# Non-perturbative topological strings
from resurgence

Murad<EMAIL_ADDRESS>
Maxwell Institute for Mathematical Sciences, Edinburgh EH14 4AS, UK
Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AP, UK
Department of Mathematics, University of Hamburg, Bundesstr. 55, 20146,
Hamburg, Germany

###### Abstract

The partition function of topological string theory on any family of Calabi-
Yau threefolds is defined perturbatively as an asymptotic series in the
topological string coupling and encodes, in a holomorphic limit, higher genus
Gromov-Witten as well as Gopakumar-Vafa invariants. We prove that the
partition function of topological strings of any CY in this limit can be
written as a product, where each factor is given by the partition function of
the resolved conifold with shifted arguments, raised to the power of certain
sheaf invariants. We use this result to put forward an expression for the non-
perturbative topological string partition function in this limit, as a product
over analytic functions in the topological string coupling which correspond to
the Borel sums for the resolved conifold found previously. The non-
perturbative corrections to the partition function are identified with Stokes
jumps of a Borel summation. They depend only on genus zero GV invariants and
can be expressed entirely in terms of a single function which is introduced as
a deformation of the prepotential.

###### Contents

  1. 1 Introduction and summary
    1. 1.1 Introduction
    2. 1.2 Summary
  2. 2 Topological strings and enumerative geometry
    1. 2.1 The Gromov-Witten potential
    2. 2.2 The Gopakumar-Vafa resummation
    3. 2.3 Finiteness of GV invariants
    4. 2.4 GW potential of the resolved conifold
  3. 3 Building blocks of non-perturbative topological strings
    1. 3.1 The resolved conifold
    2. 3.2 Constant maps contributions
  4. 4 Non-perturbative topological strings
    1. 4.1 Sheaf invariants
    2. 4.2 Free energy in terms of the resolved conifold
    3. 4.3 Non-perturbative free energy
    4. 4.4 Deformation of the prepotential
    5. 4.5 Non-perturbative topological string partition function
  5. 5 Resurgence
    1. 5.1 Asymptotic series
    2. 5.2 Stokes factors
    3. 5.3 Singularities in the Borel plane
  6. 6 Discussion and Conclusions
  7. A Multiple sine functions

##  1 Introduction and summary

###  1.1 Introduction

Topological string theory is defined perturbatively and its partition function
is given by an asymptotic series in the topological string coupling. The
perturbative definition as well as the asymptotic nature of the expansion of
its partition function are common traits of quantum field and string theories,
see e. g. [1] and references therein. The clear mathematical formulation of
topological strings as well as its connections to enumerative geometry make
however both perturbative computations as well as the study of its non-
perturbative structure highly accessible.

Perturbative computations within topological strings are accessed by the
holomorphic anomaly equations of [2] and their recursive solution in terms of
Feynman diagrams. The feasibility of higher genus computation was
significantly enhanced due to the discovery of a polynomial structure of the
free energies [3, 4] 222See also [5] and references therein  which can be used
together with expected boundary conditions to access higher genus computations
[6], see also [7] for a recent study. The anomaly equations were also used to
access the non-perturbative structure by making a suitable ansatz using
transseries in [8, 9], see also the recent [10]. The non-perturbative
structure of topological strings has been addressed in many works, using
connections and dualities of topological string theory with other areas of
physics and mathematics such as the study of black holes partition functions
[11], large order behavior [12], matrix models [13, 14], superconformal
theories [15], quantum mechanical spectral theory [16] and Chern-Simons
theories [17] to name a few, see [18, 1] and references therein.

The focus of this work is on the non-perturbative structure of topological
string theory as addressed via resurgence, see [19] for an introduction to the
subject as well as [1] for applications in physics. Borel summation and
resurgence of topological strings for geometries which can be related to
matrix models including the resolved conifold was studied in [14], the Borel
resummation for the resolved conifold was further studied in [20]. The study
of Borel resummation for the resolved conifold was revisited in [21] building
on methods and results of [22, 23, 24, 25, 26] and obtaining exact analytical
results for the Borel transform and Stokes jumps and identifying the Stokes
jump coefficients with BPS (or Dondaldson-Thomas (DT)) invariants using the
link to the Riemann-Hilbert problem studied in [27, 28]. The resurgence of
open and closed as well as refined topological strings on this geometry was
further studied in [29, 30]. Further recent studies addressing resurgence of
open and closed topological strings include [31, 32, 33, 34, 35, 36]

In this work we start with the general form of the topological string
partition function encoding the Gromov-Witten and Gopakumar-Vafa invariants.
By using sheaf invariants put forward by Maulik and Toda [37] to define
mathematically the GV invariants, we bring the partition function into a form
which can be recognised as an infinite product of a fundamental building block
for every sheaf invariant which is evaluated at a shifted argument and raised
to the power of the corresponding sheaf invariant. The building block is the
topological string partition function of the resolved conifold. We then use
the analytic functions in the topological string coupling which were obtained
as Borel sums in the resurgence analysis of [21] of the resolved conifold to
put forward a non-perturbative expression for the topological string partition
function in the holomorphic limit in the general case. Similar to the case of
the resolved conifold, the latter also has a distinguished expression where
all the non-perturbative corrections are contained in a single function which
plays the analogous role as the topological string partition function in the
Nekrasov-Shatashvili limit [38] in the case of the resolved conifold. This
function is identified with an $\varepsilon-$deformation of the prepotential
which we introduce.

###  1.2 Summary

Before proceeding we give a brief summary of the results. The main result of
this paper is an expression for the non-perturbative topological string
partition function on any Calabi-Yau manifold $X$ which is expressed, up to
the contributions from constant maps at genus $0,1$, as:

$Z_{\textrm{top},np}(\lambda,t)=Z_{c,np}(\lambda)\prod_{n\in\mathbb{Z}}\prod_{\beta>0}\left(Z^{con}_{np}(\lambda,t^{\beta}+n\,\check{\lambda})\right)^{\Omega_{\beta,n}}\,,$
(1.1)

where

$Z^{con}_{np}(\lambda,t)=\prod_{m=0}^{\infty}(1-Q\,q^{m+1})^{m+1}\cdot\exp\left(-\frac{1}{2\pi
i}\textrm{Li}_{2}(Q^{\prime}q^{\prime m})\right)\cdot(1-Q^{\prime}q^{\prime
m})^{-\frac{1}{\check{\lambda}}\left(t+m\right)}\,,$
$Z^{con}_{c,np}(\lambda)=\left(\prod_{m=1}^{\infty}(1-q^{m})^{m}\cdot\exp\left(-\frac{1}{2\pi
i}\textrm{Li}_{2}(q^{\prime m})\right)\cdot(1-q^{\prime
m})^{-\frac{m}{\check{\lambda}}}\right)^{-\chi(X)/2}\,,$

where $Q=\exp(2\pi it),\quad q=e^{\textnormal{i}\lambda}\,,\quad
Q^{\prime}=\exp(2\pi it/\check{\lambda})\,,\quad q^{\prime}=\exp(2\pi
i/\check{\lambda})\,,\quad\check{\lambda}=\frac{\lambda}{2\pi}$ and where the
subscript $c$ in $Z_{c}$ stands for the constant map contribution. $\beta\in
H_{2}(X,\mathbb{Z})$ refers to a curve class, and $\beta>0$ denotes the non-
constant curves. $\Omega_{\beta,n}$ are up to a sign the sheaf invariants
defined in terms of the higher genus Gopakumar-Vafa invariants as follows [37]
333We follow the notation of [39]

$\sum_{n\in\mathbb{Z}\,}\Omega_{\beta,n}y^{n}=\sum_{g\geq
0}[\textrm{GV}]_{\beta,g}\,(-1)^{g}(y^{1/2}-y^{-1/2})^{2g}\,.$ (1.2)

The expression can be given in terms of special functions as follows:

$\begin{split}Z_{\textrm{top},np}(\lambda,t)=\left(\mathcal{S}_{3}(0|\check{\lambda},1)\right)^{-\chi(X)/2}\prod_{n\in\mathbb{Z}}\prod_{\beta>0}\left(\mathcal{S}_{3}(t^{\beta}+n\check{\lambda}|\check{\lambda},1)\right)^{\Omega_{\beta,n}}\,,\end{split}$
(1.3)

where

$\mathcal{S}_{3}(z\,|\,\omega_{1},\omega_{2}):=\exp\left(\frac{\pi{\mathrm{i}}}{6}\cdot
B_{3,3}(z+\omega_{1}\,|\,\omega_{1},\omega_{1},\omega_{2})\right)\cdot\sin_{3}(z+\omega_{1}\,|\,\omega_{1},\omega_{2},\omega_{3}),$

and the relevant definitions for the Bernoulli polynomials and the triple sine
function are given in the appendix.

This proposal is put forward by using the main theorem of this paper 4.1,
stating that the generating function of GW invariants, excluding the classical
terms as well as the contributions from constant maps for any Calabi-Yau
threefold:

$\tilde{F}_{GW}(\lambda,t)=\sum_{g=0}^{\infty}\sum_{\beta>0}\lambda^{2g-2}[GW]_{\beta,g}Q^{\beta}\,,$
(1.4)

can be written as:

$\displaystyle\tilde{F}_{GW}(\lambda,t)$
$\displaystyle=\sum_{n\in\mathbb{Z}}\sum_{\beta>0}\Omega_{\beta,n}\widetilde{F}^{con}(\lambda,t^{\beta,n})\,$
(1.5)
$\displaystyle=\sum_{n\in\mathbb{Z}}\sum_{\beta>0}\Omega_{\beta,n}\left(\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(Q^{\beta,n})+\sum_{g=1}^{\infty}\lambda^{2g-2}\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\mathrm{Li}_{3-2g}(Q^{\beta,n})\,\right)\,,$
(1.6)

where
$t^{\beta,n}:=t^{\beta}+n\check{\lambda}\,,Q^{\beta,n}:=\exp(2\pi\textnormal{i}t^{\beta,n})$
and then using the Borel summation analysis of [21].

Furthermore the non-perturbative partition function is written as:

$\displaystyle Z_{\textrm{top},np}(\lambda,t)=Z_{c,np}(\lambda)\times
Z_{top,wk}(\lambda,t)\times Z_{top,sg}(\lambda,t)$ (1.7)

where

$\displaystyle Z_{top,wk}(\lambda,t)$
$\displaystyle:=\prod_{n\in\mathbb{Z}}\prod_{\beta>0}\prod_{m=1}^{\infty}\left(1-Q^{\beta}\exp(\textnormal{i}(n+m)\,\lambda)\right)^{m\,\Omega_{\beta,n}}\,,$
(1.8) $\displaystyle Z_{top,sg}(\lambda,t)$
$\displaystyle:=\prod_{\beta>0}\exp\left(-\frac{[\textrm{GV}]_{\beta,0}}{2\pi
i}\textrm{Li}_{2}(Q^{\prime\beta}q^{\prime
m})\right)\cdot(1-Q^{\prime\beta}q^{\prime
m})^{-\frac{[\textrm{GV}]_{\beta,0}}{\check{\lambda}}\left(t^{\beta}+m\right)}\,,$
(1.9)

where the subscripts $wk$ and $sg$ stand for weak and strong respectively and
denote the factors of the non-perturbative partition function which are
expressions in $q=\exp(\textnormal{i}\lambda)$ and
$q^{\prime}=\exp(2\pi\textnormal{i}/\check{\lambda})$ respectively. The
surprising result is that the non-perturbative corrections only depend on
genus $0$ GV invariants which is due do cancellations leading to a dependence
on only the sum of the sheaf invariants for each curve class $\beta$, see
(4.4):

$\sum_{n\in\mathbb{Z}}\Omega_{\beta,n}=[\textrm{GV}]_{\beta,0}\,,$ (1.10)

where the sum on the l. h. s. has only finitely many non-zero terms due to the
finiteness of GV invariants, see sec.2.3.

