free energy, which is a formal
series in the topological string coupling $\lambda$. More precisely, the
A-model closed string free energy of a Calabi-Yau (CY) threefold $X$ (in the
holomorphic limit) is a series of the form

$F(\lambda,Q)=\sum_{g\geq 0}F_{g}(Q)\,\lambda^{2g-2},$ (1.1)

where $F_{g}(Q)$ coincides with the generating function for genus $g$ Gromov-
Witten (GW) invariants of $X$. In studying non-perturbative effects it has
been very fruitful to consider, where possible, the Borel sum of the free
energy [3, 18, 14, 15, 16]. The latter depends on a choice of a ray
$r\subset\mathbb{C}^{*}$, with the Borel sum for different choices being
related by Stokes jumps. These jumps are conjecturally related to the
Donaldson-Thomas (DT) invariants of $X$.

One (non-compact) CY threefold that has been much-studied from this point of
view is the resolved conifold. The Borel summability of its A-model free
energy was established in [22, 17, 3]. The resulting non-perturbative free
energy is closely related to the log of the triple sine function
$\sin_{3}(z\,|\,\omega_{1},\omega_{2},\omega_{3})$111The multiple sine
functions are in turn defined in terms of the multiple gamma functions of
Barnes [4]. [20]. The Stokes behaviour of the Borel sums was completely
described in [3] and shown to be controlled by the corresponding DT
invariants.

These ideas make contact with a closely related area of research which aims to
use the DT invariants of a CY3 category to construct a geometric structure on
the space of stability conditions [6, 11, 7, 10, 8]. The geometric structure
goes by the name of a Joyce structure, and is built from solutions to a
collection of non-linear Riemann-Hilbert (RH) problems. These problems involve
piecewise holomorphic functions on $\mathbb{C}^{*}$, with discontinuities
prescribed by the DT invariants along a collection of rays, and fixed
asymptotics at $0$ and $\infty$. The existence and uniqueness of solutions for
such problems is not known in general, but several examples have been worked
out in detail [6, 10, 9].

In the case of the resolved conifold the RH problems have unique solutions [9,
1], which are closely related to the Borel sums of the free energy. More
precisely, this statement holds after appropriately fixing the constant term
in the asymptotics at $\epsilon=0$. The solutions are then given by functions
$Y_{i}^{r}(\epsilon\,|\,\omega_{1},\omega_{2})$, with $i=1,2$, where
$r=\mathbb{R}_{>0}\cdot\zeta\subset\mathbb{C}^{*}$ is a ray, and
$\epsilon\in\mathbb{H}_{r}$ lies in the open half-plane
$\mathbb{H}_{r}\subset\mathbb{C}$ centered on $r$. They can be repackaged in
terms of functions $\tau^{r}(\epsilon\,|\,\omega_{1},\omega_{2})$ satisfying
the equations

$\frac{\partial}{\partial\omega_{i}}\log\tau^{r}(\epsilon\,|\,\omega_{1},\omega_{2})=\frac{1}{2\pi\mathrm{i}}\cdot\frac{\partial}{\partial\epsilon}\log
Y_{i}^{r}(\epsilon\,|\,\omega_{1},\omega_{2}).$ (1.2)

It is these functions $\tau^{r}(\epsilon\,|\,\omega_{1},\omega_{2})$ which are
closely related to the Borel summation of the free energy along the ray $r$
[9, 2, 3].

The goal of this paper is to address similar questions for compact CY
threefolds with elliptic fibrations. We study the Borel sums of the A-model
free energy and its relation to the RH problem defined by the DT invariants.
We only consider the part of the free energy of $X$ corresponding to fibre
classes, i.e. classes $\beta\in H_{2}(X,\mathbb{Z})$ satisfying
$\pi_{*}(\beta)=0$. Similarly we only consider DT invariants for coherent
sheaves supported on the fibres of $\pi$.

The rest of the introduction contains a detailed summary of our main results.
In general we find a similar situation to that of the resolved conifold,
although our results are not as complete. One significant additional
difficulty is that whereas in the case of the resolved conifold the set of
Stokes directions is a closed subset of the circle, for the Borel sums
considered in this paper the Stokes directions are everywhere dense.
Nonetheless, we find that the free energy is again Borel summable at least
along almost all non-Stokes rays, and we construct natural solutions to a weak
version of the RH problem which ignores the asymptotics at $\infty$. Moreover,
the two stories are again related by the equation (1.2).

###  1.1. Borel sum of the free energy

In order to state our results in more detail let us briefly recall the basics
of Borel summation. Consider a formal power series $F(\epsilon)=\sum_{i\geq
1}a_{i}\epsilon^{i}$. The Borel transform is the series $f(\eta)=\sum_{i\geq
1}a_{i}\eta^{i-1}/(i-1)!$. For simplicity let us assume that $f(\eta)$ is the
Taylor expansion of a meromorphic function on $\mathbb{C}$ which we also
denote by $f(\eta)$. We consider rays $r\subset\mathbb{C}^{*}$ of the form
$r=\mathbb{R}_{>0}\cdot\zeta$ with $\zeta\in\mathbb{C}^{*}$. Such a ray is
called a Stokes ray if it contains a pole of $f(\eta)$, otherwise it is a non-
Stokes ray. For a given $\epsilon\in\mathbb{C}^{*}$ the series $F(\epsilon)$
is said to be Borel summable along a non-Stokes ray $r\subset\mathbb{C}^{*}$
if the integral

$F^{r}(\epsilon)=\int_{r}e^{-\eta/\epsilon}f(\eta)\mathrm{d}\eta$ (1.3)

exists. The Borel sum is then defined to be the value of this integral. In
practice, given a non-Stokes ray $r\subset\mathbb{C}^{*}$, we shall only
consider the Borel sum for $\epsilon$ lying in the open half-plane
$\mathbb{H}_{r}=\\{z\in\mathbb{C}:\operatorname{Re}(z/\zeta)>0\\}$ centered on
$r$.

We will be interested in applying Borel summation to the fibre-class free
energy of the A-model topological string on an elliptic CY threefold $X$.

######  Assumptions .

By an elliptic CY threefold we mean a smooth projective threefold $X$, with
trivial canonical bundle, equipped with a flat map $\pi\colon X\to B$ whose
general fibre is a genus 1 curve. We always assume that $B$ is smooth, that
the singular fibers of $\pi$ are reduced and irreducible, and that $\pi$ has a
section. We further assume that the DT/GW correspondence holds for $X$.

Under these assumptions the GW invariants of $X$ in the fibre classes were
computed in [21, Section B.3]. See Appendix A for a more detailed discussion.
This leads to an expression

$F_{\text{GW}}(\lambda\,|\,\tau)=-e(X)\cdot\sum_{g\geq
2}\frac{B_{2g}\,G_{2g-2}(\tau)}{4g(2g-2)}\left(\frac{\lambda}{2\pi}\right)^{2g-2}$
(1.4)

for the $g\geq 2$ part of the fibre-class free energy. It is a formal series
in $\lambda$ whose coefficients depend on a Kähler parameter
$\tau\in\mathbb{C}$ satisfying $\operatorname{Im}(\tau)>0$. More precisely,
$\tau$ is the pairing of the complexified Kähler class $B+\mathrm{i}\omega\in
H^{2}(X,\mathbb{C})$ with the fundamental class $\beta\in H_{2}(X,\mathbb{Z})$
of a smooth fibre of $\pi$. The expression involves the Bernoulli numbers
$B_{2g}$, the Eisenstein series $G_{2g-2}(\tau)$, and the topological Euler
characteristic $e(X)$.

It will be convenient to set $2\pi\mathrm{i}\epsilon=\lambda/2\pi$ and view
$F_{\text{GW}}(\lambda\,|\,\tau)$ as a formal series in $\epsilon$.
Furthermore, in order to relate the Borel summations of
$F_{\text{GW}}(\epsilon\,|\,\tau)$ to the RH problem below, we consider
instead

$F_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2}):=F_{\text{GW}}(\epsilon/\omega_{1}\,|\,\omega_{2}/\omega_{1})\,,$
(1.5)

where $\epsilon,\omega_{1},\omega_{2}\in\mathbb{C}^{*}$ and
$\text{Im}(\omega_{2}/\omega_{1})>0$. All our results about
$F_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})$ then specialize to results
about (1.4) by simply taking $\omega_{1}=1$ and $\omega_{2}=\tau$. Our main
result concerning the Borel summation of
$F_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})$ is as follows:

######  Theorem 1.1.

Fix $\omega_{1},\omega_{2}\in\mathbb{C}^{*}$ with
$\operatorname{Im}(\omega_{2}/\omega_{1})>0$.

  * (i)

The Borel transform of $F_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})$ is
the Taylor expansion of a meromorphic function on $\mathbb{C}$ with double
poles at the points $a_{1}\omega_{1}+a_{2}\omega_{2}$ with
$(a_{1},a_{2})\in\mathbb{Z}^{2}\setminus\\{0\\}$ and no other poles.

  * (ii)

For almost all non-Stokes rays $r\subset\mathbb{C}^{*}$ the series
$F_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})$ is Borel summable along
$r$ provided $\epsilon$ lies in the corresponding open half-plane
$\mathbb{H}_{r}\subset\mathbb{C}^{*}$.∎

More precisely, given a non-Stokes ray $r\subset\mathbb{C}^{*}$, there is a
unique real number $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ such that
$\pm(\omega_{1}+\alpha\omega_{2})\in r$. We show that the Borel sum
$F^{r}_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})$ exists and defines a
holomorphic function of $\epsilon\in\mathbb{H}_{r}$ whenever $\alpha$ does not
lie in the measure-zero subset of $\mathbb{R}\setminus\mathbb{Q}$ consisting
of Liouville irrationals. For a general non-Stokes ray
$r\subset\mathbb{C}^{*}$ we can still associate a meaningful holomorphic
function of $\epsilon\in\mathbb{H}_{r}$ by using integrals along certain
detour paths (see Section 2.4). These integrals reduce to the Borel sums from
Theorem 1.1 whenever the ray $r$ corresponds to an element
$\alpha\in\mathbb{R}\setminus\mathbb{Q}$ which is not a Liouville irrational.

###  1.2. DT invariants and the RH problem

Let $\pi\colon X\to B$ be an elliptic CY threefold satisfying Assumptions 1.1.
We consider the full triangulated subcategory $\mathcal{D}(\pi)\subset
D^{b}\operatorname{Coh}(X)$ of the bounded derived category of coherent
sheaves consisting of objects whose set-theoretic support is contained in a
finite union of fibres of $\pi$. The Chern character defines a homomorphism

$\operatorname{ch}\colon K_{0}(\mathcal{D}(\pi))\to N(\pi)\subset
H^{*}(X,\mathbb{Z}),$ (1.6)

whose image $N(\pi)=\mathbb{Z}\gamma_{1}\oplus\mathbb{Z}\gamma_{2}$ is a free
abelian group of rank 2. It is convenient to choose the generators
$\gamma_{1},\gamma_{2}\in N(\pi)$ so that if $E$ is a rank $r$, degree $d$
vector bundle supported on a smooth fibre of $\pi$ then
$\operatorname{ch}(E)=-d\gamma_{1}+r\gamma_{2}$.

Given a pair of complex numbers $\omega_{1},\omega_{2}\in\mathbb{C}^{*}$ with
$\operatorname{Im}(\omega_{2}/\omega_{1})>0$ there is a natural stability
condition on the category $\mathcal{D}(\pi)$, uniquely defined up to the
action of the even shifts, whose central charge $Z\colon
K_{0}(\mathcal{D}(\pi))\to\mathbb{C}$ is the composition of the Chern
character (1.6) with the map

$Z\colon N(\pi)\to\mathbb{C},\qquad
Z(a_{1}\gamma_{1}+a_{2}\gamma_{2})=a_{1}\omega_{1}+a_{2}\omega_{2}.$ (1.7)

A calculation of Toda [23, Thm. 6.9] shows that the corresponding DT
invariants are

$\Omega(a_{1}\gamma_{1}+a_{2}\gamma_{2})=-e(X),\qquad(a_{1},a_{2})\in\mathbb{Z}^{2}\setminus\\{0\\}.$
(1.8)

In [5] it was explained how to associate a RH problem to the data of the
lattice $N(\pi)$, the central charge (1.7), and the DT invariants (1.8). We
will recall the details of this construction in Section 3 below. Here we will
simply state the resulting RH problem and discuss its solution. A ray
$r\subset\mathbb{C}^{*}$ will be called a Stokes ray if it contains a point of
the form $Z(\gamma)$ with $0\neq\gamma\in N(\pi)$, otherwise $r$ will be
called non-Stokes. As before, given a ray $r\subset\mathbb{C}^{*}$, we denote
by $\mathbb{H}_{r}\subset\mathbb{C}^{*}$ the open half-plane centered on it.

######  Problem .

For each non-Stokes ray $r\subset\mathbb{C}^{*}$ find holomorphic functions
$Y^{r}_{i}\colon\mathbb{H}_{r}\to\mathbb{C}^{*}$ for $i=1,2$ such that the
following statements hold.

  * (RH1)

If $\Delta\subset\mathbb{C}^{*}$ be a convex sector whose boundary consists of
non-Stokes rays $r_{1},r_{2}$ taken in clockwise order then

$Y^{r_{2}}_{i}(\epsilon)=Y^{r_{1}}_{i}(\epsilon)\cdot\prod_{\gamma=a_{1}\gamma_{1}+a_{2}\gamma_{2}\in
Z^{-1}(\Delta)}\left(1-e^{-Z(\gamma)/\epsilon}\right)^{-a_{i}\cdot e(X)}$
(1.9)

for $\epsilon\in\mathbb{H}_{r_{1}}\cap\mathbb{H}_{r_{2}}$ with
$0<|\epsilon|\ll 1$.

  * (RH2)

As $\epsilon\to 0$ in any closed subsector of $\mathbb{H}_{r}$ we have
$Y^{r}_{i}(\epsilon)\to 1$.

  * (RH3)

There is an $N>0$ such that as $\epsilon\to\infty$ in $\mathbb{H}_{r}$ there
is a bound $|\epsilon|^{-N}<|Y^{r}_{i}(\epsilon)|<|\epsilon|^{N}$.

It is easy to see that if this problem has a solution then it is unique. We
shall instead consider what we call the weak RH problem in which we drop
condition (RH3). The resulting solution is then unique up to simultaneous
multiplication of the functions $Y^{r}_{i}$ for all rays
$r\subset\mathbb{C}^{*}$ by an arbitrary pair of holomorphic functions
$P_{i}\colon\mathbb{C}\to\mathbb{C}^{*}$ satisfying $P_{i}(0)=1$.

