# Mathematical structures of non-perturbative topological string theory:

from GW to DT invariants

Murad Alim  Arpan Saha  Jörg Teschner  Iván Tulli  M.A., A.S., I.T.:
Fachbereich Mathematik, Universität Hamburg, Bundesstr. 55, 20146, Hamburg;
J.T.: Fachbereich Mathematik, Universität Hamburg, Bundesstr. 55, 20146 and
DESY Theory, Notkestr. 85, 22607 Hamburg

###### Abstract

We study the Borel summation of the Gromov–Witten potential for the resolved
conifold. The Stokes phenomena associated to this Borel summation are shown to
encode the Donaldson–Thomas invariants of the resolved conifold, having a
direct relation to the Riemann–Hilbert problem formulated by T. Bridgeland.
There exist distinguished integration contours for which the Borel summation
reproduces previous proposals for the non-perturbative topological string
partition functions of the resolved conifold. These partition functions are
shown to have another asymptotic expansion at strong topological string
coupling. We demonstrate that the Stokes phenomena of the strong-coupling
expansion encode the DT invariants of the resolved conifold in a second way.
Mathematically, one finds a relation to Riemann–Hilbert problems associated to
DT invariants which is different from the one found at weak coupling. The
Stokes phenomena of the strong-coupling expansion turn out to be closely
related to the wall-crossing phenomena in the spectrum of BPS states on the
resolved conifold studied in the context of supergravity by D. Jafferis and G.
Moore.

###### Contents

  1. 1 Introduction
  2. 2 Borel summations of the resolved conifold partition function
    1. 2.1 Motivation: Topological string theory on the resolved conifold
    2. 2.2 Statement of results for the Borel sum and its Stokes phenomena
    3. 2.3 Connection to topological string theory
    4. 2.4 Previous results
  3. 3 Proofs of the results of Section 2.2
    1. 3.1 The Borel transform
    2. 3.2 The Borel sum along $\mathbb{R}_{>0}$
    3. 3.3 Stokes phenomena of the Borel sum
    4. 3.4 The limits $\lim_{k\to\pm\infty}F_{\rho_{k}}(\lambda,t)$ and $F_{\mathrm{GV}}(\lambda,t)$
  4. 4 Relation to the Riemann–Hilbert problem and line bundles
    1. 4.1 Bridgeland’s Riemann–Hilbert problem and its solution
    2. 4.2 Relation of the partition function to the Riemann–Hilbert problem
    3. 4.3 The line bundle defined by the normalized partition function
    4. 4.4 Relation to the Rogers dilogarithm and hyperholomorphic line bundles
  5. 5 The strong-coupling expansion and its Stokes phenomena
    1. 5.1 Borel summation of the strong-coupling expansion
    2. 5.2 Stokes jumps
    3. 5.3 Relation to framed BPS states
  6. 6 S-duality
    1. 6.1 Global aspects
    2. 6.2 Relation to string-theoretic S-duality
    3. 6.3 Langlands modular duality
  7. A Alternative proof for the Borel sum
  8. B Asymptotic series from Borel transforms

##  1 Introduction

The study of the geometric structures associated to quantum field and string
theories has been extremely fruitful in revealing connections between
different areas of mathematics as well as in putting forward organizing
principles and relations for mathematical structures and invariants.

The focus of this work is on the connection of two types of invariants
associated to families of Calabi–Yau (CY) threefolds. On the one hand, the
Gromov–Witten invariants are characteristics of the enumerative geometry of
maps into the CY. Their generating function is closely related to the
partition function of topological string theory. The latter is a formal power
series which is asymptotic in the topological string coupling constant. On the
other hand, the Donaldson–Thomas or BPS invariants associated to the same
geometry can be defined using the enumeration of coherent sheaves supported on
holomorphic submanifolds on the same CY subject to a stability condition.
Physically, the latter correspond to BPS states, which are realized by
D-branes supported on subspaces of the CY geometry. The generating functions
of BPS invariants are expected to correspond to physical partition functions
of black holes. In physical terms, the topological string theory is obtained
from a perturbative formulation of the underlying string theory, while the BPS
invariants represent data representing non-perturbative effects in string
theory. Relations between the two very different types of data and
mathematical invariants have long been expected both from the points of view
of physics [INOV08, DVV06, OSV04] as well as mathematics [MNOP06a, MNOP06b].

The link between GW and DT invariants is thus expected to be intimately
related to the non-perturbative structure of topological string theory. Since
the latter is defined by an asymptotic series in the topological string
coupling, the most canonical path to its non-perturbative structure is to
consider the theory of resurgence and Borel resummation; see [Mn14] and
references therein for an overview. This has indeed been applied to
topological string theory in connection with Chern–Simons theory and matrix
models in [PS10] as well as for the resolved conifold in [HO15]. In
particular, [HO15] used a generalization of the Borel resummation and produced
via Borel resummation a partition function which matched the expectations of a
proposal for the non-perturbative structure of topological string theory on
non-compact CY manifolds put forward earlier in [HMnMO14, GHMn16]. A non-
perturbative definition of the topological string free energy for general
toric CY has been proposed in [GHMn16] in terms of the spectral determinants
of the finite difference operators obtained by quantising the mirror curves.
In [CSESV15], techniques of resurgence and transseries were applied to the
study of topological string theory via the holomorphic anomaly equations of
BCOV [BCOV94]; see also [CS14] and references therein. These techniques have
been applied to the study of the proposal of [GHMn16] in [CSMnS17]. The link
to BPS structures started to emerge more clearly recently [GGMn20b, GMn21]
where connections between Stokes phenomena and BPS invariants have been
investigated. See also [KS20, GGMn20a] for works in related directions.

On the side of DT invariants and BPS structures, exciting insights are coming
from the study of wall-crossing phenomena. The wall-crossing formulas of
Kontsevich and Soibelman [KS08] as well as Joyce and Song [JS12] have led to a
lot of progress on wall-crossing phenomena of BPS states. In [GMN10, GMN13b,
GMN13a], Gaiotto, Moore and Neitzke (GMN) provided a physical interpretation
of these developments as well as new geometric constructions of hyperkähler
manifolds having metrics determined by the BPS spectra; see e. g. [Nei14].
More recent developments are concerned with the analytic and integrable
structures behind wall-crossing phenomena. The emerging links indicate new
connections between DT invariants and GW invariants, going substantially
beyond the scope of the MNOP relation [MNOP06a, MNOP06b]. Bridgeland [Bri19]
formulated a Riemann–Hilbert associated to the Donaldson–Thomas invariants of
a given derived category and defined an associated potential called Tau-
function in [Bri19]. In simple examples including the resolved conifold
[Bri20], it was shown that an asymptotic expansion of the Tau-function
reproduces the full Gromov–Witten potential. In [CLT20] it was proposed that
the topological string partition functions for a certain class of local CY
represent local sections of certain canonical holomorphic line bundles defined
by the relevant solutions to the Riemann-Hilbert problems from [Bri19].

In this paper, we will revisit the Borel summation of the resolved conifold
partition function from a new perspective. We will show, on the one hand, that
the Stokes jumps of the Borel summation of the expansion in powers of the
topological string coupling have a close relation to the jumps defining the
Riemann–Hilbert problem defined by Bridgeland using DT invariants as input
data in [Bri20]. The Stokes jumps serve as certain types of potentials for the
jumps of the Darboux coordinates defining the Riemann–Hilbert problem in
[Bri20].

The Borel summations along different rays $\rho$ are found to have the
following structure

$F_{\rho}(\lambda,t)=F_{\mathrm{GV}}(\lambda,t)+F_{\mathrm{D}}(\lambda,t;\rho),$
(1.1)

where $\lambda$ is the topological string coupling, and $t$ the complexified
Kähler parameter. The contribution denoted $F_{\mathrm{GV}}(\lambda,t)$ is the
canonical re-organisation known from the work of Gopakumar and Vafa of the
formal series in powers of $\lambda$ as a series in powers of
$Q=e^{2\pi\mathrm{i}t}$ which is convergent for $\mathrm{Im}(t)>0$.
$F_{\mathrm{GV}}(\lambda,t)$ does not depend on the ray $\rho$. The second
part, $F_{\mathrm{D}}(\lambda,t;\rho)$ strongly depends on the choice of a ray
$\rho$. $F_{\mathrm{D}}(\lambda,t;\rho)$ can be represented as functions of
the variables $Q^{\prime}=e^{4\pi^{2}\mathrm{i}t/\lambda}$ and
$q^{\prime}=e^{4\pi^{2}\mathrm{i}/\lambda}$, suggesting an interpretation in
terms of non-perturbative effects associated to D-branes in type II string
theory. It is known that there exist non-perturbative effects in string theory
represented by disk amplitudes associated to stable D-branes. Closely related
effects have recently been identified with non-perturbative corrections to the
metric on the hypemultiplet moduli space in type II string theory on CY three-
folds [ASS21]. It seems natural to interpret the jumps of
$F_{\rho}(\lambda,t)$ across Stokes rays as the consequences of changes of the
set of stable objects contributing to the non-perturbative effects in the
partition functions. The explicit results for the jumps take a particularly
simple form, having a direct relation to the Riemann-Hilbert problems
associated to DT-invariants in [Bri20] further discussed below.

The results associated to different rays $\rho$ interpolate between two
special functions which had previously been proposed as candidates for non-
perturbative definitions of the topological string partition functions:
Integration along the imaginary axis yields the Gopakumar–Vafa resummation
$F_{\mathrm{GV}}(\lambda,t)$ on the one hand, while choosing $\rho$ to be the
positive real axis, $\rho={\mathbb{R}}_{>0}$, yields a function closely
related to the triple sine function. In the case $\rho={\mathbb{R}}_{>0}$, we
find that the function $F_{\mathrm{D}}(\lambda,t;\rho)$ appearing in equation
(1.1) specialises to the previously known function
$F_{\mathrm{NS}}(\lambda,t)$ which can be obtained from the refined version of
$F_{\mathrm{GV}}$ introduced in [IKV09] in the limits studied by Nekrasov and
Shatashvili [NS09]. The combination

$F_{\rm
np}(\lambda,t):=F_{\mathrm{GV}}(\lambda,t)+F_{\mathrm{NS}}(\lambda,t),$ (1.2)

appearing on the right side of (1.1) in the case $\rho={\mathbb{R}}_{>0}$ has
been studied before as a candidate for a non-perturbative completion of the
topological string partition function [HMnMO14]. Relations to previous work
studying the function $F_{\rm np}(\lambda,t)$ in connection to topological
string theory are further discussed in Section 2.4.

It turns out that there is an appealing way to encode the Stokes data
geometrically, in line with the previous suggestions made in [CLT20]. It will
be shown that the Stokes jumps can be interpreted as transition functions of a
certain line bundle canonically associated to the solution of the
Riemann–Hilbert problem considered by Bridgeland. We will show that this line
bundle is closely related to the hyperholomorphic line bundles studied in
relation to hyperkähler geometry in [APP11b, Nei11]. The Borel summations of
the topological string partition functions represent local sections of this
line bundle. In the previous work [CPT18, CLT20], it had been demonstrated
that the Fourier transforms of the topological string partition functions
associated to a certain class of local CY are related to the isomonodromic
tau-functions which represent local sections of this line bundle. Due to the
absence of compact four-cycles, the tau-functions simply coincide with the
topological string partition functions for the case at hand.

The Borel summation along the positive real axis appears to be distinguished
in some ways. This function also has an asymptotic expansion for
$\lambda\to\infty$, referred to as the strong-coupling expansion in the
following. The Borel summations of the strong-coupling expansion along
different rays $\rho^{\prime}$ are found to have the following structure:

${F}_{\rho^{\prime}}^{\prime}(\lambda,t)=F_{\rm
BPS}(\lambda,t;\rho^{\prime})+F_{\rm NS}(\lambda,t).$ (1.3)

The contribution $F_{\rm NS}(\lambda,t)$ is now independent of
$\rho^{\prime}$, while $F_{\rm BPS}(\lambda,t;\rho^{\prime})$ exhibits jumps
when $\rho^{\prime}$ crosses certain rays in the complex plane of the variable
$1/\lambda$. We find that $Z_{\rm BPS}(\lambda,t;\rho^{\prime}):=e^{F_{\rm
BPS}(\lambda,t;\rho^{\prime})}$ is closely related to the counting functions
for BPS states previously studied in the context of supergravity by Jafferis
and Moore [JM08]. The Stokes jumps of
$Z_{\mathrm{BPS}}(\lambda,t;\rho^{\prime})$ display a precise correspondence
to the wall-crossing behaviour of the counting functions for BPS states
studied in [JM08]. In the case of $\rho^{\prime}={\mathbb{R}}_{>0}$ we recover
$F_{\mathrm{GV}}(\lambda,t)$.

Mathematically one may again observe a close relation to a Riemann–Hilbert
problem associated to DT theory. However, the jumps of
$Z_{\mathrm{BPS}}(\lambda,t;\rho^{\prime})$ now directly coincide with the
jumps of a particular coordinate function in a close relative of Bridgeland’s
Riemann–Hilbert problem, as could have been expected from previous
computations of $Z_{\mathrm{BPS}}(\lambda,t;\rho^{\prime})$ on the basis of
wall-crossing formulae [BLR19, Appendix A]. It should be stressed that both
the location of jumps, and the functional form of the jumps are different for
weak- and strong-coupling expansions. However, we find that both are
determined by Riemann–Hilbert type problems associated to DT invariants,
albeit quite remarkably in somewhat different ways.

At least in the example studied in this paper, we have identified two new ways
to extract non-perturbative information on DT invariants from the GW
invariants defining the topological string partition functions. Our results
suggest that these data are deeply encoded in the analytic structures of non-
perturbatively defined partition functions. The way this happens indicates
close connections to string-theoretic S-duality conjectures, as will be
briefly discussed in Section 6.

Acknowledgments: we have benefited from discussions with Vicente Cortés, Timo
Weigand, and Alexander Westphal around common research projects within the
Cluster of Excellence “Quantum Universe”. The authors would furthermore like
to thank Sergei Alexandrov, Tom Bridgeland, Marcos Mariño, Greg Moore, and
Boris Pioline for comments on a preliminary version of this paper. The work of
J.T. and I.T. is funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy EXC 2121 Quantum
Universe 390833306. The work of M.A and A.S. is supported through the DFG Emmy
Noether grant AL 1407/2-1.

##  2 Borel summations of the resolved conifold partition function

We are going to study the formal series

$\widetilde{F}(\lambda,t)=\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(Q)+\frac{B_{2}}{2}\mathrm{Li}_{1}(Q)+\sum_{g=2}^{\infty}\lambda^{2g-2}\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\mathrm{Li}_{3-2g}(Q),$

with $Q=\exp(2\pi{\mathrm{i}}t)$, and polylogarithms $\mathrm{Li}_{s}(z)$ and
Bernoulli numbers $B_{n}$ defined by

$\mathrm{Li}_{s}(z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{s}}\,,\quad
s\in\mathbb{C}\,,\qquad\frac{w}{e^{w}-1}=\sum_{n=0}^{\infty}B_{n}\frac{w^{n}}{n!}\,.$
(2.1)

Borel summation of this formal series will repackage the information contained
in it in an interesting way, revealing non-obvious mathematical structures.
Our goals in this section will be to state the results on the Stokes phenomena
of the Borel sums of $\widetilde{F}(\lambda,t)$, to discuss some of its
implications, and relations to previous results in the literature.

###  2.1 Motivation: Topological string theory on the resolved conifold

Topological string theory motivates the consideration of the topological
string partition functions. One expects to be able to associate such partition
functions to families of Calabi–Yau (CY) threefolds $X=X_{t}$, with
$t=(t^{1},\dots,t^{n})$ being a set of distinguished local coordinates on the
CY Kähler moduli space $\mathcal{M}$ of dimension $n=h^{1,1}(X_{t})$. The
partition function is expected to be defined by an asymptotic series in the
topological string coupling $\lambda$ of the form

$Z_{\rm
top}(\lambda,t)=\exp\left(\sum_{g=0}^{\infty}\lambda^{2g-2}{F}^{g}(t)\right).$
(2.2)

In order to provide a rigorous mathematical basis for the definition of
topological string partition functions, one may start by defining the GW
potential of a Calabi–Yau threefold $X$ as the formal power series

${\mathcal{F}}(Q,\lambda)=\sum_{g\geq
0}\lambda^{2g-2}{\mathcal{F}}^{g}(Q)=\sum_{g\geq 0}\sum_{\beta\in
H_{2}(X,\mathbb{Z})}\lambda^{2g-2}N_{\beta}^{g}\,Q^{\beta}\,,$ (2.3)

where $Q^{\beta}=\prod_{r=1}^{n}Q_{r}^{\beta_{r}}$ if
$\beta=\sum_{r=1}^{n}\beta_{r}\gamma_{r}$, with
$\\{\gamma_{1},\dots,\gamma_{n}\\}$ being an integral basis for
$H_{2}(X,{\mathbb{Z}})$, and $Q_{r}$ being formal variables for $r=1,\dots,n$.
One may note that the term associated to $\beta=0$ is independent of the
Kähler class $\beta$, motivating the decomposition

${\mathcal{F}}(Q,\lambda)={\mathcal{F}}_{0}(\lambda)+\widetilde{{\mathcal{F}}}(Q,\lambda)\,,$
(2.4)

where the contribution ${\mathcal{F}}_{0}(\lambda)$ takes the universal form
[FP98]

${\mathcal{F}}_{0}(\lambda)=\sum_{g\geq 0}\lambda^{2g-2}F_{0}^{g},\qquad
F_{0}^{g}=\frac{\chi(X)(-1)^{g-1}\,B_{2g}\,B_{2g-2}}{4g(2g-2)\,(2g-2)!}\,,\quad
g\geq 2\,,$ (2.5)

with $\chi(X)$ being the Euler characteristic of $X$. The formal series
$\widetilde{{\mathcal{F}}}(Q,\lambda)$ is defined as

$\widetilde{{\mathcal{F}}}(Q,\lambda)=\sum_{g\geq
0}\sum_{\beta\in\Gamma}\lambda^{2g-2}[\mathrm{GW}]_{\beta,g}\,Q^{\beta},$
(2.6)

where $\Gamma=\\{\beta\in H_{2}(X,{\mathbb{Z}});\beta\neq 0\\}$, with
$[\mathrm{GW}]_{\beta,g}$ being the Gromov–Witten invariants. In this way, one
arrives at a precise definition of ${\mathcal{F}}(Q,\lambda)$ as a formal
series.

There is a class of CY manifolds where the series ${\mathcal{F}}^{g}(Q)$
actually have finite radii of convergence, allowing us to define the functions
$F^{g}(t)={\mathcal{F}}^{g}(e^{2\pi{\mathrm{i}}t_{1}},...,e^{2\pi{\mathrm{i}}t_{n}})$,
where $t=(t_{1},...,t_{n})$. The resulting power series in $\lambda$ is not
expected to be convergent, in general. One may hope, however, that there can
exist analytic functions having the series $F(\lambda,t)=\sum_{g\geq
0}\lambda^{2g-2}F^{g}(t)$ as asymptotic expansion.

We are here considering a particular example of a CY manifold $X$ called the
resolved conifold. This CY threefold represents the total space of the rank
two bundle over the projective line:

$X:=\mathcal{O}(-1)\oplus\mathcal{O}(-1)\rightarrow\mathbb{P}^{1}\,,$ (2.7)

and corresponds to the resolution of the conifold singularity.

The GW potential for this geometry was determined in physics [GV98b, GV99],
and in mathematics [FP00] with the following outcome for the non-constant
maps:111See also [MnM99] for the determination of $F^{g}$ from a string theory
duality and the explicit appearance of the polylogarithm expressions.

$\widetilde{F}(\lambda,t)=\sum_{g=0}^{\infty}\lambda^{2g-2}\widetilde{F}^{g}(t)=\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(Q)+\sum_{g=1}^{\infty}\lambda^{2g-2}\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\mathrm{Li}_{3-2g}(Q)\,,$
(2.8)

using the notation $Q=e^{2\pi\mathrm{i}t}$. The constant map constribution has
the form (2.5) with $\chi(X)=2$ and $F_{0}^{0}=-\zeta(3)$. The value of
$F_{0}^{1}$ only shifts $\mathcal{F}(Q,\lambda)$ by an overall constant, and
its specific value won’t be important. Our first goal will be to study the
Borel summability of the series (2.8) and (2.5). This was first studied in
[PS10]. The results presented below complete and clarify previous work on this
subject, as will be discussed in more detail below.

###  2.2 Statement of results for the Borel sum and its Stokes phenomena

Here we state a theorem collecting the results that we wish to prove. The
proof of each part will be presented in Section 3.

Before stating the theorem, we briefly recall the definition of Borel
summation. Given a formal power series
$a(\check{\lambda})\in\check{\lambda}\mathbb{C}[[\check{\lambda}]]$, we
consider its Borel transform $\mathcal{B}(a)(\xi)$, where

$\mathcal{B}\colon\check{\lambda}\mathbb{C}[[\check{\lambda}]]\to\mathbb{C}[[\xi]],\;\;\;\;\mathcal{B}(\check{\lambda}^{n+1})=\frac{\xi^{n}}{n!}.$
(2.9)

Let $\check{\lambda}\in\mathbb{C}^{\times}$ and let $\rho$ be a ray from $0$
to $\infty$ in the complex $\xi$-plane. If $\mathcal{B}(a)(\xi)$ defines an
analytic function along $\rho$, we define the Borel sum of
$a(\check{\lambda})$ at $\check{\lambda}$, along $\rho$ by

$\int_{\rho}\mathrm{d}\xi\;e^{-\xi/\check{\lambda}}\mathcal{B}(a)(\xi)\,.$
(2.10)

If (2.10) is finite, we say $a(\check{\lambda})$ is Borel summable at
$\check{\lambda}$, along $\rho$.