The non-perturbative corrections can be furthermore packaged in terms of an
$\varepsilon-$deformation of the prepotential, which we define as follows:

$\tilde{F}^{0}_{\textrm{def}}(\varepsilon,t):=\frac{1}{2}\sum_{k,\beta>0}\frac{[\textrm{GV}]_{\beta,0}}{k^{2}}\frac{Q^{k\beta}}{\sin(k\varepsilon/2)}\,.$
(1.11)

The $q^{\prime}$ dependent factor

$Z_{top,sg}(\lambda,t)=\exp\left(F_{top,sg}(\lambda,t)\right)\,,$

can be expressed as

$F_{top,sg}(\lambda,t)=-\frac{1}{2\pi}\partial_{\lambda}\left(\lambda\,\tilde{F}^{0}_{\textrm{def}}\left(\frac{2\pi}{\check{\lambda}},\frac{t-1/2}{\check{\lambda}}\right)\right)\,.$
(1.12)

This $\varepsilon$ deformation of the prepotential which we propose agrees
with the refined topological string in the Nekrasov-Shatashvili limit in the
case of the resolved conifold.

The organization of this work is as follows. In sec. 2, the topological string
free energies are recalled as well as their Gromov-Witten and Gopakumar-Vafa
expansions. We proceed with a review of the resurgence analysis results of
[21] to introduce the building blocks of the more general topological string
analysis in later sections, we provide new infinite product expressions for
the non-perturbative partition function of topological strings on the resolved
conifold as well as for topological strings on $\mathbb{C}^{3}$ which is also
the building block for the more general constant maps contributions. The
latter also provides an interesting non-perturbative completion of the
MacMahon function.

##  2 Topological strings and enumerative geometry

To a mirror family of CY threefolds, topological string theory associates the
topological string partition function which is defined as an asymptotic series
in the topological string coupling $\lambda$, summing over the free energies
$\mathcal{F}^{g}(t)$ associated to Riemann surfaces of genus $g$:

$Z_{top}(\lambda,t)=\exp\left(\sum_{g=0}^{\infty}\lambda^{2g-2}\mathcal{F}^{g}(t)\right)\,,$
(2.1)

where $t=(t^{1},\dots,t^{n})$ is a set of distinguished local coordinates on
the underlying moduli space $\mathcal{M}$, which is of dim
$n=h^{1,1}(X_{t})=h^{2,1}(\check{X}_{t(z}))$. $X_{t}$ and $\check{X}_{t(z)}$
are a mirror pair of CY threefolds which correspond to the A-model and B-model
sides of mirror symmetry.

###  2.1 The Gromov-Witten potential

In a certain limit together with an expansion around a distinguished large
volume point in the moduli space, the topological string free energies
$\mathcal{F}^{g}(t)$ become the generating functions of higher genus Gromov-
Witten invariants $F^{g}(t)$ on the A-model side of mirror symmetry. The GW
potential of $X$ is the following formal power series:

$F_{GW}(\lambda,t)=\sum_{g\geq 0}\lambda^{2g-2}F^{g}(t)=\sum_{g\geq
0}\lambda^{2g-2}\sum_{\beta\in
H_{2}(X,\mathbb{Z})}[\textrm{GW}]_{\beta,g}\,Q^{\beta}\,,$ (2.2)

where $Q^{\beta}:=\exp(2\pi{\mathrm{i}}t^{\beta})$ is a formal variable and
$[\textrm{GW}]_{\beta,g}$ are the Gromov-Witten invariants associated to a
curve of genus $g$ and class $\beta$. Let $C_{a}\in
H_{2}(X,\mathbb{Z}),a=1,\dots,n=h^{1,1}(X)$ be a set of curve classes spanning
$H_{2}(X,\mathbb{Z})$, then in general all curve classes $\beta$ can be
written as $\beta=\sum_{a=1}^{n}d_{a}C_{a}$, where the $d$’s are the degrees
of the curve. The formal variables $t^{\beta}$ become
$t^{\beta}=\sum_{a=1}^{n}d_{a}t^{a}$ and the $t^{a}$ correspond to local
coordinates on the moduli space of complexified Kähler forms of the underlying
CY $X$. Bearing this in mind, we will in the following treat $t^{\beta}$ as
independent formal variables for the curve classes $\beta$ for ease of
exposition of the general results.

The GW potential can be furthermore written as:

$F_{GW}=F_{c}+\tilde{F}_{GW}\,,$ (2.3)

where $F_{c}$ 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 $t$ dependent and the higher genus constant map contributions take
the universal form [40]:

$F_{c}^{g}:=\frac{\chi(X)(-1)^{g-1}\,B_{2g}\,B_{2g-2}}{4g(2g-2)\,(2g-2)!}\,,\quad
g\geq 2\,,$ (2.4)

where $\chi(X)$ is the Euler characteristic of $X$ and the Bernoulli numbers
$B_{n}$ are generated by:

$\frac{w}{e^{w}-1}=\sum_{n=0}^{\infty}B_{n}\frac{w^{n}}{n!}\,.$ (2.5)

###  2.2 The Gopakumar-Vafa resummation

The Gopakumar-Vafa (GV) resummation of the GW potential [41, 42] reformulates
the non-constant part of the GW potential in terms of the Gopakumar-Vafa
invariants $[\textrm{GV}]_{\beta,g}\in\mathbb{Z}$ which are given by a count
of electrically charged $M_{2}$ branes in an M-theory setup. The GW potential
can thus be written as:

$F_{GV}(\lambda,t)=\sum_{\beta>0}\sum_{g\geq
0}[\textrm{GV}]_{\beta,g}\,\sum_{k\geq
1}\frac{1}{k}\left(2\sin\left(\frac{k\lambda}{2}\right)\right)^{2g-2}Q^{k\beta}\,.$
(2.6)

In particular $F_{GV}$ has the same asymptotic expansion at $\lambda=0$ as
$\tilde{F}_{GW}$ and

$\tilde{F}^{0}(t)=\sum_{\beta>0}[\textrm{GV}]_{\beta,0}\,\textrm{Li}_{3}(Q^{\beta})\,,\quad
Q^{\beta}=\exp(2\pi it^{\beta})\,.$

###  2.3 Finiteness of GV invariants

We remark that in the GV resummation of the GW potential the integrality of
the $[\textrm{GV}]_{\beta,g}$ was conjectured from their M-theoretic BPS
counting origin. The integrality was proven in [43]. A further conjecture
concerns the finiteness of the GV invariants, meaning that for every curve
class $\beta$, there exists a $g_{\beta}\in\mathbb{N}$ such that

$[\textrm{GV}]_{\beta,g^{\prime}}=0\,,\quad\forall g^{\prime}\geq
g_{\beta}\,,$

see [44] for a proof of this using methods from symplectic geometry. Algebraic
geometrically it is expected that the $[\textrm{GV}]_{\beta,g}$ vanish for
each curve class $\beta$ at high enough genus, this is discussed as the
Castelnuovo bound for instance in [45] and is used as a boundary condition
together with the polynomiality of [3], for higher genus GV computations in
the quintic [6].

###  2.4 GW potential of the resolved conifold

An example of a non-compact CY manifold $X$ for which the asymptotic series is
known to all orders is the resolved conifold which is given by the total space
of the rank two bundle over the projective line:

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

and corresponds to the resolution of the conifold singularity.

The GW potential for this geometry was determined in physics [41, 46], and in
mathematics [40] with the following outcome for the non-constant maps:

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

using the notation $Q=e^{2\pi\textnormal{i}t}$.

##  3 Building blocks of non-perturbative topological strings

In the following we review the resurgence study of the building blocks of
topological strings which will be used in the next sections to spell out the
generalization for topological strings on arbitrary CY manifolds. The building
blocks in this section will be topological strings on the resolved conifold as
well as on $\mathbb{C}^{3}$ which also gives the piece of the topological
string partition function stemming from constant maps. The contents of this
section build on [21] with a few new important additions.

###  3.1 The resolved conifold

The starting point is the formal series giving the non-constant and non-
classical piece of the Gromov-Witten potential or the topological string free
energy of the resolved conifold:

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

In [47] following methods of [48], it was proven that this series satisfies
the following difference equation:

$\widetilde{F}^{con}\left(\lambda,t+\check{\lambda}\right)+\widetilde{F}^{con}\left(\lambda,t-\check{\lambda}\right)-2\widetilde{F}^{con}\left(\lambda,t\right)=-\textrm{Li}_{1}(Q)\,,\quad\check{\lambda}=\frac{\lambda}{2\pi}\,.\\\
$ (3.1)

A solution in terms of the triple sine function of this difference equation
was found [49]:

$\begin{split}\widetilde{F}^{con}_{\text{np}}(\lambda,t):=\log\left(\mathcal{S}_{3}(t|\check{\lambda},1)\right)\,,\end{split}$
(3.2)

where

$\mathcal{S}_{3}(z\,|\,\omega_{1},\omega_{2}):=\exp\left(\frac{\pi{\mathrm{i}}}{6}\cdot
B_{3,3}(z+\omega_{1}\,|\,\omega_{1},\omega_{1},\omega_{2})\right)\cdot\sin_{3}(z+\omega_{1}\,|\,\omega_{1},\omega_{2},\omega_{3}),$

and the relevant definitions for the Bernoulli polynomials and the triple sine
function are given in the appendix. The non-perturbative content of this
solution was analyzed in [25] and in [21]. In [30] it was furthermore shown
that $\widetilde{F}^{con}_{\text{np}}(\lambda,t)$ satisfies a further
difference equation given by:

$\displaystyle\widetilde{F}_{\text{np}}^{con}(\lambda,t+1)-\widetilde{F}_{\text{np}}^{con}(\lambda,t)=\frac{1}{2\pi{\mathrm{i}}}\frac{\partial}{\partial\check{\lambda}}\left(\check{\lambda}\,\textrm{Li}_{2}(e^{2\pi{\mathrm{i}}t/\check{\lambda}})\right)\,.$
(3.3)

$\widetilde{F}^{con}_{\text{np}}(\lambda,t)$ was identified in [21] as the
Borel summation of the asymptotic series along a distinguished ray on the real
axis in the Borel plane, furthermore the inhomogeneous piece of the difference
equation (3.3) corresponds to the Stokes jumps of the Borel summation of the
asymptotic series. To clarify this we recall here the general story of Borel
summation.

Given a formal power series
$a(\check{\lambda})\in\check{\lambda}\mathbb{C}[[\check{\lambda}]]$, its Borel
transform $\mathcal{B}(a)(\xi)$, is given by:

$\mathcal{B}\colon\check{\lambda}\mathbb{C}[[\check{\lambda}]]\to\mathbb{C}[[\xi]],\;\;\;\;\mathcal{B}(\check{\lambda}^{n+1})=\frac{\xi^{n}}{n!}.$
(3.4)

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

$\int_{\rho}\mathrm{d}\xi\;e^{-\xi/\check{\lambda}}\mathcal{B}(a)(\xi)\,.$
(3.5)

If (3.5) is finite, we say $a(\check{\lambda})$ is Borel summable at
$\check{\lambda}$, along $\rho$.

