#  $x-y$ duality in Topological Recursion for exponential variables via
Quantum Dilogarithm

Alexander Hock  Mathematical Institute, University of Oxford, Andrew Wiles
Building, Woodstock Road, OX2 6GG, Oxford, UK email
<EMAIL_ADDRESS>
###### Abstract.

For a given spectral curve, the theory of topological recursion generates two
different families $\omega_{g,n}$ and $\omega_{g,n}^{\vee}$ of multi-
differentials, which are for algebraic spectral curves related via the
universal $x-y$ duality formula. We propose a formalism to extend the validity
of the $x-y$ duality formula of topological recursion from algebraic curves to
spectral curves with exponential variables of the form $e^{x}=F(e^{y})$ or
$e^{x}=F(y)e^{ay}$ with $F$ rational and $a$ some complex number, which was in
principle already observed in [DBPSS19, BDBKS20]. From topological recursion
perspective the family $\omega_{g,n}^{\vee}$ would be trivial for these
curves. However, we propose changing the $n=1$ sector of $\omega_{g,n}^{\vee}$
via a version of the Faddeev’s quantum dilogarithm which will lead to the
correct two families $\omega_{g,n}$ and $\omega_{g,n}^{\vee}$ related by the
same $x-y$ duality formula as for algebraic curves. As a consequence, the
$x-y$ symplectic transformation formula extends further to important examples
governed by topological recursion including, for instance, the topological
vertex curve which computes Gromov-Witten invariants of $\mathbb{C}^{3}$,
equivalently triple Hodge integrals on the moduli space of complex curves,
orbifold Hurwitz numbers, or stationary Gromov-Witten invariants of
$\mathbb{P}^{1}$. The proposed formalism is related to the issue topological
recursion encounters for specific choices of framings for the topological
vertex curve.

##  1\. Introduction

This article mainly deals with the theory of topological recursion (TR)
[EO07a] which, roughly speaking, can be seen as an algorithm that associates
to the a complex curve (vanishing locus of a polynomial in two variables
$P(x,y)=0$) a family of multi-differentials $\omega_{g,n}$ indexed by
$g\in\mathbb{Z}_{\geq 0}$ and $n\in\mathbb{Z}_{>0}$. Depending on the curve
these multi-differential can carry information which are of interest in
enumerative geometry [BM08, EO07b, DLN16, BCEGF21], random matrix theory
[CEO06, EO07a, BCGF21a, BDBKS21], topological string theory [BKMP09, Che09,
EO15], knot theory [GS12, BE15], free probability [BCGF+21b, Hoc23a], quantum
field theory [GHW19, BHW20, BGHW22, BH23], etc. Therefore, TR provides a huge
range of applications which goes back to a common algorithm starting from some
algebraic curve. The theory of TR was developed around 2007 but it is still a
current research topic by itself.

The connection of TR to all these different areas of mathematics and
mathematical physics is mostly of a structural nature. It is cumbersome to
apply the algorithm of TR to perform explicit computations due to its
algorithmic structure, which is recursive in the Euler characteristic
$-\chi=2g+n-2$. This property prevents direct computations using TR since
other techniques are more efficient. For instance, explicit formulas for
intersection numbers on the moduli space of complex curves
$\overline{\mathcal{M}}_{g,n}$ for small $n$ are well-known and can be derived
from Virasoro constraints or localization theory in algebraic geometry.
However, TR would require $2g+n-2$ computational steps.

Changing the roles of $x$ and $y$ in the polynomial, a second family of multi-
differentials can be generated by TR, denoted by $\omega_{g,n}^{\vee}$, which
clearly distinguishes it from the first family $\omega_{g,n}$. In almost all
cases where simple explicit formulas are known for $\omega_{g,n}$ for small
$n$, the corresponding dual family $\omega_{g,n}^{\vee}$ is actually trivial,
i.e. $\omega_{g,n}^{\vee}=0$ for $2g+n-2>0$. Therefore, being able to express
one family explicitly depends somehow on the other family. Very recently,
completely new insights concerning the relation between $\omega_{g,n}$ and
$\omega_{g,n}^{\vee}$ for any curve for the form $P(x,y)=0$ were understood;
see Fig. 1. Recent works in the context of this duality include [BDBKS20,
BDBKS21, BDBKS22b, ABDB+22, ABDB+23, BCGF+21b, Hoc22, Hoc23a, Hoc23c].
Understanding this relation was already considered, for instance, in [EO13].
The duality between $\omega_{g,n}$ and $\omega_{g,n}^{\vee}$ has resolved, in
an extended way, an open problem in the theory of free probability [BCGF+21b],
providing the functional relation between the generating series of higher-
order free cumulants and moments.

$P(x,y)=0$
---
$\omega_{g,n}$
---
$\omega_{g,n}^{\vee}$
---
$x-y$ duality
---
Figure 1. The two families $\omega_{g,n}$ and $\omega_{g,n}^{\vee}$ which are
generated from a curve $P(x,y)=0$ via TR are related through the universal
$x-y$ duality formula

Applying this universal duality formula between $\omega_{g,n}$ and
$\omega_{g,n}^{\vee}$ to a curve, which generates a trivial family
$\omega_{g,n}^{\vee}$, produces an explicit formula for $\omega_{g,n}$. In
other words, for a trivial family $\omega_{g,n}^{\vee}$, we overcome the
algorithmic procedure of TR, which is recursive in the Euler characteristic.
Consequently, this explains the existence of such explicit formulas, which
were generally derived in the past using problem-specific techniques.

It is known and observed, for instance, in [Hoc23c], for the Lambert curve
that the $x-y$ duality formula does not hold in general for curves of the form
$P(e^{x},e^{y})=0$ which have exponential variables. However, very important
examples are exactly of this form, such as Gromov-Witten invariants of toric
Calabi-Yau 3-folds [BKMP09] and conjectured applications in knot theory
[BE15].

This article makes progress in understanding the $x-y$ duality formula for
curves of the form $P(e^{x},e^{y})=e^{x}-F(e^{y})=0$ (or $e^{x}-F(y)e^{ay}=0$)
with exponential variables, where $F$ is rational and $a$ some complex number.
This family of spectral curves is already a very important subclass with
different applications, such as simple Hurwitz numbers (equivalently to linear
Hodge integrals) and framed Gromov-Witten invariants of $\mathbb{C}^{3}$
(equivalently to triple Hodge integrals on $\overline{\mathcal{M}}_{g,n}$).

For curves of this form, the theory of TR would predict a trivial family
$\omega_{g,n}^{\vee}=0$ for $2g+n-2$. However, we will redefine
$\omega_{g,1}^{\vee}:=\frac{B_{2g}(1/2)}{(2g)!}\partial^{2g}_{y}\omega_{0,1}^{\vee}$
through $\omega_{0,1}^{\vee}$ and keep $\omega_{g,n}^{\vee}=0$ for $n>1$ and
$2g+n-2>0$, see Fig. 2. The coefficient $B_{n}(x)$ is the Bernoulli polynomial
and appears exactly in this form in Faddeev’s quantum dilogarithm. This
proposed redefinition of $\omega_{g,n}^{\vee}$ makes the $x-y$ duality formula
work again, which is tested on several examples.

$e^{x}-F(e^{y})=0$
---
$\omega_{g,n}$
---
$\omega_{g,n}^{\vee}:=\delta_{n,1}\frac{B_{2g}(1/2)}{(2g)!}\partial^{2g}_{y}\omega_{0,1}^{\vee}$
---
$x-y$ duality
---
Figure 2. For a curve of the form $e^{x}-F(e^{y})=0$ (or $e^{x}-F(y)e^{ay}=0$)
the family $\omega_{g,n}^{\vee}$ has to be redefined, then the family
$\omega_{g,n}$ generated by TR is related to the redefined family
$\omega_{g,n}^{\vee}$ via the universal $x-y$ duality formula

The idea of using the form of $\omega_{g,1}^{\vee}$ comes from the quantum
spectral curve $\hat{P}(\hat{x},\hat{y})$ observed by Gukov and Sulkowski
[GS12], which will be explained along the way in this article.

The paper is organised in the following way:

In Sec. 2, we review facts and definitions about TR, starting in Sec. 2.1 with
TR itself. In Sec. 2, we will use a notation with a ”tilde”
($\tilde{\omega}_{g,n}$ and $\tilde{\omega}^{\vee}_{g,n}$) which are the
multi-differentials defined by TR directly to distinguish them from the later
redefined $\omega_{g,n}^{\vee}$ related to curves of the form
$e^{x}-F(e^{y})=0$ or $e^{x}-F(y)e^{ay}=0$. In Sec. 2.2, we review some
background on the quantum spectral curve $\hat{P}(\hat{x},\hat{y})$ and how to
construct the wave function from TR, which is annihilated by the quantum
spectral curve. We also discuss the interplay of the $x-y$ duality with the
wave function, which is heuristically the Fourier/Laplace transform of the
wave function. Then, we recall how the multi-differentials can be
reconstructed from the wave function through the kernel $K(x_{i},x_{j})$ in
Sec. 2.3. This reconstruction formula is known as the determinantal formula,
which will be later compared with a special case of the $x-y$ duality formula.
Then, we review the $x-y$ duality formula in general in Sec. 2.4, which gives
the duality between $\tilde{\omega}_{g,n}$ and $\tilde{\omega}^{\vee}_{g,n}$
for an algebraic curve $P(x,y)=0$. The special case where $y$ is unramified is
discussed separately in Sec. 2.5. We prove, in this case, a new version of the
formula which highly resembles the determinantal formula, represented via a
summation over permutations $\sigma\in S_{n}$ consisting just of $n$-cycles.

Sec. 3 considers the curve $e^{x}-F(e^{y})=0$ (or $e^{x}-F(y)e^{ay}=0$), where
we use the ordinary notation without ”tilde”. We start with a motivating
example in Sec. 3.1, which has, as a wave function, the Euler $\Gamma$
function. In Sec. 3.2, we explain and motivate the general construction of the
two families $\omega_{g,n}$ and $\omega_{g,n}^{\vee}$ for these specific
curves. The family $\omega_{g,n}^{\vee}$ will be defined by
$\omega_{g,n}^{\vee}=\delta_{n,1}\frac{B_{2g}(1/2)}{(2g)!}\partial^{2g}_{y}\omega_{0,1}^{\vee}$,
and the family $\omega_{g,n}$ through the $x-y$ duality formula. We propose
that for this definition, $\tilde{\omega}_{g,n}$ generated by TR and
$\omega_{g,n}$ generated by the $x-y$ formula coincide. In Sec. 3.3, we
discuss the example of Faddeev’s quantum dilogarithm as the wave function.
Based on the work of Garoufalidis and Kashaev, we discuss some general
constructions for performing Borel resummation for the wave function for
curves of the form $e^{x}-F(e^{y})=0$ in Sec. 3.4. An explicit formula is
given for the Borel transform.

In Sec. 4, we apply our proposed construction to curves with important
enumerative meaning. We start with the Lambert curve in Sec. 4.1, providing
(to our knowledge) new formulas for simple Hurwitz numbers or equivalently by
the ELSV formula for linear Hodge integrals. In Sec. 4.2, the framed
topological vertex curve is discussed, providing Gromov-Witten invariants of
$\mathbb{C}^{3}$ or equivalently triple Hodge integrals. The last example
discusses stationary Gromov-Witten invariants of $\mathbb{P}^{1}$ where
explicit formulas of Pandharipande are reproduced as examples. We also
discuss, for this specific curve, that taking limits for the $x-y$ formula is
much more convenient than taking limits for TR. This comes from the fact that
TR is much more sensitive, where, for instance, ramification points can
collide or run to infinity. Therefore, the $x-y$ duality formula seems to be a
much more universal duality than just the $x-y$ symplectic transformation in
TR.

### Acknowledgement

I would like to thank Nitin Chidambaram, Alessandro Giacchetto, Motohico
Mulase and Federico Zerbini for discussions. I am especially grateful to
Alexander Alexandrov, Boris Bychkov, Petr Dunin-Barkwoski, Maxim Kazarian and
Sergey Shadrin on comments on my unpublished draft and explaining that the
proposed construction was, in principle, already appearing in some their
earlier papers [DBPSS19, BDBKS20]. This work was supported through the Walter-
Benjamin fellowship111“Funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) – Project-ID 465029630.

##  2\. Background on Topological Recursion

This section will provide the necessary background on the theory of
topological recursion including the quantum spectral curve, determinantal
formula and $x-y$ symplectic duality. We will set the notation used throughout
the article focusing on $x-y$ duality in conjunction with the spectral curve
and the quantum spectral curve. The explicit result for the $x-y$ duality in
case of an algebraic spectral curve will be cited and explained in details to
be able to understand the extension to spectral curves of the for
$e^{x}=F(e^{y})$ (or $e^{x}=F(y)e^{ay}$, respectively) in Sec. 3. We will
define the corresponding objects in this section with ”tilde” (e.g.
$\tilde{\omega}_{g,n}$, $\tilde{\omega}_{g,n}^{\vee}$, $\tilde{W}_{g,n}$,
$\tilde{\Phi}_{g,n}$, etc) through the algorithmic definition of topological
recursion.

###  2.1. Topological Recursion

We understand by the term topological recursion an algorithm which associates
to a given initial data, the so-called spectral curve, a family of multi-
differentials. To be more precise, the spectral curve is a tuple
$(\Sigma,x,y,B)$, where $\Sigma$ is a not necessarily compact Riemann surface
with $x,y:\Sigma\to\mathbb{C}$ are complex functions such that $dx$ and $dy$
are meromorphic, i.e. $x,y$ can have logarithms. Both functions $x,y$ should
have at most simple algebraic ramification points on $\Sigma$ which are
distinct (higher order ramifications are excluded due to technical reasons,
see [BE13] for the definition with higher order ramification points). Then,
topological recursion associates to the spectral curve $(\Sigma,x,y,B)$ the
multi-differentials $\tilde{\omega}_{g,n}$ living on $\Sigma^{n}$ with
$\tilde{\omega}_{0,1}=y\,dx$ and $\tilde{\omega}_{0,2}=B$, where $B$ is
symmetric with double pole on the diagonal and no residue, bi-residue 1 and
normalised such that the $A$-periods vanish.

For negative Euler characteristic $\chi=-2g-n+2<0$, all $\tilde{\omega}_{g,n}$
are defined recursively via [EO07a]

$\displaystyle\tilde{\omega}_{g,n+1}(I,z):=\sum_{\beta_{i}}\operatorname{Res}\displaylimits_{q\to\beta_{i}}K_{i}(z,q)\bigg{(}\tilde{\omega}_{g-1,n+2}(I,q,\sigma_{i}(q))$
(2.1)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\sum_{\begin{subarray}{c}g_{1}+g_{2}=g\\\
I_{1}\uplus I_{2}=I\\\
(g_{i},I_{i})\neq(0,\emptyset)\end{subarray}}\tilde{\omega}_{g_{1},|I_{1}|+1}(I_{1},q)\tilde{\omega}_{g_{2},|I_{2}|+1}(I_{2},\sigma_{i}(q))\\!\bigg{)}.$

The following notation is used:

  * •

$I=\\{z_{1},\dots,z_{n}\\}$ is a collection of $n$ local coordinates $z_{j}$
on $\Sigma$, and $I_{1},I_{2}\subset I$ any subset of $I$

  * •

the ramification points $\beta_{i}$ of $x$ are defined by $dx(\beta_{i})=0$

  * •

the local Galois involution $\sigma_{i}\neq\mathrm{id}$ with
$x(q)=x(\sigma_{i}(q))$ is defined in the vicinity of $\beta_{i}$ with fixed
point $\beta_{i}$

  * •

the recursion kernel $K_{i}(z,q)$ is also locally defined in the vicinity of
$\beta_{i}$ by

$\displaystyle
K_{i}(z,q)=\frac{\frac{1}{2}\int^{q}_{\sigma_{i}(q)}B(z,\bullet)}{\tilde{\omega}_{0,1}(q)-\tilde{\omega}_{0,1}(\sigma_{i}(q))}.$

Properties which follow (more or less directly) from the definition are

  * •

All $\tilde{\omega}_{g,n}$ are symmetric

  * •

For $2g+n-2>0$, $\tilde{\omega}_{g,n}$ has just poles located at the
ramification points of $x$

  * •

$\tilde{\omega}_{g,n}$ is homogeneous of degree $2-2g-n$, i.e. changing
$y\to\lambda y$ or $x\to\lambda x$ by some scalar $\lambda$ transforms
$\tilde{\omega}_{g,n}\to\lambda^{2-2g-n}\tilde{\omega}_{g,n}$

  * •

For $2g+n-2>0$, all $\tilde{\omega}_{g,n}$ are invariant under symplectic
transformations of the symplectic form $dx\wedge dy$ which leaves the
ramification points $\beta_{i}$ invariant. These transformations are generated
by

    * $(x,y)\to\big{(}\frac{ax+b}{cx+d},\frac{(cx+d)^{2}}{ad-bc}y\big{)}$, where $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in PSL_{2}(\mathbb{C})$

    * $(x,y)\to\big{(}x,y+R(x)\big{)}$, where $R(x)$ is any rational function.