In order to motivate our solution of the weak RH problem, consider again the
holomorphic functions $F_{\text{GW}}^{r}(\epsilon\,|\,\omega_{1},\omega_{2})$
from Section 1.1 and define

$\tau_{\text{GW}}^{r}(\epsilon\,|\,\omega_{1},\omega_{2}):=\exp(F_{\text{GW}}^{r}(\epsilon\,|\,\omega_{1},\omega_{2}))\,.$
(1.10)

As before, for a general non-Stokes ray $r$ it is understood that the above
expression is defined via detour paths. Furthermore, let
$H_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})$ be the formal series in
$\epsilon$ without constant term satisfying

$\frac{\partial}{\partial\epsilon}H_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})=F_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})\,.$
(1.11)

When looking for solutions of the RH problem related to
$\tau_{\text{GW}}^{r}(\epsilon\,|\,\omega_{1},\omega_{2})$ via (1.2), it is
then natural to consider Borel summations of
$2\pi\mathrm{i}\cdot\frac{\partial}{\partial\omega_{i}}H_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})$.
Our second main result is then as follows:

######  Theorem 1.2.

Fix $\omega_{1},\omega_{2}\in\mathbb{C}^{*}$ with
$\text{Im}(\omega_{2}/\omega_{1})>0$. Then there exists a solution
$Y^{r}_{i}(\epsilon\,|\,\omega_{1},\omega_{2})$ of the weak RH problem such
that $\log Y_{i}^{r}(\epsilon\,|\,\omega_{1},\omega_{2})$ is the Borel sum of
$2\pi\mathrm{i}\frac{\partial}{\partial\omega_{i}}H_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})$
along $r$ for almost all non-Stokes rays $r$. Furthermore,

$\frac{\partial}{\partial\omega_{i}}\log\tau^{r}_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})=\frac{1}{2\pi\mathrm{i}}\cdot\frac{\partial}{\partial\epsilon}\log
Y^{r}_{i}(\epsilon\,|\,\omega_{1},\omega_{2})$ (1.12)

for all non-Stokes rays $r$. ∎

###  1.3. Further remarks

Our results leave several natural challenges and questions for future
research. For almost all rays $r\subset\mathbb{C}^{*}$ our solution to the
weak RH problem can be expressed (3.14) as an integral

$Y_{i}^{r}(\epsilon)=\exp\left(-\frac{e(X)}{2\pi\mathrm{i}}\int_{r}\mathrm{Li}_{1}(e^{-\eta/\epsilon})\frac{\partial}{\partial\omega_{i}}\mathscr{h}(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta\right)\,,$
(1.13)

where $\mathscr{h}(\eta\,|\,\omega_{1},\omega_{2})$ is closely related to the
log of the Jacobi theta function, and is defined in terms of the Weierstrass
sigma function by the equation

$\mathscr{h}(u\,|\,\omega_{1},\omega_{2})=\log\sigma(u\,|\,\omega_{1},\omega_{2})-\log(u)-\tfrac{1}{2}G_{2}(\omega_{1},\omega_{2})u^{2}.$

An obvious challenge is to upgrade Theorem 1.2 by constructing a solution to
the full Riemann-Hilbert problem. This would involve understanding the
behaviour of the integral (1.13) in the limit $\epsilon\to\infty$.

In the case of the resolved conifold, the Borel sum of the free energy along a
particular ray can be re-expressed [3, Theorem 2.1] in terms of the Barnes
triple sine function. It is natural to ask whether the integral (1.13) can
also be re-expressed in some more convenient form, and whether it can be
related to known special functions.

A very interesting property of the solution to the RH problem in the case of
the resolved conifold is an unexpected symmetry exchanging the parameter
$\epsilon\in\mathbb{C}^{*}$ with the central charge parameter corresponding to
the class of a point. A possible relation to S-duality in string theory was
discussed in [3, Section 6]. For the RH problem considered in this paper the
analogous symmetry would exchange the parammeters $\epsilon$ and $\omega_{2}$.
Since the solutions to the RH problem already have an obvious
$\operatorname{SL}_{2}(\mathbb{Z})$ symmetry acting on the parameters
$\omega_{1},\omega_{2}$, this perhaps hints at a possible connection with
modular forms for $\operatorname{SL}_{3}(\mathbb{Z})$.

###  1.4. Structure of the paper

In Section 2 we introduce some modified Weierstrass elliptic functions and
summarize the results from complex analysis that we will need. In Section 3 we
apply the contents of Section 2 to prove our main results, Theorems 1.1 and
1.2. The proofs of the results in Section 2 can be found in Sections 4 and 5.
Section 4 is concerned with the properties of the Borel transforms, whereas
Section 5 deals with the proof of Borel summability and the existence of
closely related integrals along detour paths.

###  1.5. Acknowledgements

The authors are grateful to Georg Oberdieck who suggested applying the
techniques of [3, 9] to the case of elliptic CY threefolds.

##  2\. Summary of the relevant complex analysis

In this section we collect the precise statements of the results from complex
analysis which will be applied to prove Theorems 1.1 and 1.2 in Section 3. The
proofs for the results in this section can be found in Sections 4 and 5.

###  2.1. Elliptic functions

Define the region

$\mathcal{R}=\\{(\omega_{1},\omega_{2})\in(\mathbb{C}^{*})^{2}:\operatorname{Im}(\omega_{2}/\omega_{1})>0\\}.$
(2.1)

A point $(\omega_{1},\omega_{2})\in\mathcal{R}$ defines a lattice

$\Lambda(\omega_{1},\omega_{2})=\mathbb{Z}\omega_{1}+\mathbb{Z}\omega_{2}\subset\mathbb{C}.$
(2.2)

We set
$\Lambda^{*}(\omega_{1},\omega_{2})=\Lambda(\omega_{1},\omega_{2})\setminus\\{0\\}$.
For an even integer $n\geq 2$ we introduce the Eisenstein series

$G_{n}(\omega_{1},\omega_{2})=\sum_{\omega\in\Lambda^{*}(\omega_{1},\omega_{2})}\frac{1}{\omega^{n}}=\sum_{0\neq(a_{1},a_{2})\in\mathbb{Z}^{2}}\frac{1}{(a_{1}\omega_{1}+a_{2}\omega_{2})^{n}}.$
(2.3)

This series is absolutely convergent for $n>2$, while for $n=2$ it is only
conditionally convergent. We define $G_{2}$ by the Eisenstein summation

$G_{2}(\omega_{1},\omega_{2}):=\sum_{a_{1}\in\mathbb{Z}\setminus\\{0\\}}\frac{1}{(a_{1}\omega_{1})^{2}}+\sum_{a_{2}\in\mathbb{Z}\setminus\\{0\\}}\sum_{a_{1}\in\mathbb{Z}}\frac{1}{(a_{1}\omega_{1}+a_{2}\omega_{2})^{2}}\,.$
(2.4)

The resulting functions $G_{n}(\omega_{1},\omega_{2})$ are holomorphic on
$\mathcal{R}$ for all $n\geq 2$. These are related to the $G_{n}(\tau)$
appearing in (1.4) via $G_{n}(\tau)=G_{n}(1,\tau)$.

We recall the Weierstrass elliptic functions. The functions
$\wp(u\,|\,\omega_{1},\omega_{2})$ and $\zeta(u\,|\,\omega_{1},\omega_{2})$
are meromorphic functions of $u\in\mathbb{C}$ with poles of order 2 and 1
respectively at the lattice points $\Lambda(\omega_{1},\omega_{2})$. The
function $\sigma(u\,|\,\omega_{1},\omega_{2})$ is an entire function of
$u\in\mathbb{C}$ with simple zeroes at the lattice points. There are relations

$\zeta(u\,|\,\omega_{1},\omega_{2})=\frac{\partial}{\partial
u}\log\sigma(u\,|\,\omega_{1},\omega_{2}),\qquad\wp(u\,|\,\omega_{1},\omega_{2})=-\frac{\partial^{2}}{\partial
u^{2}}\log\sigma(u\,|\,\omega_{1},\omega_{2}),$ (2.5)

and a Laurent expansion at $u=0$

$\log\sigma(u\,|\,\omega_{1},\omega_{2})-\log(u)=-\sum_{g\geq
3}\frac{G_{2g-2}(\omega_{1},\omega_{2})}{2g-2}\cdot u^{2g-2}.$ (2.6)

We introduce minor modifications

$\mathscr{h}(u\,|\,\omega_{1},\omega_{2})=\log\sigma(u\,|\,\omega_{1},\omega_{2})-\log(u)-\tfrac{1}{2}G_{2}(\omega_{1},\omega_{2})u^{2},$
(2.7) $\tilde{\zeta}(u\,|\,\omega_{1},\omega_{2})=\frac{\partial}{\partial
u}\mathscr{h}(u\,|\,\omega_{1},\omega_{2})=\zeta(u\,|\,\omega_{1},\omega_{2})-u^{-1}-G_{2}(\omega_{1},\omega_{2})\,u,$
(2.8) $\tilde{\wp}(u\,|\,\omega_{1},\omega_{2})=-\frac{\partial^{2}}{\partial
u^{2}}\mathscr{h}(u\,|\,\omega_{1},\omega_{2})=\wp(u\,|\,\omega_{1},\omega_{2})-u^{-2}+G_{2}(\omega_{1},\omega_{2}),$
(2.9)

which are holomorphic near $u=0$. Below we shall need the related functions

$\mathscr{f}(u\,|\,\omega_{1},\omega_{2})=2\tilde{\zeta}(u\,|\,\omega_{1},\omega_{2})-u\tilde{\wp}(u\,|\,\omega_{1},\omega_{2}),$
(2.10)
$\mathscr{k}_{i}(u\,|\,\omega_{1},\omega_{2})=\frac{\partial}{\partial\omega_{i}}\mathscr{h}(u\,|\,\omega_{1},\omega_{2}),$
(2.11)

which have poles precisely at the nonzero lattice points
$\Lambda^{*}(\omega_{1},\omega_{2})$. These are double poles in the case of
$\mathscr{f}$ and simple poles in the case of $\mathscr{k}_{i}$. Later we will
need the parity properties

$\mathscr{f}(-u\,|\,\omega_{1},\omega_{2})=-\mathscr{f}(u\,|\,\omega_{1},\omega_{2}),\qquad\mathscr{k}_{i}(-u\,|\,\omega_{1},\omega_{2})=\mathscr{k}_{i}(u\,|\,\omega_{1},\omega_{2})$
(2.12)

which follow immediately from the expansion (2.6).

###  2.2. Borel transforms

Our starting point is the following formal power series in $\epsilon$

$H(\epsilon\,|\,\omega_{1},\omega_{2})=\sum_{g\geq
2}\frac{B_{2g}\,G_{2g-2}(\omega_{1},\omega_{2})\,(2\pi\mathrm{i})^{2g}\,\epsilon^{2g-1}}{4g(2g-1)(2g-2)}.$
(2.13)

The coefficients are holomorphic functions of
$(\omega_{1},\omega_{2})\in\mathcal{R}$ involving the Bernoulli numbers
$B_{2g}$ and the Eisenstein series (2.3). We then consider the formal power
series

$F(\epsilon\,|\,\omega_{1},\omega_{2})=\frac{\partial}{\partial\epsilon}H(\epsilon\,|\,\omega_{1},\omega_{2}),\qquad
K_{i}(\epsilon\,|\,\omega_{1},\omega_{2})=\frac{\partial}{\partial\omega_{i}}H(\epsilon\,|\,\omega_{1},\omega_{2}).$
(2.14)

We denote by $f(\eta\,|\,\omega_{1},\omega_{2})$ and
$k_{i}(\eta\,|\,\omega_{1},\omega_{2})$ the Borel transforms of these series.
They are power series in $\eta$ with coefficients which are holomorphic
functions of $(\omega_{1},\omega_{2})\in\mathcal{R}$.

Note that $F(\epsilon\,|\,\omega_{1},\omega_{2})$ is related to the previous
$F_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})$ from (1.5) by

$F_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})=-\frac{e(X)}{(2\pi\mathrm{i})^{2}}\cdot
F(\epsilon\,|\,\omega_{1},\omega_{2})\,.$ (2.15)

In particular, for $F_{\text{GW}}(\lambda\,|\,\tau)$ given in (1.4)

$F_{\text{GW}}(\lambda\,|\,\tau)=-\frac{e(X)}{(2\pi\mathrm{i})^{2}}\cdot
F(\epsilon\,|\,1,\tau),\qquad 2\pi\mathrm{i}\epsilon=\frac{\lambda}{2\pi}\,.$
(2.16)

We choose to work with $F(\epsilon\,|\,\omega_{1},\omega_{2})$ rather than
directly with $F_{\text{GW}}(\lambda\,|\,\tau)$ for two reasons. On the one
hand, the change of variables from $\lambda$ to $\epsilon$ and the rescaling
by $-e(X)/(2\pi\mathrm{i})^{2}$ eliminates certain awkward factors from the
Borel sums and the positions of the poles of the Borel transform. On the other
hand, the introduction of the variables $\omega_{1},\omega_{2}$ facilitates
the relation with the RH problem.

Let us fix a point $(\omega_{1},\omega_{2})\in\mathcal{R}$. The following
result is proved in Section 4.

######  Proposition .

  * (i)

The Borel transforms $f(\eta\,|\,\omega_{1},\omega_{2})$ and
$k_{i}(\eta\,|\,\omega_{1},\omega_{2})$ have positive radius of convergence
and hence define holomorphic functions in a neighbourhood of $\eta=0$.

  * (ii)

These functions extend to meromorphic functions of $\eta\in\mathbb{C}$ with
poles precisely at the nonzero lattice points
$\Lambda^{*}(\omega_{1},\omega_{2})$. The poles are double poles in the case
of $f$ and simple poles in the case of $k_{i}$.

  * (iii)

There are explicit expressions

$f(\eta\,|\,\omega_{1},\omega_{2})=\sum_{m\geq
1}\frac{1}{m^{3}}\,\mathscr{f}\Big{(}\frac{\eta}{m}\,|\,\omega_{1},\omega_{2}\Big{)},$
(2.17) $k_{i}(\eta\,|\,\omega_{1},\omega_{2})=\sum_{m\geq
1}\frac{1}{m^{2}}\,\mathscr{k}_{i}\Big{(}\frac{\eta}{m}\,|\,\omega_{1},\omega_{2}\Big{)},$
(2.18)

which converge absolutely and uniformly for $\eta$ in compact subsets of
$\mathbb{C}$. ∎

The Borel transform of $H(\epsilon\,|\,\omega_{1},\omega_{2})$ also has
positive radius of convergence, but the holomorphic continuation of the
resulting function $h(\eta\,|\,\omega_{1},\omega_{2})$ is more complicated due
to the presence of logarithmic singularities, and we will not directly
consider this function here.

###  2.3. Irrationality measure

To define the Borel sum of the series (2.14) we must consider a Laplace-type
integral of the form (1.3). Note that a ray $r\subset\mathbb{C}^{*}$ is non-
Stokes precisely if it contains no points of the lattice
$\Lambda(\omega_{1},\omega_{2})\subset\mathbb{C}$. Since a non-Stokes ray
still comes arbitrarily close to points of $\Lambda(\omega_{1},\omega_{2})$,
when trying to control the growth of such integrals we encounter some basic
notions from Diophantine approximation which we now recall.

