††institutetext:
aLaboratoire Charles Coulomb (L2C), Université de Montpellier, CNRS, F-34095,
Montpellier, France
bDépartement de Physique Théorique et Section de Mathématiques
Université de Genève, Genève, CH-1211 Switzerland
c Laboratoire de Physique Théorique et Hautes Energies (LPTHE), UMR 7589,
CNRS-Sorbonne Université, Campus Pierre et Marie Curie, 4 place Jussieu,
F-75005 Paris, France

# Resurgence of refined topological strings and dual partition functions

Sergei Alexandrova, Marcos Mariñob and Boris Piolinec
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the resurgent structure of the refined topological string partition
function on a non-compact Calabi-Yau threefold, at large orders in the string
coupling constant $g_{s}$ and fixed refinement parameter ${\mathsf{b}}$. For
${\mathsf{b}}\neq 1$, the Borel transform admits two families of simple poles,
corresponding to integral periods rescaled by ${\mathsf{b}}$ and
$1/{\mathsf{b}}$. We show that the corresponding Stokes automorphism is
expressed in terms of a generalization of the non-compact quantum dilogarithm,
and we conjecture that the Stokes constants are determined by the refined DT
invariants counting spin-$j$ BPS states. This jump in the refined topological
string partition function is a special case (unit five-brane charge) of a more
general transformation property of wave functions on quantum twisted tori
introduced in earlier work by two of the authors. We show that this property
follows from the transformation of a suitable refined dual partition function
across BPS rays, defined by extending the Moyal star product to the realm of
contact geometry.

##  1 Introduction

As a mathematically well-defined subsector of type II superstring theories,
topological string theory provides a prime arena for exploring the non-
perturbative completion of the asymptotic series predicted by the perturbative
genus expansion in string theory. In cases where the target Calabi-Yau
threefold $X$ admits a large $N$ dual, a non-perturbative formulation is
available, which resums the perturbative series to all orders in the
topological string coupling $g_{s}=1/N$, for fixed values of the moduli
$\boldsymbol{t}$ (corresponding to t’Hooft couplings in the dual gauge theory)
at least for integer values of $N$. Non-perturbative corrections of order
$e^{-1/g_{s}}$ typically arise from tunneling (or instanton) effects in the
dual matrix model. In general however, one has only access to the genus-$g$
free energy ${\cal F}_{g}(\boldsymbol{t})$ occurring at order $g_{s}^{2g-2}$
in the perturbative expansion, and non-perturbative corrections are ambiguous,
without further assumptions.

Assuming that the putative, non-perturbative topological string partition
function ${\cal F}(\boldsymbol{t};g_{s})$ belongs to a suitable class of
resurgent functions, one can however extract much information about non-
perturbative corrections from the growth of the coefficients ${\cal
F}_{g}(\boldsymbol{t})$ at large genus. This method was proposed early on in
Shenker:1990uf  for general string theory models, and first applied to the
topological string in mm-open ; msw ; mmnp ; ps09 . In particular, the
instanton action ${\cal A}$ controlling the size of the leading instanton
effects $e^{-{\cal A}/g_{s}}$ can be read off from the location of the
singularity closest to the origin in the Borel plane.

While the explicit computation of the free energies ${\cal
F}_{g}(\boldsymbol{t})$ is usually impractical at large genus, they are
strongly constrained by the holomorphic anomaly equations (HAE) bcov .
Following cesv1 ; cesv2 , much progress has been made recently in constructing
the trans-series solution to the HAE gm-multi ; gkkm ; Iwaki:2023cek ; ms . In
particular, it was shown that the instanton actions ${\cal A}$ entering in the
trans-series are equal (up to overall normalization) to the central charge
$Z(\gamma)$ of vectors $\gamma$ in the charge lattice (the Grothendieck
lattice $K_{0}(X)$ in the $A$-model, or the homology lattice
$H_{3}(X,{\mathbb{Z}})$ in the B-model). Moreover, the Stokes automorphism
$\mathfrak{S}$ controlling the jump in the Borel resummation when a radial
integration contour crosses a sequence of singularities was determined in
Iwaki:2023cek : the jump induces the following transformation in the partition
function ${\cal Z}=e^{{\cal F}}$,

${\cal Z}\ \mapsto\ \mathfrak{S}{\cal
Z}=\exp\left[\frac{\omega(\gamma)}{2\pi{\rm i}}\left({\rm Li}_{2}(e^{2\pi
x})+2\pi x\log(1-e^{2\pi x})\right)\right]{\cal Z},$ (1.1)

where $\omega(\gamma)$ is (up to normalization) the so-called Stokes constant,
${\rm Li}_{2}(z)=\sum_{k\geq 1}z^{k}/k^{2}$ is the dilogarithm function and
$x$ is suitable operator (equal to $\frac{{\rm
i}g_{s}}{2\pi}c^{a}\partial_{t^{a}}$ in the simplest case considered in
Iwaki:2023cek , which we also restrict to here for simplicity; here $c^{a}$
are the components of $\gamma$ in $H_{4}(X,{\mathbb{Z}})$, interpreted as
D4-brane charges in type IIA set-up, or D3-brane charges in type IIB). As
observed in Iwaki:2023cek , the transformation (1.1) is exactly such that the
so-called dual partition function

$\tau(\boldsymbol{t},\boldsymbol{u};g_{s})=\sum_{\boldsymbol{n}\in{\mathbb{Z}}^{n}}e^{2\pi\boldsymbol{n}\cdot\boldsymbol{u}/g_{s}}\,{\cal
Z}(\boldsymbol{t}-{\rm i}g_{s}\boldsymbol{n};g_{s})$ (1.2)

transforms by a simple factor, up to a $\boldsymbol{u}$-dependent shift in
$\boldsymbol{t}$,

$\mathfrak{S}\tau(\boldsymbol{t},\boldsymbol{u};g_{s})=e^{\frac{\omega(\gamma)}{2\pi{\rm
i}}\,{\rm Li}_{2}({\rm
e}^{2\pi{\boldsymbol{c}}\cdot\boldsymbol{u}/g_{s}})}\,\tau\left(\boldsymbol{t}+\frac{\omega(\gamma)}{2\pi}\,g_{s}{\boldsymbol{c}}\,\log\left(1-{\rm
e}^{2\pi{\boldsymbol{c}}\cdot\boldsymbol{u}/g_{s}}\right),\boldsymbol{u};g_{s}\right).$
(1.3)

We note that dual partition functions (which are obtained as so-called Zak
transforms of the conventional partition functions) have appeared in many
different contexts, including supersymmetric gauge theories no2 , topological
string theory adkmv ; dhsv , topological recursion em ; Eynard:2021sxg , the
spectral theory/topological string correspondence ghm  and isomonodromic tau
functions GIL-12 ; GIL-13 ; cpt ; CLT .

Quite remarkably, the transformations (1.1) and (1.3) have appeared before in
yet a different context, namely the study of five-brane instanton corrections
to the hypermultiplet moduli space in type II strings compactified on a
Calabi-Yau threefold APP ; Alexandrov:2015xir . The key idea of APP ;
Alexandrov:2015xir  is that S-duality relates NS5-brane instanton corrections
in type IIB string theory to D5-brane instanton corrections, which (for unit
D5-brane charge) are governed by the topological string partition function by
virtue of gw-dt ; gw-dt2 . More specifically, for fixed unit D5-brane charge,
the sum over D3-D1-D(-1) brane charges leads to a non-Gaussian theta series of
the form (1.2) (with $\boldsymbol{n}$ being the D3-brane charge). Due to wall-
crossing behavior of the Donaldson-Thomas (DT) invariants $\Omega(\gamma)$
counting D5-D3-D1-D(-1) instantons in type IIB (or equivalently D6-D4-D2-D0
black holes in type IIA), the theta series becomes a section of a certain line
bundle ${\cal L}$ over the (twistor space of) the hypermultiplet moduli space,
with gluing conditions given by (1.3), which in turn imply the transformation
(1.1) for the kernel of the theta series. This suggests in particular that the
Stokes constant $\omega(\gamma)$ should be equated with the DT invariant
$\Omega(\gamma)$ counting BPS states associated to the same ray, as proposed
in different contexts in mm-s2019 ; gm-peacock ; astt ; gm-multi ; gkkm ; gu-
res . These considerations were extended in Alexandrov:2015xir  in two
directions: first, to $k>1$ units of five-brane charge, where the theta series
becomes a section of the $k$-th power of the line bundle ${\cal L}$, and
second, to include the effect of the refinement parameter $y$ in Donaldson-
Thomas theory, conjugate to angular momentum, which induces a non-commutative
deformation of the (twistor space of the) hypermultiplet moduli space.

This brings us to the main goal of the present paper, which is to extend the
results (1.1)–(1.3) to the refined topological string partition function. The
latter can be viewed as a one-parameter deformation ${\cal
F}(\boldsymbol{t};g_{s},{\mathsf{b}})$ of the usual topological free energy
${\cal F}(\boldsymbol{t};g_{s})$, which exists on non-compact Calabi-Yau
threefolds with a ${\mathbb{C}}^{\times}$ action.111It was suggested in hkatzk
that the refined topological string could also be defined on compact CY
threefolds, but the resulting free energies are no longer independent of
complex structure moduli, and it is unclear if the HAE still holds. While the
worldsheet definition of refined topological strings for ${\mathsf{b}}\neq 1$
remains obscure (see Antoniadis:2013bja  for an early attempt), in space-time
it is interpreted as the partition function of the five-dimensional gauge
theory engineered by M-theory compactified on $X$ times a five-dimensional
$\Omega$-background n ; Nekrasov:2010ka  with parameters

$\epsilon_{1}=g_{s}{\mathsf{b}},\qquad\epsilon_{2}=-g_{s}{\mathsf{b}}^{-1}.$
(1.4)

The resulting partition function leads to integer BPS invariants
$N_{\boldsymbol{d}}^{j_{L},j_{R}}$ which refine the usual Gopakumar–Vafa and
Donaldson-Thomas invariants hiv ; ckk ; no . When $X$ is a toric CY
singularity, ${\cal F}(\boldsymbol{t};g_{s},{\mathsf{b}})$ can be computed by
a refined version ikv  of the topological vertex formalism akmv . Otherwise,
the main computational tool is the refined version hk  of the standard
holomorphic anomaly equations (HAE) bcov , which can be integrated inductively
with sufficient control on the behavior at boundaries of moduli space.

In this work, we extend the analysis of gm-multi ; Iwaki:2023cek  and
construct the general trans-series solution to the refined HAE.222The trans-
series solution in the NS limit of the refined theory ns  requires a
particular treatment and was studied in gm-qp . In particular, we find that
there are now two instanton actions ${\mathsf{b}}Z(\gamma)$ and
${\mathsf{b}}^{-1}Z(\gamma)$ contributing to the trans-series for each
$\gamma$ in the charge lattice.333The occurrence of two instanton actions
${\cal A}/{\mathsf{b}}$ and ${\mathsf{b}}{\cal A}$ was already observed in an
earlier study of the resurgent structure of the refined topological string on
the resolved conifold Grassi:2022zuk . Our main claim is that this feature
arises for refined topological strings on arbitrary non-compact CY threefolds.
Moreover, in the simplest case where the singularity corresponds to a BPS
state with unit Stokes constant $\omega(\gamma)=1$ (as befits states which
become massless at a conifold singularity), we find that the Stokes
automorphism (1.1) is deformed to

$\mathfrak{S}_{{\mathsf{b}}}{\cal Z}=\Phi_{{\mathsf{b}}}^{-1}(x)\,{\cal Z},$
(1.5)

where $\Phi_{{\mathsf{b}}}(x)$ is the Faddeev quantum dilogarithm defined in
(A.132), reducing to (1.1) at ${\mathsf{b}}=1$ due to (A.135). More generally,
the Stokes automorphism associated to a singularity supported at $Z(\gamma)$
is given by

$\mathfrak{S}_{{\mathsf{b}}}{\cal
Z}=\prod_{j}\left[\prod_{m=-j}^{j}\Phi_{\mathsf{b}}\left(x+{\rm
i}m({\mathsf{b}}-{\mathsf{b}}^{-1})\right)\right]^{-\omega_{[j]}}{\cal Z},$
(1.6)

where the product runs over half integer $m$ such that $m+j$ is integer, and
$\omega_{[j]}$ are Stokes constants. This has a natural interpretation in
terms of motivic or refined DT invariants, which can be decomposed as
$\Omega(\gamma,y)=\sum_{j}\Omega_{[j]}(\gamma)\chi_{j}(y)$, where

$\chi_{j}(y):=\sum_{m=-j}^{j}y^{2m}=\frac{y^{2j+1}-y^{-2j-1}}{y-y^{-1}}$ (1.7)

is the character of the spin $j$ representation of $SU(2)$. We conjecture that
the Stokes constants $\omega_{[j]}$ associated to the singularity at
$Z(\gamma)$ are equal to the coefficients $\Omega_{[j]}(\gamma)$ appearing in
this decomposition.

It is then natural to ask whether there exists a deformation of the dual
partition function (1.2), with kernel given by the refined topological string
partition function ${\cal Z}$, whose transformation rules across rays would be
guaranteed by the Stokes automorphism (1.5), or more generally (1.6). In fact,
the Stokes automorphism (1.5) is a special case $k=1$ of an operator ${\bf
A}_{{\mathsf{b}}^{2}}^{(k)}$ acting on
$L^{2}({\mathbb{R}})\otimes{\mathbb{C}}^{k}$ which was introduced in
(Alexandrov:2015xir, , §4) in a rather ad hoc fashion, for the purpose of
producing a doubly-quantized version of the standard 5-term relation for the
quantum dilogarithm. In this work, we provide a rationale for the construction
of (Alexandrov:2015xir, , §4), by explicitly constructing a generalized theta
series valued in a degree $k$ line bundle ${\cal L}_{{\mathsf{b}}}$ over the
quantum torus, which is well defined across BPS rays provided the kernel of
the theta series transforms according to ${\bf A}_{{\mathsf{b}}^{2}}^{(k)}$,
or more generally by a $j$-dependent extension of ${\bf
A}_{{\mathsf{b}}^{2}}^{(k)}$. For $k=1$, this provides a refined version of
the dual partition function (1.2). Both the theta series and its
transformation under wall-crossing are defined using a certain Moyal-type non-
commutative product on twistor space. In fact, the non-commutativity makes it
possible to introduce two different refined dual partition functions which are
both related to the refined topological string partition function, but with
one of the twistor space coordinates identified with (the inverse of) either
of the two deformation parameters $\epsilon_{1,2}$ from (1.4).

These results raise several obvious questions: first, can one prove the
equality between Stokes constants and Donaldson-Thomas invariants, perhaps in
the framework of the Riemann-Hilbert problem proposed in Bri-19 ;
Barbieri:2019yya ; Bridgeland:2020zjh ; Alexandrov:2021wxu ; Bri-23 ? Second,
what is the physical interpretation of the refined dual partition function,
for example in terms of free fermions, spectral determinants, tau functions,
partition functions of line defects, or otherwise? Third, does there exists a
rank-$k$ version of the topological string (refined or unrefined) whose large
order behavior would be governed by the Stokes automorphism ${\bf
A}_{{\mathsf{b}}^{2}}^{(k)}$, and which would be related to the partition
function of $k$ five-branes constructed in (Alexandrov:2010ca, , §5) by a
rank-$k$ version of the GV/DT correspondence gw-dt ? We hope to return to some
of these challenging problems in the near future.

The remainder of this work is organized as follows. In §2 we review basic
properties of the refined topological string free energy. In §3 we construct
the trans-series solution of the holomorphic anomaly equations, determine the
boundary conditions at the conifold locus and in the large volume limit,
deduce the Stokes automorphism associated to the singularities in the Borel
plane, and perform numerical checks in the case of $X=K_{{\mathbb{P}}^{2}}$.
In §4, we recover the same automorphism and generalize it to any integer
$k\geq 1$ by constructing a class of dual partition functions on quantum
twisted tori. Definitions and properties of several variants of the quantum
dilogarithm function, which plays a central role in this work, are collected
in Appendix §A.