There are several further properties which will not play any role in this
article. We refer a curious reader for instance to [EO07a, Eyn16].

Now, we want to fix further notation and definitions for functions used
throughout the article. We define

$\displaystyle\tilde{W}_{g,n}(x(z_{1}),...,x(z_{n}))dx(z_{1})...dx(z_{n}):=\tilde{\omega}_{g,n}(z_{1},...,z_{n})$
(2.2)
$\displaystyle\tilde{W}_{n}(x(z_{1}),...,x(z_{n})):=\sum_{g=0}^{\infty}\hbar^{2g+n-2}\tilde{W}_{g,n}(x(z_{1}),...,x(z_{n}))$
(2.3)
$\displaystyle\tilde{\Phi}_{g,n}(x(z_{1}),...,x(z_{n})):=\int^{z_{1}}...\int^{z_{n}}\tilde{\omega}_{g,n}(z_{1},...,z_{n})$
(2.4)
$\displaystyle\tilde{\Phi}_{n}(x(z_{1}),...,x(z_{n})):=\sum_{g=0}^{\infty}\hbar^{2g+n-2}\tilde{\Phi}_{g,n}(x(z_{1}),...,x(z_{n})),$
(2.5)

where the $\hbar$-series is understood as a formal power series and the
integration constants for $\tilde{\Phi}$ will not play any role.

The invariance property of the $\tilde{\omega}_{g,n}$ under specific
symplectic transformations plays an important role in the theory of TR and
also in the theories where TR finds its applications. A third symplectic
transformation which transforms the $\tilde{\omega}_{g,n}$’s and does not
leave them invariant is the $x-y$ symplectic transformation

$\displaystyle(x,y)\mapsto(y,x).$ (2.6)

Strictly speaking, this is a symplectic transformation up to a sign $-1$ which
can be restored by the homogeneity property. To clearly distinguish between
$\tilde{\omega}_{g,n}$ generated by the spectral curve $(\Sigma,x,y,B)$ or by
the curve $(\Sigma,y,x,B)$, we will use the notation $\vee$ as superscript.
Therefore, we define the $\tilde{\omega}_{g,n}^{\vee}$ to be generated by the
spectral curve $(\Sigma,y,x,B)$ by the same TR algorithm (2.1) but $x$ and $y$
interchanged. For instance, we have $\tilde{\omega}_{0,1}^{\vee}=x\,dy$ and
$\tilde{\omega}_{0,2}^{\vee}=B=\tilde{\omega}_{0,2}$.

The families $\tilde{\omega}_{g,n}$ and $\tilde{\omega}^{\vee}_{g,n}$ of
multi-differentials are dual to each over. If the family
$\tilde{\omega}_{g,n}$ can be reconstructed from the family
$\tilde{\omega}^{\vee}_{g,n}$ through a specific formula, then the family
$\tilde{\omega}^{\vee}_{g,n}$ can also be reconstructed from the family
$\tilde{\omega}_{g,n}$ via the same formula but $x$ and $y$ interchanged.

Similar to the previous definitions, we will use throughout the article the
following functions

$\displaystyle\tilde{W}^{\vee}_{g,n}(y(z_{1}),...,y(z_{n}))dy(z_{1})...dy(z_{n}):=\tilde{\omega}^{\vee}_{g,n}(z_{1},...,z_{n})$
(2.7)
$\displaystyle\tilde{W}^{\vee}_{n}(y(z_{1}),...,y(z_{n})):=\sum_{g=0}^{\infty}\hbar^{2g+n-2}\tilde{W}^{\vee}_{g,n}(y(z_{1}),...,y(z_{n}))$
(2.8)
$\displaystyle\tilde{\Phi}^{\vee}_{g,n}(y(z_{1}),...,y(z_{n})):=\int^{z_{1}}...\int^{z_{n}}\tilde{\omega}^{\vee}_{g,n}(z_{1},...,z_{n})$
(2.9)
$\displaystyle\tilde{\Phi}^{\vee}_{n}(y(z_{1}),...,y(z_{n})):=\sum_{g=0}^{\infty}\hbar^{2g+n-2}\tilde{\Phi}^{\vee}_{g,n}(y(z_{1}),...,y(z_{n})).$
(2.10)

The relation between all $\tilde{W}_{g,n}$ and $\tilde{W}^{\vee}_{g,n}$ will
be presented in Sec. 2.4 for spectral curves with meromorphic
$x,y:\Sigma\to\mathbb{C}$, i.e. there exists an irreducible polynomial
$P(x(z),y(z))=0$ for $z\in\Sigma$. However, the relation for the first few
examples can be read off by the definitions. For $(g,n)=(0,1)$, we have

$\displaystyle\tilde{W}_{0,1}(x(z))=y(z),\qquad\tilde{W}^{\vee}_{0,1}(y(z))=x(z).$

It implies that $\tilde{W}_{0,1}$ is the formal inverse of
$\tilde{W}^{\vee}_{0,1}$, i.e.
$\tilde{W}_{0,1}(\tilde{W}^{\vee}_{0,1}(y(z)))=y(z)$ and
$\tilde{W}^{\vee}_{0,1}(\tilde{W}_{0,1}(x(z)))=x(z)$. For $(g,n)=(0,2)$, we
have

$\displaystyle\tilde{W}_{0,2}(x(z_{1}),x(z_{2}))=\frac{B(z_{1},z_{2})}{dx(z_{1})dx(z_{2})},\qquad\tilde{W}^{\vee}_{0,2}(y(z_{1}),y(z_{2}))=\frac{B(z_{1},z_{2})}{dy(z_{1})dy(z_{2})}$

from which one concludes

$\displaystyle\tilde{W}_{0,2}(x(z_{1}),x(z_{2}))=\tilde{W}^{\vee}_{0,2}(y(z_{1}),y(z_{2}))\frac{dy(z_{1})}{dx(z_{1})}\frac{dy(z_{2})}{dx(z_{2})}.$

Just as a side remark, the functional relation for $\tilde{W}_{0,2}$ is the
second order moment-cumulant functional relation in the theory of free
probability, see [CMSS07, BCGF+21b, Hoc23a] for more information about the
relation to the theory of free probability.

The relation between $\tilde{W}_{g,n}$ and $\tilde{W}^{\vee}_{g,n}$ can thus
be interpreted as a generalization of an inversion formula graded by the Euler
characteristic $2g+n-2$ of the two integers $g$ and $n$. In the case of $g=0$
and $n=1$, this simply corresponds to the classical inversion of a function
$\tilde{W}^{\vee}_{0,1}(y)=x(y)$ and $\tilde{W}_{0,1}(x)=y(x)$, respectively.

###  2.2. Quantum Spectral Curve

The quantum spectral curve is an important application of TR to ordinary
differential equations and its solutions, suggested by Gukov and Sulkowski
[GS12]. For an accessible review on quantum spectral curve see [Nor15]. The
WKB solution is reconstructed from TR as explained later, this is proved for
genus zero curves in [BE16]. For higher genus algebraic curves, new insight
was gained in [Iwa20] and extended to hyperelliptic curves in [MO22, EGF21]
and further generalised in [EGFMO21]. The perturbative WKB solution has to be
extended to a non-perturbative wave function. However, we will just review
some facts about the WKB solution via TR, and refer the reader to the
literature mentioned above for higher genus spectral curves.

A spectral curve $(\Sigma,x,y,B)$ with meromorphic $x,y:\Sigma\to\mathbb{C}$
can also be represented as the vanishing locus of a polynomial $P(x,y)=0$ as
mentioned before. Being more precise, we have a complex curve defined by

$\displaystyle\\{P(x(z),y(z))=0|\forall z\in\Sigma\\}\subset\mathbb{P}^{2}.$

Now, we are interested in an operator-valued quantisation of the polynomial in
a quantum mechanical sense. The polynomial $P$ is quantised with
$x\to\hat{x}=x$ and $y\to\hat{y}=\hbar\frac{d}{dx}$ to
$\hat{P}(\hat{x},\hat{y})$, together with the semi-classical limit
$\lim_{\hbar\to 0}\hat{P}(x,y)=P(x,y)$. The quantisation of $P$ to $\hat{P}$
is obviously not unique due to ambiguities by ordering the operators $\hat{x}$
and $\hat{y}$ and additional $\hbar$ terms which vanish in the semi-classical
limit. Nevertheless, having such an differential operator $\hat{P}$ one might
ask for the wave function $\tilde{\Psi}(x)$ which is annihilated by it, i.e.

$\displaystyle\hat{P}(\hat{x},\hat{y})\tilde{\Psi}(x)=0.$ (2.11)

The algorithm of TR provides a way to compute a wave function
$\tilde{\Psi}(x)$. Let $(\Sigma,x,y,B)$ be a given spectral curve of genus
zero represented as $P(x,y)=0$, then there exists a quantisation
$\hat{P}(\hat{x},\hat{y})$ which annihilates the (perturbative) wave function

$\displaystyle\tilde{\Psi}(x):=\exp\bigg{(}\sum_{g\geq 0,n\geq
1}\frac{\hbar^{2g+n-2}}{n!}\tilde{\Phi}_{g,n}(x,...,x)\bigg{)},$ (2.12)

where $\tilde{\Phi}_{g,n}$ is defined in (2.4), and the special case
$\tilde{\Phi}_{0,2}$ is regularised by
$\int\int\tilde{\omega}_{0,2}(z_{1},z_{2})-\frac{dx(z_{1})\,dx(z_{2})}{(x(z_{1})-x(z_{2}))^{2}}$.
Here, the integration constants of $\tilde{\Phi}$ actually play some role, but
we do not go into the details. The explicit form of the quantum spectral curve
can be construct from the structure of the singularities (see [BE16,
EGFMO21]).

In the special case, where the polynomial is not algebraic but takes the form
$e^{x}=F(e^{y})$ (or $e^{x}=F(y)e^{ay}$, respectively), where $F$ is rational,
a quantum spectral curve was conjectured in [GS12, eq. (3.20)] which
annihilates the wave function constructed by TR via (2.12) to be of the form

$\displaystyle\hat{P}(e^{\hat{x}},e^{\hat{y}})=e^{\hat{x}+\frac{\hbar}{2}}-F(e^{\hat{y}-\frac{\hbar}{2}}).$
(2.13)

However, it was already observed in [BS12, GS12] that there are some issues if
the spectral curve $e^{x}=F(e^{y})$ does not have ramification points in $x$,
which includes the framed topological vertex curve with framing $f=0$. These
observations and comments in the literature seem not to be resolved in the
meantime, but they are highly related to the construction in Sec. 3 in
conjunction with the $x-y$ duality.

The quantum spectral curve and the wave function can of course also be studied
from the $x-y$ duality perspective This means there is a second quantum
spectral curve of the form $\hat{P}^{\vee}(\hat{x},\hat{y})$ with operators
$\hat{x}=\hbar\frac{d}{dy}$ and $\hat{y}=y$ and the semi-classical limit
$\lim_{\hbar\to 0}\hat{P}^{\vee}(x,y)=P(x,y)$. The dual wave function
$\tilde{\Psi}^{\vee}(y)$ is construction again via TR but with interchanged
roles of $x$ and $y$, i.e.

$\displaystyle\tilde{\Psi}^{\vee}(y):=\exp\bigg{(}\sum_{g\geq 0,n\geq
1}\frac{\hbar^{2g+n-2}}{n!}\tilde{\Phi}^{\vee}_{g,n}(y,...,y)\bigg{)},$ (2.14)

where $\tilde{\Phi}^{\vee}_{g,n}$ is defined in (2.9), and the special case
$\tilde{\Phi}^{\vee}_{0,2}$ is regularised by
$\int\int\tilde{\omega}_{0,2}^{\vee}(z_{1},z_{2})-\frac{dy(z_{1})\,dy(z_{2})}{(y(z_{1})-y(z_{2}))^{2}}$.

So what is now the relation between the two wave function $\tilde{\Psi}(x)$
and $\tilde{\Psi}^{\vee}(y)$? The naive suggestion would be of course that
both wave function are related via Fourier/Laplace transformation, which from
an other perspective implies that the $x-y$ symplectic transformation in TR is
deeply related to a Fourier/Laplace transformation.

Let us give two examples for the quantum spectral curve, the $x-y$ duality and
its interrelation, where the first one is almost too simple:

######  Example 2.1.

Take the trivial spectral curve
$(\mathbb{C},x=z,y=z,\frac{dz_{1}\,dz_{2}}{(z_{1}-z_{2})^{2}})$ which is
represented by the polynomial $P(x,y)=x-y=0$. For this example, $x$ and $y$
have no ramification points which implies that all $\tilde{\omega}_{g,n}$ and
$\tilde{\omega}^{\vee}_{g,n}$ vanish for $2g+n-2>0$. However,
$\tilde{\omega}_{0,1}$ and $\tilde{\omega}^{\vee}_{0,1}$ are not trivial. Just
applying the previous definitions, we find

$\displaystyle\tilde{\Phi}_{0,1}(x)=\frac{x^{2}}{2}\quad\rightarrow\quad\tilde{\Psi}(x)=\exp\left(\frac{x^{2}}{2\hbar}\right)$
$\displaystyle\tilde{\Phi}^{\vee}_{0,1}(y)=\frac{y^{2}}{2}\quad\rightarrow\quad\tilde{\Psi}^{\vee}(y)=\exp\left(\frac{y^{2}}{2\hbar}\right).$

Both wave functions are the Gaussian function, where it is very well known
that it Fourier transform is again a Gaussian function. The wave functions are
annihilated by the quantum spectral curves
$\hat{P}(\hat{x},\hat{y})=x-\hbar\frac{d}{dx}$ and
$\hat{P}^{\vee}(\hat{x}^{\vee},\hat{y}^{\vee})=\hbar\frac{d}{dy}-y$,
respectively.

######  Example 2.2.

Take the simplest nontrivial example, the Airy spectral curve
$(\mathbb{C},x=z^{2},y=z,\frac{dz_{1}\,dz_{2}}{(z_{1}-z_{2})^{2}})$ which is
represented by the polynomial $P(x,y)=x-y^{2}=0$. In this example, $y$ has no
ramification points implying the all $\tilde{\omega}^{\vee}_{g,n}=0$ with
$2g+n-2>0$. From the previous definitions, we find

$\displaystyle\tilde{\Phi}^{\vee}_{0,1}(y)=\frac{y^{3}}{3}\quad\rightarrow\quad\tilde{\Psi}^{\vee}(y)=\exp\left(\frac{y^{3}}{3\hbar}\right)$

satisfying the differential equation

$\displaystyle\left(\hbar\frac{d}{dy}-y^{2}\right)\tilde{\Psi}^{\vee}(y)=0.$

On the other hand, all $\tilde{\omega}_{g,n}$ do not vanish since $x$ has a
ramification point at $z=0$. It is very well-known that $\tilde{\omega}_{g,n}$
computes the $\psi$-class intersection numbers on
$\overline{\mathcal{M}}_{g,n}$ the moduli space of complex curves with $n$
marked points [EO07a], which was essentially observed by Kontsevich [Kon92] in
proving Witten’s conjecture [Wit91]. Furthermore, the wave function generated
by these $\tilde{\omega}_{g,n}$’s gives actually the asymptotic expansion of
the Airy function (or Bairy function respectively) [BE16, EGFG+23]

$\displaystyle\tilde{\Psi}_{\pm}(x)=\frac{e^{\mp\frac{2}{3\hbar}x^{3/2}}}{\sqrt{2\pi}x^{1/4}}\sum_{k=0}^{\infty}\frac{(6k)!}{1296^{k}(2k)!(3k)!}\left(\mp\frac{\hbar}{x^{3/2}}\right)^{k},$

where $\tilde{\Psi}_{+}(x)$ corresponds to the Airy function and
$\tilde{\Psi}_{-}(x)$ to the Bairy function. The wave functions satisfy

$\displaystyle\left(\left(\hbar\frac{d}{dx}\right)^{2}-x\right)\tilde{\Psi}_{\pm}(x)=0.$

The Airy (or Bairy function, respectively) can be obtained by a
Fourier/Laplace transform of the dual wave function $\tilde{\Psi}^{\vee}(y)$,
where the contour depends on the complex argument of $\hbar$

$\displaystyle\tilde{\Psi}_{\pm}(x)=\frac{1}{\sqrt{2\pi\hbar}}\int_{C_{\pm}}e^{xy/\hbar}\tilde{\Psi}^{\vee}(y)dy.$

###  2.3. Reconstruction of $\tilde{\omega}_{g,n}$ from the wave function

The reconstruction of $\tilde{\omega}_{g,n}$ or equivalently $\tilde{W}_{g,n}$
is given by the determinantal formula [BE09] through the kernel
$K(x_{1},x_{2})$, which by itself is constructed from all wave functions (this
means from all linearly independent solutions of the differential equation
(2.11))

$\displaystyle
K(x_{1},x_{2})\,=\,\frac{\exp\left(\sum_{n=1}^{\infty}\frac{\int_{x_{2}}^{x_{1}}...\int_{x_{2}}^{x_{1}}\tilde{W}_{n}(x_{1}^{\prime},...,x_{n}^{\prime})dx_{1}^{\prime}...dx_{n}^{\prime}}{n!}\right)}{x_{1}-x_{2}}$
(2.15)

where one has to be very careful since the pullback to the $x$-space form the
$z$-space chooses a specific branch, or equivalently a specific solution of
the of the quantum spectral curve. The determinantal formula is therefore an
other way of representing the formal power series
$\tilde{W}_{n}=\sum_{g}\hbar^{2g+n-2}\tilde{W}_{g,n}$ by an explicit formula
if the kernel is known. The determinantal formula was recently used in
combination with resurgence to derive the large genus asympotitcs for
intersection numbers on $\overline{\mathcal{M}}_{g,n}$ at subleading
order[EGFG+23].