Writing

$\displaystyle\widetilde{F}^{con}(\lambda,t)=\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(Q)+\frac{B_{2}}{2}\mathrm{Li}_{1}(Q)+\Phi^{con}(\check{\lambda},t)\,,\quad\check{\lambda}=\frac{\lambda}{2\pi}\,,\quad
Q=e^{2\pi{\mathrm{i}}t}\,,$

the following was proven in [21] for the Borel resummation of the series
$\Phi(\check{\lambda},t)$ :

  * (i)

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

$G^{con}(\xi,t)=\frac{1}{(2\pi)^{2}}\sum_{m\in\mathbb{Z}\setminus\\{0\\}}\frac{1}{m^{3}}\frac{1}{2\xi}\frac{\partial}{\partial\xi}\bigg{(}\frac{\xi^{2}}{1-e^{-2\pi{\mathrm{i}}t+\xi/m}}-\frac{\xi^{2}}{1-e^{-2\pi{\mathrm{i}}t-\xi/m}}\bigg{)},$
(3.6)

which can be used to analytically continue $G^{con}(\xi,t)$ in the $\xi$
variable to a meromorphic function with poles at $\xi=2\pi{\mathrm{i}}(t+k)m$
for $k\in\mathbb{Z}$ and $m\in\mathbb{Z}\setminus\\{0\\}$.

  * (ii)

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

$F^{con}_{\rho}(\lambda,t):=\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(Q)+\frac{B_{2}}{2}\mathrm{Li}_{1}(Q)+\int_{\rho}\mathrm{d}\xi\,e^{-\xi/\check{\lambda}}G^{con}(\xi,t)\,.$
(3.7)

  * (iii)

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

$\displaystyle\phi_{\pm l_{k}}(\lambda,t)$
$\displaystyle:=F^{con}_{\pm\rho_{k+1}}(\lambda,t)-F^{con}_{\pm\rho_{k}}(\lambda,t)=\frac{1}{2\pi{\mathrm{i}}}\partial_{\check{\lambda}}\Big{(}\check{\lambda}\,\mathrm{Li}_{2}\big{(}e^{\pm
2\pi{\mathrm{i}}(t+k)/\check{\lambda}}\big{)}\Big{)}\;,\,$ (3.8)
$\displaystyle=\frac{1}{2\pi{\mathrm{i}}}\left(\mathrm{Li}_{2}\big{(}e^{\pm
2\pi{\mathrm{i}}(t+k)/\check{\lambda}}\big{)}\pm
2\pi{\mathrm{i}}\frac{(t+k)}{\check{\lambda}}\log\left(1-e^{\pm
2\pi{\mathrm{i}}(t+k)/\check{\lambda}}\right)\right)\;.$ (3.9)

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

  * (iv)

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

$\lim_{k\to\infty}F^{con}_{\rho_{k}}(\lambda,t)=\lim_{k\to\infty}F^{con}_{-\rho_{k}}(-\lambda,t)=\sum_{k=1}^{\infty}\frac{e^{2\pi{\mathrm{i}}kt}}{k\big{(}2\sin\big{(}\frac{\lambda
k}{2}\big{)}\big{)}^{2}}\,.$ (3.10)

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

$\sum_{k=0}^{\infty}\phi_{l_{k}}(\lambda,t)=\frac{1}{2\pi{\mathrm{i}}}\partial_{\lambda}\Big{(}\lambda\sum_{m=1}^{\infty}\frac{Q^{\prime
l}}{m^{2}(1-q^{\prime
m})}\Big{)},\;\;\;\;Q^{\prime}:=e^{2\pi{\mathrm{i}}t/\check{\lambda}},\;\;q^{\prime}:=e^{2\pi{\mathrm{i}}/\check{\lambda}}\;.$
(3.11)

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

$\lim_{k\to-\infty}F^{con}_{\rho_{k}}(\lambda,t)=\lim_{k\to-\infty}F^{con}_{-\rho_{k}}(-\lambda,t)=\sum_{k=1}^{\infty}\frac{e^{2\pi{\mathrm{i}}kt}}{k\big{(}2\sin\big{(}\frac{\lambda
k}{2}\big{)}\big{)}^{2}}.$ (3.12)

$\lim_{k\to\infty}F^{con}_{\rho_{k}}(\lambda,t)$ differs from
$F^{con}_{\mathbb{R}_{>0}}({\lambda},t)$ by the sum over all jumps
$\phi_{l_{k}}(\lambda,t)$ for $k\geq 0$, leading to the decomposition, see
[25, 21]:

$F^{con}_{{\mathbb{R}}_{>0}}({\lambda},t)=\sum_{k=1}^{\infty}\frac{e^{2\pi{\mathrm{i}}kt}}{k\big{(}2\sin\big{(}\frac{\lambda
k}{2}\big{)}\big{)}^{2}}-\frac{1}{2\pi{\mathrm{i}}}\partial_{\lambda}\Big{(}\lambda\sum_{m=1}^{\infty}\frac{Q^{\prime
l}}{m^{2}(1-q^{\prime m})}\Big{)}.$ (3.13)

$F^{con}_{\mathbb{R}_{>0}}({\lambda},t)$ could furthermore be identified with
the solution $\widetilde{F}^{con}_{\text{np}}(\lambda,t)$ of the difference
equation 3.1.

$\begin{split}F^{con}_{\mathbb{R}_{>0}}({\lambda},t)=\widetilde{F}^{con}_{\text{np}}(\lambda,t):=\left(\frac{\pi{\mathrm{i}}}{6}\,B_{3,3}\left(t+\check{\lambda}\,|\,\check{\lambda},\check{\lambda},1\right)\right)+\log\left(\sin_{3}\left(t+\check{\lambda}\,|\,\check{\lambda},\check{\lambda},1\right)\right).\end{split}$
(3.14)

We proceed by giving the product structure of the partition functions obtained
from the Borel sums corresponding to the free energies, these follow from the
results of [21] but were not contained in that paper.

######  Proposition 3.1.

Product form of the partition functions. For the partition functions

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

we have the following product forms:

$\displaystyle Z^{con}_{GV}(\lambda,t)$
$\displaystyle:=Z_{\rho_{\infty}}(\lambda,t)=\prod_{m=0}^{\infty}(1-Qq^{m+1})^{m+1}\,,$
(3.15) $\displaystyle Z^{con}_{np}(\lambda,t)$
$\displaystyle:=Z_{\rho_{0}}(\lambda,t)=\prod_{m=0}^{\infty}(1-Q\,q^{m+1})^{m+1}\cdot\exp\left(-\frac{1}{2\pi
i}\textrm{Li}_{2}(Q^{\prime}q^{\prime m})\right)\cdot(1-Q^{\prime}q^{\prime
m})^{-\frac{1}{\check{\lambda}}\left(t+m\right)}\,,$ (3.16) $\displaystyle
Z^{con}_{\rho_{k}}(\lambda,t)$
$\displaystyle=\prod_{m=0}^{\infty}(1-Q\,q^{m+1})^{m+1}\cdot\prod_{m=k}^{\infty}\exp\left(-\frac{1}{2\pi
i}\textrm{Li}_{2}(Q^{\prime}q^{\prime m})\right)\cdot(1-Q^{\prime}q^{\prime
m})^{-\frac{1}{\check{\lambda}}\left(t+m\right)}\,,$ (3.17)

where $Q=\exp(2\pi it),\quad q=e^{\textnormal{i}\lambda}\,,\quad
Q^{\prime}=\exp(2\pi it/\check{\lambda})\,,\quad q^{\prime}=\exp(2\pi
i/\check{\lambda})$ .

###### Proof.

To prove the first product form we use the following expansion:

$\frac{1}{\left(2\sin\left(k\lambda/2\right)\right)^{2}}=-\sum_{m=1}^{\infty}me^{\textnormal{i}mk\lambda}\,,$
(3.18)

valid for $\textrm{Im}\lambda>0$ and which can be obtained from the geometric
series by a derivation in $\lambda$. We have

$F_{GV}(\lambda,t):=F_{\infty}(\lambda,t)=\sum_{k=1}^{\infty}\frac{e^{2\pi{\mathrm{i}}kt}}{k\big{(}2\sin\big{(}\frac{\lambda
k}{2}\big{)}\big{)}^{2}}\,,$ (3.19)

which can be re-written using the identity 3.18 as:

$F_{GV}(\lambda,t)=-\sum_{m=1}^{\infty}m\sum_{k=1}^{\infty}\frac{e^{2\pi{\mathrm{i}}kt}e^{\textnormal{i}mk\lambda}}{k}=\sum_{m=1}^{\infty}m\,\log(1-e^{2\pi{\mathrm{i}}t}e^{\textnormal{i}m\lambda})\,,$
(3.20)

the product form then follows by taking
$Z^{con}_{GV}(\lambda,t)=\exp(F_{GV}(\lambda,t))$ and shifting the summation
index. The product form for $Z_{\rho_{0}}$ follows from the form:

$F_{\rho_{0}}({\lambda},t)=\sum_{k=1}^{\infty}\frac{e^{2\pi{\mathrm{i}}kt}}{k\big{(}2\sin\big{(}\frac{\lambda
k}{2}\big{)}\big{)}^{2}}-\sum_{k=0}^{\infty}\phi_{l_{k}}(\lambda,t)\,,$ (3.21)

and the expressions

$\displaystyle\phi_{l_{k}}(\lambda,t)=\frac{1}{2\pi{\mathrm{i}}}\left(\mathrm{Li}_{2}\big{(}e^{2\pi{\mathrm{i}}(t+k)/\check{\lambda}}\big{)}+2\pi{\mathrm{i}}\frac{(t+k)}{\check{\lambda}}\log\left(1-e^{2\pi{\mathrm{i}}(t+k)/\check{\lambda}}\right)\right)\;,$
(3.22)

the product expression then follows by exponentiation, the product form for
$Z_{\rho_{k}}$ follows from taking into account the Stokes factors.

∎

###  3.2 Constant maps contributions

For the formal series giving the constant map contributions to the GW
potential for any CY we have

$\displaystyle F_{c}(\lambda)=\sum_{g=0}^{\infty}F^{g}_{c}$
$\displaystyle=F_{c}^{0}(t)+F_{c}^{1}(t)+\chi(X)\sum_{g=2}^{\infty}\frac{(-1)^{g-1}\,B_{2g}\,B_{2g-2}}{4g(2g-2)\,(2g-2)!}\lambda^{2g-2}\,,\quad\check{\lambda}=\frac{\lambda}{2\pi}\,\,$
(3.23)
$\displaystyle=F_{c}^{0}(t)+F_{c}^{1}(t)+\Phi_{c}(\check{\lambda})\,,\quad\check{\lambda}=\frac{\lambda}{2\pi}\,\,,$
(3.24)

the terms at genus $0$ and $1$ are $t$ dependent and encode also classical
topological data of the CY such as the triple intersections and second Chern
class, see [50] for details.

