Exponential Networks, WKB and Topological String
We propose a connection between 3d-5d exponential networks and exact WKB for difference equations associated to five-dimensional Seiberg-Witten curves, or equivalently, to quantum mirror curves to toric Calabi-Yau threefolds : the singularities in the Borel planes of local solutions to such differen...
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| description | We propose a connection between 3d-5d exponential networks and exact WKB for difference equations associated to five-dimensional Seiberg-Witten curves, or equivalently, to quantum mirror curves to toric Calabi-Yau threefolds : the singularities in the Borel planes of local solutions to such difference equations correspond to central charges of 3d-5d BPS KK-modes. It follows that there should be distinguished local solutions of the difference equation in each domain of the complement of the exponential network, and these solutions jump at the walls of the network. We verify and explore this picture in two simple examples of 3d-5d systems, corresponding to taking the toric Calabi-Yau to be either ℂ³ or the resolved conifold. We provide the full list of local solutions in each sector of the Borel plane and in each domain of the complement of the exponential network, and find that local solutions in disconnected domains correspond to non-perturbative open topological string amplitudes on with insertions of branes at different positions of the toric diagram. We also study the Borel summation of the closed refined topological string free energy on and the corresponding non-perturbative effects, finding that central charges of 5d BPS KK-modes are related to the singularities in the Borel plane.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 064, 44 pages
Exponential Networks, WKB and Topological String
Alba GRASSI ab, Qianyu HAO c and Andrew NEITZKE d
a) Section de Mathématiques, Université de Genève, 1211 Genève 4, Switzerland
E-mail: alba.grassi@cern.ch
b) Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland
E-mail: alba.grassi@cern.ch
c) Department of Physics, University of Texas at Austin,
2515 Speedway, C1600, Austin, TX 78712-1992, USA
E-mail: qhao@utexas.edu
d) Department of Mathematics, Yale University,
PO Box 208283, New Haven, CT 06520-8283, USA
E-mail: andrew.neitzke@yale.edu
Received March 07, 2023, in final form August 23, 2023; Published online September 13, 2023
https://doi.org/10.3842/SIGMA.2023.064
Abstract. We propose a connection between 3d-5d exponential networks and exact WKB
for difference equations associated to five dimensional Seiberg–Witten curves, or equivalently,
to quantum mirror curves to toric Calabi–Yau threefolds X: the singularities in the Borel
planes of local solutions to such difference equations correspond to central charges of 3d-5d
BPS KK-modes. It follows that there should be distinguished local solutions of the difference
equation in each domain of the complement of the exponential network, and these solutions
jump at the walls of the network. We verify and explore this picture in two simple examples
of 3d-5d systems, corresponding to taking the toric Calabi–Yau X to be either C3 or the
resolved conifold. We provide the full list of local solutions in each sector of the Borel
plane and in each domain of the complement of the exponential network, and find that local
solutions in disconnected domains correspond to non-perturbative open topological string
amplitudes onX with insertions of branes at different positions of the toric diagram. We also
study the Borel summation of the closed refined topological string free energy on X and the
corresponding non-perturbative effects, finding that central charges of 5d BPS KK-modes
are related to the singularities in the Borel plane.
Key words: difference equation; Stokes phenomenon; BPS states; topological string; expo-
nential network
2020 Mathematics Subject Classification: 39A70; 40G10; 81T30; 81T60
1 Introduction
This paper is motivated by the necessity to deepen our understanding of exact WKB methods
for difference equations, non-perturbative effects in (refined) open and closed topological string
amplitudes, and their relation to 5d/3d-5d BPS states and exponential networks.
We begin with the open string. In recent years, there has been considerable progress in the
study of four-dimensional N = 2 supersymmetric gauge theories and their connection to linear
differential equations. One of the basic geometric objects in the story is the spectral network,
introduced in the context of 4d N = 2 theories in [44, 45, 46, 47, 48]. The spectral network
captures the BPS spectrum of a surface defect in the 4d N = 2 theory on the one hand, while on
the other hand it is identified with the Stokes graph of the corresponding differential equation.
Various parts of this structure are modified when we lift from four-dimensional theories to
five-dimensional ones (compactified on S1). In five-dimensional theories there is an analog of
mailto:alba.grassi@cern.ch
mailto:alba.grassi@cern.ch
mailto:qhao@utexas.edu
mailto:andrew.neitzke@yale.edu
https://doi.org/10.3842/SIGMA.2023.064
2 A. Grassi, Q. Hao and A. Neitzke
the spectral network, namely the exponential network introduced in [9, 38], which captures the
BPS spectrum of the 5d theory coupled to a 3d defect. (See also [10, 11, 25, 81] for some other
developments in this direction and [14, 16, 31] for connections between q-Painlevé equations,
exponential networks and 5d BPS quivers.) Moreover, it is expected that these 3d-5d systems
should be related to q-difference equations, replacing the differential equations which appeared in
the 4d case. In particular, it was pointed out in [1], built upon [87, 91], that the WKB expansion
of a certain class of difference equations, known as higher genus quantum mirror curves, is closely
related to the Nekrasov–Shatashvili (NS) limit of refined topological strings.12 Since topological
strings can compute observables of 5d field theories (obtained via compactification of M-theory
on a non-compact Calabi–Yau threefold) this leads to the expectation that some observables of
5d field theories should obey q-difference equations.
The five-dimensional case has one important new feature: the WKB expansion for the differ-
ence equations has to be augmented by a new type of non-perturbative effects [26, 54, 84], which
vanish when we implement the four-dimensional limit [74, 75] leading to differential equations.
See also [42, 58, 63, 97] for other applications to the quantization conditions of relativistic inte-
grable systems, and [17] for connections between 5d quantum mirror curves and tau functions
of q-Painlevé equations [15].
In this paper, we clarify the relations between five-dimensional gauge theories, exponential
networks, and difference equations:
� In Section 2, we point out a direct connection between exponential networks and exact
WKB-type solutions of difference equations.3 We study the singularities in the Borel plane
of local solutions to difference equations associated with Seiberg–Witten curves of 3d-5d
systems, or equivalently, quantum mirror curves to toric Calabi–Yau (CY) threefolds X.
We focus on the first singularity in any given direction, and propose that these singularities
correspond to BPS particles living on the S1 compactification of the 3d defect, with the
positions of the singularities matching the central charges of the BPS particles. It then
follows from the definition of exponential network that there should be distinguished local
solutions of the difference equation in each domain of the complement of the exponential
network, and these solutions should jump at the walls of the exponential network.
All this is closely parallel to the story for more conventional 2d-4d systems [45, 47], where
it was conjectured in [53] that the positions of Borel plane singularities for local solutions
of the differential equations are the central charges of BPS particles lying on the surface
defect. The 3d-5d setting however brings a few new features; in particular, in addition to
the usual walls which carry labels ij, there are new walls carrying labels ii, corresponding
to BPS particles charged under the flavor symmetry.
� In Sections 3 and 4, we study this proposal in two specific examples of 3d-5d systems. These
examples correspond to takingX to be either C3 or the resolved conifold. In these examples
the desired local solutions can be described explicitly in terms of quantum dilogarithms,
1We should however stress that such WKB-like methods based on the refined topological strings involve a second
series expansion in Q = e−t where, schematically, t denotes the Kähler parameters of the underlying Calabi–Yau
threefold. More precisely, at each order in ℏ, we have a series expansion in Q which converges only in the large
radius region of the moduli space. Hence this method does not allow us to have access to a standard WKB
expansion. This is very similar to what happens in four-dimensional theories: see discussion in [53, pp. 37–38]
and references there.
2For difference equations associated with genus zero mirror curves, the WKB expansion of local solutions is
usually expressed by using unrefined open topological strings [2, 72], also known as the Gopakumar–Vafa (GV)
limit of refined topological strings. This might seem like a puzzle since above we stated that it is the NS limit
which is related to difference equations. The resolution is that, in the genus zero case, the refined open topological
string partition functions in the NS and GV limits are related in a very simple way; see for instance [23, 35, 69, 76].
3Similar ideas have been explored by Fabrizio Del Monte and Pietro Longhi; we thank them for a discussion
about this.
Exponential Networks, WKB and Topological String 3
for which the Borel plane structure is completely known thanks to the recent work [49].
Using the techniques and results of [49] we show that the proposed picture indeed holds.
(We expect this structure holds also for other local CY manifolds with higher genus mirror
curves. We comment more on this aspect in Section 5.)
In these two examples, we give closed form expressions for the Borel transform and Borel
summation of the local solutions in each domain of the complement of the exponential net-
work. In the C3 case the various solutions are described in Section 3.5 and summarized in
Figure 4, while the example of the resolved conifold is discussed in Section 4.4. In addition,
we relate the resulting expressions to the open topological string partition function: local
solutions in different domains of the exponential network correspond to open topological
string partition functions with brane insertion at different positions (e.g., on the internal
or on the external leg).
One interesting feature which appears in the C3 case is that, for generic phase ϑ, the
complement of the spectral network is actually simply connected; thus the jump of the
local solution which occurs at a wall can also be obtained by analytic continuation of the
solution along a path. We discuss this point in more detail in Section 3.6.
In the conifold case, for each domain we give an analytic computation of the non-perturba-
tive effects and compare with some available results in the topological string literature [84]:
see Section 4.4.
Now let us discuss the closed string. As in the open string case, our analysis is organized
around the theme of Borel plane singularities and their relation to BPS particles — now for the
bulk theory rather than the theory with a defect. Indeed, in 4d N = 2 theories the positions of
Borel plane singularities for quantum periods are central charges of bulk BPS particles [52, 53].
The closed topological string amplitudes are analogues of quantum periods, now associated with
difference equations rather than differential equations [1, 87]; with this in mind, we expect that
the singularities in the Borel plane of the closed topological string amplitudes should be related
to the central charges of 5d BPS KK modes. This is the prediction which we investigate.
For the unrefined limit of the resolved conifold, several studies in this direction have already
been performed, for example in [62, 77, 80, 94]. Another interesting approach based on the
Mellin–Barnes representation of the spectral zeta function can be found in [57]. The closed
topological string partition function on the resolved conifold in the ϵ1 + ϵ2 = 0 phase was also
discussed in [19] as a solution to a certain Riemann–Hilbert problem, and in [17] from the point
of view of q-Painlevé equations. See also [5, 8] for other interesting related work.
In this paper, we adopt an analytic approach and go beyond the unrefined case. We can
summarise our results for the closed sector as follows:
� For the C3 example our approach simply translates to the study of the resurgence properties
of the McMahon function: see Section 3.7. It is nice to see that even in this toy model,
as we go away from the imaginary axis, we have non-perturbative effects which are in fact
encoded in the NS limit of the refined McMahon function, very much in line with what
was found originally in the context of ABJM theory [59].
� For the resolved conifold we compute analytically the Borel transform and Borel sum-
mation, both in the unrefined and refined cases. We give a detailed description of the
non-perturbative effects in each sector, and compare with some previous results in the
literature. In particular, for the unrefined case we can relate our picture to [62], where
numerical studies have been performed, while for the refined case our results are new. See
Section 4.5.
� We find that there is a correspondence between the singularities in the Borel transform of
the refined closed topological string free energy and the central charges of 5d BPS KK-
4 A. Grassi, Q. Hao and A. Neitzke
modes. For the GV (ϵ1 = −ϵ2) and the NS (ϵ2 = 0) phases of the Ω background, the
singularities lie precisely at the central charges of 5d BPS KK-modes. However, for more
generic phases of the Ω background (ϵ1 = αϵ2) the Borel plane has an additional series
of poles: see (4.19) and Section 4.5. When α → 0 these extra poles go to infinity, while
when α→ −1 they merge with the original series of poles. This behavior suggests that in
the refined topological string a BPS particle of central charge Z gives rise to two distinct
nonperturbative effects, of sizes e−|2πRZ/ϵ1| and e−|2πRZ/ϵ2|.
We conclude this introduction by listing some open problems and future directions:
� In the case of 2d-4d systems, the theory of spectral networks and exact WKB are use-
ful tools for studying the hyperkähler geometry of moduli spaces of solutions of Hitchin
equations [44, 48]. The basic reason why exact WKB has something to do with Hitchin
equations is that solutions of Hitchin equations can be identified with Higgs bundles and
also with differential equations.
It is natural to imagine that this theory can be extended to 3d-5d systems. In this ex-
tension, Higgs bundles would be replaced by group-valued Higgs bundles [39], differential
equations would be replaced by difference equations (q-difference modules), and solutions of
Hitchin equations would be replaced by doubly periodic monopoles [22, 24]. The correspon-
dence between q-difference modules and doubly periodic monopoles is carefully developed
in [88].
The results in this paper can be regarded as a step in this direction. It would be very
interesting to go further and give a twistorial construction of moduli spaces of periodic
monopoles in terms of central charge data and BPS degeneracies, in parallel to [44].
� The local solutions which we consider are closely related to objects discussed in the
literature on boundary conditions and holomorphic blocks in 3d N = 2 theories, e.g.,
[12, 22, 34, 99]. There is also closely related work in 3d N = 4 theories such as [20, 21].
We do not develop this point of view much in the current paper; however, it seems likely
that in future developments this will be an important perspective.
� In a single 5d theory there are many different possible 3d defects which can be added. For
example, the defect we consider in the C3 theory sits naturally in a family parameterized
by f ∈ Z, with corresponding Seiberg–Witten curves
XY f − Y + 1 = 0.
The quantity f is often called the “framing” following [2, 3]. The framing we are using
in this paper is f = 0; for this defect there is only a single vacuum, and thus a unique
local solution to the difference equation up to the flavor ambiguities, which substantially
simplifies the analysis. It would be interesting to extend our considerations to more general
framings. Some of the relevant exponential networks have already been described — e.g.,
see [9] for the case f = −1.
� Spectral/exponential networks drawn on a surface C can be used to study even BPS
particles whose central charges do not vary along C. These particles are detected indirectly:
the network Wϑ depends on a phase ϑ = arg ℏ, and when there is a BPS particle whose
central charge has arg (−Z) = ϑ, the network Wϑ degenerates. From these degenerations
one can try to read out the BPS spectrum, via wall-crossing methods described in [9, 47].
One important instance of this is the use of spectral/exponential networks attached to
coupled 2d-4d or 3d-5d systems, to study BPS particles in the 4d or 5d bulk which carry
electromagnetic charge.
Exponential Networks, WKB and Topological String 5
In the cases we consider here, there are BPS particles in the bulk (corresponding in the
Type IIA language to D0-branes or D0-D2 bound states), and there are corresponding de-
generations of the exponential network. It would be very interesting to understand whether
it is possible to compute the bulk BPS degeneracies directly by wall-crossing methods from
these degenerations. For X = C3 and the framing f = −1, such a computation was given
in [9].