The irrationality measure $\mu(\alpha)$ of a real number $\alpha\in\mathbb{R}$
[12, Definition E.1] is defined to be the infimum $\mu(\alpha)=\inf R(\alpha)$
of the set

$R(\alpha)=\big{\\{}d\in\mathbb{R}_{>0}\;|\;0<\left|\alpha-p/q\right|<1/q^{d}\text{
for at most finitely many }p,q\in\mathbb{Z},q>0\big{\\}}.$ (2.19)

If $R(\alpha)=\emptyset$ we set $\mu(\alpha)=\infty$. In this case $\alpha$ is
known as a Liouville irrational. We will use the following well-known
properties of $\mu(\alpha)$.

######  Theorem 2.1.

  * (i)

if $\alpha\in\mathbb{Q}$ then $\mu(\alpha)=1$,

  * (ii)

if $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ then $\mu(\alpha)\geq 2$,

  * (iii)

if $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ then

$\mu\bigg{(}\frac{a\alpha+b}{c\alpha+d}\bigg{)}=\mu(\alpha)\text{ for all
}\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\operatorname{GL}_{2}(\mathbb{Z}),$
(2.20)

  * (iv)

the subset $\\{\alpha\in\mathbb{R}:\mu(\alpha)>2\\}$ has measure zero.

###### Proof.

If $\alpha\in\mathbb{Q}$ then it is easy to check that $\mu(\alpha)\geq 1$,
while $\mu(\alpha)\leq 1$ follows from Liouville’s theorem, which states that
algebraic numbers of degree $n$ satisfy $\mu(\alpha)\leq n$. Part (ii) follows
immediately from the Dirichlet approximation theorem, while (iv) is a Theorem
due to Khinchin [19], whose proof is essentially an application of the Borel-
Cantelli Lemma. We could not find a direct reference for part (iii) so we
include a proof in Appendix B. ∎

Given a point $(\omega_{1},\omega_{2})\in\mathcal{R}$ we can define the
irrationality measure $\mu(r)\in[1,\infty]$ of a ray $r\subset\mathbb{C}^{*}$
as follows. If $\pm\omega_{2}\in r$ we define $\mu(r)=1$. Otherwise there is a
unique $\alpha\in\mathbb{R}$ such that
$\pm(\omega_{1}+\alpha\cdot\omega_{2})\in r$ and we define
$\mu(r)=\mu(\alpha)$. Part (iii) of Theorem 2.1 ensures that the resulting
notion depends only on the lattice
$\Lambda(\omega_{1},\omega_{2})\subset\mathbb{C}$ rather than the specific
generators $\omega_{1},\omega_{2}$. Part (i) shows that $\mu(r)=1$ precisely
if $r$ contains a lattice point, and part (iv) that almost all rays have
$\mu(r)=2$.

###  2.4. Borel sums and integrals along detour paths

Let us again fix a point $(\omega_{1},\omega_{2})\in\mathcal{R}$. Recall that
a ray $r\subset\mathbb{C}^{*}$ is non-Stokes precisely if it contains no
lattice points. The following results about Borel summation are proved in
Section 5.2.

######  Theorem 2.2.

Let $r\subset\mathbb{C}^{*}$ be a non-Stokes ray with $\mu(r)<\infty$ and take
$\epsilon\in\mathbb{H}_{r}$.

  * (i)

The integrals

$F^{r}(\epsilon\,|\,\omega_{1},\omega_{2})=\int_{r}e^{-\eta/\epsilon}f(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta,\qquad
K_{i}^{r}(\epsilon)=\int_{r}\
e^{-\eta/\epsilon}k_{i}(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta,$ (2.21)

are absolutely convergent and depend holomorphically on
$\epsilon\in\mathbb{H}_{r}$. In particular, the series
$F(\epsilon\,|\,\omega_{1},\omega_{2})$ and
$K_{i}(\epsilon\,|\,\omega_{1},\omega_{2})$ are Borel summable along the ray
$r$.

  * (ii)

The Borel sums can be re-expressed as absolutely convergent integrals

$F^{r}(\epsilon\,|\,\omega_{1},\omega_{2})=\int_{r}\mathrm{Li}_{2}(e^{-\eta/\epsilon})\,\mathscr{f}(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta,$
(2.22)
$K_{i}^{r}(\epsilon\,|\,\omega_{1},\omega_{2})=\int_{r}\operatorname{Li}_{1}(e^{-\eta/\epsilon})\,\mathscr{k}_{i}(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta,$
(2.23)

where $\mathrm{Li}_{k}(z)$ denotes the $k$-th polylogarithm. ∎

Note that equation (2.22) follows from (2.17) and the following formal
rearrangements, which are justified in the proof of Theorem 2.2:

$\begin{split}\int_{r}e^{-\eta/\epsilon}\,\sum_{m\geq
1}\frac{1}{m^{3}}\,\mathscr{f}\Big{(}\frac{\eta}{m}\Big{)}\,\mathrm{d}\eta&=\sum_{m\geq
1}\,\int_{r}e^{-\eta/\epsilon}\,\frac{1}{m^{3}}\,\mathscr{f}\Big{(}\frac{\eta}{m}\Big{)}\,\mathrm{d}\eta=\sum_{m\geq
1}\,\int_{r}\frac{1}{m^{2}}\,e^{-m\eta/\epsilon}\mathscr{f}(\eta)\,\mathrm{d}\eta\\\
&=\int_{r}\,\sum_{m\geq
1}\frac{1}{m^{2}}\,e^{-m\eta/\epsilon}\mathscr{f}(\eta)\,\mathrm{d}\eta=\int_{r}\operatorname{Li}_{2}(e^{-\eta/\epsilon})\mathscr{f}(\eta)\,\mathrm{d}\eta.\end{split}$
(2.24)

Similar remarks apply to (2.23).

Consider now an arbitrary non-Stokes ray $r\subset\mathbb{C}^{*}$. For any
$0<\delta\ll\text{min}\\{|\omega_{1}|,|\omega_{2}|\\}$ there is a uniquely-
defined detour path $r(\delta)$ which combines the ray $r$ with arcs of angle
$<\pi$ taken from discs of radius $\delta$ centered on points of
$\Lambda^{*}(\omega_{1},\omega_{2})$ (see Figure 1). The following proposition
is proved in Section 5.3.

Figure 1. The vectors denote the generators $\omega_{1}$ and $\omega_{2}$ of
the lattice $\Lambda(\omega_{1},\omega_{2})$. The discs are centered at the
points in $\Lambda(\omega_{1},\omega_{2})^{*}$ and have radius $\delta>0$
small enough that they do not intersect. The bold path $r(\delta)$ is
determined by the direction of the non-Stokes ray $r$, and takes a detour
along the boundary of any disc intersected by $r$. These detours traverse arcs
of the boundary of angle $<\pi$.

######  Proposition .

Let $r\subset\mathbb{C}^{*}$ be a non-Stokes ray with respect to the lattice
$\Lambda(\omega_{1},\omega_{2})$. Then there is $D>0$ such that for all
$0<\delta<D$ the integrals

$F^{r(\delta)}(\epsilon\,|\,\omega_{1},\omega_{2})=\int_{r(\delta)}\mathrm{Li}_{2}(e^{-\eta/\epsilon})\,\mathscr{f}(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta,$
(2.25)
$K_{i}^{r(\delta)}(\epsilon\,|\,\omega_{1},\omega_{2})=\int_{r(\delta)}\operatorname{Li}_{1}(e^{-\eta/\epsilon})\,\mathscr{k}_{i}(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta,$
(2.26)

are absolutely convergent for all $\epsilon\in\mathbb{H}_{r}$. The resulting
integrals depend holomorphically on $\epsilon\in\mathbb{H}_{r}$, and are
independent of $\delta$. Moreover

$F^{r(\delta)}(\epsilon\,|\,\omega_{1},\omega_{2})=F^{r}(\epsilon\,|\,\omega_{1},\omega_{2}),\qquad
K_{i}^{r(\delta)}(\epsilon\,|\,\omega_{1},\omega_{2})=K_{i}^{r}(\epsilon\,|\,\omega_{1},\omega_{2})$
(2.27)

whenever $\mu(r)<\infty$. ∎

Thus in the case of a non-Stokes ray $r\subset\mathbb{C}^{*}$ satisfying
$\mu(r)=\infty$ we can use (2.25) and (2.26) to define substitutes for the
functions (2.21), although these are no longer directly related to the Borel
sums of the series $F(\epsilon\,|\,\omega_{1},\omega_{2})$ and
$K_{i}(\epsilon\,|\,\omega_{1},\omega_{2})$.

Finally, we record how the integrals corresponding to different rays are
related to each other. This proposition is proved at the end of Section 5.4.

######  Proposition .

Let $\Delta\subset\mathbb{C}^{*}$ be a convex sector whose boundary consists
of two rays $r_{1}$ and $r_{2}$ taken in clockwise order. Assume the rays
$r_{1},r_{2}$ are non-Stokes with respect to the lattice
$\Lambda(\omega_{1},\omega_{2})$. Then for
$\epsilon\in\mathbb{H}_{r_{1}}\cap\mathbb{H}_{r_{2}}$ and small enough
$\delta>0$ we have

$\begin{split}K_{i}^{r_{2}(\delta)}(\epsilon\,|\,\omega_{1},\omega_{2})-K_{i}^{r_{1}(\delta)}(\epsilon\,|\,\omega_{1},\omega_{2})&=2\pi\mathrm{i}\sum_{\omega\in\Delta\cap\Lambda^{*}(\omega_{1},\omega_{2})}a_{i}\cdot\log(1-e^{-\omega/\epsilon})\\\
F^{r_{2}(\delta)}(\epsilon,\omega_{1},\omega_{2})-F^{r_{1}(\delta)}(\epsilon,\omega_{1},\omega_{2})&=2\pi\mathrm{i}\sum_{\omega\in\Delta\cap\Lambda^{*}(\omega_{1},\omega_{2})}\frac{\partial}{\partial\epsilon}\left(\epsilon\,\mathrm{Li}_{2}(e^{-\omega/\epsilon})\right)\,,\end{split}$
(2.28)

where we write $\omega=a_{1}\omega_{1}+a_{2}\omega_{2}$. ∎

##  3\. Free energy, DT invariants and the RH problem

In this section we use the analytic results of the previous section to prove
our main results. As in the introduction we consider a smooth projective CY
threefold with an elliptic fibration $\pi\colon X\to B$ satisfying Assumptions
1.1.

###  3.1. Free energy and its Borel sums

Recall from (1.4) that the $g\geq 2$ part of the GW generating function in the
fibre classes is given by the formal power series in $\lambda$

$F_{\text{GW}}(\lambda\,|\,\tau)=-e(X)\cdot\sum_{g\geq
2}\frac{B_{2g}\,G_{2g-2}(\tau)}{4g(2g-2)}\cdot\left(\frac{\lambda}{2\pi}\right)^{2g-2},$
(3.1)

whose coefficients depend on a Kähler parameter $\tau\in\mathbb{C}$ satisfying
$\operatorname{Im}(\tau)>0$. As before, we set
$2\pi\mathrm{i}\epsilon=\lambda/2\pi$, and via the change of variables (1.5)
consider $F_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})$ as a function of
$\epsilon\in\mathbb{C}^{*}$ and $(\omega_{1},\omega_{2})$ lying in the region

$\mathcal{R}=\\{(\omega_{1},\omega_{2})\in(\mathbb{C}^{*})^{2}:\operatorname{Im}(\omega_{2}/\omega_{1})>0\\}.$
(3.2)

Recall from (2.15) that $F_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})$
and $F(\epsilon\,|\,\omega_{1},\omega_{2})$ are related by a rescaling by
$-e(X)/(2\pi\mathrm{i})^{2}$. The following result then follows immediately by
combining Proposition 2.2 and Theorem 2.2.

######  Theorem 3.1.

Fix $(\omega_{1},\omega_{2})\in\mathcal{R}$.

  * (i)

The Borel transform of the series
$F_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})$ is a meromorphic function
on $\mathbb{C}$ with double poles at the non-zero lattice points
$\Lambda^{*}(\omega_{1},\omega_{2})$ and no other poles.

  * (ii)

Suppose a non-Stokes ray $r\subset\mathbb{C}^{*}$ satisfies $\mu(r)<\infty$
with respect to the lattice $\Lambda(\omega_{1},\omega_{2})$. Then the Borel
sum $F^{r}_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})$ exists for all
$\epsilon\in\mathbb{H}_{r}$. ∎

Theorem 3.1 together with part (iv) of Theorem 2.1 implies that for almost all
non-Stokes rays $r\subset\mathbb{C}^{*}$ the Borel sum of
$F_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})$ exists for
$\epsilon\in\mathbb{H}_{r}$. Combining Proposition 2.4 with (2.15) gives the
following result relating the Borel sums along different rays.

######  Proposition .

Fix $(\omega_{1},\omega_{2})\in\mathcal{R}$. Let $\Delta\subset\mathbb{C}^{*}$
be a convex sector whose boundary consists of two rays $r_{1}$ and $r_{2}$
taken in clockwise order. Assume the rays $r_{1},r_{2}$ are non-Stokes and
satisfy $\mu(r_{i})<\infty$ with respect to the lattice
$\Lambda(\omega_{1},\omega_{2})$. Then

$F^{r_{2}}_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})-F^{r_{1}}_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})=-\frac{e(X)}{2\pi\mathrm{i}}\cdot\sum_{\omega\in\Delta\cap\Lambda^{*}(\omega_{1},\omega_{2})}\frac{\partial}{\partial\epsilon}\left(\epsilon\,\mathrm{Li}_{2}(e^{-\omega/\epsilon})\right)\,$
(3.3)

for all $\epsilon\in\mathbb{H}_{r_{1}}\cap\mathbb{H}_{r_{2}}$. ∎

This matches previous results on the Stokes jumps of the Borel sum of free
energies and their relation to DT invariants. See for example [3, Equation
4.55] or [18, Equation 1.1 and 1.4].

###  3.2. Stability conditions and DT invariants

We consider the full triangulated subcategory $\mathcal{D}(\pi)\subset
D^{b}\operatorname{Coh}(X)$ of the bounded derived category of coherent
sheaves consisting of objects whose set-theoretic support is contained in a
finite union of fibres of $\pi$. The Chern characters of such objects can be
viewed as elements

$\operatorname{ch}(E)=(\operatorname{ch}_{2}(E),\operatorname{ch}_{3}(E))\in
N(\pi)=N_{1}(\pi)\oplus N_{0}(X),$ (3.4)

where $N_{1}(\pi)\subset N_{1}(X)$ consists of curve classes contracted by
$\pi$. The group $N_{0}(X)$ is freely generated by the class of a point, which
it is convenient to denote by $-\gamma_{1}$. The assumption that $\pi$ has
integral fibres implies that $N_{1}(\pi)$ is freely generated by the class
$\gamma_{2}$ of a fibre. Then

$\operatorname{ch}\colon K_{0}(\mathcal{D}(\pi))\to
N(\pi)=\mathbb{Z}\gamma_{1}\oplus\mathbb{Z}\gamma_{2},$ (3.5)

sends a rank $r$, degree $d$ bundle supported on a smooth fibre of $\pi$ to
the class $-d\gamma_{1}+r\gamma_{2}$. The Riemann-Roch theorem shows that for
any objects $A,B\in\mathcal{D}(\pi)$ we have

$\chi(A,B):=\sum_{i\in\mathbb{Z}}\operatorname{dim}_{\mathbb{C}}\operatorname{Hom}_{X}(A,B[i])=0.$

Thus the Euler form for the category $\mathcal{D}(\pi)$ is identically zero,
and we therefore also equip the group $N(\pi)$ with the zero form
$\langle-,-\rangle=0$.