Acknowledgements: The authors would like to thank Murad Alim, Tom Bridgeland,
Alba Grassi, Jie Gu and Joerg Teschner for valuable discussions. The research
of MM is supported in part by the ERC-SyG project “Recursive and Exact New
Quantum Theory” (ReNewQuantum), which received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation program, grant agreement No. 810573. The research of BP is
supported by Agence Nationale de la Recherche under contract number
ANR-21-CE31-0021. SA and BP would like to thank the Isaac Newton Institute for
Mathematical Sciences in Cambridge (supported by EPSRC grant EP/R014604/1),
for hospitality during the programme Black holes: bridges between number
theory and holographic quantum information where work on this project was
undertaken.

##  2 The refined topological string

In this section, we review some basic facts about the refined topological
string and how to calculate its free energy.

As mentioned above, the refined topological string is parametrized by two
complex parameters $\epsilon_{1,2}$. Setting

$\epsilon_{1}=g_{s}{\mathsf{b}},\qquad\epsilon_{2}=-g_{s}{\mathsf{b}}^{-1},$
(2.1)

the parameter $g_{s}$ can be identified as the topological string coupling
constant, while ${\mathsf{b}}$ can be regarded as a deformation parameter.
When ${\mathsf{b}}=1$ one recovers the standard topological string. We note
that many references (like kw ) use a parameter $\beta$, which is related to
ours simply by ${\mathsf{b}}=\beta^{1/2}$. The refined topological string free
energy is a formal power series hk

${\cal F}({\boldsymbol{t}};\epsilon_{1},\epsilon_{2})=\sum_{i,j\geq
0}(\epsilon_{1}+\epsilon_{2})^{2i}\,(-\epsilon_{1}\epsilon_{2})^{j-1}{\cal
F}^{(i,j)}(\boldsymbol{t}),$ (2.2)

where the coefficients ${\cal F}^{(i,j)}(\boldsymbol{t})$ are sections of an
appropriate line bundle on the moduli space of $X$ (of Kähler structures in
the A-model, or complex structures in the B-model), parametrized by the flat
coordinates ${\boldsymbol{t}}$. For the purposes of this paper, we regard the
free energy as a formal power series in $g_{s}$ whose coefficients depend on
the moduli and on the deformation parameter ${\mathsf{b}}$,

${\cal F}({\boldsymbol{t}};g_{s},{\mathsf{b}})=\sum_{g\geq
0}g_{s}^{2g-2}\,{\cal F}_{g}({\boldsymbol{t}};{\mathsf{b}}),$ (2.3)

where

${\cal F}_{g}({\boldsymbol{t}};{\mathsf{b}})=\sum_{k=0}^{g}{\cal
F}^{(k,g-k)}({\boldsymbol{t}})\,({\mathsf{b}}-{\mathsf{b}}^{-1})^{2k}.$ (2.4)

The deformed free energies ${\cal F}_{g}({\boldsymbol{t}};{\mathsf{b}})$ are
Laurent polynomials in ${\mathsf{b}}$, of degree $2g$, and invariant under the
exchange ${\mathsf{b}}\leftrightarrow{\mathsf{b}}^{-1}$. When
${\mathsf{b}}=1$,

${\cal F}_{g}({\boldsymbol{t}};1)={\cal F}^{(0,g)}({\boldsymbol{t}})={\cal
F}_{g}({\boldsymbol{t}})$ (2.5)

is the conventional, unrefined topological string free energy at genus $g$.
The amplitudes ${\cal F}^{(k,g-k)}({\boldsymbol{t}})$ are related to the ones
defined in hk  by an overall sign $(-1)^{k}$.

The refined free energies ${\cal F}_{g}({\boldsymbol{t}};{\mathsf{b}})$ can be
calculated on certain local CY geometries using instanton calculus n , and
more generally with the refined topological vertex of ikv . These A-model-like
calculations are intrinsically attached to the large radius of the geometry.
In the refined case, the B-model approach to the free energies is based on the
HAE of bcov 444In the unrefined case, and in local CY geometries, the B-model
is described by the topological recursion of eo , as proposed in mm-open ;
bkmp . To our knowledge a refined version of the formalism for generic local
CY geometries is not available yet, see however Kidwai:2022kxx  for recent
progress.. An extension of the HAE to the refined case was proposed in hk
(see also kw ), and tested in detail in e.g. hkk ; ckk . In the framework of
the HAE, one extends the free energies ${\cal
F}_{g}({\boldsymbol{t}};{\mathsf{b}})$ to non-holomorphic functions, and the
HAE control their non-holomorphic dependence. We will use Roman capital
letters for the non-holomorphic free energies obtained from the HAE, and curly
capital letters for their holomorphic limit, as in gm-multi ; gkkm .

To write the HAE, we need some ingredients from special geometry (see e.g.
klemm  for more details). In the local case, the mirror CY is encoded in an
algebraic curve usually called the mirror curve. We recall that the moduli
space ${\cal M}$ of complex structures of the mirror CY is a special Kähler
manifold of complex dimension $n$. We will denote by $z^{a}$, with
$a=1,\cdots,n$ a set of algebraic complex coordinates on this moduli space.
The Kähler metric (here $\partial_{a}=\partial_{z^{a}}$)

$G_{a\bar{b}}=\partial_{a}\partial_{\bar{b}}K$ (2.6)

derives from a Kähler potential $K$, which is determined from the prepotential
${\cal F}_{0}$, equal to the genus zero free energy (the latter does not
depend on ${\mathsf{b}}$, as it clear from (2.4)). We introduce the covariant
derivative $D_{a}$ for the Levi–Civita connection associated to the metric,

$\Gamma^{a}_{bc}=G^{a\bar{k}}\partial_{b}G_{c\bar{k}}.$ (2.7)

In the case of compact CY threefolds, the covariant derivative involves as
well a connection on the Hodge bundle ${\cal L}$ over ${\cal M}$, but since
this connection vanishes in the holomorphic limit bcov  we can take it to be
zero from the very beginning, as noted in kz . The Yukawa couplings are
determined from the genus 0 free energy via

$C_{abc}=\partial_{a}\partial_{b}\partial_{c}{\cal F}_{0}.$ (2.8)

Finally, we have to introduce an anti-holomorphic version of the Yukawa
coupling, defined by

${\overline{C}}_{\bar{c}}^{ab}={\rm
e}^{2K}\,G^{a\bar{d}}G^{b\bar{e}}{\overline{C}}_{\bar{a}\bar{d}\bar{e}}.$
(2.9)

We can now write the refined HAE, following hk . In terms of the free energies
$F^{(k,g-k)}$, they read

$\partial_{\bar{c}}F^{(k,g-k)}={1\over
2}\,\overline{C}^{ab}_{\bar{c}}\left(D_{a}D_{b}F^{(k,g-k-1)}+\sum_{m,h}D_{a}F^{(m,h)}D_{b}F^{(k-m,g-k-h)}\right),$
(2.10)

where $g\geq 2$ and the sum over $m,h$ is such that $m+h\geq 1$. However, it
is easy to check that, in terms of the combinations $F_{g}({\mathsf{b}})$
defined in (2.4), the HAE take the same form as in the unrefined case of bcov
:

$\partial_{\bar{c}}F_{g}({\mathsf{b}})={1\over
2}\,\overline{C}^{ab}_{\bar{c}}\left(D_{a}D_{b}F_{g-1}({\mathsf{b}})+\sum_{h=1}^{g-1}D_{a}F_{h}({\mathsf{b}})D_{b}F_{g-h}({\mathsf{b}})\right),\qquad
g\geq 2.$ (2.11)

(Here, we only indicate the dependence of the refined free energies on the
deformation parameter, but they of course also depend on $z^{a}$.) Therefore,
the refined free energies satisfy the same HAE as the unrefined ones, but the
starting point of the recursion, given by the free energies at $g=1$, is
different and given by

$F_{1}({\mathsf{b}})=F_{1}({\mathsf{b}}=1)+({\mathsf{b}}-{\mathsf{b}}^{-1})^{2}F^{(1,0)}.$
(2.12)

Here, $F_{1}({\mathsf{b}}=1)=F^{(0,1)}$ is the conventional, unrefined free
energy at genus one. The free energy $F^{(1,0)}$ turns out to be given by a
holomorphic function of the moduli. It has the form hk ; hkk

$F^{(1,0)}=-{1\over
24}\log\left(f(\boldsymbol{z})\Delta(\boldsymbol{z})\right),$ (2.13)

where $\Delta(\boldsymbol{z})$ is the discriminant of the mirror curve, and
$f(\boldsymbol{z})$ is a rational function of the moduli, explicitly known in
many cases. The fact that $F^{(1,0)}$ is holomorphic will be important later
on.

As in the unrefined local case, it is useful to introduce propagators
$S^{ab}$, defined by the condition that bcov

$\partial_{\bar{c}}S^{ab}=\overline{C}^{ab}_{\bar{c}}.$ (2.14)

The propagators contain the full anti-holomorphic dependence of the free
energies. As a result, the HAE (2.11) can be rewritten in the form

${\partial F_{g}({\mathsf{b}})\over\partial S^{ab}}={1\over
2}\left(D_{a}D_{b}F_{g-1}({\mathsf{b}})+\sum_{h=1}^{g-1}D_{a}F_{h}({\mathsf{b}})D_{b}F_{g-h}({\mathsf{b}})\right),\qquad
g\geq 2.$ (2.15)

The initial condition (2.12) has to be also re-expressed in terms of
propagators. Since $F^{(1,0)}$ is purely holomorphic, it does not depend on
the propagators, whereas for $F^{(0,1)}$ one has bcov

$\partial_{a}F^{(0,1)}={1\over 2}\,C_{abc}S^{bc}+f_{a}(\boldsymbol{z}).$
(2.16)

In this equation, $f_{a}(\boldsymbol{z})$ is a function of the moduli only.
The equation (2.15), together with the initial conditions (2.12), (2.16) and
(2.13), provides a recursive procedure to obtain the free energies
$F_{g}({\mathsf{b}})$ as polynomials in the propagators, which is sometimes
called “direct integration” yy ; hk06 ; gkmw . At each genus $g$ one has to
fix an integration constant, independent of the propagators, but dependent on
the moduli, and called the holomorphic ambiguity (the functions
$f_{a}(\boldsymbol{z})$ in (2.16) can be regarded as examples thereof). In the
case of local CY manifolds this ambiguity can be fixed in many cases by using
the behavior of the free energies at special points in moduli space hkr . Let
us review this behavior in the case of the refined topological string.

We first consider the behavior at the conifold locus, defined by the vanishing
of a flat coordinate $t_{c}\to 0$. In the adapted symplectic frame, the free
energy has the following behavior kw ; hk

${\cal F}_{g}(t_{c};{\mathsf{b}})={c_{g}({\mathsf{b}})\over
t_{c}^{2g-2}}+{\cal O}(1),$ (2.17)

where the coefficient $c_{g}({\mathsf{b}})$ is given by

$c_{g}({\mathsf{b}})=-(2g-3)!\sum_{m=0}^{g}\widehat{B}_{2m}\widehat{B}_{2g-2m}{\mathsf{b}}^{2(2m-g)},$
(2.18)

where

$\widehat{B}_{m}=(2^{1-m}-1)\,{B_{m}\over m!}$ (2.19)

and $B_{m}$ is the Bernoulli number. In the unrefined limit ${\mathsf{b}}=1$
one has

$c_{g}(1)={B_{2g}\over 2g(2g-2)},\qquad g\geq 2,$ (2.20)

and one recovers the well-known universal conifold behavior of the standard
topological string gv-conifold .

Another universal result concerns the behavior of the refined free energies at
large radius. It is possible to generalize the Gopakumar–Vafa integrality
structure gv  to a refined version hiv ; hk . Let us introduce

$\epsilon_{L,R}={\epsilon_{1}\mp\epsilon_{2}\over 2}\,.$ (2.21)

In terms of the characters $\chi_{j}(y)$ defined in (1.7), the total free
energy is then given, up to a cubic polynomial in $\boldsymbol{t}$, by a sum
of the form

${\cal
F}({\boldsymbol{t}};g_{s},{\mathsf{b}})=\sum_{w,\boldsymbol{d}}\sum_{j_{L},j_{R}}\frac{\chi_{j_{L}}\left(e^{{\rm
i}w\epsilon_{L}}\right)\chi_{j_{R}}\left(e^{{\rm
i}w\epsilon_{R}}\right)}{4\sin\left({g_{s}w{\mathsf{b}}\over
2}\right)\sin\left({g_{s}w\over
2{\mathsf{b}}}\right)}\,N^{j_{L},j_{R}}_{\boldsymbol{d}}\,Q_{\boldsymbol{d}}^{w}\,,$
(2.22)

where $Q_{\boldsymbol{d}}={\rm e}^{-\boldsymbol{d}\cdot\boldsymbol{c}}$ and
$N^{j_{L},j_{R}}_{\boldsymbol{d}}$ are integers (here,
$j_{L},j_{R}\in{\mathbb{Z}}^{+}/2$). The integers
$N^{j_{L},j_{R}}_{\boldsymbol{d}}$ count BPS states with charge
${\boldsymbol{d}}$ transforming with spin $(j_{L},j_{R})$ under the little
group $SU(2)_{L}\times SU(2)_{R}$ in 5 dimensions, and are sometimes called
BPS invariants. When ${\mathsf{b}}=1$, we recover the integrality structure of
the standard topological string free energy. In particular, we have the
following relation between the genus zero Gopakumar–Vafa invariants
$n^{(0)}_{\boldsymbol{d}}$ and the BPS invariants

$n_{\boldsymbol{d}}^{(0)}=\sum_{j_{L},j_{R}}d_{L}d_{R}N^{j_{L},j_{R}}_{\boldsymbol{d}},\qquad
d_{L,R}=2j_{L,R}+1.$ (2.23)

The refined (or motivic) Donaldson-Thomas invariants
$\Omega({\boldsymbol{d}},y)$ in the large volume limit are characters for the
diagonal $SU(2)$ action:

$\Omega({\boldsymbol{d}},y)=\sum_{j_{L},j_{R}}\chi_{j_{L}}(y)\,\chi_{j_{R}}(y)\,N^{j_{L},j_{R}}_{\boldsymbol{d}}.$
(2.24)

In particular, they reduce to (2.23) in the unrefined limit $y\to 1$. Unlike
the BPS invariants $N^{j_{L},j_{R}}_{\boldsymbol{d}}$, which are only defined
when ${\boldsymbol{d}}$ is a curve class, the refined DT invariants
$\Omega(\gamma,y)$ are defined for any vector $\gamma\in K_{0}(X)$, i.e. for
arbitrary D6-D4-D2-D0 brane charge in type IIA, or D5-D3-D1-D(-1) charge in
type IIB. For later purposes, it will be convenient to decompose
$\Omega(\gamma,y)$ as a sum of $SU(2)$ characters,

$\Omega(\gamma,y):=\sum_{j}\chi_{j}(y)\,\Omega_{[j]}(\gamma),$ (2.25)

where the integers $\Omega_{[j]}(\gamma)$ count BPS multiplets of angular
momentum $j\in{\mathbb{Z}}/2$, and $y$ is the corresponding fugacity
parameter. We note that the representation (2.25) implies that
$\Omega(\gamma,y)$ is a Laurent polynomial in $y$, invariant under Poincaré
duality $y\mapsto 1/y$. As emphasized in Mozgovoy:2020has , for the standard
definition of DT invariants using cohomology with compact supports, Poincaré
duality is broken when $X$ is non-compact, primarily due to the non-invariance
of the attractor indices counting D0-branes. Here we assume that the DT
invariants which are pertinent for the refined topological string are in fact
invariant under Poincaré duality.

##  3 Trans-series solutions and resurgent structure

In order to understand the resurgent structure of the refined topological
string, we follow the strategy of cesv1 ; cesv2 ; cms ; gm-multi ; gkkm  and
look for trans-series solutions to the HAE. These define non-perturbative
sectors of the theory, but they require boundary conditions, as in the
perturbative case. As first pointed out in cesv1 ; cesv2 , we can obtain these
boundary conditions by looking at the large order behavior of the genus
expansions near special points in moduli space. It turns out that the first
aspect of this procedure, namely the construction of formal trans-series
solutions to the HAE, is essentially identical to the unrefined case.
Therefore, we will be rather succinct and refer the reader to gm-multi ; gkkm
. The second aspect of the procedure, namely the analysis of the boundary
conditions, has some new ingredients, and we will analyze it in more detail.

