Quantum Curves, Resurgence and Exact WKB

Quantum Curves, Resurgence and Exact WKB

Murad ALIM a, Lotte HOLLANDS b and Iván TULLI a

M. Alim, L. Hollands and I. Tulli

a) Fachbereich Mathematik, Universität Hamburg, Bundesstr. 55, 20146 Hamburg,
Germany<EMAIL_ADDRESS><EMAIL_ADDRESS>
b) Department of Mathematics at Heriot-Watt University,
b) Maxwell Institute for Mathematical Sciences, Edinburgh EH14 4AS, UK
<EMAIL_ADDRESS>
Received June 09, 2022, in final form February 08, 2023; Published online
March 06, 2023

We study the non-perturbative quantum geometry of the open and closed
topological string on the resolved conifold and its mirror. Our tools are
finite difference equations in the open and closed string moduli and the
resurgence analysis of their formal power series solutions. In the closed
setting, we derive new finite difference equations for the refined partition
function as well as its Nekrasov–Shatashvili (NS) limit. We write down a
distinguished analytic solution for the refined difference equation that
reproduces the expected non-perturbative content of the refined topological
string. We compare this solution to the Borel analysis of the free energy in
the NS limit. We find that the singularities of the Borel transform lie on
infinitely many rays in the Borel plane and that the Stokes jumps across these
rays encode the associated Donaldson–Thomas invariants of the underlying
Calabi–Yau geometry. In the open setting, the finite difference equation
corresponds to a canonical quantization of the mirror curve. We analyze this
difference equation using Borel analysis and exact WKB techniques and identify
the 5d BPS states in the corresponding exponential spectral networks. We
furthermore relate the resurgence analysis in the open and closed setting.
This guides us to a five-dimensional extension of the
Nekrasov–Rosly–Shatashvili proposal, in which the NS free energy is computed
as a generating function of $q$-difference opers in terms of a special set of
spectral coordinates. Finally, we examine two spectral problems describing the
corresponding quantum integrable system.

resolved conifold; topological string theory; Borel summation; difference
equations; exponential spectral networks

40G10; 39A70; 81T30

###### Contents

  1. 1 Introduction
  2. 2 Topological string free energies
    1. 2.1 GW and GV expansions
    2. 2.2 Refinement of topological string theory
    3. 2.3 Topological string free energies of the resolved conifold
    4. 2.4 Mirror geometry
    5. 2.5 Picard–Fuchs equation and classical terms
    6. 2.6 Geometric engineering
  3. 3 WKB analysis of quantum mirror curves
    1. 3.1 The quantum mirror curve
    2. 3.2 All-order WKB solution
    3. 3.3 Exact solutions
  4. 4 Quantum curves of closed string moduli
    1. 4.1 The refined difference equation
    2. 4.2 A non-perturbative solution
    3. 4.3 Difference equations in the NS limit
    4. 4.4 Unrefined topological string
    5. 4.5 Relations between free energies through difference equations
    6. 4.6 A quantum curve for closed string moduli
    7. 4.7 Quantum Picard–Fuchs operator
  5. 5 Borel sums and Stokes phenomena
    1. 5.1 Main results
    2. 5.2 Borel transform
    3. 5.3 Borel sum
    4. 5.4 Stokes jumps
    5. 5.5 Limits to $\pm l_{\infty}$
    6. 5.6 A potential for the Borel sums
    7. 5.7 Relation between $\Psi_{\rho}$ and $W_{\rho}$
  6. 6 DT invariants and line bundles
    1. 6.1 Normalized $W_{\rho}$ and the DT invariants
    2. 6.2 Line bundle defined by $\hat{W}_{\rho}$
  7. 7 Gauge theory, exact WKB and integrable systems
    1. 7.1 Resurgence and BPS states
    2. 7.2 Exponential spectral networks
    3. 7.3 Exact WKB for difference operators
    4. 7.4 Spectral coordinates
    5. 7.5 5d NRS proposal
    6. 7.6 Spectral problems and exact quantization conditions
  8. 8 Discussion and outlook
  9. A Some technical or long proofs
    1. A.1 Proof of the Borel transform
    2. A.2 The Borel sum along $\mathbb{R}_{>0}$
    3. A.3 Proof of the limits to $l_{\infty}$
  10. B Special functions
    1. B.1 Multiple gamma and multiple sine functions
    2. B.2 Quantum dilogarithm and its properties
    3. B.3 Product forms
    4. Acknowledgements

##  1 Introduction

A recurring fascinating insight of the interaction of mathematics and physics
is that mathematical invariants and structures are most naturally encoded in
terms of the data of physical theories. This is in particular the case for
supersymmetric quantum field theories and string theories whose partition
functions often have interpretations as generating functions of invariants of
manifolds and whose parameter spaces become mathematical moduli spaces.

The physical partition functions which encode the mathematical invariants
often correspond however to asymptotic formal power series with zero radius of
convergence which is a common feature of series obtained in quantum field and
string theories using a perturbative approach. A theme which has generated a
considerable amount of excitement recently is the realization that the
systematic mathematical treatment of the asymptotic series using the ideas of
resurgence leads to the uncovering of further deep mathematical structures as
well as physical insights, see, e.g., [11, 78] and references therein.

Mathematically, when the Borel resummation of the asymptotic series is
considered it turns out that other sets of invariants are encoded in the
Stokes jumps of different Borel resummations, see for example [31, 44, 45,
55], see also [47] for a study of resurgence for the quantum dilogarithm
function, which is a building block of many interesting objects in
mathematical physics and in particular in Chern–Simons theories. In [9], the
techniques of [47] were used to study the Borel resummation of the
Gromov–Witten potential $F^{\mathrm{top}}(\lambda,t)$ for the resolved
conifold. Earlier results on the Borel resummation for the resolved conifold
with different techniques and scope were obtained in [59, 91]. The Borel
transform has infinitely many singularities organized along rays coinciding
with the rays $\pm\mathbb{R}_{<0}Z_{\gamma}$, where $Z_{\gamma}$ denotes the
central charge of a BPS state of charge $\gamma$. Different Borel resummations
were defined along rays which avoid the singularities, and it was found that
they experience Stokes jumps across the rays $\pm\mathbb{R}_{<0}Z_{\gamma}$,
with the BPS charge $\gamma\in\Gamma$ contributing to the jump by [9]:

$\Delta_{\gamma}F^{\rm
top}_{\textrm{Borel}}(\lambda,t)=\frac{\Omega(\gamma)}{2\pi\mathrm{i}}\partial_{\check{\lambda}}\big{(}\check{\lambda}{\rm
Li}_{2}\big{(}{\rm
e}^{Z_{\gamma}/\check{\lambda}}\big{)}\big{)},\qquad\check{\lambda}=\frac{\lambda}{2\pi},$
(1.1)

where $\Omega(\gamma)$ correspond to the Donaldson–Thomas (DT) invariant. The
identification of the DT invariants was established by providing the link to a
Riemann–Hilbert problem put forward by Bridgeland in [20] and applied to the
resolved conifold in [21].

The Borel analysis furthermore allowed to connect to previous proposals for
definitions of non-perturbative topological string theory and elucidate their
overlaps of validity. The Borel summation along a distinguished ray for
instance gave an expression previously proposed in [58, 59], while a limiting
expression obtained from the latter through infinitely many jumps gave the
Gopakumar–Vafa expression for the resummation of the free energies. In the
work of Bridgeland it was suggested that a tau-function, obtained as a
solution of a Riemann–Hilbert problem defined from the wall-crossing structure
of Donaldson–Thomas invariants, provides a non-perturbative completion of the
Gromov–Witten potential. In further developments, a difference equation was
obtained from the asymptotic expansion of the Gromov–Witten free energy of the
resolved conifold in [6], while it was found in [8] that this difference
equation admits an analytic solution in $\lambda$ related to Bridgeland’s tau-
function, whose non-perturbative content was shown in [7] to match earlier
expectations of [58, 59].

Moreover, following the ideas of [26, 27], it was found in [9] that the
exponentials of the Borel summations
$\exp\big{(}F^{\mathrm{top}}_{\rho}\big{)}$ along the ray $\rho$ form a
collection of local sections of the conformal limit of a certain
hyperholomorphic line bundle previously considered in [5, 84], whose
transition functions are the exponentials of the above Stokes factors, leading
to a new geometrical, non-perturbative picture of the topological string.

In parallel developments in a different context, it was realized in [60, 63]
that Borel resummation plays a central role in the geometric formulation of
the effective twisted superpotential $\mathcal{W}^{\mathrm{eff}}$ of a four-
dimensional $\mathcal{N}=2$ theory $T_{\mathrm{4d}}$ of class S in the
$\frac{1}{2}\Omega$-background
$\mathbb{R}^{2}_{\epsilon}\times\mathbb{R}^{2}$. This was motivated by a
conjecture of Nekrasov, Rosly and Shatashvili [87], which says that the
superpotential $\mathcal{W}^{\mathrm{eff}}$ may be obtained as a generating
function of opers in terms of a special kind of Darboux coordinates on the
associated moduli space of complexified flat connections. It was found that
the NRS Darboux coordinates $\\{x_{i},y^{i}\\}$ can be expressed in terms of
the Borel summation $\mathcal{B}_{\theta}$ of the quantum periods
$\oint_{\gamma}\lambda^{\mathrm{qu}}(\epsilon)$ in a certain critical
direction with phase $\vartheta$.111To be more precise, this critical
direction with phase $\vartheta$ corresponds to a ray in the Borel plane with
infinitely many singularities. The Borel summation $\mathcal{B}_{\vartheta}$
is not defined precisely in this critical direction. Instead we refer to the
median of the so-called lateral Borel summation, which averages the result of
the two Borel summations $\mathcal{B}_{\vartheta^{+}}$ and
$\mathcal{B}_{\vartheta^{-}}$. That is,

$x_{i}=\frac{\epsilon}{\pi\mathrm{i}}\log\bigg{(}\mathcal{B}_{\vartheta}\oint_{A^{i}}\lambda^{\mathrm{qu}}(\epsilon)\bigg{)}\qquad\mathrm{and}\qquad
y^{i}=\frac{1}{2\epsilon}\log\bigg{(}\mathcal{B}_{\vartheta}\oint_{B_{i}}\lambda^{\mathrm{qu}}(\epsilon)\bigg{)}.$
(1.2)

In terms of these Borel sums, the superpotential $\mathcal{W}^{\mathrm{eff}}$
may be obtained as the generating function

$y^{i}=\frac{1}{\epsilon}\frac{\partial\mathcal{W}^{\mathrm{eff}}(x,\epsilon)}{\partial
x_{i}}.$ (1.3)

For a four-dimensional gauge theory $T_{\mathrm{4d}}$ it was observed that the
critical phase $\vartheta$ agrees with the phase of the central charge of the
W-boson. In this case the corresponding NRS Darboux coordinates are a
complexified version of the well-known Fenchel–Nielsen coordinates (see also
[61]). For non-Lagrangian theories $T_{\mathrm{4d}}$ of class S, such as the
$E_{6}$ Minahan–Nemeschansky theory, there is a discrete set of critical
phases $\vartheta_{\gamma}$ corresponding to 4d BPS particles whose electro-
magnetic charge is a multiple of $\gamma$ [62].

Similar to [9], it was noted that there exists a natural generalization
$\mathcal{W}^{\mathrm{eff}}_{\vartheta^{\prime}}$ of the effective twisted
superpotential $\mathcal{W}^{\mathrm{eff}}$. The superpotential
$\mathcal{W}^{\mathrm{eff}}_{\vartheta^{\prime}}(x,\epsilon)$ is defined in
the same way as in equations (1.2) and (1.3), but now in terms of quantum
periods that are Borel summed in an arbitrary direction $\vartheta^{\prime}$.
As for $F^{\mathrm{top}}_{\rho}$, the generalized superpotential
$\mathcal{W}^{\mathrm{eff}}_{\vartheta^{\prime}}$ is piece-wise constant in
$\vartheta^{\prime}$ and jumps along a discrete set of Stokes rays. In the
$\mathcal{N}=2$ context these Stokes rays are naturally labeled by 4d BPS
states in the theory $T$. A BPS hypermultiplet with mass $x$ would for
instance induce a Stokes factor of the form

$\Delta\mathcal{W}^{\mathrm{eff}}_{\vartheta^{\prime}}(\epsilon)=\frac{\epsilon}{2\pi\mathrm{i}}\mathrm{Li}_{2}\big{(}{\rm
e}^{\frac{\pi\mathrm{i}x}{\epsilon}}\big{)}.$

Again, the exponentials of the Borel summations
$\mathcal{W}^{\mathrm{eff}}_{\vartheta^{\prime}}$ can be interpreted as a
collection of local sections of a distinguished line bundle.

It was proposed in [62, 63] that the generalized superpotential
$\mathcal{W}^{\mathrm{eff}}_{\vartheta^{\prime}}$ has a natural physical
interpretation in terms of the 4d $\mathcal{N}=2$ theory $T$. It was argued
that each Stokes sector corresponds to a certain IR boundary condition of the
4d theory $T$ in the $\frac{1}{2}\Omega$-background
$\mathbb{R}_{\epsilon}^{2}\times\mathbb{R}^{2}$ labeled by
$\vartheta^{\prime}$, which can be explicitly described by coupling a 3d
$\mathcal{N}=2$ theory of class R to the boundary $S^{1}\times\mathbb{R}^{2}$
of the $\frac{1}{2}\Omega$-background.

Our goal in this work is to establish the relation between the two occurrences
of Borel resummation and the almost identical Stokes jumps and transition
functions for $F^{\rm top}$ and $\mathcal{W}^{\mathrm{eff}}$. To do so, we
note that both the topological string partition function as well as the
effective twisted superpotential can be obtained as two distinguished limits
of a refinement of topological string theory, which has two deformation
parameters $\epsilon_{1}$ and $\epsilon_{2}$. The refinement of the
topological string partition function is motivated by Nekrasov’s computation
of the instanton partition function in the $\Omega$-background for four-
dimensional $\mathcal{N}=2$ gauge theories as well as their five-dimensional
lifts [85], and the fact that the K-theoretic instanton partition function
agrees with the topological string partition function on a class of non-
compact Calabi–Yau threefolds that geometrically engineer the afore-mentioned
gauge theories [74, 73], when the limit $\epsilon_{1}=-\epsilon_{2}=\lambda$
is taken.

Topological string theory on any Calabi–Yau manifold $\mathbf{X}$ has a
connection to quantization, since the topological string partition function
may be interpreted as a wave function obtained by quantizing the space of
real-valued 3-forms on $\mathbf{X}$ in a complex polarization [96]. In this
interpretation the holomorphic anomaly equations [18] correspond to a
projectively flat connection on the associated bundle of Hilbert spaces that
describes the independence of infinitesimal changes in the polarization. In
the context of topological string theory on non-compact CY manifolds, the
authors of [2] provided a further link to quantization and moreover to
integrability. It was proposed that the mirror curves of non-compact
Calabi–Yau manifolds can be intepreted as the analogs of Hamiltonians of a
quantum mechanical system and the closed string moduli as the analogs of
energy levels in the quantum mechanical phase space. This interpretation has
in particular led to the notion of a quantum curve, which depending on whether
the curve variables are in $\mathbb{C}$ or $\mathbb{C}^{*}$, are described by
a differential or finite difference equation.

A relation between the two occurrences of a quantum mechanical system was
expected, but could not be made precise. One reason for this is that the
quantum mechanical setup of the flatness equation of [96] contains derivatives
with respect to the closed string moduli, while the quantum curves contains
derivatives with respect to the open string moduli. In [1], it was furthermore
realized that the quantum curves are naturally associated to the refined
topological string partition function in the Nekrasov–Shatashvili (NS) limit
[88]. More insights on the relation between the two occurrences of
quantization have been obtained through the study of spectral properties of
the difference operators describing the mirror curves of non-compact toric
Calabi–Yau manifolds [53]. See also [79] and references therein.

To achieve the goal of this paper, namely relating the resurgence of $F^{\rm
top}$ and $\mathcal{W}^{\mathrm{eff}}$, we are naturally led to re-consider
the precise relations between appearances of quantum mechanical setups
occurring in the open and closed string moduli. This will in particular lead
to an understanding of the non-perturbative completion of the quantum geometry
considered in [1].

The organization of this work is as follows: In Section 2, we introduce the
main players of our work, such as the topological string free energy (in
Gromov–Witten and Gopakumar–Vafa form), its refinement and the
Nekrasov–Shatashvili limit. We also recollect the definition of the resolved
conifold geometry, its mirror and the associated mirror curve, as well as the
relation of topological string theory to five-dimensional gauge theory.

In Section 3, we briefly summarize how to obtain a Schrödinger operator
$\mathbf{D}$ from quantizing the mirror curve, in particular for the resolved
conifold geometry. Using the WKB approximation, we find the all-order-
in-$\epsilon$ formal power series solution for the Schrödinger operator
$\mathbf{D}$. We furthermore identify two particular exact solutions. The
first exact solution is a ratio of quantum dilogarithm functions that was
previously considered in for example [1]. The other exact solution is given in
terms of the Faddeev quantum dilogarithm. In Section 5, it will become clear
that these two exact solutions correspond to two distinguished Borel
summations of the formal WKB solution.222See also the related discussion in
[52, Section 4], where the all-order-in-$\epsilon$ solution and its Borel
analysis is discussed as well.

Generalizing the approaches of [6, 68], in Section 4, we use the perturbative
expansion of the refined free energies for the resolved conifold to derive a
finite difference equation in the closed string modulus. We will show in
Section 4.2 that this difference equation (4.1) is solved by a special
function with pleasant analytical properties. This solution provides a natural
candidate for the non-perturbative completion of the perturbative expansion
that we started with. We show that it is furthermore nicely related to
previous proposals for non-perturbative refined partition functions for the
resolved conifold, such as the one obtained by Lockhart and Vafa from the
relation to superconformal field theory [77], as well as the one obtained in
[57] from ABJM theory and spectral analysis and the proposal of [75] obtained
from universal Chern–Simons theory. We furthermore find that this non-
perturbative solution satisfies an additional difference equation that
involves integer shifts of the closed string modulus. This new difference
equation (the third equation in (4.4)) is invisible to the perturbative
expansion of the refined free energy, which is periodic in the closed modulus.
We show in (4.15) that this new difference equation encodes the Stokes jumps
of the unrefined topological string, obtained previously in [9]. In Section
4.3, we write down the analogous statements in the Nekrasov–Shatashvili (NS)
limit. We will see in Section 5 that the NS difference equation (4.14)
corresponding to integer shifts in the closed string modulus encodes the
Stokes phenomena of the Borel summations. In Section 4.5, we establish a
difference equation relating the refined topological free energy to the NS
free energy, and in Section 4.6, we interpret this difference equation as the
quantization of an algebraic curve in the closed string modulus.

In Section 5, we complete a systematic study of the Borel summation of the NS
free energy of the resolved conifold, as well as the formal WKB solution to
the Schrödinger operator $\mathbf{D}$ from Section 3, and establish a relation
between the two. Similar to [9], we prove in Theorem 5.2 that the Borel
transform of the NS free energy has infinitely many singularities organized
along rays that may be labeled by the charges of 5d BPS states, and that
accumulate along the imaginary axis in the $\epsilon$-plane. We find that the
non-perturbative solution of the finite difference equation found in Section 4
agrees with the Borel summation along the ray $\mathbb{R}_{>0}$ in the
$\epsilon$-plane. We compute the Stokes jumps along these rays in equation
(5.2) and the limit of the Borel summed free energy along the imaginary axis
in equation (5.3). We find that this last limit agrees with the Gopakumar–Vafa
formulation of the NS free energy. We complete a similar analysis in Theorem
5.5 for the Borel sums of the formal solution of the Schrödinger operator
$\mathbf{D}$. In particular, we find that the two non-perturbative solutions
found in Section 3 correspond to the Borel sums along
$\mathrm{i}\mathbb{R}_{>0}$ and $\mathbb{R}_{>0}$. We finish the section with
Theorem 5.7, relating the Borel sums of the free energy with the Borel sums of
the formal solution of the Schrödinger operator.

In Section 6, we extract the BPS structure from the previous analysis. We
furthermore use this structure to formulate the Borel summation results
geometrically as defining a section of a certain holomorphic line bundle
$\mathcal{L}\to\mathbb{C}^{\times}\times M$, where $M$ denotes the Kähler
parameter space, and $\mathbb{C}^{\times}$ the parameter space for $\epsilon$.
We furthermore show that $\mathcal{L}$ is actually the same line bundle as
defined by $\exp\big{(}F^{{\rm top}}_{\rho}\big{)}$ and studied in [9].