In [21], the case of the resolved conifold was treated for which $\chi(X)=2$
and the constant map contribution has the form:

$\displaystyle F^{con}_{c}(\lambda)$
$\displaystyle=F_{c}^{0}(t)+F_{c}^{1}(t)+\sum_{g=2}^{\infty}\frac{(-1)^{g-1}\,B_{2g}\,B_{2g-2}}{2g(2g-2)\,(2g-2)!}\lambda^{2g-2}\,,\quad\check{\lambda}=\frac{\lambda}{2\pi}\,$
(3.25)
$\displaystyle=F_{c}^{0}(t)+F_{c}^{1}(t)+\Phi_{c}^{con}(\check{\lambda})\,,$
(3.26)

it was shown that the Borel transform of $\Phi_{c}^{con}(\check{\lambda})$ is
given by:

$G_{c}^{con}(\xi)=-G^{con}(\xi,0)-\frac{1}{12\xi}\,,$ (3.27)

and that correspondingly the Borel sums of the constant map contributions are
given by

$F^{con}_{c,\rho}(\lambda)=-F^{con}_{\rho}(\lambda,0)-\frac{1}{12}\log\check{\lambda}+C\,,$
(3.28)

with $C$ a constant which does not play a role. Using this result we obtain
immediately that the Borel transform of $\Phi_{c}(\check{\lambda})$ is given
by:

$G_{c}(\xi)=-\frac{\chi(X)}{24\xi}-\frac{\chi(X)}{2(2\pi)^{2}}\sum_{m\in\mathbb{Z}\setminus\\{0\\}}\frac{1}{m^{3}}\frac{1}{2\xi}\frac{\partial}{\partial\xi}\bigg{(}\frac{\xi^{2}}{1-e^{\xi/m}}-\frac{\xi^{2}}{1-e^{-\xi/m}}\bigg{)},$
(3.29)

which can be analytically continued in the $\xi$ variable to a meromorphic
function with poles at $\xi=2\pi{\mathrm{i}}k$ for $k\in\mathbb{Z}$ .

Furthermore given any ray $\rho$ from $0$ to $\infty$ different from $\\{\pm
l_{\infty}\\}$, and $\lambda$ in the half-plane $\mathbb{H}_{\rho}$ centered
at $\rho$, the Borel sum of $F_{c}(\lambda)$ along $\rho$ is given by

$F_{c,\rho}(\lambda):=\frac{1}{\lambda^{2}}F^{0}_{c}(t)+F^{1}_{c}(t)+\int_{\rho}\mathrm{d}\xi\,e^{-\xi/\check{\lambda}}G_{c}(\xi)\,.$
(3.30)

######  Proposition 3.2.

Product form of the partition function. For the partition functions

$Z^{con}_{c,\rho}(\lambda,t):=\exp(F^{con}_{c,\rho}(\lambda))$

we have the following product form:

$\displaystyle Z^{con}_{c,np}(\lambda)$
$\displaystyle:=Z^{con}_{\rho_{0}}(\lambda,0)^{-1}\check{\lambda}^{-1/12}e^{C}$
(3.31)
$\displaystyle=\left(e^{C}\check{\lambda}^{-1/12}\exp\left(\frac{\textnormal{i}\pi}{12}\right)\prod_{m=1}^{\infty}(1-q^{m})^{m}\cdot\exp\left(-\frac{1}{2\pi
i}\textrm{Li}_{2}(q^{\prime m})\right)\cdot(1-q^{\prime
m})^{-\frac{m}{\check{\lambda}}}\right)^{-1}\,,$ (3.32)

where $q=\exp(i\lambda)\quad q^{\prime}=\exp(2\pi i/\check{\lambda})$ ,

###### Proof.

This follows from evaluating the $t\rightarrow 0$ limit of the conifold
partition function. ∎

We note that

$\prod_{m=1}^{\infty}(1-q^{m})^{-m}$

corresponds to the MacMahon function and that $Z^{con}_{c,np}(\lambda)$ is a
non-perturbative completion thereof which is likely interesting in its own
right, this has potentially an interpretation as a quantum modular form [51].

For the constant map contributions for Gromov-Witten theory on an arbitrary CY
manifold $X$, we introduce the following non-perturbative generating function
which is valid up to the genus $0$ and $1$ terms which should be included
separately.

$Z^{con}_{c,np}(\lambda)=\left(\prod_{m=1}^{\infty}(1-q^{m})^{m}\cdot\exp\left(-\frac{1}{2\pi
i}\textrm{Li}_{2}(q^{\prime m})\right)\cdot(1-q^{\prime
m})^{-\frac{m}{\check{\lambda}}}\right)^{-\chi(X)/2}$ (3.33)

##  4 Non-perturbative topological strings

###  4.1 Sheaf invariants

Our first goal is to re-write the topological string free energy on an
arbitrary CY threefold in terms of the free energy of the resolved conifold.
This will be possible using a re-arrangement of the GV invariants in terms of
a set of sheaf invariants $\Omega_{\beta,n}$ defined in the following way:

$\sum_{n\in\mathbb{Z}\,}\Omega_{\beta,n}y^{n}=\sum_{g\geq
0}[\textrm{GV}]_{\beta,g}\,(-1)^{g}(y^{1/2}-y^{-1/2})^{2g}\,,$ (4.1)

since the r. h. s. is invariant under $y\rightarrow y^{-1}$ we have:

$\Omega_{\beta,n}=\Omega_{\beta,-n}\,,\forall n\in\mathbb{Z}\,,$ (4.2)

these are up to a sign the sheaf invariants put forward by Maulik and Toda
[37], following works of Hosono-Saito-Takahashi [52] and Kiem-Li [53] to
define the Gopakumar-Vafa invariants in terms of moduli spaces of sheaves, we
are using the notation of [39] which should also be consulted for further
details and references.

It is useful to note the relation of how each $\Omega_{\beta,k}\,$ can be
obtained from the GV invariants. We recall the finiteness of the GV invariants
discussed in 2.3, stating that for every curve class $\beta$, there exists a
$g_{\beta}\in\mathbb{N}$ such that

$[\textrm{GV}]_{\beta,g^{\prime}}=0\,,\quad\forall g^{\prime}\geq
g_{\beta}\,,$

by comparing the $y^{k}$ coefficient on both sides of 4.1 we obtain:

$\Omega_{\beta,k}=\left\\{\begin{array}[]{ll}(-1)^{k}\sum_{g=k}^{g_{\beta}-1}[\textrm{GV}]_{\beta,g}\binom{2g}{g-k}\,,\quad
k\geq 0\,,&\mbox{if }k<g_{\beta}\\\ 0&\mbox{if }k\geq
g_{\beta}\end{array}\right.$ (4.3)

the $k<0$ invariants are obtained by 4.2.

We further note that accordingly the sum
$\sum_{n\in\mathbb{Z}}\Omega_{\beta,n}$ is finite. This sum is equal to the
limit $y\rightarrow 1$ of the l. h. s. of

$\sum_{n\in\mathbb{Z}\,}\Omega_{\beta,n}y^{n}=\sum_{g\geq
0}[\textrm{GV}]_{\beta,g}\,(-1)^{g}(y^{1/2}-y^{-1/2})^{2g}\,,$

evaluating the limit on the r. h. s. we obtain:

$\sum_{n\in\mathbb{Z}}\Omega_{\beta,n}=[\textrm{GV}]_{\beta,0}\,.$ (4.4)

###  4.2 Free energy in terms of the resolved conifold

######  Theorem 4.1.

The generating function of GW invariants, excluding the classical terms as
well as the contributions from constant maps for any Calabi-Yau threefold:

$\tilde{F}_{GW}(\lambda,t)=\sum_{g=0}^{\infty}\sum_{\beta>0}\lambda^{2g-2}[GW]_{\beta,g}Q^{\beta}\,,$
(4.5)

can be written as:

$\displaystyle\tilde{F}_{GW}(\lambda,t)$
$\displaystyle=\sum_{n\in\mathbb{Z}}\sum_{\beta>0}\Omega_{\beta,n}\widetilde{F}^{con}(\lambda,t^{\beta,n})\,$
(4.6)
$\displaystyle=\sum_{n\in\mathbb{Z}}\sum_{\beta>0}\Omega_{\beta,n}\left(\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(Q^{\beta,n})+\sum_{g=1}^{\infty}\lambda^{2g-2}\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\mathrm{Li}_{3-2g}(Q^{\beta,n})\,\right)\,,$
(4.7)

where
$t^{\beta,n}:=t^{\beta}+n\check{\lambda}\,,Q^{\beta,n}:=\exp(2\pi\textnormal{i}t^{\beta,n})$
and $\Omega_{\beta,n}$ are the integers defined in (4.1). A similar relation
holds for the generating function of GV invariants:

$F_{GV}(\lambda,t)=\sum_{\beta>0}\sum_{g\geq
0}[\textrm{GV}]_{\beta,g}\,\sum_{k\geq
1}\frac{1}{k}\left(2\sin\left(\frac{k\lambda}{2}\right)\right)^{2g-2}Q^{k\beta}\,,$
(4.8)

namely:

$\displaystyle F_{GV}(\lambda,t)$
$\displaystyle=\sum_{n\in\mathbb{Z}}\sum_{\beta>0}\Omega_{\beta,n}\sum_{m=1}^{\infty}\log\left(1-Q^{\beta}\exp(\textnormal{i}(n+m)\,\lambda)\right)^{m}\,,$
(4.9)
$\displaystyle=\sum_{n\in\mathbb{Z}}\sum_{\beta>0}\Omega_{\beta,n}F_{GV}^{con}(\lambda,t^{\beta}+n\check{\lambda})\,.$
(4.10)

###### Proof.

We start with the Gopakumar-Vafa form of the topological string free energy:

$F_{GV}(\lambda,t)=\sum_{\beta>0}\sum_{k\geq
1}\frac{Q^{k\beta}}{k\left(2\sin\left(\frac{k\lambda}{2}\right)\right)^{2}}\sum_{g\geq
0}[\textrm{GV}]_{\beta,g}\,\left(2\sin\left(\frac{k\lambda}{2}\right)\right)^{2g}\,,$
(4.11)

using (4.1) with $y=\exp(\textnormal{i}k\lambda)$ this becomes:

$F_{GV}(\lambda,t)=\sum_{\beta>0}\sum_{k\geq
1}\frac{Q^{k\beta}}{k\left(2\sin\left(\frac{k\lambda}{2}\right)\right)^{2}}\sum_{n\in\mathbb{Z}\,}\Omega_{\beta,n}\exp(\textnormal{i}nk\lambda)\,,$
(4.12)

using the following identity which can be derived from the geometric series:

$\frac{1}{\left(2\sin\left(k\lambda/2\right)\right)^{2}}=-\frac{1}{\textnormal{i}k}\sum_{m=1}^{\infty}m\exp(\textnormal{i}mk\lambda)\,,$
(4.13)

we obtain:

$F_{GV}(\lambda,t)=-\sum_{m=1}^{\infty}\sum_{n\in\mathbb{Z}}\sum_{\beta>0}\sum_{k\geq
1}\frac{m\,Q^{k\beta}}{k}\Omega_{\beta,n}\exp(\textnormal{i}(n+m)k\,\lambda)\,,$
(4.14)

we can further re-write this as:

$F_{GV}(\lambda,t)=\sum_{n\in\mathbb{Z}}\sum_{\beta>0}\Omega_{\beta,n}\sum_{m=1}^{\infty}\log\left(1-Q^{\beta}\exp(\textnormal{i}(n+m)\,\lambda)\right)^{m}\,,$
(4.15)

which can be expressed in terms of the free energy of the resolved conifold
as:

$\displaystyle
F_{GV}(\lambda,t)=\sum_{n\in\mathbb{Z}}\sum_{\beta>0}\Omega_{\beta,n}F_{GV}^{con}(\lambda,t^{\beta}+n\check{\lambda})\,,$
(4.16)

in this last expression we introduce $t^{\beta,n}:=t^{\beta}+n\check{\lambda}$
and consider this as a formal variable and look at the asymptotic expansion of
each summand in $\lambda$ keeping $t^{\beta,n}$ fixed, obtaining the following
formal series, which is asymptotic in $\lambda$:

$\tilde{F}_{GW}(\lambda,t)=\sum_{n\in\mathbb{Z}}\sum_{\beta>0}\Omega_{\beta,n}\left(\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(Q^{\beta,n})+\sum_{g=1}^{\infty}\lambda^{2g-2}\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\mathrm{Li}_{3-2g}(Q^{\beta,n})\,\right)\,,$
(4.17)

where $Q^{\beta,n}:=\exp(2\pi\textnormal{i}t^{\beta,n}).$

∎

###  4.3 Non-perturbative free energy

Having expressed the generating function of GW invariants in an arbitrary CY
as a sum over generating functions of the GW invariants of the resolved
conifold, it is now natural to use the previously obtained analytic functions
in the topological string coupling to put forward a non-perturbative
expression for the topological string free energy in the holomorphic limit.