######  Theorem 2.1.

Consider the formal series

$\displaystyle\widetilde{F}(\lambda,t)$
$\displaystyle=\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(Q)+\frac{B_{2}}{2}\mathrm{Li}_{1}(Q)+\sum_{g=2}^{\infty}\lambda^{2g-2}\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\,\mathrm{Li}_{3-2g}(Q)\,$
$\displaystyle=\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(Q)+\frac{B_{2}}{2}\mathrm{Li}_{1}(Q)+\Phi(\check{\lambda},t)\,,\quad\check{\lambda}=\frac{\lambda}{2\pi}\,,\quad
Q=e^{2\pi{\mathrm{i}}t}\,.$ (2.11)

Then we have the following:

  * (i)

(Borel transform) For $t\in\mathbb{C}^{\times}$ with $|\mathrm{Re}(t)|<1/2$,
let $G(\xi,t):=\mathcal{B}(\Phi(-,t))(\xi)$ denote the Borel transform of
$\Phi(\check{\lambda},t)$. Then $G(\xi,t)$ converges for $|\xi|<2\pi|t|$.
Furthermore, $G(\xi,t)$ admits a series representation of the form

$G(\xi,t)=\frac{1}{(2\pi)^{2}}\sum_{m\in\mathbb{Z}\setminus\\{0\\}}\frac{1}{m^{3}}\frac{1}{2\xi}\frac{\partial}{\partial\xi}\bigg{(}\frac{\xi^{2}}{1-e^{-2\pi{\mathrm{i}}t+\xi/m}}-\frac{\xi^{2}}{1-e^{-2\pi{\mathrm{i}}t-\xi/m}}\bigg{)}.$
(2.12)

We can use the above series representation to analytically continue $G(\xi,t)$
in the $\xi$ variable to a meromorphic function with poles at
$\xi=2\pi{\mathrm{i}}(t+k)m$ for $k\in\mathbb{Z}$ and
$m\in\mathbb{Z}\setminus\\{0\\}$.

  * (ii)

(Borel sum) For $k\in\mathbb{Z}$ let $l_{k}:=\mathbb{R}_{<0}\cdot
2\pi{\mathrm{i}}(t+k)$ and $l_{\infty}:={\mathrm{i}}\mathbb{R}_{<0}$. Given
any ray $\rho$ from $0$ to $\infty$ different from $\\{\pm
l_{k}\\}_{k\in\mathbb{Z}}\cup\\{\pm l_{\infty}\\}$, and $\lambda$ in the half-
plane $\mathbb{H}_{\rho}$ centered at $\rho$, we define the Borel sum of
$\widetilde{F}(\lambda,t)$ along $\rho$ as

$F_{\rho}(\lambda,t):=\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(Q)+\frac{B_{2}}{2}\mathrm{Li}_{1}(Q)+\int_{\rho}\mathrm{d}\xi\,e^{-\xi/\check{\lambda}}G(\xi,t)\,.$
(2.13)

Taking $\rho=\mathbb{R}_{>0}$, and assuming that $\mathrm{Im}(t)>0$ and
$0<\mathrm{Re}(t)<1$, we have the following identity whenever
$\mathrm{Re}(t)<\mathrm{Re}(\check{\lambda}+1)$:

$F_{\mathbb{R}_{>0}}(\lambda,t)=-\int_{\mathbb{R}+{\mathrm{i}}0^{+}}\frac{\mathrm{d}u}{8u}\frac{e^{u(t-1/2)}}{\sinh(u/2)(\sinh(\check{\lambda}u/2))^{2}}\,.$
(2.14)

  * (iii)

(Stokes jumps) Let $\rho_{k}$ be a ray in the sector determined by the Stokes
rays $l_{k}$ and $l_{k-1}$. Then if $\mathrm{Im}(t)>0$, on the overlap of
their domains of definition in the $\lambda$ variable we have

$\phi_{\pm
l_{k}}(\lambda,t):=F_{\pm\rho_{k+1}}(\lambda,t)-F_{\pm\rho_{k}}(\lambda,t)=\frac{1}{2\pi{\mathrm{i}}}\partial_{\check{\lambda}}\Big{(}\check{\lambda}\,\mathrm{Li}_{2}\big{(}e^{\pm
2\pi{\mathrm{i}}(t+k)/\check{\lambda}}\big{)}\Big{)}\;.$ (2.15)

If $\mathrm{Im}(t)<0$, then the previous jumps also hold provided $\rho_{k+1}$
is interchanged with $\rho_{k}$ in the above formula.

  * (iv)

(Limits to $\pm{\mathrm{i}}\mathbb{R}_{>0}$) Let $\rho_{k}$ denote any ray
between the rays $l_{k}$ and $l_{k-1}$. Furthermore, assume that
$0<\mathrm{Re}(t)<1$, $\mathrm{Im}(t)>0$, $\mathrm{Re}(\lambda)>0$,
$\mathrm{Im}(\lambda)<0$, and $\mathrm{Re}\,t<\mathrm{Re}(\check{\lambda}+1)$.
Then

$\lim_{k\to\infty}F_{\rho_{k}}(\lambda,t)=\lim_{k\to\infty}F_{-\rho_{k}}(-\lambda,t)=\sum_{k=1}^{\infty}\frac{e^{2\pi{\mathrm{i}}kt}}{k\big{(}2\sin\big{(}\frac{\lambda
k}{2}\big{)}\big{)}^{2}}\,.$ (2.16)

Furthemore, we can write the sum of the Stokes jumps along $l_{k}$ for $k\geq
0$ as

$\sum_{k=0}^{\infty}\phi_{l_{k}}(\lambda,t)=\frac{1}{2\pi{\mathrm{i}}}\partial_{\lambda}\Big{(}\lambda\sum_{l=1}^{\infty}\frac{w^{l}}{l^{2}(1-\widetilde{q}^{l})}\Big{)},\;\;\;\;w:=e^{2\pi{\mathrm{i}}t/\check{\lambda}},\;\;\widetilde{q}:=e^{2\pi{\mathrm{i}}/\check{\lambda}}\;.$
(2.17)

If, on the other hand, we take $0<\mathrm{Re}(t)<1$, $\mathrm{Im}(t)>0$,
$\mathrm{Re}(\lambda)>0$, $\mathrm{Im}(\lambda)>0$,
$\mathrm{Re}(t)<\mathrm{Re}(\check{\lambda}+1)$ and furthermore assume that
$|e^{2\pi{\mathrm{i}}t/\check{\lambda}}|<1$, then we also have

$\lim_{k\to-\infty}F_{\rho_{k}}(\lambda,t)=\lim_{k\to-\infty}F_{-\rho_{k}}(-\lambda,t)=\sum_{k=1}^{\infty}\frac{e^{2\pi{\mathrm{i}}kt}}{k\big{(}2\sin\big{(}\frac{\lambda
k}{2}\big{)}\big{)}^{2}}.$ (2.18)

$\textrm{Re}(\xi)$$\textrm{Im}(\xi)$$l_{0}$$-l_{0}$$\rho_{1}$$-l_{1}$$l_{1}$$l_{-1}$$-l_{-1}$$l_{\infty}$
Figure 1: Illustration of the Stokes rays $l_{k}=\mathbb{R}_{<0}\cdot
2\pi{\mathrm{i}}(t+k)$ in the Borel plane, plotted for
$t=\frac{1}{\pi}\left(1+\frac{{\mathrm{i}}}{2}\right)$ and $k=-10,\dots,10$,
as well as a possible integration ray $\rho_{1}$.

Let us note that under the assumptions of the first part of (iv),
$\lim_{k\to\infty}F_{\rho_{k}}(\lambda,t)$ differs from
$F_{\mathbb{R}_{>0}}(\lambda,t)$ by the sum over all jumps
$\phi_{l_{k}}(\lambda,t)$ for $k\geq 0$, leading to the decomposition

$F_{{\mathbb{R}}_{>0}}(\lambda,t)=\lim_{k\to\infty}F_{\rho_{k}}(\lambda,t)-\frac{1}{2\pi\mathrm{i}}\frac{\partial}{\partial\lambda}\Big{(}\lambda\sum_{l=1}^{\infty}\frac{w^{l}}{l^{2}(1-\widetilde{q}^{l})}\Big{)}.$
(2.19)

As we will see in Proposition 3.15, this decomposition can be obtained by
evaluating the integral on the right of (2.14) as a sum over residues. Part
(iv) of the theorem above identifies the second term on the right of (2.19)
with the sum over the Stokes jumps in the lower right quadrant of the Borel
plane.

It will turn out that the Borel summation $F_{0,\rho}(\lambda)$ of the formal
series (2.5) is closely related to the value of the function
$F_{\rho}(\lambda,0)$ defined in Theorem 2.1. The relation will be found to be
of the form

$F_{0,\rho}(\lambda)=-F_{\rho}(\lambda,0)-\frac{1}{12}\log\check{\lambda}+C,$
(2.20)

where $C$ is a constant independent of $\rho$ which won’t be of interest for
us. Equation (2.20) finally allows us to represent the Borel summation
$\widehat{F}_{\rho}(\lambda,t)$ of the full free energy
$F(\lambda,t)=F_{0}(\lambda)+\widetilde{F}(\lambda,t)$ of the topological
string theory by the formula

$\widehat{F}_{\rho}(\lambda,t)=F_{\rho}(\lambda,t)-F_{\rho}(\lambda,0)-\frac{1}{12}\log\check{\lambda}+C.$
(2.21)

In the following two subsections we will first discuss the interpretation of
Theorem 2.1 in the context of topological string theory. This will be followed
by a discussion of the relation to previous results in this direction.

###  2.3 Connection to topological string theory

In the case of the resolved conifold, non-perturbative definitions of the
topological string partition functions should be analytic functions of
$\lambda$ and $t$ such that (2.8) gives an asymptotic series expansion for
$\lambda\to 0$ of the corresponding free energy $\widetilde{F}(\lambda,t)$.

#### 2.3.1

The Gopakumar–Vafa (GV) resummation of the GW potential [GV98a] re-organises
the non-constant part $\widetilde{F}(\lambda,t)$ of the GW potential in the
following form:

$\sum_{g\geq
0}\lambda^{2g-2}\sum_{\beta\in\Gamma}[\mathrm{GW}]_{\beta,g}\,Q^{\beta}=\sum_{\beta\in\Gamma}\sum_{g\geq
0}\,[\mathrm{GV}]_{\beta,g}\,\sum_{k\geq
1}\frac{1}{k}\left(2\sin\big{(}{\textstyle\frac{k\lambda}{2}}\big{)}\right)^{2g-2}Q^{k\beta}\,.$
(2.22)

Equation (2.22) can be understood as an equality of formal power series in
$Q^{\beta}$ with coefficients being Laurent series in $\lambda$. One can
thereby regard (2.22) as a definition of the GV invariants
$[\mathrm{GV}]_{\beta,g}$ in terms of the Gromov–Witten invariants
$[\mathrm{GW}]_{\beta,g}$.

Using the known results for the invariants $\mathrm{GW}_{\beta,g}$ of the
conifold, one finds that the right-hand side of (2.22) simplifies to

$F_{\mathrm{GV}}(\lambda,t)=\sum_{k=1}^{\infty}\frac{e^{2\pi{\mathrm{i}}kt}}{k\big{(}2\sin\big{(}\frac{\lambda
k}{2}\big{)}\big{)}^{2}}\,.$ (2.23)

This has also been derived using the topological vertex formalism [AKMnV05].
Assuming $\mathrm{Im}(t)>0$, one may notice that the series defining
$F_{\mathrm{GV}}(\lambda,t)$ in (2.23) is convergent for
$\mathrm{Im}(\lambda)>0$ or $\mathrm{Im}(\lambda)<0$. One may regard
$F_{\mathrm{GV}}(\lambda,t)$ as a minimal summation of the divergent series
(2.1), in the sense that it is obtained by a rearrangement of the formal
series $\widetilde{F}(\lambda,t)$ into a convergent series in powers of
$Q=e^{2\pi\mathrm{i}t}$ that defines functions analytic in $\lambda$ away from
the real line ${\mathbb{R}}$.

Our results relate $F_{\mathrm{GV}}(\lambda,t)$ to the limits of the Borel
summations along rays $\rho_{k}$ for $k\to\pm\infty$ when the rays $\rho_{k}$
approach the imaginary axis.

#### 2.3.2

As mentioned above, the function $F_{\mathrm{GV}}(\lambda,t)$ is not well-
defined for $\lambda\in{\mathbb{R}}$. This is one of the motivations to look
for analytic functions having the same asymptotic expansion, but larger
domains of definition, as candidates for non-perturbative definitions of the
topological string partition functions.

A general proposal has been made in [HMnMO14] for non-perturbative definitions
of topological string partition functions. This proposal was motivated by the
observation [HMO13] that one can systematically add functions of
$e^{(2\pi)^{2}\frac{\mathrm{i}}{\lambda}}$ to the function
$F_{\mathrm{GV}}(\lambda,t)$ cancelling all the singularities that
$F_{\mathrm{GV}}(\lambda,t)$ develops on the real $\lambda$-axis. The function
of $e^{(2\pi)^{2}\frac{\mathrm{i}}{\lambda}}$ having this property can be
interpreted as certain non-perturbative corrections in string theory.

Specialised to the conifold, the proposal made in [HMnMO14] yields the
following function:

$F_{\mathrm{np}}(\lambda,t):=F_{\mathrm{GV}}(\lambda,t)+\frac{1}{2\pi{\mathrm{i}}}\frac{\partial}{\partial\lambda}\lambda
F_{\mathrm{NS}}\big{(}{\textstyle\frac{4\pi^{2}}{\lambda},\frac{2\pi}{\lambda}(t-\frac{1}{2})}\big{)},$
(2.24)

using the notations

$F_{\mathrm{GV}}(\lambda,t):=\sum_{k=1}^{\infty}\frac{e^{2\pi{\mathrm{i}}kt}}{k\big{(}2\sin\big{(}{\textstyle\frac{\lambda
k}{2}}\big{)}\big{)}^{2}}\,,\qquad
F_{\mathrm{NS}}(g,t):=\frac{1}{2{\mathrm{i}}}\sum_{k=1}^{\infty}\frac{e^{2\pi{\mathrm{i}}kt}}{k^{2}\sin\big{(}{\textstyle\frac{gk}{2}}\big{)}}\,.$
(2.25)

It is easy to see that the right side of (2.24) coincides with the expression
on the right of (2.19) (i.e. that $F_{\mathbb{R}_{>0}}=F_{\mathrm{np}}$).

#### 2.3.3

Using Borel summation is another natural approach to finding non-perturbative
definitions of the topological string partition functions, as previously
investigated in [PS10] and in [HO15]. A formula for the Borel transform had
been first proposed in [PS10], and in [HO15] it was conjectured that the Borel
transform along the real axis is equal to (2.24). Extensive numerical studies
provided convincing evidence for these proposals.

Our Theorem 2.1 offers a more complete picture. It shows that the Borel
summations $F_{\rho}(\lambda,t)$ interpolate between
$F_{\mathrm{GV}}(\lambda,t)$ and $F_{\mathrm{np}}(\lambda,t)$. All the
functions $F_{\rho}(\lambda,t)$ defined by different choices of the ray $\rho$
can be regarded as different re-packagings of the same information, contained
in the formal series (2.8). Any of these summations can serve as a candidate
for a non-perturbative definition of the topological string partition function
of the resolved conifold. Additional requirements have to be imposed to
distinguish a particular choice among others.

Defining the topological string partition functions by Borel summation
whenever this possibility exists seems to be the most canonical way to
associate actual functions to the divergent series (2.8). The price to pay is
that the resulting function is only piecewise analytic, having jumps across
the rays $\pm l_{k}$. However, as will be explained in the rest of the paper,
there is interesting information contained in these jumps. We are going to
demonstrate that the jump functions encode information on the spectrum of BPS
states on the resolved conifold in a particularly simple and transparent way
by relating them to the Riemann–Hilbert problem formulated in [Bri20] which
takes as input data the generalised DT invariants for the resolved conifold.

The Borel summations $F_{\rho_{k}}(\lambda,t)$ each have natural domains of
definition, bounded by the rays $l_{k}$ and $l_{k-1}$. It seems important to
note, however, that the functions $F_{\rho_{k}}(\lambda,t)$ can be
analytically continued in $\lambda$ to larger domains of definition containing
$l_{k}$ and $l_{k-1}$. This suggests to regard the analytically continued
functions $F_{\rho_{k}}(\lambda,t)$ as local sections of a line bundle defined
by taking exponentials of the jumps
$\phi_{l_{k}}(\lambda,t)=F_{\rho_{k+1}}(\lambda,t)-F_{\rho_{k}}(\lambda,t)$ as
transition functions. This line bundle, together with the collection of
distinguished local sections $F_{\rho_{k}}(\lambda,t)$ is a natural geometric
object canonically associated to the formal series (2.8) by Borel summation.
We will see that it is a natural analog of the line bundle proposed in [CLT20]
for the case of the resolved conifold.

#### 2.3.4

Let us note that the differences
$F_{\mathrm{D},k}(\lambda,t):=F_{\rho_{k}}(\lambda,t)-F_{\mathrm{GV}}(\lambda,t)$
can be represented as sums of terms which are all proportional to an
exponential function having dependence with respect to the topological string
coupling $\lambda$ of the form $e^{(const.)/\lambda}$. It is therefore natural
to associate the differences $F_{\mathrm{D},k}(\lambda,t)$ with non-
perturbative effects in string theory. They can be represented as a sum over
the Stokes jumps across the rays $l_{k}$ enclosed by $\rho_{k}$ and
$\mathrm{i}{\mathbb{R}}$. We will see that these jumps are in a one-to-one
correspondence with D-branes in type II string theory on the resolved
conifold.

A dependence of the form of the form $e^{(const.)/\lambda}$ is characteristic
for non-perturbative effects in string theory having a world-sheet description
through disk amplitudes with boundaries associated to D-branes. Such disk
amplitudes can represent central charge functions of D-branes in type II
string theory [HIV00]. We will see that the constants in the exponential
functions $e^{(const.)/\lambda}$ appearing in the differences
$F_{\mathrm{D},k}(\lambda,t)$ have a simple relation to the central charge
functions of the D-branes associated to the jumps. The functions
$F_{\mathrm{D},k}(\lambda,t)$ can be represented as sums over all terms which
are exponentially suppressed in the wedge of the $\lambda$-plane bounded by
$l_{k}$ and $l_{k-1}$.

These observations suggest that the non-perturbative effects represented by
the functions $F_{\mathrm{D},k}(\lambda,t)$ may have a world-sheet description
in terms of disk amplitudes with boundaries associated to stable D-branes
representing states in the BPS-spectrum of the resolved conifold. The set of
D-branes contributing to the non-perturbatively defined partition functions
would then depend on the phase of $\lambda$, and jump across the rays $l_{k}$.
It would be interesting to verify this interpretation more directly.

###  2.4 Previous results

Previous work on this subject had obtained several important partial results.
The first study of the Borel summability of the series
$\widetilde{F}(\lambda,t)$ was performed in [PS10], where an explicit formula
for the Borel transform was found. While the direct comparison of the formula
derived in [PS10] with (2.12) is not completely straightforward, it is easy to
see that the poles and residues agree.

Another approach to the summation of the formal series
$\widetilde{F}(\lambda,t)$ has been proposed in [HO15]. The summation
considered in [HO15] is an analytic function $F^{\rm resum}_{\rm
coni}(\lambda,t)$ defined through an explicit integral representation.
Numerical evidence has been presented for the conjecture that $F^{\rm
resum}_{\rm coni}(\lambda,t)$ is equal to the Borel summation
$F_{{\mathbb{R}}_{>0}}(\lambda,t)$ along $\rho={\mathbb{R}}_{>0}$ in our
notations. We will later in Section 3.4.1 explicitly establish the relation
between $F^{\rm resum}_{\rm coni}(\lambda,t)$ and
$F_{{\mathbb{R}}_{>0}}(\lambda,t)$ considered in our paper. It was furthermore
proposed in [HO15] that the function $F^{\rm resum}_{\rm coni}(\lambda,t)$
admits the decomposition (2.24). This conjecture has been extensively checked
numerically.

Interesting relations with spectral determinants of finite difference
operators along the lines of [GHMn16] have been found in [BGT19]. Further
exploration of the relations to our results should be illuminating.

It has been demonstrated in [Bri20] that a special function closely related to
the triple sine function has (2.1) as its asymptotic expansion. The relation
between the triple sine function and the formal series
$\widetilde{F}(\lambda,t)$ has stimulated the work [Ali20, AS21, Ali21]
studying the function defined on the right side of (2.14) as a promising
candidate for a non-perturbative definition of the topological string
partition function. It was identified in [AS21] as a solution with pleasant
analytic properties of a difference equation [Ali20] which governs the
topological string free energy. In [Ali21], the non-perturbative content of
this function was extracted demonstrating that this function admits the
decomposition (2.24) and matching in particular with the results of [HO15].

Further work on the function $F_{\rm np}(\lambda,t)$ in connection with the
non-perturbative structure of topological strings can be found in [LV18,
KM15].

##  3 Proofs of the results of Section 2.2

In this section we prove each of the points of Theorem 2.1. Our approach is
strongly inspired by the paper [GK20] which has studied the analogous problem
for the non-compact quantum dilogarithm function. Each of the four subsections
below corresponds to each of the four points of the Theorem.

###  3.1 The Borel transform

We start by proving the first part of Theorem 2.1, concerning the Borel
transform of of $\widetilde{F}(\lambda,t)$. We remark that an alternative
expression of the Borel transform was previously given in [PS10], which we
recall in Section 3.1.1.

Recall the asymptotic expansion of the topological string free energy for the
resolved conifold, which is given by (2.8):

$\begin{split}\widetilde{F}(\lambda,t)&=\sum_{g=0}^{\infty}\lambda^{2g-2}\widetilde{F}^{g}(t)=\frac{1}{\lambda^{2}}\textrm{Li}_{3}(Q)+\frac{B_{2}}{2}\mathrm{Li}_{1}(Q)+\sum_{g=2}^{\infty}\lambda^{2g-2}\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\,\textrm{Li}_{3-2g}(Q)\,\\\
&=\frac{1}{\lambda^{2}}\textrm{Li}_{3}(Q)+\frac{B_{2}}{2}\mathrm{Li}_{1}(Q)+\Phi(\check{\lambda},t)\,,\quad\check{\lambda}=\frac{\lambda}{2\pi}\,,\quad
Q=e^{2\pi it}\,,\end{split}$ (3.1)

We use the property

$\theta_{Q}\textrm{Li}_{s}(Q)=\textrm{Li}_{s-1}(Q)\,,\quad\theta_{Q}:=Q\,\frac{d}{dQ}\,,$
(3.2)

to write

$\widetilde{F}^{g}=\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\,\theta_{Q}^{2g}\textrm{Li}_{3}(Q)\,,\quad
g\geq 2\,.$ (3.3)

Furthermore, using that $\theta_{Q}=\frac{1}{2\pi{\mathrm{i}}}\partial_{t}$ we
obtain

$\widetilde{F}^{g}=\frac{(-1)B_{2g}}{2g(2g-2)!(2\pi)^{2g}}\,\partial_{t}^{2g}\textrm{Li}_{3}(Q)\,,\quad
g\geq 2\,.$ (3.4)

We thus have

$\Phi(\check{\lambda},t)=-\frac{1}{4\pi^{2}}\sum_{g=2}^{\infty}\frac{B_{2g}}{2g(2g-2)!}\,\check{\lambda}^{2g-2}\partial_{t}^{2g}\textrm{Li}_{3}(Q)\,.$
(3.5)

We now wish to compute the Borel transform of $\Phi(\check{\lambda},t)$ and
specify its domain of convergence. The Borel transform is defined as the
formal power series $G(\xi,t):=\mathcal{B}(\Phi(-,t))(\xi)$, where

$\mathcal{B}\colon\check{\lambda}\mathbb{C}[[\check{\lambda}]]\to\mathbb{C}[[\xi]],\;\;\;\;\mathcal{B}(\check{\lambda}^{n+1})=\frac{\xi^{n}}{n!}.$
(3.6)

Namely, we wish to study

$G(\xi,t)=-\frac{1}{4\pi^{2}}\sum_{g=2}^{\infty}\frac{B_{2g}}{2g(2g-2)!(2g-3)!}\xi^{2g-3}\,\partial_{t}^{2g}\textrm{Li}_{3}(Q)\,.$
(3.7)

In order to do this, it will be convenient to first recall the Hadamard
product and a certain integral representation thereof. The techniques used
below follow the lines of [GK20].