The definition of the subcategory $\mathcal{D}(\pi)$ ensures that the standard
t-structure on $\mathcal{D}^{b}\operatorname{Coh}(X)$ induces a t-structure on
$\mathcal{D}(\pi)$. The heart $\mathcal{A}(\pi)\subset\mathcal{D}(\pi)$
consists of coherent sheaves on $X$ whose set-theoretic support is contained
in a finite union of fibres of $\pi$. Fix an element $\tau\in\mathbb{C}$ with
$\operatorname{Im}(\tau)>0$. Then, as in [23, Example 2.3 (iii)], there is a
unique stability condition on the category $\mathcal{D}(\pi)$ whose heart is
the subcategory $\mathcal{A}(\pi)\subset\mathcal{D}(\pi)$, and whose central
charge $Z\colon K_{0}(\mathcal{D}(\pi))\to\mathbb{C}$ is the composition of
the Chern character (3.5) with the map

$Z\colon N(\pi)\to\mathbb{C},\qquad
Z(a_{1}\gamma_{1}+a_{2}\gamma_{2})=a_{1}+a_{2}\tau.$

There is a standard action of the group $\mathbb{C}$ on the space of stability
conditions which rotates the central charge and shifts the phases of the
semistable objects. Applying this to the stability conditions constructed
above we obtain for each point $(\omega_{1},\omega_{2})\in\mathcal{R}$, a
stability condition, unique up to the action of even shifts, whose central
charge is the composition of (3.5) with the map

$Z\colon N(\pi)\to\mathbb{C},\qquad
Z(a_{1}\gamma_{1}+a_{2}\gamma_{2})=a_{1}\omega_{1}+a_{2}\omega_{2}.$ (3.6)

Since rotating stability conditions does not effect the subcategories of
semistable objects, the calculation of Toda [23, Theorem 6.9] shows that the
DT invariants for any of these stability conditions are given by

$\Omega(a_{1}\gamma_{1}+a_{2}\gamma_{2})=-e(X),\qquad(a_{1},a_{2})\in\mathbb{Z}^{2}\setminus\\{0\\},$
(3.7)

where $e(X)$ is the topological Euler characteristic of the complex projective
variety $X$.

###  3.3. Riemann-Hilbert problem

Fix a point $(\omega_{1},\omega_{2})\in\mathcal{R}$. The data introduced in
the previous section defines what is called a BPS structure in [5]. Namely we
have a finite-rank free abelian group $N(\pi)$ equipped with a skew-symmetric
form $\langle-,-\rangle$, a group homomorphism $Z\colon N(\pi)\to\mathbb{C}$,
and a map of sets $\Omega\colon N(\pi)\to\mathbb{Z}$ which encodes the DT
invariants. Following [5], and exactly as in [9], we now explain the steps to
go from this data to a RH problem.

Since the skew-symmetric form $\langle-,-\rangle$ on $N(\pi)$ is identically
zero, in order to obtain a non-trivial RH problem we must first perform the
doubling procedure of [5, Section 2.8]. To do this we introduce the dual
abelian group
$N(\pi)^{\vee}=\operatorname{Hom}_{\mathbb{Z}}(N(\pi),\mathbb{Z})$ and
consider the lattice $\Gamma=N(\pi)\oplus N(\pi)^{\vee}$ equipped with the
canonical skew-symmetric pairing

$\langle-,-\rangle\colon\Gamma\times\Gamma\to\mathbb{Z},\qquad\langle(\gamma,\lambda),(\gamma^{\prime},\lambda^{\prime})\rangle=\lambda(\gamma^{\prime})-\lambda^{\prime}(\gamma).$
(3.8)

We denote by $\gamma^{\vee}_{i}$ the basis element of $N(\pi)^{\vee}$ dual to
$\gamma_{i}$. Thus $\langle\gamma^{\vee}_{i},\gamma_{j}\rangle=\delta_{ij}$.
We extend the central charge map $Z\colon N(\pi)\to\mathbb{C}$ defined by
(3.6) arbitrarily to a homomorphism $Z\colon\Gamma\to\mathbb{C}$. The choice
of this extension will play no significant role below. We also extend the map
of sets $\Omega\colon N(\pi)\to\mathbb{Z}$ to $\Gamma$ by insisting that
$\Omega(\gamma)=0$ unless $\gamma\in N(\pi)\subset\Gamma$.

The resulting doubled BPS structure $(\Gamma,Z,\Omega)$ has several special
properties identified in [5]. It is convergent because for large enough $R>0$

$\sum_{(a_{1},a_{2})\in\mathbb{Z}^{2}\setminus\\{0\\}}\exp({-R|a_{1}\omega_{1}+a_{2}\omega_{2}|})\,<\infty.$
(3.9)

It is moreover uncoupled since $\\{\gamma\in\Gamma:\Omega(\gamma)\neq
0\\}\subset N(\pi)$ and $\langle\gamma_{1},\gamma_{2}\rangle=0$ for
$\gamma_{1},\gamma_{2}\in N(\pi)$. We can then formulate a RH problem exactly
as in [9]. As well as the BPS structure $(\Gamma,Z,\Omega)$ it depends on an
element $\xi$ of the twisted torus

$\\{\xi\colon\Gamma\to\mathbb{C}^{*}:\xi(\gamma_{1}+\gamma_{2})=(-1)^{\langle\gamma_{1},\gamma_{2}\rangle}\xi(\gamma_{1})\xi(\gamma_{2})\\}$
(3.10)

called the constant term.

Recall from Section 1.2 that a ray $r\subset\mathbb{C}^{*}$ is called a Stokes
ray if it contains a point of $\Lambda(\omega_{1},\omega_{2})$, and otherwise
a non-Stokes ray. The RH problem involves holomorphic functions
$X_{\gamma}^{r}\colon\mathbb{H}_{r}\to\mathbb{C}^{*}$ for each non-Stokes ray
$r$ and each class $\gamma\in\Gamma$, where as before
$\mathbb{H}_{r}\subset\mathbb{C}^{*}$ denotes the half-plane centered on $r$.
Arguing as in [5, Section 5.1] we can use the fact that $(\Gamma,Z,\Omega)$ is
an uncoupled BPS structure to write for each $i=1,2$

$X_{\gamma_{i}}^{r}(\epsilon)=\exp(-Z(\gamma_{i})/\epsilon)\cdot\xi(\gamma_{i}),\qquad
X_{\gamma^{\vee}_{i}}^{r}=\exp(-Z(\gamma^{\vee}_{i})/\epsilon)\cdot\xi(\gamma^{\vee}_{i})\cdot
Y_{i}^{r}(\epsilon),$ (3.11)

with $Y_{i}^{r}\colon\mathbb{H}_{r}\to\mathbb{C}^{*}$ holomorphic. For $i=1,2$
we define

$\ell_{i}\colon\Lambda(\omega_{1},\omega_{2})\to\mathbb{Z},\qquad\ell_{i}(a_{1}\omega_{1}+a_{2}\omega_{2})=a_{i}.$
(3.12)

Choose a constant term $\xi\colon\Gamma\to\mathbb{C}^{*}$ satisfying
$\xi(\gamma_{i})=1$ for $i=1,2$. Then the RH problem can be formulated as
follows:

######  Problem .

For each non-Stokes ray $r\subset\mathbb{C}^{*}$ find holomorphic functions
$Y_{i}^{r}\colon\mathbb{H}_{r}\to\mathbb{C}^{*}$ with $i=1,2$ such that the
following statements hold.

  * (RH1)

If $\Delta\subset\mathbb{C}^{*}$ is a convex sector whose boundary consists of
non-Stokes rays $r_{1},r_{2}$ taken in clockwise order then

$Y_{i}^{r_{2}}(\epsilon)=Y_{i}^{r_{1}}(\epsilon)\cdot\prod_{\omega\in\Delta(r_{1},r_{2})\cap\Lambda^{*}}\left(1-e^{-\omega/\epsilon}\right)^{-\ell_{i}(\omega)\cdot
e(X)},$ (3.13)

for $\epsilon\in\mathbb{H}_{r_{1}}\cap\mathbb{H}_{r_{2}}$ with
$0<|\epsilon|\ll 1$.

  * (RH2)

As $\epsilon\to 0$ in any closed subsector of $\mathbb{H}_{r}$ we have
$Y^{r}(\epsilon)\to 1$.

  * (RH3)

There is an $N>0$ such that as $\epsilon\to\infty$ in $\mathbb{H}_{r}$ there
is a bound $|\epsilon|^{-N}<|Y^{r}(\epsilon)|<|\epsilon|^{N}$.

If this problem has a solution then it is unique [9]. We shall instead
consider what we call the weak RH problem in which we drop condition (RH3).
The resulting solutions are unique up to multiplication of $Y_{i}^{r}$ by
arbitrary holomorphic functions $P_{i}\colon\mathbb{C}\to\mathbb{C}^{*}$
satisfying $P_{i}(0)=1$.

###  3.4. Solution to the weak RH problem

We again fix a point $(\omega_{1},\omega_{2})\in\mathcal{R}$. Recall the
functions $K_{i}^{r(\delta)}(\epsilon\,|\,\omega_{1},\omega_{2})$ defined in
Proposition 2.4. For each non-Stokes ray $r\subset\mathbb{C}^{*}$ we define a
function $Y_{i}^{r}\colon\mathbb{H}_{r}\to\mathbb{C}^{*}$ by

$\begin{split}Y_{i}^{r}(\epsilon)&:=\exp\Big{(}-\frac{e(X)}{2\pi\mathrm{i}}\cdot
K_{i}^{r(\delta)}(\epsilon\,|\,\omega_{1},\omega_{2})\Big{)}\\\
&=\exp\left(-\frac{e(X)}{2\pi\mathrm{i}}\int_{r(\delta)}\mathrm{Li}_{1}(e^{-\eta/\epsilon})\,\mathscr{k}_{i}(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta\right)\,.\end{split}$
(3.14)

The integral is absolutely convergent, holomorphic in
$\epsilon\in\mathbb{H}_{r}$, and does not depend on $0<\delta\ll 1$ by
Proposition 2.4. As before, we remark that for almost all non-Stokes rays $r$
we have $\mu(r)<\infty$, and for such rays it follows from Proposition 2.4 and
Theorem 2.2 that

$Y_{i}^{r}(\epsilon)=\exp\left(-\frac{e(X)}{2\pi\mathrm{i}}\int_{r}e^{-\eta/\epsilon}\,k_{i}(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta\right)\,.$
(3.15)

######  Theorem 3.2.

The functions $Y_{i}^{r}\colon\mathbb{H}_{r}\to\mathbb{C}^{*}$ give a solution
to the weak RH problem.

###### Proof.

Let $\Delta\subset\mathbb{C}^{*}$ be a convex sector whose boundary consists
of non-Stokes rays $r_{1},r_{2}$ taken in clockwise order. By Proposition 2.4
it follows that for $\epsilon\in\mathbb{H}_{r_{1}}\cap\mathbb{H}_{r_{2}}$

$\begin{split}Y_{i}^{r_{2}}(\epsilon)&=Y_{i}^{r_{1}}(\epsilon)\cdot\exp\left(-e(X)\sum_{\omega\in\Delta\cap\Lambda^{*}}\ell_{i}(\omega)\cdot\log(1-e^{-\omega/\epsilon})\right)\\\
&=Y_{i}^{r_{1}}(\epsilon)\prod_{\omega\in\Delta\cap\Lambda^{*}}(1-e^{-\omega/\epsilon})^{-e(X)\cdot\ell_{i}(\omega)}\end{split}$
(3.16)

so property (RH1) holds. Property (RH2) follows from the following lemma. ∎

######  Lemma .

Fix a non-Stokes ray $r\subset\mathbb{C}^{*}$ and a closed subsector
$S_{r}\subset\mathbb{H}_{r}$. Then

$\lim_{\epsilon\to 0,\;\epsilon\in
S_{r}}K^{r(\delta)}_{i}(\epsilon\,|\,\omega_{1},\omega_{2})=0.$ (3.17)

###### Proof.

Given the closed sector $S_{r}$, we can assume that $\delta>0$ is small enough
such that

$\text{Re}(\eta/\epsilon)>0\text{ for all $\eta\in r(\delta)$ and all
$\epsilon\in S_{r}$\,.}$ (3.18)

More precisely, given $K>0$ there exists a constant $C>0$ such that

$\text{Re}(\eta/\epsilon)=\left|\frac{\eta}{\epsilon}\right|\cos(\arg(\eta/\epsilon))>C\cdot\left|\frac{\eta}{\epsilon}\right|>\frac{C}{K}\cdot|\eta|>0$
(3.19)

for all $\eta\in r(\delta)$ and $\epsilon\in S_{r}$ with $|\epsilon|<K$. This
in particular implies (using that $|\log(1-z)|<-\log(1-|z|)$ for $|z|<1|$)
that in the same range of parameters

$|\mathrm{Li_{1}}(e^{-\eta/\epsilon})\,\mathscr{k}_{i}(\eta)|<-\log(1-|e^{-\eta/\epsilon}|)\,|\mathscr{k}_{i}(\eta)|<-\log(1-e^{-C|\eta|/K})\,|\mathscr{k}_{i}(\eta)|\,.$
(3.20)

Finally, by the same argument as in Proposition 5.3 one can show that

$-\int_{r(\delta)}\log(1-e^{-C|\eta|/K})|\mathscr{k}_{i}(\eta)||\mathrm{d}\eta|<\infty,$
(3.21)

so by applying the dominated convergence theorem, we can interchange limits
and integrals and obtain

$\begin{split}\lim_{\epsilon\to 0,\;\epsilon\in
S_{r}}K_{i}^{r(\delta)}&=\int_{r(\delta)}\mathrm{Li}_{1}(0)\mathscr{k}_{i}(\eta,\omega_{1},\omega_{2})\mathrm{d}\eta=0\,.\\\
\end{split}$ (3.22)

∎

Fix a non-Stokes ray $r\subset\mathbb{C}^{*}$. For $\epsilon\in\mathbb{H}_{r}$
we define

$\tau^{r}_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2}):=\exp\Big{(}-\frac{e(X)}{(2\pi\mathrm{i})^{2}}\cdot
F^{r(\delta)}(\epsilon\,|\,\omega_{1},\omega_{2})\Big{)}\,,$ (3.23)

where $0<\delta\ll 1$ and $F^{r(\delta)}(\epsilon\,|\,\omega_{1},\omega_{2})$
is as in Proposition 2.4. Note that by (2.15)

$\tau^{r}_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})=\exp\Big{(}F^{r(\delta)}_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})\Big{)}\,.$
(3.24)

In the case that $\mu(r)<\infty$ the following result relates the Borel sum of
the free energy to the solution to the weak RH problem constructed above.