The expression of the asymptotic series of GW invariants of an arbitrary CY
threefold in terms of the resolved conifold:

$\tilde{F}_{GW}(\lambda,t)=\sum_{n\in\mathbb{Z}}\sum_{\beta>0}\Omega_{\beta,n}F_{GW}^{con}(\lambda,t^{\beta}+n\check{\lambda})\,,$
(4.18)

suggests the use of the resurgence analysis of the resolved conifold to define
the following:

$\displaystyle\widetilde{F}_{top,np}(\lambda,t)$
$\displaystyle=\sum_{n\in\mathbb{Z}}\sum_{\beta>0}\Omega_{\beta,n}F^{con}_{np}(\lambda,t^{\beta}+n\check{\lambda})\,$
(4.19)
$\displaystyle=\sum_{n\in\mathbb{Z}}\sum_{\beta>0}\Omega_{\beta,n}\log\left(\mathcal{S}_{3}(t+n\check{\lambda}|\check{\lambda},1)\right)\,.$
(4.20)

On the r. h. s. of this equation each summand has good analytic properties in
$\lambda$ from the triple sine function, to be able to say whether this sum
also defines a good analytic function one must however be careful with the
summation coefficients, which are, especially for compact Calabi-Yau manifolds
generically unknown. We note however that for each curve class $\beta$ the
summation of the coefficients $\Omega_{\beta,n}$ is given by 4.4:

$\sum_{n\in\mathbb{Z}}\Omega_{\beta,n}=[\textrm{GV}]_{\beta,0}\,,$ (4.21)

although now there are generically infinitely many $[\textrm{GV}]_{\beta,0}$
which are non-vanishing for compact CY manifolds, it is known by mirror
symmetry that all these invariants are generated by rational functions on the
moduli space of the mirror manifolds, the Yukawa couplings, see [54] for more
details. It seems thus reasonable to expect that the proposed non-perturbative
expression could correspond to an actual function. We will later see however
that although the issue with potential growth of the $\Omega_{\beta,n}$
remains the same as for the perturbative expansion, the non-perturbative
corrections obtained in this way nicely assemble into a function which will be
given in sec. 4.4.

We then have the following:

######  Proposition 4.2.

Define

$\tilde{Z}_{top,np}(\lambda,t)=\exp\left(\tilde{F}_{top,np}(\lambda,t)\right)$

then we have the following product expansion:

$\displaystyle\tilde{Z}_{top,np}(\lambda,t)$
$\displaystyle:=Z_{top,wk}(\lambda,t)\times Z_{top,sg}(\lambda,t)$ (4.22)

where

$\displaystyle Z_{top,wk}(\lambda,t)$
$\displaystyle:=\prod_{n\in\mathbb{Z}}\prod_{\beta>0}\prod_{m=1}^{\infty}\left(1-Q^{\beta}\exp(\textnormal{i}(n+m)\,\lambda)\right)^{m\,\Omega_{\beta,n}}\,,$
(4.23) $\displaystyle Z_{top,sg}(\lambda,t)$
$\displaystyle:=\prod_{\beta>0}\exp\left(-\frac{[\textrm{GV}]_{\beta,0}}{2\pi
i}\textrm{Li}_{2}(Q^{\prime\beta}q^{\prime
m})\right)\cdot(1-Q^{\prime\beta}q^{\prime
m})^{-\frac{[\textrm{GV}]_{\beta,0}}{\check{\lambda}}\left(t^{\beta}+m\right)}\,,$
(4.24)

$t^{\beta,n}:=t^{\beta}+n\check{\lambda}\,,t^{\beta,0}:=t^{\beta}\,,Q^{\beta,n}:=\exp(2\pi\textnormal{i}t^{\beta,n})\,,Q^{\beta}:=Q^{\beta,0}$
and $q=e^{\textnormal{i}\lambda}\,,\quad Q^{\prime}=\exp(2\pi
it/\check{\lambda})\,,\quad q^{\prime}=\exp(2\pi i/\check{\lambda})$ . The
subscripts $wk$ and $sg$ stand for weak and strong and refer to the pieces of
the partition function which are expansions in $q$ or $q^{\prime}$
respectively.

###### Proof.

From the definition

$\displaystyle\tilde{F}_{top,np}(\lambda,t)$
$\displaystyle=\sum_{n\in\mathbb{Z}}\sum_{\beta>0}\Omega_{\beta,n}F^{con}_{np}(\lambda,t^{\beta}+n\check{\lambda})\,,$
(4.25)

we obtain:

$\displaystyle
Z_{top,np}(\lambda,t)=\prod_{n\in\mathbb{Z}}\prod_{\beta>0}Z^{con}_{np}(\lambda,t^{\beta}+n\check{\lambda})^{\Omega_{\beta,n}}\,,$
(4.26)

we can now use the product expansion of 3.1:

$Z^{con}_{np}(\lambda,t)=\prod_{m=0}^{\infty}(1-Q\,q^{m+1})^{m+1}\cdot\exp\left(-\frac{1}{2\pi
i}\textrm{Li}_{2}(Q^{\prime}q^{\prime m})\right)\cdot(1-Q^{\prime}q^{\prime
m})^{-\frac{1}{\check{\lambda}}\left(t+m\right)}\,,$

to obtain:

$\displaystyle
Z_{top,np}(\lambda,t)=\prod_{n\in\mathbb{Z}}\prod_{\beta>0}\left(\prod_{m=0}^{\infty}(1-Q^{\beta,n}\,q^{m+1})^{m+1}\cdot\exp\left(-\frac{1}{2\pi
i}\textrm{Li}_{2}(Q^{\prime\beta}q^{\prime
m})\right)\cdot(1-Q^{\prime\beta}q^{\prime
m})^{-\frac{1}{\check{\lambda}}\left(t^{\beta,n}+m\right)}\,\right)^{\Omega_{\beta,n}}\,,$
(4.27)

where
$t^{\beta,n}:=t^{\beta}+n\check{\lambda}\,,t^{\beta,0}:=t^{\beta}\,,Q^{\beta,n}:=\exp(2\pi\textnormal{i}t^{\beta,n})\,,Q^{\beta}:=Q^{\beta,0}$
as before, and the first surprise is that the primed variables are independent
of $n$ since:

$Q^{\prime\beta,n}:=\exp(2\pi\textnormal{i}t^{\beta,n}/\check{\lambda})=\exp(2\pi\textnormal{i}t^{\beta}/\check{\lambda})\cdot\exp(2\pi\textnormal{i}n)=Q^{\prime\beta}\,,\quad\forall
n\in\mathbb{Z}\,,$

we define

$Z_{np,sg}(\lambda,t):=\prod_{n\in\mathbb{Z}}\prod_{\beta>0}\prod_{m=0}^{\infty}\left(\exp\left(-\frac{1}{2\pi
i}\textrm{Li}_{2}(Q^{\prime\beta}q^{\prime
m})\right)\cdot(1-Q^{\prime\beta}q^{\prime
m})^{-\frac{1}{\check{\lambda}}\left(t^{\beta,n}+m\right)}\,\right)^{\Omega_{\beta,n}}\,,$
(4.28)

and continue to show that it is of the desired form by first writing:

$Z_{np,sg}(\lambda,t):=\prod_{\beta>0}\prod_{m=0}^{\infty}\left(\exp\left(-\frac{1}{2\pi
i}\sum_{n\in\mathbb{Z}}\Omega_{\beta,n}\,\textrm{Li}_{2}(Q^{\prime\beta}q^{\prime
m})\right)\cdot(1-Q^{\prime\beta}q^{\prime
m})^{-\frac{1}{\check{\lambda}}\sum_{n\in\mathbb{Z}}\Omega_{\beta,n}\left(t^{\beta,n}+m\right)}\,\right)\,,$
(4.29)

noting that the argument of the dilogartithm does not depend on $n$, the
prefactor becomes $\sum_{n\in\mathbb{Z}}\Omega_{\beta,n}$. This sum is equal
equal to 4.4

$\sum_{n\in\mathbb{Z}}\Omega_{\beta,n}=[\textrm{GV}]_{\beta,0}\,.$ (4.30)

We now consider the exponent of the second factor:

$\displaystyle-\frac{1}{\check{\lambda}}\sum_{n\in\mathbb{Z}}\Omega_{\beta,n}\left(t^{\beta,n}+m\right)$
$\displaystyle=-\frac{1}{\check{\lambda}}\sum_{n\in\mathbb{Z}}\Omega_{\beta,n}\left(t^{\beta}+n\check{\lambda}+m\right)$
(4.31)
$\displaystyle=-\frac{1}{\check{\lambda}}[\textrm{GV}]_{\beta,0}\,\left(t^{\beta}+m\right)-\sum_{n\in\mathbb{Z}}n\,\Omega_{\beta,n}$
(4.32)
$\displaystyle=-\frac{1}{\check{\lambda}}[\textrm{GV}]_{\beta,0}\,\left(t^{\beta}+m\right)\,.$
(4.33)

the summand $\sum_{n\in\mathbb{Z}}n\,\Omega_{\beta,n}=0$ since
$\Omega_{\beta,n}=\Omega_{\beta,-n}\,\forall n\in\mathbb{Z}$. These
cancellations yield the desired form:

$\displaystyle
Z_{top,sg}(\lambda,t):=\prod_{\beta>0}\exp\left(-\frac{[\textrm{GV}]_{\beta,0}}{2\pi
i}\textrm{Li}_{2}(Q^{\prime\beta}q^{\prime
m})\right)\cdot(1-Q^{\prime\beta}q^{\prime
m})^{-\frac{[\textrm{GV}]_{\beta,0}}{\check{\lambda}}\left(t^{\beta}+m\right)}\,.$
(4.34)

∎

We pause to comment on this result, it seems rather remarkable that, due to
some non-trivial cancellation only the genus zero GV invariants appear in the
non-perturbative corrections to the topological string partition function. We
will now proceed to show that these non-perturbative corrections can be cast
in a rather simple function which we introduce as a deformation of the genus
zero prepotential.