� As we mentioned above, it would be desirable to understand in detail the relation between
exponential networks and exact WKB in more complicated examples, involving higher-
genus mirror curves. One of the main technical obstacles here is to develop an efficient
way to compute the WKB expansion. One can try to do this directly by writing a WKB
ansatz like one writes for differential equations, and then solving a Riccati-type equation
order by order in ℏ, as discussed, e.g., in [36, 73, 100]; applying this method to the simple
cases we consider in this paper indeed gives the correct series. Alternatively, we could
use the refined holomorphic anomaly equation in the NS limit [27, 67, 78], but then we
still have to deal with the quantum mirror map, which at present we can compute only
in a large radius expansion; see footnote 1. We comment more on this open direction in
Section 5.
Related work
While this paper was in preparation, the independent work [7] appeared; among other things,
this paper gives a clear and careful treatment of the Borel summation for the closed topological
string in the conifold in the GV phase, using substantially the same techniques we used. The
results in Section 4.5.2 match with [7]. We understand that more recently Murad Alim, Lotte
Hollands and Ivan Tulli have also studied the NS phase and independently obtained, among
other things, results overlapping with Section 4.5.1 [6]. We thank them for discussions about
this.
2 Stokes phenomena and BPS particles
2.1 Stokes phenomena in 2d-4d systems
Studying the large-order behavior of perturbation theory in quantum mechanics, quantum field
theory, or string theory has often given insight into the nature of discrete objects in the theory
such as instantons, particles, or branes. (For one remarkable example, see [95] where it was
argued that D-branes are responsible for effects of order e−1/gs in string perturbation theory.
We refer to [82] for a review and list of references.)
An interesting class of examples comes from supersymmetric coupled 2d-4d systems, consist-
ing of a 4d N = 2 theory and a BPS surface defect preserving 2d N = (2, 2) supersymmetry. In
this case one has observables ψi in the defect theory, interpreted as boundary states associated
to vacua, with the index i labeling the choice of vacuum. The ψi depend on a parameter ℏ ∈ C×
which can be interpreted as an Ω background parameter in the NS phase. They admit an
asymptotic series expansion as ℏ → 0, and can be computed directly by Borel summation of
that series. This Borel summation suffers from Stokes phenomena, directly associated with BPS
particles in the defect theory. Indeed, one has a sharp relation [53]
ξ = −Z, (2.1)
where ξ is the position of a singularity in the Borel plane for ψi, and Z is the central charge
of a BPS particle which is in vacuum i at −∞. In general all of the quantities we consider —
6 A. Grassi, Q. Hao and A. Neitzke
ψi, ξ and Z — are holomorphic functions of other parameters x, which represent couplings and
moduli in the 2d-4d system.4
Because of the Borel plane singularities, the observable ψi is only piecewise analytic as a func-
tion of parameters (ℏ, x); it jumps when arg ℏ = arg ξ(x), which using (2.1) means it jumps at
the loci
arg ℏ = arg (−Z(x)) , (2.2)
where Z ranges over the central charges of BPS particles in vacuum i at −∞.
2.2 Chiral couplings and spectral networks
The best-explored examples of this story arise in UV complete 2d-4d systems, such as Lagrangian
theories and theories of class S [45]. In these examples, one considers a surface defect with
a parameter space C of chiral couplings, where perturbation along C is accomplished by adding
the descendant of a chiral operator on the defect. Thus we now specialize to let x denote a point
of C, holding other moduli fixed.
The observables ψi = ψi(ℏ, x) are flat sections of a connection over C; more concretely, they
are solutions of a linear ordinary differential equation over C (e.g., a meromorphic Schrödinger
equation). This equation can be viewed as a quantization of the Seiberg–Witten curve Σ ⊂ T ∗C
determined by the 2d-4d system; it has been discussed from many different points of view, e.g.,
[1, 2, 43, 64, 65, 70, 86, 90, 91]. The perturbation series in ℏ is the usual WKB series representing
solutions of the differential equation, and the Borel plane singularities are responsible for Stokes
phenomena, as familiar in the exact WKB theory.
Now suppose we consider the Borel summation for fixed ℏ, as a function of x. Then (2.2) says
that Stokes phenomena occur at codimension-1 walls on C. These walls make up the “Stokes
graph” or “spectral network”. Each wall corresponds to a particular BPS particle on the surface
defect, and carries a label ij, where i is the vacuum at −∞ and j ̸= i is the vacuum at +∞. At
a wall with label ij, the solution ψi jumps by adding some multiple of the solution ψj .
2.3 Flavor masses and exponential networks
A variant of the above story arises for 2d-4d systems in which the surface defect supports a flavor
symmetry. For simplicity let us discuss only the case of a U(1) symmetry. In this case the defect
theory can be deformed by a complex flavor mass x, parameterizing the space C = C. This
situation is similar to the previous one, with a few new features:
� The Seiberg–Witten curve Σ ⊂ C × C× rather than T ∗C. Correspondingly, the observ-
ables ψi are solutions of a difference equation in x involving shifts x 7→ x − 2πiℏ, rather
than a differential equation.
� Each ψi depends on an additional Z-fold choice; changing this choice multiplies ψi by enx/ℏ
for some n ∈ Z. This operation corresponds to modifying the boundary condition by
adding a supersymmetric flavor Wilson line with charge n.
� As before, we expect that the local solutions ψi experience Stokes phenomena at walls
in C determined by the equation (2.2). These walls make up a generalized kind of spectral
network, which is a very simple example of the notion of “exponential network” considered
in [9, 10, 11, 38, 81]. In these examples, any BPS particle which is charged under the
4More precisely, in the Borel plane, there can be multiple singularities along any given ray emanating from
the origin, and there could be multiple BPS particles with central charges along a given ray as well. In the
relation (2.1), and in similar relations we write below, ξ is to be interpreted as the first pole along a given ray,
and Z similarly as the first central charge along a given ray.
Exponential Networks, WKB and Topological String 7
defect flavor symmetry will have mass depending on the parameter x. In particular, this
can include BPS particles which sit in a single vacuum i rather than interpolating from
one vacuum to another. Thus exponential networks generally include walls with labels ii,
as well as the more familiar ones with labels ij.
For a simple example, we could consider the case where the 4d system is actually trivial,
and take the 2d-4d system to be a 2d N = (2, 2) theory with one chiral multiplet. We turn
on a complex flavor mass x for the U(1) flavor symmetry. This theory has a Landau-Ginzburg
model description, and one can get its Seiberg–Witten curve by minimizing the potential, found
for example in [9, 66, 92, 96]. The Seiberg–Witten curve is simply
x = Y, Y ∈ C×,
where Y = ey. The corresponding difference equation is
xψ(x) = ψ(x− 2πiℏ),
which has a solution involving the gamma function (note that if −2πiℏ = 1 this equation becomes
exactly the functional equation of the gamma function).
In this paper, we will not explore this kind of example in detail. Rather we move on directly
to the next case.
2.4 Compactified 3d-5d systems and exponential networks
Now we come to the type of examples we consider in this paper. We start with a 3d-5d system,
consisting of a 5dN = 1 theory coupled to a defect preserving 3dN = 2, with a U(1)f flavor sym-
metry on the defect. After compactification on S1 we obtain a 2d-4d system with U(1)f × U(1)K
flavor symmetry, with U(1)K coming from shifts along the compactification circle. Then, as
above, we consider deforming by a flavor mass for U(1)f . In this case the imaginary part of
the flavor mass comes from the log-holonomy of a background U(1)f connection around the
compactification circle, and invariance under large gauge transformations of the background
field implies that the theory with mass x/R is equivalent to the theory with mass (x+ 2πi)/R.
Said otherwise, the parameter space of inequivalent theories is actually C = C×
X , parameterized
by X = exp(x).5 Then, the picture we expect is
� The Seiberg–Witten curve Σ ⊂ C×
X×C×
Y . The observables ψi are solutions of a q-difference
equation in X, involving shifts X 7→ qX with q = eiℏ. These difference equations have
again been studied from various points of view, e.g., [2, 12, 33, 83].
� Each ψi depends on an additional Z2-fold choice; changing this choice multiplies ψi
by e(2πn(x+iπ)+4π2im)/ℏ for some (n,m) ∈ Z2. This operation corresponds to modifying
the boundary condition by adding a supersymmetric U(1)f × U(1)K Wilson line with
charges (n,m).
� At phase ϑ = ±π
2 we have some special features (arising ultimately from the fact that
the supersymmetric boundary conditions at this phase descend from Lorentz invariant
boundary conditions in 3d [22].) For example, under analytic continuation x → x + 2πi,
each ψi is multiplied by e2πki(x+πi)/ℏ, for some ki ∈ Z. The constant ki is the effective
flavor Chern–Simons level of the 3d theory in vacuum i; this transformation law reflects
the fact that a supersymmetric domain wall in which x shifts by 2πi is equivalent to
5In passing from pure 2d-4d systems to 3d-5d systems on a circle, we have renormalized our parameters:
the dictionary is ℏ3d-5d = 2πRℏ2d-4d and x3d-5d = Rx2d-4d. This renormalization is not really necessary, but it
matches the conventions in the literature, which tends to use dimensionless parameters in the 3d-5d context. The
difference between the renormalizations of x and ℏ accounts for some shifts in factors of 2π.
8 A. Grassi, Q. Hao and A. Neitzke
a supersymmetric U(1)f Wilson line with charge ki.
6 The ki can depend on Rex, since
the effective Chern–Simons level can jump as we vary the flavor mass parameter of the 3d
theory.
� The ψi suffer Stokes phenomena associated to BPS particles living on the compactified
defect. Precisely, we expect the positions ξ of the Borel plane poles to be given by
ξ = −2πRZ.
Since R is real, this would lead to the jump locus for the ψi being given by (2.2) just as
in the 2d-4d case. Thus again we expect that the ψi jump at the walls of the exponential
network determined by the 3d-5d BPS spectrum. Said otherwise, we expect that the
exponential network plays the role of a Stokes graph for the difference equation obeyed by
the ψi.
This is the picture we will check below, in two simple examples, where the defect theory has
only a single vacuum. These examples isolate one of the key new phenomena in the cases with
flavor mass, namely the walls of type ii — indeed they have only the walls of type ii!
In the examples we consider, the local solutions ψi are combinations of variants of the quantum
dilogarithm function. The key technical advance which makes our study possible is the work [49],
where the Borel poles for the ℏ-expansion of this function are determined.
The difference equations we consider also arise in a different context, that of A model topo-
logical strings on a Calabi–Yau threefold X with a D-brane placed on a Lagrangian submani-
fold L ⊂ X, e.g., [1, 2, 32]. In this language Σ is the mirror curve of X. One can try to connect
this directly to a 3d-5d system by considering M-theory on X×R5 with an M5-brane on L×R3,
with some appropriate regularization to take care of the non-compactness of X and L; this setup
has been used often in the literature beginning with [93]. We will not try to make the connection
between the two pictures directly here, but freely use both languages.
3 A simple model: C3
A simple example of the difference equations that we study in this paper arises as the quantized
mirror curve of C3. We first remind the readers of the basic setup. The mirror curve is
Σ = {X − Y + 1 = 0} ⊂ C×
X × C×
Y .
Σ is a thrice-punctured sphere, with the punctures at
{(X,Y )}sing = {(0, 1), (−1, 0), (∞,∞)} .
We will also use the logarithmic variables
x = log(X), y = log(Y ).
Below we will sometimes need to pick a specific branch, e.g., in writing explicit formulas for
local solutions; we will always take the principal branch, i.e.,
−π < Im(x) ≤ π. (3.1)
6We can roughly understand this as follows. We consider the 3-dimensional theory in D×S1, where D denotes
the disc, with a domain-wall background, where
∮
S1 A shifts by 2π in a small neighborhood of a loop ℓ ⊂ D.
This shift requires that the background field has nontrivial curvature, and in the presence of this curvature the
Chern–Simons term k
∫
S1×D
A∧F becomes k
∮
ℓ
A, a flavor Wilson line with charge k. The statement we are after
is a supersymmetrization of this one.
Exponential Networks, WKB and Topological String 9
In these variables the mirror curve becomes
ex − ey + 1 = 0.
We choose C×
X to be the base of Σ, and will later introduce the exponential network on this
base.
Quantization of this curve gives rise to the quantum mirror curve. Our convention for the
quantum mirror,(
ep̂ − 1− ex̂q−
1
2
)
ψ(x, ℏ) = 0, [x̂, p̂] = iℏ, (3.2)
is the same as the one used in [49].7
3.1 All-orders WKB expansion of local solutions
For convenience, rather than studying local solutions ψ of (3.2) directly, we study ϕ(x, ℏ) =
logψ(x, ℏ) which satisfies the difference equation
ϕ
(
x+ i
ℏ
2
, ℏ
)
− ϕ
(
x− i
ℏ
2
, ℏ
)
= − log (1 + ex) .
This equation is solved by a formal series (see, for example, [49, Section 2.1]):8
ϕ(x, ℏ) ∼ 1
iℏ
Li2(−ex) +
∑
k≥1
B2k
(
1
2
)
(iℏ)2k−1
(2k)!
Li2−2k (−ex) . (3.3)
There is a Z × Z ambiguity here associated with the choice of branch for Li2. The resulting
ambiguity of the local solutions is the one discussed in Section 2.4. When we write explicit
formulas, we will always resolve this ambiguity by choosing the principal branch of Li2.
The formal series is not the actual analytic solution that we seek, but the Borel summation
of it is. The Borel transform of (3.3), as defined in (A.2), can be rewritten as [49]
Bϕ(x, ξ) = − i
4π2
∞∑
n=1
(−1)n
n2
(
1
1 + e
ξ
2πn
−x
+
1
1 + e−
ξ
2πn
−x
)
. (3.4)
We can see that Bϕ(x, ξ) has singularities in the Borel plane located at
ξ∗(x,m, n) ≡ 2πn (x+ πi(2m+ 1)) , m ∈ Z, n ∈ Z/{0}, (3.5)
with residues
−(−1)n
2πin
. (3.6)
Hence when
arg(ℏ) = arg(ξ∗(x,m, n)),
7If we had chosen the alternative convention
(
ep̂ − 1 − ex̂
)
ψ(x, ℏ) = 0, we would get slightly different series,
but all the structure we consider in this paper, particularly the positions of the Borel plane singularities, would
be the same.
8We did not obtain the series solution (3.3) by a direct WKB-type expansion of a solution of the difference
equation; rather we just lifted it from [49]. However, using the difference equation and the boundary condition
obeyed by (3.3), one can show that it does match with the result of a WKB-type expansion.