######  Definition 3.1.

Consider two formal power series
$\sum_{n=0}^{\infty}a_{n}z^{n},\sum_{n=0}^{\infty}b_{n}z^{n}\in\mathbb{C}[[z]]$.
Then the Hadamard product
$\oast\colon\mathbb{C}[[z]]\times\mathbb{C}[[z]]\to\mathbb{C}[[z]]$ is defined
by

$\bigg{(}\sum_{n=0}^{\infty}a_{n}z^{n}\bigg{)}\oast\bigg{(}\sum_{n=0}^{\infty}b_{n}z^{n}\bigg{)}=\sum_{n=0}^{\infty}a_{n}b_{n}z^{n}\,.$
(3.8)

######  Lemma 3.2.

Consider two holomorphic functions near $z=0$ having series expansions

$f_{1}(z)=\sum_{n=0}^{\infty}a_{n}z^{n},\;\;\;f_{2}(z)=\sum_{n=0}^{\infty}b_{n}z^{n}$
(3.9)

with radius of convergence $r_{1}>0$ and $r_{2}>0$, respectively. Then
$(f_{1}\oast f_{2})(z)$ converges for $|z|<r_{1}r_{2}$, and for any
$\rho\in(0,r_{1})$ the following holds for $|z|<\rho r_{2}$:

$(f_{1}\oast f_{2})(z)=\frac{1}{2\pi
i}\int_{|s|=\rho}\frac{\mathrm{d}s}{s}f_{1}(s)f_{2}\Big{(}\frac{z}{s}\Big{)}\,.$
(3.10)

###### Proof.

By using the limsup definition of the radius of convergence, one can easily
check that the radius of convergence of $(f_{1}\oast f_{2})(z)$ must be bigger
or equal than $r_{1}r_{2}$. On the other hand, we have that for $|z|<\rho
r_{2}<r_{1}r_{2}$,

$\begin{split}(f_{1}\oast
f_{2})(z)&=\sum_{n=0}^{\infty}a_{n}b_{n}z^{n}=\sum_{n=0}^{\infty}\bigg{(}\frac{1}{2\pi{\mathrm{i}}}\int_{|s|=\rho}\mathrm{d}s\,\frac{f_{1}(s)}{s^{n+1}}\bigg{)}b_{n}z^{n}\\\
&=\frac{1}{2\pi{\mathrm{i}}}\int_{|s|=\rho}\frac{\mathrm{d}s}{s}f_{1}(s)\sum_{n=0}^{\infty}b_{n}\Big{(}\frac{z}{s}\Big{)}^{n}=\frac{1}{2\pi{\mathrm{i}}}\int_{|s|=\rho}\frac{\mathrm{d}s}{s}f_{1}(s)f_{2}\Big{(}\frac{z}{s}\Big{)}\\\
\end{split}$ (3.11)

where the interchange of sum and integrals is justfied by the Fubini–Tonelli
theorem and the absolute convergence of $(f_{1}\oast f_{2})(z)$. ∎

The idea is to write

$G(\xi,t)=(f_{1}\oast f_{2}(-,t))(\xi)$ (3.12)

for two functions $f_{1}(\xi)$, $f_{2}(\xi,t)$ which are holomorphic near
$\xi=0$, and then use the first part of the previous lemma. We will take
$f_{1}(\xi)$, $f_{2}(\xi,t)$ to be the following:

$\begin{split}f_{1}(\xi)&=-\frac{1}{4\pi^{2}}\sum_{g=2}^{\infty}\frac{(2g-1)\,B_{2g}}{(2g)!}\xi^{2g-3}\\\
f_{2}(\xi,t)&=\sum_{g=2}^{\infty}\frac{\xi^{2g-3}}{(2g-3)!}\,\partial_{t}^{2g}\textrm{Li}_{3}(Q)=\sum_{g=2}^{\infty}\frac{\xi^{2g-3}}{(2g-3)!}\,(2\pi{\mathrm{i}})^{2g}\textrm{Li}_{3-2g}(Q)\,.\end{split}$
(3.13)

######  Proposition 3.3.

Let $t\in\mathbb{C}^{\times}$ with $|\mathrm{Re}(t)|<\frac{1}{2}$. Then
$G(\xi,t)$ converges for $|\xi|<2\pi|t|$.

###### Proof.

Using the fact that

$B_{2g}\sim(-1)^{g+1}4\sqrt{\pi g}\Big{(}\frac{g}{\pi
e}\Big{)}^{2g}\;\;\;\text{as}\;\;\;g\to\infty,$ (3.14)

we find that the radius of convergence for $f_{1}(\xi)$ is $2\pi$. On the
other hand, using the fact that for $|\text{Re}(t)|<1/2$, we have

$\mathrm{Li}_{3-2g}(e^{2\pi{\mathrm{i}}t})\sim\Gamma(1-3+2g)(-2\pi{\mathrm{i}}t)^{3-2g-1}\,\;\;\;\text{as}\;\;\;g\to\infty,$
(3.15)

we find that the radius of convergence of $f_{2}(\xi,t)$ is $r_{2}(t)=|t|$.

By the use of Lemma 3.2, we find that provided $t\in\mathbb{C}^{\times}$
satisfies $|\text{Re}(t)|<\frac{1}{2}$, we have that $G(\xi,t)=(f_{1}\oast
f_{2}(-,t))(\xi)$ converges for $|\xi|<r_{1}r_{2}(t)=2\pi|t|$. ∎

We now wish to use the integral representation of the Hadamard product to find
a more convenient representation of $G(\xi,t)$.

######  Proposition 3.4.

With the same hypothesis as in Proposition 3.3, we have

$G(\xi,t)=\frac{1}{(2\pi)^{2}}\sum_{m\in\mathbb{Z}\setminus\\{0\\}}\frac{1}{m^{3}}\bigg{(}1+\frac{\xi}{2}\frac{\partial}{\partial\xi}\bigg{)}\bigg{(}\frac{1}{1-e^{-2\pi{\mathrm{i}}t+\xi/m}}-\frac{1}{1-e^{-2\pi{\mathrm{i}}t-\xi/m}}\bigg{)}$
(3.16)

In particular, for fixed $t$, the expression on the right allows us to
analytically continue $G(\xi,t)$ to a meromorphic function in $\xi$ with poles
at $\xi=2\pi{\mathrm{i}}(t+k)m$ for $k\in\mathbb{Z}$ and
$m\in\mathbb{Z}\setminus\\{0\\}$.

###### Proof.

The idea is to now use the integral representation of the Hadamard product in
Lemma 3.2, together with the results in the proof of Proposition 3.3. In
particular, for $t\in\mathbb{C}^{\times}$ with $|\text{Re}(t)|<1/2$ and
$\rho\in(0,2\pi)$, we have for $|\xi|<\rho|t|$

$G(\xi,t)=\frac{1}{2\pi\mathrm{i}}\int_{|s|=\rho}\frac{\mathrm{d}s}{s}f_{1}(s)f_{2}\Big{(}\frac{\xi}{s},t\Big{)}$
(3.17)

where $f_{1}(\xi)$ and $f_{2}(\xi,t)$ are as in Proposition 3.3.

We have, on the one hand,

$\begin{split}f_{1}(\xi)&=-\frac{1}{4\pi^{2}}\sum_{g=2}^{\infty}\frac{(2g-1)\,B_{2g}}{(2g)!}\xi^{2g-3}=-\frac{1}{4\pi^{2}}\frac{1}{\xi}\partial_{\xi}\left(\frac{1}{\xi}\sum_{g=2}^{\infty}\frac{\,B_{2g}}{(2g)!}\xi^{2g}\right)\\\
&=-\frac{1}{4\pi^{2}}\frac{1}{\xi}\partial_{\xi}\left(\frac{1}{\xi}\left(\sum_{g=0}^{\infty}\frac{\,B_{g}}{g!}\xi^{g}-1+\frac{\xi}{2}-\frac{\xi^{2}}{12}\right)\right)\\\
&=-\frac{1}{4\pi^{2}}\frac{1}{\xi}\partial_{\xi}\left(\frac{1}{\xi}\left(\frac{\xi}{e^{\xi}-1}-1+\frac{\xi}{2}-\frac{\xi^{2}}{12}\right)\right)\\\
&=-\frac{1}{4\pi^{2}}\left(\frac{1}{\xi^{3}}-\frac{1}{\xi(e^{\xi/2}-e^{-\xi/2})^{2}}-\frac{1}{12\xi}\right)\,,\end{split}$
(3.18)

where we have used the expression for the generating function of the Bernoulli
numbers

$\frac{w}{e^{w}-1}=\sum_{n=0}^{\infty}B_{n}\frac{w^{n}}{n!}\,,$ (3.19)

and the fact that except $B_{1}=-\frac{1}{2}$, all odd Bernoulli numbers
vanish. From the final expression we see that $f_{1}(\xi)$ admits an analytic
continuation to a meromorphic function with double poles at
$\xi=2\pi{\mathrm{i}}\mathbb{Z}\setminus\\{0\\}$.

On the other hand, for $f_{2}(\xi,t)$, we have

$\displaystyle f_{2}(\xi,t)$
$\displaystyle=\sum_{g=2}^{\infty}\frac{\xi^{2g-3}}{(2g-3)!}\,\partial_{t}^{2g}\textrm{Li}_{3}(Q)=\partial_{\xi}^{3}\,\left(\sum_{g=1}^{\infty}\frac{\partial_{t}^{2g}\textrm{Li}_{3}(Q)}{(2g)!}\xi^{2g}\right)$
$\displaystyle=\partial_{\xi}^{3}\left(\frac{1}{2}(\textrm{Li}_{3}(e^{2\pi{\mathrm{i}}(t+\xi)})+\textrm{Li}_{3}(e^{2\pi{\mathrm{i}}(t-\xi)}))-\textrm{Li}_{3}(e^{2\pi{\mathrm{i}}t})\right)$
$\displaystyle=\frac{(2\pi{\mathrm{i}})^{3}}{2}\left(\mathrm{Li}_{0}\big{(}e^{2\pi{\mathrm{i}}(t+\xi)}\big{)}-\mathrm{Li}_{0}\big{(}e^{2\pi{\mathrm{i}}(t-\xi)}\big{)}\right)$
$\displaystyle=\frac{(2\pi{\mathrm{i}})^{3}}{2}\bigg{(}\frac{e^{2\pi{\mathrm{i}}(t+\xi)}}{1-e^{2\pi{\mathrm{i}}(t+\xi)}}-\frac{e^{2\pi{\mathrm{i}}(t-\xi)}}{1-e^{2\pi{\mathrm{i}}(t-\xi)}}\bigg{)}\,,$
(3.20)

so that $f_{2}(\xi,t)$ admits an analytic continuation in $\xi$ with simple
poles at $\pm t+\mathbb{Z}$. The integral representation then becomes

$G(\xi,t)=\frac{1}{2}\int_{|s|=\rho}\frac{\mathrm{d}s}{s}\left(\frac{1}{s^{3}}-\frac{e^{s}}{s(e^{s}-1)^{2}}-\frac{1}{12s}\right)\bigg{(}\frac{e^{2\pi{\mathrm{i}}(t+\xi/s)}}{1-e^{2\pi{\mathrm{i}}(t+\xi/s)}}-\frac{e^{2\pi{\mathrm{i}}(t-\xi/s)}}{1-e^{2\pi{\mathrm{i}}(t-\xi/s)}}\bigg{)}\,.$
(3.21)

Notice that $f_{2}(\xi/s,t)$ as a function of $s$ has simple poles at
$s=\pm\xi/(t+k)$ for all $k\in\mathbb{Z}$. By our assumption that
$|\xi|<\rho|t|$ and $|\mathrm{Re}(t)|<\frac{1}{2}$, we have

$\Big{|}\frac{\pm\xi}{t+k}\Big{|}<\rho\frac{|t|}{|t+k|}\leq\rho\;,$ (3.22)

so that all the poles of $f_{2}(\xi/s,t)$ lie inside the contour. Furthermore,
since $\rho<2\pi$, all the poles of $f_{1}(s)$ lie outside the contour.

If, for each $k\in\mathbb{Z}$ with $k>1$, we denote as $\gamma_{k}$ the
contour given by $|s|=\pi(2k+1)$, then between $|s|=\rho$ and $\gamma_{k}$, we
have the poles at $\pm 2\pi{\mathrm{i}}n$ for $n=1,...,k$ due to $f_{1}(s)$.
We can therefore write the following for any $k>1$:

$\displaystyle G(\xi,t)$
$\displaystyle=\frac{1}{2}\int_{|s|=\rho}\frac{\mathrm{d}s}{s}\left(\frac{1}{s^{3}}-\frac{e^{s}}{s(e^{s}-1)^{2}}-\frac{1}{12s}\right)\bigg{(}\frac{e^{2\pi{\mathrm{i}}(t+\xi/s)}}{1-e^{2\pi{\mathrm{i}}(t+\xi/s)}}-\frac{e^{2\pi{\mathrm{i}}(t-\xi/s)}}{1-e^{2\pi{\mathrm{i}}(t-\xi/s)}}\bigg{)}$
(3.23) $\displaystyle=2\pi{\mathrm{i}}\sum_{m\in\mathbb{Z}:-k<m<k,m\neq
0}\frac{1}{2}\left.\frac{\mathrm{d}}{\mathrm{d}s}\left(\frac{e^{s}(s-2\pi{\mathrm{i}}m)^{2}}{(e^{s}-1)^{2}s^{2}}\bigg{(}\frac{e^{2\pi{\mathrm{i}}(t+\xi/s)}}{1-e^{2\pi{\mathrm{i}}(t+\xi/s)}}-\frac{e^{2\pi{\mathrm{i}}(t-\xi/s)}}{1-e^{2\pi{\mathrm{i}}(t-\xi/s)}}\bigg{)}\right)\right|_{s=2\pi{\mathrm{i}}m}$
$\displaystyle\;\;\;+\frac{1}{2}\int_{\gamma_{k}}\frac{\mathrm{d}s}{s}\left(\frac{1}{s^{3}}-\frac{e^{s}}{s(e^{s}-1)^{2}}-\frac{1}{12s}\right)\bigg{(}\frac{e^{2\pi{\mathrm{i}}(t+\xi/s)}}{1-e^{2\pi
i(t+\xi/s)}}-\frac{e^{2\pi{\mathrm{i}}(t-\xi/s)}}{1-e^{2\pi{\mathrm{i}}(t-\xi/s)}}\bigg{)}\,,$

where the terms in the sum come from the contribution of the (clockwise)
contours around the poles between the two contours.

One can check that $f_{2}(\xi/s,t)=\mathcal{O}(1/s)$ as $s\to\infty$, while
$f_{1}|_{\gamma_{k}}=\mathcal{O}(1/k)$ as $k\to\infty$. Hence, taking the
limit $k\to\infty$ in (3.23) we obtain the following expression:

$\displaystyle G(\xi,t)$
$\displaystyle=2\pi{\mathrm{i}}\sum_{m\in\mathbb{Z}-\\{0\\}}\frac{1}{2}\left.\frac{\mathrm{d}}{\mathrm{d}s}\left(\frac{e^{s}(s-2\pi{\mathrm{i}}m)^{2}}{(e^{s}-1)^{2}s^{2}}\bigg{(}\frac{e^{2\pi{\mathrm{i}}(t+\xi/s)}}{1-e^{2\pi{\mathrm{i}}(t+\xi/s)}}-\frac{e^{2\pi{\mathrm{i}}(t-\xi/s)}}{1-e^{2\pi{\mathrm{i}}(t-\xi/s)}}\bigg{)}\right)\right|_{s=2\pi{\mathrm{i}}m}$
$\displaystyle=-\sum_{m\in\mathbb{Z}\setminus\\{0\\}}\frac{1}{(2\pi{\mathrm{i}})^{2}}\left(\frac{1}{m^{3}}\left(\frac{e^{2\pi{\mathrm{i}}t+\xi/m}}{1-e^{2\pi{\mathrm{i}}t+\xi/m}}-\frac{e^{2\pi{\mathrm{i}}t-\xi/m}}{1-e^{2\pi{\mathrm{i}}t-\xi/m}}\right)\right.$
$\displaystyle\qquad\qquad\qquad+\left.\frac{\xi}{2m^{4}}\left(\frac{e^{2\pi{\mathrm{i}}t+\xi/m}}{(1-e^{2\pi{\mathrm{i}}t+\xi/m})^{2}}+\frac{e^{2\pi{\mathrm{i}}t-\xi/m}}{(1-e^{2\pi{\mathrm{i}}t-\xi/m})^{2}}\right)\right).$
(3.24)

The result then follows by a simple rewriting of the summands. ∎

A direct check that $G(\xi,t)$ is the Borel transform of
$\widetilde{F}(\lambda,t)$ can be found in Lemma B.1 from Appendix B.

####  3.1.1 Previous expression for the Borel transform

In the following we review an expression for the Borel transform of the
topological string free energy for the resolved conifold obtained in [PS10],
starting again with

$\begin{split}\widetilde{F}(\lambda,t)&=\sum_{g=0}^{\infty}\lambda^{2g-2}\widetilde{F}^{g}(t)=\frac{1}{\lambda^{2}}\textrm{Li}_{3}(q)+\frac{B_{2}}{2}\mathrm{Li}_{1}(q)+\sum_{g=2}^{\infty}\lambda^{2g-2}\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\textrm{Li}_{3-2g}(q)\,\\\
&=\frac{1}{\lambda^{2}}\textrm{Li}_{3}(q)+\frac{B_{2}}{2}\mathrm{Li}_{1}(q)+\Phi(\check{\lambda},t)\,,\quad\check{\lambda}=\frac{\lambda}{2\pi}\,.\end{split}$
(3.25)

Using the series representation of the polylogarithm

$\textrm{Li}_{s}(e^{2\pi it})=\Gamma(1-s)\,\sum_{k\in\mathbb{Z}}(2\pi
i)^{s-1}(k-t)^{s-1}\,,$ (3.26)

valid for $\textrm{Re}(s)<0$ and $t\notin\mathbb{Z}$, we can write

$\begin{split}\Phi(\check{\lambda},t)&=\sum_{g=2}^{\infty}\lambda^{2g-2}\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\,\textrm{Li}_{3-2g}(q)=\sum_{g=2}^{\infty}\check{\lambda}^{2g-2}\frac{B_{2g}}{2g(2g-2)}\sum_{k\in\mathbb{Z}}(k-t)^{2-2g}.\end{split}$
(3.27)

Taking the Borel transform of $\Phi(\check{\lambda},t)$, we find

$\begin{split}G(\xi,t)&=\sum_{g=2}^{\infty}\frac{B_{2g}}{2g\,(2g-2)!}\xi^{2g-3}\sum_{k\in\mathbb{Z}}(k-t)^{2-2g}=\sum_{k\in\mathbb{Z}}\frac{1}{\xi}\left(\frac{(k-t)^{2}}{\xi^{2}}-\frac{e^{\xi/(k-t)}}{(e^{\xi/(k-t)}-1)^{2}}-\frac{1}{12}\right),\\\
\end{split}$ (3.28)

where the second equality follows from

$\frac{e^{w}}{(e^{w}-1)^{2}}=\frac{1}{w^{2}}-\frac{1}{12}-\sum_{g=2}^{\infty}\frac{B_{2g}}{2g(2g-2)!}w^{2g-2}\,,$
(3.29)

which can be obtained by taking a derivative of the generating function of
Bernoulli numbers (3.19) and rearranging the outcome. We note here that this
expression for the Borel transform, which was previously obtained in [PS10]
has the same set of poles at

$\xi=2\pi{\mathrm{i}}m(t+k)\,,\quad m\in\mathbb{Z}\setminus\\{0\\}\,,\quad
k\in\mathbb{Z}\,,$

as the one we have obtained in Theorem 2.1. We will later show that it also
has the same Stokes jumps.

One advantage of the expression (3.16) compared to (3.28) is that the first
gives a well-defined expression for $t\in\mathbb{Z}$. As we will see below,
this will allow us to express the Borel transform of constant map contribution
of (2.4) in terms of $G(\xi,0)$ (see Corollary 3.13).

######  Remark 3.5.

It seems one might be able to obtain (3.28) from the integral representation
(3.21) by deforming the contour to $0$ instead of $\infty$, and summing over
the residues of the poles inside the contour. The only technical issue is that
it seems harder to show that the contour limiting to $s=0$ limits to a zero
contribution.

###  3.2 The Borel sum along $\mathbb{R}_{>0}$

We now prove the second point of Theorem 2.1. Hence, we wish to study the
Borel sum of $\widetilde{F}(\lambda,t)$ along $\mathbb{R}_{>0}$. More
generally, we define the following:

######  Definition 3.6.

Given $t\in\mathbb{C}$ with $\text{Im}(t)\neq 0$, and a ray
$\rho\subset\mathbb{C}^{\times}$ from $0$ to $\infty$ avoiding the poles
$2\pi{\mathrm{i}}(t+k)m$ of $G(\xi,t)$ and which is different from
$\pm{\mathrm{i}}\mathbb{R}_{>0}$, we define

$F_{\rho}(\lambda,t):=\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(Q)+\frac{B_{2}}{2}\mathrm{Li}_{1}(Q)+\int_{\rho}\mathrm{d}\xi\,e^{-\xi/\check{\lambda}}G(\xi,t)\,,\quad
Q=e^{2\pi{\mathrm{i}}t}\,,$ (3.30)

for $\lambda$ in the half-plane $\mathbb{H}_{\rho}$ centered at $\rho$. We
call $F_{\rho}(\lambda,t)$ the Borel sum of $\widetilde{F}(\lambda,t)$ along
$\rho$.