In Section 7, we interpret the Borel analysis of the previous sections in
terms of 5d gauge theory and quantum integrable systems. In Section 7.2, we
study the spectral networks (or Stokes graphs) corresponding to the
Schrödinger operator $\mathbf{D}$ for the resolved conifold geometry in the
spirit of the exact WKB analysis, and identify the above 5d BPS states as
degenerate trajectories. We identify two distinguished spectral networks and
define the associated spectral coordinates (on the moduli space of the
corresponding multiplicative Hitchin system) in the language of [61]. In
Section 7.5, we interpret the relation between the Borel sum of the formal
solution to the Schrödinger operator $\mathbf{D}$ and the Borel sum of the NS
free energy as a non-perturbative completion of the relation between the open
and closed topological string found in [1]. We furthermore interpret this
relation along the ray $\rho=\mathrm{i}\mathbb{R}_{>0}$ as a lift of the
Nekrasov–Rosly–Shatashvili conjecture [87] to five dimensions. That is, we
argue that the NS free energy is equal to the generating function of the space
of $q$-difference opers in the relevant spectral coordinates. We conclude this
section with a description of two associated spectral problems.

Finally, we speculate about a possible generalization of our results to mirror
curves of higher genus in the discussion Section 8.

##  2 Topological string free energies

Let $\mathbf{X}$ and $\check{\mathbf{X}}$ be a mirror pair of Calabi–Yau
threefolds with $h^{1,1}(\mathbf{X})=h^{2,1}(\check{\mathbf{X}})$ and
$h^{2,1}(\mathbf{X})=h^{1,1}(\check{\mathbf{X}})$. Consider a distinguished
set $\mathbf{t}=\big{(}t^{1},\dots,t^{n}\big{)}$ of local coordinates on the
moduli space $\mathcal{M}_{\mathrm{symp}}(\mathbf{X})$ of symplectic
structures on $\mathbf{X}$. These coordinates are related by the mirror map
$\mathbf{t}(z)$ to a set of complex structure coordinates $z$ on the
isomorphic moduli space $\mathcal{M}_{\mathrm{comp}}(\check{\mathbf{X}})$ of
complex structures on $\check{\mathbf{X}}$.

Then topological string theory associates a topological string partition
function to the mirror family $\mathbf{X}_{\mathbf{t}}$ and
$\check{\mathbf{X}}_{\mathbf{t}(z)}$, which is defined as an asymptotic series
in the topological string coupling $\lambda$ and sums over the free energies
associated to the world-sheets of genus $g$:

$Z^{\mathrm{top}}(\lambda,\mathbf{t})=\exp\bigg{(}\sum_{g=0}^{\infty}\lambda^{2g-2}\mathcal{F}_{g}(\mathbf{t})\bigg{)}.$

This topological string partition function may either be computed in terms of
the topological A-model on $\mathbf{X}_{\mathbf{t}}$, or equivalently in terms
of the topological B-model on $\check{\mathbf{X}}_{\mathbf{t}(z)}$, which are
related by mirror symmetry.

###  2.1 GW and GV expansions

In an expansion around a distinguished “large volume” point in the moduli
space $\mathcal{M}_{\mathrm{symp}}(\mathbf{X})$, the topological string free
energies become the generating functions of higher genus Gromov–Witten (GW)
invariants on the A-model side of mirror symmetry. The GW potential of
$\mathbf{X}$ is the formal power series

$F^{\mathrm{top}}(\lambda,\mathbf{t})=\sum_{g\geq
0}\lambda^{2g-2}F_{g}(\mathbf{t})=\sum_{g\geq
0}\lambda^{2g-2}\sum_{\mathbf{\beta}\in
H_{2}(X,\mathbb{Z})}N^{g}_{\mathbf{\beta}}\mathbf{Q}^{\mathbf{\beta}},$

where
$\mathbf{Q}^{\mathbf{\beta}}:=\exp(2\pi\mathrm{i}\langle\mathbf{t},\mathbf{\beta}\rangle)$
is a formal variable living in a suitable completion of the effective cone in
the group ring of $H_{2}(\mathbf{X},\mathbb{Z})$.

The GW potential can be furthermore written as

$F^{\mathrm{top}}=F^{\mathrm{top}}_{\mathbf{\beta}=0}+\tilde{F}^{\mathrm{top}},$

where $F_{\beta=0}$ denotes the contribution from constant maps and
$\tilde{F}$ the contribution from non-constant maps. The constant map
contribution at genus 0 and 1 are $\mathbf{t}$-dependent and the higher genus
constant map contributions take the universal form [40]

$F^{\mathrm{top}}_{g,\beta=0}=\frac{\chi(\mathbf{X})(-1)^{g-1}B_{2g}B_{2g-2}}{4g(2g-2)(2g-2)!},$

for $g\geq 2$, where $\chi(\mathbf{X})$ is the Euler characteristic of
$\mathbf{X}$ and the Bernoulli numbers $B_{n}$ are generated by

$\frac{w}{{\rm e}^{w}-1}=\sum_{n=0}^{\infty}B_{n}\frac{w^{n}}{n!}.$

The non-constant part of the GW potential may be resummed in terms of the
Gopakumar–Vafa invariants $n_{g}^{\beta}\in\mathbb{Z}$ that “count”
electrically charged $M2$ branes wrapping two-cycles of degree $\beta$ in an
M-theory compactification on the Calabi–Yau $\mathbf{X}$ [49, 50]. The GW
potential in GV form reads

$\tilde{F}^{\mathrm{top}}(\lambda,\mathbf{t})=\sum_{\beta>0}\sum_{g\geq
0}n_{g}^{\beta}\sum_{k\geq
1}\frac{1}{k}\left(2\sin\left(\frac{k\lambda}{2}\right)\right)^{2g-2}\mathbf{Q}^{k\mathbf{\beta}},$

with in particular:

$\tilde{F}_{0}(\mathbf{t})=\sum_{\mathbf{\beta}>0}n^{\beta}_{0}{\rm
Li}_{3}\big{(}\mathbf{Q}^{\mathbf{\beta}}\big{)}.$

###  2.2 Refinement of topological string theory

The refinement of topological string theory is motivated by Nekrasov’s
instanton calculation in the $\Omega$-background [85]. A topological string
interpretation of the refinement can be given in terms of the refined
Gopakumar–Vafa invariants $N_{j_{L},j_{R}}^{\beta}$. These GV invariants
compute an index of five-dimensional BPS states with charge $\beta$ and spin
$(j_{L},j_{R})$ with respect to the rotation group ${\rm SU}(2)_{L}\times{\rm
SU}(2)_{R}$ of $\mathbb{R}^{4}$ [67]. This index is an invariant of non-
compact CY manifolds and can be used to define refined topological string
theory.

The refined topological string free energy is an expression in the moduli
$\mathbf{t}$ as well as the two parameters $\epsilon_{1}$, $\epsilon_{2}$. It
can be written as (see, e.g., [58, 66])

$F^{{\rm
ref}}(\epsilon_{1},\epsilon_{2},\mathbf{t})=\sum_{\beta}\sum_{j_{L},j_{R}\geq
0}N_{j_{L},j_{R}}^{\beta}\sum_{k\geq
1}\frac{1}{k}\frac{\chi_{j_{L}}\big{(}q^{k}_{L}\big{)}\chi_{j_{R}}\big{(}q^{k}_{R}\big{)}}{\big{(}q_{1}^{k/2}-q_{1}^{-k/2}\big{)}\big{(}q_{2}^{k/2}-q_{2}^{-k/2}\big{)}}\mathbf{Q}^{k\beta},$

in terms of the ${\rm SU}(2)$ characters

$\chi_{j}(q)=\frac{q^{2j-1}-q^{-2j-1}}{q-q^{-1}},$

where $q_{j}={\rm e}^{\mathrm{i}\epsilon_{j}}$ for $j=1,2,L,R$ and

$\epsilon_{L}=\frac{\epsilon_{1}-\epsilon_{2}}{2},\qquad\epsilon_{R}=\frac{\epsilon_{1}+\epsilon_{2}}{2}.$

The unrefined topological string may be obtained from the refined one by
setting

$\epsilon_{1}=-\epsilon_{2}=\lambda.$

Another limit of refined topological string theory, which was put forward in
[88], is given by sending one of the parameters $\epsilon_{1}$, $\epsilon_{2}$
to zero while the other one is kept finite. E.g.,

$\epsilon_{1}=\epsilon,\qquad\epsilon_{2}\rightarrow 0.$

Since the refined topological string free energy has a simple pole in this
limit, the free energy in this Nekrasov–Shatashvili (NS) limit is defined as

$F^{\mathrm{NS}}(\epsilon,{\bf{Q}}):=\lim_{\epsilon_{2}\rightarrow
0}\epsilon_{2}F^{\rm
ref}(\epsilon_{1},\epsilon_{2},{\bf{Q}})|_{\epsilon_{1}=\epsilon}.$

###  2.3 Topological string free energies of the resolved conifold

The CY threefold given by the total space of the rank two bundle

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

over the projective line corresponds to the resolution of the conifold
singularity in $\mathbb{C}^{4}$ and is known as the resolved conifold. This
geometry is defined on the A-model side of mirror symmetry and $t$ corresponds
to

$t=\int_{C}B+\mathrm{i}\omega,$

where $B\in H^{2}(\mathbf{X},\mathbb{R})/H^{2}(\mathbf{X},\mathbb{Z})$ is the
B-field, while $\omega$ is the Kähler form and $C$ corresponds to the
$\mathbb{P}^{1}$ class in this example. The GW potential for this geometry was
determined, in physics [49, 51] as well as in mathematics [40], with the
outcome333See also [80] for the determination of $F^{g}$ from a string theory
duality and the explicit appearance of the polylogarithm expressions.

$\tilde{F}^{\mathrm{top}}(\lambda,t)=\sum_{g=0}^{\infty}\lambda^{2g-2}\tilde{F}_{g}(t)=\frac{1}{\lambda^{2}}{\rm
Li}_{3}(Q)+\sum_{g=1}^{\infty}\lambda^{2g-2}\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}{\rm
Li}_{3-2g}(Q)$ (2.1)

for the non-constant maps, where $Q=\exp(2\pi\mathrm{i}t)$ and the
polylogarithm is defined by

${\rm Li}_{s}(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{n^{s}}$

for $s\in\mathbb{C}$ and $|z|<1$.

The refined topological string free energy for the resolved conifold geometry
is given by [67]

$\begin{split}F^{{\rm
ref}}(\epsilon_{1},\epsilon_{2},t)&=-\sum_{k=1}^{\infty}\frac{Q^{k}}{k(2\sin(k\epsilon_{1}/2))(2\sin(k\epsilon_{2}/2))}.\end{split}$

From this expression it is easy to find formal expansions in $\epsilon_{1}$
and $\epsilon_{2}$ provided we consider shifts of the form $F^{\rm
ref}\big{(}\epsilon_{1},\epsilon_{2},t\pm\frac{\check{\epsilon}_{1}}{2}\pm\frac{\check{\epsilon}_{2}}{2}\big{)}$,
where $\check{\epsilon}_{i}=\epsilon_{i}/2\pi$. Indeed, using the expression
for the generating function of the Bernoulli numbers

$\frac{w}{{\rm e}^{w}-1}=\sum_{n=0}^{\infty}B_{n}\frac{w^{n}}{n!},$ (2.2)

we can write the expansion of, for example, $F^{\rm
ref}\big{(}\epsilon_{1},\epsilon_{2},t-\frac{\check{\epsilon}_{1}+\check{\epsilon}_{2}}{2}\big{)}$
as

$\displaystyle F^{\rm
ref}\left(\epsilon_{1},\epsilon_{2},t-\frac{\check{\epsilon}_{1}+\check{\epsilon}_{2}}{2}\right)$
$\displaystyle=-\sum_{k=1}^{\infty}\frac{{\rm
e}^{2\pi\mathrm{i}k(t-\frac{\check{\epsilon}_{1}+\check{\epsilon}_{2}}{2})}}{k(2\sin(k\epsilon_{1}/2))(2\sin(k\epsilon_{2}/2))}$
$\displaystyle=\sum_{k=1}^{\infty}\frac{Q^{k}}{k({\rm
e}^{\mathrm{i}k\epsilon_{1}}-1)({\rm e}^{\mathrm{i}k\epsilon_{2}}-1)}$
$\displaystyle=\sum_{k=1}^{\infty}\sum_{m,n=0}^{\infty}\frac{Q^{k}}{k^{3-m-n}}\frac{B_{n}B_{m}}{n!m!}\mathrm{i}^{m+n-2}\epsilon_{1}^{m-1}\epsilon_{2}^{n-1}$
$\displaystyle=-\sum_{m,n=0}^{\infty}{\rm
Li}_{3-m-n}(Q)\frac{B_{m}B_{n}}{m!n!}\mathrm{i}^{m+n}\epsilon_{1}^{m-1}\epsilon_{2}^{n-1}.$
(2.3)

A similar computation shows that

$\displaystyle F^{\rm
ref}\left(\epsilon_{1},\epsilon_{2},t+\frac{\check{\epsilon}_{1}+\check{\epsilon}_{2}}{2}\right)$
$\displaystyle=\sum_{m,n=0}^{\infty}{\rm
Li}_{3-m-n}(Q)\frac{B_{m}B_{n}}{m!n!}(-1)^{m+n-1}\mathrm{i}^{m+n}\epsilon_{1}^{m-1}\epsilon_{2}^{n-1},$
$\displaystyle F^{\rm
ref}\left(\epsilon_{1},\epsilon_{2},t+\frac{\check{\epsilon}_{1}-\check{\epsilon}_{2}}{2}\right)$
$\displaystyle=\sum_{m,n=0}^{\infty}{\rm
Li}_{3-m-n}(Q)\frac{B_{m}B_{n}}{m!n!}(-1)^{m-1}\mathrm{i}^{m+n}\epsilon_{1}^{m-1}\epsilon_{2}^{n-1},$
$\displaystyle F^{\rm
ref}\left(\epsilon_{1},\epsilon_{2},t-\frac{\check{\epsilon}_{1}-\check{\epsilon}_{2}}{2}\right)$
$\displaystyle=\sum_{m,n=0}^{\infty}{\rm
Li}_{3-m-n}(Q)\frac{B_{m}B_{n}}{m!n!}(-1)^{n-1}\mathrm{i}^{m+n}\epsilon_{1}^{m-1}\epsilon_{2}^{n-1}.$
(2.4)

On the other hand, the NS limit of $F^{\rm ref}(\epsilon_{1},\epsilon_{2},t)$
in the case of the resolved conifold is given by

$\displaystyle F^{\rm NS}(\epsilon,t)$
$\displaystyle:=\lim_{\epsilon_{2}\rightarrow 0}\epsilon_{2}\cdot F^{\rm
ref}(\epsilon_{1},\epsilon_{2},t)\big{|}_{\epsilon_{1}=\epsilon}$
$\displaystyle=\lim_{\epsilon_{2}\rightarrow
0}-\epsilon_{2}\sum_{k=1}^{\infty}\frac{Q^{k}}{k(2\sin(k\epsilon_{1}/2))(2\sin(k\epsilon_{2}/2))}\bigg{|}_{\epsilon_{1}=\epsilon}$
$\displaystyle=-\frac{1}{2}\sum_{k=1}^{\infty}\frac{Q^{k}}{k^{2}\sin(k\epsilon/2)}.$
(2.5)

As before with $F^{\rm ref}(\epsilon_{1},\epsilon_{2},t)$, one can easily find
a formal expansion in $\epsilon$ for the shifts $F^{\rm
NS}(\epsilon,t\pm\check{\epsilon}/2)$ using the generating function (2.2).
That is,

$\displaystyle F^{\rm NS}(\epsilon,t-\check{\epsilon}/2)$
$\displaystyle=-\mathrm{i}\sum_{n=0}^{\infty}{\rm
Li}_{3-n}(Q)\frac{B_{n}}{n!}(\mathrm{i}\epsilon)^{n-1}=-\frac{1}{2\pi}\sum_{n=0}^{\infty}\partial_{t}^{n}{\rm
Li}_{3}(Q)\frac{B_{n}}{n!}\check{\epsilon}^{n-1}$ $\displaystyle F^{\rm
NS}(\epsilon,t+\check{\epsilon}/2)$
$\displaystyle=\mathrm{i}\sum_{n=0}^{\infty}\mathrm{Li}_{3-n}(Q)\frac{B_{n}}{n!}(-\mathrm{i}\epsilon)^{n-1}=\frac{1}{2\pi}\sum_{n=0}^{\infty}\partial_{t}^{n}{\rm
Li}_{3}(Q)\frac{B_{n}}{n!}(-1)^{n-1}\check{\epsilon}^{n-1}.$ (2.6)

###  2.4 Mirror geometry

Mirrors of non-compact CY threefolds are described in [24, 64, 74], see also
[65]. We focus here on toric examples. The non-compact CY threefolds in these
cases are given by

$\mathbf{X}=\frac{\mathbb{C}^{3+k}\setminus S}{(\mathbb{C}^{*})^{k}},$

where the $k$ algebraic tori $\mathbb{C}^{*}$ act on the space by

$(\mathbb{C}^{*})^{a}\colon\
(z_{1},\dots,z_{j},\dots,z_{3+k})\mapsto\big{(}\lambda^{l^{(a)}_{1}}z_{1},\dots,\lambda^{l^{(a)}_{j}}z_{j},\dots,\lambda^{l^{(a)}_{3+k}}z_{3+k}\big{)},\qquad
a=1,\dots,k.$

Here, $\lambda\in\mathbb{C}^{*}$, whereas $l^{(a)}_{i}\in\mathbb{Z}$ are the
toric charges and $S$ is a subset which is fixed by a subgroup of
$(\mathbb{C}^{*})^{k}$. The resolved conifold geometry corresponds to the
toric variety associated to the toric charge vector

$l=(1,1,-1,-1).$

To define the mirror of $\mathbf{X}$ we consider the variables
$w_{i}\in\mathbb{C}^{*}$, for $i=0,\dots,3$, subject to the constraint

$\prod_{i=0}^{3}w_{i}^{l_{i}}=\frac{w_{0}w_{1}}{w_{2}w_{3}}=1,$

and the polynomial

$P(a,w)=\sum_{i=0}^{3}a_{i}w_{i},$

with $a_{i}\in\mathbb{C}$ for $i=0,\ldots,3$. This polynomial enters the
definition of the Landau–Ginzburg potential of the mirror, which is given by

$W=-v^{+}v^{-}+P(a,w).$ (2.7)

The additional variables $v^{+},v^{-}\in\mathbb{C}$ are an artifact of local
mirror symmetry, see, e.g., [64]. There is a freedom to rescale $W$ by a
nonzero complex number, which we can use to set one of the $w_{i}$ variables
to 1. Without loss of generality we set $w_{0}=1$.

The mirror $\check{\mathbf{X}}$ of the resolved conifold geometry is then
defined as the variety

$\check{\mathbf{X}}=\big{\\{}(v^{+},v^{-},w_{1},w_{2})\in\mathbb{C}^{2}\times(\mathbb{C}^{*})^{2}\mid
v^{+}v^{-}=a_{0}+a_{1}w_{1}+a_{2}w_{2}+a_{3}w_{1}w_{2}^{-1}\big{\\}}.$

The $a_{i}$ are complex parameters which determine the complex structure of
$\check{\mathbf{X}}$. The rescaling of $W$ and $w_{1}$, $w_{2}$ can be further
used to show that the complex structure of $\check{\mathbf{X}}$ only depends
on the combination

$Q=\frac{a_{2}a_{3}}{a_{0}a_{1}}.$ (2.8)

The equation $W=0$ can be written in the form444The term $v^{+}v^{-}$ has been
rescaled here compared to (2.7)

$v^{+}v^{-}={\rm e}^{2\pi\mathrm{i}x}+{\rm e}^{2\pi\mathrm{i}y}+1+Q{\rm
e}^{2\pi\mathrm{i}(y-x)},$

with $w_{1}=\exp(2\pi\mathrm{i}x)$, $w_{2}=\exp(2\pi\mathrm{i}y)$ and
$Q=\exp(2\pi\mathrm{i}t)$. The equation

$\Sigma\colon\ {\rm e}^{2\pi\mathrm{i}x}+{\rm e}^{2\pi\mathrm{i}y}+1+Q{\rm
e}^{2\pi\mathrm{i}(y-x)}=0,$ (2.9)

defines the so-called mirror curve $\Sigma$.

It turns out to be convenient to redefine the variables $x$ and $y$ by mapping
$x\mapsto x+t+1/2$ and $y\mapsto y+x$, so that $\Sigma$ is now parametrized by
the equation

$\Sigma\colon\
(1-X)Y-(1-QX)=0\subset\mathbb{C}^{*}_{X}\times\mathbb{C}^{*}_{Y},$ (2.10)

in terms of the $\mathbb{C}^{*}$-variables $X={\rm e}^{2\pi\mathrm{i}x}$ and
$Y={\rm e}^{2\pi\mathrm{i}y}$. Topologically, $\Sigma$ is a four-punctured
sphere, with punctures at

$\displaystyle(X,Y)\in\big{\\{}(0,1),(1,\infty),\big{(}Q^{-1},0\big{)},(\infty,Q)\big{\\}}.$

That is, its punctures are located at the points where the curve intersects
the lines $X=0,\infty$ or $Y=0,\infty$ (see Figure 1).