######  Example 2.3.

The Airy kernel $K_{Airy}$ is constructed by the Airy function
$\tilde{\Psi}_{+}(x)$ and the B-Airy function $\tilde{\Psi}_{-}(x)$ of Example
2.2 which are the two independent solutions of the differential equation
$\hbar^{2}\tilde{\Psi}^{\prime\prime}(x)-x\tilde{\Psi}=0$:

$\displaystyle
K_{Airy}(x_{1},x_{2})=\frac{1}{\hbar}\frac{\tilde{\Psi}_{+}(x_{1})\tilde{\Psi}_{-}^{\prime}(x_{2})-\tilde{\Psi}^{\prime}_{+}(x_{1})\tilde{\Psi}_{-}(x_{2})}{x_{1}-x_{2}}.$

An other way is to construct the kernel from its differential system (see
[BE09] for details), but this is beyond the scope of the article.

We want to emphasise the structure of the determinantal formula, and compare
the different structures of the determinantal formula and the $x-y$ duality
formula (2.18). Let $K(x_{1},x_{2})$ be the kernel (2.15) (also used in
[ABDB+23, eq. (3.8)]), then it was conjectured in [BE09] that the correlators

$\displaystyle\tilde{W}_{n}(x_{1},...,x_{n})=$
$\displaystyle(-1)^{n-1}\sum_{\begin{subarray}{c}\sigma\in S_{n}\\\
\sigma=n\text{-cycle}\end{subarray}}K(x_{i},x_{\sigma_{(i)}})-\frac{\delta_{n,2}}{(x_{1}-x_{2})^{2}}$
(2.16) $\displaystyle\tilde{W}_{1}(x_{1})=$ $\displaystyle\lim_{x^{\prime}\to
x}\bigg{(}K(x,x^{\prime})-\frac{1}{x-x^{\prime}}\bigg{)}$ (2.17)

are equal to $\tilde{W}_{n}$ generated by TR (2.1). The sum over all
permutations $\sigma\in S_{n}$ is restricted to permutations with one cycle of
length $n$.

###  2.4. $x-y$ Duality for Algebraic Curves

Now, we want to recall the relation between $\tilde{\omega}_{g,n}$ generated
by the spectral curve $(\Sigma,x,y,B)$ and $\tilde{\omega}^{\vee}_{g,n}$
generated by the spectral curve $(\Sigma,y,x,B)$ with interchanged $x$ and
$y$. We assume $x,y$ to be meromorphic and have simple distinct ramification
points. It turns out that it is much more convenient to represent this
functional relation in terms of $\tilde{W}_{n}$ and $\tilde{W}^{\vee}_{n}$
defined in (2.3) and (2.8).

We use in this article the $x-y$ formula as it appeared in [Hoc23a, ABDB+22]
or in [Hoc22] for the special case of genus $g=0$. For this, we will need
first to define the the following graphs:

######  Definition 2.4.

Let $\mathcal{G}_{n}$ be the set of connected bicoloured graphs $\Gamma$ with
$\bigcirc$-vertices and $\bullet$-vertices, where the number of
$\bigcirc$-vertices is $n$. A graph $\Gamma$ satisfies the following
conditions:

  * -

the $\bigcirc$-vertices are labelled from $1,...,n$

  * -

edges are only connecting $\bullet$-vertices with $\bigcirc$-vertices

  * -

$\bullet$-vertices have valence $\geq 2$.

For a graph $\Gamma\in\mathcal{G}_{n}$, let $r_{i}(\Gamma)$ be the valence of
the $i^{\text{th}}$ $\bigcirc$-vertex.

Let $I\subset\\{1,...,n\\}$ be the set associated to a $\bullet$-vertex, where
$I$ is the set of labels of $\bigcirc$-vertices connected to this
$\bullet$-vertex. Let $\mathcal{I}(\Gamma)$ be the set of all sets $I$ for a
given graph $\Gamma\in\mathcal{G}_{n}$.

We will abuse the notation $x_{i}=x_{i}(z_{i})=x_{i}(y_{i})=x(z_{i})$ and
$y_{i}=y_{i}(z_{i})=y_{i}(x_{i})=y(z_{i})$ for convenience. Then, the
functional relation between the two families $\tilde{W}_{n}$ and
$\tilde{W}^{\vee}_{n}$ is the following sum over graphs (see[Hoc23a, ABDB+22]
for more details and examples):

######  Theorem 2.5.

Let $(\Sigma,x,y,B)$ be an algebraic spectral curve, and let
$\tilde{W}^{\vee}_{n}(y_{1},...,y_{n}):=\sum_{g=0}^{\infty}\hbar^{2g+n-2}\tilde{W}^{\vee}_{g,n}(y_{1},...,y_{n})$
be generated by TR with the spectral curve $(\Sigma,y,x,B)$. Let further be
$S(u)=\frac{e^{u/2}-e^{-u/2}}{u}$ and for $I=\\{i_{1},...,i_{n}\\}$

$\displaystyle\hat{c}^{\vee}(u_{I},y_{I}):=$ $\displaystyle\bigg{(}\prod_{i\in
I}\hbar u_{i}S(\hbar
u_{i}\partial_{y_{i}})\bigg{)}\big{(}\tilde{W}_{n}^{\vee}(y_{I})\bigg{)},$

and for $I=\\{j,j\\}$ the special case

$\displaystyle\hat{c}^{\vee}(u_{I},y_{I}):=(\hbar u_{j}S(\hbar
u_{j}\partial_{y_{j}}))(\hbar u_{j}S(\hbar
u_{j}\partial_{y}))\bigg{(}\tilde{W}_{2}^{\vee}(y_{j},y)-\frac{1}{(y_{j}-y)^{2}}\bigg{)}\bigg{|}_{y=y_{j}}.$

Define the following differential operator acting from the left

$\displaystyle\hat{O}^{\vee}(y_{i}(x_{i})):=$ $\displaystyle\sum_{m\geq
0}\big{(}-\partial_{x_{i}}\big{)}^{m}\bigg{(}-\frac{dy_{i}}{dx_{i}}\bigg{)}[u_{i}^{m}]\frac{\exp\bigg{(}\hbar
u_{i}S(\hbar
u_{i}\partial_{y_{i}(x_{i})})\tilde{W}_{1}^{\vee}(y_{i}(x_{i}))-x_{i}u_{i}\bigg{)}}{\hbar
u_{i}}.$

Then, all $\tilde{W}_{g,n}$ generated by TR by the spectral curve
$(\Sigma,x,y,B)$ are related to the $\tilde{W}^{\vee}_{g^{\prime},n^{\prime}}$
via formal power series in $\hbar$ of

$\displaystyle\tilde{W}_{n}(x_{1},...,x_{n})=\sum_{\Gamma\in\mathcal{G}_{n}}\frac{1}{|\mathrm{Aut}(\Gamma)|}\prod_{i=1}^{n}\hat{O}^{\vee}(x_{i})\prod_{I\in\mathcal{I}(\Gamma)}\hat{c}^{\vee}(u_{I},y_{I}),$
(2.18)

where the graphs $\mathcal{G}_{n}$ are defined in Definition 2.4.

This theorem is proved in [ABDB+22] and provides a solution for an open
problem in the theory of TR. Actually, the theorem gives a duality between two
families of correlators $\tilde{W}_{g,n}$ and $\tilde{W}^{\vee}_{g,n}$, since
the same formula holds by interchanging $x$ and $y$ as well as
$\tilde{W}_{g,n}$ with $\tilde{W}^{\vee}_{g,n}$. The functional relation of
the theorem has solved simultaneously an open problem in the theory of free
probability [CMSS07] related higher order free cumulants and moments
[BCGF+21b]. In the same vein, it relates from a combinatorial perspective
ordinary maps and fully simple maps [BCGF21a, BDBKS21].

Remember that the family $\tilde{W}_{g,n}$ gives rise to the wave function
$\tilde{\Psi}(x)$ and the dual family $\tilde{W}^{\vee}_{g,n}$ to the wave
function $\tilde{\Psi}^{\vee}(y)$ via (2.12) and (2.14), respectively. Since
both wave function $\tilde{\Psi}(x)$ and $\tilde{\Psi}^{\vee}(y)$ are related
via Fourier/Laplace transformation, we can conclude that the functional
relation of Theorem 2.5 is the corresponding duality formula for each $g$ and
$n$. Therefore, the $x-y$ duality formula (2.18) is, casually speaking, a way
to generated asymptotically a Fourier/Laplace transformation which is graded
by the Euler characteristic. At each grading the functions
$\tilde{W}_{g,n}^{\vee}$ are functions in several variables. The Fig. 3 shows
the interplay between the different dualities, the Fourier/Laplace
transformation between the wave function $\tilde{\Psi}$ and
$\tilde{\Psi}^{\vee}$ as well as the $x-y$ duality between $\tilde{W}_{g,n}$
and $\tilde{W}^{\vee}_{g,n}$. The dashed lines show the reconstructions
mentioned before. The remaining duality between the two kernels $K$ and
$K^{\vee}$ was recently considered in [ABDB+23] for KP integrable systems.

$\overline{\Psi}(x)$
---
$\overline{\Psi}^{\vee}(y)$
---
$\overline{W}_{g,n}$
---
$\overline{W}^{\vee}_{g,n}$
---
$K(x_{1},x_{2})$
---
$K^{\vee}(y_{1},y_{2})$
---
$x-y$ duality
---
Fourier/Laplace
---
transformation
Figure 3. Duality Interplay

###  2.5. $x-y$ Duality for Algebraic Curves with unramified $y$

The formula becomes particularly nice and applicable if one of the families,
for instance $\tilde{W}^{\vee}_{g,n}$, is trivial. An important subclass of
genus zero spectral curves are curves of the form $x=F(y)$, where $F$ is
rational which implies that $y$ is unramified. Therefore, the family
$\tilde{W}^{\vee}_{g,n}$ vanishes identically. In this case Theorem 2.5 breaks
down to:

######  Corollary 2.6.

Consider the situation where $y$ is unramified, i.e. $dy(z)\neq 0$ for all
$z\in\Sigma$. In particular, we can set $y(z)=z$ or $y(z)=\frac{1}{z}$. Then,
all $\tilde{W}_{g,n}^{\vee}=0$ for $2g+n-2>0$, and therefore all
$\hat{c}^{\vee}(u_{I},y_{I})=0$ for $|I|>2$. In this specific case, the $x-y$
formula reduces to (see [ABDB+22, Hoc23c] for further details)

$\displaystyle\tilde{W}_{n}(x_{1}(z_{1}),...,x_{1}(z_{n}))=\prod_{i=1}^{n}\hat{O}^{\vee}(y_{i}(z_{i}))\sum_{\Gamma\in\mathcal{G}_{n}^{2}}\prod_{\\{i,j\\}\in\mathcal{I}(\Gamma)}\frac{\hbar^{2}u_{i}u_{j}}{(y_{i}-y_{j})^{2}-\frac{\hbar^{2}}{4}(u_{i}+u_{j})^{2}},$
(2.19)

where $\mathcal{G}_{n}^{2}\subset\mathcal{G}_{n}$ is the set of graphs defined
in Definition 2.4 with only 2-valent $\bullet$-vertices just connecting two
different $\bigcirc$-vertices and at most one $\bullet$-vertex connects the
same pair of $\bigcirc$-vertices, or equivalently (and much simpler)
$\mathcal{G}_{n}^{2}$ is the set of connected labeled graphs with $n$ vertices
(A001187).

With the formula (2.19) of Corollary 2.6, one can compute all
$\tilde{W}_{g,n}$ for any $g$ directly. It is not necessary any more to follow
the algorithmic procedure of TR recursively in the Euler characteristic.
Examples which are included in the class of spectral curves considered in
Corollary 2.6 are the enumeration of $\psi$-class intersection numbers [Wit91,
Kon92], $r$-spin intersection numbers [BH07, BCEGF21], $\Theta$-class
intersection numbers and $r$-spin $\Theta$-class intersection numbers on
$\tilde{\mathcal{M}}_{g,n}$ [Nor23, CGFG22] or the enumeration of genus
permutations [Hoc23b].

Interestingly, the $x-y$ formula of Corollary 2.6 can be brought into a
formula of exactly the same shape as the determinantal formula (2.16) for
algebraic curves with unramified $y$. The sum over all connected labeled
graphs in (2.19) can indeed by turned into a sum over permutations $\sigma\in
S_{n}$ where $\sigma$ is an $n$-cycle. As a consequence we get an other
explicit formula with significantly less terms. The number of terms is reduced
since the number of $n$-cycle permutations in $S_{n}$ is $(n-1)!$, whereas the
number of connected graphs with $n$ labeled vertices grow much faster.
Actually, the asymptotic growth of connected graphs with $n$ labeled vertices
is $2^{\binom{n}{2}}=2^{\frac{n(n-1)}{2}}$ since almost all graphs with $n$
labeled vertices are connected (see [FS09, p. 138]).

######  Proposition 2.7.

Consider the situation where $y$ is unramified, i.e. $dy(z)\neq 0$ for all
$z\in\Sigma$. In particular, we can set $y(z)=z$ or $y(z)=\frac{1}{z}$. Then,
all $\tilde{W}_{g,n}$ generated by TR from this spectral curve are

$\displaystyle\tilde{W}_{g,n}(x_{1}(z_{2}),...,x_{n}(z_{n}))$ (2.20)
$\displaystyle=$
$\displaystyle[\hbar^{2g+n-2}]\prod_{i=1}^{n}\hat{O}^{\vee}(y_{i}(z_{i}))\hbar
u_{i}\sum_{\begin{subarray}{c}\sigma\in S_{n}\\\
\sigma=n\text{-cycle}\end{subarray}}\prod_{i=1}^{n}\frac{1}{y_{i}(z_{i})+\frac{\hbar
u_{i}}{2}-y_{\sigma(i)}(z_{\sigma(i)})+\frac{\hbar u_{\sigma(i)}}{2}}$

and in the special case $n=1$, the sum over all $n$-cycles is defined to be
$\frac{1}{\hbar u_{1}}$.

###### Proof.

The aim is to transform the rhs of (2.19) into the rhs of (2.20). The first
step is to rewrite

$\displaystyle\frac{\hbar^{2}u_{i}u_{j}}{(y_{i}-y_{j})^{2}-\frac{\hbar^{2}}{4}(u_{i}+u_{j})^{2}}=\frac{(y_{i}-y_{j})^{2}-\frac{\hbar^{2}}{4}(u_{i}-u_{j})^{2}}{(y_{i}-y_{j})^{2}-\frac{\hbar^{2}}{4}(u_{i}+u_{j})^{2}}-1,$

where the subtraction of 1 corresponds to just taking connected correlators
which was in principle already used to derive Corollary 2.6 (see [ABDB+22,
Hoc23c]).