###  4.4 Deformation of the prepotential

Recall that the non-classical part of the prepotential of a Calabi-Yau
threefold has an expression in terms of GV invariants given by:

$\tilde{F}^{0}(t)=\sum_{\beta>0}[\textrm{GV}]_{\beta,0}\textrm{Li}_{3}(Q^{\beta})=\sum_{\beta>0}\sum_{k=1}^{\infty}\frac{[\textrm{GV}]_{\beta,0}}{k^{3}}Q^{k\beta}\,,\quad
Q^{\beta}=\exp(2\pi it^{\beta})\,.$ (4.35)

We introduce the following $\varepsilon-$deformation of the prepotential:

$\tilde{F}^{0}_{\textrm{def}}(\varepsilon,t):=\frac{1}{2}\sum_{\beta>0}\sum_{k=1}^{\infty}\frac{[\textrm{GV}]_{\beta,0}}{k^{2}}\frac{Q^{k\beta}}{\sin(k\varepsilon/2)}\,,$
(4.36)

this is chosen such that

$\lim_{\varepsilon\rightarrow
0}\varepsilon\,\tilde{F}^{0}_{\textrm{def}}(\varepsilon,t)=\tilde{F}^{0}(t)\,.$
(4.37)

For the case of the resolved conifold with only one curve class $\beta$ and a
nonvanishing GV invariant only at degree $1$, this deformed prepotential is:

$\tilde{F}^{0}_{\textrm{def}}(\varepsilon,t):=\frac{1}{2}\sum_{k=1}^{\infty}\frac{1}{k^{2}}\frac{Q^{k\beta}}{\sin(k\varepsilon/2)}=-F^{NS,con}(\varepsilon,t)\,,$
(4.38)

where $F^{NS,con}(\varepsilon,t)$ denotes the refined topological string in
the Nekrasov-Shatashvili limit in the normalization of [30]. Note that here
the deformation does not require a refined topological string origin and can
therefore be applied to both non-compact as well as compact Calabi-Yau
geometries. We can now express the non-perturbative corrections to the
topological string as follows:

######  Proposition 4.3.

The non-perturbative corrections to the topological string

$\displaystyle
Z_{top,sg}(\lambda,t):=\prod_{\beta>0}\prod_{m=0}^{\infty}\exp\left(-\frac{[\textrm{GV}]_{\beta,0}}{2\pi
i}\textrm{Li}_{2}(Q^{\prime\beta}q^{\prime
m})\right)\cdot(1-Q^{\prime\beta}q^{\prime
m})^{-\frac{[\textrm{GV}]_{\beta,0}}{\check{\lambda}}\left(t^{\beta}+m\right)}\,,$
(4.39)

can be expressed as

$Z_{top,sg}(\lambda,t)=\exp(F_{top,sg}(\lambda,t))\,,$ (4.40)

where

$F_{top,sg}(\lambda,t)=-\frac{1}{2\pi}\partial_{\lambda}\left(\lambda\,\tilde{F}^{0}_{\textrm{def}}\left(\frac{2\pi}{\check{\lambda}},\frac{t-1/2}{\check{\lambda}}\right)\right)\,,$
(4.41)

where

$\tilde{F}^{0}_{\textrm{def}}(\varepsilon,t):=\frac{1}{2}\sum_{k,\beta>0}\frac{[\textrm{GV}]_{\beta,0}}{k^{2}}\frac{Q^{k\beta}}{\sin(k\varepsilon/2)}\,.$
(4.42)

###### Proof.

From $Z_{top,sg}(\lambda,t)=\exp(F_{top,sg}(\lambda,t))$ and (4.39) we obtain

$\displaystyle
F_{top,sg}(\lambda,t)=-\sum_{\beta>0}\sum_{m=0}^{\infty}\frac{[\textrm{GV}]_{\beta,0}}{2\pi
i}\left(\textrm{Li}_{2}(Q^{\prime\beta}q^{\prime
m})+\frac{2\pi\textnormal{i}}{\check{\lambda}}\left(t^{\beta}+m\right)\,\log(1-Q^{\prime\beta}q^{\prime
m})\right)\,,$ (4.43)

which can be written as

$\displaystyle
F_{top,sg}(\lambda,t)=-\sum_{\beta>0}\sum_{m=0}^{\infty}\frac{[\textrm{GV}]_{\beta,0}}{2\pi
i}\partial_{\check{\lambda}}\left(\check{\lambda}\,\textrm{Li}_{2}(e^{2\pi\textnormal{i}(t^{\beta}+m)/\check{\lambda}})\right)\,,$
(4.44)

writing out the series expansion for the dilogarithm, we obtain

$\displaystyle
F_{top,sg}(\lambda,t)=-\sum_{\beta>0}\sum_{m=0}^{\infty}\sum_{k=1}^{\infty}\frac{[\textrm{GV}]_{\beta,0}}{2\pi
i}\partial_{\check{\lambda}}\left(\check{\lambda}\,\frac{e^{2\pi\textnormal{i}(t^{\beta}+m)k/\check{\lambda}}}{k^{2}}\right)\,,$
(4.45)

summing over $m$ using the geometric series gives

$\displaystyle F_{top,sg}(\lambda,t)$
$\displaystyle=-\partial_{\check{\lambda}}\left(\check{\lambda}\sum_{\beta>0}\sum_{k=1}^{\infty}\frac{[\textrm{GV}]_{\beta,0}}{2\pi
i}\,\frac{e^{2\pi\textnormal{i}t^{\beta}k/\check{\lambda}}}{k^{2}(1-e^{2\pi\textnormal{i}k/\check{\lambda}})}\right)\,$
(4.46)
$\displaystyle=\partial_{\check{\lambda}}\left(\check{\lambda}\sum_{\beta>0}\sum_{k=1}^{\infty}\frac{[\textrm{GV}]_{\beta,0}}{2\pi
i}\,\frac{e^{2\pi\textnormal{i}t^{\beta}k/\check{\lambda}}e^{-\pi\textnormal{i}k/\check{\lambda}}}{k^{2}(e^{\pi\textnormal{i}k/\check{\lambda}}-e^{-\pi\textnormal{i}k/\check{\lambda}})}\right)\,$
(4.47)
$\displaystyle=-\frac{1}{2\pi}\partial_{\check{\lambda}}\left(\check{\lambda}\frac{1}{2}\sum_{\beta>0}\sum_{k=1}^{\infty}\frac{[\textrm{GV}]_{\beta,0}}{2\pi
i}\,\frac{e^{2\pi\textnormal{i}(t^{\beta}-1/2)k/\check{\lambda}}}{k^{2}\,\sin(\pi
k/\check{\lambda})}\right)\,$ (4.48)
$\displaystyle=-\frac{1}{2\pi}\partial_{\lambda}\left(\lambda\,\tilde{F}^{0}_{\textrm{def}}\left(\frac{2\pi}{\check{\lambda}},\frac{t-1/2}{\check{\lambda}}\right)\right)\,.$
(4.49)

∎

###  4.5 Non-perturbative topological string partition function

We can now include the contributions from constant maps to the topological
string free energies and use the analysis of sec. 3 to obtain the following:

######  Proposition 4.4.

The non-perturbative partition function of topological string theory on any CY
threefold, up to the contributions from constant maps at genus $0,1$, can be
expressed as:

$Z_{\textrm{top},np}(\lambda,t)=Z_{c,np}(\lambda)\prod_{n\in\mathbb{Z}}\prod_{\beta>0}\left(Z^{\textrm{con}}_{np}(\lambda,t^{\beta}+n\,\check{\lambda})\right)^{\Omega_{\beta,n}}\,,$
(4.50)

where

$Z^{con}_{np}(\lambda,t)=\prod_{m=0}^{\infty}(1-Q\,q^{m+1})^{m+1}\cdot\exp\left(-\frac{1}{2\pi
i}\textrm{Li}_{2}(Q^{\prime}q^{\prime m})\right)\cdot(1-Q^{\prime}q^{\prime
m})^{-\frac{1}{\check{\lambda}}\left(t+m\right)}\,,$

$Z^{con}_{c,np}(\lambda)=\left(\prod_{m=1}^{\infty}(1-q^{m})^{m}\cdot\exp\left(-\frac{1}{2\pi
i}\textrm{Li}_{2}(q^{\prime m})\right)\cdot(1-q^{\prime
m})^{-\frac{m}{\check{\lambda}}}\right)^{-\chi(X)/2}$

###### Proof.

This follows from the definition of the free energies by exponentiation as
well as using the product form 3.1 and the analysis of sec. 3.2.

∎

##  5 Resurgence

###  5.1 Asymptotic series

We now wish to interpret the non-perturbative corrections to the topological
string partition function as products over Stokes factors as was done for the
resolved conifold in [21]. To do this we will use the form of the free energy
of topological strings as a sum of free energies of resolved conifolds
obtained in 4.1 where

$\tilde{F}_{GW}(\lambda,t)=\sum_{g=0}^{\infty}\sum_{\beta>0}\lambda^{2g-2}[GW]_{g,\beta}Q^{\beta}\,,$
(5.1)

was written as:

$\displaystyle\tilde{F}_{GW}(\lambda,t)$
$\displaystyle=\sum_{n\in\mathbb{Z}}\sum_{\beta>0}\Omega_{\beta,n}\left(\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(Q^{\beta,n})+\sum_{g=1}^{\infty}\lambda^{2g-2}\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\mathrm{Li}_{3-2g}(Q^{\beta,n})\,\right)\,,$
(5.2)

where
$t^{\beta,n}:=t^{\beta}+n\check{\lambda}\,,Q^{\beta,n}:=\exp(2\pi\textnormal{i}t^{\beta,n})$
and $\Omega_{\beta,n}$ are the integers defined in (4.1). Each summand in
(5.2) can be treated individually as an asymptotic series in $\lambda$ with a
fixed parameter $t^{\beta}+n\check{\lambda}$. However to give an
interpretation of the non-perturbative corrections to the free energy in terms
of Stokes factors we would like to apply resurgence techniques to a single
asymptotic series in $\lambda$. We will therefore first use the difference
equation of [47]:

$\widetilde{F}^{con}\left(\lambda,t+\check{\lambda}\right)+\widetilde{F}^{con}\left(\lambda,t-\check{\lambda}\right)-2\widetilde{F}^{con}\left(\lambda,t\right)=-\textrm{Li}_{1}(Q)\,,\quad\check{\lambda}=\frac{\lambda}{2\pi}\,.\\\
$ (5.3)

to express the free energies whose arguments are shifted by $\check{\lambda}$
in terms of unshifted ones. We obtain the following:

######  Proposition 5.1.

The Gromov-Witten generating function of any CY threefold

$\tilde{F}_{GW}(\lambda,t)=\sum_{g=0}^{\infty}\sum_{\beta>0}\lambda^{2g-2}[GW]_{g,\beta}Q^{\beta}\,,$
(5.4)

can be written as:

$\displaystyle\tilde{F}_{GW}(\lambda,t)=\tilde{F}_{asymp}(\lambda,t)+F_{\textrm{rest}}(\lambda,t)\,,$
(5.5)

where

$\displaystyle\tilde{F}_{asymp}(\lambda,t)$
$\displaystyle=\sum_{\beta>0}[GV]_{\beta,0}\left(\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(Q^{\beta})+\sum_{g=1}^{\infty}\lambda^{2g-2}\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\mathrm{Li}_{3-2g}(Q^{\beta})\,\right)\,$
(5.6) $\displaystyle F_{\textrm{rest}}(\lambda,t)$
$\displaystyle=\sum_{\beta>0}\sum_{n=1}^{\infty}\Omega_{\beta,n}\sum_{k=-(n-1)}^{n-1}(n-|k|)\log(1-Q^{\beta}q^{k})\,.$
(5.7)