10 A. Grassi, Q. Hao and A. Neitzke
the Borel summation (A.3) is not defined. Nevertheless, it is analytic for sufficiently small
variations of ℏ, those for which the contour of integration in the Borel summation does not go
through any of the poles in the Borel plane.
For example, it has been proved in [49] that
s(ϕ(x, ℏ)) = log (Φ(x, ℏ)) , arg(ℏ) = 0, −π < Im(x) ≤ π, (3.7)
where we define
Φ(x, ℏ) ≡ Φb
( x
2πb
)
, (3.8)
with Φb(x) the quantum dilogarithm function of Faddeev [40], and
ℏ = 2πb2.
More about the quantum dilogarithm can be found in Appendix B. For Im(ℏ) ̸= 0, we can
express the Faddeev quantum dilogarithm in terms of q-Pochhammer symbols; for example we
use (B.2) for Im(ℏ) > 0. For Im(ℏ) = 0, we need to use the integral expression (B.1).
Likewise, it was shown in [49] that
s(ϕ(x, ℏ)) = log
((
−q
1
2 ex; q
)
∞
)
, (3.9)
for arg(ℏ) = π
2 , −π < Im(x) ≤ π, Re(x) < 0.
We will discuss the Borel summation and its analytic structure in the full Borel plane in
Section 3.5.
3.2 Field theory and BPS states
Now we recall that C×
X is not only the base of the 5d Seiberg–Witten curve: it also plays the
role of a parameter space of flavor mass couplings in the S1 compactification of a 3d-5d system.
In the example we are discussing now, the 5d system is actually trivial, so rather than a 3d
defect we are just considering a 3d N = 2 field theory with a U(1) flavor symmetry. The 3d
theory is the “tetrahedron” theory of [33], which can be described as the Lagrangian field theory
of a single 3d chiral multiplet with charge 1 under the U(1) flavor symmetry, plus a background
Chern–Simons coupling at level −1
2 .
We compactify the theory on S1 and consider the spectrum of BPS particles in the compact-
ified theory. Since the theory is free, this spectrum can be described simply: the single chiral
multiplet of the 3d theory gives rise to an infinite Kaluza-Klein tower of chiral multiplets in 2d,
and each one of these leads to a single BPS particle and its corresponding antiparticle. The
central charges are
Z = ±R−1(x+ πi(2m+ 1)), (3.10)
where the integer m keeps track of the KK momentum.
3.3 Exponential network
The exponential network Wϑ on C×
X is defined as the set of points X ∈ C×
X , such that in the
theory with parameter X there exists a BPS particle satisfying
arg(−Z(X)) = ϑ. (3.11)
Combining (3.10) and (3.11), we see that X ∈ Wϑ if and only if
X = −e±seiϑ (3.12)
Exponential Networks, WKB and Topological String 11
(a) ϑ = 0 (b) ϑ = π/2 (c) ϑ = 2π
5 (d) ϑ = 3π
5
Figure 1. The exponential networks Wϑ on C×
X at various phases ϑ. The blue dot at X = 0
and purple dot at X = −1 represent punctures of Σ. The orange wall is the locus X = −ese
iϑ
and green wall is the locus X = −e−seiϑ . The degenerate wall at phase ϑ = π/2 is painted
in red.
for some s ∈ R≥0. In Figure 1, we show the networks Wϑ for several choices of ϑ.9 Three
distinct shapes occur:
(1) ϑ = 0 or ϑ = π: Wϑ consists of 2 straight walls ending at X = 0 and X = ∞ respectively.
See Figure 1a.
(2) ϑ = π
2 or ϑ = −π
2 : Wϑ consists of a degenerate wall which is a circle of radius 1
around X = 0. See Figure 1b.
(3) ϑ ̸= nπ
2 , n ∈ Z: Wϑ consists of two spirals. One spiral ends at X = 0 and is contained
in the region 0 < |X| < 1. The other spiral ends at X = ∞ and is contained in the
region |X| > 1. The correspondence between walls in (3.12) and spirals, as well as the
orientation of the spirals, depend on the phase ϑ. Examples are shown in Figures 1c
and 1d.
3.4 The exponential network and exact WKB
Now recall the basic picture we proposed in Section 2: there are canonical formal WKB solutions,
and the poles ξ in the Borel plane for these formal solutions are related to the central charges Z
of BPS particles in the 3d-5d system on S1, via the formula
ξ = −2πRZ. (3.13)
More precisely, in the Borel plane, there can be multiple poles in each direction, and in the
relation (3.13), ξ is to be interpreted as the first pole in any given direction.
Let us see whether this relation holds in the C3 example. In (3.5), we see infinite sequences
of poles, distinguished by the multiplicity n ∈ Z\{0}; we consider the first pole in each se-
quence, i.e.,
ξ = ξ∗(x,m, n = ±1) = ±2π(x+ πi(2m+ 1)).
On the other hand, (3.10) says that
Z = ±R−1(x+ πi(2m+ 1)).
Thus we see directly that the relation (3.13) indeed holds in this example.
A graphical interpretation of this statement is that, for any m ∈ Z, the truncated exponential
network Warg(ξ∗(x,m,±1))(|ξ∗(x,m,±1)|) ends at the point X = ex. We illustrate this in two
examples in Figure 2 by plotting the truncated networks directly.
9See [9, 38] for previous studies in another framing.
12 A. Grassi, Q. Hao and A. Neitzke
Figure 2. Left: Singularities of the Borel transform in the Borel plane when x = 1 + i.
The two points in red are ξ∗1 = ξ∗(x, 0, 1) and ξ∗2 = ξ∗(x, 1, 1). Right: Truncated exponential
network Warg(ξ∗i )(|ξ∗i |), i = 1, 2, on C×
X . The red dots are at the point X(x).
3.5 Local solutions in each sector
In this section we review the Borel summation (A.3) of the local solution (3.3) and discuss the
corresponding jumps as we move in the Borel plane. Ultimately the reason for these jumps is
due to the fact that the integration contour in equation (A.3) depends on ℏ. Consequently, when
we change the contour, we cannot do so continuously because of the existence of poles (3.5) in
the Borel transform (3.4). In addition, since these poles ξ∗(x,m, n) depend on x, m, n, we need
to separate the discussion by quadrants in the Borel plane and the sign of Re(x).
Borel resummation of local solutions for the case Re(x) < 0 and arg(ℏ) ∈
[
0, π2
]
was discussed
in [49]. Here we complete the analysis by exploring other regions as well. This extended analysis
will also be useful in the study of the resolved conifold: see Section 4.
We will elaborate only on the case of the first quadrant; all other cases can be found in
Appendix C and are summarised in Figure 4.
A sample Borel plane is shown in Figure 3. We can see that there is an infinite number of
rays containing poles, with phases
ϑ±x,m = arg (± (x+ πi (2m+ 1))) , m ∈ Z.
These rays divide the Borel plane into sectors. We define the sector containing the positive real
axis as
I iO
0 ,
where iO denotes the quadrant.10
3.5.1 First quadrant of the Borel plane and Re(x) < 0
In this case the relevant sectors of the Borel plane are
I 1O
x,m =
(
ϑ−x,m;ϑ−x,m−1
)
, m ≤ −1.
For convenience, we neglect the subscript x and simply use
I iO
m ≡ I iO
x,m.
10I 1O
0 and I 4O
0 refer to the same sector. Same for I 2O
0 and I 3O
0 .
Exponential Networks, WKB and Topological String 13
Figure 3. The Borel plane of the local solution (3.3) for x = −1 + i. The rays of poles in the
first quadrant separate the Borel plane into sectors I 1O
m , m ≤ 0.
To calculate the jump of the solution in the first quadrant, we assume that ϑ is independent
of ℏ and study the two contour integrals along rays whose phases are given by
ϑ− ∈ I 1O
m+1, ϑ+ ∈ I 1O
m ,
respectively.
The jump of the solution when crossing the ray of phase ϑ−x,m is
− 2πi
∞∑
n=1
Res
(
Bϕ(x, ξ)e−
ξ
ℏ ,−2πn(x+ (2m+ 1)πi)
)
= −2πi
∞∑
n=1
(−1)n
2πin
e
2πn(x+(2m+1)πi)
ℏ = log
(
1 + e
2π(x+(2m+1)πi)
ℏ
)
, m ≤ −1,
where we have used (3.6).
Using the explicit expressions of the jumps and the solution along the positive real axis (3.7),
the Borel summation of (3.3) for arg(ℏ) ∈ I 1O
m is given by
s(ϕ)(x, ℏ) = log
( (
−q
1
2 ex; q
)
∞(
−q̃
1
2 e
2πx
ℏ ; q̃
)
∞
−(m+1)∏
i=0
(
1 + e
2πx
ℏ q̃
1
2 q̃i
))
= log
( (
−q
1
2 ex; q
)
∞(
−q̃
1
2 e
2πx
ℏ ; q̃
)
∞
(
−e
2πx
ℏ q̃
1
2 , q̃
)
−m
)
,
where the q-Pochhammer symbol is defined as
(a; q)n =
n−1∏
i=0
(
1− aqi
)
. (3.14)
Using (3.8) and (B.8), this can also be written as
s(ϕ)(x, ℏ) = log Φ(x+ 2πim, ℏ), arg(ℏ) ∈ I 1O
m , m ≤ −1. (3.15)
We will discuss the appearance of Φ(x+ 2πim) from the point of view of analytic continuation
in Section 3.6. Note that
lim
m→−∞
(
−q
1
2 ex; q
)
∞(
−q̃
1
2 e
2πx
ℏ ; q̃
)
∞
(
−e
2πx
ℏ q̃
1
2 , q̃
)
−m
=
(
−q
1
2 ex; q
)
∞. (3.16)
14 A. Grassi, Q. Hao and A. Neitzke
Hence we get (3.9)
s(ϕ)(x, ℏ) = log
(
−q
1
2 ex; q
)
∞, ℏ ∈ iR+, Re(x) < 0. (3.17)
So far this was as in [49]. Let us now look instead at Re(x) > 0.
3.5.2 First quadrant of the Borel plane and Re(x) > 0
In this case the relevant sectors of the Borel plane are
I 1O
m =
(
ϑ+m−1;ϑ
+
m
)
, m ≥ 1,
with the understanding that I 1O
0 is the sector containing the real axis.
We use the sum of residues to get the jump
(−2πi)
∞∑
n=1
m−1∑
k=0
−(−1)n
2πin
e−
n(2π(x+(1+2k)πi))
ℏ = log
(
1(
−q̃
1
2 e−
2πx
ℏ ; q̃
)
m
)
, m ≥ 1,
for I 1O
m from the real axis. The local solution in I 1O
m is thus(
−q
1
2 ex; q
)
∞(
−q̃
1
2 e
2πx
ℏ ; q̃
)
∞
1(
−q̃
1
2 e−
2πx
ℏ ; q̃
)
m
, m ≥ 1. (3.18)
Alternatively, we can express (3.18) as
s(ϕ)(x, ℏ) = logΦ(x+ 2πim, ℏ) +
2πm(x+mπi)
ℏ
, arg(ℏ) ∈ I 1O
m , m ≥ 1. (3.19)
We will discuss more about this expression from the point of view of analytic continuation in
Section 3.6.
The solution along positive imaginary axis for Re(x) > 0 is obtained from
lim
m→∞
(
−q
1
2 ex; q
)
∞(
−q̃
1
2 e
2πx
ℏ ; q̃
)
∞
1(
−q̃
1
2 e−
2πx
ℏ ; q̃
)
m
=
(
−q
1
2 ex; q
)
∞(
−q̃
1
2 e
2πx
ℏ ; q̃
)
∞
1(
−q̃
1
2 e−
2πx
ℏ ; q̃
)
∞
=
e
x2i
2ℏ q
1
24
q̃
1
24
1(
−q
1
2 e−x; q
)
∞
.
Hence
s(ϕ)(x, ℏ) = log
e
x2i
2ℏ q
1
24
q̃
1
24
1
(−q
1
2 e−x; q)∞
, ℏ ∈ iR+, Re(x) > 0 . (3.20)
3.5.3 First quadrant of the Borel plane and Re(x) = 0
When Re(x) = 0 the situation is special since all the poles in the Borel plane lie on the imaginary
axis. We found that in this case the median summation is
s(ϕ)(x, ℏ) =
1
2
(
i
(
12x2+ ℏ2+ 4π2
)
24ℏ
− log
(
−e−xq
1
2 ; q
)
∞(
−exq
1
2 ; q
)
∞
)
, Re(x) = 0, ℏ ∈ iR+.(3.21)
Note also that (3.21) = 1
2 ((3.17) + (3.20)).
For all the other values of ℏ, the Borel summation matches with Φ(x):
s(ϕ)(x, ℏ) = logΦ(x, ℏ), Re(x) = 0, Re(ℏ) > 0, Im(ℏ) ≥ 0.
Exponential Networks, WKB and Topological String 15
Figure 4. A summary of local solutions for all the sectors in the Borel plane. The angle
represents arg(ℏ). The circle is cut into sectors I iO
m where m is listed for each sector. If the
function is written in the interior or exterior of the red circle, it represents the solution for
Re(x) < 0 or Re(x) > 0 respectively. In I iO
0 , there is a unique solution for both Re(x) < 0 and
Re(x) > 0. For I iO
0 , i = 1, 4, this unique solution is Φ(x, ℏ) and for I iO
0 , i = 2, 3, this solution
is Φ̃(x, ℏ). The functions Φ(x, ℏ) and Φ̃(x, ℏ) are defined in (B.4) and (B.7), respectively. For
practical reasons the solution at Re(x) = 0 is not shown on the figure but can be found in the
main text, see, e.g., (3.21) and item C.7.
3.5.4 Summary and comments
Calculations for all the other quadrants can be found in Appendix C; we summarize all the local
solutions in the whole Borel plane in Figure 4. It is interesting to note that for generic ℏ there
are two kinds of solutions depending on whether Re(x) < 0 or Re(x) > 0, which nevertheless
coincide when ℏ ∈ R. Let us look at the imaginary axis ℏ ∈ iR+: the two solutions are (3.17)
and (3.20). Physically q-Pochhammer in (3.17) gives the open topological string partition func-
tion on C3 corresponding to an anti-brane where iℏ = gs, see for instance [72, p. 24]. The other
solution (3.20) can be schematically obtained from (3.17) using an S transformation, up to an
overall
(
qq̃−1
) 1
24 and a shift in the argument.
3.6 Jumps of local solutions via analytic continuation
In this subsection, we use the exponential network to explain why the jumps of local solutions
have the form of an analytic continuation, for example as in (3.15). The discussion depends
on which quadrant of the Borel plane we are in. However, they are all similar, so we will only
consider the first quadrant, i.e., ϑ ∈
(
0, π2
)
.