Figure 1: Plot of a cross-section of the mirror curve $\Sigma$ (in blue), with
$Q=1/2$, as well as its projection to $C=\mathbb{C}^{*}_{X}$ (in orange), and
its logarithmic punctures at $X=1$ and $X=Q^{-1}$ (in orange) and regular
punctures at $X=0$ and $X=\infty$ (in black).

The mirror curve $\Sigma$ comes equipped with the tautological 1-form

$\lambda^{\mathrm{cl}}=y\,{\rm d}x=-\frac{1}{4\pi^{2}}\log Y\,{\rm d}\log X,$
(2.11)

which is a reduction from the holomorphic three-form on $\check{\mathbf{X}}$
to $\Sigma$. The genus zero free energy $F^{\mathrm{top}}_{0}(\mathbf{t})$ at
large radius is then simply determined by the relations

$t_{i}=\oint_{\gamma_{A^{i}}}\lambda^{\mathrm{cl}},\qquad\frac{1}{2\pi\mathrm{i}}\frac{\partial
F^{\mathrm{top}}_{0}}{\partial
t_{i}}=\frac{4\pi^{2}}{\epsilon}\oint_{\gamma_{B_{i}}}\lambda^{\mathrm{cl}},$
(2.12)

where $\gamma_{A^{i}}$ and $\gamma_{B_{i}}$ are a suitable basis of 1-cycles
on $\Sigma$, illustrated in Figure 2 for the resolved conifold
geometry.555Since the B-cycle $\gamma_{B}$ is non-compact in this example, the
corresponding period requires a regularization. This will be discussed in
Section 2.5. Note that the periods of $\lambda^{\mathrm{cl}}$ are invariant
under ${\rm SL}(2,\mathbb{Z})$-transformations acting on the variables $x$ and
$y$. This is known as the framing ambiguity.

Figure 2: Plot of 1-cycles $\gamma_{A}$ (in blue) and $\gamma_{B}$ (in red) on
the mirror curve $\Sigma$ with $Q=1/2$. The 1-cycle $\gamma_{A}$ is compact
and oriented in the clockwise direction. The 1-cycle $\gamma_{B}$ runs from
the puncture at $X=1$ to the puncture at $X=0$ and is non-compact.

Because of the logarithms in the definition (2.11) of $\lambda^{\mathrm{cl}}$,
it is sometimes needed to consider a $\mathbb{Z}$-covering
$\widetilde{\Sigma}$ of $\Sigma$ that resolves the logarithmic ambiguities of
$\lambda^{\mathrm{cl}}$. This covering has additional branching at the
punctures at $X=1$ and $X=Q^{-1}$ where $\lambda^{\mathrm{cl}}$ is multi-
valued. We choose a trivialization of the covering by connecting these
“logarithmic” punctures by a logarithmic branch-cut and labeling the sheets
away from the cut by an integer $N$.

The $\mathbb{Z}$-covering $\widetilde{\Sigma}$ has an additional 1-cycle
$\gamma_{0}$ compared to $\Sigma$. In contrast to the 1-cycles $\gamma_{A}$
and $\gamma_{B}$, the 1-cycle $\gamma_{0}$ crosses the logarithmic cut twice
(with opposite orientations). This 1-cycle is for instance illustrated in [13,
Figure 2].

###  2.5 Picard–Fuchs equation and classical terms

The periods of the classical differential $\lambda^{\text{cl}}$ can be
computed explicitly as outlined in the previous subsection. Alternatively they
can be determined as solutions of a differential equation, called the
Picard–Fuchs equation. The latter stems from the fact that the differential
$\lambda^{\text{cl}}$ descends from the holomorphic three-form on the mirror
CY threefolds. The holomorphic three-form can be used to capture the variation
of Hodge structure in the middle dimensional cohomology of the threefold. The
flatness of the associated Gauss-Manin connection leads to differential
equations which annihilate the periods of the three-form and correspondingly
the periods of the meromorphic differential $\lambda^{\text{cl}}$ on the
curve. To derive the Picard–Fuchs equation, rescalings of the defining
equation of the mirror curve can be used which we review for the case of the
mirror of the resolved conifold.

As mentioned in the previous section, the $a_{i}$ appearing in (2.7) are
complex parameters which determine the complex structure of
$\check{\mathbf{X}}$, and the rescaling of $W$ and $w_{1}$, $w_{2}$ can be
used to show that the complex structure of $\check{\mathbf{X}}$ only depends
on the combination (2.8). Keeping the explicit dependence on the $a_{i}$ is
however more convenient for the derivation of the Picard–Fuchs equations from
a GKZ [48] system of differential equations annihilating periods of the unique
holomorphic $(3,0)$ form on $\check{\mathbf{X}}$. The latter is given by

$\Omega=\textrm{Res}_{W=0}\frac{1}{W}\,{\rm d}v^{+}\,{\rm d}v^{-}\frac{{\rm
d}w_{1}}{w_{1}}\frac{{\rm d}w_{2}}{w_{2}}.$

The periods of $\Omega$ are annihilated by the GKZ operator

$\frac{\partial}{\partial a_{0}}\frac{\partial}{\partial
a_{1}}-\frac{\partial}{\partial a_{2}}\frac{\partial}{\partial a_{3}},$

which translates into the Picard–Fuchs operator expressed in $Q$, namely

$L=(1-Q)\theta^{2},\qquad\theta=Q\frac{\mathrm{d}}{\mathrm{d}Q}.$

This operator has the following solutions:

$\varpi^{0}=1,\qquad\varpi^{1}=\frac{1}{2\pi\mathrm{i}}\log(Q).$

These correspond to periods of $\Omega$ over appropriately defined compact
three-cycles in $H_{3}(\check{\mathbf{X}},\mathbb{Z})$. The mirror map is
identified as

$t=\frac{1}{2\pi\mathrm{i}}\log(Q),$

we note that the mirror of the resolved conifold is a special case where the
parameter appearing in the definition of the curve equation $Q$ is given by
$\exp(2\pi\mathrm{i}t)$ without further corrections, so that $Q$ is the
exponentiated mirror map.

A familiar phenomenon of mirror symmetry for local CY is that the Picard–Fuchs
system of the mirror does not have enough solutions to recover the expected
ingredients of a special Kähler geometry. Generically it is missing
expressions for periods of non-compact three-cycles. One way to recover these
is to carefully define non-compact three-cycles on the geometry as was done in
[65]. Alternatively one may extend the PF operators, guided by the expectation
of its general form in compact CY, when it is formulated in terms of the
distinguished coordinates corresponding to the mirror map. This was done in
[41], which we will outline here.

The guiding principle is the expected form of the PF operator in terms of the
special (flat) coordinate $t$ in the case when the moduli space is complex
one-dimensional. It is given by

$L=\partial_{t}^{2}C_{ttt}^{-1}\partial_{t}^{2},$

where $C_{ttt}:=\frac{\partial^{3}}{\partial t^{3}}F_{0}$, see, e.g., [23,
32]. This leads to the extended PF operator

$L=\theta^{2}(1-Q)\theta^{2}$ (2.13)

in the $Q$ coordinate [41]. This operator has the solutions

$\displaystyle\varpi^{0}=1,$
$\displaystyle\varpi^{1}=\frac{1}{2\pi\mathrm{i}}\log(Q),$
$\displaystyle\varpi^{2}=\frac{1}{(2\pi\mathrm{i})^{2}}\left(\frac{1}{2}(\log(Q))^{2}+{\rm
Li}_{2}(Q)\right),$
$\displaystyle\varpi^{3}=\frac{1}{(2\pi\mathrm{i})^{3}}\left(-\frac{1}{6}(\log(Q))^{3}-\log(Q){\rm
Li}_{2}(Q)+2{\rm Li}_{3}(Q)\right).$

We can identify the additional solutions with

$\varpi^{2}=:F_{t},\qquad\varpi^{3}=2F_{0}-tF_{t},$

where $F_{t}:=\partial_{t}F_{0}$ and where the prepotential $F_{0}$ reads

$F_{0}=\frac{1}{(2\pi\mathrm{i})^{3}}\left(\frac{1}{3!}(\log Q)^{3}+{\rm
Li}_{3}(Q)\right).$

This matches with the expected generating function of the GW invariants of the
resolved conifold.

Note that the first term of the prepotential was not included in the
discussion of the topological string free energies in the previous subsections
and in particular in equation (2.1). This term is known as the classical
contribution to prepotential and should reflect the triple intersection number
of three divisors in the CY threefold $\mathbf{X}$. These intersection numbers
are however ill-defined geometrically for non-compact CY manifolds. The term
at hand, and more generally the intersection numbers for any non-compact CY
threefold, can be thought of as being obtained through a careful
regularization of an appropriate decompactification limit, in which the non-
compact CY is considered as a local geometry that is embedded in a compact
geometry, see, e.g., [24].

We will later in particular need the expression for the period for $F_{t}$,
including the classical term, which is given by

$F_{t}=\frac{t^{2}}{2}+\frac{1}{(2\pi\mathrm{i})^{2}}{\rm Li}_{2}(Q).$ (2.14)

###  2.6 Geometric engineering

Non-compact toric Calabi–Yau threefolds $\mathbf{X}$ (or more generally dot
diagrams) are related to five-dimensional $\mathcal{N}=1$ field theories on
$\mathbb{R}^{4}\times S_{R}^{1}$ by geometric engineering [17, 73]. In this
correspondence the mirror curve $\Sigma$ gets interpreted as the
Seiberg–Witten curve of the five-dimensional field theory. In most of this
paper we have set $R=1$. If we want to re-introduce $R$ we roughly need to
scale all $\mathbb{C}$-valued variables and parameters by a factor of $1/R$.

The field theory may be interpreted as a theory of class S once we choose a
projection $\Sigma\subset\mathbb{C}_{X}^{*}\times\mathbb{C}_{Y}^{*}$ to
$C=\mathbb{C}_{X}^{*}$. For instance, the resolved conifold geometry with
mirror curve $\Sigma$ as in equation (2.10), projected to
$\mathbb{C}^{*}_{X}$, engineers the five-dimensional pure ${\rm U}(1)$ theory
on $\mathbb{R}^{4}\times S_{R}^{1}$ with $Q=R^{2}\Lambda^{2}$. On the other
hand, the projection of $\Sigma$, as parametrized in (2.9), to
$\mathbb{C}^{*}_{Y}$ geometrically engineers a massive five-dimensional ${\rm
SU}(2)$ hypermultiplet with $Q=\exp(2\pi\mathrm{i}m)$. In the limit $R\to 0$
the five-dimensional $\mathcal{N}=1$ field theory reduces to a four-
dimensional $\mathcal{N}=2$ field theory whose Seiberg–Witten curve is given
by the $R\to 0$ limit of $\Sigma$.

The refined partition function $Z^{\rm ref}=\exp\big{(}F^{\rm ref}\big{)}$ of
the non-compact Calabi–Yau threefold $\mathbf{X}$ corresponds to the
K-theoretic version of the Nekrasov instanton partition function of the five-
dimensional field theory. For instance, the refined partition function for the
resolved conifold equals666Remark that by using (B.10), we can equivalently
write $Z^{\rm
ref}(\epsilon_{1},\epsilon_{2},t)=\prod_{k,l=1}^{\infty}\big{(}1-Qq_{1}^{k-1/2}q_{2}^{-l+1/2}\big{)}$.
Going to the unrefined limit where $\epsilon_{1}=-\epsilon_{2}=\lambda$ sets
$q=q_{1}=q_{2}^{-1}$, so that $Z^{\rm
ref}(\lambda,-\lambda,t)=\prod_{k,l=1}\big{(}1-Qq^{k+l-1}\big{)}=\prod_{s=1}\big{(}1-Qq^{s}\big{)}^{s}$,
recovering the usual expression for the partition function of the resolved
conifold found in [67].

$\displaystyle Z^{\rm ref}(\epsilon_{1},\epsilon_{2},t)=\exp
F_{\mathrm{GV}}^{\rm
ref}(\epsilon_{1},\epsilon_{2},t)=\prod_{k,l=1}^{\infty}\big{(}1-Qq_{1}^{k-1/2}q_{2}^{l-1/2}\big{)}^{-1},$
(2.15)

(where we recall that $q_{j}={\rm e}^{\mathrm{i}\epsilon_{j}}$ for $j=1,2$)
and either computes the instanton partition function for the pure ${\rm U}(1)$
gauge theory or the massive hypermultiplet, which both only receive a 1-loop
contribution.

Finally, there are various kinds of BPS states corresponding to the
topological string set-up. In particular, in the context of Type IIA string
theory one may wrap D0-branes around the 0-cycle mirror to the 1-cycle
$\gamma_{0}\subset\widetilde{\Sigma}$, D2-branes around 2-cycles mirror to the
1-cycles $\gamma_{A^{i}}\subset\Sigma$, D4-branes around 4-cycles mirror to
the 1-cycles $\gamma_{B_{i}}\subset\Sigma$ and D6-branes on $X$. These may be
interpreted as BPS states in the corresponding five-dimensional field theory
through the technique of exponential spectral networks [12, 37]. Particularly
relevant for this paper is the study of exponential spectral networks for the
resolved conifold geometry in [13].

##  3 WKB analysis of quantum mirror curves

In this section we study the WKB asymptotics of the quantum mirror curve of
the resolved conifold, in preparation for a more advanced exact WKB analysis
in the forthcoming sections. For general background in exact WKB analysis for
differential operators we refer to [69, 94]. These techniques were first
applied to difference operators in [35, 36]. The WKB analysis in relation to
quantum mirror curves was studied in many places including [1, 72]. Our goal
in this section is to derive an all-order expression for the asymptotic
expansion of local solutions to $q$-difference equations corresponding to
quantum mirror curves. This will be useful later in the study of the Borel
resummation of these solutions, as well as in relation of these solutions to
closed partition functions.

###  3.1 The quantum mirror curve

The period integral relations (2.12)

$t_{i}=\oint_{\gamma_{A^{i}}}\lambda^{\mathrm{cl}},\qquad\frac{1}{2\pi\mathrm{i}}\frac{\partial
F^{\mathrm{top}}_{0}}{\partial
t_{i}}=\frac{4\pi^{2}}{\epsilon}\oint_{\gamma_{B_{i}}}\lambda^{\mathrm{cl}},$

with $\lambda^{\mathrm{cl}}=y(x)\mathrm{d}x$ reveal that the “classical”
topological string has the structure of an classical integrable system, with
spectral curve $\Sigma$ and Liouville form $\lambda^{\mathrm{cl}}$. This is a
structure that is argued to persists to all orders [2], perhaps most
transparent in the NS limit [88].

The mirror curve is given by an algebraic equation in two $\mathbb{C}^{*}$
variables $X=\exp(2\pi\mathrm{i}x)$ and $Y=\exp(2\pi\mathrm{i}y)$, and has the
form

$P\big{(}{\rm e}^{2\pi\mathrm{i}x},{\rm
e}^{2\pi\mathrm{i}y},\mathbf{t}\big{)}=0,$

where $\mathbf{t}$ are the closed string moduli.

The quantum mirror curve is a quantization of the mirror curve obtained by
promoting its variables to operators which act on states in a corresponding
Hilbert space. In our case we will replace $x$ and $y$ with the operators

$\hat{x}\Psi(\epsilon,x,\mathbf{t})=x\Psi(\epsilon,x,\mathbf{t}),\qquad\hat{y}\Psi(\epsilon,x,\mathbf{t})=\frac{\epsilon}{4\pi^{2}\mathrm{i}}\partial_{x}\Psi(\epsilon,x,\mathbf{t}),$

that obey the canonical quantization relation

$\left[\hat{x},\hat{y}\right]=\frac{\mathrm{i}\epsilon}{4\pi^{2}}.$

For instance, quantizing the curve $\Sigma$ defined in (2.10) by

$(1-\exp(2\pi\mathrm{i}x))\exp(2\pi\mathrm{i}y)-(1-Q\exp(2\pi\mathrm{i}x))=0,$

we obtain the Schrödinger operator

$\mathbf{D}_{\Sigma}(\epsilon,x,t)=\big{(}1-{\rm
e}^{2\pi\mathrm{i}x}\big{)}{\rm
e}^{\check{\epsilon}\partial_{x}}-\big{(}1-Q{\rm e}^{2\pi\mathrm{i}x}\big{)}.$
(3.1)

acting on wave-functions $\Psi(\epsilon,x,t)$ in the relevant Hilbert space.
Since

${\rm e}^{\check{\epsilon}\partial_{x}}\Psi(X)=\Psi(qX)$

with $q=\exp(\mathrm{i}\epsilon)$, the Schrödinger operator
$\mathbf{D}_{\Sigma}$ is also known as a $q$-difference operator.

Note that there is no unique difference operator $\mathbf{D}_{\Sigma}$
associated to $\Sigma$, because of the ${\rm SL}(2,\mathbb{Z})$ symmetry
mentioned above. Up to some ambiguities in quantization, that may be absorbed
in the redefinition of some of the parameters, there is a unique difference
operator $\mathbf{D}_{\Sigma}$ for $\Sigma$ together with a choice of
projection to $\mathbb{C}^{*}$. As we explained in Section 2.6, this choice of
projection is relevant in determining which five-dimensional gauge theory the
Calabi–Yau threefold $\mathbf{X}$ engineers. For instance, the projection
$\Sigma\to\mathbb{C}^{*}_{X}$, leading to (3.1), geometrically engineers the
pure ${\rm U}(1)$ theory in five dimensions.

Consider a B-brane extended in the $v^{+}$-direction of the mirror
$\check{\mathbf{X}}$, while at $v^{-}=0$, and concentrated at a point
$x\in\Sigma$. If $y(x)$ is the local function determined by the equation
$(\ref{eqn:mirrorcurve1p})$ for $\Sigma$, then it is known that the classical
vev of this brane is given by [3]

$\Psi^{\text{cl}}(\epsilon,x,t)=\exp\left(\frac{4\pi^{2}\mathrm{i}}{\epsilon}\int^{z}\lambda^{\text{cl}}\right),$

whereas its quantum vev satisfies the time-independent Schrödinger equation
[2]

$\mathbf{D}_{\Sigma}(\epsilon,x,t)\Psi(\epsilon,x,t)=0.$ (3.2)

Hence, while the curve $\Sigma$ has the physical interpretation as the moduli
space of a particular class of branes in the open topological string theory,
the solutions $\Psi(\epsilon,x,t)$ to the difference equation (3.2) have the
physical interpretation as vevs for these branes [1, 88]. In the corresponding
five-dimensional gauge theory these have the interpretation as surface defect
vevs [34]. For example, the brane corresponding to (3.1) is ending on the
compact toric leg of the toric diagram of the resolved conifold. In the
corresponding ${\rm U}(1)$ theory this engineers a simple abelian surface
defect [34].

It is also known that the NS partition function $Z^{\mathrm{NS}}(\epsilon,t)$,
in a series expansion in $\epsilon$, may be recovered from an all-order WKB
analysis of $\mathbf{D}_{\Sigma}(\epsilon,x,t)$ [1, 88]. If we write the
general solution $\Psi(\epsilon,x,t)$ to the difference equation (3.2) in the
form

$\Psi(\epsilon,x,t)=\exp\left(\frac{4\pi^{2}\mathrm{i}}{\epsilon}\int^{x}\mathrm{d}z\,\lambda^{\mathrm{qu}}(\epsilon,z,t)\right),$

then it follows that the quantum Liouville form
$\lambda^{\mathrm{qu}}(\epsilon,x,t)$ has an $\epsilon$-expansion

$\lambda^{\mathrm{qu}}(\epsilon)=\lambda^{\mathrm{cl}}+\sum_{n=1}^{\infty}\lambda^{\mathrm{qu}}_{n}\epsilon^{n},$

starting with the classical Liouville form
$\lambda^{\mathrm{cl}}=y\,\mathrm{d}x$. The all-order-in-$\epsilon$-expansion
of the NS free energy $F^{\mathrm{NS}}(\epsilon,t)$ may then be obtained from
the relations

$t^{\mathrm{qu}}_{i}:=\oint_{\gamma_{A^{i}}}\lambda^{\mathrm{qu}},\qquad\frac{1}{2\pi\mathrm{i}}\frac{\partial
F^{\rm NS}\big{(}t^{\mathrm{qu}}_{i};\epsilon\big{)}}{\partial
t^{\mathrm{qu}}_{i}}=\frac{4\pi^{2}}{\epsilon}\oint_{\gamma_{B_{i}}}\lambda^{\mathrm{qu}}.$
(3.3)

Note that, somewhat confusingly, the relations on the left-hand side of (3.3)
define new parameters $t^{\mathrm{qu}}_{i}$ such that the function $F^{\rm
NS}\big{(}t^{\mathrm{qu}}_{i},\epsilon\big{)}$, as defined by the relations on
the right-hand side of (3.3), equals the NS free energy $F^{\rm
NS}(\epsilon,t)$ when simply replacing the $t^{\mathrm{qu}}_{i}$ by the
symbols $t_{i}$. This was first verified in the four-dimensional pure ${\rm
SU}(2)$ gauge theory in [82].