###  3.1 Trans-series solutions

As in gm-multi ; gkkm , we want to construct multi-instanton trans-series
solutions for the reduced topological string partition function

$Z_{\rm r}=Z/Z^{(0)},$ (3.1)

where $Z$ and $Z^{(0)}$ are respectively the full partition function
(including non-perturbative sectors) and the perturbative partition function.
As shown in cesv1 ; cesv2 ; gm-multi ; gkkm , the reduced partition function
satisfies a linear, homogeneous version of the HAE, therefore any (possibly
infinite) linear combination of solutions will be also a solution. We will
consider a special class of solutions of the form

$\exp(\Sigma_{\lambda}),$ (3.2)

where

$\Sigma_{\lambda}=-{\lambda\over g_{s}}\,{\cal A}+{\cal O}(g_{s}^{0}),$ (3.3)

so that (3.2) is a non-perturbative, exponentially small quantity. In this
equation, ${\cal A}$ is independent of the propagators, and $\lambda$ is an
appropriate normalization constant. We call ${\cal A}$ the instanton action.
It was argued in dmp-np ; cesv1 ; cesv2 ; gkkm  that ${\cal A}$ is an integer
period of the mirror CY manifold.

As explained in coms ; gm-multi ; gkkm , in order to obtain these
exponentially small solutions with behavior (3.3), it is very convenient to
write the HAE (2.15) in terms of a pair of ${\cal A}$-dependent operators,
$\mathsf{W}$ and $\mathsf{D}$. In the local case these operators are defined
as follows. Let us introduce

$T^{a}=g_{s}\partial_{b}{\cal A}\left(S^{ab}-{\cal S}_{{\cal A}}^{ab}\right).$
(3.4)

Here, ${\cal S}_{{\cal A}}^{ab}$ is the holomorphic limit of the propagator in
a so-called ${\cal A}$-frame, i.e. a symplectic frame in which the action
${\cal A}$ is a linear combination of the flat coordinates $t^{a}$. Then

$\mathsf{D}=T^{a}{\partial\over\partial z^{a}}\,.$ (3.5)

To define the operator $\mathsf{W}$, we first introduce the derivation
$\omega_{S}$ defined as

$\omega_{S}={1\over g_{s}^{2}}\,T^{a}T^{b}\,{\partial\over\partial S^{ab}}\,.$
(3.6)

Then $\mathsf{W}$ is given by

$\mathsf{W}=\omega_{S}-\sum_{g\geq
1}\left[\mathsf{D}\left(g_{s}^{2g-2}F_{g}\right)\right]\mathsf{D}.$ (3.7)

Since the structure of the HAE in the refined case is identical to the
conventional case, the operators to be used in the refined topological string
are the same. As explained in gm-multi ; gkkm , the HAE leads to the following
operator equation for $\Sigma_{\lambda}$:

$\mathsf{W}\Sigma_{\lambda}=\frac{1}{2}\left(\mathsf{D}^{2}\Sigma_{\lambda}+(\mathsf{D}\Sigma_{\lambda})^{2}\right).$
(3.8)

In gm-multi ; gkkm , this equation was solved as follows. Consider the formal
series

$G={{\cal A}\over g_{s}}+\sum_{g\geq
1}\mathsf{D}\left(g_{s}^{2g-2}F_{g}\right),$ (3.9)

where ${\cal A}$ is the action appearing in (3.3). Let us now assume that $G$
satisfies the equation

$\mathsf{W}G={1\over 2}\,\mathsf{D}^{2}G.$ (3.10)

Then

$\Sigma_{\lambda}=\mathsf{O}^{(\lambda)}G,$ (3.11)

where

$\mathsf{O}^{(\lambda)}=\sum_{k\geq 1}{(-\lambda)^{k}\over
k!}\,\mathsf{D}^{k-1},$ (3.12)

satisfies (3.8). Therefore, it suffices to check that (3.10) is still true in
the refined case. After using the HAEs, one finds that (3.10) holds if and
only if

$\omega_{S}\left(\mathsf{D}F_{1}\right)-\mathsf{D}\left({{\cal A}\over
g_{s}}\right)\mathsf{D}F_{1}-{1\over 2}\,\mathsf{D}^{2}\left({{\cal A}\over
g_{s}}\right)=0.$ (3.13)

This equation is true in the conventional topological string, when
$F_{1}=F_{1}({\mathsf{b}}=1)$, as one can check by using (2.16). In
particular, it holds for any choice of holomorphic ambiguity
$f^{(1)}_{i}(\boldsymbol{z})$. It is easily checked to be true in the refined
case as well: since $F_{1}({\mathsf{b}})$ differs from $F_{1}({\mathsf{b}}=1)$
in a purely holomorphic function of the moduli, as noted after (2.12), it
follows immediately that (3.13) must also be true for $F_{1}({\mathsf{b}})$.
This can also be checked by a direct calculation.

Boundary conditions for trans-series solutions to the HAE are obtained by
considering their holomorphic limit in an ${\cal A}$-frame. It follows from
the properties of the operator $\mathsf{D}$ discussed in gm-multi ; gkkm  that
$\Sigma_{\lambda}$, when evaluated in the ${\cal A}$-frame, is given by

$\Sigma_{\lambda,\,{\cal A}}=-\lambda\,{{\cal A}\over g_{s}}.$ (3.14)

As we will see, this class of solutions will be enough to construct the
relevant trans-series for the refined topological string. The solution (3.11)
involves the full, non-holomorphic propagators, but in practice one wants to
understand its holomorphic limit. This goes as follows. First, we write the
action ${\cal A}$ as a linear combination of periods555For compact CY, there
is an additional contribution $-\frac{c^{0}}{2\pi}({\cal
F}_{0}-2t^{a}\partial_{a}{\cal F}_{0})$, corresponding to the D5-brane charge
in IIB, or D6-brane charge in IIA, but this term is absent when $X$ is non-
compact.

${\cal A}=-{\rm i}c^{a}{\partial{\cal F}_{0}\over\partial t^{a}}+2\pi
d_{a}t^{a}+4\pi^{2}{\rm i}d_{0},$ (3.15)

where $t^{a}=-\log(z^{a})+\cdots$ are the complexified Kähler parameters,
${\cal F}_{0}$ is the genus zero free energy in the large radius frame,
behaving as666Here $\kappa_{abc}$ is a rational number, which plays the rôle
of the triple intersection number for a non-compact CY threefold. Note that in
our conventions, monodromies around the large radius point induce shifts
$t^{a}\mapsto t^{a}+2\pi{\rm i}\epsilon^{a}$.

${\cal F}_{0}={\kappa_{abc}\over 3!}\,t^{a}t^{b}t^{c}+{\cal O}({\rm
e}^{-t^{a}}),$ (3.16)

as $t^{a}\to+\infty$, and $(c^{a},d_{a},d_{0})$ are integers, corresponding to
D3-D1-D(-1) charges in type IIB set-up, or D4-D2-D0 charges in type IIA. The
action ${\cal A}$ is then identified with the central charge $Z(\gamma)$, up
to an overall factor $4\pi^{2}{\rm i}$.

When not all $c^{a}$ vanish at once, one defines a new prepotential by

${\cal A}=-{\rm i}c^{a}\,{\partial\widetilde{\cal F}_{0}\over\partial t^{a}}.$
(3.17)

It differs from the conventional prepotential at most in a quadratic
polynomial in the flat coordinates $t^{a}$. Then, in the holomorphic limit,
one has

$\mathsf{D}\rightarrow-{\rm i}g_{s}c^{a}{\partial\over\partial t^{a}}\,,$
(3.18)

and $\Sigma_{\lambda}$ becomes

$\Sigma_{\lambda}\rightarrow\tilde{\cal F}(\boldsymbol{t}+{\rm i}\lambda
g_{s}\boldsymbol{c};g_{s},{\mathsf{b}})-\tilde{\cal
F}(\boldsymbol{t};g_{s},{\mathsf{b}}),$ (3.19)

where

$\tilde{\cal F}({\boldsymbol{t}};g_{s},{\mathsf{b}})=g_{s}^{-2}\tilde{\cal
F}_{0}({\boldsymbol{t}})+\sum_{g\geq 1}g_{s}^{2g-2}{\cal
F}_{g}({\boldsymbol{t}};{\mathsf{b}}),$ (3.20)

i.e. it differs from the conventional free energy only in the genus zero part.

###  3.2 Boundary conditions from the conifold

Let us now discuss the boundary conditions for the trans-series. These follow
from the behavior of the free energies, in appropriate frames, at special
points in the moduli space. This behavior determines the large order
asymptotics of the free energies, and this leads in turn to multi-instanton
trans-series.

Let us first review how the conifold boundary condition is determined in the
case of the standard topological string with ${\mathsf{b}}=1$ cesv1 ; cesv2 .
The free energies ${\cal F}_{g}$ in the conifold frame have the following
behavior

${\cal F}_{g}(t_{c};{\mathsf{b}}=1)={B_{2g}\over 2g(2g-2)}\,t_{c}^{2-2g}+{\cal
O}(1),$ (3.21)

where $B_{2g}$ are Bernoulli numbers, and $t_{c}$ is an appropriately
normalized flat coordinate vanishing at the conifold locus. The singular
behavior in (3.21) also determines the growth of the free energies at large
genus close to the conifold point where $t_{c}=0$. To compute the large order
behavior, one can use the following representation of the Bernoulli numbers

$B_{2g}=(-1)^{g-1}{2(2g)!\over(2\pi)^{2g}}\sum_{\ell\geq 1}\ell^{-2g}$ (3.22)

to obtain the exact asymptotic formula:

${B_{2g}\over 2g(2g-2)}\,t_{c}^{2-2g}={1\over
2\pi^{2}}\,\Gamma(2g-1)\sum_{\ell\geq 1}\left(\ell{\cal A}\right)^{1-2g}{{\cal
A}\over\ell}\left(1+{1\over 2g-2}\right),$ (3.23)

where

${\cal A}=2\pi{\rm i}t_{c}.$ (3.24)

In local CY manifolds, the conifold flat coordinate $t_{c}$ is conjugate to
the large radius flat coordinate $t$, in the sense that

$t_{c}={r\over 2\pi}{\partial{\cal F}_{0}\over\partial t},$ (3.25)

where ${\cal F}_{0}$ is the prepotential in the large radius frame, and $r$ is
an integer. Therefore, the Borel singularity located at (3.24) corresponds to
a D4 BPS state. By using the standard correspondence between large order
behavior and exponentially small corrections (see e.g. mmbook ), one sees that
the $\ell$-th term in (3.23) corresponds to an $\ell$-th instanton amplitude
of the Pasquetti–Schiappa form ps09 ,

${\cal F}_{\cal A}^{(\ell)}={1\over 2\pi}\left[{1\over\ell}\left({{\cal
A}\over g_{s}}\right)+{1\over\ell^{2}}\right]{\rm e}^{-\ell{\cal A}/g_{s}}.$
(3.26)

This gives the $\ell$-th order trans-series which will provide boundary
conditions in the conifold frame.

Another way of finding the trans-series corresponding to the large genus
asymptotics of (3.23) is to write an integral formula for the coefficient, of
the form

$\displaystyle(-1)^{g-1}{B_{2g}\over 2g(2g-2)}$ $\displaystyle={1\over
2\pi^{2}}\int_{0}^{\infty}{{\rm d}z\over z^{2g-1}}\left\\{{\rm Li}_{2}({\rm
e}^{-2\pi/z})-{2\pi\over z}\log\left(1-{\rm e}^{-2\pi/z}\right)\right\\}$
(3.27) $\displaystyle=-{{\rm i}\over\pi}\int_{0}^{\infty}{{\rm d}z\over
z^{2g-1}}\log\,\operatorname{\Phi}_{1}\left(-{1\over z}\right),$

where $\operatorname{\Phi}_{\mathsf{b}}(z)$ is Faddeev’s quantum dilogarithm
(see Appendix A). Up to the overall factor of $1/\pi$, the integrand in the
last line of (3.27) gives the sum over all the multi-instantons trans-series

$\sum_{\ell\geq 1}{\cal F}_{\cal A}^{(\ell)}=-{\rm
i}\log\operatorname{\Phi}_{1}\left(-{{\cal A}\over 2\pi g_{s}}\right).$ (3.28)

This function determines the structure of the Stokes automorphism, as
explained in Iwaki:2023cek , and in line with what was obtained in
Alexandrov:2015xir .

Let us now consider the generalization of the above result to the refined
topological string.777See Grassi:2022zuk ; Alim:2022oll  for earlier studies
of the resurgent structure of the refined topological string on the resolved
conifold. We need an exact asymptotic formula for the coefficient
$c_{g}({\mathsf{b}})$ appearing in (2.17). This formula turns out to be

$c_{g}({\mathsf{b}})={1\over 2\pi}\Gamma(2g-2)\sum_{\ell\geq
1}\left[{(-1)^{\ell-1}\over\ell}{1\over\sin\left({\pi\ell\over{\mathsf{b}}^{2}}\right)}\left({2\pi{\rm
i}\ell\over{\mathsf{b}}}\right)^{2-2g}+\left({\mathsf{b}}\leftrightarrow{\mathsf{b}}^{-1}\right)\right].$
(3.29)

This can be derived by using Borel transform techniques, as follows. Let us
define the more convenient set of coefficients:

$\displaystyle\tilde{c}_{0}({\mathsf{b}})$ $\displaystyle=1,$ (3.30)
$\displaystyle\tilde{c}_{1}({\mathsf{b}})$ $\displaystyle={1\over
12}({\mathsf{b}}^{2}+{\mathsf{b}}^{-2}),$
$\displaystyle\tilde{c}_{g}({\mathsf{b}})$
$\displaystyle=(-1)^{g-1}{(2g)!\over(2g-3)!}\,c_{g}({\mathsf{b}}),\quad g\geq
2,$

and its generating function

$\varphi(x)=\sum_{g\geq 0}\tilde{c}_{g}({\mathsf{b}})x^{2g}.$ (3.31)

The Borel transform of $\varphi(x)$ can be obtained in closed form, as

$\widehat{\varphi}(\zeta)={\zeta^{2}\over 4\sin\left({\zeta{\mathsf{b}}\over
2}\right)\sin\left({\zeta{\mathsf{b}}^{-1}\over 2}\right)}\,.$ (3.32)

When ${\mathsf{b}}^{2}$ is not a rational number, this function has simple
pole singularities at

$\zeta={2\pi\ell\over{\mathsf{b}}}\quad\mbox{and}\quad
2\pi\ell{\mathsf{b}},\qquad\ell\in{\mathbb{Z}},$ (3.33)

with residues

${(2\pi\ell)^{2}(-1)^{\ell}\over{\mathsf{b}}^{3}}{2\over\sin\left({\pi\ell\over{\mathsf{b}}^{2}}\right)}$
(3.34)

at the first set of poles. The residues at the second set are obtained by
exchanging ${\mathsf{b}}\leftrightarrow{\mathsf{b}}^{-1}$. The standard
connection between singularities of the Borel transform and large order
asymptotics gives then (3.29). In our derivation we have assumed that
${\mathsf{b}}^{2}$ is not rational. When ${\mathsf{b}}^{2}\in\mathbb{Q}$, we
have singularities in the denominators in (3.29). One can verify though that
the singularities cancel between the two summands related by
${\mathsf{b}}\leftrightarrow{\mathsf{b}}^{-1}$, as noted in ho . The resulting
expression for Faddeev’s quantum dilogarithm when ${\mathsf{b}}^{2}$ is
rational can be found in garkas .

We can now use the expression (A.137) for Faddeev’s quantum dilogarithm to
obtain the following generalization of (3.27)

$(-1)^{g-1}c_{g}({\mathsf{b}})=-{{\rm i}\over\pi}\int_{0}^{\infty}{{\rm
d}z\over z^{2g-1}}\log\,\operatorname{\Phi}_{\mathsf{b}}\left(-{1\over
z}\right).$ (3.35)

Using (A.137), we conclude that the relevant trans-series are given by

$\displaystyle{\cal F}_{{\cal A},{\mathsf{b}}}^{(\ell)}$
$\displaystyle={(-1)^{\ell-1}\over\ell}{{\rm e}^{-{\ell{\cal
A}\over{\mathsf{b}}g_{s}}}\over
2\sin\left({\pi\ell\over{\mathsf{b}}^{2}}\right)}\,,$ (3.36)
$\displaystyle{\cal F}_{{\cal A},{\mathsf{b}}^{-1}}^{(\ell)}$
$\displaystyle={(-1)^{\ell-1}\over\ell}{{\rm e}^{-{\ell{\cal
A}{\mathsf{b}}\over g_{s}}}\over
2\sin\left({\pi\ell{\mathsf{b}}^{2}}\right)}\,,$

and

$\sum_{\ell\geq 1}\left({\cal F}_{{\cal A},{\mathsf{b}}}^{(\ell)}+{\cal
F}_{{\cal A},{\mathsf{b}}^{-1}}^{(\ell)}\right)=-{\rm
i}\log\operatorname{\Phi}_{\mathsf{b}}\left(-{{\cal A}\over 2\pi
g_{s}}\right).$ (3.37)

In particular, the trans-series involves two different actions ${\cal
A}/{\mathsf{b}}$ and ${\mathsf{b}}{\cal A}$, as first observed in
Grassi:2022zuk .