######  Theorem 3.3.

For each $i=1,2$ there is a relation

$\frac{\partial}{\partial\omega_{i}}\log\tau^{r}_{\text{GW}}(\epsilon\,|\,\omega_{1},\omega_{2})=\frac{1}{2\pi\mathrm{i}}\cdot\frac{\partial}{\partial\epsilon}\log
Y_{i}^{r}(\epsilon\,|\,\omega_{1},\omega_{2})\,.$ (3.25)

###### Proof.

Using (3.23) and (3.15), showing that (3.25) holds reduces to showing that

$\frac{\partial}{\partial\omega_{i}}F^{r(\delta)}(\epsilon\,|\,\omega_{1},\omega_{2})=\frac{\partial}{\partial\epsilon}K_{i}^{r(\delta)}(\epsilon\,|\,\omega_{1},\omega_{2})\,.$
(3.26)

This easily follows by differentiating under the integral sign and integrating
by parts twice using (2.8), (2.9), (2.10), and (2.11). ∎

##  4\. Borel transforms

In this section we prove Proposition 2.2 concerning the Borel transforms of
our series. We fix a point $(\omega_{1},\omega_{2})\in\mathcal{R}$ throughout.

###  4.1. Hadamard product

Consider again the power series

$H(\epsilon)=\sum_{g\geq
2}\frac{B_{2g}\,G_{2g-2}\,(2\pi\mathrm{i})^{2g}\epsilon^{2g-1}}{4g(2g-1)(2g-2)}\in\epsilon\mathbb{C}[[\epsilon]].$
(4.1)

from (2.13). We have suppressed the dependence on $\omega_{1},\omega_{2}$ from
the notation. The Borel transform is the series

$h(\eta)=\sum_{g\geq
2}\frac{B_{2g}\,G_{2g-2}\,(2\pi\mathrm{i})^{2g}\eta^{2g-2}}{2(2g-2)(2g)!}\in\mathbb{C}[[\eta]].$
(4.2)

Following the approach of [3, Section 3.1] we can write $h(\eta)$ as a
Hadamard product of series $h_{1}(\eta)$ and $h_{2}(\eta)$, where

$h_{1}(\eta)=-\sum_{g\geq
2}\frac{B_{2g}\,(2\pi\mathrm{i})^{2g}\eta^{2g-2}}{2(2g)!},\qquad
h_{2}(\eta)=-\sum_{g\geq 2}\frac{G_{2g-2}\,\eta^{2g-2}}{2g-2}.$ (4.3)

Using the defining generating series for the Bernoulli numbers we find that
$h_{1}(\eta)$ is the Taylor expansion at the origin of a meromorphic function
on $\mathbb{C}$ with poles only at the points
$m\in\mathbb{Z}\setminus\\{0\\}$. Indeed

$h_{1}(\eta)=(2\pi\mathrm{i})^{2}\left(\frac{1}{2(2\pi\mathrm{i}\eta)(1-e^{2\pi\mathrm{i}\eta})}+\frac{1}{2(2\pi\mathrm{i}\eta)^{2}}-\frac{1}{4(2\pi\mathrm{i}\eta)}+\frac{1}{24}\right).$
(4.4)

Moreover $h_{2}(\eta)$ is the Taylor expansion at $\eta=0$ of the function

$\mathscr{h}(\eta)=\log\sigma(\eta)-\log(\eta)-\tfrac{1}{2}G_{2}\eta^{2}$
(4.5)

introduced in Section 2.1. It follows from [3, Lemma 3.2] that $h(\eta)$ is
the Taylor expansion at $\eta=0$ of the function $h(\eta)$ given by the anti-
clockwise contour integral

$h(\eta)=\frac{1}{2\pi\mathrm{i}}\int_{|s|=\tfrac{1}{2}}h_{1}(s)\,h_{2}(\eta/s)\,\frac{\mathrm{d}s}{s}.$
(4.6)

This expression is valid providing $|\eta|<\tfrac{1}{2}|\omega|$ for all
nonzero lattice points $\omega\in\Lambda^{*}(\omega_{1},\omega_{2})$. This
ensures that the singularities of $h_{2}(\eta/s)$ all lie inside the contour
$|s|=\tfrac{1}{2}$. Note that the poles of $h_{1}(s)$ always lie outside this
contour.

###  4.2. Borel transforms

Recall the formal power series $F(\epsilon)$ and $K_{i}(\epsilon)$ defined by
(2.14). Reall also the functions

$\mathscr{f}(\eta\,|\,\omega_{1},\omega_{2})=2\tilde{\zeta}(\eta\,|\,\omega_{1},\omega_{2})-\eta\tilde{\wp}(\eta\,|\,\omega_{1},\omega_{2}),$
(4.7)
$\mathscr{k}_{i}(\eta\,|\,\omega_{1},\omega_{2})=\frac{\partial}{\partial\omega_{i}}\mathscr{h}(\eta\,|\,\omega_{1},\omega_{2}).$
(4.8)

defined in Section 2.1. They are holomorphic in a neighbourhood of $\eta=0$.

######  Proposition .

The Borel transforms of the series $F(\epsilon)$ and $K_{i}(\epsilon)$ are the
Taylor expansions of holomorphic functions $f(\eta)$ and $k_{i}(\eta)$ defined
near $\eta=0$ by the expressions

$f(\eta)=\frac{1}{2\pi\mathrm{i}}\int_{|s|=\tfrac{1}{2}}h_{1}(s)\,\mathscr{f}(\eta/s)\,\frac{\mathrm{d}s}{s^{2}},\qquad
k_{i}(\eta)=\frac{1}{2\pi\mathrm{i}}\int_{|s|=\tfrac{1}{2}}h_{1}(s)\,\mathscr{k}_{i}(\eta/s)\,\frac{\mathrm{d}s}{s}.$
(4.9)

###### Proof.

The expression for $k_{i}(\eta)$ follows immediately by differentiating (4.6)
under the integral with respect to $\omega_{i}$. To obtain the expression for
$f(\eta)$ note first that if formal series
$F(\epsilon),H(\epsilon)\in\epsilon\mathbb{C}[[\epsilon]]$ have Borel
transforms $f(\eta),h(\eta)\in\mathbb{C}[[\eta]]$ respectively, then

$F(\epsilon)=\frac{\mathrm{d}}{\mathrm{d}\epsilon}H(\epsilon)\implies
f(\eta)=\frac{1}{\eta}\frac{\mathrm{d}}{\mathrm{d}\eta}\bigg{(}\eta^{2}\,\frac{\mathrm{d}}{\mathrm{d}\eta}h(\eta)\bigg{)}.$
(4.10)

Indeed, it is enough to check the case $H(\epsilon)=\epsilon^{n}$ when the
relation becomes

$\frac{n\,\eta^{n-2}}{(n-2)!}=\frac{1}{\eta}\,\frac{\mathrm{d}}{\mathrm{d}\eta}\bigg{(}\eta^{2}\frac{\mathrm{d}}{\mathrm{d}\eta}\bigg{(}\frac{\eta^{n-1}}{(n-1)!}\bigg{)}\bigg{)}.$
(4.11)

Next note that (2.7) - (2.9) give

$\frac{1}{\eta}\frac{\mathrm{d}}{\mathrm{d}\eta}\bigg{(}\eta^{2}\,\frac{\mathrm{d}}{\mathrm{d}\eta}\mathscr{h}\left(\frac{\eta}{s}\right)\bigg{)}=\frac{1}{s}\left(2\tilde{\zeta}\left(\frac{\eta}{s}\right)-\frac{\eta}{s}\tilde{\wp}\left(\frac{\eta}{s}\right)\right)=\frac{1}{s}\mathscr{f}\left(\frac{\eta}{s}\right).$
(4.12)

The result then follows by differentiating (4.6) under the integral with
respect to $\eta$. ∎

###  4.3. Explicit expressions

The following result completes the proof of Proposition 2.2.

######  Proposition .

The functions $f(\eta)$ and $k_{i}(\eta)$ extend to meromorphic functions of
$\eta\in\mathbb{C}$ with poles precisely at the nonzero lattice points
$\Lambda^{*}(\omega_{1},\omega_{2})$. These poles are double poles in the case
of $f$ and simple poles in the case of $k_{i}$. There are explicit expressions

$f(\eta)=\sum_{m\geq
1}\frac{1}{m^{3}}\cdot\mathscr{f}\Big{(}\frac{\eta}{m}\Big{)},\qquad
k_{i}(\eta)=\sum_{m\geq
1}\frac{1}{m^{2}}\cdot\mathscr{k}_{i}\Big{(}\frac{\eta}{m}\Big{)},$ (4.13)

where the two series converge uniformly and absolutely in $\eta$ on compact
subsets of $\mathbb{C}$.

###### Proof.

We show the result for $f$, since a similar argument applies to $k_{i}$. For
each integer $N>0$ we consider the square contour

$C_{N}=\big{\\{}s\in\mathbb{C}:\max\big{(}|\text{Re}(s)|,|\text{Im}(s)|\big{)}=N+\tfrac{1}{2}\big{\\}},$
(4.14)

taken with the anti-clockwise orientation. Take $\eta\in\mathbb{C}$ such that
$|\eta|<\tfrac{1}{2}|\omega|$ for all nonzero lattice points
$\omega\in\Lambda^{*}(\omega_{1},\omega_{2})$.

Note that the function $h_{1}(s)$ has a simple pole at each point
$m\in\mathbb{Z}\setminus\\{0\\}$ with residue $-1/2m$. Moving the contour in
(4.9) and using $\mathscr{f}(-\eta)=-\mathscr{f}(\eta)$ therefore shows that
for any integer $N>0$

$f(\eta)-\frac{1}{2\pi\mathrm{i}}\int_{C_{N}}h_{1}(s)\,\mathscr{f}(\eta/s)\,\frac{\mathrm{d}s}{s^{2}}=\sum_{m=1}^{N}\frac{1}{m^{3}}\cdot\mathscr{f}\Big{(}\frac{\eta}{m}\Big{)}.$
(4.15)

On the other hand, one easily checks using the power expansion of
$\mathscr{f}(\eta)$ at $\eta=0$ that

$\mathscr{f}(\eta/s)=\mathcal{O}(1/s),\quad\text{as $|s|\to\infty$},$ (4.16)

while

$\left|\frac{1}{2(2\pi\mathrm{i}s)(1-e^{2\pi\mathrm{i}s})}\right|<\frac{C}{N+\tfrac{1}{2}},\quad\text{for
$s\in C_{N}$,}$ (4.17)

for some $C>0$ independent of $N$. It then follows from (4.4) that there is
some constant $D>0$ such that for all $N$ sufficiently large

$\left|\,\int_{C_{N}}h_{1}(s)\,\mathscr{f}(\eta/s)\,\frac{\mathrm{d}s}{s^{2}}\,\right|<\frac{D}{(N+\tfrac{1}{2})^{2}}\,.$
(4.18)

Thus the integral on the left-hand side of (4.15) tends to $0$ as
$N\to\infty$, and (4.13) holds.

It remains to show that the series (4.13) defines a meromorphic function on
$\mathbb{C}$ with double poles exactly at the points of
$\Lambda^{*}(\omega_{1},\omega_{2})$. Note that $\mathscr{f}(\eta)$ has poles
only at the points $\Lambda^{*}(\omega_{1},\omega_{2})$ and in particular is
holomorphic at $\eta=0$.

Let $D\subset\mathbb{C}$ be small disc centered at $0$ such that
$D\cap\Lambda^{*}(\omega_{1},\omega_{2})=\emptyset$ and $K\subset\mathbb{C}$ a
compact subset. For $\eta\in K$ and $M>0$ sufficiently large we have
$\eta/m\in D$ for $m\geq M$. In particular, the functions
$\mathscr{f}(\eta/m)$ are holomorphic and uniformly bounded for $m\geq M$ and
$\eta\in K$. It follows that the tail of the first sum in (4.13) (i.e. the sum
for $m\geq M$) converges uniformly and absolutely on $K$, and hence to a
holomorphic function on $K$. Since any lattice point
$\omega\in\Lambda^{*}(\omega_{1},\omega_{2})$ is a double pole of the function
$\mathscr{f}(\eta/m)$ for a finite but non-empty set of positive integers $m$,
the resulting $f(\eta)$ is a meromorphic function with double poles at
$\Lambda^{*}(\omega_{1},\omega_{2})$. ∎

##  5\. Borel summability and detour integrals

In this section we collect the results needed to show that the series
$K_{i}(\epsilon\,|\,\omega_{1},\omega_{2})$ and
$F(\epsilon\,|\,\omega_{1},\omega_{2})$ are Borel summable along almost all
non-Stokes rays $r\subset\mathbb{C}^{*}$. More precisely, the Borel sum exists
for non-Stokes rays $r$ that, in the sense of Section 2.4, have finite
irrationality measure $\mu(r)<\infty$ with respect to the lattice
$\Lambda(\omega_{1},\omega_{2})$. For a general non-Stokes ray $r$, we define
integrals along certain detour paths $r(\delta)$ which coincide with the Borel
sums when $\mu(r)<\infty$.

###  5.1. Key lemmas

We begin with the following useful lemmas:

######  Lemma .

Fix $(\omega_{1},\omega_{2})\in\mathcal{R}$ and take
$\omega=a_{1}\omega_{1}+a_{2}\omega_{2}\in\Lambda(\omega_{1},\omega_{2})$.
Then the functions $\wp(\eta\,|\,\omega_{1},\omega_{2})$,
$\zeta(\eta\,|\,\omega_{1},\omega_{2})$ and
$\rho_{i}(\eta\,|\,\omega_{1},\omega_{2}):=\partial_{\omega_{i}}\log(\sigma(\eta\,|\,\omega_{1},\omega_{2}))$
satisfy

$\begin{split}\wp(\eta+\omega\,|\,\omega_{1},\omega_{2})&=\wp(\eta\,|\,\omega_{1},\omega_{2}),\\\
\zeta(\eta+\omega\,|\,\omega_{1},\omega_{2})&=\zeta(\eta\,|\,\omega_{1},\omega_{2})+2(a_{1}\eta_{1}+a_{2}\eta_{2}),\quad\text{where}\quad\eta_{i}=\zeta(\omega_{i}/2\,|\,\omega_{1},\omega_{2}),\\\
\rho_{i}(\eta+\omega\,|\,\omega_{1},\omega_{2})&=\rho_{i}(\eta\,|\,\omega_{1},\omega_{2})-a_{i}\zeta(\eta\,|\,\omega_{1},\omega_{2})+\sum_{j=1,2}a_{j}(-a_{i}\eta_{j}+(2\eta+\omega)\partial_{\omega_{i}}\eta_{j})\,.\\\
\end{split}$ (5.1)

###### Proof.

The first and second identity follow from the well-known periodicity of $\wp$
and quasi-periodicity of $\zeta$. On the other hand, the $\sigma$ function
satisfies

$\sigma(\eta+\omega\,|\,\omega_{1},\omega_{2})=(-1)^{a_{1}+a_{2}+a_{1}a_{2}}e^{(2\eta+\omega)(a_{1}\eta_{1}+a_{2}\eta_{2})}\sigma(\eta\,|\,\omega_{1},\omega_{2})\,,$
(5.2)

from which the last identity follows by taking logs and derivatives with
respect to $\omega_{i}$. Indeed, differentiating the left-hand side of (5.2)
gives

$\partial_{\omega_{i}}\log(\sigma(\eta+\omega\,|\,\omega_{1},\omega_{2}))=a_{i}\zeta(\eta+\omega\,|\,\omega_{1},\omega_{2})+\rho_{i}(\eta+\omega\,|\,\omega_{1},\omega_{2})\,,$
(5.3)

while differentiating the right hand side gives

$\partial_{\omega_{i}}\left((2\eta+\omega)(a_{1}\eta_{1}+a_{2}\eta_{2})+\log(\sigma(\eta\,|\,\omega_{1},\omega_{2}))\right)=\rho_{i}(\eta\,|\,\omega_{1},\omega_{2})+\sum_{j=1}^{2}a_{j}(a_{i}\eta_{j}+(2\eta+\omega)\partial_{\omega_{i}}\eta_{j})\,.$
(5.4)

The last identity of (5.1) then follows by applying the second identity and
reorganizing terms. ∎

Recall the notion of the irrationality measure $\mu(\alpha)$ of a real number
$\alpha\in\mathbb{R}$, and of the irrationality measure $\mu(r)$ of a ray
$r\subset\mathbb{C}^{*}$ with respect to a lattice
$\Lambda(\omega_{1},\omega_{2})$ introduced in Section 2.4. In the following,
we will use that if $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ and
$n>\mu(\alpha)$ then

$|\alpha-p/q|\geq 1/q^{n}$ (5.5)

for all $p,q\in\mathbb{Z}$ with $q$ sufficiently large.