Then, we go from connected correlators $\tilde{W}_{g,n}(x_{1},...,x_{n})$ to
disconnected $\overset{\circ}{\tilde{W}}_{g,n}(x_{1},...,x_{n})$ defined as
sum over all partitions

$\displaystyle\overset{\circ}{\tilde{W}}_{n}(x_{I}):=\sum_{\lambda\vdash
I}\prod_{i=1}^{l(\lambda)}\tilde{W}_{|\lambda_{i}|}(x_{\lambda_{i}}).$ (2.21)

We achieve an equation for the disconnected
$\overset{\circ}{\tilde{W}}_{g,n}$, where the sum over all connected graphs
turns into the product

$\displaystyle\overset{\circ}{\tilde{W}}_{n}(x_{1}(z_{1}),...,x_{1}(z_{n}))=\prod_{i=1}^{n}\hat{O}^{\vee}(y_{i}(z_{i}))\prod_{1\leq
i<j\leq
1}\frac{(y_{i}-y_{j})^{2}-\frac{\hbar^{2}}{4}(u_{i}-u_{j})^{2}}{(y_{i}-y_{j})^{2}-\frac{\hbar^{2}}{4}(u_{i}+u_{j})^{2}}.$
(2.22)

Combining the factor $\frac{1}{\hbar u_{i}}$ from
$\hat{O}^{\vee}(y_{i}(z_{i}))$ and the product $\prod_{1\leq i<j\leq n}$, we
can write it in the form of the determinant of the Cauchy matrix

$\displaystyle\prod_{i=1}^{n}\frac{1}{a_{i}+b_{i}}\prod_{i<j}\frac{(a_{i}-b_{j})(a_{j}-b_{i})}{(a_{i}+b_{j})(a_{j}-b_{i})}=\mathrm{det}\bigg{(}\frac{1}{a_{i}+b_{j}}\bigg{)}=\sum_{\sigma\in
S_{n}}\mathrm{sign}(\sigma)\prod_{i}\frac{1}{a_{i}+b_{\sigma(i)}},$ (2.23)

with $a_{i}=-y_{i}+\frac{\hbar u_{i}}{2}$ and $b_{j}=y_{j}+\frac{\hbar
u_{j}}{2}$. Restricting to the connected part, which is nothing than the
Möbius function for the partially ordered set of partitions of
$\\{1,...,n\\}$, all terms coming from permutations $\sigma\in S_{n}$ with
more than one cycle vanish. The remaining $n$-cycle permutations $\sigma\in
S_{n}$ have parity $n-1$ which gives an overall factor of
$\mathrm{sign}(\sigma)=(-1)^{n-1}$. This proves the the assertion. ∎

Applying this formula to some known interesting example like the Airy curve,
$r$-spin Airy curve and Lambert curve will be provided:

######  Example 2.8.

For the Airy curve
$(\mathbb{C},x=z^{2},y=z,\frac{dz_{1}\,dz_{2}}{(z_{1}-z_{2})^{2}})$,
Proposition 2.7 yields the following representation for $\tilde{W}_{g,n}$
which are the same as computed by TR

$\displaystyle\tilde{W}_{g,n}(x_{1}(z_{1}),...,x_{n}(z_{n}))$ (2.24)
$\displaystyle=$
$\displaystyle[\hbar^{2g+n-2}]\prod_{i=1}^{n}\left(\sum_{m_{i}}\bigg{(}-\frac{\partial}{\partial(z_{i})^{2}}\bigg{)}^{m_{i}}[u_{i}^{m_{i}}]\frac{e^{\frac{\hbar^{2}u_{i}^{3}}{12}}}{2z_{i}}\right)\sum_{\begin{subarray}{c}\sigma\in
S_{n}\\\
\sigma=n\text{-cycle}\end{subarray}}\prod_{i=1}^{n}\frac{1}{z_{i}+\frac{\hbar
u_{i}}{2}-z_{\sigma(i)}+\frac{\hbar u_{\sigma(i)}}{2}}.$

The $\psi$-class intersection numbers are extracted from $\tilde{W}_{g,n}$ via

$\displaystyle\tilde{W}_{g,n}(x_{1}(z_{1}),...,x_{n}(z_{n}))=\sum_{k_{1},...,k_{n}=3g+n-3}\langle\psi_{1}^{k_{1}}...\psi_{n}^{k_{n}}\rangle_{g,n}\prod_{i=1}^{n}\frac{(2k_{i}+1)!!}{2z_{i}^{2k_{i}+3}}.$
(2.25)

An explicit formula for the intersection numbers can be extracted by taking
the Laplace transformation of (2.25) with
$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dx(z_{i})\frac{e^{-\mu_{i}z_{i}^{2}}}{z_{i}^{2k_{i}+3}}=\frac{\mu_{i}^{k_{i}+1/2}}{(2k_{i}+1)!!}$
for the rhs, which can further be represented as a geometric series. For the
lhs, we insert (2.24) and perform integration by parts $m_{i}$ times (see a
detailed discussion for this in [Hoc23c]) to finally get

$\displaystyle\bigg{\langle}\prod_{i=1}^{n}\frac{\sqrt{\mu_{i}}}{1-\mu_{i}\psi_{i}}\bigg{\rangle}_{g,n}$
$\displaystyle=$
$\displaystyle[\hbar^{2g+n-2}]\prod_{i=1}^{n}\left(\frac{e^{\frac{\hbar^{2}\mu_{i}^{3}}{12}}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dz_{i}e^{-\mu_{i}z^{2}}\right)\sum_{\begin{subarray}{c}\sigma\in
S_{n}\\\
\sigma=n\text{-cycle}\end{subarray}}\prod_{i=1}^{n}\frac{1}{z_{i}+\frac{\hbar\mu_{i}}{2}-z_{\sigma(i)}+\frac{\hbar\mu_{\sigma(i)}}{2}},$

where the contours are on the real line with a small semi circle above the
origin. This formula coincides with the formula which can be found in [BG16,
AHIS21].

######  Example 2.9.

The $r$-spin Ariy spectral curve is a generalisation of the form
$(\mathbb{C},x=z^{r},y=z,\frac{dz_{1}\,dz_{2}}{(z_{1}-z_{2})^{2}})$. This
spectral curve has for positive integer $r$ greater than 2 higher order
ramification. The corresponding version of TR which keeps track of higher
order ramification was formulated in [BE13]. A new cohomological class
$c_{W}(a_{1},...,a_{n})\in
H^{\bullet}(\overline{\mathcal{M}}_{g,n},\mathbb{Q})$ of degree
$s=\frac{(r-2)(g-1)+\sum_{i}(a_{i}-1)}{r}$ (which has to be a positive integer
$s\in\mathbb{N}$) was defined by Witten in [Wit93] for the moduli space
complex curves associated with an $r$-spin structure. A conjecture of Witten
concerning the intersection number
$\langle\tau_{k_{1},a_{1}}...\tau_{k_{n},a_{n}}\rangle_{g,n}=\langle
c_{W}(a_{1},...,a_{n})\psi_{1}^{k_{1}}...\psi_{n}^{k_{n}}\rangle_{g,n}$ of
this class $c_{W}(a_{1},...,a_{n})$ with $\psi$-classes relates to the $r$-KdV
hierarchy, which was proved [FSZ10]. We use the notation of [EGFG+23], where
the relation of these intersection numbers to TR and the determinantal formula
was discussed. The correlators $\tilde{W}_{g,n}$ computed by higher order TR
are related to the intersection numbers via

$\displaystyle\tilde{W}_{g,n}(x_{1}(z_{1}),...,x_{n}(z_{n}))$ (2.26)
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}0\leq
k_{1},...,k_{n}\\\ 1\leq a_{1},...,a_{n}\leq
r-1\end{subarray}}\prod_{i=1}^{n}(-r)^{g-1-|k|}\langle\tau_{k_{1},a_{1}}...\tau_{k_{n},a_{n}}\rangle_{g,n}\prod_{i=1}^{n}\frac{(rk_{i}+a_{i})!_{(r)}}{rz_{i}^{r(k_{i}+1)+a_{i}}},$
(2.27)

where $m!_{(r)}$ is the $r$-factorial defined recursively for $m>r$ by
$m!_{(r)}=m(m-r)!_{(r)}$ and $m!_{(r)}=m$ for $0<m\leq r$.

On the other hand, Proposition 2.7 yields the following representation for
$\tilde{W}_{g,n}$ (note that applying the higher order generalisation of TR
from [BE13] is quite tedious to use)

$\displaystyle\tilde{W}_{g,n}(x_{1}(z_{1}),...,x_{n}(z_{n}))$ (2.28)
$\displaystyle=$
$\displaystyle[\hbar^{2g+n-2}]\prod_{i=1}^{n}\left(\sum_{m_{i}}\bigg{(}-\frac{\partial}{\partial(z_{i})^{r}}\bigg{)}^{m_{i}}[u_{i}^{m_{i}}]\frac{e^{\frac{(z_{i}+\frac{\hbar
u_{i}}{2})^{r+1}-(z_{i}-\frac{\hbar
u_{i}}{2})^{r+1}}{\hbar(r+1)}-z_{i}^{r}u_{i}}}{rz_{i}^{r-1}}\right)\sum_{\begin{subarray}{c}\sigma\in
S_{n}\\\
\sigma=n\text{-cycle}\end{subarray}}\prod_{i=1}^{n}\frac{1}{z_{i}+\frac{\hbar
u_{i}}{2}-z_{\sigma(i)}+\frac{\hbar u_{\sigma(i)}}{2}}.$

Again, taking the Laplace transformation of (2.26) with
$\frac{1}{\Gamma(1-\frac{a_{i}}{r})}\int_{0}^{\infty}dx_{i}\frac{e^{-\mu_{i}x_{i}}}{x_{i}^{k_{i}+a_{i}/r+1}}=(-1)^{k_{i}}\frac{r^{k_{i}}\mu_{i}^{k_{i}+a_{i}/r}}{(rk_{i}+a_{i})!_{(r)}}$
for each $i$, we extract the intersection number. The lhs behaves nicely under
the Laplace transformation as described in [Hoc23c] such that we conclude

$\displaystyle\sum_{1\leq a_{1},...,a_{n}\leq
r-1}\bigg{\langle}c_{W}(a_{1},...,a_{n})\prod_{i=1}^{n}\frac{\Gamma(1-\frac{a_{i}}{r})\mu_{i}^{a_{i}/r}}{1-\mu_{i}\psi_{i}}\bigg{\rangle}_{g,n}(-r)^{g-1}$
(2.29) $\displaystyle=$
$\displaystyle[\hbar^{2g+n-2}]\prod_{i=1}^{n}\left(\int_{0}^{\infty}dz_{i}e^{-\frac{(z_{i}+\frac{\hbar\mu_{i}}{2})^{r+1}-(z_{i}-\frac{\hbar\mu_{i}}{2})^{r+1}}{\hbar(r+1)}}\right)\sum_{\begin{subarray}{c}\sigma\in
S_{n}\\\
\sigma=n\text{-cycle}\end{subarray}}\prod_{i=1}^{n}\frac{1}{z_{i}+\frac{\hbar\mu_{i}}{2}-z_{\sigma(i)}+\frac{\hbar\mu_{\sigma(i)}}{2}}.$
(2.30)

This formula recovers the formula of Brezin and Hikami [BH07, BH16], which was
can be used in the $n=1$ to derive more explicit formulas [LVX17, DYZ23].

Note that the formula (2.29) depends on the rhs continuously on $r$, whereas
using higher order TR [BE13] to derive explicit values for these intersection
numbers needs a fixed integer $r$ corresponding to the order of ramification.
When $n>1$, the integration contours are shifted slightly relative to each
other.

######  Example 2.10.

The Lambert spectral curve [BM08, EMS09] computes simple Hurwitz numbers which
due to the ELSV formula to linear Hodge integrals on
$\overline{\mathcal{M}}_{g,n}$. The spectral curve takes the form
$(\mathbb{C},x=z-\log z,y=\log z,\frac{dz_{1}\,dz_{2}}{(z_{1}-z_{2})^{2}}$,
where $y$ was shifted by $x$ to make the $x-y$ formula applicable, see
[Hoc23c] for details. Let $c_{k}(\mathbb{E})$ be the $k$th Chern class of the
Hodge bundle $\mathbb{E}$ and
$\Lambda(\alpha)=1+\sum_{k=1}^{g}(-1)^{k}\alpha^{-k}c_{k}(\mathbb{E})$. Then,
the correlators generated by TR compute the intersection numbers of the Hodge
classes in the following way

$\displaystyle\tilde{W}_{g,n}(x_{1}(z_{1}),...,x_{n}(z_{n}))=\sum_{k_{1},...,k_{n}\geq
0}\prod_{i=1}^{n}\frac{k_{i}^{k_{i}+1}}{k_{i}!}\bigg{\langle}\frac{\Lambda(1)}{\prod_{i=1}^{n}(1-k_{i}\psi_{i})}\bigg{\rangle}_{g,n}e^{k_{i}x_{i}(z_{i})}.$
(2.31)

Since $y=\log z$, we can not use the formula from Corollary 2.6 and
Proposition 2.7. However, analogous computations provide the formula (Cauchy
determinant has to be used with $a_{i}=z_{i}e^{\hbar k_{i}/2}$ and
$b_{j}=-z_{j}e^{-\hbar k_{i}/2}$)

$\displaystyle\tilde{W}_{g,n}(x_{1}(z_{1}),...,x_{n}(z_{n}))$ (2.32)
$\displaystyle=$
$\displaystyle[\hbar^{2g+n-2}]\prod_{i=1}^{n}\left(\sum_{m_{i}}\bigg{(}-\frac{\partial}{\partial
x_{i}(z_{i})}\bigg{)}^{m_{i}}[u_{i}^{m_{i}}]\frac{e^{u_{i}(S(\hbar
u_{i})-1)z_{i}}}{1-z_{i}}\right)\sum_{\begin{subarray}{c}\sigma\in S_{n}\\\
\sigma=n\text{-cycle}\end{subarray}}\prod_{i=1}^{n}\frac{1}{z_{i}e^{\frac{\hbar
k_{i}}{2}}-z_{\sigma(i)}e^{-\frac{\hbar k_{\sigma(i)}}{2}}}.$

Taking now the Laplace transformation of (2.31) with contour around the origin
by
$\operatorname{Res}_{z_{i}}e^{(u_{i}-\mu_{i})x_{i}(z_{i})}dx_{i}(z_{i})=\delta_{\mu_{i},u_{i}}$
extracts the intersection number. On the other hand, applying this Laplace
transformation to (2.32) as described in [Hoc23c] yields finally

$\displaystyle\prod_{i=1}^{n}\frac{k_{i}^{k_{i}+1}}{k_{i}!}\bigg{\langle}\frac{\Lambda(1)}{\prod_{i=1}^{n}(1-k_{i}\psi_{i})}\bigg{\rangle}_{g,n}$
$\displaystyle=$
$\displaystyle\operatorname{Res}_{z_{i}=0}[\hbar^{2g+n-2}]\prod_{i=1}^{n}\frac{dz_{i}e^{k_{i}z_{i}S(\hbar
k_{i})}}{z_{i}^{k_{i}}}\sum_{\begin{subarray}{c}\sigma\in S_{n}\\\
\sigma=n\text{-cycle}\end{subarray}}\prod_{i=1}^{n}\frac{1}{z_{i}e^{\frac{\hbar
k_{i}}{2}}-z_{\sigma(i)}e^{-\frac{\hbar k_{\sigma(i)}}{2}}}.$

The difference to the formula already appearing in [Hoc23c] is that the sum is
over $n$-cycle permutations rather than over connected labelled graphs.

##  3\. $x-y$ Duality for Curves of the form $e^{x}=F(e^{y})$ or
$e^{x}=F(y)e^{ay}$

It was already observed in [BS12, GS12] that TR possess some issues with
curves of the form $e^{y}=F(e^{x})$ which are for instance the framed
topological vertex curve with framing $f=0$. To understand on a very simple
example that the $x-y$ symplectic transformation formula is directly related
to this observation, we will present an almost trivial example related to the
Euler $\Gamma$-function as wave function of the quantum spectral curve. This
example carries already all information needed to understand the general
formalism which makes the $x-y$ formula applicable to the curves under
consideration.

In this section we switch to a notation without ”tilde”, and will explain in
Sec. (3.2) how these objects are defined in detail. Assume $\omega_{g,n}$ and
$\omega_{g,n}^{\vee}$ are given (the explicit definition how these objects are
defined will be given in Sec. (3.2)), then the relation to $W_{g,n}$, $W_{n}$,
$W_{g,n}^{\vee}$, $W_{n}^{\vee}$, $\Phi_{g,n}$, $\Phi_{n}$,
$\Phi_{g,n}^{\vee}$, $\Psi(x)$ and $\Psi^{\vee}(y)$ is given as before
(2.2)-(2.5), (2.7)-(2.10), (2.12) and (2.14) by just removing ”tilde”.

###  3.1. The $\Gamma$-wave function

Just for this subsection (due to pedagogical reasons), we consider the
spectral curve $(\mathbb{P}^{1},x,y,B)$ of the form $e^{y}=...$ (rather than
$e^{x}=....$) with

$\displaystyle e^{y}=x$ (3.1)

with the parametrisation

$\displaystyle x=z,\qquad y=\log z.$

Quantising naively this spectral curve via $x\to\hat{x}=x$ and
$y\to\hat{y}=\hbar\frac{\partial}{\partial x}$ leads to the formal
differential operator $e^{\hbar\frac{\partial}{\partial x}}-x$ which
annihilates formally the Euler $\Gamma$ function

$\displaystyle(e^{\hbar\frac{\partial}{\partial
x}}-x)\Gamma\bigg{(}\frac{x}{\hbar}\bigg{)}=0,$

where we have used the action $e^{\hbar\frac{\partial}{\partial
x}}f(x)=f(x+\hbar)$ and $\Gamma(1+x)=x\Gamma(x)$. Note this is just a formal
observation and all functions are understood in terms of formal expansions in
$\hbar$. The asymptotic222Note an asymptotic expansion at $x=0$ for the form
$f(x)\sim\sum_{n}a_{n}x^{n}$ satisfies
$|f(x)-\sum_{n=0}^{N}a_{n}x^{n}|\in\mathcal{O}(x^{N+1})$ where $a_{n}$ can
have factorial growth (not convergent) expansion of the logarithm of the
$\Gamma$ function at infinity is very well-known in terms of Bernoulli numbers

$\displaystyle\log\Gamma\bigg{(}\frac{x}{\hbar}\bigg{)}\sim\frac{x}{\hbar}\log\frac{x}{\hbar}-\frac{x}{\hbar}-\frac{1}{2}\log\frac{x}{\hbar}+\frac{1}{2}\log
2\pi+\sum_{k=2}^{\infty}\frac{B_{k}\hbar^{k-1}}{k(k-1)x^{k-1}}.$

Comparing this to the construction of the wave function $\tilde{\Psi}(x)$ via
$\tilde{\omega}_{g,n}$ as defined in Sec. 2.2, we find heuristic arguments
that the $\tilde{\omega}_{g,n}$ should actually not vanish for $2g+n-2>0$ if
the wave function is constructed as suggested in Sec.2.2 . However, applying
TR to the curve defined in (3.1) has vanishing $\tilde{\omega}_{g,n}=0$ for
all $2g+n-2>0$, since $x(z)=z$ is unramified, i.e. $dx(z)\neq 0$ for all
$z\in\mathbb{P}^{1}$.