The integral appearing in (2.14) corresponds to an integral representation of
a certain function $G_{3}(z\,|\,\omega_{1},\omega_{2})$ related to the triple
sine function. Hence, before studying the Borel sum along $\mathbb{R}_{>0}$,
we recall how this function is defined and some of its properties. For a
convenient review of the special functions appearing below and their
properties, see for example [Bri20, Section 4] and the references cited
therein.

######  Definition 3.7.

For $z\in\mathbb{C}$ and $\omega_{1},\omega_{2}\in\mathbb{C}^{\times}$, we
define

$G_{3}(z\,|\,\omega_{1},\omega_{2}):=\exp\Big{(}\frac{\pi{\mathrm{i}}}{6}B_{3,3}(z+\omega_{1}\,|\,\omega_{1},\omega_{1},\omega_{2})\Big{)}\cdot\sin_{3}(z+\omega_{1}\,|\,\omega_{1},\omega_{1},\omega_{2})\,,$
(3.31)

where $\sin_{3}(z\,|\,\omega_{1},\omega_{2},\omega_{3})$ denotes the triple
sine function, and $B_{3,3}(z\,|\,\omega_{1},\omega_{2},\omega_{3})$ is the
multiple Bernoulli polynomial.

What will be most important for us are the following properties:

######  Proposition 3.8.

[Bri20, Prop. 4.2][Nar04, Prop. 2] $G_{3}(z\,|\,\omega_{1},\omega_{2})$ is a
single-valued meromorphic function under the assumption
$\omega_{1}/\omega_{2}\not\in\mathbb{R}_{<0}$. Furthermore, we have

  * •

It is regular everywhere, and vanishes only at the points

$z=a\omega_{1}+b\omega_{2},\quad a,b\in\mathbb{Z}\,,$ (3.32)

with $a<0$ and $b\leq 0$, or $a>0$ and $b>0$.

  * •

Let $\mathrm{Re}(\omega_{i})>0$ and
$-\mathrm{Re}(\omega_{1})<\mathrm{Re}(z)<\mathrm{Re}(\omega_{1}+\omega_{2})$.
Then

$G_{3}(z\,|\,\omega_{1},\omega_{2})=\exp\bigg{(}-\int_{\mathbb{R}+\mathrm{i}0^{+}}\frac{\mathrm{d}u}{8u}\frac{e^{u(z-\omega_{2}/2)}}{\sinh(\omega_{2}u/2)(\sinh(\omega_{1}u/2))^{2}}\;\bigg{)}.$
(3.33)

######  Definition 3.9.

For $\mathrm{Re}\,\check{\lambda}>0$ and
$-\mathrm{Re}(\check{\lambda})<\mathrm{Re}(t)<\mathrm{Re}(\check{\lambda}+1)$,
we define

$F_{\mathrm{np}}(\lambda,t):=\log
G_{3}(t\,|\,\check{\lambda},1)=-\int_{\mathbb{R}+\mathrm{i}0^{+}}\frac{\mathrm{d}u}{8u}\frac{e^{u(t-1/2)}}{\sinh(u/2)(\sinh(\check{\lambda}u/2))^{2}}\;,$
(3.34)

We now wish to relate $F_{\mathbb{R}_{>0}}(\lambda,t)$ to
$F_{\mathrm{np}}(\lambda,t)$. For this, we will need the following lemma,
giving a “Woronowicz form” for $F_{\mathrm{np}}$. We remark that a similar
form for the triple-sine function was conjectured in [AP21, Equation B.17].

######  Lemma 3.10.

Let $t\in\mathbb{C}$ be such that $0<\mathrm{Re}(t)<1$, $\mathrm{Im}(t)>0$,
and let $\lambda$ be in the sector determined by $l_{0}=\mathbb{R}_{<0}\cdot
2\pi{\mathrm{i}}t$ and $l_{-1}=\mathbb{R}_{<0}\cdot 2\pi{\mathrm{i}}(t-1)$.
Furthermore, assume that $\mathrm{Re}(t)<\mathrm{Re}(\check{\lambda}+1)$. Then
$F_{\mathrm{np}}(\lambda,t)$ admits the following Woronowicz form:

$F_{\mathrm{np}}(\lambda,t)=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}+{\mathrm{i}}0^{+}}\mathrm{d}v\frac{v}{1-e^{v}}\log(1-e^{\check{\lambda}v+2\pi{\mathrm{i}}t})\,.$
(3.35)

###### Proof.

We will follow the method of [GK20], based on the unitarity of the Fourier
transform:

$\langle f,g\rangle=\langle Ff,Fg\rangle,\qquad\langle
f,g\rangle=\int_{\mathbb{R}}\mathrm{d}x\;f(x)\overline{g(x)},\qquad(F\psi)(x)=\int_{\mathbb{R}}\mathrm{d}y\;e^{2\pi\mathrm{i}\,xy}\psi(y).$

We start by defining for sufficiently small $\epsilon>0$,

$f_{\epsilon}(x):=e^{-\epsilon
x}\log\big{(}1-e^{\check{\lambda}x+2\pi\mathrm{i}\,t}\big{)},\;\;\;\;g_{\epsilon}(x):=e^{+\epsilon
x}\frac{1}{1-e^{x+\mathrm{i}\epsilon}},\;\;\;\;G_{\epsilon}(x):=e^{+\epsilon
x}\frac{x}{1-e^{x+\mathrm{i}\epsilon}}.$ (3.36)

We then easily see that

$\lim_{\epsilon\to 0^{+}}\frac{1}{(2\pi)^{2}}\langle
f_{\epsilon},G_{\epsilon}\rangle=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}+{\mathrm{i}}0^{+}}dv\frac{v}{1-e^{v}}\log(1-e^{\check{\lambda}v+2\pi{\mathrm{i}}t})\;.$
(3.37)

We now compute the Fourier transform of $f_{\epsilon}$, $g_{\epsilon}$, and
$G_{\epsilon}$. Setting $\zeta=2\pi x+{\mathrm{i}}\epsilon$, we find that

$Ff_{\epsilon}(x)=\int_{\mathbb{R}}\mathrm{d}y\;e^{{\mathrm{i}}y\zeta}\log(1-e^{\check{\lambda}y+2\pi{\mathrm{i}}t})=\frac{{\mathrm{i}}\check{\lambda}}{\zeta}\int_{\mathbb{R}}\mathrm{d}y\;\frac{e^{{\mathrm{i}}y\zeta}}{1-e^{-\check{\lambda}y-2\pi{\mathrm{i}}t}}\;,$
(3.38)

where we have integrated by parts, and used that the boundary terms vanish.
The last integral has simple poles at
$y=2\pi{\mathrm{i}}(k-t)/\check{\lambda}$, and under our assumptions for the
parameters $t$ and $\lambda$, it is easy to check that the poles on the upper
half-plane correspond to $k>0$, while those in the lower half-plane correspond
to $k\leq 0$. If $\text{Re}(x)>0$, by an application of Jordan’s lemma and the
residue theorem, we can compute $Ff_{\epsilon}(x)$ by summing up the residues
in the upper half-plane, obtaining

$\begin{split}\frac{{\mathrm{i}}\check{\lambda}}{\zeta}\int_{\mathbb{R}}\mathrm{d}y\;\frac{e^{{\mathrm{i}}y\zeta}}{1-e^{-\check{\lambda}y-2\pi{\mathrm{i}}t}}&=2\pi{\mathrm{i}}\frac{{\mathrm{i}}\check{\lambda}}{\zeta}\sum_{k=1}^{\infty}\frac{e^{{\mathrm{i}}y\zeta}}{\check{\lambda}}\Big{|}_{y=2\pi{\mathrm{i}}(k-t)/\check{\lambda}}=-\frac{2\pi}{\zeta}e^{2\pi
t\zeta/\check{\lambda}}\sum_{k=1}^{\infty}e^{-2\pi k\zeta/\check{\lambda}}\\\
&=-\frac{\pi}{\zeta}e^{\pi\zeta(2t-1)/\check{\lambda}}\Big{(}2e^{-\pi\zeta/\check{\lambda}}\sum_{k=0}^{\infty}e^{-2\pi\zeta
k/\check{\lambda}}\Big{)}=-\frac{\pi}{\zeta}\frac{e^{\pi\zeta(2t-1)/\check{\lambda}}}{\sinh(\pi\zeta/\check{\lambda})}\;,\end{split}$
(3.39)

where in the last equality we have used the Dirichlet series representation of
$1/\sinh(z)$. Similarly, if $\text{Re}(x)<0$, we can compute $Ff_{\epsilon}$
summing up the residues in the lower half-plane, obtaining

$\displaystyle\frac{{\mathrm{i}}\check{\lambda}}{\zeta}\int_{\mathbb{R}}\mathrm{d}y\;\frac{e^{{\mathrm{i}}y\zeta}}{1-e^{-\check{\lambda}y-2\pi{\mathrm{i}}t}}$
$\displaystyle=-2\pi{\mathrm{i}}\frac{{\mathrm{i}}\check{\lambda}}{\zeta}\sum_{k=0}^{-\infty}\frac{e^{{\mathrm{i}}y\zeta}}{\check{\lambda}}\Big{|}_{y=2\pi{\mathrm{i}}(k-t)/\check{\lambda}}=\frac{2\pi}{\zeta}e^{2\pi
t\zeta/\check{\lambda}}\sum_{k=0}^{\infty}e^{2\pi k\zeta/\check{\lambda}}$
$\displaystyle=-\frac{\pi}{\zeta}e^{\pi\zeta(2t-1)/\check{\lambda}}\Big{(}-2e^{\pi\zeta/\check{\lambda}}\sum_{k=0}^{\infty}e^{2\pi\zeta
k/\check{\lambda}}\Big{)}=-\frac{\pi}{\zeta}\frac{e^{\pi\zeta(2t-1)/\check{\lambda}}}{\sinh(\pi\zeta/\check{\lambda})}\,,$
(3.40)

so that $Ff_{\epsilon}(x)$ exists for $x\neq 0$ and

$Ff_{\epsilon}(x)=-\frac{\pi}{\zeta}\frac{e^{\pi\zeta(2t-1)/\check{\lambda}}}{\sinh(\pi\zeta/\check{\lambda})}\;.$
(3.41)

The computation of $Fg_{\epsilon}(x)$ is simpler, and follows similar lines.
One obtains that for $x\neq 0$,

$\overline{Fg_{\epsilon}(x)}=\pi{\mathrm{i}}\frac{e^{(\epsilon-\pi)\zeta}}{\sinh(\pi\zeta)}\,.$
(3.42)

On the other hand, since $G_{\epsilon}(x)=xg_{\epsilon}(x)$, we find

$\overline{FG_{\epsilon}(x)}=-\frac{1}{2\pi{\mathrm{i}}}\frac{\partial}{\partial
x}\overline{Fg_{\epsilon}(x)}=\frac{\partial}{\partial\zeta}\Big{(}\frac{2\pi
e^{\epsilon\zeta}}{1-e^{2\pi\zeta}}\Big{)}\,.$ (3.43)

We then have that

$\displaystyle\lim_{\epsilon\to 0^{+}}\frac{1}{(2\pi)^{2}}\langle
f_{\epsilon},G_{\epsilon}\rangle$ $\displaystyle=\lim_{\epsilon\to
0^{+}}\frac{1}{(2\pi)^{2}}\langle Ff_{\epsilon},FG_{\epsilon}\rangle$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}+{\mathrm{i}}0^{+}}\mathrm{d}x\bigg{(}-\frac{\pi}{2\pi
x}\frac{e^{\pi(2\pi
x)(2t-1)/\check{\lambda}}}{\sinh(2\pi^{2}x/\check{\lambda})}\bigg{)}\bigg{(}\frac{(2\pi)^{2}}{4(\sinh(2\pi^{2}x))^{2}}\bigg{)}$
$\displaystyle=-\int_{\check{\lambda}^{-1}\cdot(\mathbb{R}+{\mathrm{i}}0^{+})}\frac{\mathrm{d}v}{8v}\frac{e^{v(t-1/2)}}{\sinh(v/2)(\sinh(\check{\lambda}v/2))^{2}}$
$\displaystyle=-\int_{\mathbb{R}+{\mathrm{i}}0^{+}}\frac{\mathrm{d}v}{8v}\frac{e^{v(t-1/2)}}{\sinh(v/2)(\sinh(\check{\lambda}v/2))^{2}}=F_{\mathrm{np}}(\lambda,t)\,,$
(3.44)

where we used the fact that the range of the parameter $\lambda$ allows us to
deform the contour back to $\mathbb{R}+i0^{+}$. The result then follows. ∎

We are now ready to prove the second point of Theorem 2.1. Another proof is
given in Appendix A.

######  Proposition 3.11.

Under the same assumptions as in Lemma 3.10, we have

$F_{\mathrm{np}}(\lambda,t)=\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(Q)+\frac{B_{2}}{2}\mathrm{Li}_{1}(Q)+\int_{0}^{\infty}\mathrm{d}\xi\,e^{-\xi/\check{\lambda}}G(\xi,t)\,,$
(3.45)

where $G(\xi,t)$ is the Borel transform (3.16) obtained in the previous
section, $Q=e^{2\pi{\mathrm{i}}t}$, and $\check{\lambda}=\lambda/2\pi$.

###### Proof.

We start by performing the change of variables $y=\lambda v/2\pi$ on (3.35),
obtaining

$\begin{split}F_{\mathrm{np}}(\lambda,t)&=\frac{1}{\lambda^{2}}\int_{\lambda(\mathbb{R}+{\mathrm{i}}0^{+})}\mathrm{d}y\;\;\frac{y}{1-e^{2\pi
y/\lambda}}\log(1-e^{y+2\pi{\mathrm{i}}t})\\\
&=\frac{1}{\lambda^{2}}\int_{\mathbb{R}+{\mathrm{i}}0^{+}}\mathrm{d}y\;\;\frac{y}{1-e^{2\pi
y/\lambda}}\log(1-e^{y+2\pi{\mathrm{i}}t})\\\ &=\lim_{\epsilon\to
0^{+}}\frac{1}{\lambda^{2}}\int_{\mathbb{R}}\mathrm{d}y\,\frac{y}{1-e^{2\pi
y/\lambda-i\epsilon}}\log(1-e^{y+2\pi{\mathrm{i}}t})\,,\end{split}$ (3.46)

where in the second equality we have used that the range of $\lambda$ allows
us to deform the contour back to $\mathbb{R}+{\mathrm{i}}0^{+}$. Now using

$\frac{\mathrm{d}}{\mathrm{d}y}(-\log(1-e^{-2\pi
y/\lambda+{\mathrm{i}}\epsilon}))=\frac{2\pi}{\lambda(1-e^{2\pi
y/\lambda-{\mathrm{i}}\epsilon})}$ (3.47)

and integration by parts, we find

$\begin{split}F_{\mathrm{np}}(\lambda,t)=\lim_{\epsilon\to
0}\Big{[}&-\frac{y}{2\pi\lambda}\log(1-e^{-2\pi
y/\lambda+i\epsilon})\log(1-e^{y+2\pi it})\Big{|}_{y=-\infty}^{\infty}\\\
&+\frac{1}{2\pi\lambda}\int_{\mathbb{R}}\mathrm{d}y\;\;\log(1-e^{-2\pi
y/\lambda+i\epsilon})\Big{(}\log(1-e^{y+2\pi it})+\frac{y}{1-e^{-y-2\pi
it}}\Big{)}\Big{]}\,.\end{split}$ (3.48)

Because $\text{Re}(\lambda)>0$, we obtain that the boundary terms vanish.
Furthermore, splitting the integration over the left and right half intervals,
one then obtains

$\begin{split}F_{\mathrm{np}}(\lambda,t)=\lim_{\epsilon\to
0}&\Big{[}\frac{1}{2\pi\lambda}\int_{0}^{\infty}\mathrm{d}y\;\;\log(1-e^{-2\pi
y/\lambda+{\mathrm{i}}\epsilon})\Big{(}\log(1-e^{y+2\pi{\mathrm{i}}t})+\frac{y}{1-e^{-y-2\pi{\mathrm{i}}t}}\Big{)}\\\
&+\frac{1}{2\pi\lambda}\int_{0}^{\infty}\mathrm{d}y\;\;\log(1-e^{2\pi
y/\lambda+{\mathrm{i}}\epsilon})\Big{(}\log(1-e^{-y+2\pi{\mathrm{i}}t})-\frac{y}{1-e^{y-2\pi{\mathrm{i}}t}}\Big{)}\Big{]}\\\
=\widetilde{H}(&\lambda,t)+\lim_{\epsilon\to
0}H(\lambda,t,\epsilon)\;,\end{split}$ (3.49)

where we have defined

$\displaystyle\widetilde{H}(\lambda,t):=\frac{1}{2\pi\lambda^{2}}\int_{0}^{\infty}\mathrm{d}y\;\;(2\pi
y+\pi{\mathrm{i}}\lambda)$
$\displaystyle\Big{(}\log(1-e^{-y+2\pi{\mathrm{i}}t})-\frac{y}{1-e^{y-2\pi
it}}\Big{)}$ $\displaystyle
H(\lambda,t,\epsilon):=\frac{1}{2\pi\lambda}\int_{0}^{\infty}\mathrm{d}y\;\;\log(1-e^{-2\pi
y/\lambda+{\mathrm{i}}\epsilon})\Big{(}$
$\displaystyle\log(1-e^{y+2\pi{\mathrm{i}}t})+\log(1-e^{-y+2\pi{\mathrm{i}}t})$
$\displaystyle+\frac{y}{1-e^{-y-2\pi{\mathrm{i}}t}}-\frac{y}{1-e^{y-2\pi{\mathrm{i}}t}}\Big{)}\;.$
(3.50)

One can compute $\widetilde{H}(\lambda,t)$ explicitly by performing an
integration by parts to get rid of the log term:

$\begin{split}\widetilde{H}(\lambda,t)&=\frac{1}{2\pi\lambda^{2}}\bigg{(}(\pi
y^{2}+\pi{\mathrm{i}}\lambda
y)\log(1-e^{-y+2\pi{\mathrm{i}}t})\Big{|}_{y=0}^{\infty}-\int_{0}^{\infty}\mathrm{d}y\;\;(\pi
y^{2}+\pi{\mathrm{i}}\lambda y)\frac{-1}{1-e^{y-2\pi{\mathrm{i}}t}}\bigg{)}\\\
&\;\;\;\;-\frac{1}{2\pi\lambda^{2}}\int_{0}^{\infty}\mathrm{d}y\;\;(2\pi
y+\pi{\mathrm{i}}\lambda)\frac{y}{1-e^{y-2\pi{\mathrm{i}}t}}\\\
&=\frac{1}{2\lambda^{2}}\int_{0}^{\infty}\mathrm{d}y\frac{y^{2}}{e^{y-2\pi{\mathrm{i}}t}-1}\;.\end{split}$
(3.51)

Since $\text{Im}(t)>0$, we find that $|e^{2\pi{\mathrm{i}}t}|<1$, so that the
last integral in (3.51) corresponds to an integral representation of
$\mathrm{Li}_{3}(e^{2\pi{\mathrm{i}}t})/\lambda^{2}$. Hence, we conclude that

$\widetilde{H}(\lambda,t)=\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(e^{2\pi{\mathrm{i}}t})\;.$
(3.52)

On the other hand, by expanding the first log term of $H$ and applying the
Fubini–Tonelli theorem, we find that

$\begin{split}H(\lambda,t,\epsilon)=-\sum_{n=1}^{\infty}\frac{1}{2\pi\lambda}\int_{0}^{\infty}\mathrm{d}y\;\;\frac{{\mathrm{e}}^{-2\pi
ny/\lambda+{\mathrm{i}}n\epsilon}}{n}\Big{(}&\log(1-e^{y+2\pi{\mathrm{i}}t})+\log(1-e^{-y+2\pi{\mathrm{i}}t})\\\
&+\frac{y}{1-e^{-y-2\pi{\mathrm{i}}t}}-\frac{y}{1-e^{y-2\pi{\mathrm{i}}t}}\Big{)}\;.\end{split}$
(3.53)

Performing a change of variables in each integral, and interchanging integral
and summations again, we obtain

$\begin{split}H(\lambda,t,\epsilon)=-\frac{1}{2\pi\lambda}\int_{0}^{\infty}\mathrm{d}y\;\;{\mathrm{e}}^{-2\pi
y/\lambda}\sum_{n=1}^{\infty}\frac{e^{in\epsilon}}{n^{2}}\Big{(}&\log(1-e^{y/n+2\pi{\mathrm{i}}t})+\log(1-e^{-y/n+2\pi{\mathrm{i}}t})\\\
&+\frac{y/n}{1-e^{-y/n-2\pi{\mathrm{i}}t}}-\frac{y/n}{1-e^{y/n-2\pi{\mathrm{i}}t}}\Big{)}\\\
\end{split}$ (3.54)

Letting $H(\lambda,t):=\lim_{\epsilon\to 0}H(\lambda,t,\epsilon)$, we get

$\begin{split}H(\lambda,t)=-\frac{1}{2\pi\lambda}\int_{0}^{\infty}\mathrm{d}y\;\;{\mathrm{e}}^{-2\pi
y/\lambda}\sum_{n=1}^{\infty}\frac{1}{n^{2}}\Big{(}&\log(1-e^{y/n+2\pi{\mathrm{i}}t})+\log(1-e^{-y/n+2\pi{\mathrm{i}}t})\\\
&+\frac{y/n}{1-e^{-y/n-2\pi{\mathrm{i}}t}}-\frac{y/n}{1-e^{y/n-2\pi{\mathrm{i}}t}}\Big{)}\;.\end{split}$
(3.55)

Finally, using that $-\frac{2\pi}{\lambda}e^{-2\pi
y/\lambda}=\frac{\mathrm{d}}{\mathrm{d}y}e^{-2\pi y/\lambda}$ and integrating
by parts yields

$\displaystyle H(\lambda,t)=\Bigg{[}\frac{e^{-2\pi
y/\lambda}}{(2\pi)^{2}}\sum_{n=1}^{\infty}\frac{1}{n^{2}}\Big{(}\log(1-e^{y/n+2\pi{\mathrm{i}}t})+\log(1-e^{-y/n+2\pi{\mathrm{i}}t})$
$\displaystyle\hskip
156.49014pt+\frac{y/n}{1-e^{-y/n-2\pi{\mathrm{i}}t}}-\frac{y/n}{1-e^{y/n-2\pi{\mathrm{i}}t}}\Big{)}\Bigg{]}\Bigg{|}_{y=0}^{\infty}$
$\displaystyle\hskip 42.67912pt-\int_{0}^{\infty}dy\;\;\frac{e^{-2\pi
y/\lambda}}{(2\pi)^{2}}\frac{\mathrm{d}}{\mathrm{d}y}\Bigg{[}\sum_{n=1}^{\infty}\frac{1}{n^{2}}\Big{(}\log(1-e^{y/n+2\pi{\mathrm{i}}t})+\log(1-e^{-y/n+2\pi{\mathrm{i}}t})$
$\displaystyle\hskip
156.49014pt+\frac{y/n}{1-e^{-y/n-2\pi{\mathrm{i}}t}}-\frac{y/n}{1-e^{y/n-2\pi{\mathrm{i}}t}}\Big{)}\Bigg{]}\,.$
(3.56)

Using that the boundary term at $\infty$ vanishes, and interchanging the
derivative with the sum, we obtain

$\begin{split}H(\lambda,t)&=-\frac{2}{(2\pi)^{2}}\log(1-Q)\sum_{n=1}^{\infty}\frac{1}{n^{2}}+\int_{0}^{\infty}\mathrm{d}y\;\;{\mathrm{e}}^{-2\pi
y/\lambda}G(y,t)\\\
&=\frac{1}{2}\mathrm{Li}_{1}(Q)B_{2}+\int_{0}^{\infty}\mathrm{d}y\;\;{\mathrm{e}}^{-2\pi
y/\lambda}G(y,t)\;,\end{split}$ (3.57)

where we used that $\sum_{n=1}^{\infty}\frac{1}{n^{2}}=\pi^{2}B_{2}$,
$\mathrm{Li}_{1}(Q)=-\log(1-Q)$.