Let us now use these boundary conditions to obtain appropriate trans-series in
an arbitrary frame. After exponentiating, we find

$Z_{{\rm r},{\cal A}}=\exp\left[\sum_{\ell\geq 1}\left({\cal F}_{{\cal
A},{\mathsf{b}}}^{(\ell)}+{\cal F}_{{\cal
A},{\mathsf{b}}^{-1}}^{(\ell)}\right)\right]=\sum_{n,m\geq
0}C_{n,m}\exp\left(-{n{\cal A}\over{\mathsf{b}}g_{s}}-{m{\mathsf{b}}{\cal
A}\over g_{s}}\right),$ (3.38)

where $C_{n,m}$ are constants (depending on ${\mathsf{b}}$) which can be read
from the representation (A.137) of
$\log\operatorname{\Phi}_{\mathsf{b}}\left(-{1\over z}\right)$. We have, for
example,

$C_{1,0}={1\over 2\sin\left({\pi\ell\over{\mathsf{b}}^{2}}\right)}\,,\qquad
C_{0,1}={1\over 2\sin\left({\pi\ell{\mathsf{b}}^{2}}\right)}\,.$ (3.39)

We note that each of the terms in the sum of the r.h.s. in (3.38) is of the
form $\exp(\Sigma_{\lambda})$, with

$\lambda={n\over{\mathsf{b}}}+m{\mathsf{b}}.$ (3.40)

Therefore, we find by linearity

$Z_{\rm r}=\sum_{n,m\geq
0}C_{n,m}\exp\left(\Sigma_{{n\over{\mathsf{b}}}+m{\mathsf{b}}}\right),$ (3.41)

with holomorphic limit

${\cal Z}_{\rm r}=\sum_{n,m\geq 0}C_{n,m}\exp\left[\tilde{\cal
F}\left(\boldsymbol{t}+{\rm
i}g_{s}\boldsymbol{c}\left(n/{\mathsf{b}}+m{\mathsf{b}}\right)\right)-\tilde{\cal
F}(\boldsymbol{t})\right].$ (3.42)

For example, the one-instanton amplitude is

${\cal F}^{(1)}={1\over
2\sin\left({\pi\over{\mathsf{b}}^{2}}\right)}\exp\left[\tilde{\cal
F}(\boldsymbol{t}+{\rm i}g_{s}\boldsymbol{c}/{\mathsf{b}})-\tilde{\cal
F}(\boldsymbol{t})\right]+\left({\mathsf{b}}\leftrightarrow{\mathsf{b}}^{-1}\right).$
(3.43)

When expanded in $g_{s}$, we find

$\displaystyle{\cal F}^{(1)}=$ $\displaystyle\,{{\rm e}^{-{{\cal
A}\over{\mathsf{b}}g_{s}}}\over
2\sin\left({\pi\over{\mathsf{b}}^{2}}\right)}\exp\left(-{c^{a}c^{b}\over
2{\mathsf{b}}^{2}}\,\tau_{ab}\right)\left[1-{\rm
i}\left({c^{a}\over{\mathsf{b}}}{\partial{\cal
F}_{1}({\mathsf{b}})\over\partial t^{a}}-{1\over
6{\mathsf{b}}^{3}}\,c^{a}c^{b}c^{e}C_{abe}\right)g_{s}+\cdots\right]$ (3.44)
$\displaystyle\,+\left({\mathsf{b}}\leftrightarrow{\mathsf{b}}^{-1}\right),$

where

$\tau_{ab}={\partial^{2}\widetilde{\cal F}_{0}\over\partial t^{a}\partial
t^{b}}\,,\qquad C_{abe}={\partial^{2}{\cal F}_{0}\over\partial t^{a}\partial
t^{b}\partial t^{e}}\,.$ (3.45)

Note that ${\cal F}_{1}({\mathsf{b}})$ is the holomorphic limit of the full,
${\mathsf{b}}$-dependent genus one amplitude (2.12). We expect that the trans-
series (3.44) will control the asymptotic behavior of ${\cal
F}_{g}({\boldsymbol{t}};{\mathsf{b}})$ at large genus, not far from the
conifold locus. We will test this numerically in section 3.5.

###  3.3 Boundary conditions from large radius

As we have seen in the last section, the behavior of the refined topological
string predicts the existence of a Borel singularity given (up to a
normalization) by the conifold flat coordinate, and leads to an explicit form
for the corresponding multi-instanton trans-series. This can be transformed to
an arbitrary frame, leading to (3.42). In gkkm  it was shown that, by using
the Gopakumar–Vafa formula gv , one can obtain information on the Borel
singularities and their trans-series near the large radius point. We will now
generalize the argument in gkkm  to the refined case.

The starting point is the representation (2.22) of the topological free energy
in terms of the BPS invariants. Let us introduce

${\mathsf{b}}_{\pm}={\mathsf{b}}\pm{\mathsf{b}}^{-1},$ (3.46)

and the coefficients $s_{g}^{j_{L},j_{R}}({\mathsf{b}})$ defined in terms of
the generating function

$\frac{\chi_{j_{L}}\left(e^{\frac{{\rm
i}x{\mathsf{b}}_{+}}{2}}\right)\chi_{j_{R}}\left(e^{\frac{{\rm
i}x{\mathsf{b}}_{-}}{2}}\right)}{4\sin\left({x{\mathsf{b}}\over
2}\right)\sin\left({x\over 2{\mathsf{b}}}\right)}=\sum_{g\geq
0}s_{g}^{j_{L},j_{R}}({\mathsf{b}})\,x^{2g-2}.$ (3.47)

We have, for example,

$\displaystyle s_{0}^{j_{L},j_{R}}({\mathsf{b}})$ $\displaystyle=d_{L}d_{R},$
(3.48) $\displaystyle s_{1}^{j_{L},j_{R}}({\mathsf{b}})$
$\displaystyle={d_{L}d_{R}\over
24}\,\Bigl{[}{\mathsf{b}}^{2}+{\mathsf{b}}^{-2}+(1-d_{L}^{2}){\mathsf{b}}_{+}^{2}+(1-d_{R}^{2}){\mathsf{b}}_{-}^{2}\Bigr{]}.$

By expanding both sides of (2.22) in powers of $g_{s}$ we obtain the refined
multicovering formula

${\cal
F}_{g}({\boldsymbol{t}};{\mathsf{b}})=\sum_{\boldsymbol{d}}\sum_{j_{L},j_{R}}N^{j_{L},j_{R}}_{\boldsymbol{d}}s_{g}^{j_{L},j_{R}}({\mathsf{b}})\,{\rm
Li}_{3-2g}(Q_{\boldsymbol{d}}).$ (3.49)

Now, as in gkkm , we use

$\sum_{n\in{\mathbb{Z}}}{1\over(2\pi n-{\rm i}t)^{m}}={{\rm
i}^{m}\over(m-1)!}\,{\rm Li}_{-m+1}({\rm e}^{-t}),\qquad m\geq 2,$ (3.50)

to write the free energy as

${\cal
F}_{g}({\boldsymbol{t}};{\mathsf{b}})=\sum_{\boldsymbol{d}}\sum_{n\in{\mathbb{Z}}}\sum_{j_{L},j_{R}}N^{j_{L},j_{R}}_{\boldsymbol{d}}r_{g}^{j_{L},j_{R}}({\mathsf{b}}){\cal
A}_{{\boldsymbol{d}},n}^{2-2g},\qquad g\geq 2,$ (3.51)

where

$r_{g}^{j_{L},j_{R}}({\mathsf{b}})=(-1)^{g-1}(2\pi{\rm
i})^{2g-2}(2g-3)!\,s_{g}^{j_{L},j_{R}}({\mathsf{b}})$ (3.52)

and

${\cal
A}_{{\boldsymbol{d}},n}=2\pi{\boldsymbol{d}}\cdot{\boldsymbol{t}}+4\pi^{2}{\rm
i}n\,.$ (3.53)

We want to obtain the large genus behavior of (3.51), and extract the
corresponding trans-series.888In the case of the resolved conifold, the Borel
transform of the total free energy was calculated in Grassi:2022zuk . To do
this, we need to know the large order behavior of the coefficients
$r_{g}^{j_{L},j_{R}}({\mathsf{b}})$. This can be obtained by the same argument
we used in obtaining (3.29), since (3.47) can be regarded as the Borel
transform of the divergent series with coefficients
$(2g)!\,s_{g}^{j_{L},j_{R}}$. The singularities are still at (3.33), but the
residues now depend on the BPS invariants. Introducing the variables

$y_{\mathsf{b}}=-{\rm e}^{\pi{\rm
i}{\mathsf{b}}^{2}},\qquad{\tilde{y}}_{{\mathsf{b}}}=-{\rm e}^{-\pi{\rm
i}/{\mathsf{b}}^{2}},$ (3.54)

one finds the following asymptotic expansion for $g\gg 1$

$r_{g}^{j_{L},j_{R}}({\mathsf{b}})\sim{1\over 2\pi}\Gamma(2g-2)\sum_{\ell\geq
1}\left[{(-1)^{\ell-1}\over\ell}\,\frac{\chi_{j_{L}}\big{(}{\tilde{y}}_{\mathsf{b}}^{\ell}\big{)}\chi_{j_{R}}\big{(}{\tilde{y}}_{\mathsf{b}}^{\ell}\big{)}}{\sin\left({\pi\ell\over{\mathsf{b}}^{2}}\right)}\left({\ell\over{\mathsf{b}}}\right)^{-2g+2}+\left({\mathsf{b}}\rightarrow{\mathsf{b}}^{-1}\right)\right].$
(3.55)

We conclude that there is a sequence of Borel singularities at

${\ell\over{\mathsf{b}}}\,{\cal
A}_{{\boldsymbol{d}},n}\quad\mbox{and}\quad{\ell{\mathsf{b}}}{\cal
A}_{{\boldsymbol{d}},n},\qquad\ell\in{\mathbb{Z}}_{>0},\quad
n\in{\mathbb{Z}},$ (3.56)

and the corresponding trans-series are

$\displaystyle{\cal F}_{{\cal A},{\mathsf{b}},j_{L},j_{R}}^{(\ell)}$
$\displaystyle={(-1)^{\ell-1}\over\ell}\,{\chi_{j_{L}}\big{(}{\tilde{y}}_{\mathsf{b}}^{\ell}\big{)}\chi_{j_{R}}\big{(}{\tilde{y}}_{\mathsf{b}}^{\ell}\big{)}\over
2\sin\left({\pi\ell\over{\mathsf{b}}^{2}}\right)}\,{\rm e}^{-{\ell{\cal
A}\over{\mathsf{b}}g_{s}}},$ (3.57) $\displaystyle{\cal F}_{{\cal
A},{\mathsf{b}}^{-1},j_{L},j_{R}}^{(\ell)}$
$\displaystyle={(-1)^{\ell-1}\over\ell}\,{\chi_{j_{L}}\big{(}y_{\mathsf{b}}^{\ell}\big{)}\chi_{j_{R}}\big{(}y_{\mathsf{b}}^{\ell}\big{)}\over
2\sin\left({\pi\ell{\mathsf{b}}^{2}}\right)}\,{\rm e}^{-{\ell{\mathsf{b}}{\cal
A}\over g_{s}}},$

where ${\cal A}={\cal A}_{{\boldsymbol{d}},n}$. When $j_{L}=j_{R}=0$, we
recover (3.36). The sum over all multi-instanton sectors gives

$\sum_{\ell\geq 1}\left({\cal F}_{{\cal
A},{\mathsf{b}},j_{L},j_{R}}^{(\ell)}+{\cal F}_{{\cal
A},{\mathsf{b}}^{-1},j_{L},j_{R}}^{(\ell)}\right)=-{\rm
i}\log\operatorname{\Phi}_{\mathsf{b}}^{[j_{L},j_{R}]}\left(-{{\cal A}\over
2\pi g_{s}}\right),$ (3.58)

where the series on the r.h.s. is the following generalization of Faddeev’s
quantum dilogarithm

$\log\operatorname{\Phi}_{\mathsf{b}}^{[j_{L},j_{R}]}\left(-{{\cal A}\over
2\pi g_{s}}\right)={1\over{\rm i}}\sum_{\ell\geq
1}{(-1)^{\ell}\over\ell}\left[{\chi_{j_{L}}\big{(}{\tilde{y}}_{\mathsf{b}}^{\ell}\big{)}\chi_{j_{R}}\big{(}{\tilde{y}}_{\mathsf{b}}^{\ell}\big{)}\over
2\sin\left({\pi\ell\over{\mathsf{b}}^{2}}\right)}\,{\rm e}^{-{\ell{\cal
A}\over{\mathsf{b}}g_{s}}}+{\chi_{j_{L}}\big{(}y_{\mathsf{b}}^{\ell}\big{)}\chi_{j_{R}}\big{(}y_{\mathsf{b}}^{\ell}\big{)}\over
2\sin\left({\pi\ell{\mathsf{b}}^{2}}\right)}\,{\rm e}^{-{\ell{\cal
A}{\mathsf{b}}\over g_{s}}}\right].$ (3.59)

When $j_{L}=j_{R}=0$, we recover the conventional quantum dilogarithm. In
fact, (3.59) can be expressed in terms of a more elementary function
$\operatorname{\Phi}_{\mathsf{b}}^{[j]}(z)$ defined in (A.141). Indeed, from
the representation of $\operatorname{\Phi}_{\mathsf{b}}^{[j]}(z)$ given in the
first line of (A.142), one finds the following identity

$\log\operatorname{\Phi}_{\mathsf{b}}^{[j_{L},j_{R}]}(z)=\sum_{j=|j_{L}-j_{R}|}^{j_{L}+j_{R}}\log\operatorname{\Phi}_{\mathsf{b}}^{[j]}(z),$
(3.60)

where the sum runs over half-integer $j$ such that $j-j_{L}-j_{R}$ is integer.
The third line in (A.142) also shows that
$\operatorname{\Phi}_{\mathsf{b}}^{[j]}(z)$ can be expressed as a product of
the ordinary quantum dilogarithms with shifted arguments.

The full trans-series corresponding to all the singularities near large radius
is a sum of trans-series of the form (3.59),

$-{\rm
i}\sum_{\boldsymbol{d}}\sum_{n\in{\mathbb{Z}}}\sum_{j_{L},j_{R}}N_{\boldsymbol{d}}^{j_{L},j_{R}}\log\operatorname{\Phi}_{\mathsf{b}}^{[j_{L},j_{R}]}\left(-{{\cal
A}_{{\boldsymbol{d}},n}\over 2\pi g_{s}}\right)=-{\rm
i}\sum_{\boldsymbol{d}}\sum_{n\in{\mathbb{Z}}}\sum_{j}\Omega_{[j]}(\boldsymbol{d})\log\operatorname{\Phi}_{\mathsf{b}}^{[j]}\left(-{{\cal
A}_{{\boldsymbol{d}},n}\over 2\pi g_{s}}\right)$ (3.61)

where

$\Omega_{[j]}(\boldsymbol{d})=\sum_{j_{L},j_{R}\atop|j_{L}-j_{R}|\leq j\leq
j_{L}+j_{R}}N_{\boldsymbol{d}}^{j_{L},j_{R}}$ (3.62)

in line with (2.25). The refined DT invariants $\Omega_{[j]}(\boldsymbol{d})$
play the role of Stokes constants, and are common to all singularities (3.56)
with different values of $\ell$ and $n$. In the limit ${\mathsf{b}}\rightarrow
1$, upon using (2.23), we recover the result of gkkm

$-{\rm
i}\sum_{\boldsymbol{d}}\sum_{n\in{\mathbb{Z}}}\left(\sum_{j_{L},j_{R}}d_{L}d_{R}N_{\boldsymbol{d}}^{j_{L},j_{R}}\right)\log\operatorname{\Phi}_{1}\left(-{{\cal
A}_{{\boldsymbol{d}},n}\over 2\pi g_{s}}\right)=-{\rm
i}\sum_{\boldsymbol{d}}\sum_{n\in{\mathbb{Z}}}n_{\boldsymbol{d}}^{(0)}\log\operatorname{\Phi}_{1}\left(-{{\cal
A}_{{\boldsymbol{d}},n}\over 2\pi g_{s}}\right).$ (3.63)

###  3.4 Stokes automorphism in the refined case

From the above results, and using the methods of Iwaki:2023cek , one can
easily obtain the Stokes automorphism corresponding to the trans-series
(3.37), (3.58). We refer e.g. to Iwaki:2023cek  for background on the Stokes
automorphism and related issues in the theory of resurgence.