Having the $x-y$ symplectic transformation in mind, we apply TR for the same
curve we exchanged $x$ and $y$. In this situation, we find also vanishing
$\tilde{\omega}_{g,n}^{\vee}=0$ for all $2g+n-2>0$ since $y(z)=\log z$ has no
algebraic ramification point. After having swapped $x$ and $y$, consider
heuristically the quantum spectral curve with the quantisation
$x\to\hat{x}=\hbar\frac{\partial}{\partial y}$ and $y\to\hat{y}=y$. Direct
computation shows

$\displaystyle\bigg{(}e^{y}-\hbar\frac{\partial}{\partial
y}\bigg{)}e^{\frac{e^{y}}{\hbar}}=0,$

where the logarithms of the wave function has no contributions at order
$\hbar^{n}$ and $n\geq 0$, which aligns with the observation that all
$\tilde{\omega}_{g,n}^{\vee}=0$ for $2g+n-2>0$.

We conclude an obvious mismatch by naively trying to construct the wave
function from the correlators $\tilde{\omega}_{g,n}$ generated by TR from the
curve (3.1). However, we show that the $x-y$ symplectic transformation formula
resolves this problem in the following way.

Now we change the notation to $\omega_{g,n}$ without ”tilde”. Consider the
spectral curve (3.1) with $\omega_{g,n}^{\vee}\equiv 0$ for $2g+n-2>0$, then
we define all $\omega_{g,n}$ through the $x-y$ symplectic transformation
formula (2.18) by replacing $\tilde{\omega}^{\vee}_{g,n}$ by
$\omega^{\vee}_{g,n}$, which means replacing $\tilde{W}^{\vee}_{g,n}$ by
$W^{\vee}_{g,n}$. The first examples already shows that
$W_{g,n}=\frac{\omega_{g,n}}{dx_{1}...dx_{n}}$ does not vanish for $2g+n-2>0$
with $n=1$. It turns out that the corresponding wave function constructed as
in Sec. 2.2 from this $\omega_{g,n}$ is the $\Gamma$-function. More precisely,
we have

$\displaystyle\Psi(x)=\frac{\Gamma(\frac{x}{\hbar}+\frac{1}{2})\cdot\hbar^{\frac{x}{\hbar}}}{\sqrt{2\pi}}$

which is annihilated by the operator

$\displaystyle\hat{P}(x,\hbar\partial_{x})=e^{\hbar\partial_{x}}-x-\frac{\hbar}{2}$
(3.2)

which is in accordance with the conjecture by Gukov and Sulkowski (2.13).

To see this, take $x=z$ and $y=\log z$. We assume that all $W_{g,n}^{\vee}=0$
for $2g+n-2>0$ since $y$ is unramified. Consider the cases $(g,n)=(g,1)$.
Apply the $x-y$ duality formula of Corollary 2.6 for $W_{g,1}$

$\displaystyle
W_{g,1}(x)=[\hbar^{2g-1}]\sum_{k}(-\partial_{x})^{k}[u^{k}]\frac{\exp(xu(S(\hbar
u)-1))}{\hbar uS(\hbar u)}\frac{(-1)}{x},$

where $S(x)=\frac{e^{x/2}-e^{-x/2}}{x}=1+\frac{x^{2}}{24}+...$. Integrating
once wrt $x$ gives $\Phi_{g,1}(x)=\int W_{g,1}(x)dx$ which changes just a
factor of $u$. We have as a formal power series

$\displaystyle\Phi_{g,1}(x)=[\hbar^{2g-1}]\sum_{k}(-\partial_{x})^{k}[u^{k}]\frac{\exp(xu(S(\hbar
u)-1))}{\hbar u^{2}S(\hbar u)}\frac{(-1)}{x}.$

Note expanding the exponential in $\hbar$ has a higher power of $u$ than of
$x$. This means that after acting with $(-\partial_{x})^{k}[u^{k}]$ on it,
this term vanishes. In summary, the exponential does only contribute as a 1.
We find

$\displaystyle\Phi_{g,1}(x)=$
$\displaystyle[\hbar^{2g-1}]\sum_{k}(-\partial_{x})^{k}[u^{k}]\frac{1}{\hbar
u^{2}S(\hbar u)}\frac{(-1)}{x}$ $\displaystyle=$
$\displaystyle[\hbar^{2g-1}]\sum_{k}(-\partial_{x})^{k}[u^{k}]\frac{1}{\hbar
S(\hbar u)}(x-x\log x)$ $\displaystyle=$
$\displaystyle[\hbar^{2g-1}]\frac{e^{\hbar/2\partial_{x}}\hbar\partial_{x}}{\hbar(e^{\hbar\partial_{x}}-1)}(x-x\log
x)$ $\displaystyle=$
$\displaystyle[\hbar^{2g-1}]e^{\hbar/2\partial_{x}}\bigg{(}x/\hbar\log
x/\hbar-x/\hbar-\frac{1}{2}\log
x/\hbar+\sum_{k=2}^{\infty}\frac{B_{k}}{k(k-1)(x/\hbar)^{k-1}}\bigg{)}+\frac{x}{\hbar}\log\hbar$
$\displaystyle=$
$\displaystyle[\hbar^{2g-1}]\log\Gamma\bigg{(}\frac{x}{\hbar}+\frac{1}{2}\bigg{)}+\frac{x}{\hbar}\log\hbar-\frac{1}{2}\log
2\pi$

which coincides with the $\Gamma$ function. Next, we argue that all $W_{g,n}$
with $n>2$ and $2g+n-2>0$ vanish. The poles at the diagonal vanish due to the
argument in [ABDB+22, Proposition 4.6] (nothing changes by including
logarithms). The only possible pole would be at $x_{i}=0$ which is generated
by the term $\frac{dy_{i}}{dx_{i}}$, however the expansion of the exponential
yields factors of $x_{i}$ together with factors of $u_{i}^{2+n}$. These
generate higher derivatives wrt $x_{i}$. So, we use the same argument as
before which means we have terms of the form
$\partial_{x_{i}}^{n+k}x_{i}^{n}$, where $k>0$. An explicit computation is
very cumbersome and there is no new insight.

In conclusion, the wave function is constructed by just $W_{g,1}(x)$

$\displaystyle\log\Psi(x)=\sum_{g=0}^{\infty}\hbar^{2g-1}\Phi_{g,1}(x)=\log\Gamma\bigg{(}\frac{x}{\hbar}+\frac{1}{2}\bigg{)}+\frac{x}{\hbar}\log\hbar-\frac{1}{2}\log
2\pi.$

Via the functional equation of the $\Gamma$ function, the quantum spectral
curve (3.2) is also derived.

######  Remark 3.1.

The dual wave function is

$\displaystyle\Psi^{\vee}(y)=\exp\left(\sum_{g,n}\frac{\hbar^{2g+n-2}}{n!}\Phi^{\vee}_{g,n}(y,...,y)\right)=\exp\left(\frac{e^{y}}{\hbar}-\frac{1}{2}y\right),$

since $\Phi^{\vee}_{0,2}(y,y)=-y$. It is annihilated by
$\hat{P}^{\vee}(\hbar\partial_{y},y)=e^{y}-\hbar\partial_{y}-\frac{\hbar}{2}$.
Its formal Fourier/Laplace transform with an appropriate contour and $\hbar$
chosen in an appropriate region in the complex plane is equal to the wave
function $\Psi(x)$

$\displaystyle\frac{1}{\sqrt{\hbar
2\pi}}\int\underbrace{e^{\left(\frac{e^{y}}{\hbar}-\frac{1}{2}y\right)}}_{=\Psi^{\vee}(y)}e^{-xy/\hbar}dy=\frac{1}{\sqrt{\hbar
2\pi}}\int
e^{-t}t^{x/\hbar-1/2}\hbar^{x/\hbar+1/2}dt=\underbrace{\frac{\Gamma(\frac{x}{\hbar}+\frac{1}{2})\hbar^{x/\hbar}}{\sqrt{2\pi}}}_{=\Psi(x)},$

where we have substituted $\frac{e^{y}}{\hbar}=-t$.

We summarise the previous observation in the following way. TR implies that
both families $\tilde{\omega}_{g,n}$ and $\tilde{\omega}_{g,n}^{\vee}$ vanish
for the spectral curve (3.1). However, constructing the wave functions from
these trivial families $\tilde{\omega}_{g,n}$ and
$\tilde{\omega}_{g,n}^{\vee}$ gives wrong results. Now, assuming that the
family $\omega_{g,n}^{\vee}=\tilde{\omega}_{g,n}^{\vee}$ is trivial and
letting $\omega_{g,n}$ be defined by the $x-y$ duality formula produces a non-
vanishing family $\omega_{g,n}$ which generates the correct wave function.
Therefore, one of the families should actually not be trivial even though TR
implies it.

###  3.2. General Construction

We propose the following extension of the $x-y$ formula for spectral curves of
the form $e^{x}=F(e^{y})$ (or $e^{x}=F(y)e^{ay}$) with exponential variables
having, a priori from TR perspective, one trivial family of correlators
$\tilde{\omega}_{g,n}^{\vee}=0$. The formalism extends the trivial family
$\tilde{\omega}_{g,n}^{\vee}$ to $\omega_{g,n}^{\vee}$ by a version of the
asymptotic series of Faddeev’s quantum dilogarithm [FK94]. The connection to
Faddeev’s quantum dilogarithm will be explained in the next subsection.

We define for the spectral curve $e^{x}=F(e^{y})$ (or $e^{x}=F(y)e^{ay}$) the
correlators

$\displaystyle\boxed{W^{\vee}_{g,1}(y):=\frac{B_{2g}(1/2)}{(2g)!}\partial_{y}^{2g}(W^{\vee}_{0,1}(y)),}$
(3.3)

where $W_{0,1}(y)=x(y)=\log F(e^{y})$. The coefficient $B_{n}(x)$ is the $n$th
Bernoulli polynomial, and more precisely the coefficient appearing (3.3) is
the $2g$ coefficient of the function
$\frac{1}{S(x)}=\frac{x}{e^{x/2}-e^{-x/2}}$, i.e.
$[x^{2g}]\frac{x}{e^{x/2}-e^{-x/2}}=\frac{B_{2g}(1/2)}{(2g)!}$. Note that this
coefficient already appeared in expansion of the $\Gamma$ function before.

Now we define all $W_{g,n}$ through the $x-y$ symplectic transformation
formula (2.18) which are in general nontrivial with poles at the ramification
points of $x$ for $2g+n-2>0$.

Now, we have the following proposal:

If we have a spectral curve of the form $e^{x}=F(e^{y})$ (or
$e^{x}=F(y)e^{ay}$) with $F$ rational and simple algebraic ramification points
for $x$, then the correlators $\tilde{\omega}_{g,n}$ computed by TR via (2.1)
coincide with the correlators $\omega_{g,n}$ from the $x-y$ formula (2.18)
where $W_{g,1}^{\vee}$ is defined by (3.3), i.e.
$\displaystyle\tilde{\omega}_{g,n}=\omega_{g,n}.$

######  Remark 3.2.

When this paper was finished, I was informed by Alexander Alexandrov, Boris
Bychkov, Petr Dunin-Barkowski, Maxim Kazarian, and Sergey Shadrin that this
proposal is in some sense a special case of the results in [DBPSS19, BDBKS20].
However, it was not formulated in this way directly in the past. An equivalent
version of the explicit formulas (3.4) or (3.5) were instead directly derived
and proved to satisfy TR, and not taken as a consequence of the proposed
extension of the $x-y$ duality formula. Therefore, this paper still provides a
new perspective on the topic revolving around the $x-y$ duality in general.

Considering (3.3), we find as a consequence of the $x-y$ duality formula,
Corollary 2.6 and Proposition 2.7 the explicit form

$\displaystyle W_{n}(x_{1},...,x_{n})$ (3.4) $\displaystyle=$
$\displaystyle\prod_{i=1}^{n}\sum_{m_{i}\geq 0}\left(-\frac{\partial}{\partial
x_{i}}\right)\left(-\frac{\partial y_{i}}{\partial
x_{i}}\right)[u_{i}^{m_{i}}]\exp\left(\sum_{g=0}^{\infty}\hbar^{2g-1}(\Phi^{\vee}_{g,1}(y_{i}+\hbar
u_{i}/2)-\Phi^{\vee}_{g,1}(y_{i}-\hbar u_{i}/2))-x_{i}u_{i}+y_{i}\right)$
$\displaystyle\times\sum_{\begin{subarray}{c}\sigma\in S_{n}\\\
\sigma=n\text{-cycle}\end{subarray}}\prod_{i=1}^{n}\frac{1}{e^{y_{i}+\frac{\hbar
k_{i}}{2}}-e^{y_{\sigma(i)}-\frac{\hbar k_{\sigma(i)}}{2}}}$

for curves of the form $e^{x_{i}}=F(e^{y_{i}})$.

In case of curves of the form $e^{x_{i}}=F(y_{i})e^{ay_{i}}$, we have the
minor change

$\displaystyle W_{n}(x_{1},...,x_{n})$ (3.5) $\displaystyle=$
$\displaystyle\prod_{i=1}^{n}\sum_{m_{i}\geq 0}\left(-\frac{\partial}{\partial
x_{i}}\right)\left(-\frac{\partial y_{i}}{\partial
x_{i}}\right)[u_{i}^{m_{i}}]\exp\left(\sum_{g=0}^{\infty}\hbar^{2g-1}(\Phi^{\vee}_{g,1}(y_{i}+\hbar
u_{i}/2)-\Phi^{\vee}_{g,1}(y_{i}-\hbar u_{i}/2))-x_{i}u_{i}\right)$
$\displaystyle\times\sum_{\begin{subarray}{c}\sigma\in S_{n}\\\
\sigma=n\text{-cycle}\end{subarray}}\prod_{i=1}^{n}\frac{1}{y_{i}+\frac{\hbar
k_{i}}{2}-y_{\sigma(i)}+\frac{\hbar k_{\sigma(i)}}{2}},$

where $\Phi^{\vee}_{0,1}(y)=\int x\,dy=\int\log F(y)dy$ and
$\Phi^{\vee}_{g,1}(y)=\frac{B_{2g}(1/2)}{(2g)!}\partial_{y}^{2g}(\Phi^{\vee}_{0,1}(y))$.

Note that the argument of the exponential in both expressions has the
following equivalent representation:

$\displaystyle\sum_{g=0}^{\infty}\hbar^{2g-1}(\Phi^{\vee}_{g,1}(y+\hbar
u/2)-\Phi^{\vee}_{g,1}(y-\hbar
u/2))-xu=\left(\frac{S(u\hbar\partial_{y})}{S(\hbar\partial_{y})}-1\right)u\,x(y),$

where we remind $S(x)=\frac{e^{x/2}-e^{-x/2}}{x}$. The form is very similar to
the original formulation of the $x-y$ duality formula appearing in [BDBKS20,
BDBKS22a].