Hence, putting (3.49), (3.52) and (3.57) together gives us (3.45). ∎

We finish this section with the following corollaries:

######  Corollary 3.12.

Let $l_{k}=\mathbb{R}_{<0}\cdot 2\pi{\mathrm{i}}(t+k)$, and let $\rho_{k}$ be
a ray between $l_{k}$ and $l_{k-1}$. Then the following holds for
$n\in\mathbb{Z}$:

$F_{\rho_{k-n}}(\lambda,t+n)=F_{\rho_{k}}(\lambda,t)\,.$ (3.58)

In particular, if $0<\mathrm{Re}(t)<1$ and $\mathrm{Im}(t)>0$, we have on
their common domains of definition

$F_{\rho_{-n}}(\lambda,t+n)=\log(G_{3}(t\,|\,\check{\lambda},1))$ (3.59)

###### Proof.

Note that the labels $l_{k}$ (and hence also $\rho_{k}$) depend on $t$. In the
following, we denote $l_{k}(t)$ and $\rho_{k}(t)$ to emphasize the $t$
dependence. In particular, we have the relations $l_{k}(t+n)=l_{k+n}(t)$ and
$\rho_{k}(t+n)=\rho_{k+n}(t)$ for $n\in\mathbb{Z}$.

Using the fact that $G(\xi,t)=G(\xi,t+n)$ for any $n\in\mathbb{Z}$, we thus
obtain

$\displaystyle F_{\rho_{k-n}}(\lambda,t+n)$
$\displaystyle=\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(e^{2\pi{\mathrm{i}}(t+n)})+\frac{B_{2}}{2}\mathrm{Li}_{1}(e^{2\pi{\mathrm{i}}(t+n)})+\int_{\rho_{k-n}(t+n)}\mathrm{d}\xi\;e^{-\xi/\check{\lambda}}G(\xi,t+n)$
$\displaystyle=\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(e^{2\pi{\mathrm{i}}t})+\frac{B_{2}}{2}\mathrm{Li}_{1}(e^{2\pi{\mathrm{i}}t})+\int_{\rho_{k-n}(t+n)}\mathrm{d}\xi\;e^{-\xi/\check{\lambda}}G(\xi,t)$
$\displaystyle=\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(e^{2\pi{\mathrm{i}}t})+\frac{B_{2}}{2}\mathrm{Li}_{1}(e^{2\pi{\mathrm{i}}t})+\int_{\rho_{k}(t)}\mathrm{d}\xi\;e^{-\xi/\check{\lambda}}G(\xi,t)$
$\displaystyle=F_{\rho_{k}}(\lambda,t)\,.$ (3.60)

The final result then follows from Proposition 3.11 and 3.8. ∎

######  Corollary 3.13.

Let $\rho$ be a ray different from $\pm i\mathbb{R}_{>0}$. The Borel-transform
$F_{0,\rho}(\lambda)$ of the formal series $F_{0}(\lambda)$ defined in (2.5)
can be represented in the form

$F_{0,\rho}(\lambda)=-\frac{1}{\lambda^{2}}\zeta(3)+F_{0}^{1}-\int_{\rho}\mathrm{d}\xi\;e^{-\xi/\check{\lambda}}\left(G(\xi,0)+\frac{1}{12\xi}\right).$
(3.61)

It is related to the Borel transform $F_{\rho}(\lambda,t)$ of
$\tilde{F}(\lambda,t)$ by the equation (2.20).

###### Proof.

We will consider a limit of $F_{\rho}(\lambda,t)$ as $t\to 0$, where $t$ is
taken to satisfy $\text{Re}(t)>0$, $\text{Im}(t)>0$; and such that along the
limit, $\rho$ is always between $l_{-1}$ and $l_{0}$ (resp. $-l_{-1}$ and
$-l_{0}$) if $\rho$ is on the right (resp. left) Borel half-plane. Let us
first assume that $\rho$ is taken on the right half plane. The limit requires
some care as $G(\xi,0)$ has a simple pole with residue
$-\frac{B_{2}}{2}=-\frac{1}{12}$ at $\xi=0$. We may observe, however, that

$F_{\rho}(\lambda,t)=\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(e^{2\pi{\mathrm{i}}t})+\int_{\rho}\mathrm{d}\xi\;\Big{(}e^{-\xi/\check{\lambda}}G(\xi,t)+\frac{1}{12}\frac{1}{e^{\xi-2\pi{\mathrm{i}}t}-1}\Big{)}$

has a well-defined limit for $t\to 0$, as the poles at $\xi=0$ in the
integrand cancel each other. This limit can be represented as

$F_{\rho}(\lambda,0)=\frac{1}{\lambda^{2}}\zeta(3)+\int_{\rho}\mathrm{d}\xi\;e^{-\xi/\check{\lambda}}\bigg{(}G(\xi,0)+\frac{1}{12\xi}\bigg{)}+\frac{1}{12}\int_{\rho}\mathrm{d}\xi\;\bigg{(}\frac{1}{e^{\xi}-1}-\frac{e^{-\xi/\check{\lambda}}}{\xi}\bigg{)}.$
(3.62)

The derivative of the second integral with respect to $\check{\lambda}$ is
easily found to be equal to $-\frac{1}{12}\frac{1}{\check{\lambda}}$. It
follows that this integral is equal to $-\frac{1}{12}\log{\check{\lambda}}+C$,
with $C$ being an undetermined constant. The integrand of the first integral
in (3.62), on the other hand, is analytic at $\xi=0$. By straightforward
computation of the Taylor series one may check that
$G_{0}(\xi):=-G(\xi,0)-\frac{1}{12\xi}$ is the Borel transform of the formal
series $F_{0}(\lambda)+\zeta(3)/\lambda^{2}-F_{0}^{1}$ (see Lemma B.2 for the
computation), so the relation (2.20) then follows. On the other hand, if
$\rho$ is on the left half plane, one can apply the same argument from before
by using the relation $F_{\rho}(\lambda,t)=F_{-\rho}(-\lambda,t)$ (see Lemma
3.16 below). ∎

###  3.3 Stokes phenomena of the Borel sum

In the previous section, we studied $F_{\rho}(\lambda,t)$ for
$\rho=\mathbb{R}_{>0}$. However, the ray $\mathbb{R}_{>0}$ is a choice, and
any other ray $\rho$ that avoids the poles of $G(\xi,t)$ in principle is an
equally valid choice to perform the Borel sum. In this section we study the
dependence on this choice.

######  Proposition 3.14.

Assume that $\mathrm{Im}(t)>0$ and for $k\in\mathbb{Z}$ let
$l_{k}=\mathbb{R}_{<0}\cdot 2\pi{\mathrm{i}}(t+k)$. Furthermore let $\rho$ be
a ray in the sector determined by the Stokes rays $l_{k+1}$ and $l_{k}$, and
$\rho^{\prime}$ a ray in the sector determined by $l_{k}$ and $l_{k-1}$. Then
for $\lambda\in\mathbb{H}_{\rho}\cap\mathbb{H}_{\rho^{\prime}}$ (resp.
$\lambda\in\mathbb{H}_{-\rho}\cap\mathbb{H}_{-\rho^{\prime}}$) we have

$F_{\pm\rho}(\lambda,t)-F_{\pm\rho^{\prime}}(\lambda,t)=\frac{1}{2\pi
i}\partial_{\check{\lambda}}\Big{(}\check{\lambda}\,\mathrm{Li}_{2}\big{(}e^{\pm
2\pi{\mathrm{i}}(t+k)/\check{\lambda}})\Big{)}\,.$ (3.63)

If $\mathrm{Im}(t)<0$, then the previous jumps also hold provided $\rho$ is
interchanged with $\rho^{\prime}$ in the above formulas.

###### Proof.

Notice that

$F_{\rho}(\lambda,t)-F_{\rho^{\prime}}(\lambda,t)=\int_{\mathcal{H}(l_{k})}\mathrm{d}\xi\,e^{-\xi/\check{\lambda}}G(\xi,t)\,,$
(3.64)

where $\mathcal{H}(l_{k})$ is a Hankel contour around
$l_{k}=\mathbb{R}_{<0}\cdot 2\pi{\mathrm{i}}(t+k)$.

To compute this, notice that for our range of parameters, $G(\xi,t)$ has
double poles at $\xi=2\pi{\mathrm{i}}m(t+k)$ for all $k\in\mathbb{Z}$ and
$m\in\mathbb{Z}\setminus\\{0\\}$ with generalized residues

$\left.\frac{\mathrm{d}}{\mathrm{d}\xi}(e^{-\xi/\check{\lambda}}(\xi-2\pi{\mathrm{i}}m(t+k))^{2}G(\xi,t))\right|_{\xi=2\pi{\mathrm{i}}m(t+k)}=-\frac{e^{-2\pi{\mathrm{i}}m(t+k)/\check{\lambda}}}{(2\pi)^{2}m^{2}}\left(1+\frac{2\pi{\mathrm{i}}m(t+k)}{\check{\lambda}}\right).$
(3.65)

In particular, we have that

$\displaystyle\int_{\mathcal{H}(l_{k})}\mathrm{d}\xi\,e^{-\xi/\check{\lambda}}G(\xi,t)$
$\displaystyle=2\pi{\mathrm{i}}\sum_{m=-1}^{-\infty}\left.\frac{\mathrm{d}}{\mathrm{d}\xi}(e^{-\xi/\check{\lambda}}(\xi-2\pi\mathrm{i}m(t+k))^{2}G(\xi,t))\right|_{\xi=2\pi{\mathrm{i}}m(t+k)}$
$\displaystyle=-\frac{{\mathrm{i}}}{2\pi}\sum_{m=1}^{\infty}\frac{e^{2\pi{\mathrm{i}}m(t+k)/\check{\lambda}}}{m^{2}}\left(1-\frac{2\pi{\mathrm{i}}m(t+k)}{\check{\lambda}}\right)$
$\displaystyle=-\frac{{\mathrm{i}}}{2\pi}\Big{(}\mathrm{Li}_{2}\big{(}e^{2\pi{\mathrm{i}}(t+k)/\check{\lambda}}\big{)}+\frac{2\pi{\mathrm{i}}(t+k)}{\check{\lambda}}\log\big{(}1-e^{2\pi\mathrm{i}(t+k)/\check{\lambda}}\big{)}\Big{)}$
$\displaystyle=\frac{1}{2\pi\mathrm{i}}\partial_{\check{\lambda}}\Big{(}\check{\lambda}\mathrm{Li}_{2}\big{(}e^{2\pi\mathrm{i}(t+k)/\check{\lambda}})\Big{)}\;,$
(3.66)

where we have used the series representation of $\mathrm{Li}_{s}(z)$ for
$s=1,2$; and the fact that for $\lambda$ in a sufficiently small sector
containing $l_{k}$, we have $|e^{2\pi\mathrm{i}(t+k)/\check{\lambda}}|<1$.

A similar argument follows for the rest of the cases in the statement of the
proposition. ∎

####  3.3.1 Stokes jumps of the other Borel transform

Recall the expression for the Borel transform which was obtained previously in
[PS10], given in (3.28):

$G(\xi,t)=\sum_{k\in\mathbb{Z}}\frac{1}{\xi}\left(\frac{(k-t)^{2}}{\xi^{2}}-\frac{e^{\xi/(k-t)}}{(e^{\xi/(k-t)}-1)^{2}}-\frac{1}{12}\right),\\\
$ (3.67)

Similarly to the previous discussion, if $y_{m}=\xi-2\pi\mathrm{i}m(t+k)$ near
the pole given by $\xi=2\pi\mathrm{i}m(t+k)$, then by Taylor expanding the
integrand near $y_{m}=0$ we obtain

$e^{-\xi/\check{\lambda}}G(\xi,t)=e^{-2\pi\mathrm{i}m(t+k)/\check{\lambda}}\left(\frac{1}{y_{m}^{2}}\frac{\mathrm{i}(t+k)}{2\pi
m}+\frac{1}{y_{m}}\left(-\frac{1}{4\pi^{2}m^{2}}-\frac{\mathrm{i}}{2\pi
m\check{\lambda}}(t+k)\right)+\mathcal{O}(1)\right)\,.$ (3.68)

Hence, by following the argument of Proposition 3.14, we obtain the same
Stokes jumps.

###  3.4 The limits $\lim_{k\to\pm\infty}F_{\rho_{k}}(\lambda,t)$ and
$F_{\mathrm{GV}}(\lambda,t)$

To finish the proof of Theorem 2.1, we study the limits of
$F_{\rho}(\lambda,t)$, discussed in point (iv).

######  Proposition 3.15.

Let $\rho_{k}$ denote any ray between the Stokes rays $l_{k}$ and $l_{k-1}$.
Furthermore, assume that $0<\mathrm{Re}(t)<1$, $\mathrm{Im}(t)>0$,
$\mathrm{Re}(\lambda)>0$, $\mathrm{Im}(\lambda)<0$, and
$\mathrm{Re}(t)<\mathrm{Re}(\check{\lambda}+1)$. Then

$\lim_{k\to\infty}F_{\rho_{k}}(\lambda,t)=\sum_{k=1}^{\infty}\frac{e^{2\pi\mathrm{i}kt}}{k\big{(}2\sin\big{(}\frac{\lambda
k}{2}\big{)}\big{)}^{2}}=F_{\mathrm{GV}}(\lambda,t)\,.$ (3.69)

Furthermore, we can write the sum of the Stokes jumps along $l_{k}$ for $k\geq
0$ as

$\sum_{k=0}^{\infty}\phi_{l_{k}}(\lambda,t)=\frac{1}{2\pi
i}\partial_{\lambda}\Big{(}\lambda\sum_{l=1}^{\infty}\frac{w^{l}}{l^{2}(1-\widetilde{q}^{l})}\Big{)},\;\;\;\;w:=e^{2\pi\mathrm{i}t/\check{\lambda}},\;\;\widetilde{q}:=e^{2\pi\mathrm{i}/\check{\lambda}}\;.$
(3.70)

###### Proof.

By Proposition 3.14 we find that

$F_{\rho_{k}}-F_{\rho_{k+1}}=\frac{\mathrm{i}}{2\pi}\Big{(}\mathrm{Li}_{2}\big{(}e^{2\pi\mathrm{i}(t+k)/\check{\lambda}}\big{)}+\log(e^{2\pi\mathrm{i}\frac{t+k}{\check{\lambda}}})\log\big{(}1-e^{2\pi\mathrm{i}(t+k)/\check{\lambda}}\big{)}\Big{)}\,.$
(3.71)

Denoting $w=e^{2\pi\mathrm{i}t/\check{\lambda}}$ and
$\widetilde{q}=e^{2\pi\mathrm{i}/\check{\lambda}}$, we find

$\begin{split}F_{\rho_{0}}(\lambda,t)-\lim_{k\to\infty}F_{\rho_{k}}(\lambda,t)&=\sum_{k=0}^{\infty}F_{\rho_{k}}(\lambda,t)-F_{\rho_{k+1}}(\lambda,t)\\\
&=\frac{\mathrm{i}}{2\pi}\sum_{k=0}^{\infty}\Big{(}\mathrm{Li}_{2}\big{(}w\widetilde{q}^{k}\big{)}+\log\big{(}w\widetilde{q}^{k}\big{)}\log\big{(}1-w\widetilde{q}^{k}\big{)}\Big{)}\,.\end{split}$
(3.72)

We now use the following identities:

$\displaystyle\sum_{k=0}^{\infty}\log(1-w\widetilde{q}^{k})=-\sum_{l=1}^{\infty}\frac{1}{l}\frac{w^{l}}{1-\widetilde{q}^{l}}\,,$
(3.73a)
$\displaystyle\sum_{k=0}^{\infty}k\log(1-w\widetilde{q}^{k})=-\sum_{l=1}^{\infty}\frac{\widetilde{q}^{l}}{l}\frac{w^{l}}{(1-\widetilde{q}^{l})^{2}}\,,$
(3.73b)
$\displaystyle\sum_{k=0}^{\infty}\mathrm{Li}_{2}(w\widetilde{q}^{k})=\sum_{l=1}^{\infty}\frac{1}{l^{2}}\frac{w^{l}}{1-\widetilde{q}^{l}}\,.$
(3.73c)

The first two identities (3.73a) and (3.73b) are easily established by using
the Taylor expansion of the logarithm function; using that $|\widetilde{q}|<1$
and $|w|<1$; and exchanging the two summations. In order to verify (3.73c),
one can first act on it with $w\frac{\mathrm{d}}{\mathrm{d}w}$. The left side
of the resulting equation is easily seen to be equal to

$-\sum_{k=0}^{\infty}\log(1-w\widetilde{q}^{k})=\sum_{l=1}^{\infty}\frac{1}{l}\frac{w^{l}}{1-\widetilde{q}^{l}}\,,$

using (3.73a). It follows that (3.73c) holds up to addition of a term which is
constant with respect to $w$. In order to fix this freedom, it suffices to
note that (3.73c) holds for $w=0$.

Using the previous identities, we obtain

$\begin{split}F_{\rho_{0}}(\lambda,t)-\lim_{k\to\infty}F_{\rho_{k}}(\lambda,t)&=\frac{\mathrm{i}}{2\pi}\sum_{l=1}^{\infty}\frac{w^{l}}{l(1-\widetilde{q}^{l})}\bigg{(}\frac{1}{l}-\frac{\widetilde{q}^{l}\log\widetilde{q}}{1-\widetilde{q}^{l}}-\log
w\bigg{)}\\\
&=-\frac{\mathrm{i}}{2\pi}\sum_{l=1}^{\infty}\frac{\partial}{\partial
l}\bigg{(}\frac{w^{l}}{l(1-\widetilde{q}^{l})}\bigg{)}\,.\end{split}$ (3.74)

Now notice that under our assumptions on $t$ and $\lambda$, we have
$F_{\rho_{0}}=F_{\mathrm{np}}$ by Proposition 3.11. We now show that
$F_{\rho_{0}}$ admits the following representation as sum over residues:

$F_{\rho_{0}}(\lambda,t)=\frac{1}{2\pi\mathrm{i}}\sum_{l=1}^{\infty}\frac{\partial}{\partial
l}\bigg{(}\frac{w^{l}}{l(1-\widetilde{q}^{l})}\bigg{)}+\sum_{k=1}^{\infty}\frac{e^{2\pi\mathrm{i}kt}}{k\big{(}2\sin\big{(}\frac{\lambda
k}{2}\big{)}\big{)}^{2}}\,.$ (3.75)

In order to see this, let us recall that by Proposition 3.8, we have

$\displaystyle F_{\mathrm{np}}(\lambda,t)$
$\displaystyle=-\int_{\mathbb{R}+\mathrm{i}0^{+}}\frac{\mathrm{d}u}{8u}\,\frac{e^{u(t-\frac{1}{2})}}{\sinh(u/2)(\sinh(\lambda
u/4\pi))^{2}}=\int_{\mathbb{R}+\mathrm{i}0^{+}}\frac{\mathrm{d}u}{u}\,\frac{e^{ut}}{1-e^{u}}\frac{1}{(2\sinh(\lambda
u/4\pi))^{2}}\,.$

The integrand has two series of poles, one at
$u=u_{l}:=(2\pi)^{2}\frac{\mathrm{i}}{\lambda}l$, $l\in\mathbb{Z}$ and the
other at $u=\widetilde{u}_{k}:=2\pi\mathrm{i}k$, $k\in\mathbb{Z}$. We can
compute the previous integral by closing the contour in the upper half-plane.
The contributions from the poles at $u=u_{l}$ are calculated as

$\displaystyle
2\pi\mathrm{i}\left(\frac{4\pi}{2\lambda}\right)^{2}\frac{\partial}{\partial
u}\frac{e^{ut}}{(1-e^{u})u}\bigg{|}_{u=(2\pi)^{2}\frac{\mathrm{i}}{\lambda}l}$
$\displaystyle=2\pi\mathrm{i}\,\frac{(2\pi)^{2}}{\lambda^{2}}\bigg{(}\frac{\lambda}{\mathrm{i}(2\pi)^{2}}\bigg{)}^{2}\frac{\partial}{\partial
l}\frac{w^{l}}{(1-\widetilde{q}^{l})l}$
$\displaystyle=\frac{1}{2\pi\mathrm{i}}\frac{\partial}{\partial
l}\frac{w^{l}}{(1-\widetilde{q}^{l})l}\,,$

while the contributions of the poles at $u=2\pi\mathrm{i}k$ give the remaining
term.