###  3.2 All-order WKB solution

In the following we will apply the WKB analysis to the solution of the quantum
curve. The outcome of the analysis will provide the asymptotic series whose
non-perturbative completion will be studied in Section 5 using Borel
analysis.777We remark that the all-order WKB solution has also appeared in the
recent work [52, Section 4].

We will derive a formal power series solution, to all orders in $\epsilon$, of
the Schrödinger equation (3.2). We could do this slightly more generally,
namely for the cases in which the Schrödinger operator is a $q$-difference
operator of the form

$\mathbf{D}_{\Sigma}(\epsilon,x,t)=\prod_{j=1}^{K}(\exp(\check{\epsilon}\partial_{x})-\exp(2\pi\mathrm{i}y_{j}(x,t))).$
(3.4)

Note that, in general, the $q$-difference operator $\mathbf{D}_{\Sigma}$
requires a specification of an operator ordering.888Which choice of operator
ordering one chooses is not very relevant, since the operator ordering
ambiguities may be absorbed in a redefinition of parameters. Below we
specialize to the case $K=1$ where no such ordering ambiguity occurs.

The operator $\mathbf{D}_{\Sigma}$ in equation (3.4) may be obtained from a
general mirror curve $\Sigma$ given by an equation

$P\big{(}{\rm e}^{2\pi\mathrm{i}x},{\rm e}^{2\pi\mathrm{i}y},t\big{)}=0$

of degree $K$ in $Y={\rm e}^{2\pi\mathrm{i}y}$, and thus defining a (ramified)
covering of degree $K$ over $\mathbb{C}^{*}_{X}$. Indeed, we obtain the
difference operator (3.4) after expressing $y$ in terms of $x$ and $t$ and
then writing the equation $P$ in the form

$P\big{(}{\rm e}^{2\pi\mathrm{i}x},{\rm
e}^{2\pi\mathrm{i}y},t\big{)}=\prod_{k=1}^{K}(\exp(y)-\exp(2\pi\mathrm{i}y_{j}(x,t))),$

where $K$ is the degree of the covering and $y_{j}(x,t)$ the solutions to the
curve equation.

In the following proposition will restrict ourselves to $K=1$. This will
however also be the building block of the solution in the more general case.

######  Proposition 3.1.

A formal power series solution of the equation

$(\exp(\check{\epsilon}\partial_{x})-\exp(2\pi\mathrm{i}y(x,t)))\Psi(\epsilon,x,t)=0$
(3.5)

is given by the formal power series

$\Psi(\epsilon,x,t)=\exp S(\epsilon,x,t)\qquad\mathrm{with}\quad
S(\epsilon,x,t)=\sum_{n=0}^{\infty}\epsilon^{n-1}S_{n}(x,t),$

where

$S(\epsilon,x,t)=\frac{4\pi^{2}\mathrm{i}}{\epsilon}\int^{x}y(z,t)\,{\rm
d}z+2\pi\mathrm{i}\sum_{n=1}^{\infty}\frac{B_{n}}{n!}\check{\epsilon}^{n-1}\partial_{x}^{n-1}y(x,t),$
(3.6)

and as before $\check{\epsilon}=\epsilon/2\pi$.

###### Proof.

We act on (3.5) on the left with $\partial_{x}$ to obtain

$\displaystyle(\exp(\check{\epsilon}\partial_{x})-\exp(2\pi\mathrm{i}y(x,t)))\cdot\partial_{x}S\cdot\Psi-2\pi\mathrm{i}\partial_{x}y(x,t)\cdot\exp(2\pi\mathrm{i}y(x,t))\cdot\Psi=0.$

Then we use that

$\exp(\check{\epsilon}\partial_{x})(\partial_{x}S\cdot\Psi)=(\exp(\check{\epsilon}\partial_{x})\partial_{x}S)\cdot(\exp(\check{\epsilon}\partial_{x})\Psi),$

and the original equation (3.5) to obtain

$\big{(}{\rm
e}^{\check{\epsilon}\partial_{x}}-1\big{)}\partial_{x}S=2\pi\mathrm{i}\partial_{x}y(x,t),$
(3.7)

assuming that $\exp(2\pi\mathrm{i}y(x,t))\cdot\Psi\neq 0$.

From the generating function

$\frac{w}{{\rm e}^{w}-1}=\sum_{n=0}^{\infty}\frac{B_{n}}{n!}w^{n}$

of Bernoulli numbers, after rearranging and replacing
$w\rightarrow\check{\epsilon}\partial_{x}$, we obtain the identity

$\big{(}{\rm
e}^{\check{\epsilon}\partial_{x}}-1\big{)}\sum_{n=0}^{\infty}\frac{B_{n}}{n!}(\check{\epsilon}\partial_{x})^{n}y(x,t)=\check{\epsilon}\partial_{x}y(x,t),$

when acting on $y(x,t)$ with both sides. Comparing this equation with (3.7)
yields

$\partial_{x}S=\frac{2\pi\mathrm{i}}{\check{\epsilon}}\sum_{n=0}^{\infty}\frac{B_{n}}{n!}\check{\epsilon}^{n}\partial_{x}^{n}y(x,t),$

which can be integrated to

$S(\epsilon,x,t)=\frac{4\pi^{2}\mathrm{i}}{\epsilon}\int^{x}y(z,t)\,{\rm
d}z+2\pi\mathrm{i}\sum_{n=1}^{\infty}\frac{B_{n}}{n!}\check{\epsilon}^{n-1}\partial_{x}^{n-1}y(x,t).$
∎

######  Corollary 3.2.

The formal power series solution for the Schrödinger equation

$\mathbf{D}_{\Sigma}(\epsilon,x,t)\Psi(\epsilon,x,t)=\big{(}\big{(}1-{\rm
e}^{2\pi\mathrm{i}x}\big{)}{\rm
e}^{\check{\epsilon}\partial_{x}}-\big{(}1-Q{\rm
e}^{2\pi\mathrm{i}x}\big{)}\big{)}\Psi(\epsilon,x,t)=0$ (3.8)

is given by $\Psi(\epsilon,x,t)=\exp(S(\epsilon,x,t))$ with

$S(\epsilon,x,t)=-\sum_{n=0}^{\infty}\frac{B_{n}}{n!}(\mathrm{i}\epsilon)^{n-1}\left({\rm
Li}_{2-n}(QX)-{\rm Li}_{2-n}(X)\right)$ (3.9)

and $X=\exp(2\pi\mathrm{i}x)$. In other words,

$\lambda^{\text{qu}}(\epsilon,x,t)=\left(-\frac{1}{2\pi\mathrm{i}}\sum_{n=0}^{\infty}\frac{B_{n}}{n!}(\mathrm{i}\epsilon)^{n}\left({\rm
Li}_{1-n}(QX)-{\rm Li}_{1-n}(X)\right)\right)\mathrm{d}x.$

###### Proof.

The operator (3.1) corresponds to

$y(x,t)=\frac{1}{2\pi\mathrm{i}}\log\left(\frac{1-Q{\rm
e}^{2\pi\mathrm{i}x}}{1-{\rm e}^{2\pi\mathrm{i}x}}\right).$

Using the statement of Proposition 3.1 as well as the formulae

${\rm Li}_{1}(z)=-\log(1-z),\qquad{\rm Li}_{s-1}\big{(}{\rm
e}^{2\pi\mathrm{i}x}\big{)}=\frac{1}{2\pi\mathrm{i}}\partial_{x}{\rm
Li}_{s}({\rm e}^{2\pi\mathrm{i}x}),$

we find

$S=\frac{1}{\check{\epsilon}}\int^{x}\log\left(\frac{1-Q{\rm
e}^{2\pi\mathrm{i}z}}{1-{\rm e}^{2\pi\mathrm{i}z}}\right){\rm
d}z-\sum_{n=1}^{\infty}\frac{B_{n}}{n!}(\mathrm{i}\epsilon)^{n-1}\left({\rm
Li}_{2-n}(QX)-{\rm Li}_{2-n}(X)\right).$

We obtain the statement of the corollary by further integrating the first
term, and obtain the expression for $\lambda^{\text{qu}}$ by noting that

$\lambda^{\text{qu}}(\epsilon,x,t)=\frac{\epsilon}{4\pi^{2}\mathrm{i}}\partial_{x}S(\epsilon,x,t)\,\mathrm{d}x.$
∎

###  3.3 Exact solutions

Exact solutions to difference operators corresponding to quantum mirror curves
have been found in many places, using multiple methods. For instance, from the
perspective of 5d instanton calculus [22, 42] and spectral theory [81]. Also
Borel summation has played an important role in the literature. What is new in
this paper is that we consider the Borel summation along any ray with phase
$\vartheta$. In this subsection we write down a few known exact solutions for
the Schrödinger equation (3.1), corresponding to our resolved conifold
example, and make the connection with the Borel analysis performed in Section
5.

Reference [1] wrote down the exact solution

$\Psi_{{\rm GV}}(\epsilon,x,t)=\frac{L(x,\epsilon)}{L(x+t,\epsilon)},$ (3.10)

for $\operatorname{Im}(\epsilon)>0$, which expresses the quantum vev as a
quotient of quantum dilogarithms

$L(x,\epsilon)=\prod_{j=0}^{\infty}(1-\exp(2\pi\mathrm{i}(x+j\check{\epsilon}))).$
(3.11)

We have given this expression for $\Psi$ the subscript GV since it is the open
topological string expectation value for the brane corresponding to $\Psi$ in
Gopakumar–Vafa form. It is also the quantum vev that appears in the gauge
theory context, computing the expectation value of an abelian surface defect
in the ${\rm U}(1)$ gauge theory (see for instance [34]).

In Section 5, we find that (3.10) may be obtained by Borel summing (3.9) along
the positive imaginary axis. For this it is convenient to note that
$L(x,\epsilon)$, with $\operatorname{Im}(x)>0$ and
$\operatorname{Im}(\epsilon)>0$, may be rewritten as

$\displaystyle\log(L(x,\epsilon))$
$\displaystyle=\sum_{j=0}^{\infty}\log(1-\exp(2\pi\mathrm{i}(x+j\check{\epsilon})))$
$\displaystyle=-\sum_{l=1}^{\infty}\sum_{j=0}^{\infty}\frac{{\rm
e}^{2\pi\mathrm{i}l(x+j\check{\epsilon})}}{l}=\sum_{l=1}^{\infty}\frac{{\rm
e}^{2\pi\mathrm{i}lx}}{l({\rm e}^{2\pi\mathrm{i}l\check{\epsilon}}-1)},$
(3.12)

where the last expression is well-defined in the larger domain
$\operatorname{Im}(x)>0$ and $\operatorname{Im}(\epsilon)\neq 0$.

Another solution with better analytical properties is given by the ratio

$\Psi_{{\rm
np}}(\epsilon,x,t)=\frac{\mathcal{S}_{2}(x|\check{\epsilon},1)}{\mathcal{S}_{2}(x+t|\check{\epsilon},1)}$
(3.13)

of Faddeev’s quantum dilogarithms $\mathcal{S}_{2}$, in the conventions
reviewed in Appendix B.2. Using the difference equations (B.4) satisfied by
$\mathcal{S}_{2}$, it can be verified that $\Psi_{{\rm np}}$ is a solution of
the Schrödinger equation (3.1), and from equation (B.7) it can be checked that
$\Psi_{{\rm np}}$ has an asymptotic expansion given by (3.9). We will see in
Section 5 that this solution corresponds to Borel summing (3.9) along the
positive real axis.

Using (B.5), the expression (3.13) can also be written as the product

$\displaystyle\Psi_{{\rm
np}}(\epsilon,x,t)=\prod_{k=0}^{\infty}\frac{(1-\exp(2\pi\mathrm{i}(x+k\check{\epsilon}))}{(1-\exp(2\pi\mathrm{i}(x+t+k\check{\epsilon}))}\frac{(1-\exp(2\pi\mathrm{i}((x+t)/\check{\epsilon}-(k+1)/\check{\epsilon}))}{(1-\exp(2\pi\mathrm{i}(x/\check{\epsilon}-(k+1)/\check{\epsilon}))},$

which is valid for $\operatorname{Im}(\epsilon)>0$.

##  4 Quantum curves of closed string moduli

In this section we study difference equations obeyed by the refined
topological string free energy on the resolved conifold, its limits and the
connections between the resulting objects. The approach is similar to [6, 68]
where difference equations were derived starting from the asymptotic expansion
of the free energy. We will then show that these difference equations obtained
from the asymptotic expansion also admit natural analytic functions in the
perturbative expansion parameter as in [7, 8], thus providing candidates for
the non-perturbative completion. From the analytic solutions we will be able
to moreover derive new difference equations which are invisible to the
asymptotic expansion, these correspond to shifts of the closed string moduli
by integers.

###  4.1 The refined difference equation

We first derive a refined version of the difference equations for the free
energies of the resolved conifold starting from its asymptotic expansion.

######  Proposition 4.1.

The refined free energy for the resolved conifold geometry satisfies

$\displaystyle F^{{\rm ref}}$
$\displaystyle\left(\epsilon_{1},\epsilon_{2},t+\frac{\check{\epsilon}_{1}}{2}-\frac{\check{\epsilon}_{2}}{2}\right)+F^{{\rm
ref}}\left(\epsilon_{1},\epsilon_{2},t-\frac{\check{\epsilon}_{1}}{2}+\frac{\check{\epsilon}_{2}}{2}\right)$
$\displaystyle-F^{{\rm
ref}}\left(\epsilon_{1},\epsilon_{2},t+\frac{\check{\epsilon}_{1}}{2}+\frac{\check{\epsilon}_{2}}{2}\right)-F^{{\rm
ref}}\left(\epsilon_{1},\epsilon_{2},t-\frac{\check{\epsilon}_{1}}{2}-\frac{\check{\epsilon}_{2}}{2}\right)=-{\rm
Li}_{1}(Q),$ (4.1)

with

$\check{\epsilon}^{a}=\frac{\epsilon^{a}}{2\pi},\qquad a=1,2.$

###### Proof.

The above proposition can be verified by an explicit computation, since the
above shifts of $F^{\rm ref}$ can all be expanded in $\epsilon_{1}$ and
$\epsilon_{2}$ with $Q$-dependent coefficients as in (2.3) and (2.4). We then
find

$\displaystyle F^{{\rm
ref}}\left(\epsilon_{1},\epsilon_{2},t+\frac{\check{\epsilon}_{1}}{2}-\frac{\check{\epsilon}_{2}}{2}\right)+F^{{\rm
ref}}\left(\epsilon_{1},\epsilon_{2},t-\frac{\check{\epsilon}_{1}}{2}+\frac{\check{\epsilon}_{2}}{2}\right)$
$\displaystyle\qquad\quad{}-F^{{\rm
ref}}\left(\epsilon_{1},\epsilon_{2},t+\frac{\check{\epsilon}_{1}}{2}+\frac{\check{\epsilon}_{2}}{2}\right)-F^{{\rm
ref}}\left(\epsilon_{1},\epsilon_{2},t-\frac{\check{\epsilon}_{1}}{2}-\frac{\check{\epsilon}_{2}}{2}\right)$
$\displaystyle\qquad{}=\sum_{m,n=0}^{\infty}{\rm
Li}_{3-m-n}(Q)\frac{B_{m}B_{n}}{m!n!}\mathrm{i}^{m+n}\epsilon_{1}^{m-1}\epsilon_{2}^{n-1}\left(1+(-1)^{m+n}+(-1)^{m-1}+(-1)^{n-1}\right).$

When $m$ and $n$ are either even or of different parity, the last factor in
the summands gives $0$, while for $m$ and $n$ odd, only $m=n=1$ survives (due
to the Bernoulli numbers vanishing), giving
$4\mathrm{i}^{2}(B_{1})^{2}\mathrm{Li}_{1}(Q)=-\mathrm{Li}_{1}(Q)$. The
desired result then follows.

In the following we also supplement a proof based on the techniques of [68],
which were used in [6]. Consider

$\frac{w_{1}w_{2}}{({\rm e}^{w_{1}}-1)({\rm
e}^{w_{2}}-1)}=\sum_{m,n=0}^{\infty}\frac{B_{n}B_{m}}{n!m!}w_{1}^{m}w_{2}^{n},$

obtained from (2.2). This gives

$\big{(}{\rm e}^{w_{1}}-1\big{)}\big{(}{\rm
e}^{w_{2}}-1\big{)}\sum_{m,n=0}^{\infty}\frac{B_{n}\cdot B_{m}}{n!\cdot
m!}w_{1}^{m-1}w_{2}^{n-1}=1.$ (4.2)

Next, note that the asymptotic expansion of the refined topological string
free energy can be written as

$\displaystyle F^{{\rm
ref}}\left(\epsilon_{1},\epsilon_{2},t-\frac{\epsilon_{1}+\epsilon_{2}}{4\pi}\right)$
$\displaystyle=-\sum_{m,n=0}^{\infty}{\rm
Li}_{3-m-n}(Q)\frac{B_{m}B_{n}}{m!n!}i^{m+n}\epsilon_{1}^{m-1}\epsilon_{2}^{n-1}$
$\displaystyle=\sum_{m,n=0}^{\infty}\frac{B_{m}B_{n}}{m!n!}\left(\partial_{t}^{m+n-2}{\rm
Li}_{1}(Q)\right)\check{\epsilon}_{1}^{m-1}\check{\epsilon}_{2}^{n-1},$

where we used

$2\pi\mathrm{i}{\rm Li}_{s-1}(Q)=\partial_{t}{\rm
Li}_{s}(Q)\qquad\mathrm{and}\qquad\check{\epsilon}_{i}=\frac{\epsilon_{i}}{2\pi}.$

The negative powers of $\partial_{t}$ correspond to the indefinite integration
while setting the integration constant to zero, i.e.,

$2\pi\mathrm{i}\partial_{t}^{-1}{\rm Li}_{s-1}(Q)={\rm Li}_{s}(Q).$

The difference equation then follows after turning (4.2) into an operator
identity, where we replace $w_{i}$ on the left-hand side with the derivative
$\rightarrow\check{\epsilon}_{i}\partial_{t}$ for $i=1,2$, while the right-
hand side becomes the identity operator. Subsequently, we act with both sides
on ${\rm Li}_{1}(Q)$. ∎

###  4.2 A non-perturbative solution

In this section we write down a particularly interesting solution of the
difference equation (4.1) and describe its non-perturbative content. The
unrefined limit of this solution equals the Borel sum along the real axis of
the GW potential, which was obtained in [9], whereas the NS limit of this
solution matches the Borel sum of the NS limit of the GW potential along the
real axis, as we will find in Theorem 5.2.

Define the function999We have given this function $F^{{\rm ref}}_{{\rm np}}$ a
subscript “np” because it represents a non-perturbative completion of the free
energy $F^{{\rm ref}}$. Later in this paper we will find that there are many
other non-perturbative completions of $F^{{\rm ref}}$, which we label using
the subscript $\rho$. The function $F^{{\rm ref}}_{{\rm np}}$ with subscript
“np” will turn out to play a special role among them.

$\displaystyle F^{{\rm ref}}_{{\rm
np}}(\epsilon_{1},\epsilon_{2},t):=\left(\frac{\pi\mathrm{i}}{6}B_{3,3}\left(t+\frac{\check{\epsilon}_{1}-\check{\epsilon}_{2}}{2}\mid\check{\epsilon}_{1},-\check{\epsilon}_{2},1\right)\right)$
$\displaystyle\hphantom{F^{{\rm ref}}_{{\rm
np}}(\epsilon_{1},\epsilon_{2},t):=}{}+\log\left(\sin_{3}\left(t+\frac{\check{\epsilon}_{1}-\check{\epsilon}_{2}}{2}\mid\check{\epsilon}_{1},-\check{\epsilon}_{2},1\right)\right),$
(4.3)

where $B_{3,3}(z\mid\omega_{1},\omega_{2})$ is a multiple Bernoulli polynomial
and $\sin_{3}(z\mid\omega_{1},\omega_{2})$ the triple sine function, both
defined in Appendix B.101010Multiple sine functions in the context of the
quantum Riemann–Hilbert problem determined by refined Donaldson–Thomas theory
on the resolved conifold have recently appeared in [25].