We start by noticing that varying ϑ rotates the integral contour in the Laplace transform,
while varying x shifts the poles. However, as long as the poles passing through the integral
contour are the same, the jumps obtained by varying ϑ or x are equivalent. Therefore, we can
equally well study the behavior of the solutions on C×
X\Wϑ, instead of in the Borel plane.
16 A. Grassi, Q. Hao and A. Neitzke
Figure 5. Two paths tracking the change of solutions on C×
X . The blue dashed line is the
branch cut for Li2(−X). The short red path going through the green wall corresponds to a
jump of the solution. The dashed red path corresponds to analytic continuation. Since the two
paths have the same starting and ending points, the transformations of the solution obtained
from the two paths must be the same.
The exponential network on C×
X encodes information on the solution Φ(X, ℏ)11 to the q-
difference equation. For ϑ ∈
(
0, π2
)
, the exponential networks have two constituent walls in the
region |X| < 1 and |X| > 1, respectively; see green and orange walls in Figure 1c or in Figure 5.
There are some subtleties in the discussion of |X| > 1; thus we first discuss the case |X| < 1.
For ϑ ∈
(
0, π2
)
, C×
X\Wϑ is simply connected. By analytic continuation, we assign a single
solution Φ(X, ℏ) to the whole complement of the exponential network for |X| < 1. Now, when we
follow the short solid red path in Figure 5, the solution Φ(X, ℏ) jumps; the jump is captured by
changingm→ m−1 in Figure 5. On the other hand, this jump must be equivalent to performing
analytic continuation along the dashed red path in Figure 5. In terms of the variable x, this
continuation is x→ x− 2πi; indeed, in (3.15) we see that changing m→ m− 1 is equivalent to
shifting x→ x− 2πi.
For |X| > 1, the situation is slightly more complicated: we need to consider the branch cut
of the dilogarithm Li2(−X) from X = −1 to X = ∞, plotted as a dashed blue line in Figure 5.
Every time a path on C×
X crosses this branch cut, in order to match with the Borel resummation
which chooses the principal branch of quantum dilogarithm, the solution acquires a factor12
e
2π(log(X)+iπ)
ℏ ,
where log(X) also has a branch cut along the negative real axis, fixed as in (3.1). This factor
is also subject to analytic continuation. Taking these factors into account we see that the
shift m→ m+ 1 in (3.19) is indeed equivalent to continuing x→ x+ 2πi.
When ϑ = π
2 , the exponential network consists of a degenerate wall lying on the unit circle.
It separates C×
X into two domains: |X| < 1 and |X| > 1. So there are two solutions for the two
domains; they can be thought of as m → ∓∞ limits of the exponentials of (3.15) and (3.19).
The explicit form of the solutions is(
−q
1
2 ex; q
)
∞, ℏ ∈ iR+, Re(x) < 0,
e
x2i
2ℏ q
1
24
q̃
1
24
1(
−q
1
2 e−x; q
)
∞
, ℏ ∈ iR+, Re(x) > 0.
In the low energy effective theory of the defect described in Section 3.2, integrating out the
chiral shifts the flavor Chern–Simons coupling level by ±1
2 for ∓x > 0. Thus the effective
11We are considering the solution on C×
X directly, rather than writing it in terms of the logarithmic variable x.
12The jump factor can be checked by
lim
x→log(X)+πi
Li2(−X) = lim
x→Re(log(X))−πi
Li2(−X) − 2πi Re(log(X)).
Exponential Networks, WKB and Topological String 17
defect theory for x < 0 does not have the effective flavor Chern–Simons term and the one
with x > 0 has an effective flavor Chern–Simons term with level −1.13 Thus that under analytic
continuation x→ x+ 2πi, (3.15) stays fixed and (3.19) is multiplied by e−
2π(x+πi)
ℏ is exactly what
we expected according to Section 2.4. The solutions in different domains and their connections
to the Chern–Simons terms have also been discussed in [12].
3.7 The closed sector and the McMahon function
Let us now briefly discuss the resurgence structure of the closed string free energies associated
to C3. This is parallel to the study of exact WKB of quantum periods in 4d. We have
FC3
(ℏ) =
∑
g≥0
Fgℏ2g−2, (3.22)
where
F0 = −ζ(3), F1 =
1
12
log(−iℏ) + ζ ′(−1), Fg = − (−1)gB2gB2g−2
2g(2g − 2)(2g − 2)!
, g ≥ 2.
Note that
Fg ∼ (2g − 3)!, g ≫ 1.
Hence (3.22) is obviously divergent. We compute the Borel transform using the definition (A.2)
and get
BFC3
(ξ) = −
∑
n≥1
ξ2 csch2
( ξ
4πn
)
+ 8πn
(
2πn− ξ coth
( ξ
4πn
))
32π4ξn4
,
which has poles on the imaginary axis at
ξ = 4π2imn, m ∈ Z/{0}.
Physically these poles correspond to BPS states arising from D0-branes in type IIA description
(their position can be identified with the central charge associated to such an object), see
Section 1.
We now consider the Borel summation as defined in (A.3). The jump obtained from the sum
over all the residues at the poles on positive imaginary axis with m ≥ 1 and n ≥ 1 are
∆(ℏ) = 2πi
∑
m>0
Li2
(
e−
4imπ2
ℏ
)
2π2
−
2im log
(
1− e−
4iπ2m
ℏ
)
ℏ
. (3.23)
We can express (3.23) using the NS limit of the refined McMahon function, namely [69]
McMNS(ℏ) =
∑
n≥1
ie−
1
2
iℏn csc
(ℏn
2
)
2n2
.
13These background Chern–Simons levels can be conveniently understood in terms of a Type IIB (p, q)-fivebrane
construction, as discussed in [22]. The defect comes from a D3-brane with one end on a spectator brane and the
other end on the (p, q)-fivebrane web. The two domains Re(x) < 0 and Re(x) > 0 correspond to D3-branes ending
on different legs of the (p, q)-fivebrane web. The Chern–Simons levels obtained from the (p, q) charges of the legs
and the spectator brane are as above.
18 A. Grassi, Q. Hao and A. Neitzke
In particular, it is easy to check that
∆(ℏ) =
1
iπ
∂
∂ℏ
(
ℏMcMNS
(
4π2
ℏ
))
.
This is in line with the expectation from non-perturbative strings of [54, 59]. We have following
results:
(1) If ℏ ∈ iR+, the Borel summation of (3.22) reproduces the logarithm of the McMahon
function, namely
logMcM(ℏ) = s
(
FC3)
(ℏ), ℏ ∈ iR+,
where
McM(ℏ) =
(∏
k≥1
(
1− eiℏk
)−k
)
.
(2) If ℏ is not imaginary, then we have to take into account the contribution of the poles along
the imaginary axes, namely (3.23). We then find
logMcM(ℏ) +
∆(ℏ)
2
= s
(
FC3)
(ℏ), Re(ℏ) > 0.
We can also check that the r.h.s. in the above equation matches
Ac(ℏ) =
8
ℏ2
∫ ∞
0
dx
x
e
4πx
ℏ − 1
log
(
1− e−2x
)
dx− ζ(3)
ℏ2
+
ℏζ(3)
4π3
− log(i)
12
,
as in [61, 62].14
(3) By using
s
(
FC3)
(ℏ) = s
(
FC3)
(−ℏ) +
π
12
,
we can reach the rest of the ℏ plane which is not discussed in items (1) and (2) above.
4 The resolved conifold
We now move to our second example which is the resolved conifold. The Seiberg–Witten curve
of the resolved conifold is
Σ = {1− Y +X −QXY = 0} ⊂ C×
X × C×
Y ,
where Q = e−t and t is the Kähler parameter of the resolved conifold. Σ is a four-punctured
sphere with punctures at
{(X,Y )}sing =
{
(0, 1), (−1, 0),
(
− 1
Q
,∞
)
,
(
∞,
1
Q
)}
.
In this paper, we choose the following quantum mirror curve for the resolved conifold(
1− ep̂ + q−1/2ex̂ − q−1/2Qex̂ep̂
)
Ψ(x, ℏ, t) = 0, (4.1)
14In these references Ac(ℏ) was used to resumm the constant map contribution in a particular slice of lo-
cal P1 × P1.
Exponential Networks, WKB and Topological String 19
where again, q = eiℏ. Our convention here is such that the resolved conifold behaves as two
copies of C3 in the convention we used in Section 3, with the variable x shifted by t in one copy.
We recall that formal solutions to (4.1) and their connection with open topological strings
were discussed previously in the literature, for example in [1, 68, 72]. Here we are interested in
Borel summation of formal solutions and the corresponding non-perturbative effects in the open
string amplitudes. The relation with exponential networks will also play an important role in
our analysis.
4.1 All-orders WKB expansion of local solutions
We can work out the formal series expansion for log of the solution using the same technique as
in [49]. We find
φ(x, t, ℏ) = logΨ(x, t, ℏ) = −iℏ
∑
k≥0
B2k
(
1
2
)
(iℏ)2k−1
(2k)!
(
Li2−2k (−ex)− Li2−2k
(
−ex−t
))
.(4.2)
The series (4.2) is the difference of two pieces: one is the series (3.3) for the C3 introduced in
Section 3 and the other is (3.3) for the C3 with the shift x→ x− t. Hence the Borel summation
of the local solution also decouples into two pieces. Parallel to (3.1), we assume
− π < Im(x) ≤ π, −π < Im(x− t) ≤ π.
Therefore the Borel transform of (4.2) is simply
Bφ(x, t, ξ) = Bϕ(x, ξ)− Bϕ(x− t, ξ), (4.3)
where Bϕ(x, ξ) is defined in (3.4). Hence (4.3) has two sets of singularities coming from Bϕ(x, ξ)
and Bϕ(x− t, ξ), respectively,
ξ∗(x,m, n) = n(2π(x+ (2m+ 1)πi)), m ∈ Z, n ∈ Z/{0},
ξ∗t (x, t,m, n) = n(2π((x− t) + (2m+ 1)πi)), m ∈ Z, n ∈ Z/{0}. (4.4)
These singularities correspond to the central charges of the 3d-5d BPS KK-modes as we discuss
below.
4.2 BPS states in 3d-5d system
We have not studied the BPS spectrum in this case directly from a 3d-5d field theory description;
instead we use the M-theory point of view, along the lines of [50, 93]. This leads to the prediction
that there are two 3d particles, corresponding to two M2-brane discs ending on the M5-brane
(Ooguri–Vafa invariants). The areas of these two discs sum to the area of the compact CP1
in X, and there should be one 5d particle corresponding to an M2-brane wrapping this cycle
(Gopakumar–Vafa invariant).
Thus we expect that there should be an effective description of the 3d-5d system in which
the field content on the defect is two chiral multiplets with charges (+1, 0) and (−1,+1) under
a U(1)×U(1) flavor symmetry. When the system is reduced on S1, the theory has an extra U(1)
flavor symmetry coming from the rotation of the circle. These two fields give rise to two infinite
towers of KK modes corresponding to the third U(1), with central charges
±R−1(x+ (2m+ 1)πi), m ∈ Z,
±R−1((x− t) + (2m+ 1)πi), m ∈ Z. (4.5)
20 A. Grassi, Q. Hao and A. Neitzke
(a) ϑ = 0 (b) ϑ = π/2 (c) ϑ = 2π
5 (d) ϑ = 3π
5
Figure 6. The exponential networks Wϑ on C×
X at different ϑ’s for t = 1
2 +
i
10 . The blue, purple
and magenta dots represent X = 0, X = −1 and X = − 1
Q respectively. The orange, green,
red and purple walls are given by X = −ese
iϑ
, X = −e−seiϑ , X = − 1
Qese
iϑ
and X = − 1
Qe−seiϑ
respectively. The degenerate walls at phase ϑ = π/2 are painted in red.
Here (x, t) are the two complex flavor masses, complexifying the two flavor masses of the 3d
theory. In addition t is identified with the complexified vev of the scalar in the 5d vector
multiplet. When the system is reduced on S1 the 5d particle gives rise to two towers of KK
modes with central charges
±R−1(t+ 2mπi), m ∈ Z. (4.6)
4.3 The exponential network and exact WKB
Once again, we note that the central charges of the BPS KK-modes (4.5) are related to posi-
tions (4.4) of the first poles along each ray, by the relation
ξ = −2πRZ.
Recall that the exponential network Wϑ consists of those X such that there are KK-modes
obeying
arg(−Z(X)) = ϑ.
In this case X ∈ Wϑ if and only if
X = −e±seiϑ or X = − 1
Q
e±seiϑ
for some s ∈ R. Thus, for a generic phase ϑ, the exponential network for the conifold consists
of two copies of the exponential network for C3, as shown in Figure 6. One copy emanates
from X = −1, and the other copy emanates from X = − 1
Q . A new phenomenon for the resolved
conifold is that there can exist degenerate walls with two ends at X = −1 and X = − 1
Q , which
occur when ϑ is the phase of one of the 5d BPS KK-modes (4.6). Examples of such degenerate
walls can be found in Figure 8.
The graphical interpretation of that the truncated exponential network
Warg(ξ∗t (x,t,m,±1))(|ξ∗t (x, t,m,±1)|)
∀m ∈ Z should end at the point X = ex for some examples is shown in Figure 7.
Exponential Networks, WKB and Topological String 21
Figure 7. Left: Poles (4.4) of the Borel transform (4.3). The four points in red correspond
to ξ1 = ξ∗t
(
1 + i, 12 + i
10 , 1, 1
)
, ξ2 = ξ∗(1 + i, 1, 1), ξ3 = ξ∗t
(
1 + i, 12 + i
10 , 0, 1
)
, ξ4 = ξ∗(1 + i, 0, 1).
Right: Truncated exponential networksWarg(ξj)(|ξj |), j = 1, . . . , 4, corresponding to the poles ξj
in the Borel plane. The red dots are the point X(x) = e1+i on C×
X .
(a) ϑ = ArcTan(2( 1
10 − 2π)) (b) ϑ = ArcTan( 15 ) (c) ϑ = ArcTan(2( 1
10 + 2π))
Figure 8. Examples of exponential networks Wϑ’s on C×
X with a degenerate wall (painted in
red) at different ϑ’s for t = 1
2 + i
10 . The blue, purple and magenta dots are punctures of Σ
at X = 0, X = −1, X = − 1
Q , respectively. In each of the figures, there is a degenerate wall
connecting X = −1 and X = − 1
Q .
4.4 Local solutions in each sector
The rays of poles as shown in Figure 9 divide the Borel plane of the resolved conifold into sectors.
We define the sector
I iO
x,t,m1,m2
= I iO
x,m1
∩ I iO
x−t,m2
, (4.7)
where I iO
x,m1 are as in Section 3.5. For convenience, we neglect the subscripts x and t and label
the sectors by the 2 integers m1 and m2. Nevertheless the dependence on x and t is important
to keep in mind, especially in relation to the various domains of the exponential network.