In particular, we conclude that

$\displaystyle\lim_{k\to\infty}F_{\rho_{k}}(\lambda,t)$
$\displaystyle=\lim_{k\to\infty}(F_{\rho_{k}}(\lambda,t)-F_{\rho_{0}}(\lambda,t))+F_{\rho_{0}}(\lambda,t)$
$\displaystyle=-\frac{1}{2\pi\mathrm{i}}\sum_{l=1}^{\infty}\frac{\partial}{\partial
l}\bigg{(}\frac{w^{l}}{l(1-\widetilde{q}^{l})}\bigg{)}+\frac{1}{2\pi\mathrm{i}}\sum_{l=1}^{\infty}\frac{\partial}{\partial
l}\bigg{(}\frac{w^{l}}{l(1-\widetilde{q}^{l})}\bigg{)}+\sum_{k=1}^{\infty}\frac{e^{2\pi\mathrm{i}kt}}{k\big{(}2\sin\big{(}\frac{\lambda
k}{2}\big{)}\big{)}^{2}}$
$\displaystyle=\sum_{k=1}^{\infty}\frac{e^{2\pi\mathrm{i}kt}}{k\big{(}2\sin\big{(}\frac{\lambda
k}{2}\big{)}\big{)}^{2}}\,.$ (3.76)

The last statement follows easily by noticing that

$\frac{\mathrm{i}}{2\pi}\sum_{l=1}^{\infty}\frac{\partial}{\partial
l}\bigg{(}\frac{w^{l}}{l(1-\widetilde{q}^{l})}\bigg{)}=\frac{1}{2\pi\mathrm{i}}\partial_{\lambda}\Big{(}\lambda\sum_{l=1}^{\infty}\frac{w^{l}}{l^{2}(1-\widetilde{q}^{l})}\Big{)}\,.$
(3.77)

∎

To study the other limits to the imaginary rays, we use the following lemma:

######  Lemma 3.16.

For $\rho$ any ray not in $\\{\pm l_{k}\\}\cup\\{\pm l_{\infty}\\}$ and
$\lambda\in\mathbb{H}_{\rho}$, we have

$F_{\rho}(\lambda,t)=F_{-\rho}(-\lambda,t)\,.$ (3.78)

###### Proof.

The main thing to notice is that

$G(\xi,t)=-G(-\xi,t)\,.$ (3.79)

Using the earlier relation, we obtain

$\begin{split}F_{\rho}(\lambda,t)&=\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(Q)+\frac{B_{2}}{2}\mathrm{Li}_{1}(Q)-\int_{\rho}\mathrm{d}\xi\,e^{-\xi/\check{\lambda}}G(-\xi,t)\\\
&=\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(Q)+\frac{B_{2}}{2}\mathrm{Li}_{1}(Q)+\int_{-\rho}\mathrm{d}\xi\,e^{\xi/\check{\lambda}}G(\xi,t)\\\
&=F_{-\rho}(-\lambda,t)\,.\end{split}$ (3.80)

∎

As an immediate corollary from Proposition 3.15 and Lemma 3.16, we obtain:

######  Corollary 3.17.

With the same notation as in Proposition 3.15, assume that $0<\text{Re}(t)<1$,
$\mathrm{Im}(t)>0$, $\mathrm{Re}(\lambda)<0$, $\mathrm{Im}(\lambda)>0$ and
$\mathrm{Re}(t)<\mathrm{Re}(-\check{\lambda}+1)$. Then

$\lim_{k\to\infty}F_{-\rho_{k}}(\lambda,t)=F_{\mathrm{GV}}(\lambda,t)\,.$
(3.81)

######  Proposition 3.18.

With the same notation as in Proposition 3.15, assume that $0<\text{Re}(t)<1$,
$\text{Im}(t)>0$, $\text{Re}(\lambda)<0$, $\text{Im}(\lambda)<0$,
$\mathrm{Re}\,t<\mathrm{Re}(-\check{\lambda}+1)$ and that $|w^{-1}|<1$. Then

$\begin{split}\lim_{k\to-\infty}F_{-\rho_{k}}(\lambda,t)=F_{\mathrm{GV}}(\lambda,t)\end{split}$
(3.82)

######  Remark 3.19.

For fixed $\lambda$ satisfying $\mathrm{Re}(\lambda)<0$,
$\mathrm{Im}(\lambda)<0$, the condition $|w^{-1}|<1$ can be satisfied by
picking $t$ such that $0<\mathrm{Re}(t)<1$, $\mathrm{Im}(t)>0$, and
$\mathrm{Im}(t)$ is sufficiently large. Similarly, for fixed $t$ with
$0<\mathrm{Re}(t)<1$, $\mathrm{Im}(t)>0$, $|w^{-1}|<1$ can be satisfied by
picking $\lambda$ such that $\mathrm{Re}(\lambda)<0$, $\mathrm{Im}(\lambda)<0$
and $|\mathrm{Im}(\lambda)|<<|\mathrm{Re}(\lambda)|$ .

###### Proof.

Using the jumps along the Stokes rays $l_{k}$ for $k<0$, we find that

$\displaystyle F_{-\rho_{0}}-\lim_{k\to-\infty}F_{-\rho_{k}}$
$\displaystyle=\sum_{k=-1}^{-\infty}F_{-\rho_{k+1}}-F_{-\rho_{k}}$
$\displaystyle=-\frac{\mathrm{i}}{2\pi}\sum_{k=-1}^{-\infty}\Big{(}\mathrm{Li}_{2}\big{(}e^{-2\pi\mathrm{i}(t+k)/\check{\lambda}}\big{)}+\log(e^{-2\pi\mathrm{i}\frac{t+k}{\check{\lambda}}})\log\big{(}1-e^{-2\pi\mathrm{i}(t+k)/\check{\lambda}}\big{)}\Big{)}$
$\displaystyle=-\frac{\mathrm{i}}{2\pi}\sum_{k=0}^{\infty}\Big{(}\mathrm{Li}_{2}\big{(}w^{-1}\widetilde{q}^{k}\big{)}+\log(w^{-1}\widetilde{q}^{k})\log\big{(}1-w^{-1}\widetilde{q}^{k}\big{)}\Big{)}$
$\displaystyle\;\;\;\;+\frac{\mathrm{i}}{2\pi}\Big{(}\mathrm{Li}_{2}\big{(}w^{-1}\big{)}+\log(w^{-1})\log\big{(}1-w^{-1}\big{)}\Big{)}\,.$
(3.83)

Using the constraints on $t$ and $\lambda$, we find that $|w^{-1}|<1$ and
$|\widetilde{q}|<1$, so that we can expand in series as in Proposition 3.15
and write

$\displaystyle F_{-\rho_{0}}-\lim_{k\to-\infty}F_{-\rho_{k}}$
$\displaystyle=-\frac{\mathrm{i}}{2\pi}\sum_{l=1}^{\infty}\frac{w^{-l}}{l(1-\widetilde{q}^{l})}\bigg{(}\frac{1}{l}-\frac{\widetilde{q}^{l}\log\widetilde{q}}{1-\widetilde{q}^{l}}-\log
w^{-1}\bigg{)}$
$\displaystyle\;\;\;\;+\frac{\mathrm{i}}{2\pi}\Big{(}\mathrm{Li}_{2}\big{(}w^{-1}\big{)}+\log(w^{-1})\log\big{(}1-w^{-1})\Big{)}$
$\displaystyle=-\frac{1}{2\pi\mathrm{i}}\sum_{l=1}^{\infty}\frac{\partial}{\partial
l}\bigg{(}\frac{w^{-l}}{l(1-\widetilde{q}^{l})}\bigg{)}+\frac{\mathrm{i}}{2\pi}\Big{(}\mathrm{Li}_{2}\big{(}w^{-1}\big{)}+\log(w^{-1})\log\big{(}1-w^{-1})\Big{)}\,.$
(3.84)

On the other hand, under our conditions on the parameters $t$ and $\lambda$,
and by Lemma 3.16, we have that

$F_{-\rho_{0}}(\lambda,t)=F_{\mathrm{np}}(-\lambda,t)\,.$ (3.85)

Following the same argument as in Proposition 3.15 using the integral
representation of $F_{\mathrm{np}}$, we find that

$\displaystyle
F_{\mathrm{np}}(-\lambda,t)=\frac{1}{2\pi\mathrm{i}}\sum_{l=1}^{\infty}\frac{\partial}{\partial
l}\bigg{(}\frac{w^{-l}}{l(1-\widetilde{q}^{-l})}\bigg{)}+\sum_{k=1}^{\infty}\frac{e^{2\pi\mathrm{i}kt}}{k\big{(}2\sin\big{(}\frac{\lambda
k}{2}\big{)}\big{)}^{2}}.$

Hence, we find that for $\lambda$ and $t$ as in the hypothesis

$F_{-\rho_{0}}(\lambda,t)=\frac{1}{2\pi\mathrm{i}}\sum_{l=1}^{\infty}\frac{\partial}{\partial
l}\bigg{(}\frac{w^{-l}}{l(1-\widetilde{q}^{-l})}\bigg{)}+F_{\mathrm{GV}}(\lambda,t)$
(3.86)

Joining our results together, we conclude that

$\begin{split}\lim_{k\to-\infty}F_{-\rho_{k}}(\lambda,t)&=\frac{1}{2\pi\mathrm{i}}\sum_{l=1}^{\infty}\Big{(}\frac{\partial}{\partial
l}\bigg{(}\frac{w^{-l}}{l(1-\widetilde{q}^{-l})}\bigg{)}+\frac{\partial}{\partial
l}\bigg{(}\frac{w^{-l}}{l(1-\widetilde{q}^{l})}\bigg{)}\Big{)}+F_{\mathrm{GV}}(\lambda,t)\\\
&\;\;\;\;+\frac{1}{2\pi\mathrm{i}}\Big{(}\mathrm{Li}_{2}\big{(}w^{-1}\big{)}+\log(w^{-1})\log\big{(}1-w^{-1})\Big{)}\,.\end{split}$
(3.87)

Finally, notice that

$\begin{split}\sum_{l=1}^{\infty}\Big{(}\frac{\partial}{\partial
l}\bigg{(}\frac{w^{-l}}{l(1-\widetilde{q}^{-l})}+\frac{w^{-l}}{l(1-\widetilde{q}^{l})}\bigg{)}\Big{)}&=\sum_{l=1}^{\infty}\Big{(}\frac{\partial}{\partial
l}\bigg{(}\frac{w^{-l}}{l}\bigg{)}\Big{)}\\\
&=\log(w^{-1})\sum_{l=1}^{\infty}\frac{w^{-l}}{l}-\sum_{l=1}^{\infty}\frac{w^{-l}}{l^{2}}\\\
&=-(\mathrm{Li}_{2}(w^{-1})+\log(w^{-1})\log(1-w^{-1}))\,,\\\ \end{split}$
(3.88)

where in the last equality we used that $|w^{-1}|<1$ under our hypotheses.
Hence, we conclude that

$\lim_{k\to-\infty}F_{-\rho_{k}}(\lambda,t)=F_{\mathrm{GV}}(\lambda,t)\,.$
(3.89)

∎

By using Lemma 3.16 and Proposition 3.18, we get the following immediate
corollary:

######  Corollary 3.20.

With the same notation as in Proposition 3.15, assume that
$0<\mathrm{Re}(t)<1$, $\mathrm{Im}(t)>0$, $\mathrm{Re}(\lambda)>0$,
$\mathrm{Im}(\lambda)>0$, $\mathrm{Re}(t)<\mathrm{Re}(\check{\lambda}+1)$ and
such that $|w|<1$. Then

$\lim_{k\to-\infty}F_{\rho_{k}}(\lambda,t)=F_{\mathrm{GV}}(\lambda,t)\,.$
(3.90)

The limits studied above can be informally interpreted as the relation between
$F_{\mathrm{GV}}(\lambda,t)$ and the Borel summations along the imaginary
axes.

####  3.4.1 Relation between $F_{\mathbb{R}_{>0}}$ and
$F_{\mathrm{coni}}^{\mathrm{resum}}$

In this subsection, we briefly explain how $F_{\mathbb{R}_{>0}}=F_{np}$
relates to $F_{\mathrm{coni}}^{\mathrm{resum}}$ from [HO15].

On the one hand, from part (iv) of Theorem 2.1 together with the comments of
Section 2.3, we find that

$F_{\mathbb{R}_{>0}}(\lambda,t)=F_{\mathrm{GV}}(\lambda,t)+\frac{1}{2\pi
i}\frac{\partial}{\partial\lambda}\lambda
F_{NS}\Big{(}\frac{4\pi^{2}}{\lambda},\frac{2\pi}{\lambda}\Big{(}t-\frac{1}{2}\Big{)}\Big{)}\,.$
(3.91)

On the other hand, in [HO15] the following function is considered

$F_{\mathrm{coni}}^{\mathrm{resum}}(\lambda,t)=\frac{\mathrm{Li}_{3}(Q)}{\lambda^{2}}+\int_{0}^{\infty}dv\,\frac{v}{1-e^{2\pi
v-i0^{+}}}\log(1+Q^{2}-2Q\cosh(\lambda v)),\;\;\;\;Q=e^{2\pi it}\,,$ (3.92)

and it is conjectured that

$F_{\mathrm{coni}}^{\mathrm{resum}}(\lambda,t)=F_{\mathrm{GV}}(\lambda,t)+\frac{1}{2\pi
i}\frac{\partial}{\partial\lambda}\lambda
F_{NS}\Big{(}\frac{4\pi^{2}}{\lambda},\frac{2\pi}{\lambda}\Big{(}t-\frac{1}{2}\Big{)}\Big{)}\,,$
(3.93)

as explained in Section 2.4.

We show that this in indeed the case, by the use of the Woronowicz form of
$F_{\mathbb{R}_{>0}}$ of Lemma 3.10.

######  Proposition 3.21.

Let $t\in\mathbb{C}$ be such that $0<\mathrm{Re}(t)<1$, $\mathrm{Im}(t)>0$,
and let $\lambda$ be in the sector determined by $l_{0}=\mathbb{R}_{<0}\cdot
2\pi{\mathrm{i}}t$ and $l_{-1}=\mathbb{R}_{<0}\cdot 2\pi{\mathrm{i}}(t-1)$.
Then $F_{\mathbb{R}_{>0}}=F_{\mathrm{coni}}^{\mathrm{resum}}$ on their common
domains of definition.

###### Proof.

First notice that since

$1+Q^{2}-2Q\cosh(\lambda x)=(1-e^{\lambda x}Q)(1-e^{-\lambda x}Q)\,,$ (3.94)

we can rewrite (3.92) as follows

$\begin{split}F_{\mathrm{coni}}^{\mathrm{resum}}&=\frac{\mathrm{Li}_{3}(Q)}{\lambda^{2}}+\int_{\mathbb{R}+\mathrm{i}0^{+}}\mathrm{d}v\frac{v}{1-e^{2\pi
v}}\log(1-e^{\lambda v}Q)\\\
&\quad-\int_{-\infty}^{0}\mathrm{d}v\frac{v}{1-e^{2\pi
v-\mathrm{i}0^{+}}}\log(1-e^{\lambda
v}Q)+\int_{0}^{\infty}\mathrm{d}v\frac{v}{1-e^{2\pi
v-\mathrm{i}0^{+}}}\log(1-e^{-\lambda v}Q)\\\
&=\frac{\mathrm{Li}_{3}(Q)}{\lambda^{2}}+\int_{\mathbb{R}+\mathrm{i}0^{+}}\mathrm{d}v\frac{v}{1-e^{2\pi
v}}\log(1-e^{\lambda v}Q)+\int_{0}^{\infty}\mathrm{d}v\;v\log(1-e^{-\lambda
v}Q)\,.\end{split}$ (3.95)

On the other hand, notice that we can rewrite the last term in the above
expression as follows:

$\displaystyle\int_{0}^{\infty}\mathrm{d}v\;v\log(1-e^{-\lambda v}Q)$
$\displaystyle=\frac{1}{\lambda^{2}}\int_{\lambda\cdot\mathbb{R}_{>0}}\mathrm{d}v\;v\log(1-e^{-v}Q)$
$\displaystyle=\frac{1}{\lambda^{2}}\int_{0}^{\infty}\mathrm{d}v\;v\log(1-e^{-v}Q)$
$\displaystyle=\frac{1}{2\lambda^{2}}\int_{0}^{\infty}\mathrm{d}v\,\frac{v^{2}}{1-e^{v}Q^{-1}}=-\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(Q)$
(3.96)

where in the second equality we have used that the range of $\lambda$ allows
us to deform back the contour to $\mathbb{R}_{>0}$; in the third equality we
have integrated by parts; and in the last one we have used that
$\text{Im}(t)>0$ implies that $|e^{2\pi\mathrm{i}t}|<1$, and hence we can use
the integral representation of $\mathrm{Li}_{3}$.

Hence,

$F_{\mathrm{coni}}^{\mathrm{resum}}=\int_{\mathbb{R}+\mathrm{i}0^{+}}\mathrm{d}v\frac{v}{1-e^{2\pi
v}}\log(1-e^{\lambda
v}Q)=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}+\mathrm{i}0^{+}}\mathrm{d}v\frac{v}{1-e^{v}}\log(1-e^{\frac{\lambda}{2\pi}v+2\pi\mathrm{i}t})$
(3.97)

so the result follows from Lemma 3.10 and Proposition 3.11. ∎

##  4 Relation to the Riemann–Hilbert problem and line bundles

In Sections 2 and 3 we discussed the Borel sum $F_{\rho}(\lambda,t)$ of
$\widetilde{F}(\lambda,t)$ along the ray $\rho$, and its dependence on $\rho$
in terms of the Stokes jumps. Our objectives in this section are the
following:

  * •

On one hand, in [Bri20] a Riemann–Hilbert problem is associated to the BPS
spectrum of the resolved conifold. This involves finding piecewise holomorphic
functions ${\mathcal{X}}_{\gamma}$ on ${\mathbb{C}}^{\times}\times M$, with
$M$ being called the space of stability structures, related by certain Stokes
jumps along rays in ${\mathbb{C}}^{\times}$. Introducing a coordinate
$\lambda_{B}$ for ${\mathbb{C}}^{\times}$ called twistor variable one may
interpret the family of functions ${\mathcal{X}}_{\gamma}(\lambda_{B},-)$ on
$M$ as complex coordinates defining a family of complex structures on $M$. We
will show that the jumps of $F_{\rho}(\lambda,t)$ serve as “potentials” for
the Stokes jumps associated to the Riemann–Hilbert problem.

  * •

On the other hand, in thinking of $F_{\rho}(\lambda,t)$ more geometrically, it
is natural to consider the partition functions

$Z_{\rho}(\lambda,t)=\exp(F_{\rho}(\lambda,t))\;,$ (4.1)

and interpret them as defining a section of a line bundle $\mathcal{L}$,
having transition functions equal to the exponentials of the Stokes jumps.
This perspective follows the ideas of [CLT20], specialized to the case of the
resolved conifold.

  * •

We will furthermore demonstrate that the the line bundle $\mathcal{L}$ is
related to certain hyperholomorphic line bundles previously considered in
[Nei11, APP11b]. These hyperholomorphic line bundles are canonically defined
by a given BPS spectrum and represented by transition functions defined from
the Rogers dilogarithm function. We will show that the hyperholomorphic line
bundles considered in [Nei11, APP11b] are in the case of the resolved conifold
related to the line bundle $\mathcal{L}$ by performing a certain “conformal
limit” previously considered in [Gai14].

In order to facilitate comparison with [Bri20], we will represent the
parameters $\lambda$ and $t$ used in this paper in the following form

$t=v/w,\;\;\;\;\lambda=2\pi\lambda_{\mathrm{B}}/w\,,$ (4.2)

where $\lambda_{\mathrm{B}}$ is the notation used here for the variable
denoted $t$ in [Bri20], and consider a projectivized partition function

$Z_{\rho}(\lambda_{\mathrm{B}},v,w):=\exp\Big{(}F_{w^{-1}\cdot\rho}\Big{(}\frac{2\pi\lambda_{\mathrm{B}}}{w},\frac{v}{w}\Big{)}\Big{)}\,.$
(4.3)

We will show that after appropriately normalizing the partition functions
$Z_{\rho}\to\widehat{Z}_{\rho}$, the BPS spectrum of the resolved conifold
will be neatly encoded in the transition functions of the line bundle defined
by $\widehat{Z}_{\rho}$. Furthermore, we will see that the normalization
reintroduces the constant map constribution 2.5, giving a partition function
whose free energy has asymptotic expansion equal to 2.4.

###  4.1 Bridgeland’s Riemann–Hilbert problem and its solution

We begin by recalling the Riemann–Hilbert problem considered in [Bri20]. The
initial data for such Riemann–Hilbert problems is the following:

######  Definition 4.1.

A variation of BPS structures is given by a tuple $(M,\Gamma,Z,\Omega)$, where

  * •

$M$ is a complex manifold.

  * •

Charge lattice: $\Gamma\to M$ is a local system of lattices with a skew-
symmetric, covariantly constant paring
$\langle-,-\rangle\colon\Gamma\times\Gamma\to\mathbb{Z}$.

  * •

Central charge: $Z$ is a holomorphic section of
$\Gamma^{*}\otimes\mathbb{C}\to M$.

  * •

BPS indices: $\Omega\colon\Gamma\to\mathbb{Z}$ is a function satisfying
$\Omega(\gamma)=\Omega(-\gamma)$ and the Kontsevich–Soibelman wall-crossing
formula [KS08, Bri19].

The tuple $(M,\Gamma,Z,\Omega)$ should also satisfy the following conditions:

  * •

Support property: Let
$\text{Supp}(\Omega):=\\{\gamma\in\Gamma\;\;|\;\;\Omega(\gamma)\neq 0\\}$.
Given a compact set $K\subset M$ and a choice of covariantly constant norm
$|\cdot|$ on $\Gamma|_{K}\otimes_{\mathbb{Z}}\mathbb{R}$, there is a constant
$C>0$ such that for any $\text{Supp}(\Omega)\cap\Gamma|_{K}$:

$|Z_{\gamma}|>C|\gamma|\,.$ (4.4)

  * •

Convergence property: for each $p\in M$, there is an $R>0$ such that

$\sum_{\gamma\in\Gamma_{p}}|\Omega(\gamma)|e^{-R|Z_{\gamma}|}<\infty\,.$ (4.5)

The variation of BPS structures associated to the resolved conifold is then
taken to be the tuple $(M,\Gamma,Z,\Omega)$, where:

  * •

$M$ is the complex $2$-dimensional manifold

$M:=\\{(v,w)\in\mathbb{C}^{2}\;\;|\;\;w\neq 0,\;\;\;v+nw\neq 0\;\;\;\text{for
all}\;\;\;n\in\mathbb{Z}\\}\,.$ (4.6)

  * •

$\Gamma\to M$ is given by the trivial local system

$\Gamma=\mathbb{Z}\cdot\delta\oplus\mathbb{Z}\cdot\beta\oplus\mathbb{Z}\cdot\delta^{\vee}\oplus\mathbb{Z}\cdot\beta^{\vee},$
(4.7)

with pairing defined such that $(\beta^{\vee},\beta,\delta^{\vee},\delta)$ is
a Darboux frame. Namely,

$\langle\delta^{\vee},\delta\rangle=\langle\beta^{\vee},\beta\rangle=1\,,$
(4.8)

with all other pairings equal to $0$.