######  Proposition 4.2.

$F^{{\rm ref}}_{{\rm np}}(\epsilon_{1},\epsilon_{2},t)$ has the following
properties:

  1. $1.$

$F^{{\rm ref}}_{{\rm np}}(\epsilon_{1},\epsilon_{2},t)$ satisfies the
difference equations

$\displaystyle F_{{\rm np}}^{{\rm
ref}}(\epsilon_{1},\epsilon_{2},t+\check{\epsilon}_{1})-F_{{\rm np}}^{{\rm
ref}}(\epsilon_{1},\epsilon_{2},t)=-\log\left(\mathcal{S}_{2}\left(t+\frac{\check{\epsilon}_{1}-\check{\epsilon}_{2}}{2}\mid{-}\check{\epsilon}_{2},1\right)\right),$
$\displaystyle F_{{\rm np}}^{{\rm
ref}}(\epsilon_{1},\epsilon_{2},t-\check{\epsilon}_{2})-F_{{\rm np}}^{{\rm
ref}}(\epsilon_{1},\epsilon_{2},t)=-\log\left(\mathcal{S}_{2}\left(t+\frac{\check{\epsilon}_{1}-\check{\epsilon}_{2}}{2}\mid\check{\epsilon}_{1},1\right)\right),$
$\displaystyle F_{{\rm np}}^{{\rm ref}}(\epsilon_{1},\epsilon_{2},t+1)-F_{{\rm
np}}^{{\rm
ref}}(\epsilon_{1},\epsilon_{2},t)=-\log\left(\mathcal{S}_{2}\left(t+\frac{\check{\epsilon}_{1}-\check{\epsilon}_{2}}{2}\mid\check{\epsilon}_{1},-\check{\epsilon_{2}}\right)\right),$
(4.4)

where

$\mathcal{S}_{2}(z\mid\omega_{1},\omega_{2}):=\exp\left(-\frac{\pi\mathrm{i}}{2}B_{2,2}(z\mid\omega_{1},\omega_{2})\right)\sin_{2}(z\mid\omega_{1},\omega_{2})$

is Faddeev’s quantum dilogarithm in slightly different conventions, following
[20, 83], which are reviewed in Appendix B.

  2. $2.$

$F^{{\rm ref}}_{{\rm np}}(\epsilon_{1},\epsilon_{2},t)$ has the integral
representation

$F^{{\rm ref}}_{{\rm
np}}(\epsilon_{1},\epsilon_{2},t)=-\int_{\mathbb{R}+\mathrm{i}0^{+}}\frac{{\rm
e}^{\left(t+\frac{\check{\epsilon}_{1}-\check{\epsilon}_{2}}{2}\right)s}}{({\rm
e}^{s}-1)({\rm e}^{\check{\epsilon}_{1}s}-1)({\rm
e}^{-\check{\epsilon}_{2}s}-1)}\frac{{\rm d}s}{s},$ (4.5)

which is valid for $\operatorname{Re}(\check{\epsilon}_{1})>0$,
$\operatorname{Re}(-\check{\epsilon}_{2})>0$ and
$-\operatorname{Re}\big{(}\frac{\check{\epsilon}_{1}-\check{\epsilon}_{2}}{2}\big{)}<\operatorname{Re}t<\operatorname{Re}\big{(}\big{(}\frac{\check{\epsilon}_{1}-\check{\epsilon}_{2}}{2}\big{)}+1\big{)}$.
The contour $\mathbb{R}+\mathrm{i}0^{+}$ is following the real axis from
$-\infty$ to $\infty$ avoiding $0$ by a small detour in the upper half plane.

  3. $3.$

$F^{{\rm ref}}_{{\rm np}}(\epsilon_{1},\epsilon_{2},t)$ satisfies the
difference equation (4.1).

  4. $4.$

The exponential $Z^{{\rm ref}}_{{\rm
np}}(\epsilon_{1},\epsilon_{2},t):=\exp\big{(}F^{{\rm ref}}_{{\rm
np}}(\epsilon_{1},\epsilon_{2},t)\big{)}$ has the product expansion

$\displaystyle Z^{{\rm ref}}_{{\rm np}}(\epsilon_{1},\epsilon_{2},t)=Z^{{\rm
ref}}(\epsilon_{1},\epsilon_{2},t)Z^{{\rm
ref}}\big{(}4\pi^{2}/\epsilon_{1},2\pi(\epsilon_{2}/\epsilon_{1}+1),2\pi(t-1/2)/\epsilon_{1}\big{)}$
$\displaystyle\hphantom{Z^{{\rm ref}}_{{\rm
np}}(\epsilon_{1},\epsilon_{2},t)=}{}\times Z^{{\rm
ref}}\big{(}4\pi^{2}/\epsilon_{2},-2\pi(\epsilon_{1}/\epsilon_{2}+1),-2\pi(t-1/2)/\epsilon_{2}\big{)}$
(4.6)

for $\operatorname{Im}(-\epsilon_{1}/\epsilon_{2})>0$,
$\operatorname{Im}(\epsilon_{1})>0$ and $\operatorname{Im}(-\epsilon_{2})>0$,
where

$\displaystyle Z^{\rm ref}(\epsilon_{1},\epsilon_{2},t)=\exp
F_{\mathrm{GV}}^{\rm
ref}(\epsilon_{1},\epsilon_{2},t)=\prod_{k,l=1}^{\infty}\big{(}1-Qq_{1}^{k-1/2}q_{2}^{l-1/2}\big{)}^{-1}.$

###### Proof.

To prove the first difference equation in the proposition we use the
definition (4.3) and obtain

$\displaystyle F_{{\rm np}}^{{\rm
ref}}(\epsilon_{1},\epsilon_{2},t+\check{\epsilon}_{1})-F_{{\rm np}}^{{\rm
ref}}(\epsilon_{1},\epsilon_{2},t)=\left(\frac{\pi\mathrm{i}}{2}B_{2,2}\left(t+\frac{\epsilon_{1}-\epsilon_{2}}{4\pi}\mid{-}\check{\epsilon}_{2},1\right)\right)$
$\displaystyle\hphantom{F_{{\rm np}}^{{\rm
ref}}(\epsilon_{1},\epsilon_{2},t+\check{\epsilon}_{1})-F_{{\rm np}}^{{\rm
ref}}(\epsilon_{1},\epsilon_{2},t)=}{}-\log\left(\sin_{2}\left(t+\frac{\epsilon_{1}-\epsilon_{2}}{4\pi}\mid{-}\check{\epsilon}_{2},1\right)\right),$

where we have used equation (B.3) for the multiple Bernouilli polynomials, as
well as relation (B.1) for the multiple sine functions. We recognize the
right-hand side as

$-\log\left(\mathcal{S}_{2}\left(t+\frac{\epsilon_{1}-\epsilon_{2}}{4\pi}\mid{-}\check{\epsilon}_{2},1\right)\right).$

The second and third difference equation follow analogously.

The integral representation in item (ii) is the one given in [83, Proposition
2].

The proof that $F^{{\rm ref}}_{{\rm np}}$ satisfies the difference equation
(4.1) follows by successively using the first two difference equations proved
in item (i) and then using the difference equation for the quantum dilogarithm
function reviewed in equation (B.4). Indeed, after applying equation (4.4)
twice and then substituting equation (B.4), we find

$\displaystyle F^{{\rm
ref}}\left(\epsilon_{1},\epsilon_{2},t+\frac{\check{\epsilon}_{1}}{2}-\frac{\check{\epsilon}_{2}}{2}\right)+F^{{\rm
ref}}\left(\epsilon_{1},\epsilon_{2},t-\frac{\check{\epsilon}_{1}}{2}+\frac{\check{\epsilon}_{2}}{2}\right)$
$\displaystyle\qquad\quad{}-F^{{\rm
ref}}\left(\epsilon_{1},\epsilon_{2},t+\frac{\check{\epsilon}_{1}}{2}+\frac{\check{\epsilon}_{2}}{2}\right)-F^{{\rm
ref}}\left(\epsilon_{1},\epsilon_{2},t-\frac{\check{\epsilon}_{1}}{2}-\frac{\check{\epsilon}_{2}}{2}\right)$
$\displaystyle\qquad{}=-\log\left(\mathcal{S}_{2}(t-\check{\epsilon}_{2}\mid{-}\check{\epsilon}_{2},1)\right)+\log\left(\mathcal{S}_{2}(t\mid{-}\check{\epsilon}_{2},1)\right)=\log(1-Q)=-\mathrm{Li}_{1}(Q).$

Finally, the proof of the product formula in item (iv) is found using the
product formula of [83, Corollary 6], reviewed in equation (B.9) of Appendix
B, and after applying the identity (B.10) twice. Spelled out in detail, the
product formula of [83] gives

$\displaystyle Z_{{\rm np}}^{\rm
ref}=\prod_{j=k=0}^{\infty}(1-\exp(2\pi\mathrm{i}(t/\check{\epsilon}_{1}+1/2-\check{\epsilon}_{2}/(2\check{\epsilon}_{1})+(j+1)\epsilon_{2}/\epsilon_{1}-(k+1)/\check{\epsilon}_{1})))$
$\displaystyle\hphantom{Z_{{\rm np}}^{\rm
ref}=}{}\times\prod_{j=k=0}^{\infty}(1-\exp\left(2\pi\mathrm{i}\left(t+(\check{\epsilon}_{1}+\check{\epsilon}_{2})/2+j\check{\epsilon}_{1}-k\check{\epsilon}_{2}\right)\right))$
$\displaystyle\hphantom{Z_{{\rm np}}^{\rm
ref}=}{}\times\prod_{j=k=0}^{\infty}(1-\exp\left(2\pi\mathrm{i}\left(-t/\check{\epsilon}_{2}-\epsilon_{1}/(2\epsilon_{2})+1/2-j\epsilon_{1}/\epsilon_{2}+(k+1)/\check{\epsilon}_{2}\right)\right))^{-1},$

under the assumptions of point (iv). By applying (B.10) to the first product
with $q_{j}={\rm e}^{-\mathrm{i}/\epsilon_{1}}$ and the second product with
$q_{j}={\rm e}^{-\mathrm{i}\epsilon_{2}}$, we obtain

$\displaystyle Z_{{\rm np}}^{\rm
ref}=\prod_{j=k=0}^{\infty}(1-\exp\left(2\pi\mathrm{i}\left(t-1/2\right)/\check{\epsilon}_{1}+2\pi\mathrm{i}(k+1/2)/\check{\epsilon}_{1}+2\pi\mathrm{i}(j+1/2)(\epsilon_{2}/\epsilon_{1}+1)\right))^{-1}$
$\displaystyle\hphantom{Z_{{\rm np}}^{\rm
ref}=}{}\times\prod_{j=k=0}^{\infty}(1-\exp\left(2\pi\mathrm{i}t+\mathrm{i}\epsilon_{1}(j+1/2)+\mathrm{i}\epsilon_{2}(k+1/2)\right))^{-1}$
$\displaystyle\hphantom{Z_{{\rm np}}^{\rm
ref}=}{}\times\prod_{j=k=0}^{\infty}\\!(1-\exp(2\pi\mathrm{i}(-t+1/2)/\check{\epsilon}_{2}-2\pi\mathrm{i}(\epsilon_{1}/\epsilon_{2}+1)(j+1/2)+2\pi\mathrm{i}(k+1/2)/\check{\epsilon}_{2}))^{-1},$

after slightly rewriting the exponents. Finally, by shifting $j$ and $k$ by
$1$, so that the above products start from $j=k=1$, we obtain equation (4.6).
∎

Let us remark that we can also recover the product representation (4.6) from
the integral representation (4.5). Indeed, under the assumptions of item (ii),
the integrand has three infinite sets of poles in the upper half plane without
zero, given by

$s=2\pi\mathrm{i}k,\qquad
s=\frac{2\pi\mathrm{i}k}{\check{\epsilon}_{1}}\qquad\textrm{and}\qquad
s=-\frac{2\pi\mathrm{i}k}{\check{\epsilon}_{2}},$

for $k\in\mathbb{N}\setminus\\{0\\}$. These poles are simple if we assume that
$\check{\epsilon}_{1},\check{\epsilon}_{2}\notin\mathbb{Q}$, $\epsilon_{1}\neq
r\epsilon_{2}$ with $r\in\mathbb{Q}$. By closing the contour of (4.5) in the
upper half plane, and further restricting the range of parameters if
necessary, we can compute (4.5) by summing over the residues. We then find

$\displaystyle F^{\rm ref}_{{\rm np}}(\epsilon_{1},\epsilon_{2},t)=F^{{\rm
ref}}_{{\rm GV}}(\epsilon_{1},\epsilon_{2},t)+F_{{\rm GV}}^{{\rm
ref}}(4\pi^{2}/\epsilon_{1},2\pi(\epsilon_{2}/\epsilon_{1}+1),2\pi(t-1/2)/\epsilon_{1})$
$\displaystyle\hphantom{F^{\rm ref}_{{\rm
np}}(\epsilon_{1},\epsilon_{2},t)=}{}+F_{{\rm GV}}^{{\rm
ref}}(4\pi^{2}/\epsilon_{2},-2\pi(\epsilon_{1}/\epsilon_{2}+1),-2\pi(t-1/2)/\epsilon_{2}),$
(4.7)

where

$F^{{\rm ref}}_{{\rm
GV}}(\epsilon_{1},\epsilon_{2},t)=-\sum_{k=1}^{\infty}\frac{Q^{k}}{k(2\sin
k\epsilon_{1}/2)(2\sin k\epsilon_{2}/2)}$

is the Gopakumar–Vafa resummation of the free energy.

######  Remark 4.3.

  * •

The expression (4.7) for the non-perturbative refined free energy as a sum
over three perturbative pieces evaluated at different values of the arguments,
matches the proposal of Lockhart and Vafa [77] for the resolved conifold, up
to small discrepancies due to different conventions for the parameters
$\epsilon_{1}$ and $\epsilon_{2}$.

  * •

In the limit $\epsilon_{1}=-\epsilon_{2}=\lambda$, the free energy $F^{{\rm
ref}}_{{\rm np}}(\epsilon_{1},\epsilon_{2},t)$ reduces to $F^{\rm top}_{{\rm
np}}(\lambda,t)$. The latter expression was studied in [7] and was shown to
correspond to the Borel summed free energy in a distinguished region of the
Borel plane in [9]. It was furthermore shown in [7] that it can be written in
the form111111The discrepancies of the relative factor between the two
summands to [7, 9, 59] is due to different normalization conventions for
$F_{{\rm NS}}$ used in these works.

$F^{\rm top}_{{\rm np}}(\lambda,t)=F_{{\rm
GV}}(\lambda,t)+\frac{1}{2\pi}\frac{\partial}{\partial\lambda}\big{(}\lambda
F_{{\rm NS}}\big{(}1/\check{\lambda},(t-1/2)/\check{\lambda}\big{)}\big{)}.$
(4.8)

This result (4.8) matches equation (5.6) of [59], which was derived using a
generalized Borel resummation.

  * •

A similar integral representation for the refined non-perturbative topological
string partition function was obtained from Chern–Simons theory in [75].

###  4.3 Difference equations in the NS limit

In Section 2.3, we introduced the NS limit $F^{\rm NS}(\epsilon,t)$ of the
perturbative refined free energy $F^{\rm ref}(\epsilon_{1},\epsilon_{2},t)$.
In this section we define the NS limit $F^{\rm NS}_{{\rm np}}(\epsilon,t)$ of
the non-perturbative refined free energy $F^{\rm
ref}_{\mathrm{np}}(\epsilon_{1},\epsilon_{2},t)$. We find an integral
representation for $F^{\rm NS}_{{\rm np}}(\epsilon,t)$ and write down the
difference equations this free energy satisfies.

Recall the definition (2.5) of the NS limit

$\displaystyle F^{\rm
NS}(\epsilon,t):=-\frac{1}{2}\sum_{k=1}^{\infty}\frac{Q^{k}}{k^{2}\sin(k\epsilon/2)}$

of $F^{\rm ref}(\epsilon_{1},\epsilon_{2},t)$. We similarly define the NS-
limit of $F_{{\rm np}}^{\rm ref}(\epsilon_{1},\epsilon_{2},t)$ as

$F^{\rm NS}_{{\rm np}}(\epsilon,t):=\lim_{\epsilon_{2}\rightarrow
0}\epsilon_{2}\cdot F^{{\rm ref}}_{{\rm
np}}(\epsilon_{1},\epsilon_{2},t)|_{\epsilon_{1}=\epsilon}.$ (4.9)

From (4.5), we find that $F^{\rm NS}_{{\rm np}}(\epsilon,t)$ has the integral
representation

$F^{\rm NS}_{{\rm
np}}(\epsilon,t)=2\pi\int_{\mathbb{R}+\mathrm{i}0^{+}}\frac{{\rm
e}^{(t+\frac{\check{\epsilon}}{2})s}}{({\rm e}^{s}-1)({\rm
e}^{\check{\epsilon}s}-1)}\frac{{\rm d}s}{s^{2}},$ (4.10)

which is valid for $\operatorname{Re}(\check{\epsilon})>0,$ and
$-\operatorname{Re}\left(\frac{\check{\epsilon}}{2}\right)<\operatorname{Re}t<\operatorname{Re}\left(\left(\frac{\check{\epsilon}}{2}\right)+1\right)$.
The contour $\mathbb{R}+\mathrm{i}0^{+}$ is following the real axis from
$-\infty$ to $\infty$, avoiding the origin by a small detour in the upper half
plane.

######  Proposition 4.4.

Assume that $\operatorname{Re}(\check{\epsilon})>0$ and
$-\operatorname{Re}\big{(}\frac{\check{\epsilon}}{2}\big{)}<\operatorname{Re}t<\operatorname{Re}\big{(}\big{(}\frac{\check{\epsilon}}{2}\big{)}+1\big{)}$.
Then for $\operatorname{Im}(t)>0$ and $\check{\epsilon}\notin\mathbb{Q}$, the
non-perturbative NS free energy $F^{\rm NS}_{{\rm np}}(\epsilon,t)$ can be
expressed as

$F^{\rm NS}_{{\rm np}}(\epsilon,t)=F^{\rm
NS}(\epsilon,t)+\check{\epsilon}F^{\rm
NS}(2\pi(1/\check{\epsilon}+1),(t-1/2)/\check{\epsilon}).$ (4.11)

###### Proof.

In the upper half plane without zero, the integrand of the integral
representation (4.10) has two infinite sets of poles given by

$s=2\pi\mathrm{i}k,\qquad s=\frac{2\pi\mathrm{i}k}{\check{\epsilon}},$

for $k\in\mathbb{N}\setminus\\{0\\}$. These poles are simple if we assume that
$\check{\epsilon}\notin\mathbb{Q}$.

Consider a sequence $R_{n}>0$, such that $R_{n}\to\infty$ and such that the
semicircle $C_{R_{n}}$, centered at $0$ with radius $R_{n}$, in the upper
half-plane does not intersect the above sets of poles. By analyticity in $t$
and $\epsilon$, it is enough to show equation (4.11) for
$\operatorname{Re}(t)=0$. In this case, by an application of Jordan’s lemma
(which requires $\operatorname{Im}(t)>0$), we have that

$\left|\int_{C_{R_{n}}}\frac{{\rm e}^{(t+\frac{\check{\epsilon}}{2})s}}{({\rm
e}^{s}-1)({\rm e}^{\check{\epsilon}s}-1)}\frac{{\rm
d}s}{s^{2}}\right|\leq\frac{\pi}{-\mathrm{i}t}M_{R_{n}}(\epsilon),$

where

$\displaystyle
M_{R_{n}}(\epsilon)=\operatorname{Max}_{\theta\in[0,\pi]}|g(\epsilon,R_{n}{\rm
e}^{i\theta})|,\qquad\mathrm{with}\quad g(\epsilon,s)=\frac{1}{2s^{2}({\rm
e}^{s}-1)\sinh(s\check{\epsilon}/2)}.$

With the above choice of the $C_{R_{n}}$, we can ensure that
$M_{R_{n}}(\epsilon)\to 0$ as $n\to\infty$. Hence, we can compute (4.10) by
summing the residues and find the desired expression using (2.5). ∎

Furthermore, $F^{\rm NS}_{{\rm np}}(\epsilon,t)$ is invariant under an
S-duality-like transformation. Indeed, let us define

$W^{\rm NS}_{\mathrm{np}}(\epsilon,t)=F^{\rm NS}_{\rm
np}\left(\epsilon,t-\frac{\check{\epsilon}}{2}\right).$

Then we have

######  Proposition 4.5.

Under the change of variables

$\epsilon^{\prime}=\frac{4\pi^{2}}{\epsilon},\qquad t^{\prime}=\frac{2\pi
t}{\epsilon},$

we have

$W^{\rm NS}_{\rm
np}(\epsilon^{\prime},t^{\prime})=\frac{1}{\check{\epsilon}}W^{\rm NS}_{\rm
np}(\epsilon,t).$ (4.12)

###### Proof.

Equation (4.12) follows directly from the integral equation (4.10). ∎

Moreover,

######  Proposition 4.6.

$F^{\rm NS}(\epsilon,t)$ satisfies the difference equation

$F^{\rm NS}(\epsilon,t+\check{\epsilon}/2)-F^{\rm
NS}(\epsilon,t-\check{\epsilon}/2)=-\mathrm{i}{\rm Li}_{2}(Q).$ (4.13)

$F^{\rm NS}_{{\rm np}}(\epsilon,t)$ also obeys this equation, and moreover
satisfies the additional difference equation

$F^{\rm NS}_{{\rm np}}(\epsilon,t+1)-F^{\rm NS}_{{\rm
np}}(\epsilon,t)=-\mathrm{i}\check{\epsilon}\mathrm{Li}_{2}\big{(}{\rm
e}^{2\pi\mathrm{i}(t+\check{\epsilon}/2)/\check{\epsilon}}\big{)}.$ (4.14)

Note that the latter difference equation is invisible to its perturbative
expansion.