Let us justify the choice of (3.3) with a prediction for the quantum spectral
curve of the form $e^{x}=F(e^{y})$ by Gukov and Sulkowski [GS12, (3.20)],
which conjectures

$\displaystyle\hat{P}^{\vee}(\hat{x},\hat{y})=e^{\hat{x}-\frac{\hbar}{2}}-F(e^{\hat{y}+\frac{\hbar}{2}}),$

where the operators are $\hat{x}=\hbar\partial_{y}$ and $\hat{y}=y$. The
corresponding wave function $\Psi^{\vee}(y)$ will therefore satisfy the
functional relation

$\displaystyle\Psi^{\vee}(y+\hbar)e^{-\frac{\hbar}{2}}-F(e^{y+\frac{\hbar}{2}})\Psi^{\vee}(y)=0,$

or equivalently (after shifting $y\to y-\frac{\hbar}{2}$ and renormalising the
wave function $\Psi^{\vee,ren}(y)=\Psi^{\vee}(y)e^{-y/2}$, which is
essentially subtracting the contribution of $\Phi^{\vee}_{0,2}$)

$\displaystyle\Psi^{\vee,ren}\bigg{(}y+\frac{\hbar}{2}\bigg{)}-F(e^{y})\Psi^{\vee,ren}\bigg{(}y-\frac{\hbar}{2}\bigg{)}=0.$

Now, it becomes obvious that the logarithm of the renormalised wave function
together with the action of the formal derivative
$e^{\frac{\hbar}{2}\partial_{y}}-e^{\frac{\hbar}{2}\partial_{y}}$ satisfies

$\displaystyle(e^{\frac{\hbar}{2}\partial_{y}}-e^{-\frac{\hbar}{2}\partial_{y}})\log\Psi^{\vee,ren}(y)=\log\Psi^{\vee,ren}\bigg{(}y+\frac{\hbar}{2}\bigg{)}-\log\Psi^{\vee,ren}\bigg{(}y-\frac{\hbar}{2}\bigg{)}=\log
F(e^{y}).$

Inverting formally the operator
$(e^{\frac{\hbar}{2}\partial_{y}}-e^{\frac{\hbar}{2}\partial_{y}})$ and adding
on the rhs $1=\partial_{y}\partial_{y}^{-1}$ yields

$\displaystyle\log\Psi^{\vee,ren}(y)=\frac{1}{\hbar
S(\hbar\partial_{y})}\int\log F(e^{y})dy=\frac{1}{\hbar
S(\hbar\partial_{y})}\Phi^{\vee}_{0,1}(y),$ (3.6)

where $S(t)=\frac{e^{t/2}-e^{-t/2}}{t}$. Expanding both sides in a formal
power series in $\hbar$, we identify at order $\hbar^{2g-1}$ at the lhs
$\Phi_{g,1}^{\vee}(y)$ with
$\frac{B_{2g}(1/2)}{(2g)!}\partial_{y}^{2g}(\Phi^{\vee}_{0,1}(y))$ from the
rhs under the assumption that all $\Phi_{g,n}^{\vee}(y)$ with $n>1$ vanish
(except $\Phi_{0,2}^{\vee}(y)$ which contributed to the renormalisation).

###  3.3. Faddeev’s Quantum Dilogarithm and $x-y$ duality

The quantum dilogarithm [FK94] (see also [Zag07, KM16, GK20] and references
therein for further information) is a special function which plays an
important role in quantum Teichmüller theory and complex Chern-Simons theory.
The quantum dilogarithm is defined by

$\displaystyle\tilde{\varphi}_{b}(x)=\exp\left(\int_{\mathbb{R}+i\varepsilon}\frac{e^{-2ixt}}{4\sinh(tb)\sinh(t/b)}\frac{dt}{t}\right).$

The Fourier transform is given by (see for instance [AP16, eq. (A.16)])

$\displaystyle\int_{-\infty}^{\infty}\tilde{\varphi}_{b}(x)e^{2\pi
iyx}dx=\frac{e^{-\frac{\pi i}{12}(3+b^{2}+b^{-2})-\pi
y(b+b^{-1})}}{\tilde{\varphi}_{b}\bigg{(}-y-\frac{i}{2}(b+b^{-1})\bigg{)}}.$
(3.7)

For our purpose, it is convenient to use the following representation after
variable transformation

$\displaystyle\varphi_{\hbar}(x):=\tilde{\varphi}_{\sqrt{\frac{\hbar}{2\pi
i}}}(x/\sqrt{-\hbar 2\pi i})=\exp\left(\int\frac{e^{xt}}{S(t\hbar)(e^{i\pi
t}-e^{-i\pi t})}\frac{dt}{\hbar t^{2}}\right)$ (3.8)

which satisfies $\varphi_{\hbar}(x)=\frac{1}{\varphi_{-\hbar}(x)}$ and the
functional relation

$\displaystyle\varphi_{\hbar}\bigg{(}x-\frac{\hbar}{2}\bigg{)}=(1+e^{x})\,\varphi_{\hbar}\bigg{(}x+\frac{\hbar}{2}\bigg{)}$
(3.9)

and has the asymptotic expansion

$\displaystyle\log\varphi_{\hbar}(x)\sim\sum_{g=0}^{\infty}\hbar^{2g-1}\frac{B_{2g}(1/2)}{(2g)!}\partial_{x}^{2g}\mathrm{Li}_{2}(-e^{x})=\frac{1}{\hbar
S(\hbar\partial_{x})}\mathrm{Li}_{2}(-e^{x})$ (3.10)

which is exactly of the same form as the new defined family of
$W_{g,1}^{\vee}$ with $W_{0,1}^{\vee}(y)=\log(-1-e^{y})$.

Thus to make the connection to TR, we want to study the spectral curve

$\displaystyle e^{x}+e^{y}+1=0$ (3.11)

which is symmetric between $x$ and $y$. This curve is very special, since the
curve can be written as $e^{x}=F(e^{y})$ and $e^{y}=F(e^{x})$. The general
construction from Sec. 3.2 tells us that both, $W_{g,1}$ and $W_{g,1}^{\vee}$,
should not vanish and actually be

$\displaystyle\Phi_{0,1}(x)=$ $\displaystyle\int y\,dx=x\pi
i-\mathrm{Li}_{2}(-e^{x}),$ (3.12) $\displaystyle\Phi_{g,1}(x)=$
$\displaystyle\frac{B_{2g}(1/2)}{(2g)!}\partial_{x}^{2g}(\Phi_{0,1}(x)),$
(3.13) $\displaystyle\Phi_{0,1}^{\vee}(y)=$ $\displaystyle\int
x\,dy=yi\pi-\mathrm{Li}_{2}(e^{-y}),$ (3.14)
$\displaystyle\Phi_{g,1}^{\vee}(y)=$
$\displaystyle\frac{B_{2g}(1/2)}{(2g)!}\partial_{y}^{2g}(\Phi_{0,1}^{\vee}(y)).$
(3.15)

Now, we want to check if this aligns with the proposed $x-y$ duality formula.
Therefore, take (3.4) for $n=1$ which is

$\displaystyle W_{g,1}(x)=$ $\displaystyle[\hbar^{2g-1}]\sum_{m\geq
0}\left(-\frac{\partial}{\partial x}\right)^{m}\left(-\frac{\partial
y}{\partial x}\right)$ (3.16)
$\displaystyle\times[u^{m}]\frac{\exp\left(\sum_{k=0}^{\infty}\hbar^{2k-1}(\Phi^{\vee}_{k,1}(y+\hbar
u/2)-\Phi^{\vee}_{k,1}(y-\hbar u/2))-xu\right)}{\hbar uS(\hbar u)},$

where $\Phi^{\vee}_{g,1}$ is given by (3.15). Comparing now the lhs with
$W_{g,1}(x)$ of (3.14), we are expecting that the rhs computes to
$\frac{B_{2g}(1/2)}{(2g)!}\partial_{x}^{2g}(W_{0,1}(x))$. This is already
generated if we set the exponential to be 1, since

$\displaystyle[\hbar^{2g-1}]\sum_{m\geq 0}\left(-\frac{\partial}{\partial
x}\right)^{m}\left(-\frac{\partial y}{\partial x}\right)[u^{m}]\frac{1}{\hbar
uS(\hbar
u)}=\frac{B_{2g}(1/2)}{(2g)!}\partial_{x}^{2g+1}(\Phi_{0,1}(x))=\frac{B_{2g}(1/2)}{(2g)!}\partial_{x}^{2g}(W_{0,1}(x)).$

Next, we multiply (3.16) by $\hbar^{2g-1}$ and sum over $g$. Acting with
$e^{\hbar\partial_{x}/2}-e^{-\hbar\partial_{x}/2}$ on both sides generates for
the lhs (if $W_{g,1}(x)=\partial_{x}\Phi_{g,1}(x)$ of (3.13))

$\displaystyle\sum_{g=0}^{\infty}\hbar^{2g-1}(W_{g,1}(x+\hbar/2)-W_{g,1}(x-\hbar/2))=\partial_{x}\frac{e^{\hbar\partial_{x}/2}-e^{-\hbar\partial_{x}/2}}{e^{\hbar\partial_{x}/2}-e^{-\hbar\partial_{x}/2}}W_{0,1}(x)=\frac{\partial
y}{\partial x}.$

Whereas on the rhs the action of
$e^{\hbar\partial_{x}/2}-e^{-\hbar\partial_{x}/2}$ cancels against the
$S(\hbar u)$ in the denominator and we conclude the following statement:

$\displaystyle-\frac{\partial y}{\partial x}$ $\displaystyle=$
$\displaystyle[\hbar^{2g}]\sum_{m\geq 0}\left(-\frac{\partial}{\partial
x}\right)^{m}\left(-\frac{\partial y}{\partial
x}\right)[u^{m}]\exp\left(\sum_{k=0}^{\infty}\hbar^{2k-1}(\Phi^{\vee}_{k,1}(y+\hbar
u/2)-\Phi^{\vee}_{k,1}(y-\hbar u/2))-xu\right),$

with $e^{x}+e^{y}+1=0$ and $\Phi^{\vee}_{g,1}$ given by (3.15). This means
that every order $\hbar^{2g}$ with $g>1$ vanish identically. This was tested
with computer algebra system up to genus $g=5$, and it seems to be completely
non trivial.

One can summarise the previous computation in the following statement:
The $x-y$ duality formula sends coefficients of the quantum dilogarithm to
coefficients of the quantum dilogarithm in a nontrivial way.

Next, we discussion the compatibility with the quantum spectral curve. Both
wave functions $\Psi(x)$ and $\Psi^{\vee}(y)$ can directly be constructed from
$\Phi_{g,1}(x)$ and $\Phi_{g,1}^{\vee}(y)$ since all other
$\Phi_{g,n}^{(\vee)}=0$ for $n\geq 2$ except for $\Phi_{0,2}^{(\vee)}$. We
deduce

$\displaystyle\log\Psi(x)=$
$\displaystyle\frac{1}{2}x+\frac{i\pi}{2}+\frac{x\pi
i}{\hbar}+\sum_{g=0}^{\infty}(-\hbar)^{2g-1}\frac{B_{2g}(1/2)}{(2g)!}\partial_{x}^{2g}(\mathrm{Li}_{2}(-e^{x}))$
$\displaystyle\log\Psi^{\vee}(y)=$
$\displaystyle\frac{1}{2}y+\frac{i\pi}{2}+\frac{y\pi
i}{\hbar}+\sum_{g=0}^{\infty}(-\hbar)^{2g-1}\frac{B_{2g}(1/2)}{(2g)!}\partial_{y}^{2g}(\mathrm{Li}_{2}(-e^{y})),$

where the first two terms for each wave function come from
$\Phi_{0,2}^{(\vee)}$. Due to the functional relation of the quantum
dilogarithm $\varphi(x)$ (3.9) and
$\Psi(x)=\varphi_{-\hbar}(x)e^{\frac{x}{2}+\frac{x\pi i}{\hbar}}$, we find
that $\Psi(x)$ satisfies

$\displaystyle(1+e^{x+\hbar/2})\Psi(x)+\Psi(x+\hbar)e^{-\hbar/2}=0$

implying that the quantum spectral curve of (3.11) reads

$\displaystyle\hat{P}(\hat{x},\hat{y})=1+e^{\hat{x}+\hbar/2}+e^{\hat{y}-\hbar/2}.$
(3.17)

This obviously aligns with prediction of Gukov and Sulkowski (2.13). It is
straightforward to check that the dual wave function
$\Psi^{\vee}(y)=\varphi_{\hbar}(y)e^{\frac{y}{2}+\frac{y\pi i}{\hbar}}$
satisfies

$\displaystyle(1+e^{y-\hbar/2})\Psi^{\vee}(y)+\Psi^{\vee}(y-\hbar)e^{\hbar/2}=0,$

where the quantisation is $\hat{x}=-\hbar\partial_{y}$.

Last but not least, we might check if $\Psi(x)$ and $\Psi^{\vee}(y)$ are
related via Fourier/Laplace transformation. Using the previously mentioned
identities, we compute

$\displaystyle\int\Psi(x)e^{-\frac{xy}{\hbar}}dx=\int\varphi_{-\hbar}(x)e^{\frac{x}{2}+\frac{x\pi
i}{\hbar}}e^{-\frac{xy}{\hbar}}dx$ $\displaystyle=$
$\displaystyle\int\tilde{\varphi}_{\sqrt{-\hbar/(2\pi i)}}(x/\sqrt{\hbar 2\pi
i})e^{\frac{x}{\hbar}(\frac{\hbar}{2}+\pi i-y)}dx$ $\displaystyle=$
$\displaystyle\sqrt{\hbar 2\pi i}\int\tilde{\varphi}_{\sqrt{\frac{-\hbar}{2\pi
i}}}(x)e^{\frac{x}{\hbar}\sqrt{\hbar 2\pi i}(\frac{\hbar}{2}+\pi i-y)}dx$
$\displaystyle=$ $\displaystyle\sqrt{\hbar 2\pi i}\frac{e^{-\frac{\pi
i}{12}(3-\frac{\hbar}{2\pi i}-\frac{2\pi
i}{\hbar})-\pi\frac{\frac{\hbar}{2}+\pi i-y}{\sqrt{\hbar 2\pi
i}}(\sqrt{\frac{-\hbar}{2\pi i}}+\sqrt{\frac{2\pi
i}{-\hbar}})}}{\tilde{\varphi}_{\sqrt{\frac{-\hbar}{2\pi
i}}}\bigg{(}\frac{\frac{\hbar}{2}+\pi i-y}{\sqrt{\hbar 2\pi
i}}+\frac{i}{2}\bigg{(}\sqrt{\frac{-\hbar}{2\pi i}}+\sqrt{\frac{2\pi
i}{-\hbar}}\bigg{)}\bigg{)}}$ $\displaystyle=$ $\displaystyle
C\cdot\frac{e^{\frac{y}{2}+\frac{y\pi i}{\hbar}}}{\varphi_{-\hbar}(-y)}=C\cdot
e^{\frac{y}{2}+\frac{y\pi i}{\hbar}}\varphi_{\hbar}(y)=C\cdot\Psi^{\vee}(y),$

where $C$ is a constant independent of $y$ and not important for us, since it
would be related to the integration constant of $\Phi^{\vee}(y)$ which we have
not fixed. In the first step we have inserted
$\Psi(x)=\varphi_{-\hbar}(x)e^{\frac{x}{2}+\frac{x\pi i}{\hbar}}$, in the
second step we have replaced $\varphi$ through $\tilde{\varphi}$ via (3.8), in
the third step we did the variable transformation $x\to x\sqrt{\hbar 2\pi i}$,
in the fourth step we applied the Fourier transform (3.7) of the quantum
dilogarithm, in the fifth step we have used the identity
$\varphi_{\hbar}(x)=\frac{1}{\varphi_{-\hbar}(x)}$ which gave finally in the
last step the wave function $\Psi^{\vee}(y)$ up to an normalisation constant
depending on $\hbar$.

###  3.4. Resurgence for Curves of the form $e^{x}=F(e^{y})$

Computing the wave function $\Psi(x)$ or $\Psi^{\vee}(y)$ (or more precisely
the log of the wave function) from the correlator $\omega_{g,n}$ or
$\omega_{g,n}^{\vee}$ give an asymptotic expansion in $\hbar$ which is in
general not convergent. The spectral curves considered in this paper of the
form $e^{x}=F(e^{y})$ possess a particular form, for which the Borel transform
of the log of the wave function $\Psi^{\vee}(y)$ can be computed. We will
essentially just apply the result [GK20, Prop. 2.2]. The proposition states
that for a given asympotitic series of the form

$\displaystyle\phi_{f}(\tau,y)=\sum_{g=1}^{\infty}\frac{B_{2g}(1/2)}{(2g)!}f^{(2g)}(y)(2\pi
i\tau)^{2g-1}$

with $f$ analytic on $y$ for $|\mathrm{Im}(y)|<\pi$, the Borel transfroam
takes the form

$\displaystyle\mathcal{B}[\phi_{f}(\tau,y)](\xi)=\frac{i}{2\pi}\sum_{g=1}^{\infty}\frac{(-1)^{g}}{g^{2}}\left(f^{\prime\prime}\bigg{(}y+\frac{\xi}{n}\bigg{)}+f^{\prime\prime}\bigg{(}y-\frac{\xi}{n}\bigg{)}\right).$

Applying this to the previous construction of the wave function
$\Psi^{\vee}(y)$, we set $\hbar=2\pi i\tau$ and
$f(y)=\Phi_{0,1}^{\vee}(y)=\int\log F(e^{y})dy$ and derive the Borel transform

$\displaystyle\mathcal{B}[\phi_{\Phi_{0,1}^{\vee}}(\frac{\hbar}{2\pi
i},y)](\frac{\xi}{2\pi
i})=\frac{i}{2\pi}\sum_{g=1}^{\infty}\frac{(-1)^{g}}{g^{2}}\left(\frac{F^{\prime}(e^{y+\frac{\xi}{2\pi
in}})e^{y+\frac{\xi}{2\pi in}}}{F(e^{y+\frac{\xi}{2\pi
in}})}+\frac{F^{\prime}(e^{y-\frac{\xi}{2\pi in}})e^{y-\frac{\xi}{2\pi
in}}}{F(e^{y-\frac{\xi}{2\pi in}})}\right).$

We conclude that the wave function $\Psi^{\vee}(y)$ takes the form (by the
Laplace transform of the Borel transform, which is not an aymptotic expression
in $\hbar$ any more)

$\displaystyle\log\Psi^{\vee}(y)=\frac{y}{2}+\frac{i\pi}{2}+\frac{\Phi_{0,1}^{\vee}(y)}{\hbar}+\frac{1}{2\pi
i}\int_{0}^{\infty}d\xi
e^{-\xi/\hbar}\mathcal{B}[\phi_{\Phi_{0,1}^{\vee}}(\frac{\hbar}{2\pi
i},y)](\frac{\xi}{2\pi i}).$

If we want to analyse the precise analytic properties, for instance poles in
the Borel plane, the function $F$ of the spectral curve $e^{x}=F(e^{y})$
should be specified.