We found that, for the topological free energies in the large radius frame,
the leading Borel singularities near the large radius point are at ${\cal
A}={\cal A}_{{\boldsymbol{d}},n}$, corresponding to D2-D0 branes in type IIA,
or D1-D(-1) instantons in IIB. Since the corresponding frame is an ${\cal
A}$-frame, the Stokes automorphism is purely multiplicative, and is obtained
by exponentiating the sum of multi-instantons (3.58) (with an additional
factor of $-{\rm i}$ due to conventions in the definition of the Stokes
automorphism). We then obtain

$\mathfrak{S}({\cal
Z})=\prod_{j_{L},j_{R}}\left[\operatorname{\Phi}_{\mathsf{b}}^{[j_{L},j_{R}]}\left(-{{\cal
A}\over 2\pi
g_{s}}\right)\right]^{-N^{j_{L},j_{R}}_{\boldsymbol{d}}}\\!\\!\\!{\cal
Z}=\prod_{j}\left[\operatorname{\Phi}_{\mathsf{b}}^{[j]}\left(-{{\cal A}\over
2\pi g_{s}}\right)\right]^{-\Omega_{[j]}({\boldsymbol{d}})}{\cal Z}.$ (3.64)

The case (3.37) was obtained by analyzing the behavior of the topological
string free energies in the conifold frame, near the conifold point where a
D4-brane becomes massless (or a D3-instanton becomes of vanishing action). The
resulting Stokes automorphism is again purely multiplicative, and is in fact a
particular case of (3.64) in which only $j=0$ contributes with
$\Omega_{[j]}=1$.

Let us now consider what happens at an arbitrary frame, i.e. not necessarily
an ${\cal A}$-frame. In that case, not all $c^{a}$ appearing in (3.15) vanish.
As shown in Iwaki:2023cek , the Stokes automorphism can be obtained from the
multiplicative one, after promoting the exponential of the action ${\cal A}$
to a shift operator. One then finds

$\mathfrak{S}(\tilde{\cal
Z})=\prod_{j}\left[\operatorname{\Phi}_{\mathsf{b}}^{[j]}\left({\rm
i}g_{s}c^{a}{\partial\over\partial
t^{a}}\right)\right]^{-\Omega_{[j]}(\gamma)}\tilde{\cal Z},$ (3.65)

where we have denoted $\tilde{\cal Z}={\rm e}^{\tilde{\cal F}}$ the partition
function associated to the modified free energy (3.20). When ${\mathsf{b}}=1$,
(3.65) reduces to the result of Iwaki:2023cek  for the standard topological
string. Furthermore, when ${\mathsf{b}}^{2}=2$, using (A.136), we recover the
Stokes automorphism which appears in the real topological string at the
conifold point ms . The relevance of this special value for topological
strings on orientifolds was anticipated in kw .

We expect (3.64), (3.65) to give the Stokes automorphism of the refined
topological string due to arbitrary singularities at $\ell{\mathsf{b}}^{\pm
1}{\cal A}$, $\ell\in{\mathbb{Z}}_{>0}$. The Stokes constants $\Omega_{[j]}$
appearing in these formulae should be identified with the coefficients of the
character expansion of the refined DT invariant (2.25) associated to the
corresponding BPS state, as conjectured around (1.6). The formulae (3.64),
(3.65) were in fact already proposed by Alexandrov:2015xir , in a different
but related context, and we will rederive them in section 4.

###  3.5 Numerical checks for $X=K_{{\mathbb{P}}^{2}}$

As is well-known, the trans-series obtained in resurgent analysis, if correct,
should control the large order behavior of the original perturbative series.
In particular, the one-instanton series (3.44) should give the leading
asymptotic behavior in the region of moduli space in which the leading
singularity in the Borel plane is given by ${\cal A}$ (i.e. the one closer to
the origin). We will now verify that (3.44) indeed gives the correct answer in
the simplest non-trivial local CY, namely, $X=K_{{\mathbb{P}}^{2}}$, the total
space of the canonical bundle over ${\mathbb{P}}^{2}$, also known as local
${\mathbb{P}}^{2}$. We follow the notations of gm-multi  and denote by $z$ the
standard algebraic coordinate on Kähler moduli space, such that $z=0$,
$z=-\frac{1}{27}$ and $z=\infty$ correspond to the large volume, conifold and
orbifold points, respectively. For simplicity, we shall restrict ourselves to
negative values of $z$ in the large radius region of the moduli space,

$-{1\over 27}<z<0,$ (3.66)

since with our conventions the free energies in the large radius frame are
real in this interval.

From previous studies cesv2 ; cms ; gm-multi , it is known that, in most of
the range in (3.66), the leading Borel singularity corresponds to the central
charge of the D4-brane which becomes massless at the conifold point $z=-1/27$,
and is given by the conifold result (3.24). In particular, the Borel
singularity ${\cal A}_{1,0}$ in (3.53) only becomes relevant very close to the
large radius limit, when $-10^{-6}<z<0$. We will denote

${\cal A}_{c}=2\pi{\rm i}t_{c},$ (3.67)

where $t_{c}$ is the appropriately normalized conifold flat coordinate.999 We
follow the conventions of ms  for the normalization of $t_{c}$, which differs
from the one used in gm-multi  by a factor of ${\sqrt{3}}$. Therefore, in this
region the large genus behavior of the free energies ${\cal
F}_{g}({\mathsf{b}})$ in the large radius frame is determined exactly by
(3.44). Let us note that, in our conventions, the free energy $F^{(1,0)}$ is
given by

${\cal F}^{(1,0)}=-{1\over 24}\log\left(-{\Delta\over
z}\right),\qquad\Delta=1+27z.$ (3.68)

Since $b_{2}(X)=1$, there is only one coefficient $c^{a}$ in (3.44), which is
equal to $c=-3$ in the present conventions gm-multi . Writing ${\cal F}^{(1)}$
as

${\cal F}^{(1)}={\rm e}^{-{\widetilde{\cal
A}/g_{s}}}\left(\mu_{0}+\mu_{1}g_{s}+\cdots\right),$ (3.69)

standard resurgent analysis predicts

${\cal F}_{g}({\mathsf{b}})\sim{1\over\pi}\widetilde{\cal
A}^{-2g+2}\Gamma(2g-2)\left(\mu_{0}+{\mu_{1}\widetilde{\cal A}\over
2g-3}+\cdots\right),\qquad g\gg 1.$ (3.70)

We can extract numerically the value of the action $\widetilde{\cal A}$ and of
the coefficients $\mu_{0,1}$ from the perturbative free energies at
sufficiently large order, for different values of the modulus $z$ and the
parameter ${\mathsf{b}}$, and compare those to the theoretical prediction in
(3.44). This prediction implies in particular that, if $|{\mathsf{b}}|>1$, we
will have

$\widetilde{\cal A}={{\cal A}_{c}\over{\mathsf{b}}}\,.$ (3.71)

If $|{\mathsf{b}}|<1$, the action $\widetilde{\cal A}$ is obtained by
exchanging ${\mathsf{b}}\leftrightarrow{\mathsf{b}}^{-1}$ in (3.71) (note that
the sequence ${\cal F}_{g}$ is invariant under this exchange). Computing the
free energies up to $g=35$ for ${\mathsf{b}}=\pi$ (a convenient irrational
number) and several values of $z$, we already find excellent agreement, see
Fig. 1. A similar agreement is obtained for other values of ${\mathsf{b}}$,
including complex ones.

Figure 1: Values of the coefficients $\mu_{0}$ (left) and $\mu_{1}$ (right) in
(3.70), for different values of $z$ in the range $-\frac{1}{27}<z<0$ and for a
fixed value ${\mathsf{b}}=\pi$, extracted from the asymptotic behaviour of
${\cal F}_{g}({\mathsf{b}})$ up to $g=35$ (after using three Richardson
transforms to improve convergence). The numerical values are very close to the
theoretical predictions shown in black lines, with $10^{-10}$ and $10^{-9}$
accuracy for $\mu_{0}$ and $\mu_{1}$, respectively.

##  4 Refined dual partition functions

In this section, we propose an extension of the notion of dual partition
function, realized as certain generalized theta series encoding five-brane
instantons, that incorporates the refinement. The construction relies on the
use of a non-commutative Moyal star product and its extension to the realm of
contact geometry. Our main result is that the same Stokes automorphism (3.64)
that governs the refined topological string also arises as the transformation
of the kernel of the refined dual partition function induced by a non-
commutative wall-crossing transformation. Furthermore, by considering higher
five-brane charge $k>1$, we obtain a vector-valued generalization of (3.64),
which realizes the double quantization proposed in (Alexandrov:2015xir, , §4).

###  4.1 Five-brane instanton corrections and dual partition functions

Let us first revisit the construction of dual partition functions and their
behavior under wall-crossing from Alexandrov:2015xir . This construction can
be motivated by considering five-brane instanton corrections to the vector
multiplet moduli space ${\cal M}$ arising by compactifying type IIA strings on
$X\times S_{1}$ down to 3 dimensions.101010Equivalently, one can consider the
hypermultiplet moduli space of type IIB string theory compactified on $X$, or
type IIA on the mirror threefold $\widehat{X}$. When $X$ is a compact CY
threefold, ${\cal M}$ is a quaternion-Kähler (QK) manifold of real dimension
$4b_{2}(X)+4$, of the form

${\cal M}\simeq{\mathbb{R}}^{+}_{R}\times{\cal S}{\cal K}_{z^{a}}\times{\cal
T}_{\zeta^{\Lambda},\tilde{\zeta}_{\Lambda}}\times S^{1}_{\sigma},$ (4.72)

where ${\mathbb{R}}^{+}$ parametrizes the radius of the circle, ${\cal S}{\cal
K}$ the complexified Kähler structure on $X$, ${\cal T}$ the holonomies of the
Ramond gauge fields around $H_{\rm even}(X,{\mathbb{Z}})$, and $S^{1}$ the
scalar Poincaré-dual to the Kaluza-Klein gauge field in 3 dimensions, and we
have indicated in subscript the coordinates used to parametrize the various
factors (with $a$ running over $1,\dots,b_{2}:=b_{2}(X)$ and
$\Lambda=0,\dots,b_{2}$). Topologically, the level sets of $R$ are principal
bundles over ${\cal S}{\cal K}$, whose fibers are twisted tori
$\widetilde{{\cal
T}}={\mathbb{R}}^{2b_{2}+3}_{\zeta^{\Lambda},\tilde{\zeta}_{\Lambda},\sigma}/\mathds{H}\simeq{\cal
T}\times S^{1}$ where $\mathds{H}$ is the Heisenberg group of large gauge
transformations parametrized by
$\Theta=(\eta^{\Lambda},\tilde{\eta}_{\Lambda},\kappa)\in{\mathbb{Z}}^{2b_{2}+3}$
and acting by

$\mathscr{T}_{\Theta}\ :\ (\zeta^{\Lambda},\tilde{\zeta}_{\Lambda},\sigma)\
\mapsto\
(\zeta^{\Lambda}+\eta^{\Lambda},\,\tilde{\zeta}_{\Lambda}+\tilde{\eta}_{\Lambda},\,\sigma+2\kappa-\tilde{\eta}_{\Lambda}\zeta^{\Lambda}+\eta^{\Lambda}\tilde{\zeta}_{\Lambda}-\eta^{\Lambda}\tilde{\eta}_{\Lambda}).$
(4.73)

When $X$ is non-compact, three-dimensional gravity decouples and ${\cal M}$
becomes a family of hyperkähler manifolds of dimension $4b_{2}$ parametrized
by the (non-dynamical) radius $R$, of the form ${\cal S}{\cal
K}_{z^{a}}\times{\cal T}_{\zeta^{a},\tilde{\zeta}_{a}}$, obtained as rigid
limit of (4.72).

As $R\to\infty$, the QK metric on ${\cal M}$ is simply obtained from the
special Kähler metric on ${\cal S}{\cal K}$ by the $c$-map construction
Ferrara:1989ik , but for finite $R$ there are ${\cal O}(e^{-R})$ corrections
from Euclidean D-branes wrapped on even cycles in $X$ times $S^{1}$, and
${\cal O}(e^{-R^{2}})$ corrections from Euclidean five-branes.111111In the
present context, these are Kaluza-Klein five-branes of the form ${\rm
TN}_{k}\times X$, where the first factor is a Taub-NUT gravitational instanton
of charge $k$ which asymptotes to ${\mathbb{R}}^{3}\times S_{1}(R)$. Under
T-duality along the circle, this becomes a Neveu-Schwarz five-brane instanton
correcting the hypermultiplet moduli space, which is the equivalent set-up
used in Alexandrov:2010ca ; Alexandrov:2015xir . Both types of corrections
must preserve the QK property of the metric, which is equivalent MR1327157  to
the existence of a complex contact structure on the twistor space ${\cal
Z}\simeq{\mathbb{P}}^{1}_{t}\times{\cal M}$, where the first factor
parametrizes the sphere-worth of almost complex structures on ${\cal M}$. Such
a structure is guaranteed by the existence of coordinate patches parametrized
by complex Darboux coordinates
$\xi^{\Lambda},\tilde{\xi}_{\Lambda},\tilde{\alpha}$ such that the contact
one-form

${\cal X}=-\frac{1}{2}\left({\rm d}\tilde{\alpha}-\xi^{\Lambda}{\rm
d}\tilde{\xi}_{\Lambda}+\tilde{\xi}_{\Lambda}{\rm d}\xi^{\Lambda}\right)$
(4.74)

is globally well defined up to rescaling by a non-vanishing holomorphic
function. The Darboux coordinates are functions of
$t,R,z^{a},\zeta^{\Lambda},\tilde{\zeta}_{\Lambda},\sigma$, holomorphic in $t$
in the respective patches, and can be chosen such that the Heisenberg group
acts in the same way as in (4.73),

$\mathscr{T}_{\Theta}\ :\
(\xi^{\Lambda},\tilde{\xi}_{\Lambda},\tilde{\alpha})\ \mapsto\
(\xi^{\Lambda}+\eta^{\Lambda},\,\tilde{\xi}_{\Lambda}+\tilde{\eta}_{\Lambda},\,\tilde{\alpha}+2\kappa-\tilde{\eta}_{\Lambda}\xi^{\Lambda}+\eta^{\Lambda}\tilde{\xi}_{\Lambda}-\eta^{\Lambda}\tilde{\eta}_{\Lambda}).$
(4.75)

This is an example of contact transformation, i.e. preserving the contact one-
form (4.74). As a result, the twistor space ${\cal Z}$ can be obtained by
gluing together algebraic twisted tori $\widetilde{{\cal
T}}_{{\mathbb{C}}}={\mathbb{C}}^{2b_{2}+3}_{\xi^{\Lambda},\tilde{\xi}_{\Lambda},\alpha}/H$.
The latter admit a projection to algebraic tori ${{\cal
T}}_{{\mathbb{C}}}={\mathbb{C}}^{2b_{2}+2}_{\xi^{\Lambda},\tilde{\xi}_{\Lambda}}$,
with ${\mathbb{C}}^{\times}$ fiber parametrized by $e^{-\pi{\rm
i}\tilde{\alpha}}$.