###### Proof.

The first part follows from an explicit computation using the formal expansion
(2.6):

$\displaystyle F^{\rm NS}(\epsilon,t+\check{\epsilon}/2)-F^{\rm
NS}(\epsilon,t-\check{\epsilon}/2)$
$\displaystyle=\frac{1}{2\pi}\sum_{n=0}^{\infty}\partial_{t}^{n}{\rm
Li}_{3}(Q)\frac{B_{n}}{n!}\check{\epsilon}^{n-1}\big{(}(-1)^{n-1}+1\big{)},$
$\displaystyle=\frac{2B_{1}}{2\pi}\partial_{t}\mathrm{Li}_{3}(Q)=-\mathrm{i}\mathrm{Li}_{2}(Q),$

where we used that $B_{2n+1}=0$ for $n>0$.

For the second part of the proposition, we use the integral representation
(4.10) and note that

$\displaystyle F^{\rm NS}_{{\rm np}}(\epsilon,t+\check{\epsilon}/2)-F_{{\rm
np}}^{\rm NS}(\epsilon,t-\check{\epsilon}/2)$
$\displaystyle=2\pi\int_{\mathbb{R}+\mathrm{i}0^{+}}\frac{{\rm e}^{ts}}{({\rm
e}^{s}-1)}\frac{{\rm
d}s}{s^{2}}=(2\pi)^{2}\sum_{k=1}^{\infty}\frac{\mathrm{i}{\rm
e}^{ts}}{s^{2}}\Big{|}_{s=2\pi\mathrm{i}k}$
$\displaystyle=-\frac{\mathrm{i}{\rm
e}^{2\pi\mathrm{i}kt}}{k^{2}}=-\mathrm{i}{\rm Li}_{2}\left(Q\right),$

where we have computed the integral by the sum over the residues in the upper
half plane.

Similarly, we find

$\displaystyle F^{\rm NS}_{{\rm np}}(\epsilon,t+1)-F_{{\rm np}}^{\rm
NS}(\epsilon,t)$
$\displaystyle=2\pi\int_{\mathbb{R}+\mathrm{i}0^{+}}\frac{{\rm
e}^{\left(t+\frac{\check{\epsilon}}{2}\right)s}}{({\rm
e}^{\check{\epsilon}s}-1)}\frac{{\rm
d}s}{s^{2}}=\frac{(2\pi)^{2}}{\check{\epsilon}}\sum_{k=1}^{\infty}\frac{\mathrm{i}{\rm
e}^{(t+\check{\epsilon}/2)s}}{s^{2}}\Big{|}_{s=2\pi\mathrm{i}k/\check{\epsilon}}$
$\displaystyle=-\mathrm{i}\check{\epsilon}\sum_{k=1}^{\infty}\frac{{\rm
e}^{2\pi\mathrm{i}k(t+\check{\epsilon}/2)/\check{\epsilon}}}{k^{2}}=-\mathrm{i}\check{\epsilon}\mathrm{Li}_{2}\big{(}{\rm
e}^{2\pi\mathrm{i}(t+\check{\epsilon}/2)/\check{\epsilon}}\big{)}.$ ∎

###  4.4 Unrefined topological string

The limit in which the parameters of the refined topological string are set to
$\epsilon_{1}=-\epsilon_{2}=\lambda$ corresponds to the unrefined topological
string. The difference equation analogous to (4.1) was derived from the
asymptotic expansion of the Gromov–Witten potential of the resolved conifold
in [6] following methods of [68]. It is given by

$F^{\rm top}\big{(}\lambda,t+\check{\lambda}\big{)}+F^{\rm
top}\big{(}\lambda,t-\check{\lambda}\big{)}-2F^{\rm
top}\left(\lambda,t\right)=-{\rm
Li}_{1}(Q),\qquad\check{\lambda}=\frac{\lambda}{2\pi}.$

Furthermore a solution in terms of the triple sine function of this difference
equation was considered, which was given by [8]

$\displaystyle F^{\rm top}_{{\rm
np}}(\lambda,t):=\left(\frac{\pi\mathrm{i}}{6}B_{3,3}\left(t+\check{\lambda}|\check{\lambda},\check{\lambda},1\right)\right)+\log\big{(}\sin_{3}\big{(}t+\check{\lambda}|\check{\lambda},\check{\lambda},1\big{)}\big{)}.$

The non-perturbative content of this solution was analyzed in [7] and in [9].
It was identified as the Borel summation of the asymptotic series along a
distinguished ray on the real axis in the Borel plane.

Here we add a further difference equation, which is the unrefined limit of the
third difference equation in (4.4). Since the shift is integral, this
difference equation is invisible to the periodic asymptotic series. We have
that:

######  Proposition 4.7.

$F^{\rm top}_{{\rm np}}(\lambda,t)$ satisfies the difference equation

$\displaystyle F_{{\rm np}}^{\rm top}(\lambda,t+1)-F_{{\rm np}}^{\rm
top}(\lambda,t)=\frac{1}{2\pi\mathrm{i}}\frac{\partial}{\partial\check{\lambda}}\big{(}\check{\lambda}{\rm
Li}_{2}\big{(}{\rm e}^{2\pi\mathrm{i}t/\check{\lambda}}\big{)}\big{)}.$ (4.15)

###### Proof.

To prove this proposition we take the third difference equation in (4.4) and
set $\epsilon_{1}=-\epsilon_{2}=\lambda$. The right-hand side is then given in
terms of the quantum dilogarithm with repeated argument, for which we can use
the expression (B.8) in Appendix B.2. That is,

$-\log\big{(}\mathcal{S}_{2}\big{(}t+\check{\lambda}|\check{\lambda},\check{\lambda}\big{)}\big{)}=\frac{1}{2\pi\mathrm{i}}\frac{\partial}{\partial\check{\lambda}}\big{(}\check{\lambda}{\rm
Li}_{2}\big{(}{\rm e}^{2\pi\mathrm{i}t/\check{\lambda}}\big{)}\big{)}.$ ∎

######  Remark 4.8.

The right-hand side of the difference equation (4.15) equals Stokes jump of
the Borel resummation of $F^{\textrm{top}}(\lambda,t)$ obtained in [9]. The
difference equation can be given the interpretation as a relation between the
Borel resummations in the different Stokes sectors. Indeed, as was shown in
[9, Corollary 3.12],

$F^{\rm top}_{{\rho}_{k-n}}(\lambda,t+n)=F_{\rho_{k}}^{\rm top}(\lambda,t),$

where $F_{\rho}^{\rm top}$ denotes the Borel sum along the ray $\rho$, and
$\rho_{k}$ is a ray between $l_{k}$ and $l_{k-1}$, where
$l_{k}=\mathbb{R}_{<0}\cdot 2\pi\mathrm{i}(t+k)$. This relation together with
the jumps (1.1) is equivalent to the difference equation (4.15). We will find
a similar interpretations for the analogous difference equation in the NS
limit.

###  4.5 Relations between free energies through difference equations

In this section we derive further relations between the refined topological
string free energy and its various limits. These will be useful in the rest of
the paper.

######  Proposition 4.9.

The two difference equations

$\displaystyle F^{{\rm
ref}}\left(\epsilon_{1},\epsilon_{2},t-\frac{\check{\epsilon}_{1}+\check{\epsilon}_{2}}{2}+\check{\epsilon}_{1}\right)-F^{{\rm
ref}}\left(\epsilon_{1},\epsilon_{2},t-\frac{\check{\epsilon}_{1}+\check{\epsilon}_{2}}{2}\right)=\frac{1}{2\pi}\partial_{t}F^{\mathrm{NS}}\left(\epsilon_{2},t-\frac{\check{\epsilon}_{2}}{2}\right),$
$\displaystyle F^{{\rm
ref}}\left(\epsilon_{1},\epsilon_{2},t-\frac{\check{\epsilon}_{1}+\check{\epsilon}_{2}}{2}+\check{\epsilon}_{2}\right)-F^{{\rm
ref}}\left(\epsilon_{1},\epsilon_{2},t-\frac{\check{\epsilon}_{1}+\check{\epsilon}_{2}}{2}\right)=\frac{1}{2\pi}\partial_{t}F^{\mathrm{NS}}\left(\epsilon_{1},t-\frac{\check{\epsilon}_{1}}{2}\right)$
(4.16)

give a relation between $F^{{\rm ref}}$ and $F^{\mathrm{NS}}$.

###### Proof.

This proposition can again be verified by an explicit computation. We prove
the first equation here, the second one follows analogously. Since the above
shifts of $F^{\rm ref}$ can all be expanded in $\epsilon_{1}$ and
$\epsilon_{2}$ with $Q$-dependent coefficients as in (2.3) and (2.4), we find

$\displaystyle F^{{\rm
ref}}\left(\epsilon_{1},\epsilon_{2},t+\frac{\check{\epsilon}_{1}}{2}-\frac{\check{\epsilon}_{2}}{2}\right)+F^{{\rm
ref}}\left(\epsilon_{1},\epsilon_{2},t-\frac{\check{\epsilon}_{1}}{2}-\frac{\check{\epsilon}_{2}}{2}\right)$
$\displaystyle\qquad{}=\sum_{m,n=0}^{\infty}{\rm
Li}_{3-m-n}(Q)\frac{B_{m}B_{n}}{m!n!}\mathrm{i}^{m+n}\epsilon_{1}^{m-1}\epsilon_{2}^{n-1}\left((-1)^{m-1}+1\right).$

The last factor in the summand only contributes when $m=1$, since all other
odd Bernoulli numbers vanish. The right-hand side then becomes

$\sum_{n=0}^{\infty}{\rm
Li}_{2-n}(Q)\frac{B_{n}}{n!}(\mathrm{i}\epsilon_{2})^{n-1},$

which can be matched with the right-hand side of the proposition using the
asymptotic expansion (2.6). ∎

######  Corollary 4.10.

Similarly, the unrefined topological string free energies obey the difference
equation

$\displaystyle F^{\rm top}\big{(}\lambda,t+\check{\lambda}\big{)}-F^{\rm
top}(\lambda,t)=\frac{1}{2\pi}\partial_{t}F^{\mathrm{NS}}\left(\lambda,t+\frac{\check{\lambda}}{2}\right).$
(4.17)

######  Remark 4.11.

  * •

Similar relations also hold for the free energies with the subscript ${\rm
np}$. These can be easily derived from the corresponding integral
representations.

  * •

Equation (4.17) is the difference equation relating the topological string
free energy to certain Darboux coordinates, as was observed in [27] and [10].
This will become clearer in Section 7.5.

###  4.6 A quantum curve for closed string moduli

In the following, we interpret the finite difference equations found above as
the quantization of an algebraic curve that should be associated with the
closed string moduli. The motivation for this is the work of [2], in which it
was proposed that the mirror curves arising in the mirror constructions of
non-compact CY threefolds can be interpreted as the analogs of the
Hamiltonians of a quantum mechanical problem. The curve which is quantized in
the case of [2] parametrizes the open string moduli, as was put forward in
[3].

######  Proposition 4.12.

Define

$\displaystyle\Psi^{c}_{1}(\epsilon_{1},\epsilon_{2},t):=\frac{Z^{{\rm
ref}}(\epsilon_{1},\epsilon_{2},t-\frac{\check{\epsilon}_{1}+\check{\epsilon}_{2}}{2})}{Z^{{\rm
ref}}(\epsilon_{1},\epsilon_{2},t-\frac{\check{\epsilon}_{1}+\check{\epsilon}_{2}}{2}+\check{\epsilon}_{2})},$
$\displaystyle\Psi^{c}_{2}(\epsilon_{1},\epsilon_{2},t):=\frac{Z^{{\rm
ref}}(\epsilon_{1},\epsilon_{2},t-\frac{\check{\epsilon}_{1}+\check{\epsilon}_{2}}{2})}{Z^{{\rm
ref}}(\epsilon_{1},\epsilon_{2},t-\frac{\check{\epsilon}_{1}+\check{\epsilon}_{2}}{2}+\check{\epsilon}_{1})},$

where $Z^{{\rm ref}}(\epsilon_{1},\epsilon_{2},t)=\exp\big{(}F^{{\rm
ref}}(\epsilon_{1},\epsilon_{2},t)\big{)}$. Using $Z_{{\rm np}}^{{\rm
ref}}(\epsilon_{1},\epsilon_{2},t)=\exp\big{(}F_{{\rm np}}^{{\rm
ref}}(\epsilon_{1},\epsilon_{2},t)\big{)}$ we obtain analogous definitions for
$\Psi^{c}_{i,{\rm np}}(\epsilon_{1},\epsilon_{2},t)$ for $i\in\\{1,2\\}$. Then
the following holds for $i\in\\{1,2\\}$:

  1. $1.$

$\Psi^{c}_{i}(\epsilon_{1},\epsilon_{2},t)$ only depends on $\epsilon_{i}$ and

$\Psi^{c}_{i}(\epsilon_{i},t)=\exp\left(-\frac{1}{2\pi}\partial_{t}F^{\rm
NS}\left(\epsilon_{i},t-\frac{\epsilon_{i}}{4\pi}\right)\right).$

The same is true for $\Psi^{c}_{i,{\rm np}}(\epsilon_{1},\epsilon_{2},t)$.

  2. $2.$

$\Psi^{c}_{i}(\epsilon_{i},t)$ and $\Psi^{c}_{i,{\rm np}}(\epsilon_{i},t)$
satisfy the equation

$\big{(}1-Q-{\rm
e}^{\check{\epsilon}_{i}\partial_{t}}\big{)}\Psi^{c}_{i}(\epsilon_{i},t)=0.$

  3. $3.$

We have

$\Psi^{c}_{i,{\rm np}}=\mathcal{S}_{2}(t\mid\check{\epsilon}_{i},1)^{-1}.$

###### Proof.

The first part of the proposition follows from (4.16), the second part follows
by exponentiating the difference equation (4.1) and using the definitions. The
last point follows, from example, from the integral representation (4.5) of
$F_{{\rm np}}^{\rm ref}$, the definition of $\Psi_{i,{\rm np}}^{c}$, and the
integral representation (B.6) of $\mathcal{S}_{2}$. ∎

######  Remark 4.13.

We note that the equation satisfied by $\Psi^{c}_{i}$ can be interpreted as
the quantization of the curve

$1-\exp(2\pi\mathrm{i}t)-\exp(\mathrm{i}s/2\pi)=0$

in $(\mathbb{C}^{*})^{2}$, where the variables $t,s$ are promoted to operators
$\hat{t},\hat{s}$ acting on a Hilbert space and obeying the commutation
relations

$\left[\hat{t},\hat{s}\right]=\mathrm{i}\epsilon_{i},\qquad i=1,2.$

In particular, one can choose a polarization where $\hat{t}$ acts as
multiplication by $t$ and $\hat{s}$ acts as
$-\mathrm{i}\epsilon_{i}\partial_{t}$.

###  4.7 Quantum Picard–Fuchs operator

In [82] the NS limit of Nekrasov’s partition function was studied in the case
of Seiberg–Witten theory and it was argued that the NS limit corresponds to a
quantization of the classical periods of the Seiberg–Witten differential.
These classical periods were computed as explicit period integrals in the
original work [93] and were also obtained as solutions of Picard–Fuchs
equations (see [76] and references therein). A study of a quantum deformation
of the Picard–Fuchs equation governing the classical periods was initiated in
[82]. Similar ideas came up in [1], where the WKB analysis of quantum curves
was used to argue for differential Picard–Fuchs type operators that determine
the order by order corrections to the classical periods to obtain the quantum
periods. Although the arguments leading to these operators are clear it is
perhaps less clear what the interpretation of such an all-order operator
should be and how these should be derived systematically. We show in the
following that the difference equations which we have studied here provide a
clear path towards such all-order operator.

We want to derive a quantum Picard–Fuchs operator which annihilates in
particular the quantum period that we expect to be related to the derivative
of the topological string free energy in the NS limit. The asymptotic
expansion of the latter is

$F^{\rm NS}(\epsilon,t-\check{\epsilon}/2)=-\mathrm{i}\sum_{n=0}^{\infty}{\rm
Li}_{3-n}(Q)\frac{B_{n}}{n!}(\mathrm{i}\epsilon)^{n-1},$

as given in (2.6). This expansion was shown to satisfy the difference equation
(4.13)

$F^{\rm NS}(\epsilon,t+\check{\epsilon}/2)-F^{\rm
NS}(\epsilon,t-\check{\epsilon}/2)=-\mathrm{i}{\rm Li}_{2}(Q).$

It follows that $\partial_{t}F^{\rm NS}(\epsilon,t-\check{\epsilon}/2)$ has
the asymptotic expansion

$\partial_{t}F^{\rm
NS}(\epsilon,t-\check{\epsilon}/2)=2\pi\sum_{n=0}^{\infty}{\rm
Li}_{2-n}(Q)\frac{B_{n}}{n!}(\mathrm{i}\epsilon)^{n-1},$

and satisfies the difference equation

$\partial_{t}F^{\rm NS}(\epsilon,t+\check{\epsilon}/2)-\partial_{t}F^{\rm
NS}(\epsilon,t-\check{\epsilon}/2)=2\pi{\rm Li}_{1}(Q).$

Before deriving the quantum Picard–Fuchs operator from this difference
equation we would like to include the classical terms discussed in Section 2.5
in this quantum period, so that we in particular reproduce (2.14) as the
leading contribution of the quantum period in the limit $\epsilon\to 0$. The
leading contribution of $\partial_{t}F^{\rm
NS}(\epsilon,t-\check{\epsilon}/2)$ is given by

$\partial_{t}F^{\rm
NS}(\epsilon,t-\check{\epsilon}/2)=-\frac{2\pi\mathrm{i}}{\epsilon}{\rm
Li}_{2}(Q)+\mathcal{O}(1).$

We would like to modify $F^{\rm NS}$ such that the leading piece includes the
classical piece of (2.14) and hence is of the form

$-\frac{(2\pi\mathrm{i})^{3}}{\epsilon}\left(\frac{t^{2}}{2}+\frac{1}{(2\pi\mathrm{i})^{2}}{\rm
Li}_{2}(Q)\right).$

Instead of only adding the term $t^{2}/2$ with the correct prefactor, we take
advantage of the ambiguity of the linear and constant terms in $t$ in the
period expansion as well as the ambiguity in the constant term in $\epsilon$
in the quantum period, and propose to add the classical term in the guise of a
generalized Bernoulli polynomial which satisfies a difference equation on its
own. We thus use

$B_{2,2}(t\mid\check{\epsilon},1)=\frac{1}{\check{\epsilon}}\left(t^{2}-t+\frac{1}{6}\right)-t+\frac{1}{2}+\frac{\check{\epsilon}}{6},$

to obtain:

######  Proposition 4.14.

Define

$\partial_{t}F^{{\rm
NS},\sharp}(\epsilon,t-\check{\epsilon}/2):=2\pi^{2}\mathrm{i}\cdot
B_{2,2}(t\mid\check{\epsilon},1)+\partial_{t}F^{\rm
NS}(\epsilon,t-\check{\epsilon}/2).$

Then we find that:

  * •

$\partial_{t}F^{{\rm NS},\sharp}(\epsilon,t-\check{\epsilon}/2)$ satisfies the
difference equation

$\partial_{t}F^{{\rm
NS},\sharp}(\epsilon,t+\check{\epsilon}/2)-\partial_{t}F^{{\rm
NS},\sharp}(\epsilon,t-\check{\epsilon}/2)=(2\pi)\log\frac{Q}{Q-1}.$ (4.18)

  * •

$\partial_{t}F^{{\rm NS},\sharp}(\epsilon,t-\check{\epsilon}/2)$ is a solution
of the quantum Picard–Fuchs operator

$L_{\epsilon}:=\partial_{t}\circ(1-Q)\circ\partial_{t}\circ\sum_{n=1}^{\infty}\frac{\check{\epsilon}^{n}}{n!}\partial_{t}^{n}.$

###### Proof.

The difference equation follows from (4.14) as well as the difference equation
for the generalized Bernoulli polynomial of Proposition B.1:

$B_{2,2}(t+\check{\epsilon}\mid\check{\epsilon},1)-B_{2,2}(t\mid\check{\epsilon},1)=2B_{1,1}(t\mid
1)=2t-1.$

To show that $\partial_{t}F^{{\rm NS},\sharp}(\epsilon,t-\check{\epsilon}/2)$
is a solution of the quantum Picard–Fuchs operator, we write the difference
equation (4.18) as:

$\big{(}{\rm e}^{\check{\epsilon}\partial_{t}}-1\big{)}\partial_{t}F^{{\rm
NS},\sharp}(\epsilon,t-\check{\epsilon}/2)=(2\pi)\log\frac{Q}{Q-1},$

expand in $\check{\epsilon}$ and further act with
$\partial_{t}\circ(1-Q)\circ\partial_{t}$ on the result. ∎

######  Remark 4.15.

  * •

We remark that the leading contribution in $\epsilon$ to the quantum
Picard–Fuchs operator is the so-called extended classical Picard–Fuchs
operator $L_{\text{cl}}$, defined as

$\displaystyle\partial_{t}\circ(1-Q)\circ\partial_{t}\circ\sum_{n=1}^{\infty}\frac{\check{\epsilon}^{n}}{n!}\partial_{t}^{n}=\check{\epsilon}(\partial_{t}\circ(1-Q)\circ\partial_{t}\circ\partial_{t})+\mathcal{O}\big{(}\epsilon^{2}\big{)}=:\check{\epsilon}L_{\text{cl}}+\mathcal{O}\big{(}\epsilon^{2}\big{)}.$

This operator is one order less than the Picard–Fuchs operator given in (2.13)
and only has the three independent solutions $\varpi^{0}$, $\varpi^{1}$ and
$\varpi^{2}$ of Section 2.5.