  * •

If for $\gamma\in\Gamma$, we denote $Z_{\gamma}:=Z(\gamma)$, $Z$ is defined by

$Z_{n\beta+m\delta+p\beta^{\vee}+q\delta^{\vee}}=2\pi{\mathrm{i}}(nv+mw),\;\;\;\;\text{for}\;\;\;n,m,p,q\in\mathbb{Z}\,.$
(4.9)

  * •

$\Omega$ is given by the BPS spectrum of the resolved conifold [JS12], see
also [BLR19]:

$\Omega(\gamma)=\begin{cases}1&\quad\text{if
}\gamma=\pm\beta+n\delta\quad\text{for }\;\;n\in\mathbb{Z}\,,\\\
-2&\quad\text{if
}\gamma=k\delta\quad\text{for}\;\;k\in\mathbb{Z}\setminus\\{0\\}\,,\\\
0&\quad\text{otherwise}.\end{cases}$ (4.10)

To this data, the following Riemann–Hilbert problem is associated222We will
follow slightly different conventions from [Bri20]. In particular, what we
call $\mathcal{L}_{k}$ corresponds in Bridgeland’s convention to
$-\mathcal{L}_{k}$.. First, we define $\mathcal{L}_{k}:=\mathbb{R}_{<0}\cdot
2\pi{\mathrm{i}}(v+kw)$ and $\mathcal{L}_{\infty}:=\mathbb{R}_{<0}\cdot
2\pi{\mathrm{i}}w$, and assume that $(v,w)\in M$ satisfies $\text{Im}(v/w)>0$
(the case $\text{Im}(v/w)\leq 0$ is also considered in [Bri20], but we
restrict to $\text{Im}(v/w)>0$ for simplicity). Then, for each ray $\rho$ from
$0$ to $\infty$ not in
$\\{\pm\mathcal{L}_{k}\\}_{k\in\mathbb{Z}}\cup\\{\pm\mathcal{L}_{\infty}\\}$,
we should find a holomorphic function
$\mathcal{X}_{\gamma,\rho}(v,w,-)\colon\mathbb{H}_{\rho}\to\mathbb{C}^{\times}$
labeled by $\gamma\in\Gamma$ such that they satisfy the following:

  * •

Twisted homomorphism property: for $\gamma,\gamma^{\prime}\in\Gamma$ we have

$\mathcal{X}_{\gamma+\gamma^{\prime},\rho}(v,w,-)=(-1)^{\langle\gamma,\gamma^{\prime}\rangle}\mathcal{X}_{\gamma,\rho}(v,w,-)\mathcal{X}_{\gamma^{\prime},\rho}(v,w,-)\,.$
(4.11)

  * •

Stokes jumps: we denote by $\rho_{k}$ a ray between $\mathcal{L}_{k}$ and
$\mathcal{L}_{k-1}$. We then have

$\mathcal{X}_{\gamma,\pm\rho_{k+1}}(v,w,\lambda_{\mathrm{B}})=\mathcal{X}_{\gamma,\pm\rho_{k}}(v,w,\lambda_{\mathrm{B}})(1-\mathcal{X}_{\pm(\beta+k\delta)}(v,w,\lambda_{\mathrm{B}}))^{\langle\gamma,\pm(\beta+k\delta)\rangle\Omega(\beta+k\delta)}\,.$
(4.12)

On the other hand, consider $\rho_{k_{1}}$ and $\rho_{k_{2}}$ with $k_{1}\neq
k_{2}$, and let $[\rho_{k_{1}},-\rho_{k_{2}}]$ denote the smallest of the two
sectors determined by $\rho_{k_{1}}$ and $-\rho_{k_{2}}$. In the case
$\mathcal{L}_{\infty}\subset[\rho_{k_{1}},-\rho_{k_{2}}]$, for
$\lambda_{\mathrm{B}}$ in the corresponding common domains we have

$\displaystyle\mathcal{X}_{\gamma,-\rho_{k_{2}}}=\mathcal{X}_{\gamma,\rho_{k_{1}}}\cdot\Bigg{(}\prod_{k\geq
k_{1}}$
$\displaystyle(1-\mathcal{X}_{\beta+k\delta})^{\langle\gamma,\beta+k\delta\rangle\Omega(\beta+k\delta)}\prod_{k>-k_{2}}(1-\mathcal{X}_{-\beta+k\delta})^{\langle\gamma,-\beta+k\delta\rangle\Omega(\beta-k\delta)}$
$\displaystyle\prod_{k\geq
1}(1-\mathcal{X}_{k\delta})^{\langle\gamma,k\delta\rangle\Omega(k\delta)}\Bigg{)}\;,$
(4.13)

while if $-\mathcal{L}_{\infty}\subset[\rho_{k_{1}},-\rho_{k_{2}}]$, we have

$\displaystyle\mathcal{X}_{\gamma,\rho_{k_{1}}}=\mathcal{X}_{\gamma,-\rho_{k_{2}}}\cdot\Bigg{(}\prod_{k\geq
k_{2}}$
$\displaystyle(1-\mathcal{X}_{-\beta-k\delta})^{\langle\gamma,-\beta-k\delta\rangle\Omega(\beta+k\delta)}\prod_{k>-k_{1}}(1-\mathcal{X}_{\beta-k\delta})^{\langle\gamma,\beta-k\delta\rangle\Omega(\beta-k\delta)}$
$\displaystyle\prod_{k\geq
1}(1-\mathcal{X}_{-k\delta})^{\langle\gamma,-k\delta\rangle\Omega(k\delta)}\Bigg{)}\;.$
(4.14)

  * •

Asymptotics as $\lambda_{\mathrm{B}}\to 0$: For each ray $\rho$ and
$\gamma\in\Gamma$, we have

$\mathcal{X}_{\gamma,\rho}(v,w,\lambda_{\mathrm{B}})e^{-Z_{\gamma}(z,w)/\lambda_{\mathrm{B}}}\to
1,\;\;\;\text{as}\;\;\;\;\lambda_{\mathrm{B}}\to
0,\;\;\;\lambda_{\mathrm{B}}\in\mathbb{H}_{\rho}.$ (4.15)

  * •

Polynomial growth as $\lambda_{\mathrm{B}}\to\infty$: for each ray $\rho$ and
$\gamma\in\Gamma$, we have the following for some $k>0$:

$|\lambda_{\mathrm{B}}|^{-k}<|\mathcal{X}_{\gamma,\rho}(v,w,\lambda_{\mathrm{B}})|<|\lambda_{\mathrm{B}}|^{k},\;\;\;\;\text{for}\;\;\;|\lambda_{\mathrm{B}}|\gg
0\,.$ (4.16)

Such a problem is shown to admit a unique solution [Bri19, Lemma 4.9], and the
solution is given as follows. By the twisted homomorphism property, it is
enough to describe $\mathcal{X}_{\gamma,\rho}$ for
$\gamma\in\\{\beta^{\vee},\beta,\delta^{\vee},\delta\\}$. The solutions for
$\gamma=\beta$ and $\gamma=\delta$ have trivial Stokes jumps, and they are
given by

$\mathcal{X}_{\beta,\rho}(v,w,\lambda_{\mathrm{B}})=e^{2\pi{\mathrm{i}}v/\lambda_{\mathrm{B}}},\;\;\;\;\mathcal{X}_{\delta,\rho}(z,w,\lambda_{\mathrm{B}})=e^{2\pi{\mathrm{i}}w/\lambda_{\mathrm{B}}},$
(4.17)

for any ray $\rho$. On the other hand, for a ray $\rho_{k}$ between
$\mathcal{L}_{k}$ and $\mathcal{L}_{k-1}$, the functions
$\mathcal{X}_{\beta^{\vee},-\rho_{k}}(v,w,\lambda_{\mathrm{B}})$ and
$\mathcal{X}_{\delta^{\vee},-\rho_{k}}(v,w,\lambda_{\mathrm{B}})$ are given by
(see [Bri20, Equation (67)])

$\begin{split}\mathcal{X}_{\beta^{\vee},-\rho_{k}}(v,w,\lambda_{\mathrm{B}})&=F^{*}(v+kw\,|\,w,-\lambda_{\mathrm{B}})\\\
\mathcal{X}_{\delta^{\vee},-\rho_{k}}(v,w,\lambda_{\mathrm{B}})&=H^{*}(v+kw\,|\,w,-\lambda_{\mathrm{B}})(F^{*}(v+kw\,|\,w,-\lambda_{\mathrm{B}}))^{k}\,,\end{split}$
(4.18)

where $F^{*}$ is defined in terms of double sine function, and $H^{*}$ in
terms of the triple sine function (see [Bri20, Section 5] for more details on
the definitions of $F^{*}$ and $H^{*}$). On the other hand,
$\mathcal{X}_{\beta^{\vee},\rho_{k}}$ and
$\mathcal{X}_{\delta^{\vee},\rho_{k}}$ are determined by the relation

$\mathcal{X}_{\gamma,\rho_{k}}(v,w,\lambda_{\mathrm{B}})=1/\mathcal{X}_{\gamma,-\rho_{k}}(v,w,-\lambda_{\mathrm{B}})\,,$
(4.19)

that follows from uniqueness of the solutions of the Riemann–Hilbert problem.

###  4.2 Relation of the partition function to the Riemann–Hilbert problem

In this section we wish to relate the Stokes jumps of $F_{\rho}(\lambda,t)$
with the Stokes jumps of the Riemann–Hilbert problem. More specifically, we
will first consider a normalization of the partition function
$\exp(F_{\rho}(\lambda,t))$ by a factor proportional to
$\lambda^{1/12}\exp(F_{\rho}(\lambda,0))$. This normalization will not only
capture the required Stokes jumps at $\pm l_{\infty}$, but will also allow us
to recover the constant map contribution of (2.4). Indeed, by Corollary 3.13,
this normalization introduces back the Borel sum of the constant map
constribution to the free energy. We will then relate the Stokes jumps of the
normalized partition function to the jumps of the Riemann–Hilbert problem of
Section 4.1. This will in turn show us how the BPS spectrum of the resolved
conifold is encoded in the Stokes jumps of $F_{\rho}(\lambda,t)$. We will use
the notation of Section 4.1 throughout.

To establish the link with Section 4.1 more clearly, it will be convenient to
consider the projectivized parameters

$t=v/w,\;\;\;\;\check{\lambda}=\lambda_{\mathrm{B}}/w,$ (4.20)

where we recall that $\check{\lambda}=\lambda/2\pi$. We think of the tuple
$(v,w)$ as a point of $M$.

We consider the rays $\mathcal{L}_{k}=\mathbb{R}_{<0}\cdot
2\pi{\mathrm{i}}(v+kw)=\mathbb{R}_{<0}\cdot Z_{\beta+k\delta}$ for
$k\in\mathbb{Z}$ and $\mathcal{L}_{\infty}=\mathbb{R}_{<0}\cdot
2\pi{\mathrm{i}}w=\mathbb{R}_{<0}\cdot Z_{k\delta}$. The relation to the old
Stokes rays is then given by $\mathcal{L}_{k}=w\cdot l_{k}$ and
$\mathcal{L}_{\infty}=w\cdot l_{\infty}$ .

######  Definition 4.2.

Given a ray $\rho$ different from
$\\{\pm\mathcal{L}_{k}\\}_{k\in\mathbb{Z}}\cup\\{\pm\mathcal{L}_{\infty}\\}$
we define the for $\lambda_{\mathrm{B}}\in\mathbb{H}_{\rho}$ and $(v,w)\in M$
with $\text{Im}(v/w)\neq 0$,

$\mathcal{F}_{\rho}(\lambda_{\mathrm{B}},v,w):=F_{w^{-1}\cdot\rho}\bigg{(}\frac{2\pi\lambda_{\mathrm{B}}}{w},\frac{v}{w}\bigg{)}=F_{w^{-1}\cdot\rho}(\lambda,t)\,.$
(4.21)

Notice that

$\mathcal{F}_{\rho}(\lambda_{\mathrm{B}},v,1)=F_{\rho}(\lambda,t)\,.$ (4.22)

Following the same argument as in Proposition 3.14, it is easy to check the
following:

######  Proposition 4.3.

Let $\rho_{k}$ be a ray in the sector determined by the rays $\mathcal{L}_{k}$
and $\mathcal{L}_{k-1}$. Then, if $\mathrm{Im}(v/w)>0$, on the overlap of
their domains of definition in the $\lambda_{\mathrm{B}}$ variable we have

$\Phi_{\pm\mathcal{L}_{k}}(\lambda_{\mathrm{B}},v,w):=\mathcal{F}_{\pm\rho_{k+1}}(\lambda_{\mathrm{B}},v,w)-\mathcal{F}_{\pm\rho_{k}}(\lambda_{\mathrm{B}},v,w)=\frac{1}{2\pi{\mathrm{i}}}\partial_{\lambda_{\mathrm{B}}}\Big{(}\lambda_{\mathrm{B}}\mathrm{Li}_{2}\big{(}e^{\pm
2\pi{\mathrm{i}}(v+kw)/\lambda_{\mathrm{B}}}\big{)}\Big{)}\;.$ (4.23)

If $\mathrm{Im}(v/w)<0$, then the previous jumps also hold provided
$\rho_{k+1}$ is interchanged with $\rho_{k}$ in the above formulas.

###### Proof.

After a change of integration variables, we obtain

$\mathcal{F}_{\rho_{k+1}}(\lambda_{\mathrm{B}},v,w)-\mathcal{F}_{\rho_{k}}(\lambda_{\mathrm{B}},v,w)=\frac{1}{w}\int_{\mathcal{H}(\mathcal{L}_{k})}\mathrm{d}\xi\;e^{-\xi/\lambda_{\mathrm{B}}}G(\xi/w,v/w)\;,$
(4.24)

where $\mathcal{H}(\mathcal{L}_{k})$ is a Hankel contour around
$\mathcal{L}_{k}$. The result then follows by summing over residues along the
poles in $\mathcal{L}_{k}$, as in Proposition 3.14. ∎

####  4.2.1 Normalizing the partition function and the Stokes jumps at $\pm
l_{\infty}$

We would like to first make sense of a limit of the form

$F_{\rho}(\lambda,0):=\lim_{t\to 0}F_{\rho}(\lambda,t)\,,$ (4.25)

where $t$ is taken to satisfy $\text{Re}(t)>0$, $\text{Im}(t)>0$; and such
that along the limit, $\rho$ is always between $l_{-1}$ and $l_{0}$ (resp.
$-l_{-1}$ and $-l_{0}$) if $\rho$ is on the right (resp. left) Borel half-
plane.

######  Lemma 4.4.

The limit $F_{\rho}(\lambda,0)$ from above exists for
$\lambda\in\mathbb{H}_{\rho}$. In fact, if $\rho$ is on the right (resp. left)
Borel plane, is can be analytically continued to
$\lambda\in\mathbb{C}^{\times}\setminus\mathbb{R}_{\leq 0}$ (resp.
$\lambda\in\mathbb{C}^{\times}\setminus\mathbb{R}_{\geq 0}$).

###### Proof.

Let us assume that $\rho$ is on the right Borel plane. Then by Propositions
3.11 and 3.8, and our condition on $t$, we have

$F_{\rho}(\lambda,t)=F_{\mathbb{R}_{>0}}(\lambda,t)=\log(G_{3}(t\,|\,\check{\lambda},1))\,,$
(4.26)

where we recall that $G_{3}$ is the function defined in terms of the triple
sine in Definition 3.7.

By Proposition 3.8, the function $G_{3}(t\,|\,\check{\lambda},1)$ has a well
defined value at $t=0$; and $G_{3}(0\,|\,\check{\lambda},1)$ is everywhere
regular, vanishing only at the points $\check{\lambda}\in\mathbb{Q}_{\leq 0}$.

In particular, for $\check{\lambda}\in\mathbb{H}_{\rho}$, we have

$F_{\rho}(\lambda,0)=\lim_{t\to
0}F_{\rho}(\lambda,t)=\log(G_{3}(0\,|\,\check{\lambda},1))\,,$ (4.27)

and we can analytically continue $F_{\rho}(\lambda,0)$ to
$\lambda\in\mathbb{C}^{\times}\setminus\mathbb{R}_{\leq 0}$.

If $\rho$ is on the left Borel plane, the statement follows from the previous
case, together with the relation $F_{\rho}(\lambda,t)=F_{-\rho}(-\lambda,t)$
from Lemma 3.16. ∎

Notice that by Lemma 4.4, we can also write

$F_{\rho}(\lambda,0)=\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(1)+\lim_{t\to
0}\bigg{(}\frac{B_{2}}{2}\mathrm{Li}_{1}(e^{2\pi{\mathrm{i}}t})+\int_{\rho}\mathrm{d}\xi\,e^{-\xi/\check{\lambda}}G(\xi,t)\bigg{)}\,,$
(4.28)

where the limit in $t$ is assumed to satisfy the constraints from above.

The following proposition suggests that one can obtain the appropriate Stokes
jumps at $\pm l_{\infty}$ by considering a normalization involving
$F_{\rho}(\lambda,0)$:

######  Proposition 4.5.

Let $\rho$ (resp. $\rho^{\prime}$) be a ray close to
$l_{\infty}={\mathrm{i}}\mathbb{R}_{<0}$ from the left (resp. right). Then for
$\lambda$ in their common domain of definition

$F_{\pm\rho}(\lambda,0)-F_{\pm\rho^{\prime}}(\lambda,0)=\frac{1}{\pi{\mathrm{i}}}\sum_{k\geq
1}\partial_{\check{\lambda}}\Big{(}\check{\lambda}\,\mathrm{Li}_{2}\big{(}e^{\pm
2\pi{\mathrm{i}}k/\check{\lambda}}\big{)}\Big{)}-\frac{{\mathrm{i}}\pi}{12}\;.$
(4.29)

Furthermore $F_{\rho}(\lambda,0)$ only has Stokes jumps along $\pm
l_{\infty}$.

###### Proof.

First, notice that by our definition of the limit in $t$, we have

$\begin{split}F_{\rho}(\lambda,0)-F_{\rho^{\prime}}(\lambda,0)&=\lim_{t\to
0}\Big{(}\int_{\rho}\mathrm{d}\xi\,e^{-\xi/\check{\lambda}}G(\xi,t)-\int_{\rho^{\prime}}\mathrm{d}\xi\,e^{-\xi/\check{\lambda}}G(\xi,t)\Big{)}\\\
&=\lim_{t\to
0}\int_{\mathcal{H}}\mathrm{d}\xi\,e^{-\xi/\check{\lambda}}G(\xi,t)\,,\end{split}$
(4.30)

where $\mathcal{H}=\rho-\rho^{\prime}$ denotes a Hankel contour along
${\mathrm{i}}\mathbb{R}_{<0}$, containing $l_{k}$ for $k\geq 0$ and $-l_{k}$
for $k<0$. Hence, for $\lambda$ close to $l_{\infty}$, the Hankel contour just
gives the contribution of these rays that we previously computed:

$\begin{split}F_{\rho}(\lambda,0)-F_{\rho^{\prime}}(\lambda,0)&=\lim_{t\to
0}\int_{\mathcal{H}}\mathrm{d}\xi\,e^{-\xi/\check{\lambda}}G(\xi,t)\\\
&=\lim_{t\to 0}\bigg{(}\frac{1}{2\pi{\mathrm{i}}}\sum_{k\geq
1}\Big{[}\partial_{\check{\lambda}}\Big{(}\check{\lambda}\,\mathrm{Li}_{2}(e^{2\pi{\mathrm{i}}(t+k)/\check{\lambda}})\Big{)}+\partial_{\check{\lambda}}\Big{(}\check{\lambda}\,\mathrm{Li}_{2}(e^{-2\pi{\mathrm{i}}(t-k)/\check{\lambda}})\Big{)}\Big{]}\\\
&\quad\quad\quad\quad+\frac{1}{2\pi{\mathrm{i}}}\partial_{\check{\lambda}}\Big{(}\mathrm{Li}_{2}(e^{2\pi{\mathrm{i}}t/\check{\lambda}})\Big{)}\bigg{)}\\\
&=\frac{1}{\pi{\mathrm{i}}}\sum_{k\geq
1}\partial_{\check{\lambda}}\Big{(}\check{\lambda}\,\mathrm{Li}_{2}(e^{2\pi{\mathrm{i}}k/\check{\lambda}})\Big{)}+\frac{1}{2\pi{\mathrm{i}}}\mathrm{Li}_{2}(1)\\\
&=\frac{1}{\pi{\mathrm{i}}}\sum_{k\geq
1}\partial_{\check{\lambda}}\Big{(}\check{\lambda}\,\mathrm{Li}_{2}(e^{2\pi{\mathrm{i}}k/\check{\lambda}})\Big{)}-\frac{\pi{\mathrm{i}}}{12}\;.\end{split}$
(4.31)

A similar argument follows for $-l_{\infty}={\mathrm{i}}\mathbb{R}_{>0}$.
Furthermore, the fact that there are no other Stokes jumps follows from the
way we have defined the limit $F_{\rho}(\lambda,0)$. For example, if $\rho$
and $\rho^{\prime}$ are both on the right Borel plane, then along the limit
$\rho$ and $\rho^{\prime}$ are both between $l_{0}$ and $l_{-1}$, and hence
$F_{\rho}(\lambda,0)=F_{\rho^{\prime}}(\lambda,0)$. ∎

We can as before projectivize, and define

$\mathcal{F}_{\rho}(\lambda_{\mathrm{B}},0,w):=\lim_{v\to
0}\mathcal{F}_{\rho}(\lambda_{\mathrm{B}},v,w)\,,$ (4.32)

where the limit in $v$ is such that $t=v/w$ satisfies the conditions of the
previous limit in $F_{\rho}(\lambda,0)$.