###### Proof.

To prove this proposition we first re-write the difference equation (5.3) as
two equations:

$\displaystyle\widetilde{F}^{con}\left(\lambda,t+\check{\lambda}\right)-\widetilde{F}^{con}\left(\lambda,t\right)$
$\displaystyle=f(\lambda,t+\check{\lambda})\,$ (5.8) $\displaystyle
f(\lambda,t+\check{\lambda})-f(\lambda,t)$ $\displaystyle=g(t)\,,\quad
g(t)=\log(1-Q)\,.$ (5.9)

By repeated use of these equations we obtain:

$\displaystyle\widetilde{F}^{con}\left(\lambda,t+n\check{\lambda}\right)$
$\displaystyle=\widetilde{F}^{con}\left(\lambda,t\right)+\sum_{k=1}^{n}f(\lambda,t+k\check{\lambda})\,,$
(5.10)
$\displaystyle\widetilde{F}^{con}\left(\lambda,t-n\check{\lambda}\right)$
$\displaystyle=\widetilde{F}^{con}\left(\lambda,t\right)-\sum_{k=0}^{n-1}f(\lambda,t-k\check{\lambda})\,,$
(5.11) $\displaystyle f\left(\lambda,t+k\check{\lambda}\right)$
$\displaystyle=f\left(\lambda,t\right)+\sum_{j=0}^{k-1}g(t+j\check{\lambda})\,,$
(5.12) $\displaystyle f\left(\lambda,t-k\check{\lambda}\right)$
$\displaystyle=f\left(\lambda,t\right)-\sum_{j=1}^{k}g(t-j\check{\lambda})\,.$
(5.13)

We now use 4.1

$\tilde{F}_{GW}(\lambda,t)=\sum_{n\in\mathbb{Z}}\sum_{\beta>0}\Omega_{\beta,n}\widetilde{F}^{con}(\lambda,t^{\beta}+n\check{\lambda})$
(5.14)

and $\Omega_{\beta,n}=\Omega_{\beta,-n}$ to write this as

$\tilde{F}_{GW}(\lambda,t)=\sum_{\beta>0}\Omega_{\beta,0}\widetilde{F}^{con}(\lambda,t^{\beta})+\sum_{n=1}^{\infty}\sum_{\beta>0}\Omega_{\beta,n}\left(\widetilde{F}^{con}(\lambda,t^{\beta}+n\check{\lambda})+\widetilde{F}^{con}(\lambda,t^{\beta}-n\check{\lambda})\right)\,,$
(5.15)

using 5.10 and after a few steps of algebra this becomes

$\tilde{F}_{GW}(\lambda,t)=\sum_{n\in\mathbb{Z}}\sum_{\beta>0}\Omega_{\beta,n}\widetilde{F}^{con}(\lambda,t^{\beta})+\sum_{\beta>0}\sum_{n=1}^{\infty}\Omega_{\beta,n}\sum_{k=-(n-1)}^{n-1}(n-|k|)\log(1-Q^{\beta}q^{k})\,,$
(5.16)

using $\sum_{n\in\mathbb{Z}}\Omega_{\beta,n}=[GV]_{\beta,0}$ gives then the
desired result.

∎

###  5.2 Stokes factors

For

$\displaystyle\tilde{F}_{asymp}(\lambda,t)$
$\displaystyle=\sum_{\beta>0}[GV]_{\beta,0}\left(\frac{1}{\lambda^{2}}\mathrm{Li}_{3}(Q^{\beta})+\sum_{g=1}^{\infty}\lambda^{2g-2}\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\mathrm{Li}_{3-2g}(Q^{\beta})\,\right)\,$
(5.17)
$\displaystyle=\frac{1}{\lambda^{2}}\widetilde{F}^{0}(t)+\widetilde{F}^{1}(t)+\Phi(\check{\lambda},t)\,.$
(5.18)

we can now use the Borel transform of the resolved conifold found in [21],
reviewed in 3.1, to deduce that the Borel transform of
$\Phi(\check{\lambda},t)$

$G(\xi,t)=\sum_{\beta>0}\frac{[GV]_{\beta,0}}{(2\pi)^{2}}\sum_{m\in\mathbb{Z}\setminus\\{0\\}}\frac{1}{m^{3}}\frac{1}{2\xi}\frac{\partial}{\partial\xi}\bigg{(}\frac{\xi^{2}}{1-e^{-2\pi{\mathrm{i}}t^{\beta}+\xi/m}}-\frac{\xi^{2}}{1-e^{-2\pi{\mathrm{i}}t^{\beta}-\xi/m}}\bigg{)},$
(5.19)

this is a meromorphic function with poles at
$\xi=2\pi\textnormal{i}m(t^{\beta}+k)\,,k\in\mathbb{Z}\,m\in\mathbb{Z}\setminus\\{0\\}.$

Let now $\Gamma=H_{0}(X,\mathbb{Z})\oplus H_{2}(X,\mathbb{Z})$ and let the
constant $1$ be the generator of $H_{0}(X)$ and $\beta\in
H_{2}(X,\mathbb{Z})$, we denote the central charge of $\gamma\in\Gamma$ by
$Z(\gamma)$ , let $\gamma_{\beta,k}=(k,\beta)$ and its the central charge
$Z(\gamma_{\beta,k})=2\pi\textnormal{i}(t^{\beta}+k)$ and assume for the
moment that there are as many formal variables $t^{\beta}$ as there are curve
classes. We can then study the following stokes jumps, let therefore
$t^{\beta}\in\mathbb{C}-\mathbb{Z}$ and $k\in\mathbb{Z}$ let
$l_{\gamma}:=\mathbb{R}_{<0}\cdot Z(\gamma)$ and
$l_{\infty}:={\mathrm{i}}\mathbb{R}_{<0}$. Given any ray $\rho$ from $0$ to
$\infty$ different from $\\{\pm l_{\gamma}\\}_{\gamma\in\Gamma}\cup\\{\pm
l_{\infty}\\}$, and $\lambda$ in the half-plane $\mathbb{H}_{\rho}$ centered
at $\rho$, we define the Borel sum of $\widetilde{F}(\lambda,t)$ along $\rho$
as

$F_{\rho}(\lambda,t):=\
\frac{1}{\lambda^{2}}\widetilde{F}^{0}(t)+\widetilde{F}^{1}(t)+\int_{\rho}\mathrm{d}\xi\,e^{-\xi/\check{\lambda}}G(\xi,t)\,.$
(5.20)

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

$\displaystyle\phi_{\pm l_{\beta,k}}(\lambda,t)$
$\displaystyle:=F_{\pm\rho_{\beta,k+1}}(\lambda,t)-F_{\pm\rho_{\beta,k}}(\lambda,t)=\frac{[GV]_{\beta,0}}{2\pi{\mathrm{i}}}\partial_{\check{\lambda}}\Big{(}\check{\lambda}\,\mathrm{Li}_{2}\big{(}e^{\pm
2\pi{\mathrm{i}}(t^{\beta}+k)/\check{\lambda}}\big{)}\Big{)}\;,\,$ (5.21)
$\displaystyle=\frac{[GV]_{\beta,0}}{2\pi{\mathrm{i}}}\left(\mathrm{Li}_{2}\big{(}e^{\pm
2\pi{\mathrm{i}}(t^{\beta}+k)/\check{\lambda}}\big{)}\pm
2\pi{\mathrm{i}}\frac{(t+k)}{\check{\lambda}}\log\left(1-e^{\pm
2\pi{\mathrm{i}}(t^{\beta}+k)/\check{\lambda}}\right)\right)\;.$ (5.22)

######  Remark 5.2.

  * •

Equation (5.21) is the straightforward generalisation of the expressions
obtained in [21]. Via the correspondence of the resurgence and the Riemann-
Hilbert problem put forward by Bridgeland in [27, 28], discussed in [21,
Section 4]the prefactors $[GV]_{\beta,0}$ correspond to the DT invariants of
an underlying DT RH problem. The expected RH problem associated to the part of
the GW generating function which corresponds to the genus $0$ part in the GV
expansion has been addressed in [27, Section 6].

  * •

The form of the Stokes jumps (5.21) put forward in [21] was already found in a
slightly different guise in earlier work of Pasquetti and Schiappa[14], it was
furthermore identified in [30, Prop. 4.7] as the quantum dilogarithm function
with equal arguments and more recently this form was also expected to hold
more generally in topological string theory in [34].

###  5.3 Singularities in the Borel plane

The singularities of the Borel transform lie along the rays
$l_{\gamma}\,,\gamma\in\Gamma$, whose slope is obtained from the central
charges $Z(\gamma)$. In the general discussion above we have assumed that
there are as many formal variables $t^{\beta}$ as curve classes which allowed
the straightforward generalisation of [21]. From mirror symmetry however we
know that all $t^{\beta}$ should live in an $n=h^{1,1}(X)$ complex manifold,
which is the moduli space of complexified Kähler forms of $X$. Let therefore
$C_{a}\in H_{2}(X,\mathbb{Z}),a=1,\dots,n$ be a set of curve classes spanning
$H_{2}(X,\mathbb{Z})$, then in general all curve classes $\beta$ can be
written as $\beta=\sum_{a=1}^{n}d_{a}C_{a}$, where the $d$’s are the degrees
of the curve. The formal variables $t^{\beta}$ become
$t^{\beta}=\sum_{a=1}^{n}d_{a}t^{a}$ and the $t^{a}$ correspond to local
coordinates on the moduli space of complexified Kähler forms of the underlying
CY $X$. We can now see that this leads to regions in the Borel plane where
there are accumulation rays of singularities.

We first note that the singularities of the Borel transform (5.19) are at

$\xi=2\pi\textnormal{i}m(\sum_{a=1}^{n}d_{a}t^{a}+k)\,,k\in\mathbb{Z}\,,\quad
d_{a}\in\mathbb{N}\,,\quad a=1,\dots,n\,,$ (5.23)

as long as $[GV]_{\\{d_{a}\\},0}\neq 0$.

For the purpose of illustrating the loci of the singularities we focus on a
single curve class $C$ and its higher degrees given by $d\in\mathbb{N}$, the
singularities of the Borel transform coming from that curve are on the rays:

$\pm l_{d,k}:=\pm\left(-\mathbb{R}_{<0}2\pi\textnormal{i}(dt+k)\right)\,,\quad
d\in\mathbb{N},\quad k\in\mathbb{Z}\,,$ (5.24)

as long as $[GV]_{d,0}\neq 0$.

The slope of the line $l_{d,k}$ is given by:

$-\frac{\textrm{Re}(dt)+k}{\textrm{Im}dt}=-\frac{\textrm{Re}(t)}{\textrm{Im}t}-\frac{k/d}{\textrm{Im}t}\,,$
(5.25)

we observe thus, that in addition to the accumulation rays of singularities
$\pm l_{\infty}$ due to the growth in $k$, there are further accumulation rays
due to the increase in the degree around $\pm l_{1,0}$. Furthermore we can
expect generically rays of singularities along all $\pm
l_{1,0}+\mathbb{Q}\cdot\frac{1}{\textrm{Im}t}$. An illustration of the some of
the rays of singularities is give in fig. 1.