  * •

The quantum periods $\varpi^{0}_{\epsilon}$ and $\varpi^{1}_{\epsilon}$ are
equal to

$\varpi^{0}_{\epsilon}=\frac{1}{\check{\epsilon}},\qquad\varpi^{1}_{\epsilon}=\frac{t}{\check{\epsilon}},$

up to rescaling. This will be further discussed later.

  * •

We have from Proposition 4.12 (ii) and (iv) that

$\exp\left(\frac{1}{2\pi}\partial_{t}F^{\rm NS}_{{\rm
np}}\left(\epsilon_{i},t-\frac{\epsilon_{i}}{4\pi}\right)\right)=\mathcal{S}_{2}(t\mid\check{\epsilon},1),$

which suggests the non-perturbative extension

$\partial_{t}F^{{\rm NS},\sharp}_{{\rm
np}}(\epsilon,t-\check{\epsilon}/2)=2\pi(\pi\mathrm{i}B_{2,2}(t\mid\check{\epsilon},1)+\log(\mathcal{S}_{2}(t\mid\check{\epsilon},1))).$
(4.19)

One can easily check that this is also a solution of the quantum Picard–Fuchs
operator $L_{\epsilon}$.

##  5 Borel sums and Stokes phenomena

In this section we study the Borel summation and associated Stokes phenomena
of the following two objects: on one hand we consider the following shift of
$F^{\rm NS}(\epsilon,t)$:

$W(\epsilon,t):=F^{\rm
NS}(\epsilon,t-\check{\epsilon}/2)=-\mathrm{i}\sum_{k=1}^{\infty}\frac{1}{k^{2}}\frac{{\rm
e}^{2\pi\mathrm{i}tk}}{({\rm
e}^{\mathrm{i}k\epsilon}-1)}\qquad\mathrm{with}\quad\check{\epsilon}=\frac{\epsilon}{2\pi}.$

One of the main reasons to study this shifted free energy is that we will be
able to relate its Borel sums to the Borel sums and Stokes phenomena of the
topological free energy $F^{{\rm top}}(\lambda,t)$ studied in [9], where
$\lambda$ denotes the topological string coupling. More specifically, we will
see that the Borel sums of the $\epsilon$-expansion of $W(\epsilon,t)$ give an
$\epsilon$-potential for the Borel sums of the $\lambda$-expansion of $F^{{\rm
top}}(\lambda,t)$, provided we set $\lambda=\epsilon$.

Furthermore, we also study the Borel sums and Stokes phenomena of the formal
solution $S(\epsilon,x,t)$ of the Schrödinger equation (3.8), given in (3.9);
and relate the Borel sums of $S(\epsilon,x,t)$ with those of $W(\epsilon,t)$.

######  Remark 5.1.

We remark that a similar study of the Borel summability of $S(\epsilon,x,t)$
is performed in the recent work [52, Section 4]. An analogous discussion of
the Borel transform of $F^{\rm NS}$ is written down in [52, Section 4.5.1]. We
became aware of this work while this paper was being finalized.

###  5.1 Main results

We will first state the main results, and then prove them in the following
subsections. The main results we wish to prove in this section regarding
$W(\epsilon,t)$ is contained in the following theorem:

######  Theorem 5.2.

Let $W(\epsilon,t)$ be as before. Then:

  * •

(Borel sum) Let $t\in\mathbb{C}^{\times}-\mathbb{Z}$ and consider the rays in
the $\epsilon$-plane defined by $l_{k}:=\mathbb{R}_{<0}\cdot
2\pi\mathrm{i}(t+k)$ for $k\in\mathbb{Z}$ and
$l_{\infty}:=\mathrm{i}\mathbb{R}_{<0}$ $($see Figure 3 on the left$)$. Then
given any ray $\rho\in\mathbb{C}^{\times}$ from $0$ to $\infty$ different from
$\\{\pm l_{k}\\}_{k\in\mathbb{Z}}\cup\\{\pm l_{\infty}\\}$; and
$\epsilon\in\mathbb{H}_{\rho}$, where $\mathbb{H}_{\rho}$ denotes the half-
plane centered at $\rho$, the formal $\epsilon$-expansion of $W(\epsilon,t)$
is Borel summable with Borel sum:

$W_{\rho}(\epsilon,t):=-\frac{1}{\epsilon}\mathrm{Li}_{3}(Q)+\frac{\mathrm{i}}{2}\mathrm{Li}_{2}(Q)+\frac{\epsilon}{12}\mathrm{Li_{1}}(Q)+\check{\epsilon}\int_{\rho}\mathrm{d}\xi\,{\rm
e}^{-\xi/\check{\epsilon}}\widetilde{G}(\xi,t),$

where

$\widetilde{G}(\xi,t)=\frac{1}{2\pi}\sum_{m\geq
1}\frac{1}{m^{3}}\left(\frac{1}{1-{\rm
e}^{-2\pi\mathrm{i}t+\xi/m}}-\frac{1}{1-{\rm
e}^{-2\pi\mathrm{i}t-\xi/m}}\right).$

In particular, one finds that if $\operatorname{Im}(t)>0$ and
$0<\operatorname{Re}(t)<1$, then for
$\operatorname{Re}(t)<\operatorname{Re}(\check{\epsilon}+1)$:

$W_{\mathbb{R}_{>0}}(\epsilon,t)=F_{{\rm np}}^{\rm
NS}(\epsilon,t-\check{\epsilon}/2),$ (5.1)

where $F_{{\rm np}}^{\rm NS}(\epsilon,t)$ was defined in (4.10).

  * •

(Stokes jumps) Assume that $\operatorname{Im}(t)>0$. 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 $\epsilon\in\mathbb{H}_{\rho}\cap\mathbb{H}_{\rho^{\prime}}$ $($resp.
$\epsilon\in\mathbb{H}_{-\rho}\cap\mathbb{H}_{-\rho^{\prime}})$ we have

$W_{\pm\rho}(\epsilon,t)-W_{\pm\rho^{\prime}}(\epsilon,t)=-\mathrm{i}\check{\epsilon}\mathrm{Li}_{2}\big{(}{\rm
e}^{\pm 2\pi\mathrm{i}(t+k)/\check{\epsilon}}\big{)}.$ (5.2)

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

  * •

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

$\lim_{k\to\infty}W_{\rho_{k}}(\epsilon,t)=W(\epsilon,t)$ (5.3)

On the other hand, assume that $0<\operatorname{Re}(t)<1$,
$\operatorname{Im}(t)>0$, $\operatorname{Re}(\epsilon)<0$,
$\operatorname{Im}(\epsilon)<0$,
$\operatorname{Re}t<\operatorname{Re}(-\check{\epsilon}+1)$ and that
$\big{|}{\rm e}^{-2\pi\mathrm{i}t/\check{\epsilon}}\big{|}<1$. Then

$\begin{split}\lim_{k\to-\infty}W_{-\rho_{k}}(\epsilon,t)=W(\epsilon,t).\end{split}$

In other words, the limits to $l_{\infty}$ from the right and left give
$W(\epsilon,t)$.

The other limits corresponding to $-l_{\infty}$ follow from the previous
limits and the relations

$W_{\rho}(\epsilon,t)+W_{-\rho}(-\epsilon,t)=\mathrm{i}\mathrm{Li}_{2}(Q).$
(5.4)

and

$W(\epsilon,t)+W(-\epsilon,t)=\mathrm{i}\mathrm{Li}_{2}(Q).$

In particular, the limits to $-l_{\infty}$ from the right and left give
$W(\epsilon,t)$.

  * •

(Potential for the Borel sum of the topological free energy) Given
$W_{\rho}(\epsilon,t)$, we have the identity

$\partial_{\epsilon}W_{\rho}(\epsilon,t)=F_{\rho}(\epsilon,t),$

where $F_{\rho}$ denotes the Borel sum of the non-constant map contribution to
the topological free energy studied in [9].

Figure 3: On the left: plot of the rays $l_{k}=\mathrm{i}\mathbb{R}_{<0}(t+k)$
in the Borel-plane with $t=1/\pi(1+2\mathrm{i})$ for $k\in(-250,250)$. On the
right: plot of the rays $\widetilde{l}_{k}=\mathrm{i}\mathbb{R}_{<0}(x+k)$ (in
blue) as well as $\widetilde{l}_{k,t}=\mathrm{i}\mathbb{R}_{<0}(t+x+k)$ (in
orange) with $t=1/\pi(1+2\mathrm{i})$ and $z=2+\mathrm{i}/3$ for
$k\in(-250,250)$.

######  Remark 5.3.

  * •

Notice that the Stokes jumps (5.2) have the same form as the right-hand side
of the difference equation (4.14) satisfied by $F_{{\rm np}}^{\rm
NS}(\epsilon,t-\check{\epsilon}/2)$. We will use this below to establish a
relation between $W_{\rho}$ and certain shifts of $F_{{\rm np}}^{\rm
NS}(\epsilon,t-\check{\epsilon}/2)$

  * •

In the previous results, we have restricted to the case
$\operatorname{Im}(t)>0$ when discussing $W_{\mathbb{R}_{>0}}$ and the limits
to $\pm l_{\infty}$. To do an analogous treatment for the limits to $\pm
l_{\infty}$ in the case $\operatorname{Im}(t)<0$, one would need to find an
analogous relation of the form (5.1) that holds for $\operatorname{Im}(t)<0$,
since the study of the limits to $\pm l_{\infty}$ makes use of (5.1).

From the Stokes jumps of $W_{\rho}(\epsilon,t)$, together with (5.1) and the
difference equation (4.14) satisfied by $F^{\rm NS}_{{\rm np}}(\epsilon,t)$,
we obtain the following:

######  Corollary 5.4.

Assume that $\operatorname{Im}(t)>0$ and $0<\operatorname{Re}(t)<1$, and let
$\rho_{k}$ be a ray between $l_{k}$ and $l_{k-1}$. Then

$W_{\rho_{k}}(\epsilon,t)=F_{{\rm np}}^{\rm
NS}(\epsilon,t-\check{\epsilon}/2+k),$

and

$W_{-\rho_{k}}(\epsilon,t)=\mathrm{i}\mathrm{Li}_{2}(Q)-F_{{\rm np}}^{\rm
NS}(-\epsilon,t-\check{\epsilon}/2+k)$

###### Proof.

We have from $(\ref{WBSR})$ and our assumption on $t$ that

$W_{\rho_{0}}(\epsilon,t)=W_{\mathbb{R}_{>0}}(\epsilon,t)=F_{{\rm np}}^{\rm
NS}(\epsilon,t-\check{\epsilon}/2).$

On the other hand, using the Stokes jumps of $W_{\rho}$ and the difference
equation (4.14), we find

$W_{\rho_{1}}(\epsilon,t)-W_{\rho_{0}}(\epsilon,t)=-\mathrm{i}\check{\epsilon}\mathrm{Li}_{2}\big{(}{\rm
e}^{2\pi{\rm i}t/\check{\epsilon}}\big{)}=F_{{\rm np}}^{\rm
NS}(\epsilon,t-\check{\epsilon}/2+1)-F_{{\rm np}}^{\rm
NS}(\epsilon,t-\check{\epsilon}/2),$

so that $W_{\rho_{1}}(\epsilon,t)=F_{{\rm np}}^{\rm
NS}(\epsilon,t-\check{\epsilon}/2+1)$. The result then follows from general
$k\in\mathbb{Z}$ by induction. Finally, the identity for $W_{-\rho_{k}}$
follows from (5.4). ∎

On the other hand, by applying the same techniques used to compute the Borel
sum and Stokes jumps of $W(\epsilon,t)$, we can compute the Borel sum and
Stokes jumps of the formal series $S(\epsilon,x,t)$ giving a formal solution
to (3.8). More precisely, we have the following theorem

######  Theorem 5.5.

Let $S(\epsilon,x,t)$ be the formal series defined by

$S(\epsilon,x,t)=-\sum_{n=0}^{\infty}\frac{B_{n}}{n!}(\mathrm{i}\epsilon)^{n-1}({\rm
Li}_{2-n}(QX)-{\rm Li}_{2-n}(X)),\qquad X=\exp(2\pi\mathrm{i}x),$

giving a formal solution to

$\mathbf{D}_{\Sigma}(\epsilon,x,t)\Psi(\epsilon,x,t)=\big{(}\big{(}1-{\rm
e}^{2\pi\mathrm{i}x}\big{)}{\rm
e}^{\check{\epsilon}\partial_{x}}-\big{(}1-Q{\rm
e}^{2\pi\mathrm{i}x}\big{)}\big{)}\Psi(\epsilon,x,t)=0$

via

$\Psi(\epsilon,x,t)=\exp(S(\epsilon,x,t)).$

Then:

  * •

(Borel sum) Let $t,x\in\mathbb{C}^{\times}$ such that $x,t+x\notin\mathbb{Z}$
and consider the rays in the $\epsilon$-plane defined by
$\widetilde{l}_{k,t}:=\mathbb{R}_{<0}\cdot 2\pi\mathrm{i}(t+x+k)$ and
$\widetilde{l}_{k}:=\mathbb{R}_{<0}\cdot 2\pi\mathrm{i}(x+k)$ for
$k\in\mathbb{Z}$ and $l_{\infty}:=\mathrm{i}\mathbb{R}_{<0}$ $($see Figure 3
on the right$)$. Then given any ray $\rho\in\mathbb{C}^{\times}$ from $0$ to
$\infty$ different from
$\big{\\{}\pm\widetilde{l}_{k,t}\big{\\}}_{k\in\mathbb{Z}}\cup\big{\\{}\pm\widetilde{l}_{k}\big{\\}}_{k\in\mathbb{Z}}\cup\\{\pm
l_{\infty}\\}$; and $\epsilon\in\mathbb{H}_{\rho}$, where $\mathbb{H}_{\rho}$
denotes the half-plane centered at $\rho$, the formal series $S(\epsilon,x,t)$
is Borel summable with Borel sum:

$\displaystyle
S_{\rho}(\epsilon,x,t):=-\frac{1}{\mathrm{i}\epsilon}(\mathrm{Li}_{2}(QX)-\mathrm{Li}_{2}(X))+\frac{1}{2}(\mathrm{Li}_{1}(QX)-\mathrm{Li}_{1}(X))$
$\displaystyle\hphantom{S_{\rho}(\epsilon,x,t):=}{}+\int_{\rho}\mathrm{d}\xi\,{\rm
e}^{-\xi/\check{\epsilon}}G_{S}(\xi,x,t),$

where

$\displaystyle G_{S}(\xi,x,t)=\frac{1}{2\pi\mathrm{i}}\sum_{m\geq
1}\frac{1}{m^{2}}\bigg{(}\frac{1}{{\rm
e}^{-2\pi\mathrm{i}(t+x)-\xi/m}-1}+\frac{1}{{\rm
e}^{-2\pi\mathrm{i}(t+x)+\xi/m}-1}$
$\displaystyle\hphantom{G_{S}(\xi,x,t)=}{}-\frac{1}{{\rm
e}^{-2\pi\mathrm{i}x-\xi/m}-1}-\frac{1}{{\rm
e}^{-2\pi\mathrm{i}x+\xi/m}-1}\bigg{)}.$

We denote $\Psi_{\rho}(\epsilon,x,t):=\exp(S_{\rho}(\epsilon,x,t))$.

Furthermore, we have that for
$\operatorname{Min}\\{\operatorname{Im}(x),\operatorname{Im}(x+t)\\}>0$,
$\operatorname{Min}\\{\operatorname{Re}(x),\operatorname{Re}(x+t)\\}>0$,
$\operatorname{Max}\\{\operatorname{Re}(x),\operatorname{Re}(x+t)\\}<1$; the
following holds for
$\operatorname{Max}\\{\operatorname{Re}(x),\operatorname{Re}(x+t)\\}<\operatorname{Re}(\check{\epsilon}+1)$:

$\Psi_{\mathbb{R}_{>0}}=\exp(S_{\mathbb{R}_{>0}}(\epsilon,x,t))=\frac{\mathcal{S}_{2}(x\mid\check{\epsilon},1)}{\mathcal{S}_{2}(t+x\mid\check{\epsilon},1)}=\Psi_{{\rm
np}}(\epsilon,x,t).$ (5.5)

  * •

(Stokes jumps) Assume that $t$ and $x$ are such that $\widetilde{l}_{k,t}$
does not overlap $\pm\widetilde{l}_{r}$ for any $r\in\mathbb{Z}$. Furthermore
assume that $\rho$ and $\rho^{\prime}$ are two rays from $0$ to $\infty$ such
that traversing $\rho^{\prime}$ with the opposite orientation and then $\rho$
gives a positively oriented Hankel contour. Finally, assume the sector
determined by $\rho$ and $\rho^{\prime}$ contains only $\widetilde{l}_{k,t}$.
Then

$S_{\pm\rho}(\epsilon,x,t)-S_{\pm\rho^{\prime}}(\epsilon,x,t)=-\mathrm{Li}_{1}\big{(}{\rm
e}^{\pm 2\pi\mathrm{i}(k+t+x)/\check{\epsilon}}\big{)}.$

Similarly, if $\widetilde{l}_{k}$ does not overlap with
$\pm\widetilde{l}_{r,t}$ for any $r\in\mathbb{Z}$ and $\rho$ and
$\rho^{\prime}$ are given as above but now the sector they define contains
only $\widetilde{l}_{k}$, then

$S_{\pm\rho}(\epsilon,x,t)-S_{\pm\rho^{\prime}}(\epsilon,x,t)=\mathrm{Li}_{1}\big{(}{\rm
e}^{\pm 2\pi\mathrm{i}(k+x)/\check{\epsilon}}\big{)}.$

In the case one rays overlaps another, the jump on the overlap is given by the
sum of the jumps of the corresponding rays.