######  Lemma .

Fix $(\omega_{1},\omega_{2})\in\mathcal{R}$ and let $r\subset\mathbb{C}^{*}$
be a non-Stokes ray such that $\mu(r)<\infty$ with respect to the lattice
$\Lambda(\omega_{1},\omega_{2})$. Then for $\epsilon\in\mathbb{H}_{r}$ the
integrals

$\int_{r}e^{-\eta/\epsilon}\mathscr{f}(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta,\quad\int_{r}e^{-\eta/\epsilon}\mathscr{k}_{i}(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta,$
(5.6)

are absolutely convergent.

###### Proof.

The functions $\mathscr{f}(\eta)$, $\mathscr{k}_{i}(\eta)$ are meromorphic in
$\eta$ with poles only at the nonzero lattice points
$\Lambda^{*}(\omega_{1},\omega_{2})$. The fact that $r$ is a non-Stokes ray
implies that $r$ can be parameterised as

$\eta(t)=\pm t(\omega_{1}+\alpha\omega_{2})\,,$ (5.7)

with $t\in\mathbb{R}_{\geq 0}$ and $\alpha\in\mathbb{R}\setminus\mathbb{Q}$.
We assume that we are in the case with the $+$ sign, with the other case being
completely analogous.

Fix $K>0$. Since the functions $\mathscr{f}$, $\mathscr{k}_{i}$ are
holomorphic at $0\in\mathbb{C}$, the integrals over $t\in[0,K]$ are finite.
Now note that by (2.7)-(2.9), the functions $\tilde{\wp}$, $\tilde{\zeta}$ and
$\mathscr{k}_{i}$ differ from $\wp$, $\zeta$ and $\rho_{i}$ by terms that have
at most polynomial growth in $\eta(t)$ as $t\to\infty$. Due to the exponential
decay of $e^{-\eta(t)/\epsilon}$ for $\epsilon\in\mathbb{H}_{r}$ as
$t\to\infty$, to show that (5.6) holds it is then enough to check that

$\int_{r_{\infty}}|e^{-\eta/\epsilon}\wp(\eta)\mathrm{d}\eta|<\infty,\quad\int_{r_{\infty}}|e^{-\eta/\epsilon}\zeta(\eta)\mathrm{d}\eta|<\infty,\quad\int_{r_{\infty}}|e^{-\eta/\epsilon}\rho_{i}(\eta)\mathrm{d}\eta|<\infty,$
(5.8)

where $r_{\infty}$ is the segment given by $\eta(t)$ for $t\in[K,\infty)$. We
start with the first of these statements.

We work with the inner product on $\mathbb{C}$ in which $\omega_{1}$ and
$\omega_{2}$ are orthonormal and consider discs $D_{\delta}(\omega)$ of radius
$0<\delta<K$ centered at the points of $\Lambda(\omega_{1},\omega_{2})$. We
denote the norm induced by this inner product by $||\cdot||$ to distinguish it
from the canonical norm $|\cdot|$. We take $\delta>0$ sufficiently small so
that these discs do not intersect each other. Subdivide the ray $r_{\infty}$
into two sets

$r_{\infty}=r_{p}\cup r_{c}$ (5.9)

where $r_{p}$ is made up of the segments of $r_{\infty}$ inside the discs, and
$r_{c}$ is the complement of $r_{p}$ in $r_{\infty}$. In particular, we can
write

$r_{p}=\bigcup_{\omega\in\Lambda^{*}(\omega_{1},\omega_{2})}r_{\omega}$ (5.10)

where $r_{\omega}$ is the segment contained in the disc $D_{\delta}(\omega)$.

Take a non-zero lattice point
$\omega=a_{1}\omega_{1}+a_{2}\omega_{2}\in\Lambda^{*}(\omega_{1},\omega_{2})$
and consider

$\int_{r_{\omega}}|e^{-\eta/\epsilon}\wp(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta|\,.$
(5.11)

Using the fact that $\wp$ is periodic for the lattice
$\Lambda(\omega_{1},\omega_{2})$ and the Laurent expansion of $\wp(\eta)$ at
$\eta=0$ we know that when $\eta\in D_{\delta}(\omega)$

$\wp(\eta)=\wp(\eta-\omega)=\frac{1}{(\eta-\omega)^{2}}+\text{Reg}(\eta-\omega),$
(5.12)

where Reg is a holomorphic function in the disc $D_{\delta}(0)$. So in
particular we have

$|\wp(\eta)|=|\wp(\eta-\omega)|\leq\frac{1}{|\eta-\omega|^{2}}+D_{1}\leq\frac{1+C\cdot
D_{1}\delta^{2}}{|\eta-\omega|^{2}}=\frac{D_{2}}{|\eta-\omega|^{2}}\,,$ (5.13)

where $D_{1}>0$, $C>0$ is such that $|\cdot|<C||\cdot||$, and $D_{2}=1+C\cdot
D_{1}\delta^{2}>0$ are constants independent of $\omega$. Since the canonical
norm $|\cdot|$ is equivalent to $||\cdot||$, it follows that

$|\eta(t)-\omega|^{2}\geq D_{3}((t-a_{1})^{2}+(t\alpha-a_{2})^{2})$ (5.14)

for some $D_{3}>0$. Minimizing the right hand side we find

$(t-a_{1})^{2}+(t\alpha-a_{2})^{2}\geq\frac{(a_{1}\alpha-
a_{2})^{2}}{1+\alpha^{2}}\,.$ (5.15)

By picking $n>\mu(\alpha)$ and possibly increasing $K>0$, we can assume that
for all $\omega\in\Lambda^{*}(\omega_{1},\omega_{2})$ such that $r_{\omega}$
is non-empty we have

$|\alpha-a_{2}/a_{1}|\geq\frac{1}{|a_{1}|^{n}}\,,$ (5.16)

where we again wrote $\omega=a_{1}\omega_{1}+a_{2}\omega_{2}$. Hence, overall
on $r_{\omega}$ we have

$|\wp(\eta)|\leq\frac{D_{2}(1+\alpha^{2})\cdot|a_{1}|^{2n-2}}{D_{3}}\,.$
(5.17)

Recall that the discs $D_{\delta}(\omega)$ are defined with respect to the
inner product where $\omega_{1}$ and $\omega_{2}$ are orthonormal. The points
of intersection of the ray $\eta(t)$ with the boundary of $D_{\delta}(\omega)$
occur when

$t=t_{\pm}(\omega)=\frac{(a_{1}+\alpha
a_{2})\pm\sqrt{(1+\alpha^{2})\delta^{2}-(\alpha
a_{1}-a_{2})^{2}}}{(1+\alpha^{2})}\,.$ (5.18)

Note that if $r_{\omega}$ is not empty, we must have

$(1+\alpha^{2})\delta^{2}-(\alpha
a_{1}-a_{2})^{2}=(1+\alpha^{2})(\delta^{2}-\text{dist}(r,\omega)^{2})\geq
0\,,$ (5.19)

where $\text{dist}(r,\omega)$ denotes the distance between $r$ and $\omega$ in
the norm where $\omega_{1}$ and $\omega_{2}$ are orthonormal. Hence, we obtain

$\int_{r_{\omega}}|e^{-\eta/\epsilon}\wp(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta|\leq\delta_{1}|a_{1}|^{2n-2}\int_{t_{-}}^{t_{+}}e^{-t\delta_{2}}\mathrm{d}t=\delta_{1}|a_{1}|^{2n-2}\left(-\frac{1}{\delta}_{2}(e^{-t_{+}\delta_{2}}-e^{-t_{-}\delta_{2}})\right)$
(5.20)

where

$\delta_{1}=\frac{D_{2}(1+\alpha^{2})|\omega_{1}+\alpha\omega_{2}|}{D_{3}}>0,\quad\delta_{2}=\left|\frac{\omega_{1}+\alpha\omega_{2}}{\epsilon}\right|\cos\left(\text{Arg}\left(\frac{\omega_{1}+\alpha\omega_{2}}{\epsilon}\right)\right)>0\,,$
(5.21)

so $\delta_{1},\delta_{2}$ are constants depending only on
$\alpha,\omega_{1},\omega_{2},\delta,\epsilon$. Furthermore, note that

$e^{-t_{-}\delta_{2}}-e^{-t_{+}\delta_{2}}=2e^{-\delta_{2}\frac{a_{1}+\alpha
a_{2}}{1+\alpha^{2}}}\sinh\left(\delta_{2}\frac{\sqrt{\delta^{2}-\text{dist}(r,\omega)^{2}}}{(1+\alpha^{2})^{1/2}}\right)\leq
2e^{-\delta_{2}\frac{a_{1}+\alpha
a_{2}}{1+\alpha^{2}}}\sinh\left(\frac{\delta_{2}\delta}{(1+\alpha^{2})^{1/2}}\right)\,,$
(5.22)

so that

$\int_{r_{\omega}}|e^{-\eta/\epsilon}\wp(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta|\leq
2\frac{\delta_{1}}{\delta_{2}}|a_{1}|^{2n-2}\sinh\left(\frac{\delta_{2}\delta}{(1+\alpha^{2})^{1/2}}\right)e^{-\delta_{2}\frac{a_{1}+\alpha
a_{2}}{1+\alpha^{2}}}\,.$ (5.23)

Now note that if $\eta(t)=t(\omega_{1}+\alpha\omega_{2})$ intersects the disc
centered at $\omega=a_{1}\omega_{1}+a_{2}\omega_{2}$, and $\omega$ is
sufficiently large in norm, then we must have $a_{1},\alpha a_{2}>0$. In
particular, we find that

$\begin{split}\int_{r_{p}}|e^{-\eta/\epsilon}&\wp(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta|=\sum_{\omega\in\Lambda^{*}(\omega_{1},\omega_{2})}\int_{r_{\omega}}|e^{-\eta/\epsilon}\wp(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta|\\\
&<\sum_{(a_{1},a_{2})\in\mathbb{Z}^{2}\;:\;r_{a_{1}\omega_{1}+a_{2}\omega_{2}}\neq\emptyset}2\frac{\delta_{1}}{\delta_{2}}|a_{1}|^{2n-2}\sinh\left(\frac{\delta_{2}\delta}{(1+\alpha^{2})^{1/2}}\right)e^{-\delta_{2}\frac{a_{1}+\alpha
a_{2}}{1+\alpha^{2}}}<\infty\,.\\\ \end{split}$ (5.24)

On the other hand, on $r_{c}$ we simply have that due to the periodicity of
$\wp$, the factor $\wp(\eta)$ is bounded and hence the integral over $r_{c}$
is also finite. We then conclude that

$\int_{r_{\infty}}\left|e^{-\eta/\epsilon}\wp(\eta)\mathrm{d}\eta\right|=\int_{r_{c}}\left|e^{-\eta/\epsilon}\wp(\eta)\mathrm{d}\eta\right|+\int_{r_{p}}\left|e^{-\eta/\epsilon}\wp(\eta)\mathrm{d}\eta\right|<\infty$
(5.25)

The argument for the convergence of

$\int_{r_{\infty}}\left|e^{-\eta/\epsilon}\zeta(\eta)\mathrm{d}\eta\right|$
(5.26)

is similar. The only difference is that now $\zeta$ is not periodic, so we
must use the corresponding identity in Lemma 5.1. This shows that for
$\eta(t)\in r_{\omega}$ we have

$\zeta(\eta)=\zeta(\eta-\omega)+2(a_{1}\eta_{1}+a_{2}\eta_{2})=\frac{1}{\eta-\omega}+\text{Reg}(\eta-\omega)+2(a_{1}\eta_{1}+a_{2}\eta_{2})\,,$
(5.27)

where Reg as before is a holomorphic function (independent of $\omega$) on a
disc of radius $\delta$ centered at $0$, so that

$|\zeta(\eta)|\leq\frac{D_{1}}{|\eta-\omega|}+D_{2}(|a_{1}|+|a_{2}|)<\frac{D_{1}(1+\alpha^{2})^{1/2}\cdot|a_{1}|^{n-1}}{C}+D_{2}(|a_{1}|+|a_{2}|)$
(5.28)

for some constants $C,D_{1},D_{2}$ independent of
$\omega=a_{1}\omega_{1}+a_{2}\omega_{2}$. The argument for the convergence
over $r_{p}$ follows as before. For the convergence over $r_{c}$ we again use
the quasi-periodicity of $\zeta$ from Lemma 5.1 as before to show that as we
go to $\infty$ along $r_{c}$ we have

$|\zeta(\eta)|=\mathcal{O}(|\eta|)\,.$ (5.29)

Finally, to show

$\int_{r_{\infty}}\left|e^{-\eta/\epsilon}\rho_{i}(\eta)\mathrm{d}\eta\right|<\infty$
(5.30)

we use that

$\int_{r_{\infty}}\left|e^{-\eta/\epsilon}\zeta(\eta)\mathrm{d}\eta\right|<\infty$
(5.31)

together with Lemma 5.1 and a simple modification of the argument from before.
∎

###  5.2. Proof of the Borel summability

Given the previous lemmas, we now prove the Borel summability of
$K_{i}(\epsilon\,|\,\omega_{1},\omega_{2})$ and
$F(\epsilon\,|\,\omega_{1},\omega_{2})$.