D-brane instantons are labelled by charge $\gamma=(p^{\Lambda},q_{\Lambda})\in
H_{\rm even}(X,{\mathbb{Z}})$ and generate corrections to the QK metric on
${\cal M}$ that can be incorporated at the non-linear level by postulating
discontinuities of the Darboux coordinates across the so-called BPS rays
$\ell_{\gamma}$ on ${\mathbb{P}}^{1}_{t}$ (the latter being the loci where the
phase of $t$ coincides with the central charge $Z(\gamma)$). Namely, one
requires that across $\ell_{\gamma}$ they change as Alexandrov:2008gh ;
Alexandrov:2009zh ; Alexandrov:2011ac

$\mathscr{V}_{\gamma}\ :\ \begin{array}[]{rcl}\xi^{\Lambda}&\
\mapsto&\xi^{\prime\Lambda}=\xi^{\Lambda}+\frac{p^{\Lambda}}{2\pi{\rm
i}}\,\Omega(\gamma)\,\log\left(1-{\cal X}_{\gamma}\right),\\\
\tilde{\xi}_{\Lambda}&\
\mapsto&\tilde{\xi}^{\prime}_{\Lambda}=\tilde{\xi}_{\Lambda}+\frac{q_{\Lambda}}{2\pi{\rm
i}}\,\Omega(\gamma)\,\log\left(1-{\cal X}_{\gamma}\right),\\\ \tilde{\alpha}&\
\mapsto&\tilde{\alpha}^{\prime}=\tilde{\alpha}+\frac{1}{2\pi^{2}}\,\Omega(\gamma)\,L_{\sigma(\gamma)}({\cal
X}_{\gamma}).\end{array}$ (4.76)

Here $\Omega(\gamma)$ is the generalized DT invariant counting BPS states of
charge $\gamma$ in four dimensions, ${\cal
X}_{\gamma}=\sigma(\gamma)e^{-2\pi{\rm
i}(q_{\Lambda}\xi^{\Lambda}-p^{\Lambda}\tilde{\xi}_{\Lambda})}$ are the
twisted Fourier modes, with $\sigma(\gamma)$ a quadratic refinement of the
symplectic intersection pairing $\langle\gamma,\gamma^{\prime}\rangle$ on the
lattice of charges such that

${\cal X}_{\gamma}\,{\cal
X}_{\gamma^{\prime}}=(-1)^{\langle\gamma,\gamma^{\prime}\rangle}{\cal
X}_{\gamma+\gamma^{\prime}},$ (4.77)

and $L_{\sigma}(z)$ is the twisted Rogers dilogarithm

$L_{\epsilon}(z)={\rm
Li}_{2}(z)+\frac{1}{2}\,\log(\epsilon^{-1}z)\log(1-z)\,.$ (4.78)

Forgetting the action on $\tilde{\alpha}$, the transformation (4.76) can be
written in terms of the Fourier modes ${\cal X}_{\gamma}$ as

$\mathscr{U}_{\gamma}\ :\ {\cal X}_{\gamma^{\prime}}\ \mapsto\ {\cal
X}^{\prime}_{\gamma^{\prime}}={\cal X}_{\gamma^{\prime}}(1-{\cal
X}_{\gamma})^{\langle\gamma,\gamma^{\prime}\rangle\Omega(\gamma)}.$ (4.79)

This is recognized as the Kontsevich-Soibelman (KS) symplectomorphism encoding
wall-crossing transformations KS:stability ; Gaiotto:2008cd , or the
Delabaere-Dillinger-Pham (DDP) formula controlling the Stokes automorphisms
for quantum periods DDP93 . The action on the additional coordinate
$\tilde{\alpha}$ lifts $\mathscr{U}_{\gamma}$ to a contact transformation
$\mathscr{V}_{\gamma}$. The gluing conditions for $\tilde{\alpha}$ further
imply that $e^{-\pi{\rm i}\tilde{\alpha}}$ is a section of the theta line
bundle ${\cal L}$ over the algebraic tori ${\cal T}_{\mathbb{C}}$
Alexandrov:2010ca ; Neitzke:2011za . The theta line bundle descends to a
canonical hyperholomorphic line bundle over the hyperkähler space obtained
from the D-instanton corrected QK space ${\cal M}_{D}$ by the QK/HK
correspondence Alexandrov:2011ac .

In contrast, five-brane instanton corrections are poorly understood, beyond
linear order around the D-instanton corrected twistor space ${\cal Z}_{D}$
Alexandrov:2010ca  (see also Alexandrov:2014mfa ; Alexandrov:2014rca ;
Alexandrov:2023hiv ). At linear order, instanton corrections from charge $k$
five-branes are described by sections of ${\cal L}^{k}$, the $k$-th power of
the theta line bundle ${\cal L}$. In practice, this means that on local
coordinate patches they are described by functions
$H_{k}(\xi,\tilde{\xi},\tilde{\alpha}):=e^{-{\rm i}\pi
k\tilde{\alpha}}H_{k}(\xi,\tilde{\xi})$ which stay invariant under the contact
transformations $\mathscr{T}_{\Theta}$ and are mapped to each other under
$\mathscr{V}_{\gamma}$.

The invariance under the Heisenberg action (4.75) requires that

$H_{k}(\xi+\eta,\tilde{\xi}+\tilde{\eta})=(-1)^{k\eta^{\Lambda}\tilde{\eta}_{\Lambda}}e^{\pi{\rm
i}k(\eta^{\Lambda}\tilde{\xi}_{\Lambda}-\tilde{\eta}_{\Lambda}\xi^{\Lambda})}\,H_{k}(\xi,\tilde{\xi}).$
(4.80)

This implies the following ‘non-Abelian’ Fourier expansion

$H_{k}(\xi,\tilde{\xi},\tilde{\alpha})=\sum_{\ell^{\Lambda}\in\frac{{\mathbb{Z}}^{b_{2}}}{|k|{\mathbb{Z}}^{b_{2}}}}\sum_{n^{\Lambda}\in{\mathbb{Z}}^{b_{2}}+\frac{\ell^{\Lambda}}{k}}e^{-\pi{\rm
i}k(\tilde{\alpha}+\xi^{\Lambda}\tilde{\xi}_{\Lambda})+2\pi{\rm
i}kn^{\Lambda}\tilde{\xi}_{\Lambda}}\,{\cal
H}_{k,\ell^{\Lambda}}(\xi^{\Lambda}-n^{\Lambda})\,,$ (4.81)

where ${\cal H}_{k,\ell^{\Lambda}}(\xi^{\Lambda})$ is referred to as the wave-
function. The reason for this terminology is that it transforms in the
metaplectic representation under a change of symplectic basis. For example,
upon exchanging the ’position’ coordinates $\xi^{\Lambda}$ with the conjugate
’momenta’ $\tilde{\xi}_{\Lambda}$, one gets

$H_{k}(\xi,\tilde{\xi},\tilde{\alpha})=\sum_{l_{\Lambda}\in\frac{{\mathbb{Z}}}{|k|{\mathbb{Z}}}}\sum_{m_{\Lambda}\in{\mathbb{Z}}+\frac{\ell^{\Lambda}}{k}}e^{-\pi{\rm
i}k(\tilde{\alpha}-\xi^{\Lambda}\tilde{\xi}_{\Lambda})-2\pi{\rm
i}km_{\Lambda}\xi^{\Lambda}}\,{\cal
H}^{\prime}_{k,l_{\Lambda}}(\tilde{\xi}_{\Lambda}-m_{\Lambda})\,,$ (4.82)

where the wave-functions are related by Fourier transform

${\cal
H}_{k,\ell^{\Lambda}}(\xi)=\sum_{m_{\Lambda}\in\frac{{\mathbb{Z}}}{|k|{\mathbb{Z}}}}e^{-2\pi{\rm
i}m_{\Lambda}\ell^{\Lambda}/k}\int{\rm d}\tilde{\xi}_{\Lambda}\,e^{2\pi{\rm
i}k\xi^{\Lambda}\tilde{\xi}_{\Lambda}}\,{\cal
H}^{\prime}_{k,m_{\Lambda}}(\tilde{\xi})\,.$ (4.83)

For $k=1$, it was argued in Alexandrov:2010ca  using S-duality in type IIB
string theory that the wave function ${\cal H}_{1}(\xi)$ should be identified
with the topological string partition function in real polarization ${\cal
Z}_{{\mathbb{R}}}(\xi)$, analytically continued to the complex domain and
evaluated on the Darboux coordinates $\xi^{\Lambda}$. As a result, (4.81)
becomes proportional to the ‘dual partition function’:

$H_{1}(\xi,\tilde{\xi},\tilde{\alpha})=e^{2\pi{\rm
i}\alpha}\sum_{n^{\Lambda}\in{\mathbb{Z}}}e^{2\pi{\rm
i}n^{\Lambda}\tilde{\xi}_{\Lambda}}{\cal
Z}_{{\mathbb{R}}}(\xi^{\Lambda}-n^{\Lambda})\,,$ (4.84)

where we introduced

$\alpha=-\frac{1}{2}\,(\tilde{\alpha}+\xi^{\Lambda}\tilde{\xi}_{\Lambda})$
(4.85)

such that

${\cal X}={\rm d}\alpha+\xi^{\Lambda}{\rm d}\tilde{\xi}_{\Lambda}.$ (4.86)

In fact, ${\cal Z}_{{\mathbb{R}}}(\xi)$ can be shown to be proportional to the
usual holomorphic topological string partition function ${\cal
Z}(\boldsymbol{t};g_{s})$ provided one identifies121212The easiest way to
establish these relations is to compare the Fourier modes ${\cal X}_{\gamma}$
with $e^{-{\cal A}/g_{s}}$, where the instanton action ${\cal A}$ is given in
(3.15), by identifying the integer valued charges $q_{\Lambda}$ and
$(d_{a},d_{0})$. Another consistency check is that in (1.2) the argument of
${\cal Z}$ becomes $t^{a}-{\rm i}g_{s}n^{a}=2\pi{\rm
i}(\xi^{a}-n^{a})/\xi^{0}$, which is consistent with (4.84) if one sets
$n^{0}=0$. Since $n^{0}$ plays the role of D6-brane charge, this restriction
is indeed necessary in the non-compact case. Finally, one can also check that
the relations (4.87) ensure that in (1.3) the shift of $t^{a}$ in the argument
of $\tau$ is consistent with the KS transformation of $\xi^{a}$ given in
(4.76).

$t^{a}=2\pi{\rm i}\,\frac{\xi^{a}}{\xi^{0}}\,,\qquad
g_{s}=\frac{2\pi}{\xi^{0}}\,.$ (4.87)

Thus, the five-brane instantons turn out to be described by the unrefined
version of the partition function studied in the previous sections.

One of the main results of Alexandrov:2015xir  was to determine the
transformation property of the wave functions ${\cal H}_{k,\ell^{\Lambda}}$
under the contact transformations (4.76). Taking into account the
transformation of $\tilde{\alpha}$, it is immediate to see that

$\mathscr{V}_{\gamma}\ :\ H_{k}(\xi,\tilde{\xi})\ \mapsto\
H^{\prime}_{k}(\xi,\tilde{\xi})=e^{\frac{k}{2\pi{\rm
i}}\,\Omega(\gamma)\,L_{\sigma(\gamma)}({\cal
X}_{\gamma})}\,H_{k}(\xi^{\prime},\tilde{\xi}^{\prime})\,.$ (4.88)

In the pure electric case $\gamma=(0,q^{\Lambda})$, it was shown that this
implies

$\mathscr{V}_{\gamma}\ :\ {\cal H}_{k,\ell^{\Lambda}}(\xi)\ \mapsto\
\left[\mathbf{A}^{(k)}(-q_{\Lambda}\xi^{\Lambda},-q_{\Lambda}\ell^{\Lambda})\right]^{-\Omega(\gamma)}\,{\cal
H}_{k,\ell^{\Lambda}}(\xi),$ (4.89)

where131313The function defined in (4.90) is the inverse of
$\mathbf{A}^{(k)}(x,\ell)$ in Alexandrov:2015xir .

$\mathbf{A}^{(k)}(x,\ell)=\left(1-e^{2\pi{\rm
i}(x+\ell/k)}\right)^{\ell}\,\bigl{[}\Phi_{1}\left({\rm
i}(x+\ell/k)\right)\bigr{]}^{k}.$ (4.90)

The transformation property across generic BPS rays can be obtained by
conjugating (4.89) by the metaplectic representation, or by extending by a
rank 2 hyperbolic lattice, see (Alexandrov:2015xir, , (2.11)). For $k=1$,
(4.89) and (4.87) imply that the topological string amplitude ${\cal
Z}(\boldsymbol{t};g_{s})$ gets multiplied by
$\Phi_{1}\left(-(q_{a}t^{a}+2\pi{\rm i}q_{0})/g_{s}\right)$, which coincides
with the Stokes factor for the topological string partition function, as
noticed in Iwaki:2023cek .

###  4.2 Refined contact structure

Our goal is to incorporate the refinement parameter into the above
construction. In fact, a refined version of the function
$\mathbf{A}^{(k)}(x,\ell)$ was already put forward in (Alexandrov:2015xir, ,
§4). However, it was not derived nor justified by any invariance or
transformation property. To fill this gap, we have to find a proper
generalization of the dual partition function (4.84) (or more generally
(4.81)) and compute the action of a refined version of the transformation
$\mathscr{V}_{\gamma}$ (4.76) on its kernel. However, as will be discussed
shortly, while the refinement of the symplectomorphism $\mathscr{U}_{\gamma}$
is well understood, this is not so for the contact transformation
$\mathscr{V}_{\gamma}$. Therefore, the first step is to find how to lift
refined symplectomorphisms to refined contact transformations. Since the
defining property of the latter in the absence of refinement was that they
preserve the contact one-form, this can be seen as constructing a refined
version of the contact structure. This is the problem that we address in this
subsection.

Physically, the standard way to introduce a refinement is to switch on an
$\Omega$-background. As was observed in Gaiotto:2010be ; Cecotti:2014wea  in
the gauge theory context, its effect is to deform the Riemann-Hilbert problem
defining the instanton-corrected metric on the Coulomb branch into a non-
commutative one Barbieri:2019yya . This is achieved by replacing the KS
symplectomorphism (4.79) by its quantum version

$\hat{\mathscr{U}}_{\gamma}\ :\ {\cal X}_{\gamma^{\prime}}\ \mapsto\ {\cal
X}^{\prime}_{\gamma^{\prime}}={\cal U}_{\gamma}\star{\cal
X}_{\gamma^{\prime}}\star{\cal U}_{\gamma}^{-1}.$ (4.91)

Here $\star$ denotes the non-commutative Moyal product

$f\star g=f\exp\left[\frac{{\rm
i}\,\epsilon}{2\pi}\sum_{\Lambda}\left(\overleftarrow{\partial}_{\\!\xi^{\Lambda}}\overrightarrow{\partial}_{\\!\tilde{\xi}_{\Lambda}}-\overleftarrow{\partial}_{\\!\tilde{\xi}_{\Lambda}}\overrightarrow{\partial}_{\\!\xi^{\Lambda}}\right)\right]g,$
(4.92)

where $\epsilon$ is the refinement parameter to be related to ${\mathsf{b}}$
in the next subsection. It is easy to check that with respect to the Moyal
product the relation (4.77) gets deformed to

${\cal X}_{\gamma}\star{\cal
X}_{\gamma^{\prime}}=(-y)^{\langle\gamma,\gamma^{\prime}\rangle}{\cal
X}_{\gamma+\gamma^{\prime}},\qquad y=e^{2\pi{\rm i}\epsilon}.$ (4.93)

The function ${\cal U}_{\gamma}$ in (4.91), which generates the quantum KS
transformation, is defined in terms the compact quantum dilogarithm
$E_{y}(x)$, described in Appendix A, as

${\cal U}_{\gamma}=\prod_{n\in{\mathbb{Z}}}E_{y}\left(y^{n}\,{\cal
X}_{\gamma}\right)^{\Omega_{n}(\gamma)},$ (4.94)

where $\Omega_{n}(\gamma)$ are the Laurent coefficients of the refined BPS
indices (to be distinguished from the multiplicities $\Omega_{[j]}(\gamma)$
defined in (2.25))

$\Omega(\gamma,y)=\sum_{n\in{\mathbb{Z}}}\Omega_{n}(\gamma)\,y^{n}.$ (4.95)

Note that the product in (4.94) is finite, since $\Omega_{n}(\gamma)$ vanishes
for $|n|$ large enough. Substituting (4.94) into (4.91) and evaluating the
star product explicitly, one finds

$\begin{split}{\cal X}^{\prime}_{\gamma^{\prime}}=&\,{\cal
X}_{\gamma^{\prime}}\prod_{n\in{\mathbb{Z}}}\left[\frac{E_{y}\left(y^{n+\langle\gamma,\gamma^{\prime}\rangle}\,{\cal
X}_{\gamma}\right)}{E_{y}\left(y^{n-\langle\gamma,\gamma^{\prime}\rangle}\,{\cal
X}_{\gamma}\right)}\right]^{\Omega_{n}(\gamma)}\\\ =&\,{\cal
X}_{\gamma^{\prime}}\prod_{n\in{\mathbb{Z}}}\prod_{k=0}^{|\langle\gamma,\gamma^{\prime}\rangle|-1}\left(1-y^{n+2k-|\langle\gamma,\gamma^{\prime}\rangle|+1}{\cal
X}_{\gamma}\right)^{{\rm
sgn}\langle\gamma,\gamma^{\prime}\rangle\,\Omega_{n}(\gamma)}\,,\end{split}$
(4.96)

where in the second step we used the property (A.131) of the quantum
dilogarithm. In the unrefined limit $y\to 1$, this transformation reduces to
the classical symplectomorphism (4.79), as it should.