  * •

(Limits to $\pm l_{\infty}$) Assume that
$\text{min}\\{\operatorname{Im}(x),\operatorname{Im}(x+t)\\}>0$,
$\text{min}\\{\operatorname{Re}(x),\operatorname{Re}(x+t)\\}>0$, and
$\max\\{\operatorname{Re}(x),\operatorname{Re}(x+t)\\}<1$. Furthermore, let
$\rho_{k}$ for $k\in\mathbb{Z}$ be a sequence of rays, one for each of the
sectors defined by $\widetilde{l}_{n}$ and $\widetilde{l}_{m,t}$, and ordered
in a clockwise manner. Then for $\operatorname{Re}(\epsilon)>0$,
$\operatorname{Im}(\epsilon)<0$ and
$\operatorname{Max}\\{\operatorname{Re}(x),\operatorname{Re}(x+t)\\}<\operatorname{Re}(\check{\epsilon}+1)$
we have

$\lim_{k\to\infty}\Psi_{\rho_{k}}(\epsilon,x,t)=\frac{L(x,\epsilon)}{L(x+t,\epsilon)}=:\Psi_{{\rm
GV}}(x,\epsilon),$ (5.6)

where $L(x,\epsilon)$ is given in (3.12). On the other hand, if
$\operatorname{Re}(\epsilon)<0$ and $\operatorname{Im}(\epsilon)<0$, while
$\max\\{\operatorname{Re}(x),\operatorname{Re}(x+t)\\}<\operatorname{Re}(-\check{\epsilon}+1)$
and $\max\big{\\{}\big{|}{\rm
e}^{-2\pi\mathrm{i}x/\check{\epsilon}}\big{|},\big{|}{\rm
e}^{-2\pi\mathrm{i}(x+t)/\check{\epsilon}}\big{|}\big{\\}}<1$ then

$\lim_{k\to-\infty}\Psi_{-\rho_{k}}(\epsilon,x,t)=\Psi_{{\rm
GV}}(x,\epsilon).$

The limit to $-l_{\infty}$ follows from the identities

$\displaystyle
S_{\rho}(\epsilon,x,t)+S_{-\rho}(-\epsilon,x,t)=\mathrm{Li}_{1}(QX)-\mathrm{Li}_{1}(X),$
$\displaystyle
S(\epsilon,x,t)+S(-\epsilon,x,t)=\mathrm{Li}_{1}(QX)-\mathrm{Li}_{1}(X).$

In particular, this limit is also given by $\Psi_{{\rm GV}}$.

Similarly to $W_{\rho}(\epsilon,t)$, we can use (5.5) together with the
difference equation (B.4) satisfied by $\mathcal{S}_{2}$ to conclude the
following:

######  Corollary 5.6.

Assume that
$\operatorname{Min}\\{\operatorname{Im}(x),\operatorname{Im}(x+t)\\}\\!>0$,
$\operatorname{Min}\\{\operatorname{Re}(x),\operatorname{Re}(x+t)\\}\\!>0$,
$\operatorname{Max}\\{\operatorname{Re}(x),\allowbreak\operatorname{Re}(x+t)\\}<1$.
Then if $\tilde{\rho}_{m,n}$ is a ray between the rays $\widetilde{l}_{m,t}$
and $\widetilde{l}_{m-1,t}$, and the rays $\widetilde{l}_{n}$ and
$\widetilde{l}_{n-1}$, then

$\Psi_{\tilde{\rho}_{m,n}}(\epsilon,x,t)=\frac{\mathcal{S}_{2}(x+n|\check{\epsilon},1)}{\mathcal{S}_{2}(t+x+m|\check{\epsilon},1)}=\Psi_{{\rm
np}}(\epsilon,x+n,t+m-n)$

and

$\Psi_{-{\tilde{\rho}_{m,n}}}(\epsilon,x,t)=\frac{1-X}{1-QX}\left(\Psi_{{\rm
np}}(-\epsilon,x+n,t+m-n)\right)^{-1}.$

###### Proof.

The proof for the first equality follows the same reasoning as Corollary 5.4,
but now using (5.5) together with the difference equation satisfied by
$\mathcal{S}_{2}$, given in (B.4), and the Stokes jumps of $S_{\rho}$. For
example, under our hypotheses, we have

$\Psi_{\tilde{\rho}_{0,0}}(\epsilon,x,t)=\Psi_{\mathbb{R}_{>0}}(\epsilon,x,t)=\Psi_{{\rm
np}}(\epsilon,x,t).$

If the next ray to $\rho_{0,0}$ in clockwise order is $\widetilde{l}_{0,t}$
then

$\displaystyle\frac{\Psi_{\tilde{\rho}_{1,0}}(\epsilon,x,t)}{\Psi_{\tilde{\rho}_{0,0}}(\epsilon,x,t)}$
$\displaystyle=\exp\bigl{(}-\mathrm{Li}_{1}\big{(}{\rm
e}^{2\pi\mathrm{i}(t+x)/\check{\epsilon}}\big{)}\bigr{)}=1-{\rm
e}^{2\pi\mathrm{i}(t+x)/\check{\epsilon}}$
$\displaystyle=\frac{\mathcal{S}_{2}(x+t|\check{\epsilon},1)}{\mathcal{S}_{2}(x+t+1|\check{\epsilon},1)}=\frac{\Psi_{{\rm
np}}(\epsilon,x,t+1)}{\Psi_{{\rm np}}(\epsilon,x,t)},$

so that

$\Psi_{\tilde{\rho}_{1,0}}(\epsilon,x,t)=\Psi_{{\rm np}}(\epsilon,x,t+1).$

If, on the other hand, the next ray to $\tilde{\rho}_{0,0}$ in clockwise order
is $\widetilde{l}_{0}$, then

$\displaystyle\frac{\Psi_{\tilde{\rho}_{0,1}}(\epsilon,x,t)}{\Psi_{\tilde{\rho}_{0,0}}(\epsilon,x,t)}=\exp\big{(}\mathrm{Li}_{1}\big{(}{\rm
e}^{2\pi\mathrm{i}x/\check{\epsilon}}\big{)}\big{)}=\frac{1}{1-{\rm
e}^{2\pi\mathrm{i}x/\check{\epsilon}}}=\frac{\mathcal{S}_{2}(x+1|\check{\epsilon},1)}{\mathcal{S}_{2}(x|\check{\epsilon},1)}=\frac{\Psi_{{\rm
np}}(\epsilon,x+1,t-1)}{\Psi_{{\rm np}}(\epsilon,x,t)},\textbf{}$

so that

$\Psi_{\tilde{\rho}_{0,1}}(\epsilon,x,t)=\Psi_{{\rm np}}(\epsilon,x+1,t-1).$

The general identity then follows.

Last, the identity for $-\tilde{\rho}_{m,n}$ follows from the first identity
together with

$S_{\rho}(\epsilon,x,t)+S_{-\rho}(-\epsilon,x,t)=\mathrm{Li}_{1}(QX)-\mathrm{Li}_{1}(X).$
∎

Finally, we have the following theorem, relating $W_{\rho}$ with
$\Psi_{\rho}=\exp(S_{\rho})$. To state the result, we consider the paths
$X(s)={\rm e}^{2\pi\mathrm{i}x(s)}$ where $x(s)=\mathrm{i}s{\rm
e}^{\mathrm{i}\theta}$ for $s\in\mathbb{R}$ and $\theta$ fixed. These paths
connect the points $X=0$ and $X=1$ and will be important in Section 7.

######  Theorem 5.7.

Assume that $\operatorname{Im}(t)>0$, while $0<\operatorname{Re}(t)<1$, and
let $\rho_{k}$ be a ray between $l_{k}$ and $l_{k-1}$. Furthermore, pick
$x=\mathrm{i}s_{*}{\rm e}^{\mathrm{i}\theta_{*}}$ such that
$\operatorname{Im}(x)>0$, while $0<\operatorname{Re}(x)<1$, and $\rho_{k}$ is
of the form $\rho_{k,0}$ $($recall the notation of Corollary 5.6). Finally,
consider the trajectory $x(s)=\mathrm{i}s{\rm e}^{\mathrm{i}\theta_{*}}$ from
$s=0$ to $s=s_{*}$. Then, provided that $t+k\neq a\check{\epsilon}+b$ for
$a,b\in\mathbb{Z}_{\leq 0}$ or $\mathbb{Z}_{>0}$, we can analytically continue
$\mathcal{S}_{2}(x(s)\mid\check{\epsilon},1)^{-1}\cdot\Psi_{\rho_{k}}(\epsilon,x(s),t)$
along the trajectory $x(s)$ from $x(s_{*})=\mathrm{i}s_{*}{\rm
e}^{\mathrm{i}\theta_{*}}$ to $x(0)=0$, and

$\exp\left(-\frac{1}{2\pi}\partial_{t}W_{\rho_{k}}(\epsilon,t)\right)=\big{(}\mathcal{S}_{2}(x\mid\check{\epsilon},1)^{-1}\cdot\Psi_{\rho_{k}}(\epsilon,x,t)\big{)}\big{|}_{x=0}.$

In the following subsections, together with Appendix A, we give the proofs of
Theorems 5.2, 5.5 and 5.7.

###  5.2 Borel transform

Before studying the Borel summability of $W(\epsilon,t)$, we must start by
studying the Borel transform of the $\epsilon$-expansion of $W(\epsilon,t)$.
Since $W(\epsilon,t)=F^{\rm NS}(\epsilon,t-\check{\epsilon}/2)$, we have by
(2.6) that we can write the formal expansion

$W(\epsilon,t)=-\frac{1}{2\pi}\sum_{n=0}^{\infty}\partial_{t}^{n}{\rm
Li}_{3}(Q)\frac{B_{n}}{n!}\check{\epsilon}^{n-1}=-\frac{1}{\epsilon}{\rm
Li}_{3}(Q)+\frac{\mathrm{i}}{2}{\rm Li}_{2}(Q)+\frac{\epsilon}{12}{\rm
Li}_{1}(Q)+\Phi(\check{\epsilon},t),$

where (using that $B_{2n+1}=0$ for $n>0$):

$\Phi(\check{\epsilon},t):=-\frac{1}{2\pi}\sum_{n=3}^{\infty}\partial_{t}^{n}{\rm
Li}_{3}(Q)\frac{B_{n}}{n!}\check{\epsilon}^{n-1}=-\frac{1}{2\pi}\sum_{n=2}^{\infty}\partial_{t}^{2n}{\rm
Li}_{3}(Q)\frac{B_{2n}}{(2n)!}\check{\epsilon}^{2n-1}.$

We now wish to compute the Borel transform of $\Phi(\check{\epsilon},t)$ 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

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

Namely, we wish to study

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

The main result about $G(\xi,t)$ that we will prove and use is the following:

######  Proposition 5.8.

Take $t\in\mathbb{C}^{\times}$ and $\epsilon\in\mathbb{C}^{\times}$ such that
$0<|\operatorname{Re}(t)|<1/2$ and $|\epsilon/t|\in(0,2\pi)$, then $G(\xi,t)$
converges. Furthermore, we have the alternate representation

$G(\xi,t)=\frac{1}{4\pi}\sum_{m\in\mathbb{Z}-\\{0\\}}\frac{1}{m^{2}}\big{(}\mathrm{Li}_{1}\big{(}{\rm
e}^{2\pi\mathrm{i}t+\xi/m}\big{)}+\mathrm{Li}_{1}\big{(}{\rm
e}^{2\pi\mathrm{i}t-\xi/m}\big{)}-2\mathrm{Li}_{1}(Q)\big{)},$ (5.7)

where $\mathrm{Li}_{1}(z)=-\log(1-z)$ uses the principal branch. In particular
$G(\xi,t)$ admits a continuation for all $\xi>0$.

###### Proof.

The proof of this proposition is given in Appendix A.1. The techniques and
ideas are the same as the ones used in [47], which were also used in [9] to
study the Borel transform of the topological free energy of the resolved
conifold. ∎

On the other hand, the study of the Borel transform of $S(\epsilon,x,t)$
follows from the same arguments in Appendix A.1 used to study $G(\xi,t)$. One
first writes

$S(\epsilon,x,t)=-\frac{1}{\mathrm{i}\epsilon}(\mathrm{Li}_{2}(QX)-\mathrm{Li}_{2}(X))+\frac{1}{2}(\mathrm{Li}_{1}(QX)-\mathrm{Li}_{1}(X))+\Phi_{S}(\check{\epsilon},x,t),$

where

$\displaystyle\Phi_{S}(\check{\epsilon},x,t):={}$
$\displaystyle{-}\sum_{n=1}^{\infty}\frac{B_{2n}}{(2n)!}(\mathrm{i}\epsilon)^{2n-1}(\mathrm{Li}_{2-2n}(QX)-\mathrm{Li}_{2-2n}(X))$
$\displaystyle={}$
$\displaystyle{-}\frac{1}{2\pi\mathrm{i}}\sum_{n=1}^{\infty}\frac{B_{2n}}{(2n)!}\check{\epsilon}^{2n-1}\partial_{x}^{2n}(\mathrm{Li}_{2}(QX)-\mathrm{Li}_{2}(X)).$

We wish to study $G_{S}(\xi,x,t):=\mathcal{B}(\Phi_{S}(-,x,t))(\xi)$ given by

$G_{S}(\xi,x,t)=-\frac{1}{2\pi\mathrm{i}}\sum_{n=1}^{\infty}\frac{B_{2n}}{(2n)!}\frac{\xi^{2n-2}}{(2n-2)!}\partial_{x}^{2n}(\mathrm{Li}_{2}(QX)-\mathrm{Li}_{2}(X)).$
(5.8)

By using the arguments of Appendix A.1 with
$G_{S}(\xi,x,t)=(f_{1}\stackMath\mathbin{\stackinset{c}{0ex}{c}{0ex}{\ast}{\bigcirc}}f_{2}(-,x,t))(\xi)$,
and

$\displaystyle
f_{1}(\xi):=-\frac{1}{2\pi\mathrm{i}}\sum_{n=1}^{\infty}\frac{B_{2n}}{(2n)!}\xi^{2n-2},$
$\displaystyle
f_{2}(\xi,x,t):=\sum_{n=1}^{\infty}\frac{\xi^{2n-2}}{(2n-2)!}\partial_{x}^{2n}(\mathrm{Li}_{2}(QX)-\mathrm{Li}_{2}(X)),$

we find the following:

######  Proposition 5.9.

Assume $t,x\in\mathbb{C}^{\times}$ with $t\neq-x$ and
$\operatorname{Max}\\{|\operatorname{Re}(x)|,|\operatorname{Re}(t+x)|\\}<1/2$.
Then for $|\xi|<\operatorname{Min}\\{2\pi|t+x|,2\pi|x|\\}$ we have that
$G(\xi,x,t)$ in (5.8) converges. Furthermore, using the integral
representation (A.1) for
$G_{S}(\xi,x,t)=(f_{1}\stackMath\mathbin{\stackinset{c}{0ex}{c}{0ex}{\ast}{\bigcirc}}f_{2}(-,x,t))(\xi)$
we find the alternate expression

$\displaystyle G_{S}(\xi,x,t)=\frac{1}{2\pi\mathrm{i}}\sum_{m\geq
1}\frac{1}{m^{2}}\big{(}\mathrm{Li}_{0}\big{(}Q{\rm
e}^{2\pi\mathrm{i}x+\xi/m}\big{)}+\mathrm{Li}_{0}\big{(}Q{\rm
e}^{2\pi\mathrm{i}x-\xi/m}\big{)}$
$\displaystyle\hphantom{G_{S}(\xi,x,t)=\frac{1}{2\pi\mathrm{i}}\sum_{m\geq
1}}{}-\mathrm{Li}_{0}\big{(}{\rm
e}^{2\pi\mathrm{i}x+\xi/m}\big{)}-\mathrm{Li}_{0}\big{(}{\rm
e}^{2\pi\mathrm{i}x-\xi/m}\big{)}\big{)}$
$\displaystyle\hphantom{G_{S}(\xi,x,t)}{}=\frac{1}{2\pi\mathrm{i}}\sum_{m\geq
1}\frac{1}{m^{2}}\bigg{(}\frac{1}{{\rm
e}^{-2\pi\mathrm{i}(t+x)-\xi/m}-1}+\frac{1}{{\rm
e}^{-2\pi\mathrm{i}(t+x)+\xi/m}-1}$
$\displaystyle\hphantom{G_{S}(\xi,x,t)=\frac{1}{2\pi\mathrm{i}}\sum_{m\geq
1}}{}-\frac{1}{{\rm e}^{-2\pi\mathrm{i}x-\xi/m}-1}-\frac{1}{{\rm
e}^{-2\pi\mathrm{i}x+\xi/m}-1}\bigg{)},$

so that $G_{S}(\xi,x,t)$ admits an analytic continuation in $\xi$ to a
meromorphic function with poles at $2\pi\mathrm{i}(t+x+k)m$ and
$2\pi\mathrm{i}(x+k)m$ for $k\in\mathbb{Z}$ and $m\in\mathbb{Z}-\\{0\\}$.

###  5.3 Borel sum

We now study the Borel sum of the $\epsilon$-expansion of $W(\epsilon,t)$
along $\mathbb{R}_{>0}$. As before, we assume $t\in\mathbb{C}^{\times}$ such
that $0<|\operatorname{Re}(t)|<1/2$, so that $G(\xi,t)$ admits a continuation
for $\xi>0$ with the expression (5.7), and we can consider the Borel sum along
$\mathbb{R}_{>0}$.

In the following, we assume that $\epsilon\in\mathbb{H}_{\mathbb{R}_{>0}}$,
where $\mathbb{H}_{\mathbb{R}_{>0}}$ denotes the half-plane centered at
$\mathbb{R}_{>0}$. We then integrate by parts and use the fact that the
boundary terms vanish to write

$\int_{\mathbb{R}_{>0}}{\rm d}\xi\,{\rm
e}^{-\xi/\check{\epsilon}}G(\xi,t)=\frac{\check{\epsilon}}{4\pi}\int_{\mathbb{R}_{>0}}\mathrm{d}\xi{\rm
e}^{-\xi/\check{\epsilon}}\sum_{m\in\mathbb{Z}-\\{0\\}}\frac{1}{m^{3}}\left(\frac{1}{1-{\rm
e}^{-2\pi\mathrm{i}t+\xi/m}}-\frac{1}{1-{\rm
e}^{-2\pi\mathrm{i}t-\xi/m}}\right).$

The resulting expression in the integrand has poles at the points
$\xi=2\pi{\rm i}(t+k)m$. We define

$\displaystyle\widetilde{G}(\xi,t):={}$
$\displaystyle\frac{1}{4\pi}\sum_{m\in\mathbb{Z}-\\{0\\}}\frac{1}{m^{3}}\left(\frac{1}{1-{\rm
e}^{-2\pi\mathrm{i}t+\xi/m}}-\frac{1}{1-{\rm
e}^{-2\pi\mathrm{i}t-\xi/m}}\right)$ $\displaystyle={}$
$\displaystyle\frac{1}{2\pi}\sum_{m\geq 1}\frac{1}{m^{3}}\left(\frac{1}{1-{\rm
e}^{-2\pi\mathrm{i}t+\xi/m}}-\frac{1}{1-{\rm
e}^{-2\pi\mathrm{i}t-\xi/m}}\right).$

One advantage of the expression of the Borel sum with $\widetilde{G}(\xi,t)$
is that we can now integrate freely along rays avoiding the poles, provided
$\epsilon$ is in the appropriate range. More precisely, we define:

######  Definition 5.10.

For $t\in\mathbb{C}^{\times}-\mathbb{Z}$, for $\rho$ a ray from $0$ to
$\infty$ avoiding the rays $\pm l_{k}=\pm\mathbb{R}_{<0}\cdot
2\pi\mathrm{i}(t+k)$ for $k\in\mathbb{Z}$ and $\pm
l_{\infty}=\pm\mathrm{i}\mathbb{R}_{<0}$, and for
$\check{\epsilon}\in\mathbb{H}_{\rho}$ where $\mathbb{H}_{\rho}$ denotes the
half-plane centered at $\rho$, we denote the Borel sum of $W(\epsilon,t)$
along $\rho$ by

$W_{\rho}(\epsilon,t):=-\frac{1}{\epsilon}\mathrm{Li}_{3}(Q)+\frac{\mathrm{i}}{2}\mathrm{Li}_{2}(Q)+\frac{\epsilon}{12}\mathrm{Li_{1}}(Q)+\check{\epsilon}\int_{\rho}\mathrm{d}\xi\,{\rm
e}^{-\xi/\check{\epsilon}}\widetilde{G}(\xi,t).$

The arguments from Section 4.3 then motivate the consideration of the
following function:

######  Definition 5.11.

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

$W_{\rm np}(\epsilon,t):=F_{{\rm np}}^{\rm
ref}(\epsilon,t-\check{\epsilon}/2)=2\pi\int_{\mathbb{R}+\mathrm{i}0^{+}}\frac{{\rm
d}s}{s^{2}}\frac{{\rm e}^{st}}{({\rm e}^{s}-1)({\rm
e}^{\check{\epsilon}s}-1)},$

This function generates the quantum dilogarithm
$\mathcal{S}_{2}(z\mid\omega_{1},\omega_{2})$ (see Appendix B or [21]), in the
sense that

$\partial_{t}W_{{\rm
np}}(\epsilon,t)=2\pi\log\mathcal{S}_{2}(t|\check{\epsilon},1).$

Indeed, we have

$\displaystyle\partial_{t}W_{{\rm np}}(\epsilon,t)$
$\displaystyle=2\pi\partial_{t}\int_{\mathbb{R}+\mathrm{i}0^{+}}\frac{{\rm
d}s}{s^{2}}\frac{{\rm e}^{st}}{({\rm e}^{s}-1)({\rm
e}^{\check{\epsilon}s}-1)}$
$\displaystyle=2\pi\int_{\mathbb{R}+\mathrm{i}0^{+}}\frac{{\rm
d}s}{s}\frac{{\rm e}^{st}}{({\rm e}^{s}-1)({\rm