For instance, let us consider Re(t) = 1
2 . The exponential network at ϑ = π/2 is shown on
Figure 6b, where we can clearly see two degenerate walls. In particular, we have 3 domains
corresponding to Re(x) > Re (t) = 1
2 (outer domain), 0 < Re(x) < Re (t) = 1
2 (domain between
the two circles) and Re(x) < 0 (inner domain). This means that we have 3 corresponding solu-
tions which, on the physics side, correspond to insertion of branes at different locations (external
brane, internal brane, external anti-brane). The discussion of Borel summation for each local
22 A. Grassi, Q. Hao and A. Neitzke
Figure 9. We show as an example the case 0 < Re(t) < Re(x) and arg ℏ ∈
[
0, π2
]
. The red and
brown rays of poles ξ∗(1 + i,m1, n1) and ξ
∗
t
(
1 + i, 12 +
i
10 ,m2, n2
)
, where m1,m2 ≥ 0, n1, n2 > 0
separate the Borel plane into sectors I 1O
m1,m2 , which is defined in (4.7).
solution follows directly from the discussion of the C3 example in Section 3.5 and Appendix C,
as we discuss below. We focus on the first quadrant of the Borel plane for arg(ℏ) ∈
[
0, π2
]
.
4.4.1 Re(x) and Re(x − t) same sign: brane on external leg
Let us assume
Re(x) < 0, Re(x− t) < 0.
By using Section 3.5.1, we find that in the I 1O
m1,m2 sector the Borel summation of (4.2) agrees
with
s(φ)(x, t, ℏ) = logΦ(x+ 2πim1, ℏ)− log Φ(x− t+ 2πim2, ℏ),
arg ℏ ∈ Im1,m2 , m1,m2 ≤ 0. (4.8)
In the limit m1, m2 → −∞, we get
s(φ)(x, t, ℏ) = log
(∏
r≥0
(
1 + qr+1/2ex
))
− log
(∏
r≥0
(
1 + qr+1/2Qex
))
, ℏ ∈ iR+. (4.9)
This can be viewed as the log of the resolved conifold open string partition function with a brane
insertion on the external leg. Indeed, this partition function reads
Zopen
ext (z, t, ℏ) =
(
−q1/2ez, q
)
∞(
−q1/2Qez, q
)
∞
, q = eiℏ, Q = e−t, (4.10)
where “ext” is to stress that here we are considering a brane on the external leg of the toric
diagram, see for example [72]. Hence
s(φ)(x, t, ℏ) = logZopen
ext (x, t, ℏ), ℏ ∈ iR+, Re(x) < 0, Re(x− t) < 0. (4.11)
Exponential Networks, WKB and Topological String 23
Notice that the difference between the two solutions along the real and imaginary axis is
(4.9)− (4.8)
∣∣
m1=m2=0
= log
(
−e
2πx
ℏ q̃
1
2 ; q̃
)
∞(
−e
2π(x−t)
ℏ q̃
1
2 ; q̃
)
∞
. (4.12)
Keeping in mind the analogy with the open TS/ST framework of [84, 85], we may wonder how
to relate (4.12) to the NS limit of the open string partition function for the resolved conifold.
From [23, 69], it is easy to see that the refined open partition function of the resolved conifold in
the NS and GV limits are related by some simple shifts in the arguments. Hence we express (4.12)
simply by using (4.10). We have
e(4.9)
e(4.8)
∣∣
m1=m2=0
= Zopen
ext
(
2π
ℏ
x,
2π
ℏ
t,−4π2
ℏ
)
, (4.13)
which is consistent with the open TS/ST framework of [84, 85]. However here we do not have an
honest spectral theory side so this analogy is only partial. In particular, in this simple example
it all reduces to a simple transformation
(x, t, ℏ) in (4.11) vs
(
2π
ℏ
x,
2π
ℏ
t,−4π2
ℏ
)
in (4.13). (4.14)
Similar computations can be done for Re(x) > 0, Re(x− t) > 0. In this case the local solution
on the imaginary axis corresponds to a q-brane (or anti-brane) inserted on the external leg. This
means that we have a replacement Zopen
ext →
(
Zopen
ext
)−1
and some other modifications coming
from the extra polynomial factors besides the q-Pochhammer symbol in (3.20). The difference
between the solution along real and imaginary axis is then given by
log
(−e−
2π(x−t)
ℏ q̃
1
2 ; q̃
)
∞(
−e−
2πx
ℏ q̃
1
2 ; q̃
)
∞
, (4.15)
which we can express using (4.10) as
(4.15) = log
(
1
Zopen
ext
(
−2π
ℏ x,−
2π
ℏ t,−
4π2
ℏ
)) .
4.4.2 Re(x) and Re(x − t) with opposite signs: internal leg
Let us now take
0 < Re(x) < Re(t).
By using Section 3.5.1, we find that in the I 1O
m1,m2 sector the Borel summation of (4.2) agrees
with
s(φ)(x, t, ℏ) = log Φ(x+ 2πim1, ℏ) +
2πm1
ℏ
(x+m1πi)− log Φ(x− t+ 2πim2, ℏ),
where
arg ℏ ∈ I 1O
m1,m2
, m1 ≥ 0, m2 ≤ 0.
24 A. Grassi, Q. Hao and A. Neitzke
In the limit m1,−m2 → ∞ (ℏ is on positive imaginary axis), we get
s(φ)(x, t, ℏ) = − log
(∏
r≥0
(
1 + qr+1/2e−x
))
− log
(∏
r≥0
(
1 + qr+1/2Qex
))
+
1
24
log
(
qq̃−1
)
+
i
2ℏ
x2, ℏ ∈ iR+. (4.16)
Also in this case we can express (4.16) using the open topological string free energy, but in this
case the brane has to be on the internal leg. The open string partition function of the resolved
conifold with internal leg insertion can be found for example in [76] where the refined open
amplitudes are also discussed. We have
Zopen
int (tL, tR, y, ℏ) =
(
−q1/2e−tLey; q
)
∞
(
−q1/2e−tRe−y; q
)
∞, q = eiℏ. (4.17)
Hence we have (Re(x) > 0, Re(x− t) < 0)15
s(φ)(x, t, ℏ) = − logZopen
int (t, 0, x, ℏ) +
1
24
log
(
qq̃−1
)
+
i
2ℏ
x2, ℏ ∈ iR+.
In this case the difference between the solution on the imaginary and real axis is
1(
−q̃
1
2 e
2π(x−t)
ℏ ; q̃
)
∞
(
−q̃
1
2 e−
2πx
ℏ ; q̃
)
∞
, (4.18)
which can again be expressed using (4.17), we have
(4.18) =
1
Zopen
int
(
2πt
ℏ , 0,
2π
ℏ x,−
4π2
ℏ
) .
Likewise for Re(x) < 0, Re(x− t) > 0 the situation is very similar to the one we just discussed
upon replacement to Zopen
... → 1
Zopen
...
(brane → anti-brane).
4.4.3 Either Re(x) or Re(x − t) is zero
Without loss of generality, we choose the following example to elucidate this case:
Re(x) = 0, Re(t) > 0.
For this choice of parameters, the solution for ℏ ∈ iR+ is
s(φ)(x, t, ℏ) =
1
2
(
log
(
e
x2i
2ℏ q
1
24
q̃
1
24
1(
−q
1
2 e−x; q
)
∞
)
+ log
(
−q
1
2 ex; q
)
∞
)
− log
(
−q1/2Qex; q
)
∞.
Using the topological partition functions (4.10) and (4.17), this can be expressed as
s(φ)(x, t, ℏ) =
1
2
(
− logZopen
int (t, 0, x, ℏ) +
1
24
log
(
qq̃−1
)
+
i
2ℏ
x2
)
+
1
2
log
(
Zopen
ext (x, t, ℏ)
)
.
This is the average between the open string amplitude with a brane inserted in an external
leg (4.10), and the open string amplitude with a brane inserted in an internal leg (4.17).
The jump from the imaginary to the real axis is
1
2
log
( (
−e
2πx
ℏ q̃
1
2 , q̃
)
∞(
−q̃
1
2 e−
2πx
ℏ ; q̃
)
∞
)
− log
((
−e
2π(x−t)
ℏ q̃
1
2 , q̃
)
∞
)
,
which can again be expressed using both Zopen
int and Zopen
ext with a transformation of the argument
of the form (4.14).
15We chose tL = t, tR = 0, y = x but this is obviously not the only choice.
Exponential Networks, WKB and Topological String 25
4.4.4 Re(x) = 0 and Re(t) = 0
In this case all the poles are on the imaginary axis. Median summation along the imaginary
axis gives
s(φ)(x, ℏ) =− it(t− 2x)
4ℏ
+
1
2
log
( (
−exq1/2; q
)
∞(
−e−xq1/2; q
)
∞
(
−e−(x−t)q1/2; q
)
∞(
−e(x−t)q1/2; q
)
∞
)
, ℏ ∈ iR+.
In term of brane insertions, the expression inside the logarithm can be written either as a product
of two external brane or as a product of two internal brane. This is in line with the diagrammatic
picture of the toric diagram which, in this situation, is very degenerate.
The jump in the first quadrant to go from the imaginary to the real axis, is given by
1
2
log
(
−e
2πx
ℏ q̃
1
2 , q̃
)
∞(
−e−
2πx
ℏ q̃
1
2 ; q̃
)
∞
(
−e−
2π(x−t)
ℏ q̃
1
2 ; q̃
)
∞(
−e
2π(x−t)
ℏ q̃
1
2 , q̃
)
∞
.
This is in line with our expectations.
4.5 The closed sector
We now move to the study of the (refined) closed topological string free energy. This part is
analogous to the exact WKB of the 4d quantum periods. We are going to show that it is closely
related to the 5d BPS states in comparison to the relation between the open topological string
free energy and the 3d-5d BPS states in the open sector. The (refined) closed topological string
free energy quantity depends on two set of parameters: the Ω background parameters (ϵ1 and ϵ2)
and the Kähler parameter t. We will further restrict to
ϵ2 = αϵ1, ϵ1 = ℏ.
As discussed around (B.5) one has a simple symmetry ℏ → −ℏ. Hence one can study without
loss of generality the case Re(ℏ) ≥ 0. Likewise, since
Li−n(z) = (−1)n−1Li−n
(
1
z
)
, n ∈ N+
the Borel summation has the symmetry
t→ −t.
Hence we can restrict without loss of generality to the case
Re(t) ≥ 0.
In the rest of the section, we compute analytically the Borel transform and Borel summantion
of the (refined) closed topological string free energy. We also find that the singularity structure
of the Borel plane for α = 0 and α = −1 is identical and, as we predicted, the Borel singularities
correspond to the central charges of 5d BPS KK-modes for the resolved conifold (4.6). This is
the 5d generalization of [52, 53]. From the stringy perspective such 5d BPS KK-modes come
from D2±mD0 branes in the Type IIA theory compactified on X.16
When α ̸= 0,−1 we have also an additional series of poles whose positions are at
−2πn
(the central charges of 5d BPS KK-modes)
α
. (4.19)
16We do not see the purely D0 brane contribution in the Borel plane because we are not including the constant
map contribution in the (refined) free energy.
26 A. Grassi, Q. Hao and A. Neitzke
When α → 0 these poles go to infinity while when α → −1 they merge with the other series of
poles.
Note also that one should be able to obtain information on the closed sector starting from the
open sector. For example in the ϵ1 = −ϵ2 phase of the Ω background this can be done using the
topological recursion framework, see for instance [18, equation (3.9)]. For the NS phase see for
instance [1] and references there. Nevertheless our analysis of the closed sector will be carried
out independently of the discussion of the open sector.
4.5.1 The NS sector α = 0
The first case that we study is the one where ϵ1 = ℏ and ϵ2 = 0. This is the so-called NS phase
of the Ω background. The corresponding perturbative free energy is17
FNS
WKB = ℏ
∑
g≥0
ℏ2g−2FNS
g (t), (4.20)
where
FNS
g (t) =
(i)2g−1B2g
(
1
2
)
(2g)!
Li3−2g
(
e−t
)
. (4.21)
The NS free energy is [69, 91]
FNS(ℏ, t) =
1
2i
∑
k≥1
e−kt
k2
1
sin(kℏ/2)
. (4.22)
Further expansion of (4.20) with respect to Q = e−t agrees with the expansion of (4.22) with
respect to ℏ. The Borel transform of the series (4.21) is
BFNS
WKB(t, ξ) =
∑
g≥2
FNS
g (t)
ξ2g−3
(2g − 3)!
= f1(ξ) ⋆ f2(ξ, t),
where ⋆ stands for the Hadamard product, and we use
f1(ξ) =
∞∑
g=2
B2g(1/2)
(2g)!
(ξ)2g−3 =
ξ2 + 12ξ csch
( ξ
2
)
− 24
24ξ3
,
f2(ξ, t) =
∑
g≥2
Li3−2g
(
e−t
)
(i)2g−1 ξ2g−3
(2g − 3)!
= i
sin(ξ)
2 cos(ξ)− 2 cosh(t)
.
It follows from the definition of the Hadamard product that
BFNS
WKB(t, ξ) =
1
2πi
∮
γ
f1(s)f2
(
ξ
s
, t
)
ds
s
,
where γ is a contour around 0 including only poles of f2
( ξ
s
)
,18 located at
s = ± iξ
t+ 2iπm
, m ∈ Z.
17Usually there is also an overall piece which is the analogous of the ”constant map” contribution in the standard
topological string. In turn this is given by the closed C3 free energy. Here we will omit this contribution and, as
a consequence, in the Borel plane we won’t see the contribution from purely D0-branes.
18We assume t is valued in the domain such that the absolute values of poles of f2
(
ξ
s
)
are always smaller than
the absolute values of poles of f1(s). The result for generic values of t is defined by analytic continuation.
Exponential Networks, WKB and Topological String 27
Computing the integral by residues, the Borel transform can be expressed as an exact function
in ξ,
BFNS
WKB(t, ξ) =
∑
m∈Z
iξ2 + 12ξ(t+ 2πim) csch
( iξ
2(t+2πim)
)
+ 24i(t+ 2πim)2
24ξ3
.
This expression shows explicitly that the poles of the Borel transform are located at
ξ = 2πn (t+ 2πim) , m ∈ Z, n ∈ Z/{0}. (4.23)
As we discussed at the beginning of Section 4.5, these indeed correspond to the central charges
of 5d BPS KK-modes for the resolved conifold: see (4.6).