A distinguished set of singularities in the Borel plane thus corresponds to
the loci where only one $d_{a}=1$ and $m=\pm 1$ in (5.23). The corresponding
Stokes factor would then have the integer prefactor given by
$[GV]_{d_{a}=1,0}$. In the case of the quintic there is only one curve class
and the degree $d=1$ curve GV count is given by $2875$. This number was
recently seen numerically using the interplay of large order genus expansion
and resurgence in [10].

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

##  6 Discussion and Conclusions

In this work we have put forward an expression for the non-perturbative
topological string partition function on any Calabi-Yau threefold, compact or
non-compact, in the holomorphic limit which provides the asymptotic expansion
of topological strings encoding the enumerative GW as well as GV invariants of
the underlying CY threefold. A crucial step in obtaining this expression was
identifying the partition function associated to any CY manifold as a product
over basic building blocks which correspond to the constant map contributions
on the one hand while on the other hand the non-constant maps are captured by
resolved conifold partition functions with shifted arguments, raised to the
power of sheaf invariants. While the general convergence properties of the
full non-perturbative expression requires more knowledge of the sheaf
invariants which are obtained from all higher genus GV invariants, it was
found that the non-perturbative corrections are governed by only the genus 0
GV invariants, which via mirror symmetry are encoded in rational functions on
the complex structure moduli space of the mirror manifold and have thus good
convergence properties. It was furthermore shown that the non-perturbative
corrections could be entirely captured in terms of a single function which was
identified as an $\varepsilon$ deformation of the genus 0 prepotential. This
function corresponds in the case of the resolved conifold to the refined
topological string in the NS limit which was shown to capture the non-
perturbative corrections in that case [20, 25].

At the end it is perhaps possible to speculate along some future directions
and questions:

  * •

_Global properties_ The non-perturbative expression for the topological string
partition function in this work is obtained from the holomorphic limit which
contains the enumerative geometry information of the underlying CY threefold.
On the A-model side, this is the large radius region, on the mirror B-side
this is the vicinity of a point of maximal unipotent monodromy. It is known,
for instance from the computations using the topological vertex [55], that the
obtained expression does not give a good global function on the underlying
moduli space and it is in general difficult, if not impossible to analytically
continue. The route to obtain global expressions for the topological string
partition functions is using the holomorphic anomaly equations [2] together
with the recursive polynomial construction of the genus expansion put forward
in [3, 4] as was recently revisited in connection to non-perturbative
topological strings in [10]. In [34], the approach of making a transseries
ansatz and using the holomorphic anomaly as studied in [8, 9] as well as the
results of [56, 32, 10] were used to put forward an expectation for the
general form of the Stokes automorphisms acting on the topological string
partition function. The expected Stokes factors are of the form obtained from
the analytic results of [14, 21] as well as their generalizations in this
work, obtained without use of the holomorphic anomaly. One may expect that the
non-holomorphic topological string has an even richer non-perturbative
structure. In fact, in [57] the holomorphic anomaly equations were used
together with the polynomial structure of higher genus topological strings to
derive an Airy differential equation in the topological string coupling
governing a universal piece of the higher genus free energies for all CY
families, in addition to the perturbative piece captured by one solution of
the Airy differential equation, the second solution suggests corrections with
$e^{-1/\lambda^{2}}$ behaviour, this was explained by the effects of NS-5
branes in [58]. These non-perturbative effects seem to be completely missing
in the results of the current paper which is likely an artefact of starting
with the asymptotic series of topological strings in the limit leading to the
enumerative geometry content.

  * •

_Non-perturbative absence of the gap?_ Although the non-perturbative
expression of this work may not be the full non-holomorphic one, one may
however speculate at this stage, whether the _gap condition_ used in [6] and
which refers to a leading singularity in the topological string free energies
$\mathcal{F}^{g}(t)$ near a point in moduli space where finitely many physical
states become massless is an artefact of the perturbative expansion. Near the
conifold point of the resolved conifold, i. .e. $t\rightarrow 0$, it was
proven in [25] that the analytic continuation in $t$ order by order in
$\lambda$ gives a leading singular behavior of the expected form and that
moreover the constant term of this analytic continuation contained the
information of the constant map contribution. A surprise of the non-
perturbatively complete Borel sums put forward in [21] is that their limit
$t\rightarrow 0$ is well behaved and it moreover encodes the constant map
contributions, furthermore the combination of the non-constant and constant
map contributions leads to a partition function which is canonically
normalized as $t\rightarrow 0$. One may speculate if a similar phenomenon also
appears for topological strings in general. Hints that the moduli space to be
considered is a non-trivial combination of $\lambda$ and $t$ already appear in
the analysis of the natural dual, strongly coupled variables
$\frac{t}{\lambda},\frac{1}{\lambda}$. The expressions put forward in this
work in terms of the triple sine function have again good
$t^{\beta}\rightarrow 0$ limits. In general however the conifold loci
correspond to the vanishing of a linear combination of dual period which may
not be captured by the current expressions. Another speculation is whether the
topological string partition functions are normalized in a canonical way as
well or whether this was an accident of the resolved conifold, the general
case would require non-trivial relations between $\chi(X)$ the Euler character
of the CY and sums of its GV invariants.

  * •

_Quantum Picard-Fuchs equations?_ The difference equation of the asymptotic
expansion of the resolved conifold free energy of [24] was interpreted in [30]
as giving a closed expression of the quantum Picard-Fuchs equations which
refer to the differential equations annihilating the Voros symbols in the
exact WKB analysis of the quantum curves obtained from the mirrors of non-
compact CY. It was recently shown in [59] that the quantum Picard-Fuchs
equations considered by [60] in the context of quantum K-theory give, for the
resolved conifold, exactly the difference equations obtained in [24]. It would
be interesting to develop such a correspondence more generally, perhaps guided
by the expressions of the deformed prepotential put forward in this work.

  * •

_The Nekrasov-Shatshvili limit?_ The deformed prepotential introduced in this
work, which is capturing the non-perturbative corrections to the topological
string coincides with the Nekrasov-Shatashvili limit of the refined
topological string in the case of the resolved conifold. It is not entirely
clear whether such a relation can be expected more generally. What is clear
however, is that the piece coming from the deformed prepotential in the non-
perturbative expression is responsible for the cancellation of poles of the
topolgical string free energy in $\lambda$ and as such is playing the same
role as the NS limit in the topological string/spectral theory correspondence
of [16].

### Acknowledgements

I would like to thank Lotte Hollands, Arpan Saha, Ivan Tulli and Joerg
Teschner for collaboration on projects which led to this work as well as for
discussions and comments. I would like to thank Noah Dizep, Veronica Fantini,
Stavros Garoufalidis, Albrecht Klemm, Maxim Kontsevich, Andy Neitzke, Yan
Soibelman and Campbell Wheeler for discussions and comments. I would like to
furthermore thank the organizers of the Les Diablerets workshop on Quantum
Moduli in September 2023 and the organizers of the INI workshop on Number
Theory, Machine Learning and Quantum Black Holes in October 2023 for the
opportunity to announce the results of this work. This work is supported
through the DFG Emmy Noether grant AL 1407/2-1.

##  Appendix A Multiple sine functions

We introduce several special functions which are used throughout the paper
following the notation of [61]. Suppose
$\omega_{1}\,,\dots,\omega_{r}\in\mathbb{C}$ all lie on the same side of some
straight line through the origin, then the multiple zeta function is defined
by the series

$\zeta_{r}(s,z|\underline{\omega})=\sum_{n_{1},\dots,n_{r}=0}^{\infty}\frac{1}{(n_{1}\omega_{1}+\dots+n_{r}\omega_{r}+z)^{s}}\,,$
(A.1)

for $z\in\mathbb{C}$ and $\textrm{Re}(s)>r$, where the exponential is rendered
one-valued. This series is holomorphic in the domain $\\{\textrm{Re}(s)>r\\}$
and is analytically continued to $s\in\mathbb{C}$.

Since it is holomorphic at $s=0$ we can define the multiple gamma function by

$\Gamma_{r}(z|\underline{\omega})=\exp\left(\frac{\partial}{\partial
s}\zeta_{r}(s,z|\underline{\omega})|_{s=0}\right),$ (A.2)

where $|\underline{\omega}|=\omega_{1}+\dots+\omega_{r}\,,$ and the multiple
sine function is given by

$\sin_{r}(z|\underline{\omega})=\Gamma_{r}(z|\underline{\omega})^{-1}\,\Gamma_{r}(|\underline{\omega}|-z|\underline{\omega})^{(-1)^{r}}\,.$
(A.3)

This satisfies the functional equations:

$\displaystyle\sin_{r}(z+\omega_{j}|\underline{\omega})=\sin_{r-1}(z|\underline{\omega}^{-}(j))^{-1}\sin_{r}(z|\underline{\omega})\,,$
(A.4)
$\displaystyle\sin_{r}(z|\underline{\omega})\,\sin_{r}(|\underline{\omega}|-z|\underline{\omega})^{(-1)^{r}}=1\,,$
(A.5)

where

$\underline{\omega}^{-}(j)=(\omega_{1},\dots,\overset{\vee}{\omega_{j}},\dots,\omega_{r})\,,$

and $\overset{\vee}{\omega_{j}}$ means omitting $\omega_{j}$.

For $z\in\mathbb{C}$, $\underline{\omega}=(\omega_{1},\dots,\omega_{r})$,
$\omega_{j}\in\mathbb{C}\setminus\\{0\\}$ we define the multiple Bernoulli
polynomials $B_{r,n}(z,\underline{\omega})$ in terms of the generating
function

$\mathcal{B}_{r}(z,x|\,\omega_{1},\,\dots,\omega_{r})=\frac{x^{r}e^{zx}}{\prod_{j=1}^{r}(e^{\omega_{j}x}-1)}=\sum_{n=0}^{\infty}B_{r,n}(z|\underline{\omega})\frac{x^{n}}{n!}\,.$
(A.6)

The multiple sine functions are defined using the Barnes multiple Gamma
functions [62]. For a variable $z\in\mathbb{C}$ and parameters
$\omega_{1},\ldots,\omega_{r}\in\mathbb{C}^{*}$ these are defined by

$\sin_{r}(z\,|\,\omega_{1},\dots,\omega_{r}):=\Gamma_{r}(z\,|\,\omega_{1},\dots,\omega_{r})^{-1}\cdot\Gamma_{r}\left(\sum_{i=1}^{r}\omega_{i}-z\,|\,\omega_{1},\dots,\omega_{r}\right)^{(-1)^{r}}\,.$
(A.7)

For further definitions see e.g. [28, 63] and references therein.

Another special function is (related to) Fadeev’s quantum dilogarithm

$\mathcal{S}_{2}(z\,|\,\omega_{1},\omega_{2}):=\exp\left(-\frac{\pi{\mathrm{i}}}{2}\cdot
B_{2,2}(z|\,\omega_{1},\omega_{2})\right)\cdot\sin_{2}(z\,|\,\omega_{1},\omega_{2}),$
(A.8)

which we have introduced in terms of the double sine function as in [28].

The final special function is defined by

$\mathcal{S}_{3}(z\,|\,\omega_{1},\omega_{2}):=\exp\left(\frac{\pi{\mathrm{i}}}{6}\cdot
B_{3,3}(z+\omega_{1}\,|\,\omega_{1},\omega_{1},\omega_{2})\right)\cdot\sin_{3}(z+\omega_{1}\,|\,\omega_{1},\omega_{2},\omega_{3}),$
(A.9)

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