######  Proposition .

Fix $(\omega_{1},\omega_{2})\in\mathcal{R}$ and consider a non-Stokes ray $r$
such that $\mu(r)<\infty$ with respect to $\Lambda(\omega_{1},\omega_{2})$.
Then for $\epsilon\in\mathbb{H}_{r}$ the following integrals are absolutely
convergent

$K_{i}^{r}(\epsilon\,|\,\omega_{1},\omega_{2})=\int_{r}e^{-\eta/\epsilon}k_{i}(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta,\quad
F^{r}(\epsilon\,|\,\omega_{1},\omega_{2})=\int_{r}e^{-\eta/\epsilon}f(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta\,,$
(5.32)

and depend holomorphically on $\epsilon$. In particular, the formal series
$K_{i}(\epsilon\,|\,\omega_{1},\omega_{2})$ and
$F(\epsilon\,|\,\omega_{1},\omega_{2})$ are Borel summable along $r$.
Additionally, we have the alternate expressions

$\begin{split}K_{i}^{r}(\epsilon\,|\,\omega_{1},\omega_{2})&=\int_{r}\mathrm{Li}_{1}(e^{-\eta/\epsilon})\mathscr{k}_{i}(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta\,,\\\
F^{r}(\epsilon\,|\,\omega_{1},\omega_{2})&=\int_{r}\mathrm{Li}_{2}(e^{-\eta/\epsilon})\mathscr{f}(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta\,.\end{split}$
(5.33)

###### Proof.

Using that along $r$ we have $|e^{-\eta/\epsilon}|<1$ and using that

$|\mathrm{Li}_{1}(z)|=|\log(1-z)|<-\log(1-|z|),\quad|z|<1,$ (5.34)

to show the absolute convergence of the first expression of (5.33) it is
enough to show the convergence of the integral

$-\int_{r}\log(1-|e^{-\eta/\epsilon}|)|\mathscr{k}_{i}(\eta)||\mathrm{d}\eta|\,.$
(5.35)

Since near $\eta=0$ we have

$\mathscr{k}_{i}(\eta)=\mathcal{O}(\eta)$ (5.36)

the integral in (5.35) has no issue near $\eta=0$. On the other hand, as
$\eta\to\infty$ along $r$ we have

$-\log(1-|e^{-\eta/\epsilon}|)\sim|e^{-\eta/\epsilon}|\,,$ (5.37)

so by Lemma 5.1 we have that (5.35) is finite. On the other hand, by Fubini-
Tonneli and changing variables we have

$\begin{split}-\int_{r}\log(1-|e^{-\eta/\epsilon}|)&|\mathscr{k}_{i}(\eta)||\mathrm{d}\eta|=\int_{r}\sum_{m\geq
1}\frac{|e^{-m\eta/\epsilon}|}{m}|\mathscr{k}_{i}(\eta)||\mathrm{d}\eta|=\sum_{m\geq
1}\int_{r}\frac{|e^{-m\eta/\epsilon}|}{m}|\mathscr{k}_{i}(\eta)||\mathrm{d}\eta|\\\
&=\sum_{m\geq
1}\int_{r}\frac{|e^{-\eta/\epsilon}|}{m^{2}}|\mathscr{k}_{i}(\eta/m)||\mathrm{d}\eta|=\int_{r}|e^{-\eta/\epsilon}|\sum_{m\geq
1}\frac{1}{m^{2}}|\mathscr{k}_{i}(\eta/m)||\mathrm{d}\eta|\,.\end{split}$
(5.38)

In the above, we have used that along the ray $|e^{-\eta/\epsilon}|<1$, so
that the series expansion of $-\log(1-z)$ is valid along $r$. Since the first
integral is finite and $k_{i}(\eta\,|\,\omega_{1},\omega_{2})$ is given by
(2.18), it follows that the Borel sum
$K_{i}^{r}(\epsilon\,|\,\omega_{1},\omega_{2})$ is absolutely integrable. By
applying Fubini-Tonelli to the expressions without absolute values, we also
get the alternate identity in (5.33).

The argument for $F^{r}$ follows similarly. Using that along $r$ we have
$|e^{-\eta/\epsilon}|<1$ and

$|\mathrm{Li_{2}}(z)|\leq\mathrm{Li_{2}}(|z|),\quad|z|<1$ (5.39)

to show the absolute convergence of the second expression in (5.33) it is
enough to consider

$\int_{r}\mathrm{Li}_{2}(|e^{-\eta/\epsilon}|)|\mathscr{f}(\eta)||\mathrm{d}\eta|\,.$
(5.40)

Since $\mathrm{Li}_{2}(1)<\infty$ and the modified functions
$\widetilde{\zeta}$ and $\widetilde{\wp}$ are finite at $\eta=0$, the
integrand does not have any issues at $\eta=0$. Similar to the previous case,
as $\eta\to\infty$ along $r$ we have

$\mathrm{Li}_{2}(|e^{-\eta/\epsilon}|)\sim|e^{-\eta/\epsilon}|$ (5.41)

so by Lemma 5.1 we find that (5.40) is finite. By Fubini-Tonneli and
performing a change of variables as in (5.38), we find that

$\begin{split}\int_{r}\mathrm{Li}_{2}(|e^{-\eta/\epsilon}|)|\mathscr{f}(\eta)||\mathrm{d}\eta|&=\sum_{m\geq
1}\int_{r}\frac{|e^{-m\eta/\epsilon}|}{m^{2}}|\mathscr{f}(\eta)||\mathrm{d}\eta|=\sum_{m\geq
1}\int_{r}\frac{|e^{-\eta/\epsilon}|}{m^{3}}|\mathscr{f}(\eta/m)||\mathrm{d}\eta|\\\
&=\int_{r}|e^{-\eta/\epsilon}|\sum_{m\geq
1}\frac{1}{m^{3}}|\mathscr{f}(\eta/m)||\mathrm{d}\eta|\,.\\\ \end{split}$
(5.42)

As before, for the series expansion of $\mathrm{Li}_{2}(z)$ we have used that
along $r$ we have $|e^{-\eta/\epsilon}|<1$. Since the first integral is
finite, and $f$ is given by (2.17), it follows that the Borel sum
$F^{r}(\epsilon,\omega_{1},\omega_{2})$ is absolutely integrable. By applying
Fubini-Tonelli to the corresponding expressions without absolute values, we
obtain the alternate identity for $F^{r}$ in (5.33).

Finally, we show holomorphic dependence in $\epsilon\in\mathbb{H}_{r}$ for
$K^{r}_{i}$, with an identical argument for $F^{r}$. Consider any contour
$\partial\Delta\subset\mathbb{H}_{r}$. Then we clearly have

$\int_{\partial\Delta}\left(\int_{r}|e^{-\eta/\epsilon}k_{i}(\eta\,|\,\omega_{1},\omega_{2})||\mathrm{d}\eta|\right)|\mathrm{d}\epsilon|<\infty\,.$
(5.43)

By applying Fubini-Tonelli we can interchange the order of integration, and we
find

$\begin{split}\int_{\partial\Delta}K_{i}^{r}(\epsilon\,|\,\omega_{1},\omega_{2})\mathrm{d}\epsilon&=\int_{\partial\Delta}\left(\int_{r}e^{-\eta/\epsilon}k_{i}(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta\right)\mathrm{d}\epsilon\\\
&=\int_{r}\left(\int_{\partial\Delta}e^{-\eta/\epsilon}\mathrm{d}\epsilon\right)k_{i}(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta=0\,.\end{split}$
(5.44)

Hence, by Morera’s theorem it follows that
$K_{i}^{r}(\epsilon\,|\,\omega_{1},\omega_{2})$ is holomorphic in
$\epsilon\in\mathbb{H}_{r}$. ∎

###  5.3. Integrals along detour paths

When $\mu(\alpha)=\infty$, we can still define something meaningful. The idea
is as follows:

  * •

Give a non-Stokes ray $r$ with $\mu(r)=\infty$ with respect to the lattice
$\Lambda(\omega_{1},\omega_{2})$, let $r(\delta)$ be the detour path defined
in Section 2.4 for $\delta$ small enough (see figure 1).

  * •

We then define the following expressions for $\epsilon\in\mathbb{H}_{r}$

$\begin{split}K_{i}^{r(\delta)}(\epsilon\,|\,\omega_{1},\omega_{2})&=\int_{r(\delta)}\mathrm{Li}_{1}(e^{-\eta/\epsilon})\mathscr{k}_{i}(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta\,,\\\
F^{r(\delta)}(\epsilon\,|\,\omega_{1},\omega_{2})&=\int_{r(\delta)}\mathrm{Li}_{2}(e^{-\eta/\epsilon})\mathscr{f}(\eta\,|\,\omega_{1},\omega_{2})\mathrm{d}\eta\,.\end{split}$
(5.45)

  * •

We will then show the above expressions are independent of $\delta$ for
$\delta$ small enough, and they coincide with $K_{i}^{r}$ and $F^{r}$ when
$\mu(r)<\infty$.

######  Proposition .

Let $r$ be a non-Stokes ray with respect to $\Lambda(\omega_{1},\omega_{2})$.
Then there is $D>0$ such that for all $0<\delta<D$ the integrals
$F^{r(\delta)}$ and $K_{i}^{r(\delta)}$ are absolutely convergent for
$\epsilon\in\mathbb{H}_{r}$. These integrals depend holomorphically on
$\epsilon\in\mathbb{H}_{r}$, and are independent of the choice of such
$\delta$. Moreover when $\mu(r)<\infty$ we have $F^{r(\delta)}=F^{r}$ and
$K_{i}^{r(\delta)}=K_{i}^{r}$ .

###### Proof.

We take $D>0$ such that the discs of radius $0<\delta<D$ and centered at
$\Lambda(\omega_{1},\omega_{2})$ do not intersect each other. Notice that
given any parametrization $\eta(t)$ of the corresponding detour path
$r(\delta)$ and $\epsilon\in\mathbb{H}_{r}$, we have
$\text{Re}(\eta(t)/\epsilon)>0$ for $t$ sufficiently big, so we still have
exponential decay as $t\to\infty$.

On the other hand, the proof of the absolute convergence follows a simpler
argument than the one used in Lemma 5.1 and Proposition 5.2. Indeed, one first
needs a version of Lemma 5.1 for the detour paths $r(\delta)$. As in Lemma 5.1
we can focus on a segment $r_{\infty}(\delta)$ given by $\eta(t)$ for
$t\in[K,\infty)$ and $K>0$ sufficiently big, and furthemore divide
$r_{\infty}(\delta)$ into two sets

$r_{\infty}(\delta)=r_{p}\cup r_{c}$ (5.46)

where $r_{c}$ is exactly as in Lemma 5.1, and $r_{p}$ is now made of the arcs
of the detour path, belonging to circles of radius $\delta$ centered at the
poles. The argument of the absolute convergence over $r_{c}$ is exactly the
same as in Lemma 5.1, while the estimates for $r_{p}$ are easier, since we are
now always a bounded distance from the poles. For example, when dealing with
$\wp(\eta)$, using the periodicity of $\wp(\eta)$ we simply have a uniform
bound for $|\wp(\eta)|$ along $r_{p}$, while for $\zeta(\eta)$ and
$\rho_{i}(\eta)$ we use again Lemma 5.1. One can then apply the same argument
of Proposition 5.2 to show that the integrals in (5.45) are absolutely
integrable.222Note that in the case of rays $r$ with $\mu(r)=\infty$ we do not
know that the expressions (5.33) are absolutely integrable, and hence we
cannot directly relate (5.45) to Borel sums by trying to show that
$K_{i}^{r(\delta)}$ and $F^{r(\delta)}$ coincide with (5.33). This is because
the argument of Proposition 5.2 requires the absolute convergence of (5.33) to
show that $\eqref{borelalt}$ match the corresponding Borel sums.

Now let $\delta$ be small enough and $0<\delta^{\prime}<\delta$. Let
$r_{n}(\delta)$ (resp. $r_{n}(\delta^{\prime})$) be the segment of the detour
path $r(\delta)$ (resp. $r(\delta^{\prime})$) from $0$ to some point between
the $n$-th arc and the $(n+1)$-th arc. We pick the endpoint to be the same for
$r_{n}(\delta)$ and $r_{n}(\delta^{\prime})$ for all $n>0$. Similarly, we
denote by $r_{n}$ the segment of $r$ from $0$ to the endpoint of
$r_{n}(\delta)$. By a trivial argument with contour integrals using the fact
that the integrands have poles at $\Lambda^{*}(\omega_{1},\omega_{2})$

$K_{i}^{r_{n}(\delta)}=K_{i}^{r_{n}(\delta^{\prime})}=K_{i}^{r_{n}},\quad
F^{r_{n}(\delta)}=F^{r_{n}(\delta^{\prime})}=F^{r_{n}}\,,\quad\text{for all
$n>0$}\,.$ (5.47)

When $r$ is an arbitrary non-Stokes ray, we can use the existence of
$K_{i}^{r(\delta)}$ and $F^{r(\delta)}$ for all small enough $\delta$ and
(5.47) to take a limit $n\to\infty$ and obtain
$K_{i}^{r(\delta)}=K_{i}^{r(\delta^{\prime})}$,
$F^{r(\delta)}=F^{r(\delta^{\prime})}$333Note that this does not show that
$K_{i}^{r(\delta)}=K_{i}^{r}$, $F^{r(\delta)}=F^{r}$, since for a general non-
Stokes ray $r$ we do not know that $F^{r}$ and $K_{i}^{r}$ exist, and the
existence of a limit along a sequence does not guarantee the existence of the
limit.. Furthermore, when $r$ is a non-Stokes ray such that $\mu(r)<\infty$,
we can use the existence of $K_{i}^{r}$ and $F^{r}$ and (5.47) to obtain
$K_{i}^{r(\delta)}=K_{i}^{r}$, $F^{r(\delta)}=F^{r}$.

Finally, the holomorphicity in $\epsilon\in\mathbb{H}_{r}$ follows by a the
same kind of argument as in Proposition 5.2 ∎

###  5.4. Stokes jumps

Finally, we discuss how the previous integrals along different paths relate to
each other.

######  Proposition .

Let $r_{1}$ and $r_{2}$ be two non-Stokes rays ordered in clockwise order, and
assume that $\mathbb{H}_{r_{1}}\cap\mathbb{H}_{r_{2}}\neq\emptyset$.
Furthermore, let $\Delta(r_{1},r_{2})$ be the sector determined by $r_{1}$ and
$r_{2}$. Then for $\epsilon\in\mathbb{H}_{r_{1}}\cap\mathbb{H}_{r_{2}}$ and
small enough $\delta$ we have

$\begin{split}K_{i}^{r_{2}(\delta)}(\epsilon\,|\,\omega_{1},\omega_{2})-K_{i}^{r_{1}(\delta)}(\epsilon\,|\,\omega_{1},\omega_{2})&=2\pi\mathrm{i}\sum_{\omega\in\Delta(r_{1},r_{2})\cap\Lambda^{*}(\omega_{1},\omega_{2})}a_{i}\cdot\log(1-e^{-\omega/\epsilon})\\\
F^{r_{2}(\delta)}(\epsilon,\omega_{1},\omega_{2})-F^{r_{1}(\delta)}(\epsilon,\omega_{1},\omega_{2})&=2\pi\mathrm{i}\sum_{\omega\in\Delta(r_{1},r_{2})\cap\Lambda^{*}(\omega_{1},\omega_{2})}\frac{\partial}{\partial\epsilon}\left(\epsilon\mathrm{Li}_{2}(e^{-\omega/\epsilon})\right)\,,\end{split}$
(5.48)

where $\omega=a_{1}\omega_{1}+a_{2}\omega_{2}$.