Let us now look at the Borel summation as defined in (A.4). We start from the positive
imaginary axis. Along this axis we find that
s
(
FNS
WKB
)
(t, ℏ) = FNS(ℏ, t), ℏ ∈ iR+. (4.24)
Note that if Re(t) = 0 all the poles in the Borel plane are along the imaginary axis. Hence we
should understand the l.h.s. of (4.24) as median summation. To obtain the exact expression
of Borel summation in other sectors, we simply sum over the contributions coming from the
poles lying along the rays that we cross when moving from one sector to another. For example,
if we want to obtain the expression for Borel summation on the real axis, we sum over the
contributions from all the poles in the first quadrant of the Borel plane. Let assume for example
that Re(t) > 0, Im(t) ∈ (0, 2π) and ℏ ∈ R+. Since m ∈ Z the relevant poles in the first quadrant
are at
ξ =2πn(t+ 2iπm), m ≥ 0, n > 0. (4.25)
The corresponding contribution is
2πi
∑
m≥0,n≥1
Res
(
BFNS
WKB(t, ξ), 2πn(t+ 2iπm)
)
=
ℏ
4πi
∑
n≥1
(−1)n
n2 sin
(
2π2
ℏ n
)e− 2πn
ℏ (t−πi) =
ℏ
2π
FNS
(
4π2
ℏ
, e
2π
ℏ (t−iπ)+iπ)
)
.
Hence the Borel summation for ℏ ∈ R+ is19
s
(
FNS
WKB
)
(t, ℏ) =
FNS(ℏ, t) + ℏ
2πF
NS
(
4π2
ℏ , e
2π
ℏ (t−iπ)+iπ
)
, Im(t) ∈ (0, 2π), 0 < Re(t),
FNS(ℏ, t) + ℏ
2πF
NS
(
4π2
ℏ , e
2π
ℏ (t−iπ)+iπ
)
− ℏLi2
(
−e−
2πt
ℏ
)
4π , t ∈ R+.
(4.26)
This is very much expected from the point of view of the spectral theory for relativistic integrable
systems [51, 54, 97, 98].
4.5.2 The GV sector α = −1
We now study the case where ϵ1 = −ϵ2 = ℏ. This is the so-called GV phase of the Ω background
(also known as self-dual or standard topological string phase). The perturbative expansion of
the free energy is
FGV
WKB(t) =
∑
g≥0
ℏ2g−2Fg(t), (4.27)
19If Re(t) = 0 and Im(t) ∈ (0, 2π) all the poles of the Borel transform are on the imaginary axis. Hence, away
from this axis, Borel summation agrees with FNS(ℏ, t) + ℏ
2π
FNS
(
4π2
ℏ , e
2π
ℏ (t−iπ)+iπ)
)
. There is only one subtlety
which is that FNS is not well defined if Re(t) = 0 and Im(ℏ) = 0.
28 A. Grassi, Q. Hao and A. Neitzke
where
F0(t) = −Li3(exp(−t)), F1(t) = − 1
12
Li1(exp(−t)),
Fg(t) = −(−1)g−1B2g Li3−2g(exp(−t))
2g(2g − 2)!
, g ≥ 2,
and Bk is the standard Bernoulli number. The Gopakumar–Vafa free energy is
FGV(t, ℏ) = −
∑
m≥1
e−mt
m
(
2 sin
(
mℏ
2
))−2
. (4.28)
If we expand (4.27) with respect to Q = e−t and (4.28) with respect to ℏ, we find agreement
between the two series.
The Borel transform of (4.27) is
BFGV
WKB(t, ξ) =
∑
g≥1
ξ2g−1 Fg+1(t)
(2g − 1)!
= −
∑
g≥1
ξ2g−1 (−1)gB2(g+1)
2(g + 1)(2g)!
f2g−1(t)
(2g − 1)!
= −f1(ξ) ⋆ f2(ξ, t),
where ⋆ is the Hadamard product and we choose
f1(ξ) =
−ξ2 + 3ξ2 csc2
( ξ
2
)
− 12
12ξ3
, f2(ξ, t) =
1
2
(f(t+ ξ)− f(t− ξ)) ,
where f(t) = 1
1−et . By using the integral representation of the Hadamard product we get
BFGV
WKB(t, ξ) = − 1
2πi
∮
γ
f1(s)f2
(
ξ
s
, t
)
ds
s
.
As before the integral contour γ is chosen such that it only includes the contribution from the
poles of f2(ξ/s) at
s = ± ξ
t+ 2iπm
, m ∈ Z.
Hence we get
BFGV
WKB(t, ξ) = 2
∑
m∈Z
ξ2 + 3ξ2 csch2
( ξ
4πm−2it
)
− 12(2πm− it)2
24ξ3
. (4.29)
The singularities of the Borel transform (4.29) are at
ξ = 2πn (t+ 2πim) , m ∈ Z, n ∈ Z/{0}. (4.30)
This is exactly as in (4.25), and in agreement with the 5d BPS KK-modes central charges (4.6).
Let us look at the Borel summation. When ℏ is purely imaginary we find
s
(
FGV
WKB
)
(t, ℏ) = FGV(ℏ, t), ℏ ∈ iR+.
As before, to obtain the exact expression in other sectors we need to take into account the
residue contribution from the poles in the Laplace transform (A.3). For example, for ℏ ∈ R+
with a bit of algebra, we obtain
2πi
∑
m≥0,n≥1
Res
(
BFGV
WKB(t, ξ), 2πn(t+ 2iπm)
)
= − 1
2πi
∂
∂ℏ
(
ℏFNS
(
2π2
ℏ
,
2π(t− iπ)
ℏ
))
.
Exponential Networks, WKB and Topological String 29
We thus get20
s
(
FGV
WKB
)
(t, ℏ) =
FGV(ℏ, t)− 1
2πi
∂
∂ℏ
(
ℏFNS
(
2π2
ℏ
,
2π(t− iπ)
ℏ
))
,
Im(t) ∈ (0, 2π), 0 < Re(t),
FGV(ℏ, t)− 1
2πi
∂
∂ℏ
(
ℏFNS
(
2π2
ℏ
,
2π(t− iπ)
ℏ
))
−
i
(
−2πt log
(
1− e−
2πt
ℏ
)
+ ℏLi2
(
e−
2πt
ℏ
))
4ℏπ
, t ∈ R+,
(4.31)
which is in agreement with [62].
4.5.3 The refined sector ϵ2 = αϵ1, α ̸∈ Q
The refined free energy for the resolved conifold is (see, for instance, [69, equation (67)])
F ref(q1, q2, t) = −
∑
n≥1
e−nt
n
(
q
n/2
1 − q
−n/2
1
)(
q
n/2
2 − q
−n/2
2
) . (4.32)
If q1 = q−1
2 = eiℏ, we recover (4.28). If q1 = eiℏ, q2 = eiϵ2 , then we get
− lim
ϵ2→0
iϵ2F
ref
(
eiℏ, eiϵ2 , t
)
= FNS(ℏ, t),
where FNS(ℏ, t) is as in (4.22). In this section, we study the case
q1 = eiℏ, q2 = eiαℏ,
and we denote the corresponding free energy by
F (α, ℏ, t) ≡ F ref
(
eiℏ, eiαℏ, t
)
.
The perturbative refined free energy is
Fα
WKB(α, ℏ, t) =
Li3
(
e−t
)
αℏ2
−
(
α2 + 1
)
log
(
1− e−t
)
24α
+
∑
g≥2
ℏ2g−2Fα
g (t), (4.33)
where
Fα
g (t) = cg(α) Li3−2g
(
e−t
)
, cg(α) = (−1)g
g∑
k=0
B̂2kB̂2g−2k(α)
2g−2k−1,
B̂m =
(
1
2m−1
− 1
)
Bm
m!
.
The Borel transform of (4.33) is
BFα
WKB(t, ξ) =
∑
g≥2
ξ2g−3
Fα
g (t)
(2g − 3)!
=
∑
k≥0
B̂2kα
2k−1
∑
g≥k
ξ2g−3(−1)gB̂2g−2k
f2g−3(t)
(2g − 3)!
, (4.34)
20If Re(t) = 0 and Im(t) ∈ (0, 2π) all the poles of the Borel transform are on the imaginary axis. Hence, away
from this axis, Borel summation agrees with FGV(ℏ, t) − 1
2πi
∂
∂ℏ
(
ℏFNS
(
2π2
ℏ , 2π(t−iπ)
ℏ
))
. There is only one subtlety
which is that FGV is not well defined if Re(t) = 0 and Im(ℏ) = 0.
30 A. Grassi, Q. Hao and A. Neitzke
where
f(t) =
1
1− et
.
By the calculation in Appendix D, we find
BFα
WKB(t, ξ) = −
∑
n∈Z
(
α2 + 1
)
ξ2 + 6αξ2 csch
( ξ
4πn−2it
)
csch
( αξ
4πn−2it
)
− 24(2πn− it)2
24αξ3
, (4.35)
which has poles at
ξ = 2πm(t+ 2iπn), n ∈ Z, m ∈ Z/{0},
ξ =
2π
α
m(t+ 2iπn), n ∈ Z, m ∈ Z/{0}, (4.36)
in agreement with our discussion about 5d BPS KK-modes central charges, see (4.19) and (4.6).
We start from the Borel summation along the imaginary axis, we find
s
(
Fα
WKB
)
(t, ℏ) = F (α, ℏ, t), ℏ = iR+.
We now wish to go to the Borel summation along the real axis. For this we have to properly
take into account the residues of the Borel transform in the first quadrant of the Borel plane.
We find that the residue at the pole listed in (4.36) with fixed m, n is
Resm,n
(
BFα
WKB
(
t, ξeiarg(ℏ)
)
e−ξ/ℏ)
≡ Res
(
BFα
WKB
(
t, ξeiarg(ℏ)
)
e−ξ/ℏ, 2πm(t+ 2iπn)
)
+Res
(
BFα
WKB
(
t, ξeiarg(ℏ)
)
e−ξ/ℏ,
2π
α
m(t+ 2iπn)
)
=
(−1)m
4πm
(
csc
(πm
α
)
e−
2πm(t+2iπn)
αℏ + csc(παm)e−
2πm(t+2iπn)
ℏ
)
.
Note that we are considering the case α ̸∈ Q. Let us take Re(α) > 0 as an example. Since we
care about the poles in the first quadrant we take m > 0 and n ≥ 0. Hence we have to consider∑
m>0
∑
n≥0
Resm,n
(
BFα
WKB
(
t, ξeiarg(ℏ)
)
e−ξ/ℏ)
= −
∑
m>0
i(−1)m csc
(
πm
α
)
csc
(
2π2m
αℏ
)
e
2iπm(π+it)
αℏ
8πm
−
∑
m>0
i(−1)m csc(παm) csc
(
2π2m
ℏ
)
e
2iπm(π+it)
ℏ
8πm
=
1
2πi
(
F
(
2π
ℏ
,
2π
α
,
2π
αℏ
(t− iπ)
)
+ F
(
2π
αℏ
, 2πα,
2π
ℏ
(t− iπ)
))
.
Therefore for ℏ ∈ R+, we have
s
(
Fα
WKB
)
(t, ℏ) =
F (α, ℏ, t) + F
(
2π
ℏ
,
2π
α
,
2π
αℏ
(t− iπ)
)
+ F
(
2π
αℏ
, 2πα,
2π
ℏ
(t− iπ)
)
Im(t) ∈ (0, 2π), 0 < Re(t),
F (α, ℏ, t) + F
(
2π
ℏ
,
2π
α
,
2π
αℏ
(t− iπ)
)
+ F
(
2π
αℏ
, 2πα,
2π
ℏ
(t− iπ)
)
− πi
∞∑
m=1
(−1)m
(
csc
(
πm
α
)
e−
2πmt
αℏ + csc(παm)e−
2πmt
ℏ
)
4πm
0 < t ∈ R.
(4.37)
Exponential Networks, WKB and Topological String 31
We also cross-checked this result numerically. Some observations:
� Each term on the r.h.s. of equation (4.37) has a dense set of poles on the real ℏ axis.
However in the full expression these poles cancel, this is a generalization of the HMO
cancellation mechanism [60] to the refined topological string setup. After the cancellation,
the remaining regular part matches with the Borel summation.
� Even though the general structure of the r.h.s. of (4.37) resembles [59, 80], the details of
the expression are different (e.g., different shift in the Kähler parameter and in the ϵ’s).
� It would be interesting to study more in details the relation between the l.h.s. of (4.37)
and the refined CS matrix model [4] similar to what was done in [62] for the unrefined
case.
5 Comment on higher genus geometries
In this paper, we tested our proposal in two concrete examples in which the underlying mir-
ror curves have genus zero. Therefore, it is natural to ask to what extent our proposal can
be generalized to difference equations corresponding to higher genus geometries such as, say,
local P2 or local P1×P1. We expect the relation between exponential network and exact WKB,
particularly the connection between singularities in the Borel plane and BPS central charges,
still to hold also in this more general framework.21 Likewise, we also expect different domains of
the exponential network to be related to open strings with brane insertions at different places.
There are nevertheless some differences which we discuss below.
� One important difference is the fact that in genus zero geometries we can express the
local solutions to the difference equations either by using the NS or by using the GV
open topological string free partition function. Switching between these two phases is very
straightforward.
This is no longer the case for difference equations arising in quantization of mirror curves
to higher genus geometries. In this case the WKB solution to the quantum curve is
encoded in the Nekrasov–Shatashvili phase ϵ2 = 0, ϵ1 = ℏ [1] while the non-perturbative
corrections are encoded in the ϵ2 = −ϵ1 = 1
ℏ phase [26, 54, 84, 85]. One may argue
in favour of a connection between the NS and the GV phase using blowup equation as
in [13, 51, 71, 79, 89]. However this is much more subtle than for the case of the resolved
conifold.
� In the WKB solution for quantum curves of higher genus an important role is also played
by the quantum mirror map. This is a new ingredient which is absent in the resolved
conifold example (the mirror map does not get quantum corrections in this case). In
particular, even though we can compute the genus g free energy in the NS phase efficiently
via the holomorphic anomaly equation [28], we do not know an efficient way to compute
the quantum mirror map away from the large radius region of the moduli space. This
is one of the main technical obstacles we encounter when trying to construct an efficient
algorithm computing WKB for difference equations.
� Some further comments on the closed sector:
– In the case of the resolved conifold, the structure of the Borel singularities for the NS
and GV phase of the Ω background is in fact identical (see poles structure in (4.23)
21We also recall that, when ϵ1 = −ϵ2, the singularities in the Borel plane of the closed string free energy have
been studied for instance in [29, 30, 37, 56]. More precisely, it was observed that these singularities are related
to combinations of periods of the underlying CY manifold. This is of course consistent with our results since the
the central charges of 5d BPS KK-modes correspond to a particular combinations of periods.