The analysis of Alexandrov:2019rth  suggests that a similar non-commutative
deformation is induced by the refinement in full string theory as well. But in
this case we also need to understand how to extend
$\hat{\mathscr{U}}_{\gamma}$ to act on the full twistor space, including the
coordinate $\tilde{\alpha}$. In other words, we need to extend the
construction to the case of twisted quantum tori. To this end, we observe that
the contact one-form (4.86) on $\widetilde{\cal T}_{{\mathbb{C}}}$ arises by
projectivizing the symplectic form $\omega$ on a ${\mathbb{C}}^{\star}$ bundle
over $\widetilde{\cal T}_{{\mathbb{C}}}$ (which coincides at least locally
with the Swann bundle MR1096180  of the QK space ${\cal M}$), with Darboux
coordinates $(\eta^{I},\mu_{I})$, $I=\flat,0,\dots,b_{2}$, such that
Alexandrov:2010qdt

$\omega=\sum_{I}{\rm d}\eta^{I}\wedge{\rm d}\mu_{I}\,.$ (4.97)

Indeed, under the identification

$\xi^{\Lambda}=\frac{\eta^{\Lambda}}{\eta^{\flat}}\,,\qquad\tilde{\xi}_{\Lambda}=\mu_{\Lambda}\,,\qquad\alpha=\mu_{\flat}\,,$
(4.98)

we have

$\omega={\rm d}\eta^{\flat}\wedge({\rm d}\mu_{\flat}+\xi^{\Lambda}{\rm
d}\tilde{\xi}_{\Lambda})+\eta^{\flat}{\rm d}\xi^{\Lambda}\wedge{\rm
d}\tilde{\xi}_{\Lambda}={\rm d}\eta^{\flat}\wedge{\cal X}+\eta^{\flat}{\rm
d}{\cal X}.$ (4.99)

Thus, we can define a star product on functions of
$\xi^{\Lambda},\tilde{\xi}_{\Lambda},\alpha$ (or equivalently functions of
$\xi^{\Lambda},\mu_{I}$) by viewing them as functions of $(\eta^{I},\mu_{I})$
which are invariant under the ${\mathbb{C}}^{\times}$ action
$\eta^{I}\to\lambda\eta^{I}$, and using the Moyal product on
${\mathbb{C}}^{2b_{2}+4}$, with a deformation parameter that we denote by
$\epsilon^{\flat}$,

$f\star g=f\exp\left[\frac{{\rm
i}\,\epsilon^{\flat}}{2\pi}\sum_{I}\left(\overleftarrow{\partial}_{\\!\eta^{I}}\overrightarrow{\partial}_{\\!\mu_{I}}-\overleftarrow{\partial}_{\\!\mu_{I}}\overrightarrow{\partial}_{\\!\eta^{I}}\right)\right]g.$
(4.100)

In general, the resulting Moyal product is not invariant under the
${\mathbb{C}}^{\times}$ action $\eta^{I}\to\lambda\eta^{I}$, unless this
action also affects the deformation parameter via
$\epsilon^{\flat}\to\lambda\epsilon^{\flat}$. In other words, the result is
not only a function of $\xi^{\Lambda}\equiv\eta^{\Lambda}/\eta^{\flat}$ and
$\mu_{I}$ but also depends on $\epsilon\equiv\epsilon^{\flat}/\eta^{\flat}$.
Thus, the Moyal product (4.100) defines a non-commutative deformation of the
product of functions
$f(\xi^{\Lambda},\tilde{\xi}_{\Lambda},\tilde{\alpha},\epsilon)$ on
$\widetilde{T}_{{\mathbb{C}}}\times{\mathbb{C}}_{\epsilon}$, which by
construction preserves associativity.

It is easy to see that the corresponding Moyal bracket

$\\{f,g\\}_{\star}=\frac{\pi}{{\rm i}\epsilon}\left(f\star g-g\star f\right)$
(4.101)

in the unrefined limit $\epsilon\to 0$ reproduces the contact bracket
$\\{\,\cdot\,,\,\cdot\,\\}_{0,0}$ introduced in Alexandrov:2008gh .141414In
general, the contact bracket is defined on sections of ${\cal O}(2m)$ and
${\cal O}(2n)$ bundles by
$\\{f,g\\}_{m,n}=\,\partial_{\xi^{\Lambda}}f\,\partial_{\tilde{\xi}_{\Lambda}}g+\left(m-\xi^{\Lambda}\partial_{\xi^{\Lambda}}\right)f\,\partial_{\alpha}g-\partial_{\xi^{\Lambda}}g\,\partial_{\tilde{\xi}_{\Lambda}}f-\left(n-\xi^{\Lambda}\partial_{\xi^{\Lambda}}\right)g\,\partial_{\alpha}f.$
Arbitrary values of $m$ and $n$ can easily be incorporated into the above
construction since a section of ${\cal O}(2m)$ bundle is described by a
homogeneous function of degree $m$ on
$\widetilde{T}_{{\mathbb{C}}}\times{\mathbb{C}}_{\epsilon}$, i.e. it is
sufficient to postulate
$f=(\eta^{\flat})^{m}f(\eta^{\Lambda}/\eta^{\flat},\mu_{\Lambda},\mu_{\flat},\epsilon^{\flat}/\eta^{\flat})$
and evaluate the same star product (4.100). However, in this work for our
purposes it is sufficient to restrict to $m=0$.  Since the latter essentially
encodes the contact structure, e.g. it generates classical contact
transformations via exponentiation $\exp\\{h,\,\cdot\,\\}_{1,0}$
Alexandrov:2014mfa , the star product (4.100) can be thought as providing a
definition of the refined contact structure.

###  4.3 Wave functions and non-commutative wall-crossing

Equipped with the star product, we can now construct a refined analogue of the
dual partition function (4.81). To this end, we simply replace the usual
product by the non-commutative one,151515In the refined case, we define
$H^{\text{(ref)}}_{k}(\xi,\tilde{\xi},\alpha)$ to be a function of $\alpha$
rather than $\tilde{\alpha}$ because it is $\alpha$ that coincides with one of
the Darboux coordinates for the symplectic form (4.97) used to define the star
product, but one can always use (4.85) to translate between the two variables.

$H^{\text{(ref)}}_{k}(\xi,\tilde{\xi},\alpha)=\sum_{\ell^{\Lambda}\in\frac{{\mathbb{Z}}}{|k|{\mathbb{Z}}}}\sum_{n^{\Lambda}\in{\mathbb{Z}}+\ell^{\Lambda}/k}e^{2\pi{\rm
i}k(\alpha+n^{\Lambda}\tilde{\xi}_{\Lambda})}\star{\cal
H}^{\text{(ref)}}_{k,\ell^{\Lambda}}(\xi^{\Lambda}-n^{\Lambda}).$ (4.102)

In fact, the star product can be evaluated explicitly using the property

$e^{2\pi{\rm i}(k\alpha+p^{\Lambda}\tilde{\xi}_{\Lambda})}\star
f(\xi^{\Lambda},\tilde{\xi}_{\Lambda},\alpha,\epsilon)=e^{2\pi{\rm
i}(k\alpha+p^{\Lambda}\tilde{\xi}_{\Lambda})}\,f\left(\frac{\xi^{\Lambda}+\epsilon
p^{\Lambda}}{1+\epsilon
k},\tilde{\xi}_{\Lambda},\alpha,\frac{\epsilon}{1+\epsilon k}\right).$ (4.103)

In particular, it allows to see that the invariance under the Heisenberg group
still holds. Indeed, the star product changes the argument of ${\cal
H}^{\text{(ref)}}_{k,\ell^{\Lambda}}$ to $\frac{\xi^{\Lambda}+k\epsilon
n^{\Lambda}}{1+k\epsilon}-n^{\Lambda}=\frac{\xi^{\Lambda}-n^{\Lambda}}{1+k\epsilon}$
and thus the refined wave function is still a function of the difference
$\xi^{\Lambda}-n^{\Lambda}$, which ensures the invariance.

Our goal now is to determine the transformation property of the refined wave-
function ${\cal H}^{\text{(ref)}}_{k,\ell}(\xi)$ under the quantum KS
transformations $\hat{\mathscr{U}}_{\gamma}$ lifted to the twisted torus by
means of (4.100). Denoting by $\hat{\mathscr{V}}_{\gamma}$ the corresponding
lift, we arrive at the following condition

$\sum_{\ell^{\Lambda},n^{\Lambda}}e^{2\pi{\rm
i}k(\alpha+n^{\Lambda}\tilde{\xi}_{\Lambda})}\star\hat{\mathscr{V}}_{\gamma}[{\cal
H}^{\text{(ref)}}_{k,\ell^{\Lambda}}(\xi-n)]={\cal
U}_{\gamma}\star\sum_{\ell^{\Lambda},n^{\Lambda}}e^{2\pi{\rm
i}k(\alpha+n^{\Lambda}\tilde{\xi}_{\Lambda})}\star{\cal
H}^{\text{(ref)}}_{k,\ell^{\Lambda}}(\xi-n)\star{\cal U}_{\gamma}^{-1}.$
(4.104)

Restricting to the electric case $\gamma=(0,q_{\Lambda})$, such that ${\cal
U}_{\gamma}$ becomes a function of $\xi^{\Lambda}$ and $\epsilon$ only and
therefore commutes with the wave function, and using (4.103) to evaluate the
star product, we obtain that the r.h.s. of (4.104) is given by

$\sum_{\ell^{\Lambda},n^{\Lambda}}e^{2\pi{\rm
i}k(\alpha+n^{\Lambda}\tilde{\xi}_{\Lambda})}\star\frac{{\cal
U}_{\gamma}\left(\frac{\xi-2k\epsilon
n}{1-2k\epsilon};\frac{\epsilon}{1-2k\epsilon}\right)}{{\cal
U}_{\gamma}(\xi;\epsilon)}\,{\cal
H}^{\text{(ref)}}_{k,\ell^{\Lambda}}(\xi-n).$ (4.105)

Furthermore, using that

$\begin{split}e^{-2\pi{\rm i}q_{\Lambda}\xi^{\Lambda}}=&\,e^{-2\pi{\rm
i}q_{\Lambda}(\xi^{\Lambda}-n^{\Lambda}+\ell^{\Lambda}/k)},\\\ e^{-2\pi{\rm
i}q_{\Lambda}\,\frac{\xi^{\Lambda}-2k\epsilon
n^{\Lambda}}{1-2k\epsilon}}=&\,e^{-2\pi{\rm
i}q_{\Lambda}\left(\frac{\xi^{\Lambda}-n^{\Lambda}}{1-2k\epsilon}+\frac{\ell^{\Lambda}}{k}\right)},\end{split}$
(4.106)

it is easy to see that the factor generated by the transformation depends on
$\xi^{\Lambda}$ and $n^{\Lambda}$ only through their difference. Thus, the
condition (4.104) requires that the wave function should transform as

$\hat{\mathscr{V}}_{\gamma}[{\cal
H}^{\text{(ref)}}_{k,\ell^{\Lambda}}(\xi)]=\Upsilon_{k,\ell^{\Lambda}}(\xi)\,{\cal
H}^{\text{(ref)}}_{k,\ell^{\Lambda}}(\xi),\qquad\Upsilon_{k,\ell^{\Lambda}}(\xi)=\frac{{\cal
U}_{\gamma}\left(\frac{\xi}{1-2k\epsilon}+\frac{\ell}{k}\,;\,\frac{\epsilon}{1-2k\epsilon}\right)}{{\cal
U}_{\gamma}(\xi+\ell/k;\epsilon)}\,.$ (4.107)

It turns out that the function $\Upsilon_{k,\ell^{\Lambda}}$ can be expressed
through the Faddeev quantum dilogarithm or its appropriate generalization,
which are all described in Appendix A. For simplicity, let us first consider
the unit multiplicity case, $\Omega_{n}(\gamma)=\delta_{n}$, such that

${\cal U}_{\gamma}(\xi;\epsilon)=E_{y}\left(e^{-2\pi{\rm
i}q_{\Lambda}\xi^{\Lambda}}\right)$ (4.108)

with $y=e^{2\pi{\rm i}\epsilon}$. We will distinguish between cases of
positive and negative five-brane charge $k$ because they lead to different
relations between the refinement parameters $\epsilon$ and ${\mathsf{b}}$. For
$k>0$, we identify

$\epsilon=\frac{1}{2k}\left(1-{\mathsf{b}}^{-2}\right),$ (4.109)

such that $1-2k\epsilon={\mathsf{b}}^{-2}$ and

$y=e^{\pi{\rm i}/k}e^{-\pi{\rm
i}/({\mathsf{b}}^{2}k)}:={\tilde{y}}_{{\mathsf{b}},k},\qquad e^{\frac{2\pi{\rm
i}\epsilon}{1-2k\epsilon}}=e^{-\pi{\rm i}/k}e^{\pi{\rm
i}{\mathsf{b}}^{2}/k}:=y_{{\mathsf{b}},k}.$ (4.110)

which generalize the variables (3.54) to generic $k$ and satisfy
${\tilde{y}}_{{\mathsf{b}},k}=\bar{y}_{1/\bar{\mathsf{b}},k}$. Thus, the
function $\Upsilon_{k,\ell^{\Lambda}}$ in (4.107) takes the form

$\Upsilon_{k,\ell^{\Lambda}}(\xi)=\frac{E_{y_{{\mathsf{b}},k}}\left(e^{-2\pi{\rm
i}({\mathsf{b}}^{2}q_{\Lambda}\xi^{\Lambda}+\ell/k)}\right)}{E_{{\tilde{y}}_{{\mathsf{b}},k}}\left(e^{-2\pi{\rm
i}(q_{\Lambda}\xi^{\Lambda}+\ell/k)}\right)}\,,$ (4.111)

where we denoted $\ell=q_{\Lambda}\ell^{\Lambda}$. Setting also $s_{\ell}={\rm
sgn}(\ell)$ and using (A.131), the numerator can be rewritten as

$E_{y_{{\mathsf{b}},k}}\left(y_{{\mathsf{b}},k}^{2\ell}\,e^{-2\pi{\rm
i}{\mathsf{b}}^{2}(q_{\Lambda}\xi^{\Lambda}+\ell/k)}\right)=E_{y_{{\mathsf{b}},k}}\left(e^{-2\pi{\rm
i}{\mathsf{b}}^{2}(q_{\Lambda}\xi^{\Lambda}+\ell/k)}\right)\prod_{j=0}^{|\ell|-1}\left(1-y_{{\mathsf{b}},k}^{2\ell-
s_{\ell}(2j+1)}\,e^{-2\pi{\rm
i}{\mathsf{b}}^{2}(q_{\Lambda}\xi^{\Lambda}+\ell/k)}\right)^{s_{\ell}}.$
(4.112)

We can then use the property (A.130) to get

$\displaystyle\displaystyle\Upsilon_{k,\ell^{\Lambda}}(\xi)$ $\displaystyle=$
$\displaystyle\prod_{j=0}^{k-1}\frac{E_{e^{\pi{\rm
i}{\mathsf{b}}^{2}}}\left(-e^{-2\pi{\rm
i}{\mathsf{b}}\,\Xi^{+}_{k,j}}\right)}{E_{e^{-\pi{\rm
i}/{\mathsf{b}}^{2}}}\left(-e^{-2\pi{\rm
i}{\mathsf{b}}^{-1}\,\Xi^{+}_{k,j}}\right)}\prod_{j=0}^{|\ell|-1}\left(1-e^{-2\pi{\rm
i}\left({\mathsf{b}}^{2}q_{\Lambda}\xi^{\Lambda}+\frac{\ell}{k}+\frac{s_{\ell}}{k}({\mathsf{b}}^{2}-1)\left(j+\frac{1}{2}\right)\right)}\right)^{s_{\ell}}$
(4.113) $\displaystyle=$
$\displaystyle\prod_{j=0}^{k-1}\Phi_{\mathsf{b}}^{-1}(-{\rm
i}\Xi^{+}_{k,j})\prod_{j=0}^{|\ell|-1}\left(1-e^{-2\pi{\rm
i}\left({\mathsf{b}}^{2}q_{\Lambda}\xi^{\Lambda}+\frac{\ell}{k}+\frac{s_{\ell}}{k}({\mathsf{b}}^{2}-1)\left(j+\frac{1}{2}\right)\right)}\right)^{s_{\ell}},$

where

$\Xi^{+}_{k,j}={\mathsf{b}}q_{\Lambda}\xi^{\Lambda}+\frac{{\mathsf{b}}\ell}{k}-\frac{1}{k}\left({\mathsf{b}}-{\mathsf{b}}^{-1}\right)\left(j-\frac{k-1}{2}\right).$
(4.114)

For $\ell>0$ (hence $s=1$), (4.113) is recognized as the function ${\bf
A}_{\hbar}^{(k)}(t,\ell)$ defined in (Alexandrov:2015xir, , (4.1)) evaluated
at $\hbar={\mathsf{b}}^{2}$ and $t=-{\rm
i}{\mathsf{b}}^{2}q_{\Lambda}\xi^{\Lambda}$.