######  Proposition 4.6.

Let $\rho$ (resp. $\rho^{\prime}$) be a ray close to
$\mathcal{L}_{\infty}={\mathrm{i}}\mathbb{R}_{<0}$ from the left (resp.
right). Then for $\lambda_{\mathrm{B}}$ in their common domain of definition

$\mathcal{F}_{\pm\rho}(\lambda_{\mathrm{B}},0,w)-\mathcal{F}_{\pm\rho^{\prime}}(\lambda_{\mathrm{B}},0,w)=\sum_{k\geq
1}\Phi_{\pm\mathcal{L}_{\infty},k}-\frac{\pi{\mathrm{i}}}{12}\,,$ (4.33)

where

$\Phi_{\pm\mathcal{L}_{\infty},k}(\lambda_{\mathrm{B}},v,w):=\frac{1}{\pi{\mathrm{i}}}\partial_{\lambda_{\mathrm{B}}}\Big{(}\lambda_{\mathrm{B}}\mathrm{Li}_{2}\big{(}e^{\pm
2\pi{\mathrm{i}}kw/\lambda_{\mathrm{B}}}\big{)}\Big{)}$ (4.34)

Furthermore, $\mathcal{F}_{\pm\rho}(\lambda_{\mathrm{B}},0,w)$ only has Stokes
jumps along $\pm\mathcal{L}_{\infty}$.

On the other hand, Corollary 3.13 and (2.20) suggest to not only normalize by
$\mathcal{F}_{\rho}(\lambda_{B},0,w)$ but to also normalize by a
$\frac{1}{12}\log(\lambda_{B}/w)$ term as follows:

######  Definition 4.7.

Given a ray $\rho$ different from
$\\{\pm\mathcal{L}_{k}\\}_{k\in\mathbb{Z}}\cup\\{\pm\mathcal{L}_{\infty}\\}$
we define the following for $\lambda_{\mathrm{B}}\in\mathbb{H}_{\rho}$ and
$(v,w)\in M$ with $\text{Im}(v/w)>0$:

$\widehat{\mathcal{F}}_{\rho}(\lambda_{\mathrm{B}},v,w):=\mathcal{F}_{\rho}(\lambda_{\mathrm{B}},v,w)-\mathcal{F}_{\rho}(\lambda_{\mathrm{B}},0,w)-\frac{1}{12}\log\Big{(}\frac{\lambda_{B}}{w}\Big{)}\,,$
(4.35)

where we place the branch cut of the $\log$ term at
$l_{\infty}=\mathrm{i}\mathbb{R}_{<0}$. We also define the normalized
partition function as

$\widehat{Z}_{\rho}(\lambda_{\mathrm{B}},v,w):=\exp(\widehat{\mathcal{F}}_{\rho}(\lambda_{\mathrm{B}},v,w))\,.$
(4.36)

We remark that due to (2.20) and Corollary 3.13, asymptotic expansion of the
the normalized free energy $\widehat{\mathcal{F}}_{\rho}(\lambda,t,1)$ as
$\lambda\to 0$ recovers the full $\mathcal{F}(Q,\lambda)$ of (2.4) up to an
overall shift by a constant, which can be absorbed in a redefinition of
$\widehat{\mathcal{F}}_{\rho}(\lambda_{\mathrm{B}},v,w)$.

We immediately obtain the following:

######  Corollary 4.8.

The Stokes jumps of $\widehat{\mathcal{F}}(\lambda_{B},v,w)$ at
$\pm\mathcal{L}_{k}$ are given by (4.23), while (using the notation of
Proposition 4.6) we find that the Stokes jumps at $\pm\mathcal{L}_{\infty}$
are determined by

$-\mathcal{F}_{\pm\rho}(\lambda_{\mathrm{B}},0,w)+\mathcal{F}_{\pm\rho^{\prime}}(\lambda_{\mathrm{B}},0,w)-\lim_{\lambda_{B}\to\pm\mathcal{L}_{\infty}^{+}}\frac{1}{12}\log\Big{(}\frac{\lambda_{B}}{w}\Big{)}-\lim_{\lambda_{B}\to\pm\mathcal{L}_{\infty}^{-}}\frac{1}{12}\log\Big{(}\frac{\lambda_{B}}{w}\Big{)}=-\sum_{k\geq
1}\Phi_{\pm\mathcal{L}_{\infty},k}\mp\frac{\pi{\mathrm{i}}}{12}\,,$ (4.37)

where $\lim_{\lambda_{B}\to\pm\mathcal{L}_{\infty}^{+}}$ (resp.
$\lim_{\lambda_{B}\to\pm\mathcal{L}_{\infty}^{-}}$) denotes the anti-clockwise
(resp. clockwise) limit to $\pm\mathcal{L}_{\infty}$.

###### Proof.

The only new thing to notice is that because of the $\log(\check{\lambda})$
term with branch cut at $l_{\infty}$ of the normalization, we find that

$-\lim_{\lambda_{B}\to\mathcal{L}_{\infty}^{+}}\frac{1}{12}\log\Big{(}\frac{\lambda_{B}}{w}\Big{)}-\lim_{\lambda_{B}\to\mathcal{L}_{\infty}^{-}}\frac{1}{12}\log\Big{(}\frac{\lambda_{B}}{w}\Big{)}=-\frac{2\pi\mathrm{i}}{12}=-\frac{\pi\mathrm{i}}{6}\,,$
(4.38)

while

$-\lim_{\lambda_{B}\to-\mathcal{L}_{\infty}^{+}}\frac{1}{12}\log\Big{(}\frac{\lambda_{B}}{w}\Big{)}-\lim_{\lambda_{B}\to-\mathcal{L}_{\infty}^{-}}\frac{1}{12}\log\Big{(}\frac{\lambda_{B}}{w}\Big{)}=0\,.$
(4.39)

Combining these jumps with (minus) the jumps of (4.33) gives the desired
result.

∎

####  4.2.2 Relation to the Riemann–Hilbert problem

In this subsection, we wish to relate the Stokes jumps of $\widehat{Z}_{\rho}$
with the Stokes jumps of the Riemann–Hilbert problem in Section 4.1. We will
see that the jumps of $\widehat{Z}_{\rho}$ serve as “potentials” for the jumps
of the RH problem.

For the discussion below, it will be useful to note that the Stokes jumps can
be represented in terms of the double sine function, revealing some important
properties. A useful review of definition and relevant properties of the
double sine function $\sin_{2}(z\,|\,\omega_{1},\omega_{2})$ can be found in
[Bri20, Section 4] and references therein.

######  Definition 4.9.

For $z\in\mathbb{C}$ and $\omega_{1},\omega_{2}\in\mathbb{C}^{\times}$, let

$F(z\,|\,\omega_{1},\omega_{2}):=\exp\Big{(}-\frac{\pi{\mathrm{i}}}{2}B_{2,2}(z\,|\,\omega_{1},\omega_{2})\Big{)}\cdot\sin_{2}(z\,|\,\omega_{1},\omega_{2})$
(4.40)

where $B_{2,2}(z\,|\,\omega_{1},\omega_{2})$ is a multiple Bernoulli
polynomial.

We will use the following properties of the function $F$.

######  Proposition 4.10.

(See [Bri20, Proposition 4.1]) The function $F(z\,|\,\omega_{1},\omega_{2})$
is a single valued meromorphic function of the variables $z\in\mathbb{C}$ and
$\omega_{1},\omega_{2}\in\mathbb{C}^{\times}$ under the assumption
$\omega_{1}/\omega_{2}\not\in\mathbb{R}_{<0}$. It has the following
properties:

  * •

The function is regular and non-vanishing except at the points

$z=a\omega_{1}+b\omega_{2},\quad a,b\in\mathbb{Z}\,.$ (4.41)

  * •

It is invariant under simultaneous rescaling of the three arguments, and
symmetric in $\omega_{1},\omega_{2}$.

  * •

It satisfies the difference equation

$\frac{F(z+\omega_{1}\,|\,\omega_{1},\omega_{2})}{F(z\,|\,\omega_{1},\omega_{2})}=(1-e^{2\pi{\mathrm{i}}z/\omega_{2}})^{-1}\,.$
(4.42)

  * •

When $\mathrm{Re}(\omega_{i})>0$ and
$0<\mathrm{Re}(z)<\mathrm{Re}(\omega_{1}+\omega_{2})$ there is an integral
representation

$F(z\,|\,\omega_{1},\omega_{2})=\exp\Big{(}\int_{\mathbb{R}+{\mathrm{i}}0^{+}}\frac{\mathrm{d}u}{u}\frac{e^{zu}}{(e^{\omega_{1}u}-1)(e^{\omega_{2}u}-1)}\Big{)}\,.$
(4.43)

######  Definition 4.11.

Let $\Phi_{\pm\mathcal{L}_{k}}(\lambda_{\mathrm{B}},v,w)$ and
$\Phi_{\pm\mathcal{L}_{\infty},k}(\lambda_{\mathrm{B}},v,w)$ be as in (4.23)
and (4.34), respectively. We then define333The minus sign in the exponent of
the second expression in (4.44) is due to the normalization of the partition
function.

$\Xi_{\pm\mathcal{L}_{k}}(v,w,\lambda_{\mathrm{B}}):=e^{\Phi_{\pm\mathcal{L}_{k}}(\lambda_{\mathrm{B}},v,w)},\;\;\;\;\;\Xi_{\pm\mathcal{L}_{\infty},k}(v,w,\lambda_{\mathrm{B}}):=e^{-\Phi_{\pm\mathcal{L}_{\infty},k}(\lambda_{\mathrm{B}},v,w)}\;.$
(4.44)

We now give a proposition relating the exponentials of the Stokes jumps (4.44)
of $\widehat{Z}_{\rho}$ to the function $F$.

######  Proposition 4.12.

Assuming $\mathrm{Im}(v/w)>0$, we can write on their common domains of
definition

$\begin{split}\Xi_{\pm\mathcal{L}_{k}}(v,w,\lambda_{\mathrm{B}})&=(F(\pm(v+kw)/\lambda_{\mathrm{B}}+1\,|\,1,1))^{-1}\\\
\Xi_{\pm\mathcal{L}_{\infty},k}(v,w,\lambda_{\mathrm{B}})&=(F(\pm
kw/\lambda_{\mathrm{B}}+1\,|\,1,1))^{2}\,.\end{split}$ (4.45)

Furthermore, the functions on the right of (4.45) are holomorphic and non-
vanishing for $\lambda_{\mathrm{B}}\in\mathbb{H}_{\pm\mathcal{L}_{k}}$ and
$\lambda_{\mathrm{B}}\in\mathbb{H}_{\pm\mathcal{L}_{\infty}}$, respectively.

###### Proof.

First, notice that

$\begin{split}\Phi_{\mathcal{L}_{k}}(\lambda_{\mathrm{B}},v,w)&=\frac{1}{2\pi{\mathrm{i}}}\partial_{\lambda_{\mathrm{B}}}\Big{(}\lambda_{\mathrm{B}}\mathrm{Li}_{2}\big{(}e^{2\pi{\mathrm{i}}(v+kw)/\lambda_{\mathrm{B}}})\Big{)}\\\
&=\frac{1}{2\pi{\mathrm{i}}}\partial_{\check{\lambda}}\Big{(}\check{\lambda}\mathrm{Li}_{2}(e^{2\pi{\mathrm{i}}(t+k)/\check{\lambda}})\Big{)}\\\
&=\phi_{l_{k}}(\lambda,t)\,.\end{split}$ (4.46)

In particular, if $w=e^{2\pi{\mathrm{i}}t/\check{\lambda}}$ and
$\widetilde{q}=e^{2\pi{\mathrm{i}}k/\check{\lambda}}$, then for $\lambda$ near
$l_{k}$, we can expand

$\Phi_{\mathcal{L}_{k}}(\lambda_{\mathrm{B}},v,w)=\frac{1}{2\pi{\mathrm{i}}}\sum_{l=1}^{\infty}\frac{(w\widetilde{q}^{k})^{l}}{l}\Big{(}\frac{1}{l}-\log(w\widetilde{q}^{k})\Big{)}\,.$
(4.47)

On the other hand, assuming for the moment that
$0<\text{Re}(t+k)<\text{Re}(\check{\lambda})$, $\text{Re}(\check{\lambda})>0$,
and using the scaling invariance of $F$, we find by Proposition 4.10 that

$\begin{split}F((v+kw)/\lambda_{\mathrm{B}}+1\,|\,1,1)&=F((t+k)/\check{\lambda}+1\,|\,1,1)\\\
&=F(t+k+\check{\lambda}\,|\,\check{\lambda},\check{\lambda})\\\
&=\exp\Big{(}\int_{\mathbb{R}+i0^{+}}\frac{\mathrm{d}u}{u}\frac{e^{u(t+k)}}{(2\sinh(\check{\lambda}u/2))^{2}}\Big{)}\,.\end{split}$
(4.48)

Using that $\text{Im}(v/w)=\text{Im}(t)>0$, we can compute the previous
integral by closing the contour in the upper half-plane and sum over the
residues at $2\pi{\mathrm{i}}l/\check{\lambda}$ for $l\geq 1$, obtaining

$\begin{split}\int_{\mathbb{R}+i0^{+}}\frac{\mathrm{d}u}{u}\frac{e^{u(t+k)}}{(2\sinh(\check{\lambda}u/2))^{2}}&=2\pi{\mathrm{i}}\sum_{l=1}^{\infty}\Big{(}\frac{1}{\check{\lambda}}\Big{)}^{2}\frac{\partial}{\partial
u}\frac{e^{u(t+k)}}{u}\Big{|}_{u=2\pi{\mathrm{i}}l/\check{\lambda}}\\\
&=-\frac{1}{2\pi{\mathrm{i}}}\sum_{l=1}^{\infty}\frac{(w\widetilde{q}^{k})^{l}}{l}\Big{(}\frac{1}{l}-\log(w\widetilde{q}^{k})\Big{)}\,.\end{split}$
(4.49)

Comparing (4.47) with (4.49), the first result then follows, with the other
cases being analogous.

To check the second statement, we use the fact from Proposition 4.10 that the
function $F(z\,|\,\omega_{1},\omega_{2})$ is regular and non-vanishing except
at the points

$z=a\omega_{1}+b\omega_{2},\quad a,b\in\mathbb{Z}\,.$ (4.50)

We then find that $F(\pm(v+kw)/\lambda_{\mathrm{B}}+1\,|\,1,1)$ is regular
except at the points where

$(v+kw)/\lambda_{\mathrm{B}}\in\mathbb{Z}\quad\iff\lambda_{\mathrm{B}}=\frac{v+kw}{n},\quad
n\in\mathbb{Z}\,.$ (4.51)

In particular, $F(\pm(v+kw)/\lambda_{\mathrm{B}}+1\,|\,1,1)$ is regular for
$\lambda_{\mathrm{B}}$ in the half-plane centered at
$\pm\mathcal{L}_{k}=\pm\mathbb{R}_{<0}\cdot 2\pi{\mathrm{i}}(v+nw)$. The other
case follows similarly. ∎

From the previous proposition we obtain the following corollary:

######  Corollary 4.13.

Let $(v,w)\in M$ such that $\mathrm{Im}(v/w)>0$. Then
$\Xi_{\pm\mathcal{L}_{k}}$ and $\Xi_{\pm\mathcal{L}_{\infty},k}$ serve as
potentials for the jumps of the Riemann–Hilbert problem of Section 4.1, in the
sense that

$\begin{split}\frac{\Xi_{\pm\mathcal{L}_{k}}(v+\lambda_{\mathrm{B}},w,\lambda_{\mathrm{B}})}{\Xi_{\pm\mathcal{L}_{k}}(v,w,\lambda_{\mathrm{B}})}&=(1-e^{\pm
2\pi{\mathrm{i}}(v+kw)/\lambda_{\mathrm{B}}})^{\pm
1}=(1-\mathcal{X}_{\pm(\beta+k\delta)})^{\pm\langle\beta^{\vee},\beta+k\delta\rangle\Omega(\beta+k\delta)},\\\
\frac{\Xi_{\pm\mathcal{L}_{k}}(v,w+\lambda_{\mathrm{B}},\lambda_{\mathrm{B}})}{\Xi_{\pm\mathcal{L}_{k}}(v,w,\lambda_{\mathrm{B}})}&=(1-e^{\pm
2\pi{\mathrm{i}}(v+kw)/\lambda_{\mathrm{B}}})^{\pm
k}=(1-\mathcal{X}_{\pm(\beta+k\delta)})^{\pm\langle\delta^{\vee},\beta+k\delta\rangle\Omega(\beta+k\delta)},\\\
\frac{\Xi_{\pm\mathcal{L}_{\infty},k}(0,w+\lambda_{\mathrm{B}},\lambda_{\mathrm{B}})}{\Xi_{\pm\mathcal{L}_{\infty},k}(0,w,\lambda_{\mathrm{B}})}&=(1-e^{\pm
2\pi{\mathrm{i}}kw/\lambda_{\mathrm{B}}})^{\mp 2k}=(1-\mathcal{X}_{\pm
k\delta})^{\pm\langle\delta^{\vee},k\delta\rangle\Omega(k\delta)}.\\\
\end{split}$ (4.52)

###### Proof.

We use the fact from Proposition 4.10 that $F(z\,|\,1,1)$ satisfies the
difference equation

$\frac{F(z+1\,|\,1,1)}{F(z\,|\,1,1)}=\frac{1}{1-e^{2\pi{\mathrm{i}}z}}\,.$
(4.53)

Then by Proposition 4.12 we find that

$\frac{\Xi_{\mathcal{L}_{k}}(v+\lambda_{\mathrm{B}},w,\lambda_{\mathrm{B}})}{\Xi_{\mathcal{L}_{k}}(v,w,\lambda_{\mathrm{B}})}=\bigg{(}\frac{F((v+kw)/\lambda_{\mathrm{B}}+1+1|1,1)}{F((v+kw)/\lambda_{\mathrm{B}}+1,|1,1)}\bigg{)}^{-1}=1-e^{2\pi{\mathrm{i}}(v+kw)/\lambda_{\mathrm{B}}}\,.$
(4.54)

The other identities follow similarly.

∎

######  Remark 4.14.

From corollary 4.13 we see how the BPS spectrum is encoded in the jumps of the
normalized partition function. Namely, the BPS spectrum of the resolved
conifold appears in the expressions of the jumps of
$\widehat{Z}_{\rho}(\lambda_{B},v,w)$ as follows:

$\begin{split}\Xi_{\pm\mathcal{L}_{k}}(v,w,\lambda_{\mathrm{B}})&=\exp\bigg{(}\frac{\Omega(\beta+k\delta)}{2\pi{\mathrm{i}}}\partial_{\lambda_{\mathrm{B}}}\Big{(}\lambda_{\mathrm{B}}\mathrm{Li}_{2}(e^{\pm
Z_{\beta+k\delta}/\lambda_{\mathrm{B}}})\Big{)}\bigg{)}\\\
\Xi_{\pm\mathcal{L}_{\infty},k}(v,w,\lambda_{\mathrm{B}})&=\exp\bigg{(}\frac{\Omega(k\delta)}{2\pi{\mathrm{i}}}\partial_{\lambda_{\mathrm{B}}}\Big{(}\lambda_{\mathrm{B}}\mathrm{Li}_{2}\big{(}e^{\pm
Z_{k\delta}/\lambda_{\mathrm{B}}}\big{)}\Big{)}\bigg{)}\,,\end{split}$ (4.55)

making explicit how the DT-invariant are encoded in the jumps.

###  4.3 The line bundle defined by the normalized partition function

We would like to discuss how to define a line bundle
$\mathcal{L}\to\mathbb{C}^{\times}\times M$, such that the partition functions
$\widehat{Z}_{\rho}(\lambda_{\mathrm{B}},v,w)$ define a section of
$\mathcal{L}$.

To concretely define the line bundle, we restrict for simplicity to444See
Remark 4.16 for the other points of $M$.

$M_{+}:=\\{(v,w)\in M\;|\;\text{Im}(v/w)>0\\}\,.$ (4.56)

Furthermore, let $\rho_{k}$ be a ray between $\mathcal{L}_{k}$ and
$\mathcal{L}_{k-1}$. For definiteness, we pick $\rho_{k}$ to be always in the
middle of $\mathcal{L}_{k}$ and $\mathcal{L}_{k-1}$, and consider the open
sets

$\begin{split}U_{k}^{\pm}&:=\\{(\lambda_{\mathrm{B}},v,w)\in\mathbb{C}^{\times}\times
M_{+}\;|\;\lambda_{\mathrm{B}}\in\mathbb{H}_{\pm\rho_{k}}\\}\,.\end{split}$
(4.57)

We remark that the condition on $\lambda_{\mathrm{B}}$ actually depends on
$(v,w)$, since the latter specifies the rays $\mathcal{L}_{k}$ (and hence also
$\rho_{k}$). We then clearly have that
$\\{U_{k}^{+}\\}_{k\in\mathbb{Z}}\cup\\{U_{k}^{-}\\}_{k\in\mathbb{Z}}$ forms
an open cover of $\mathbb{C}^{\times}\times M_{+}$.

If $U_{k_{1}}^{+}\cap U_{k_{2}}^{+}\neq\emptyset$ for $k_{1}<k_{2}$, we then
define for $(\lambda_{\mathrm{B}},v,w)\in U_{k_{1}}^{+}\cap U_{k_{2}}^{+}$,

$g^{+}_{k_{1},k_{2}}(\lambda_{\mathrm{B}},v,w):=\prod_{k_{1}\leq
k<k_{2}}\Xi_{\mathcal{L}_{k}}(\lambda_{\mathrm{B}},v,w)\,.$ (4.58)

Notice that if $(\lambda_{\mathrm{B}},v,w)\in U_{k_{1}}^{+}\cap
U_{k_{2}}^{+}$, then
$(\lambda_{\mathrm{B}},v,w)\in\mathbb{H}_{\mathcal{L}_{k}}$ for $k_{1}\leq
k\leq k_{2}$, so by Proposition 4.12 we have that $g^{+}_{k_{1},k_{2}}$ is
$\mathbb{C}^{\times}$-valued:

$g^{+}_{k_{1},k_{2}}\colon U_{k_{1}}^{+}\cap
U_{k_{2}}^{+}\to\mathbb{C}^{\times}\,.$ (4.59)

With the assumptions $U_{k_{1}}^{+}\cap U_{k_{2}}^{+}\neq\emptyset$ for
$k_{1}<k_{2}$, we also define