32 A. Grassi, Q. Hao and A. Neitzke
and (4.30)). In higher genus geometries it could be that this relation is more com-
plicated. Nevertheless we know that these two Ω background phases are related by
blowup equations [51], see also [13, 89] and references there. Hence it should be
possible to find a relation between the two Borel planes. It would be interesting to
investigate this further.
– The topological vertex expression for the free energy of the resolved conifold is well
defined also for complex values of ϵ parameters. In particular, (4.22), (4.28) and
(4.32), as series expansion in Q = e−t, are convergent even when the ϵ’s are complex.
This is not the case for CY of higher genus. For example if we consider local P2
and we take ϵ1 = −ϵ2 to be complex, then the topological vertex expression (as series
expansion in Q = e−t) is divergent. See for example [59] for some numerical studies.22
– For the resolved conifold we saw that, on the axis where the ϵ’s are real, the Borel
summation of the Fg’s matches a suitable combination of free energies in different
phases of the Ω background, see (4.26), (4.31) and (4.37).
In the case of higher genus geometries this is no longer the case. Explicit tests have
been performed in [56]23 and further investigations were done in [30]. So in higher
genus examples the Borel summation does not match the non-perturbative completion
of topological string coming from the spectral theory of quantum mirror curves: there
are additional non-perturbative effects which are not captured by Borel summation24
(at least not in the chamber connected to the real ϵ axis). It would be interesting to
understand this using the framework of exponential networks.
A Conventions
In this work we study asymptotic series of the form
ϕ(x, ℏ) ∼ ℏa+1
∑
g≥0
cg(x)ℏ2g−b, cg ≈ (2g − b)!, g ≫ 1, (A.1)
where a, b are some fixed constants. For example, when considering the asymptotic series of
local solutions we have
(a, b) = (0, 2),
while for the closed topological string free energy we have b = 3. Our convention for the Borel
transform of (A.1) is to take
Bϕ(x, ξ) =
∑
g≥⌈b/2⌉
cg(x)
ξ2g−b
(2g − b)!
. (A.2)
Let
ℏ = eiϑ|ℏ|,
22For some geometries, like local P1×P1, one can nevertheless perform a partial resummation of the topological
vertex expression with respect to one of the Kähler parameters. This gives the Nekrasov type of expression. For
complex values of the coupling the latter is better behaving, see [15, 55] for related discussions. Such Nekrasov
expression is however not always available. For example we currently do not have it for local P2.
23To be precise in [56] the authors also consider the quantum mirror map. Here we do not consider the quantum
mirror map, but nevertheless we have checked that Borel summation does not agree with expressions like (4.31).
24This happens also in simpler quantum mechanical examples like the pure quartic oscillator [56].
Exponential Networks, WKB and Topological String 33
and let ρϑ be the ray along ϑ. We define the Borel summation of (A.1) as
LBϕ(x, ℏ) =
∫
ρϑ
Bϕ(x, ξ)e−
ξ
ℏdξ = eiϑ
∫ ∞
0
Bϕ
(
x, ξeiϑ
)
e
− ξ
|ℏ|dξ, (A.3)
with the understanding that, if Bϕ(x, ℏ) has poles along the integration contour, we take median
summation. We will also use
s(ϕ)(x, ℏ) = ℏa
⌊b/2⌋∑
g=0
cg(x)ℏ2g−b+1 + LBϕ(x, ℏ)
. (A.4)
B Quantum dilogarithm and q-Pochhammer functions
The Faddeev’s quantum dilogarithm [40, 41] admits an integral representation
Φb(x) = exp
(∫
R+iϵ
e−2ixz
4 sinh(zb) sinh
(
zb−1
) dz
z
)
, b2 /∈ R−, | Im(x)| < | Im(cb)|, (B.1)
where
cb =
i
2
(
b+ b−1
)
.
When Im
(
b2
)
> 0, −π < Im(x) ≤ π, the Faddeev’s quantum dilogarithm admit an alternative
representation as
Φb(x) =
(
e2πb(x+
i
2
(b+b−1)); q
)
∞(
e2πb
−1(x− i
2
(b+b−1)); q̃
)
∞
, Im
(
b2
)
> 0, (B.2)
where
q = eiℏ, q̃ = e−
4π2i
ℏ , ℏ = 2πb2
and we used (3.14). When Im
(
b2
)
< 0, (B.1) still makes sense, but (B.2) is not well defined.
In order to get an expression in terms of the q-Pochhammer symbol, we can use the symmetry
property
Φb(z) = Φb−1(z).
Since
Φb(z) =
(
e2πbz+πib2+πi; e2πib
2)
∞(
e2πb−1z−πi−πib−2 ; e−2πib−2
)
∞
, Φb−1(z) =
(
e2πb
−1z+πib−2+πi; e2πib
−2)
∞(
e2πbz−πi−πib2 ; e−2πib2
)
∞
for Im
(
b2
)
< 0, we should use
Φb−1
( x
2πb
)
=
(
−q̃−
1
2 e
2πx
ℏ ; q̃−1
)
∞(
−q−
1
2 ex; q−1
)
∞
, Im(b2) < 0.
We define
Φ(x, ℏ) = Φb
( x
2πb
)
. (B.3)
34 A. Grassi, Q. Hao and A. Neitzke
Hence
Φ(x, ℏ) =
(
−q
1
2 ex; q
)
∞(
−q̃
1
2 e
2πx
ℏ ; q̃
)
∞
, Im(ℏ) > 0,(
−q̃−
1
2 e
2πx
ℏ ; q̃−1
)
∞(
−q−
1
2 ex; q−1
)
∞
, Im(ℏ) < 0.
. (B.4)
When arg(ℏ) = 0, we use (B.3) and the integral expression of the Faddeev quantum dilog (B.1).
In the case of Re(ℏ) < 0, we define a similar function Φ̃(x, ℏ). Φ̃(x, ℏ) can be obtained by
taking advantage of the symmetry of Borel transform (3.4) under ℏ → −ℏ. Since
LBϕ(x,−ℏ) = −eiϑ
∫ ∞
0
Bϕ
(
x,−ξeiϑ
)
e
− ξ
|ℏ|dξ
= −eiϑ
∫ ∞
0
Bϕ
(
x, ξeiϑ
)
e
− ξ
|ℏ|dξ = −LBϕ(x, ℏ), (B.5)
and
1
i(−ℏ)
Li2
(
−ex
)
= − 1
iℏ
Li2
(
−ex
)
,
we get
Φ̃(x, ℏ) = es(ϕ)(x,ℏ) =
1
es(ϕ)(x,−ℏ) =
1
Φ(x,−ℏ)
, ℏ ∈ R−. (B.6)
Hence,
Φ̃(x, ℏ) =
(
−q
1
2 ex; q
)
∞(
−q̃
1
2 e−
2πx
ℏ ; q̃
)
∞
for Im(ℏ) > 0,(
−q̃−
1
2 e−
2πx
ℏ ; q̃−1
)
∞(
−q−
1
2 ex; q−1
)
∞
for Im(ℏ) < 0.
(B.7)
Some other useful identities of Φb are
Φb
(
x− ib
2
)
= Φb
(
x+
ib
2
)(
1 + e2πbx
)
,
Φb
(
x+ ib−1
)
= Φb(x)
1
1− q̃e2πb−1(x−cb)
.
It follows that
Φb
(
x− ib−1
)
= Φb(x)
(
1− q̃−1e2πb
−1(x−ib−1−cb)
)
.
Hence we have
Φb
(
x+
in
b
)
= Φb(x)
1∏n
k=1
(
1− q̃−1e2πb
−1(x+i k−1
b
−cb)
)
as well as
Φb
(
x− in
b
)
= Φb(x)
n∏
k=1
(
1− q̃−1e2πb
−1(x−i k
b
−cb)
)
= Φb(x)
n∏
k=1
(
1− q̃k−1e2πb
−1(x−cb)
)
. (B.8)
Exponential Networks, WKB and Topological String 35
C Calculations for the solutions in all other sectors
C.1 Fourth quadrant of the Borel plane and Re(x) < 0
In this case the relevant sectors of the Borel plane are
I 4O
m =
(
ϑ−m;ϑ−m−1
)
, m ≥ 1.
We sum over the residue contributions, so that the jump from the positive real axis solution to
the mth sector solution in the fourth quadrant is
2πi
m−1∑
k=0
∞∑
n=1
(−1)n
2πin
e
2πn(x+2πik+iπ)
ℏ = − log
((
−q̃−
1
2 e
2πx
ℏ ; q̃−1
)
m
)
.
Thus the solution in the mth sector is
s(ϕ)(x, ℏ) =
(
−q̃−
1
2 e
2πx
ℏ ; q̃−1
)
∞(
−q−
1
2 ex; q−1
)
∞
1(
−q̃−
1
2 e
2πx
ℏ ; q̃−1
)
m
, m ≥ 1. (C.1)
(C.1) can also be written as
Φ(x+ 2πim, ℏ) =
∏∞
i=0
(
1 + q̃−i− 1
2
−me
2πx
ℏ
)∏∞
i=0
(
1 + q−i− 1
2
)
=
(
−q̃−
1
2 e
2πx
ℏ ; q̃−1
)
∞(
−q−
1
2 ex; q−1
)
∞
1(
−q̃−
1
2 e
2πx
ℏ ; q̃−1
)
m
, m ≥ 1.
Hence we have
s(ϕ)(x, ℏ) = log Φ(x+ 2πim, ℏ), arg(ℏ) ∈ I 4O
m , m ≥ 0.
The solution along negative imaginary axis is obtained by taking m→ ∞:
s(ϕ)(x, ℏ) = − log
((
−q−
1
2 ex; q−1
)
∞
)
, ℏ ∈ iR−, Re(x) < 0. (C.2)
C.2 4th quadrant of the Borel plane and Re(x) > 0
In this case the relevant sector of the Borel plane is
I 4O
m =
(
ϑ+m−1;ϑ
+
m
)
, m ≤ −1.
We sum over the residue contributions, so that the jump from positive real axis solution to the
mth sector solution in the fourth quadrant is
2πi
−m−1∑
k=0
∞∑
n=1
−(−1)n
2πin
e−
2πn(x−2πik−iπ)
ℏ = log
((
−q̃−
1
2 e−
2πx
ℏ ; q̃−1
)
−m
)
, m ≤ −1.
So the solution in the mth sector is(
−q̃−
1
2 e
2πx
ℏ ; q̃−1
)
∞(
−q−
1
2 ex; q−1
)
∞
(
−q̃−
1
2 e−
2πx
ℏ ; q̃−1
)
−m
, m ≤ −1,
36 A. Grassi, Q. Hao and A. Neitzke
which can also be written as
s(ϕ)(x, ℏ) = logΦ(x+ 2πim, ℏ) +
2πm(x+mπi)
ℏ
, arg(ℏ) ∈ I 4O
m , m ≤ −1.
Taking the limit m→ −∞, the solution along negative imaginary axis is(
−q̃−
1
2 e
2πx
ℏ ; q̃−1
)
∞(
−q−
1
2 ex; q−1
)
∞
(
−q̃−
1
2 e−
2πx
ℏ ; q̃−1
)
∞
=
ϑ3
(
− iπx
ℏ , q̃
− 1
2
)
q̃
1
24 η
(
2π
ℏ
) 1(
−q−
1
2 ex; q−1
)
∞
=
e
x2i
2ℏ q
1
24
q̃
1
24
(
−q−
1
2 e−x; q−1
)
∞.
The matching with Borel summation is
s(ϕ)(x, ℏ) =
x2i
2ℏ
+ log
(
q
1
24
q̃
1
24
(
−q−
1
2 e−x; q−1
)
∞
)
, ℏ ∈ iR−, Re(x) < 0.
C.3 2nd quadrant, Re(x) < 0
In this case the relevant sector of the Borel plane is
I 2O
m =
(
ϑ+m;ϑ+m−1
)
, m ≥ 1.
We sum over the residue contributions, so that the jump from negative real axis solution to
the mth sector solution in the second quadrant is
2πi
m−1∑
k=0
∞∑
n=1
−(−1)n
2πin
e−
2πn(x+2πik+iπ)
ℏ = log
((
−q̃
1
2 e−
2πx
ℏ ; q̃
)
m
)
, m ≥ 1. (C.3)
Solution along the negative real axis can be obtained by dividing (3.16) by exponential
of (C.3) in the limit m→ ∞
Φ̃(x) ≡
(
−q
1
2 ex; q
)
∞(
−q̃
1
2 e−
2πx
ℏ ; q̃
)
∞
, Im(ℏ) > 0. (C.4)
Or we can directly using the symmetry (B.6) to get (B.7).
So the solution in the mth sector is(
−q
1
2 ex; q
)
∞(
−q̃
1
2 e−
2πx
ℏ ; q̃
)
∞
(
−q̃
1
2 e−
2πx
ℏ ; q̃
)
m
, m ≥ 1, (C.5)
which can also be written as
Φ̃(x+ 2πim, ℏ) =
(
−q
1
2 ex; q
)
∞(
−q̃
1
2 e−
2πx
ℏ ; q̃
)
m
= (C.5), m ≥ 1.
Hence
s(ϕ)(x, ℏ) = log Φ̃(x+ 2πim, ℏ), arg(ℏ) ∈ I 2O
m , m ≥ 1.
Exponential Networks, WKB and Topological String 37
C.4 2nd quadrant, Re(x) > 0
In this case the relevant sector of the Borel plane is
I 2O
m =
(
ϑ−m−1;ϑ
−
m
)
, m ≤ −1.
Summing over the residue contributions, the jump from negative real axis solution to the
mth sector solution is
2πi
−m−1∑
k=0
∞∑
n=1
(−1)n
2πin
e
2πn(x−2πik−iπ)
ℏ = log
(
1(
−q̃
1
2 e
2πx
ℏ ; q̃
)
m
)
.
Therefore, the solution in the mth sector is(
−q
1
2 ex; q
)
∞(
−q̃
1
2 e−
2πx
ℏ ; q̃
)
∞
1(
−q̃
1
2 e
2πx
ℏ ; q̃
)
−m
, m ≤ −1, (C.6)
which can also be written as
Φ̃(x+ 2πim, ℏ)e
2πm(x+mπi)
ℏ = Φ̃(x, ℏ)
e
2πm(x+mπi)
ℏ(
−e−
2πx
ℏ q̃−
1
2 ; q̃−1
)
−m
= (C.6).
Hence
s(ϕ)(x, ℏ) = log Φ̃(x+ 2πim, ℏ) +
2πm(x+mπi)
ℏ
, arg(ℏ) ∈ I 2O
m , m ≤ −1.
C.5 3rd quadrant, Re(x) < 0
In this case the relevant sector of the Borel plane is
I 3O
m =
(
ϑ+m;ϑ+m−1
)
, m ≤ −1.