##  4\. Examples and Consequences

###  4.1. Lambert Curve

The Lambert curve is a simple example which shows already how the $x-y$ should
be adjusted according to Sec. 3.2. The Lambert curve is an important example
since it enumerates simple Hurwitz numbers [BM08] or equivalently by the ELSV
formula simple Hodge integrals over $\overline{\mathcal{M}}_{g,n}$. It was
show and discussed in [Hoc23c] that the spectral curve of the form

$\displaystyle x(z)=z-\log z,\qquad\tilde{y}(z)=\log z\quad\rightarrow\quad
x=e^{\tilde{y}}-\tilde{y}$ (4.1)

can be applied to the ordinary $x-y$ duality formula and generates the same
$\tilde{W}_{g,n}$ as TR, see also Example 2.10 for this.

However, this curve differs from the curve of Bouchard and Marino by the
symplectic transformation $\tilde{y}\to y=\tilde{y}+x$ which leaves the
correlators $\tilde{W}_{g,n}$ (except for $\tilde{W}_{0,1}$) invariant (see
Sec.2.1). Note that for the curve (4.1) the logarithmic singularity of $x$ at
$z=0,\infty$ is cancelled by the logarithmic singularity of $y$ at
$z=0,\infty$, and vice versa. Therefore, the construction of Sec. 3.2 is not
needed, and the ordinary $x-y$ formula of [Hoc23a, ABDB+22] can be applied,
see also Example 2.10.

Now, we change the curve via symplectic transformation $\tilde{y}\to
y=\tilde{y}+x$ to

$\displaystyle x(z)=z-\log z,\qquad y(z)=z\quad\rightarrow\quad
e^{x}=\frac{e^{y}}{y}$ (4.2)

as it was originally formulated. The construction of Sec. 3.2 applies for this
curve. This means that the $W_{g,1}^{\vee}$’s have to be corrected as
suggested in Sec. 3.2 by equation (3.3), that is

$\displaystyle W_{g,1}^{\vee}(y)=$
$\displaystyle\frac{B_{2g}(1/2)}{(2g)!}\partial^{2g}_{y}(y-\log
y)=\delta_{g,0}y-\frac{B_{2g}(1/2)}{2g(2g-1)y^{2g}}$ $\displaystyle=$
$\displaystyle[\hbar^{2g-1}]\left[\frac{y}{\hbar}-\partial_{y}\log\Gamma\bigg{(}\frac{y}{\hbar}+\frac{1}{2}\bigg{)}-\frac{1}{\hbar}\log\hbar+\frac{1}{2}\log
2\pi\right],$

where the $\Gamma$ function is understood as asymptotic series, see Sec.3.1.
This implies that the wave function and the quantum spectral curve is of the
form

$\displaystyle\Psi^{\vee}(y)=$
$\displaystyle\frac{e^{\frac{y^{2}}{2\hbar}}\sqrt{2\pi}}{\Gamma\bigg{(}\frac{y}{\hbar}+\frac{1}{2}\bigg{)}\hbar^{y/\hbar}},$
$\displaystyle\hat{P}^{\vee}(\hbar\partial_{x},y)=$ $\displaystyle
e^{\hbar\partial_{y}}-\frac{e^{y+\hbar/2}}{y+\frac{\hbar}{2}}.$

The new ingredients for the $x-y$ formula (2.18) considering the construction
of Sec. 3.2 are

$\displaystyle\hat{O}^{\vee}(y(z))=$ $\displaystyle\sum_{m\geq
0}\bigg{(}-\frac{\partial}{\partial{x(z)}}\bigg{)}^{m}\bigg{(}-\frac{dy(z)}{dx(z)}\bigg{)}[u^{m}]\frac{z^{u}\Gamma\left(\frac{z}{\hbar}+\frac{1-u}{2}\right)}{\hbar
u\hbar^{u}\Gamma\left(\frac{z}{\hbar}+\frac{1+u}{2}\right)}$ (4.3)
$\displaystyle e^{\frac{1}{2}\hat{c}^{\vee}(u,u,y,y)}=$ $\displaystyle 1$
(4.4) $\displaystyle e^{\hat{c}^{\vee}(u_{1},u_{2},y(z_{1}),y(z_{2}))}=$
$\displaystyle\frac{\left(z_{1}-\hbar u_{1}/2-z_{2}+\hbar
u_{2}/2\right)\left(z_{1}+\hbar u_{1}/2-z_{2}-\hbar
u_{2}/2\right)}{\left(z_{1}+\hbar u_{1}/2-z_{2}+\hbar
u_{2}/2\right)\left(z_{1}-\hbar u_{1}/2-z_{2}-\hbar u_{2}/2\right)}.$ (4.5)

Showing that either (2.32) of Example 2.10, or (4.3), (4.4) and (4.5) give the
same $\tilde{W}_{g,n}=W_{g,n}$ after inserting in (2.18) is a non trivial
task, but follows after long computations which were performed in [Hoc23a] in
a slightly more general setting. We explicitly write down

$\displaystyle W_{g,n}(x_{1}(z_{1}),...,x_{n}(z_{n}))$ $\displaystyle=$
$\displaystyle\prod_{i=1}^{n}\bigg{(}\sum_{m_{i}\geq
0}\bigg{(}-\frac{\partial}{\partial{x_{i}(z_{i})}}\bigg{)}^{m_{i}}\bigg{(}-\frac{dy_{i}(z_{i})}{dx_{i}(z_{i})}\bigg{)}[u_{i}^{m_{i}}]\frac{z_{i}^{u_{i}}\Gamma\left(\frac{z_{i}}{\hbar}+\frac{1-u_{i}}{2}\right)}{\hbar^{u_{i}}\Gamma\left(\frac{z_{i}}{\hbar}+\frac{1+u_{i}}{2}\right)}\bigg{)}$
$\displaystyle\qquad\times\sum_{\begin{subarray}{c}\sigma\in S_{n}\\\
\sigma=n\text{-cycle}\end{subarray}}\prod_{i=1}^{n}\frac{1}{z_{i}+\frac{\hbar
u_{i}}{2}-z_{\sigma(i)}+\frac{\hbar u_{\sigma(i)}}{2}},$

where we have used that the sum over all graphs $\mathcal{G}^{2}_{n}$ can be
represented as sum over $n$-cycles (see Proposition 2.7).

Taking the Laplace transform of this equation (as explained in [Hoc23c]), the
following formula for the linear Hodge integrals is deduced

$\displaystyle\prod_{i=1}^{n}\frac{k_{i}^{k_{i}+1}}{k_{i}!}\bigg{\langle}\frac{\Lambda(1)}{\prod_{i=1}^{n}(1-k_{i}\psi_{i})}\bigg{\rangle}_{g,n}$
$\displaystyle=$
$\displaystyle\operatorname{Res}_{z_{i}=0}[\hbar^{2g+n-2}]\prod_{i=1}^{n}\frac{dz_{i}e^{k_{i}z_{i}}}{\hbar^{k_{i}}}\frac{\Gamma\left(\frac{z_{i}}{\hbar}+\frac{1-k_{i}}{2}\right)}{\Gamma\left(\frac{z_{i}}{\hbar}+\frac{1+k_{i}}{2}\right)}\sum_{\begin{subarray}{c}\sigma\in
S_{n}\\\
\sigma=n\text{-cycle}\end{subarray}}\prod_{i=1}^{n}\frac{1}{z_{i}+\frac{\hbar
k_{i}}{2}-z_{\sigma(i)}+\frac{\hbar k_{\sigma(i)}}{2}},$

where for $n=1$ we define $\sum_{\begin{subarray}{c}\sigma\in S_{1}\\\
\sigma=1\text{-cycle}\end{subarray}}\frac{1}{z_{i}+\frac{\hbar
k_{i}}{2}-z_{\sigma(i)}+\frac{\hbar k_{\sigma(i)}}{2}}=\frac{1}{\hbar u_{i}}$.

######  Remark 4.1.

A generalisation of these linear Hodge integrals to integrals of the form
$\bigg{\langle}\frac{\Lambda(a)}{\prod_{i=1}^{n}(1-k_{i}\psi_{i})}\bigg{\rangle}_{g,n}$
was for instance considered in [JPT11] and proved to satisfy TR with the curve
$e^{ax}=\frac{y}{e^{ay}}$ in [DLN16]. From the Hurwitz point of view, this
count of ramified coverings is called orbifold Hurwitz numbers (the
ramification profile over infinity is $\mu=(\mu_{1},...,\mu_{n})$ and over 0
of the form $(a,....,a)$, see [DLN16] for details). The corresponding spectral
curve fits perfectly in the class of curves under consideration. It is
straightforward to derive a formula for
$\bigg{\langle}\frac{\Lambda(a)}{\prod_{i=1}^{n}(1-k_{i}\psi_{i})}\bigg{\rangle}_{g,n}$
from the $x-y$ duality formula for any $a$.

###  4.2. The framed Topological Vertex Curve

The framed Topological Vertex curve encodes the Gromov-Witten invariants of
$\mathbb{C}^{3}$, more precisely the framed mirror curve of $\mathbb{C}^{3}$
is the curve

$\displaystyle e^{x}=\frac{e^{yf}}{(1-e^{-y})}$

with framing $f$. This curve is important in topological string theory. It was
conjectured in [BKMP09] more generally that if one takes the mirror curve of a
toric Calabi-Yau 3-fold to be the spectral curve, the $\tilde{\omega}_{g,n}$
generated by TR computes the $B$-model correlators. In the specific case of
$\mathbb{C}^{3}$, the conjecture was proved for instance in [Che09, Eyn11] and
in general in [EO15].

However note that for the special framing $f=0$, TR does not give the expected
results neither for the $\tilde{\omega}_{g,n}$ nor for the free energies,
which was observed for instance in [BS12].

We take the following parametrisation of the curve

$\displaystyle x(z)=-f\log z-\log(1-z),\qquad y(z)=-\log z.$

The correlators computed by TR (2.1) from this curve yield triple Hodge
integrals on $\overline{\mathcal{M}}_{g,n}$ [Eyn11, Theorem 4.1]

$\displaystyle\tilde{\omega}_{g,n}(z_{1},...,z_{n})=(f(1+f))^{g-1}\sum_{\mu_{1},...,\mu_{n}}\prod_{i=1}^{n}\frac{(\mu_{i}(1+f))!}{\mu_{i}!(f\mu_{i})!}e^{-\mu_{i}x_{i}(z_{i})}\mu_{i}dx_{i}(z_{i})\bigg{\langle}\frac{\Lambda(1)\Lambda(f)\Lambda(-1-f)}{\prod_{i=1}^{n}(1+\mu_{i}\psi_{i})}\bigg{\rangle}_{g,n},$

where $\psi_{i}$ is the $i$th $\psi$-class and
$\Lambda(\alpha)=1+\sum_{k}(-1)^{k}\alpha^{-k}c_{k}(\mathbb{E})$ with $c_{k}$
the $k$th Chern class of the Hodge bundle $\mathbb{E}$.

Taking the Laplace transform of this with contour in the $z$-plane around the
origin yields (since
$\operatorname{Res}_{z_{i}=1}e^{x_{i}(z_{i})(k_{i}-\mu_{i})}dx_{i}(z_{i})=-\delta_{\mu_{i},k_{i}}f$)

$\displaystyle\operatorname{Res}_{z_{i}=0}e^{x_{i}(z_{i})k_{i}}\tilde{\omega}_{g,n}(z_{1},...,z_{n})=(f(1+f))^{g-1}\prod_{i=1}^{n}\left(-\frac{(k_{i}(1+f))!\,fk_{i}}{k_{i}!(fk_{i})!}\right)\bigg{\langle}\frac{\Lambda(1)\Lambda(f)\Lambda(-1-f)}{\prod_{i=1}^{n}(1+k_{i}\psi_{i})}\bigg{\rangle}_{g,n}$
(4.6)

Now, we want to apply the construction of Sec.3.2 which is also valid for the
case of $f=0,-1$. We find

$\displaystyle W^{\vee}_{0,1}(y)=fy-\log(1-e^{-y}),\qquad
W^{\vee}_{g,1}(y)=-\frac{B_{2g}(1/2)}{(2g)!}\partial_{y}^{2g}\log(1-e^{-y}).$
(4.7)

By construction and the argumentation of Sec. 3.2, the wave function
$\Psi^{\vee}(y)$ and the quantum spectral curve $\hat{P}^{\vee}$ are as
expected from eq. (2.13).

The ingredients for the $x-y$ formula (2.18) considering (4.7) for
$W^{\vee}_{g,n}$ are

$\displaystyle\hat{O}^{\vee}(y(z))=$ $\displaystyle\sum_{m\geq
0}\bigg{(}-\frac{\partial}{\partial{x(z)}}\bigg{)}^{m}\bigg{(}-\frac{dy(z)}{dx(z)}\bigg{)}[u^{m}]\frac{\exp\left[\left(\frac{S(\hbar
u\partial_{y(z)})}{S(\hbar\partial_{y(z)})}-1\right)ux(z)\right]}{\hbar u}$
(4.8) $\displaystyle e^{\frac{1}{2}\hat{c}^{\vee}(u,u,y,y)}=$
$\displaystyle\frac{1}{S(\hbar u)}$ (4.9) $\displaystyle
e^{\hat{c}^{\vee}(u_{1},u_{2},y(z_{1}),y(z_{2}))}=$
$\displaystyle\frac{\left(z_{1}e^{-\hbar u_{1}/2}-z_{2}e^{-\hbar
u_{2}/2}\right)\left(z_{1}e^{\hbar u_{1}/2}-z_{2}e^{\hbar
u_{2}/2}\right)}{\left(z_{1}e^{\hbar u_{1}/2}-z_{2}e^{-\hbar
u_{2}/2}\right)\left(z_{1}e^{-\hbar u_{1}/2}-z_{2}e^{\hbar u_{2}/2}\right)}.$
(4.10)

Showing that (4.8), (4.9) and (4.10) give the same $\tilde{W}_{g,n}=W_{g,n}$
after inserting in (2.18) is again a non trivial task, but tested with
computer algebra system for several examples. We explicitly write down the
formula following from the $x-y$ transformation formula

$\displaystyle W_{g,n}(x_{1}(z_{1}),...,x_{n}(z_{n}))$ $\displaystyle=$
$\displaystyle\prod_{i=1}^{n}\left\\{\sum_{m_{i}\geq
0}\bigg{(}-\frac{\partial}{\partial{x_{i}(z_{i})}}\bigg{)}^{m_{i}}\bigg{(}-\frac{dy_{i}(z_{i})}{dx_{i}(z_{i})}\bigg{)}[u_{i}^{m_{i}}]z_{i}\exp\left[\left(\frac{S(\hbar
u_{i}\partial_{y_{i}(z_{i})})}{S(\hbar\partial_{y_{i}(z_{i})})}-1\right)u_{i}x_{i}(z_{i})\right]\right\\}$
$\displaystyle\qquad\times\sum_{\begin{subarray}{c}\sigma\in S_{n}\\\
\sigma=n\text{-cycle}\end{subarray}}\prod_{i=1}^{n}\frac{1}{z_{i}e^{\hbar
u_{i}/2}-z_{\sigma(i)}e^{-\hbar u_{\sigma(i)}/2}},$

where we have used that the sum over all graphs $\mathcal{G}^{2}_{n}$ can be
represented as sum over $n$-cycles (see Proposition 2.7) and the Cauchy matrix
in the proof Proposition 2.7 has to be chosen with $a_{i}=z_{i}e^{\hbar
u_{i}/2}$ and $b_{j}=-z_{j}e^{-\hbar u_{j}/2}$.