For $k<0$, we replace the identification (4.109) by

$\epsilon=\frac{1}{2k}(1-{\mathsf{b}}^{2}),$ (4.115)

such that $1-2k\epsilon={\mathsf{b}}^{2}$ and

$y=y_{{\mathsf{b}},-k},\qquad e^{\frac{2\pi{\rm
i}\epsilon}{1-2k\epsilon}}={\tilde{y}}_{{\mathsf{b}},-k}.$ (4.116)

Repeating the same steps as above, one arrives at the following representation
for the function $\Upsilon_{k,\ell^{\Lambda}}$ (4.107):

$\displaystyle\displaystyle\Upsilon_{k,\ell^{\Lambda}}(\xi)$ $\displaystyle=$
$\displaystyle\prod_{j=0}^{|k|-1}\Phi_{\mathsf{b}}(-{\rm
i}\Xi^{-}_{k,j})\prod_{j=0}^{|\ell|-1}\left(1-e^{-2\pi{\rm
i}\left({\mathsf{b}}^{-2}q_{\Lambda}\xi^{\Lambda}+\frac{\ell}{k}+\frac{s_{\ell}}{k}({\mathsf{b}}^{-2}-1)\left(j+\frac{1}{2}\right)\right)}\right)^{s_{\ell}},$
(4.117)

where now

$\Xi^{-}_{k,j}={\mathsf{b}}^{-1}q_{\Lambda}\xi^{\Lambda}+\frac{\ell}{{\mathsf{b}}k}-\frac{1}{|k|}\left({\mathsf{b}}-{\mathsf{b}}^{-1}\right)\left(j-\frac{|k|-1}{2}\right).$
(4.118)

For BPS rays carrying general refined BPS indices $\Omega_{n}(\gamma)$, the
wall-crossing transformation can be expressed through a generalization
$\Phi^{[j],k}_{{\mathsf{b}}}(z)$ of the Faddeev quantum dilogarithm defined in
(A.144), which also has a representation as a product of the usual quantum
dilogarithms with shifted arguments. Again, proceeding as above, it is
straightforward to show that

$\Upsilon_{k,\ell^{\Lambda}}(\xi)=\prod_{j}\left[\prod_{i=0}^{|k|-1}\left(\Phi^{[j],k}_{{\mathsf{b}}}(-{\rm
i}\Xi^{s_{k}}_{k,i})\right)^{s_{k}}\prod_{m=-j}^{j}\prod_{i=0}^{|\ell|-1}\left(1-e^{-2\pi{\rm
i}\left({\mathsf{b}}^{2s_{k}}q_{\Lambda}\xi^{\Lambda}+\frac{\ell}{k}+\frac{s_{\ell}}{k}({\mathsf{b}}^{2s_{k}}-1)\left(i-m+\frac{1}{2}\right)\right)}\right)^{-s_{\ell}}\right]^{-\Omega_{[j]}},$
(4.119)

where $s_{k}={\rm sgn}(k)$. For $k=1$, in which case $\ell=0$, this formula
reduces to

$\Upsilon_{1}(\xi)=\prod_{j}\left[\Phi_{{\mathsf{b}}}^{[j]}(-{\rm
i}{\mathsf{b}}q_{\Lambda}\xi^{\Lambda})\right]^{-\Omega_{[j]}}.$ (4.120)

This result is to be compared with (3.64) where ${\cal A}=2\pi
q_{a}t^{a}+4\pi^{2}{\rm i}q_{0}$. It s easy to see that the factor appearing
in (3.64) coincides with $\Upsilon_{1}(\xi)$ provided one relates the
variables as in (4.87) with $\xi^{\Lambda}$ replaced by
${\mathsf{b}}\xi^{\Lambda}$, i.e.

$t^{a}=2\pi{\rm i}\,\frac{\xi^{a}}{\xi^{0}}\,,\qquad
g_{s}=\frac{2\pi}{{\mathsf{b}}\xi^{0}}\,.$ (4.121)

Note that the last relation is equivalent to

$\xi^{0}=\frac{2\pi}{\epsilon_{1}}\,,$ (4.122)

where $\epsilon_{1}$ is one the deformation parameters of the
$\Omega$-background, see (2.1). The match of the Stocks factors suggests that
the wave function ${\cal H}^{\text{(ref)}}_{1}$ is proportional to the refined
topological string,

${\cal H}^{\text{(ref)}}_{1}(\xi)\sim{\cal
Z}(\boldsymbol{t};g_{s},{\mathsf{b}}),$ (4.123)

with the parameters identified as in (4.121). This generalizes a similar
relation in the unrefined case.

Note that if we had chosen the opposite ordering in the definition of the
refined dual partition function (4.102), the results for the cases of positive
and negative $k$ would effectively be swapped. Indeed, exchanging the ordering
of factors in (4.103) leads to the flip of signs in front of $k$ and
$p^{\Lambda}$ in the arguments of the function $f$. As a result, denoting by
tilde the quantities corresponding to the opposite ordering, one has

$\tilde{\Upsilon}_{k,\ell^{\Lambda}}(\xi)=\Upsilon_{-k,-\ell^{\Lambda}}^{-1}(\xi).$
(4.124)

In particular, using (4.119), one finds that

$\tilde{\Upsilon}_{1}(\xi)=\prod_{j}\left[\Phi_{{\mathsf{b}}}^{[j]}(-{\rm
i}{\mathsf{b}}^{-1}q_{\Lambda}\xi^{\Lambda})\right]^{-\Omega_{[j]}},$ (4.125)

which in turn implies the identification

$\tilde{\cal H}^{\text{(ref)}}_{1}(\xi)\sim{\cal Z}\left(2\pi{\rm
i}\,\frac{\xi^{a}}{\xi^{0}}\,;\,\frac{2\pi{\mathsf{b}}}{\xi^{0}}\,,{\mathsf{b}}\right).$
(4.126)

Note that in this case the relation (4.122) is replaced by

$\xi^{0}=-\frac{2\pi}{\epsilon_{2}}\,.$ (4.127)

We observe that the effect of the refinement is to introduce a factor of
${\mathsf{b}}$ in the relation between $\xi^{0}$ and the topological string
coupling $g_{s}$. It spoils the symmetry
${\mathsf{b}}\leftrightarrow{\mathsf{b}}^{-1}$, unless one simultaneously
changes the ordering in the definition of the refined dual partition function.
Of course, this factor could be absorbed in the definition of the Darboux
coordinates $\xi^{\Lambda}$, but the price to pay is a modification of the
quasi-periodicity conditions (4.75) and (4.80), and it would reappear anyway
in the quantization condition of charges.

##  Appendix A Quantum dilogarithms

In this Appendix, we introduce several versions of the quantum dilogarithm
function which play a role in this work.

The standard (sometimes called compact) quantum dilogarithm $E_{y}(x)$ is
defined as161616There are different conventions in the literature, e.g.
Barbieri:2019yya ; Chuang:2022uey  define the quantum dilogarithm as
$\mathbb{E}_{q}(x)=(x,q)_{\infty}$, which is related to our definition by
$E_{y}(x)=[\mathbb{E}_{y^{2}}(xy)]^{-1}$.

$E_{y}(x):=\exp\left[\sum_{k=1}^{\infty}\frac{(xy)^{k}}{k(1-y^{2k})}\right]=\prod_{n=0}^{\infty}(1-xy^{2n+1})^{-1}=(xy;y^{2})_{\infty}^{-1},$
(A.128)

where the last representation is given in terms of the $q$-Pochhammer symbol
$(a;q)_{n}:=\prod_{k=0}^{n-1}(1-aq^{k})$. The quantum dilogarithm satisfies
the following properties:

$\displaystyle\displaystyle E_{y}(xy^{2})$ $\displaystyle=$
$\displaystyle(1-xy)E_{y}(x),$ (A.129) $\displaystyle E_{y^{1/k}}(x)$
$\displaystyle=$
$\displaystyle\prod_{j=0}^{k-1}E_{y}\left(y^{\frac{2j+1}{k}-1}x\right),\qquad
k\in{\mathbb{N}}.$ (A.130)

The first property has the obvious generalization

$E_{y}(y^{2\ell}x)=E_{y}(x)\prod_{j=0}^{\ell-1}(1-xy^{2j+1}),\qquad\ell\in{\mathbb{N}}.$
(A.131)

A different version of the quantum dilogarithm (sometimes called non-compact)
is due to Faddeev Faddeev ,

$\Phi_{\mathsf{b}}(z):=\exp\left(\int_{{\mathbb{R}}+{\rm
i}\epsilon}\frac{e^{-2{\rm
i}zv}}{4\sinh(v{\mathsf{b}})\sinh(v/{\mathsf{b}})}\frac{{\rm d}v}{v}\right).$
(A.132)

It possesses several beautiful properties listed, for example, in
(Alexandrov:2015xir, , §A.2). Here we mention the quasi-periodicity

$\Phi_{\mathsf{b}}(x-{\rm i}{\mathsf{b}}^{\pm
1}/2)=\left(1+e^{2\pi{\mathsf{b}}^{\pm 1}x}\right)\Phi_{\mathsf{b}}(x+{\rm
i}{\mathsf{b}}^{\pm 1}/2),$ (A.133)

the classical limit

$\Phi_{\mathsf{b}}(x)\mathop{\sim}\limits_{{\mathsf{b}}\to
0}\exp\left(\frac{{\rm Li}_{2}(-e^{2\pi{\mathsf{b}}x})}{2\pi{\rm
i}{\mathsf{b}}^{2}}\right),$ (A.134)

the special value at ${\mathsf{b}}=1$

$\Phi_{1}(x)=\exp\left[\frac{{\rm i}}{2\pi}\left({\rm Li}_{2}(e^{2\pi x})+2\pi
x\log(1-e^{2\pi x})\right)\right],$ (A.135)

and the special value at ${\mathsf{b}}^{2}=2$

$\Phi_{\sqrt{2}}\left({x\over{\sqrt{2}}}\right)=\exp\left[\frac{{\rm
i}}{4\pi}\left({\rm Li}_{2}(-e^{2\pi x})+2\pi x\log(1+e^{2\pi
x})+2\pi\tan^{-1}({\rm e}^{\pi x})\right)\right],$ (A.136)

which follows from the more general results of garkas .

Evaluating the integral in (A.132) by residues, one finds that

$\log\Phi_{\mathsf{b}}(z)=\sum_{\ell=1}^{\infty}\frac{(-1)^{\ell}}{2{\rm
i}\ell}\left(\frac{e^{2\pi\ell
z{\mathsf{b}}}}{\sin(\ell\pi{\mathsf{b}}^{2})}+\frac{e^{2\pi\ell
z/{\mathsf{b}}}}{\sin(\ell\pi/{\mathsf{b}}^{2})}\right),$ (A.137)

which allows to establish the following relation between the two versions of
quantum dilogarithm (Faddeev:2000if, , §A)

$\Phi_{\mathsf{b}}(z)=\frac{E_{e^{-{\rm i}\pi/{\mathsf{b}}^{2}}}\left(-e^{2\pi
z/{\mathsf{b}}}\right)}{E_{e^{{\rm i}\pi{\mathsf{b}}^{2}}}\left(-e^{2\pi
z{\mathsf{b}}}\right)}=\frac{E_{{\tilde{y}}_{\mathsf{b}}}\left(e^{2\pi
z/{\mathsf{b}}}\right)}{E_{y_{\mathsf{b}}}\left(e^{2\pi
z{\mathsf{b}}}\right)}\,,$ (A.138)

where in the second representation we used the variables defined in (3.54).

Let us now introduce a generalization of the compact quantum dilogarithm,
first defined in (Dimofte:2009bv, , (2.12)), that depends on an additional
label $j\in{\mathbb{Z}}/2$

$E_{y}^{[j]}(x)=\exp\left[\sum_{k=1}^{\infty}\frac{(xy)^{k}\chi_{j}(y^{k})}{k(1-y^{2k})}\right]=\prod_{m=-j}^{j}(xy^{2m+1};y^{2})_{\infty}^{-1},$
(A.139)

where $\chi_{j}(y)$ is the character (1.7) of the $SU(2)$ representation of
spin $j$. It is easy to see that

$E_{y}^{[j]}(x)=\prod_{m=-j}^{j}E_{y}(xy^{2m}),$ (A.140)

where the product runs over half-integers $m$ such that $m-j$ is integer.
Following (A.138), we then define the corresponding generalization of the
Faddeev quantum dilogarithm

$\Phi_{{\mathsf{b}}}^{[j]}(z)=\frac{E_{{\tilde{y}}_{\mathsf{b}}}^{[j]}\left(e^{2\pi
z/{\mathsf{b}}}\right)}{E_{y_{\mathsf{b}}}^{[j]}\left(e^{2\pi
z{\mathsf{b}}}\right)}\,.$ (A.141)

From this definition and (A.139), it follows that this new function can be
written in one of the following forms

$\begin{split}\Phi_{{\mathsf{b}}}^{[j]}(z)=&\,\exp\left[\sum_{\ell=1}^{\infty}\frac{(-1)^{\ell}}{2{\rm
i}\ell}\left(\frac{\chi_{j}\left(-e^{\pi{\rm
i}\ell{\mathsf{b}}^{2}}\right)e^{2\pi\ell
z{\mathsf{b}}}}{\sin(\ell\pi{\mathsf{b}}^{2})}+\frac{\chi_{j}\left(-e^{-\pi{\rm
i}\ell/{\mathsf{b}}^{2}}\right)e^{2\pi\ell
z/{\mathsf{b}}}}{\sin(\ell\pi/{\mathsf{b}}^{2})}\right)\right]\\\
=&\,\exp\left(\int_{{\mathbb{R}}+{\rm
i}\epsilon}\frac{\chi_{j}(e^{({\mathsf{b}}-{\mathsf{b}}^{-1})v})e^{-2{\rm
i}zv}}{4\sinh(v{\mathsf{b}})\sinh(v/{\mathsf{b}})}\frac{{\rm d}v}{v}\right)\\\
=&\,\prod_{m=-j}^{j}\Phi_{\mathsf{b}}\left(z+{\rm
i}m({\mathsf{b}}-{\mathsf{b}}^{-1})\right)\end{split}$ (A.142)

in accordance with (A.140).

Finally, we incorporate one additional integer parameter $k$ corresponding to
the five-brane charge. To this end, we define

$E^{[j],k}_{y}(x)=\prod_{m=-j}^{j}E_{y}(xy^{2m/k})$ (A.143)

and

$\Phi^{[j],k}_{{\mathsf{b}}}(z):=\frac{E^{[j],k}_{{\tilde{y}}_{\mathsf{b}}}\left(e^{2\pi
z/{\mathsf{b}}}\right)}{E^{[j],k}_{y_{\mathsf{b}}}\left(e^{2\pi
z{\mathsf{b}}}\right)}=\prod_{m=-j}^{j}\Phi_{\mathsf{b}}\left(z+\frac{{\rm
i}m}{k}\,({\mathsf{b}}-{\mathsf{b}}^{-1})\right).$ (A.144)

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