We sum over the residue contributions and get the jump from negative real axis solution to the
mth sector solution is
−2πi
−m−1∑
k=0
∞∑
n=1
−(−1)n
2πin
e−
2πn(x−2πik−iπ)
ℏ = log
(
1(
−q̃−
1
2 e−
2πx
ℏ ; q̃−1
)
−m
)
, m ≤ −1. (C.7)
The solution along negative real axis can be obtained by multiply (C.2) by the inverse of e(C.7)
in the limit m→ −∞. We get
Φ̃(x, ℏ) =
(
−q̃−
1
2 e−
2πx
ℏ ; q̃−1
)
∞(
−q−
1
2 ex; q−1
)
∞
, Im(ℏ) < 0,
which is valid for Im(ℏ) < 0, compared to (C.4) which is valid for Im(ℏ) > 0. Alternatively, we
can directly using the symmetry (B.6) to get (B.7).
So the solution in the mth sector is(
−q̃−
1
2 e−
2πx
ℏ ; q̃−1
)
∞(
−q−
1
2 ex; q−1
)
∞
1(
−q̃−
1
2 e−
2πx
ℏ ; q̃−1
)
−m
, m ≤ −1,
which can also be written as
s(ϕ)(x, ℏ) = log Φ̃(x+ 2πim, ℏ), arg(ℏ) ∈ I 3O
m , m ≤ −1.
38 A. Grassi, Q. Hao and A. Neitzke
C.6 3rd quadrant, Re(x) > 0
In this case the relevant sector of the Borel plane is
I 3O
m =
(
ϑ−m−1;ϑ
−
m
)
, m ≥ 1.
We sum over the residue contributions, so that the jump from negative real axis solution to the
mth sector solution in the fourth quadrant is
−2πi
m−1∑
k=0
∞∑
n=1
(−1)n
2πin
e
2πn(x+2πik+iπ)
ℏ = log
((
−q̃−
1
2 e
2πx
ℏ ; q̃−1
)
m
)
, m ≥ 1.
Therefore the solution in the mth sector is(
−q̃−
1
2 e−
2πx
ℏ ; q̃−1
)
∞(
−q−
1
2 ex; q−1
)
∞
(
−q̃−
1
2 e
2πx
ℏ ; q̃−1
)
m
, m ≥ 1,
which can also be written as
s(ϕ)(x, ℏ) = log Φ̃(x+ 2πim, ℏ) +
2πm(x+mπi)
ℏ
, arg(ℏ) ∈ I 3O
m , m ≥ 1.
C.7 Re(x) = 0
As discussed in Section 3.5.3, along the positive imaginary axis, the solution is
s(ϕ)(x, ℏ) =
1
2
(
i
(
12x2 + ℏ2 + 4π2
)
24ℏ
− log
((
−e−x+ iℏ
2 ; eiℏ
)
∞(
−ex+
iℏ
2 ; eiℏ
)
∞
))
, Re(x) = 0, ℏ ∈ iR+.
Similarly, along the negative imaginary axis, we have
s(ϕ)(x, ℏ) =
1
2
(
i
(
12x2 + ℏ2 + 4π2
)
24ℏ
+ log
((
−e−x− iℏ
2 ; e−iℏ)
∞(
−ex−
iℏ
2 ; e−iℏ
)
∞
))
, Re(x) = 0, ℏ ∈ iR−.
For all other values of ℏ with Re(ℏ) > 0, the Borel summation matches with Φ(x):
s(ϕ)(x, ℏ) = logΦ(x, ℏ), Re(x) = 0, Re(ℏ) > 0.
And for all other values of ℏ with Re(ℏ) < 0, the Borel summation matches with Φ̃(x):
s(ϕ)(x, ℏ) = log Φ̃(x, ℏ), Re(x) = 0, Re(ℏ) < 0.
D Borel transform at generic α
In this appendix, we show the calculation for (4.35). It is convenient to write (4.34) as
Gα(t, ξ) = −
∞∑
k=0
(
1− 22k−1
)
B2k
(2k)!
Exponential Networks, WKB and Topological String 39
×
∞∑
g=k
(
1− 22g−2k−1
)
B2g−2k(i/2)
2g−2Li3−2g(Q)
(2g − 3)!
(α)2g−2k−1
(2g − 2k)!
ξ2g−3,
which we write as
Gα(t, ξ) = −
3∑
i=1
G(i)
α (t, ξ),
where
G(1)
α (t, ξ) =
∑
g≥2
4−g
(
4g − 2
)
B2gα
2g−1(iξ)2g Li3−2g
(
e−t
)
ξ3(2g)!(2g − 3)!
,
G(2)
α (t, ξ) = −
∑
g≥2
2−2g−3
(
4g − 8
)
B2g−2α
2g−3(iξ)2g Li3−2g
(
e−t
)
3ξ3(2g − 3)!(2g − 2)!
,
G(3)
α (t, ξ) =
∞∑
k=2
(
1− 22k−1
)
B2k
(2k)!
×
∞∑
g=k
(
1− 22g−2k−1
)
B2g−2k(i/2)
2g−2Li3−2g(Q)
(2g − 3)!
(α)2g−2k−1
(2g − 2k)!
ξ2g−3.
We begin with G
(3)
α (t, ξ). Let us first consider the second sum only. We have
∞∑
g=k
(
1− 22g−2k−1
)
B2g−2k(i/2)
2g−2Li3−2g(Q)
(2g − 3)!
(α)2g−2k−1
(2g − 2k)!
ξ2g−3 = fα1 (ξ) ⋆ f2(ξ, t),
where
fα1 (ξ) = ξ−1
∑
g≥k
(
1− 22g−2k−1
)
B2g−2k(iξ/2)
2g−2 (α)
2g−2k−1
(2g − 2k)!
= −
4−k(iξ)2k csc
(αξ
2
)
ξ2
,
and
f2(ξ, t, k) =
∑
g≥k
ξ2g−3 f
2g−3(t)
(2g − 3)!
=
1
2
(f(t+ ξ)− f(t− ξ))−
k−1∑
g=2
ξ2g−3 f
2g−3(t)
(2g − 3)!
= f̂2(ξ, t)−
k−1∑
g=2
ξ2g−3 f
2g−3(t)
(2g − 3)!
,
where
f(t) =
1
1− et
, f̂2(ξ, t) =
sinh(ξ)
2 cosh(t)− 2 cosh(ξ)
.
Then we have
G(3)
α (t, ξ) =
1
2πi
∮
γ
fα1 (s)f2
(
ξ
s
, t
)
ds
s
=
1
2πi
∮
γ
fα1 (s)f̂2
(
ξ
s
, t
)
ds
s
−
k−1∑
g=2
1
2πi
∮
γ
fα1 (s)
((
ξ
s
)2g−3 f2g−3(t)
(2g − 3)!
)
ds
s
,
40 A. Grassi, Q. Hao and A. Neitzke
where γ only include poles of f̂2 at s = ± ξ
t+2iπn , n ∈ Z with residue
− i
4π
∑
n∈Z
4−kξ2k−2(2πn− it)1−2k csch
(
αξ
4πn− 2it
)
.
Hence we have
G(3)
α (t, ξ) =
∞∑
k=2
(
1− 22k−1
)
B2k
(2k)!
∑
n∈Z
4−kξ2k−2(2πn− it)1−2k csch
(
αξ
4πn− 2it
)
=
∑
n∈Z
csch
( ξ
4πn−2it
)
csch
( αξ
4πn−2it
)
4ξ
+
(
ξ2 − 24(2πn− it)2
)
csch
( αξ
4πn−2it
)
48ξ2(2πn− it)
.
We now look at G
(1)
α (t, ξ). We have
G(1)
α (t, ξ) =
1
2πi
∮
fα3 (s)f̂2(ξ/s, t)
ds
s
,
where f̂2 is defined above and
fα3 (ξ) =
∑
g≥2
4−g (4g − 2)B2gα
2g−1(iξ)2g
ξ3(2g)!
=
αξ2 − 12ξ csc
(
αξ
2
)
+ 24
α
24ξ3
.
Looking at the residue we find
G(1)
α (t, ξ) =
∑
n∈Z
α2ξ2 − 12αξt csc
( αξ
2(t+2iπn)
)
+ 24παξn csch
( αξ
4πn−2it
)
− 24(2πn− it)2
24αξ3
We now look at G
(2)
α (t, ξ). We have
G(2)
α (t, ξ) =
1
2πi
∮
fα4 (s)f̂2(ξ/s, t)
ds
s
,
where f̂2 is defined above and
fα4 (ξ) = −
∑
g≥2
2−2g−3 (4g − 8)B2g−2α
2g−3(iξ)2g
3ξ3(2g − 2)!
=
1
48
(
2
αξ
− csc
(
αξ
2
))
.
Looking at the residue we find
G(2)
α (t, ξ) =
∑
n∈Z
1
48
(
2
αξ
+
csch
( αξ
4πn−2it
)
−2πn+ it
)
.
Hence by combining all together, we find
Gα(t, ξ) = −
∑
n∈Z
(
α2 + 1
)
ξ2 + 6αξ2 csch
( ξ
4πn−2it
)
csch
( αξ
4πn−2it
)
− 24(2πn− it)2
24αξ3
.
Acknowledgements
We would like to thank Murad Alim, Mat Bullimore, Fabrizio Del Monte, Lotte Hollands, Yakov
Kononov, Pietro Longhi, Marcos Mariño, Sebastian Schulz, Shamil Shakirov, Ivan Tulli and
Daniel Zhang for helpful discussion. We also thank the referees for reviewing the manuscript.
The work of AN is supported by National Science Foundation grant 2005312 (DMS). The work
of AG is partially supported by the Fonds National Suisse, Grant No. 185723 and by the NCCR
“The Mathematics of Physics” (SwissMAP).
Exponential Networks, WKB and Topological String 41
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https://arxiv.org/abs/1711.01099
1 Introduction
2 Stokes phenomena and BPS particles
2.1 Stokes phenomena in 2d-4d systems
2.2 Chiral couplings and spectral networks
2.3 Flavor masses and exponential networks
2.4 Compactified 3d-5d systems and exponential networks
3 A simple model: C^3
3.1 All-orders WKB expansion of local solutions
3.2 Field theory and BPS states
3.3 Exponential network
3.4 The exponential network and exact WKB
3.5 Local solutions in each sector
3.5.1 First quadrant of the Borel plane and Re(x)<0
3.5.2 First quadrant of the Borel plane and Re(x)>0
3.5.3 First quadrant of the Borel plane and Re(x)=0
3.5.4 Summary and comments
3.6 Jumps of local solutions via analytic continuation
3.7 The closed sector and the McMahon function
4 The resolved conifold
4.1 All-orders WKB expansion of local solutions
4.2 BPS states in 3d-5d system
4.3 The exponential network and exact WKB
4.4 Local solutions in each sector
4.4.1 Re(x) and Re(x-t) same sign: brane on external leg
4.4.2 Re(x) and Re(x-t) with opposite signs: internal leg
4.4.3 Either Re(x) or Re(x-t) is zero
4.4.4 Re(x)=0 and Re(t)=0
4.5 The closed sector
4.5.1 The NS sector alpha=0
4.5.2 The GV sector alpha=-1
4.5.3 The refined sector epsilon_2=alpha epsilon_1, alpha not in Q
5 Comment on higher genus geometries
A Conventions
B Quantum dilogarithm and q-Pochhammer functions
C Calculations for the solutions in all other sectors
C.1 Fourth quadrant of the Borel plane and Re(x)<0
C.2 4th quadrant of the Borel plane and Re(x)>0
C.3 2nd quadrant, Re(x)<0
C.4 2nd quadrant, Re(x)>0
C.5 3rd quadrant, Re(x)<0
C.6 3rd quadrant, Re(x)>0
C.7 Re(x)=0
D Borel transform at generic alpha
References
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| id | nasplib_isofts_kiev_ua-123456789-212020 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-18T21:47:42Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Grassi, Alba Hao, Qianyu Neitzke, Andrew 2026-01-22T09:21:15Z 2023 Exponential Networks, WKB and Topological String. Alba Grassi, Qianyu Hao and Andrew Neitzke. SIGMA 19 (2023), 064, 44 pages 1815-0659 2020 Mathematics Subject Classification: 39A70; 40G10; 81T30; 81T60 arXiv:2201.11594 https://nasplib.isofts.kiev.ua/handle/123456789/212020 https://doi.org/10.3842/SIGMA.2023.064 We propose a connection between 3d-5d exponential networks and exact WKB for difference equations associated to five-dimensional Seiberg-Witten curves, or equivalently, to quantum mirror curves to toric Calabi-Yau threefolds : the singularities in the Borel planes of local solutions to such difference equations correspond to central charges of 3d-5d BPS KK-modes. It follows that there should be distinguished local solutions of the difference equation in each domain of the complement of the exponential network, and these solutions jump at the walls of the network. We verify and explore this picture in two simple examples of 3d-5d systems, corresponding to taking the toric Calabi-Yau to be either ℂ³ or the resolved conifold. We provide the full list of local solutions in each sector of the Borel plane and in each domain of the complement of the exponential network, and find that local solutions in disconnected domains correspond to non-perturbative open topological string amplitudes on with insertions of branes at different positions of the toric diagram. We also study the Borel summation of the closed refined topological string free energy on and the corresponding non-perturbative effects, finding that central charges of 5d BPS KK-modes are related to the singularities in the Borel plane. We would like to thank Murad Alim, Mat Bullimore, Fabrizio Del Monte, Lotte Hollands, Yakov Kononov, Pietro Longhi, Marcos Marino, Sebastian Schulz, Shamil Shakirov, Ivan Tulli, and Daniel Zhang for helpful discussions. We also thank the referees for reviewing the manuscript. The work of AN is supported by National Science Foundation grant 2005312 (DMS). The work of AG is partially supported by the Fonds National Suisse, Grant No.185723, and by the NCCR “The Mathematics of Physics” (SwissMAP). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Exponential Networks, WKB and Topological String Article published earlier |
| spellingShingle | Exponential Networks, WKB and Topological String Grassi, Alba Hao, Qianyu Neitzke, Andrew |
| title | Exponential Networks, WKB and Topological String |
| title_full | Exponential Networks, WKB and Topological String |
| title_fullStr | Exponential Networks, WKB and Topological String |
| title_full_unstemmed | Exponential Networks, WKB and Topological String |
| title_short | Exponential Networks, WKB and Topological String |
| title_sort | exponential networks, wkb and topological string |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212020 |
| work_keys_str_mv | AT grassialba exponentialnetworkswkbandtopologicalstring AT haoqianyu exponentialnetworkswkbandtopologicalstring AT neitzkeandrew exponentialnetworkswkbandtopologicalstring |