###### Proof.

Consider a sequence $C_{n}$ with $n>0$ of discs centered at $0$ and of radius
$R_{n}$ with $R_{n}\to\infty$ as $n\to\infty$. We denote by $A_{n}$ the arc of
$C_{n}$ contained in $\Delta(r_{1},r_{2})$ and assume that $A_{n}$ does not
intersect $\Lambda^{*}(\omega_{1},\omega_{2})$ for all $n$. We orient $A_{n}$
counter-clockwise. Furthermore, consider $\delta>0$ small enough such that the
discs of radius $\delta>0$ centered at the points of
$\Lambda(\omega_{1},\omega_{2})$ do not intersect. We consider detour arcs
$A_{n}(\delta)$, defined similarly to the detour rays $r(\delta)$ by taking a
detour along the circles of radius $\delta$ centered at the points of
$\Lambda(\omega_{1},\omega_{2})$ through the shortest length arc. Furthermore,
we denote by $\Delta_{n}(r_{1},r_{2})$ the region determined by $r_{1}$,
$r_{2}$ and $A_{n}(\delta)$, and by $r_{n,1}(\delta)$ and $r_{n,2}(\delta)$
the segments of $r_{1}(\delta)$ and $r_{2}(\delta)$ from $0$ to the
intersection points with $A_{n}(\delta)$. By using that
$\mathscr{k}_{i}(\eta)$ has a simple pole at
$\omega=a_{1}\omega_{1}+a_{2}\omega_{2}\in\Lambda^{*}(\omega_{1},\omega_{2})$
with residue $-a_{i}$ we obtain using (5.45) that

$K_{i}^{r_{n,2}(\delta)}-K_{i}^{r_{n,1}(\delta)}+K_{i}^{A_{n}(\delta)}=2\pi\mathrm{i}\sum_{\omega\in\Delta_{n}(r^{1},r^{2})\cap\Lambda^{*}}a_{i}\cdot\log(1-e^{-\omega/\epsilon})$
(5.49)

where we have used that $\mathrm{Li}_{1}(z)=-\log(1-z)$. Similarly, using that
$\widetilde{\wp}=-\mathrm{d}\widetilde{\zeta}/\mathrm{d}\eta$ and the
definition of $\mathscr{f}$, we can use integration by parts on
$F^{r(\delta)}$ to write

$\begin{split}F^{r(\delta)}&=\int_{r(\delta)}\left(2\mathrm{Li}_{2}(e^{-\eta/\epsilon})-\frac{\mathrm{d}}{\mathrm{d}\eta}(\eta\mathrm{Li}_{2}(e^{-\eta/\epsilon}))\right)\widetilde{\zeta}(\eta,\omega_{1},\omega_{2})\mathrm{d}\eta\,\\\
&=\int_{r(\delta)}\left(\mathrm{Li}_{2}(e^{-\eta/\epsilon})+\frac{\eta}{\epsilon}\mathrm{Li}_{1}(e^{-\eta/\epsilon})\right)\widetilde{\zeta}(\eta,\omega_{1},\omega_{2})\mathrm{d}\eta\,\\\
\end{split}$ (5.50)

where we have used that the boundary terms of the integration by parts vanish.
Using that $\widetilde{\zeta}$ has a simple pole at
$\omega\in\Lambda^{*}(\omega_{1},\omega_{2})$ with residue $1$ then shows that

$\begin{split}F^{r_{n,2}(\delta)}-F^{r_{n,1}(\delta)}+F^{A_{n}(\delta)}&=2\pi\mathrm{i}\sum_{\omega\in\Delta_{n}(r^{1},r^{2})\cap\Lambda^{*}}\left(\mathrm{Li}_{2}(e^{-\omega/\epsilon})+\frac{\omega}{\epsilon}\mathrm{Li}_{1}(e^{-\omega/\epsilon})\right)\\\
&=2\pi\mathrm{i}\sum_{\omega\in\Delta_{n}(r^{1},r^{2})\cap\Lambda^{*}}\partial_{\epsilon}\left(\epsilon\mathrm{Li}_{2}(e^{-\omega/\epsilon})\right)\\\
\end{split}$ (5.51)

Now note that since $\epsilon\in\mathbb{H}_{r^{1}}\cap\mathbb{H}_{r^{2}}$ the
function $e^{-\eta/\epsilon}$ along $A_{n}(\delta)$ is exponentially
suppressed as $n\to\infty$. By using a similar argument to Lemma 5.1,
Proposition 5.2 and Proposition 5.3 one then finds that

$\lim_{n\to\infty}F^{A_{n}(\delta)}=\lim_{n\to\infty}K_{i}^{A_{n}(\delta)}=0\,$
(5.52)

and hence the result follows. ∎

##  Appendix A Free energy in fibre classes

Let $\pi:X\to B$ be an elliptic CY threefold satisfying the assumptions 1.1,
and consider the GW generating function in fibre classes of $\pi:X\to B$ and
for genus $g\geq 2$

$F_{\text{GW}}(\lambda\,|\,Q)=\sum_{g\geq 2}F_{g}(Q)\lambda^{2g-2}\,,\quad
F_{g}(Q)=\sum_{n=0}^{\infty}\text{GW}(g,nF)Q^{d}\,.$ (A.1)

Here $\text{GW}(g,nF)$ denotes the GW invariant of the class $nF$ at genus
$g$, and $F$ is the fiber class of $\pi:X\to B$.

In this Section we show that one can write $F_{\text{GW}}$ as in (1.4). The
expression (A.2) below is the same as the one written in [21, Section B.3]. We
nevertheless include a more detailed computation for completeness.

######  Proposition .

Assuming the GW/DT correspondence holds for $X$, we can write
$F_{\text{GW}}(\lambda\,|\,Q)$ from (A.1) as

$F_{\text{GW}}(\lambda\,|\,Q)=e(X)\sum_{g\geq
2}\frac{(-1)^{g}B_{2g}}{4g}C_{2g-2}(Q)\lambda^{2g-2}\,,$ (A.2)

where $C_{2g-2}(Q)$ is the analytic function in $Q$ for $|Q|<1$ given by

$C_{2g-2}(Q)=-\frac{B_{2g-2}}{(2g-2)\cdot(2g-2)!}+\frac{2}{(2g-2)!}\sum_{k,n\geq
1}k^{2g-3}Q^{kn}\,.$ (A.3)

Furthermore, setting $Q=e^{2\pi\mathrm{i}\tau}$ for $\text{Im}(\tau)>0$ we
have

$F_{\text{GW}}(\lambda\,|\,\tau)=-e(X)\cdot\sum_{g\geq
2}\frac{B_{2g}\,G_{2g-2}(\tau)}{4g(2g-2)}\left(\frac{\lambda}{2\pi}\right)^{2g-2}.$
(A.4)

###### Proof.

On the one hand, for $g\geq 2$ and $n=0$ we have the universal contribution of
constant maps on a CY threefold $X$ [13, Theorem 4]

$\text{GW}(g,0)=-e(X)\frac{(-1)^{g}B_{2g}B_{2g-2}}{4g(2g-2)(2g-2)!}\,\,.$
(A.5)

On the other hand, consider the Gopakumar-Vafa form of the Gromov-Witten
generating function

$\sum_{g\geq 0,\;\beta>0}\text{GW}(g,\beta)Q^{\beta}\lambda^{2g-2}=\sum_{g\geq
0,\;\beta>0,\;k>0}\frac{\text{GV}(g,\beta)}{k}\left(2\sin\left(\frac{k\lambda}{2}\right)\right)^{2g-2}Q^{k\beta}\,,$
(A.6)

where $\text{GV}(g,\beta)$ denotes the Gopakumar-Vafa invariants of the class
$\beta$ at genus $g$. Assuming the DT/GW correspondence it is shown in [23,
Section 6] that for $\beta=nF$, $n>0$, we have

$\text{GV}(0,nF)=-e(X),\qquad\text{GV}(1,nF)=e(B),\qquad\text{GV}(g,nF)=0\quad\text{for}\quad
g\geq 2\,.$ (A.7)

Hence, restricting to the sum over fiber classes (and denoting $Q^{n\cdot
F}=Q^{n}$ to simplify notation) we find

$\begin{split}\sum_{g\geq
0,\;n>0}\text{GW}(g,nF)Q^{n}\lambda^{2g-2}&=-e(X)\sum_{k,n>0}\frac{1}{k}\left(2\sin\left(\frac{k\lambda}{2}\right)\right)^{-2}Q^{kn}+e(B)\sum_{k,n>0}\frac{Q^{kn}}{k}\,,\\\
&=e(X)\sum_{k,n>0}\frac{e^{\mathrm{\mathrm{i}k\lambda}}}{k(e^{\mathrm{i}k\lambda}-1)^{2}}Q^{kn}-e(B)\sum_{n>0}\log(1-Q^{n})\,.\end{split}$
(A.8)

Using the generating function of Bernoulli numbers $B_{n}$ and the fact that
$B_{n}=0$ for odd $n>1$, one easily finds that

$\frac{e^{\mathrm{i}k\lambda}}{(e^{\mathrm{\mathrm{i}k\lambda}}-1)^{2}}=\sum_{g=0}^{\infty}\frac{(2g-1)(-1)^{g}B_{2g}(k\lambda)^{2g-2}}{(2g)!}\,\,,$
(A.9)

so when considering only terms with $g\geq 2$ we find that

$\begin{split}\sum_{g\geq
2,\;n>0}\text{GW}(g,nF)Q^{n}\lambda^{2g-2}&=e(X)\sum_{g\geq
2}\frac{(-1)^{g}B_{2g}}{(2g)\cdot(2g-2)!}\left(\sum_{k,n>0}k^{2g-3}Q^{k\cdot
n}\right)\lambda^{2g-2}\,\\\ &=e(X)\sum_{g\geq
2}\frac{(-1)^{g}B_{2g}}{4g}\left(C_{2g-2}(Q)+\frac{B_{2g-2}}{(2g-2)\cdot(2g-2)!}\right)\lambda^{2g-2}\,.\end{split}$
(A.10)

The expression (A.2) then follows by adding the constant map contribution
(A.5).

Finally, to show (A.4) note that the Eisenstein series $G_{2g-2}(\tau)$ has
the following expansion for $g\geq 2$, which is a slight rewriting of its
Fourier series444In the Fourier expansion of $G_{2g-2}(\tau)$ a sum of the
form $\sum_{m>0}\sigma_{2g-3}(m)Q^{m}$ appears, where
$\sigma_{2g-3}(m)=\sum_{d|m}d^{2g-3}$ and $Q=e^{2\pi\mathrm{i}\tau}$. We
simply use the fact that
$\sum_{m>0}\sigma_{2g-3}(m)Q^{m}=\sum_{k,n>0}k^{2g-3}Q^{k\cdot n}$.

$G_{2g-2}(\tau)=2\zeta(2g-2)\left(1-\frac{(2\pi)^{2g-2}(-1)^{g}}{(2g-3)!\zeta(2g-2)}\sum_{k,n>0}k^{2g-3}Q^{kn}\right),\quad
Q=e^{2\pi\mathrm{i}\tau}\,.$ (A.11)

In the above $\zeta(s)$ denotes the Riemann $\zeta$-function and not the
Weierstrass $\zeta$-function that is used in the rest of the paper. From

$\zeta(2g-2)=\frac{(-1)^{g}(2\pi)^{2g-2}B_{2g-2}}{2(2g-2)!}\,,\quad g\geq
2\,,$ (A.12)

it then follows that

$C_{2g-2}(Q)=-\frac{(-1)^{g}}{(2g-2)(2\pi)^{2g-2}}G_{2g-2}(\tau)\,\,.$ (A.13)

Hence (A.4) follows from (A.13) and (A.2). ∎

##  Appendix B Lemma on irrationality measure

Recall the definition of irrationality measure $\mu(\alpha)$ from Section 2.3.
Here we prove that if $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ then

$\mu\bigg{(}\frac{a\alpha+b}{c\alpha+d}\bigg{)}=\mu(\alpha)\text{ for all
}\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\operatorname{GL}_{2}(\mathbb{Z}).$
(B.1)

It it is enough to show that the result holds for the generators

$T=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix},\quad S=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix},\quad P=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}\,.$ (B.2)

of $\mathrm{GL}_{2}(\mathbb{Z})$. This is obvious for the transformations
$\alpha\mapsto\alpha+1$ and $\alpha\mapsto-\alpha$ corresponding to $T$ and
$P$. Thus it remains to prove that $\mu(\alpha)=\mu(1/\alpha)$. By the
invariance of $\mu(\alpha)$ under $\alpha\mapsto-\alpha$ we can assume that
$\alpha>0$.

Suppose that $0<r<\mu(\alpha)$. By definition of $\mu(\alpha)$, this implies
that there are infinitely many $p,q\in\mathbb{Z}$ with $q>0$ such that

$|\alpha-p/q|<1/q^{r}\,.$ (B.3)

For a given $q>0$ there can only be finitely many $p$ satisfying (B.3). So
there must be a sequence $p_{n},q_{n}\in\mathbb{Z}$ with $q_{n}>0$ satisfying
(B.3) such that $q_{n}\to\infty$ as $n\to\infty$. Then

$\frac{p_{n}}{q_{n}}\to\alpha\,.$ (B.4)

and hence $p_{n}\to\infty$ also. Passing to a subsequence we can assume that
all $p_{n}>0$. It follows from (B.3) and (B.4) that for some constant $C>0$

$\left|\frac{1}{\alpha}-\frac{q_{n}}{p_{n}}\right|<\frac{1}{q_{n}^{r-1}p_{n}\alpha}=\frac{p_{n}^{r-1}}{q_{n}^{r-1}p_{n}^{r}\alpha}<\frac{C}{p_{n}^{r}}\,.$
(B.5)

If $0<r^{\prime}<r$ then using the fact that $p_{n}\to\infty$ we can assume,
after possibly passing to another subsequence, that $p_{n}^{r-r^{\prime}}>C$
for all $n\in\mathbb{N}$, and hence that

$\left|\frac{1}{\alpha}-\frac{q_{n}}{p_{n}}\right|<\frac{1}{p_{n}^{r^{\prime}}}\,.$
(B.6)

This implies that $r^{\prime}\leq\mu(1/\alpha)$. Since this holds for all
$0<r^{\prime}<r$, it follows that $r\leq\mu(1/\alpha)$. But $0<r<\mu(\alpha)$
was chosen arbitrarily so we conclude that $\mu(\alpha)\leq\mu(1/\alpha)$.
Repeating the argument interchanging $\alpha$ and $1/\alpha$ gives
$\mu(\alpha)=\mu(1/\alpha)$.

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