Taking the Laplace transform of this equation (as explained in [Hoc23c]), the
following formula for the triple Hodge integrals is deduced

$\displaystyle(f(1+f))^{g-1}$
$\displaystyle\prod_{i=1}^{n}\left(-\frac{(k_{i}(1+f))!\,fk_{i}}{k_{i}!(fk_{i})!}\right)\bigg{\langle}\frac{\Lambda(1)\Lambda(f)\Lambda(-1-f)}{\prod_{i=1}^{n}(1+k_{i}\psi_{i})}\bigg{\rangle}_{g,n}$
$\displaystyle=$
$\displaystyle\operatorname{Res}_{z_{i}=0}[\hbar^{2g+n-2}]\prod_{i=1}^{n}dz_{i}\,e^{\frac{S(\hbar
k_{i}\partial_{y_{i}(z_{i})})}{S(\hbar\partial_{y_{i}(z_{i})})}k_{i}x_{i}(z_{i})}\sum_{\begin{subarray}{c}\sigma\in
S_{n}\\\
\sigma=n\text{-cycle}\end{subarray}}\prod_{i=1}^{n}\frac{1}{z_{i}e^{\hbar
k_{i}/2}-z_{\sigma(i)}e^{-\hbar k_{\sigma(i)}/2}},$

where we remind $S(t)=\frac{e^{t/2}-e^{-t/2}}{t}$, $x(z)=-f\log z-\log(1-z)$
and $y(z)=-\log z$, and $f$ is the framing. This formula seems to be of a very
similar nature as the original Marino-Vafa formula [MV02]. In the case $n=1$,
we have to take $\sum_{\begin{subarray}{c}\sigma\in S_{1}\\\
\sigma=1\text{-cycle}\end{subarray}}\frac{1}{z_{i}e^{\hbar
k_{i}/2}-z_{\sigma(i)}e^{-\hbar k_{\sigma(i)}/2}}=\frac{1}{z_{i}\hbar
u_{i}S(\hbar u_{i})}$.

One might further simplify the expression in the following way

$\displaystyle\exp\left(\frac{S(\hbar
k_{i}\partial_{y_{i}(z_{i})})}{S(\hbar\partial_{y_{i}(z_{i})})}k_{i}x_{i}(z_{i})\right)=\exp\left(\sum_{n=0}^{k_{i}-1}e^{\hbar(n+\frac{1-k_{i}}{2})z_{i}\partial_{z_{i}}}(-f\log
z_{i}-\log 1-z_{i})\right)$ $\displaystyle=$
$\displaystyle\frac{1}{z_{i}^{fk_{i}}\prod_{n=0}^{k_{i}-1}(1-z_{i}e^{\hbar(n+\frac{1-k_{i}}{2})})},$

where the formal action $a^{z\partial_{z}}f(z)=f(az)$ was used. For instance
for $n=1$, we find

$\displaystyle(f(1+f))^{g-1}\left(-\frac{(k(1+f))!\,fk}{k!(fk)!}\right)$
$\displaystyle\bigg{\langle}\frac{\Lambda(1)\Lambda(f)\Lambda(-1-f)}{(1+k\psi)}\bigg{\rangle}_{g,1}$
$\displaystyle=$
$\displaystyle\operatorname{Res}_{z=0}[\hbar^{2g-1}]\frac{dz}{(e^{\hbar
k/2}-e^{-\hbar
k/2})z^{fk+1}\prod_{n=0}^{k-1}(1-ze^{\hbar(n+\frac{1-k}{2})})}.$

######  Remark 4.2.

For more general triple Hodge integrals of Calabi-Yau condition of the form

$\displaystyle\bigg{\langle}\frac{\Lambda(p)\Lambda(q)\Lambda(-p-q)}{\prod_{i=1}^{n}(1+k_{i}\psi_{i})}\bigg{\rangle}_{g,n},$
(4.11)

it is also known that they can be computed by TR, see [Ale21a, Ale21b]. The
spectral curve for these integrals is also of the form $e^{x}=F(e^{y})$ such
that a formula for the more general triple Hodge integrals of the form (4.11)
can also be written down. However, formulas are getting pretty ugly and there
is no new insight for $\Lambda(p)\Lambda(q)\Lambda(-p-q)$ since
$\Lambda(1)\Lambda(f)\Lambda(-1-f)$ has the same enumerative geometric
content.

######  Remark 4.3.

Adding additional $\Theta$-classes as they are introduced by Norbury [Nor23]
to triple Hodge integrals can be performed as well, where the spectral curve
is known [Ale21b] and again of the form $e^{x}=F(e^{y})$. These are called
triple $\Theta$-Hodge integrals. More simple intersections numbers where
mixtures of a $\Theta$-class just with $\Psi$-classes have a spectral curve of
the form $P(x,y)=xy^{r}-1=0$ [CGFG22]. Results of Sec. 2.5 from the $x-y$
duality with unramified $y$ are applicable here, a particular example is the
Bessel curve of the form
$(\mathbb{P}^{1},z^{2},\frac{1}{z},\frac{dz_{1}\,dz_{2}}{(z_{1}-z_{2})^{2}}$
[Nor23].

###  4.3. Descendent Gromov-Witten invariants of $\mathbb{P}^{1}$

The stationary Gromov-Witten invariants on $\mathbb{P}^{1}$ with their
relation to integrable systems were considered by Pandharipande in [Pan00] and
Okounkov and Pandharipande in [OP06]. Later, it was conjectured that these
invariants are computable by TR in [NS14], which was proved in [DBOSS14]. The
stationary Gromov-Witten invariants are defined by (see for instance [OP06]
and references therein)

$\displaystyle\bigg{\langle}\prod_{i=1}^{n}\tau_{b_{i}}(\gamma)\bigg{\rangle}_{g}^{d}=\int_{[\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{1},d)]^{vir}}\prod_{i=1}^{n}\psi_{i}^{b_{i}}ev^{*}_{i}(\gamma),$
(4.12)

where $d$ satisfies the dimension condition $\sum_{i}b_{i}=2g-2+2d$,
$\gamma\in H^{2}(\mathbb{P}^{1})$ be the dual class of a point and
$\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{1},d)$ is the moduli space of degree
$d$ maps from a Riemann surface of genus $g$ with $n$ marked points to
$\mathbb{P}^{1}$. The corresponding spectral curve computing stationary
Gromov-Witten invariants is

$\displaystyle x=e^{y}+e^{-y}$ (4.13)

which is typically parametrised by

$\displaystyle x(z)=z+\frac{1}{z},\qquad y(z)=\log z.$

The precise statement in the notation of the article, where
$\tilde{\omega}_{g,n}(z_{1},...,z_{n})$ is derived by TR, is the following
[DBOSS14]:

$\displaystyle\tilde{\omega}_{g,n}(z_{1},...,z_{n})=\sum_{b_{1},...,b_{n}}\bigg{\langle}\prod_{i=1}^{n}\tau_{b_{i}}(\gamma)\bigg{\rangle}_{g}^{d}\prod_{i=1}^{n}\frac{(b_{1}+1)!}{x(z_{i})^{b_{i}+2}}dx(z_{i}).$

The equality holds as a Laurent expansion at $x_{i}\to\infty$.

The curve (4.13) does not fit into the class of curves considered in Sec. 3.2
(also not after interchanging $x$ and $y$). However if we add a parameter $t$
by

$\displaystyle x_{t}(z)=z+\frac{t^{2}}{z},\qquad y(z)=\log
z,\quad\rightarrow\quad x_{t}=e^{y}+t^{2}e^{-y},$ (4.14)

we get a new curve which coincides with the spectral curve (3.1) of Sec. 3.1
for $t\to 0$ and with the curve (4.13) for $t=1$.

The limit $t\to 0$ is a singular limit, since the two ramification points at
$\beta_{\pm}=\pm t$ merge and cancel out, and on the other hand the
ramification points merge with the logarithmic singularity of $y(z)$. This
geometry of the limit is not included in the recent work [BBC+23] where
several different cases of a $t\to 0$ limit are discussed.

However, taking the $x-y$ duality formula into account and the construction of
Sec. 3.2 the limit $t\to 0$ from the curve (4.14) and from the construction of
Sec. 3.2 (after changing the role of $x$ and $y$) coincide. To see this, we
observe first that the curve (4.14) does not fit into the class considered
Sec. 3.2 for $t\neq 0$. However, in both cases $t=0$ and $t\neq 0$ we have for
$\tilde{\omega}_{g,n}^{\vee}=0$ for $2g+n-2>0$ since $y(z)$ has no
ramification points. Following the $x-y$ formula (2.18), we have the
ingredients

$\displaystyle\hat{O}^{\vee}(y(z))=$ $\displaystyle\sum_{m\geq
0}\bigg{(}-\frac{\partial}{\partial{x_{t}(z)}}\bigg{)}^{m}\bigg{(}-\frac{dy(z)}{dx_{t}(z)}\bigg{)}[u^{m}]\frac{\exp\left[u\left(S(\hbar
u)-1\right)(z+\frac{t^{2}}{z})\right]}{\hbar u}$ (4.15) $\displaystyle
e^{\frac{1}{2}\hat{c}^{\vee}(u,u,y,y)}=$ $\displaystyle\frac{1}{S(\hbar u)}$
(4.16) $\displaystyle e^{\hat{c}^{\vee}(u_{1},u_{2},y(z_{1}),y(z_{2}))}=$
$\displaystyle\frac{\left(z_{1}e^{-\hbar u_{1}/2}-z_{2}e^{-\hbar
u_{2}/2}\right)\left(z_{1}e^{\hbar u_{1}/2}-z_{2}e^{\hbar
u_{2}/2}\right)}{\left(z_{1}e^{\hbar u_{1}/2}-z_{2}e^{-\hbar
u_{2}/2}\right)\left(z_{1}e^{-\hbar u_{1}/2}-z_{2}e^{\hbar u_{2}/2}\right)}.$
(4.17)

We conclude from the $x-y$ duality formula (2.18) (or more precisely a version
of the special case (2.19) where $y=\log z$ instead of $y=z$) together with
the Proposition 2.7

$\displaystyle\tilde{W}_{g,n}(x_{t}(z_{1}),...,x_{t}(z_{n}))$ (4.18)
$\displaystyle=$ $\displaystyle\prod_{i=1}^{n}\left\\{\sum_{m_{i}\geq
0}\bigg{(}-\frac{\partial}{\partial{x_{t}(z_{i})}}\bigg{)}^{m_{i}}\bigg{(}-\frac{dy(z_{i})}{dx_{t}(z_{i})}\bigg{)}[u_{i}^{m_{i}}]z_{i}\,\exp\left[u_{i}\left(S(\hbar
u_{i})-1\right)(z_{i}+\frac{t^{2}}{z_{i}})\right]\right\\}$ (4.19)
$\displaystyle\qquad\times\sum_{\begin{subarray}{c}\sigma\in S_{n}\\\
\sigma=n\text{-cycle}\end{subarray}}\prod_{i=1}^{n}\frac{1}{z_{i}e^{\hbar
u_{i}/2}-z_{\sigma(i)}e^{-\hbar u_{\sigma(i)}/2}},$ (4.20)

which was tested with computer algebra systems for small $g$ and $n$. Note
that the rhs behaves smoothly in $t$, which converges for $t\to 0$ to the case
considered in Sec. 3.1. However, the derivation of
$W_{g,n}(x_{t}(z_{1}),...,x_{t}(z_{n}))$ from TR is not smooth in $t$ since
ramification points collide and merge with the singularity of the logarithm.
This underpins again that TR needs some adaption if the curve is of the form
$e^{y}=F(e^{x})$ or $e^{y}=F(x)e^{ax}$, respectively.

Since we can extract from the correlators $\tilde{\omega}_{g,n}$ the
stationary Gromov-Witten invariants via Laplace transform

$\displaystyle(-1)^{n}\operatorname{Res}_{z_{i}=0}dx(z_{1})...dx(z_{n})e^{\mu_{1}x(z_{1})+...+\mu_{n}x(z_{n})}\tilde{\omega}_{g,n}$
$\displaystyle=$
$\displaystyle\sum_{b_{i}}\bigg{\langle}\prod_{i=1}^{n}\tau_{b_{i}}(\gamma)\bigg{\rangle}_{g}^{d}\prod_{i}\mu_{i}^{b_{i}+1}$
$\displaystyle=$
$\displaystyle\int_{[\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{1},d)]^{vir}}ev^{*}_{i}(\gamma)\prod_{i=1}^{n}\frac{\mu_{i}}{1-\mu_{i}\psi_{i}},$

we derive the following formula for the stationary Gromov-Witten invariants
(4.12) from the Laplace transform of the $x-y$ duality formula (4.18) with
$t=1$

$\displaystyle\int_{[\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{1},d)]^{vir}}ev^{*}_{i}(\gamma)\prod_{i=1}^{n}\frac{\mu_{i}}{1-\mu_{i}\psi_{i}}$
$\displaystyle=$
$\displaystyle[\hbar^{2g+n-2}]\prod_{i=1}^{n}\operatorname{Res}_{z_{i}=0}dz_{i}e^{\mu_{i}S(\hbar\mu_{i})(z_{i}+\frac{1}{z_{i}})}\sum_{\begin{subarray}{c}\sigma\in
S_{n}\\\
\sigma=n\text{-cycle}\end{subarray}}\prod_{i=1}^{n}\frac{1}{z_{i}e^{\hbar\mu_{i}/2}-z_{\sigma(i)}e^{-\hbar\mu_{\sigma(i)}/2}}.$

######  Example 4.4.

Let us consider the $n=1$ example of the above formula:

$\displaystyle\int_{[\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{1},d)]^{vir}}ev^{*}_{i}(\gamma)\frac{\mu}{1-\mu\psi}=$
$\displaystyle[\hbar^{2g-1}]\operatorname{Res}_{z=0}\frac{dze^{\mu
S(\hbar\mu)(z+\frac{1}{z})}}{z\hbar\mu S(\hbar\mu)}$ $\displaystyle=$
$\displaystyle[\hbar^{2g-1}]\operatorname{Res}_{z=0}dz\sum_{n=0}^{\infty}\frac{(\mu
S(\hbar\mu)(z+\frac{1}{z}))^{n}}{n!z\hbar\mu S(\hbar\mu)}$ $\displaystyle=$
$\displaystyle[\hbar^{2g}]\sum_{n=0}^{\infty}\frac{\mu^{n-1}S(\hbar\mu)^{n-1}}{(\frac{n}{2})!(\frac{n}{2})!}.$

Taking on the lhs due to dimensional restriction the coefficient
$\mu^{2g+2d-1}$ yields the formula of Pandharipande [Pan00, Theorem 1] with
$n=2d$.

######  Example 4.5.

The degree $d=1$ case is also an important explicit special case (considered
in by Pandharipande [Pan00, Theorem 2]) which can easily be derived from the
above formula. Due to dimensional restriction, we have after expanding the
geometric series
$\frac{1}{1-\mu_{i}\psi_{i}}=\sum_{b_{i}}\mu_{i}^{b_{i}}\psi_{i}^{b_{i}}$ that
$\sum_{i}b_{i}=2g$. Taking the $[\mu_{1}^{b_{1}+1}...\mu_{n}^{b_{n}+1}]$
coefficient, we have on the lhs

$\displaystyle\int_{[\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{1},1)]^{vir}}ev^{*}_{i}(\gamma)\prod_{i}\psi_{i}^{b_{i}}.$

Since we have to take on the rhs the coefficient $[\hbar^{2g+n-2}]$ and
$[\mu_{1}^{b_{1}+1}...\mu_{n}^{b_{n}+1}]$ with $\sum_{i}(b_{i}+1)=2g+n$, we
can redefine all $\mu_{i}\to\frac{\mu_{i}}{\hbar}$ on the rhs. The remaining
coefficient in $\hbar$ is $[\hbar^{-2}]$, which can just be generated by
expanding at most two exponentials

$\displaystyle[\hbar^{-2}][\mu_{1}^{b_{1}+1}...\mu_{n}^{b_{n}+1}]\prod_{i=1}^{n}\operatorname{Res}_{z_{i}=0}dz_{i}e^{\frac{\mu_{i}}{\hbar}S(\mu_{i})(z_{i}+\frac{1}{z_{i}})}\sum_{\begin{subarray}{c}\sigma\in
S_{n}\\\
\sigma=n\text{-cycle}\end{subarray}}\prod_{i=1}^{n}\frac{1}{z_{i}e^{\mu_{i}/2}-z_{\sigma(i)}e^{-\mu_{\sigma(i)}/2}}.$

Now, it is important that the order of the contour integrals is fixed, since
swapping order of the contour integrals gives nontrivial contributions of the
form
$\operatorname{Res}_{z_{i}=0}\operatorname{Res}_{z_{j}=0}=\operatorname{Res}_{z_{j}=0}\operatorname{Res}_{z_{i}=0}+\operatorname{Res}_{z_{j}=0}\operatorname{Res}_{z_{i}=z_{j}}$.
Therefore, we fix for all permutations $\sigma\in S_{n}$ the order of the
integrals in the consecutive order, i.e. first $z_{1}$, then $z_{2}$ etc. It
is easy to see that the residue for $z_{1}$ already vanishes except if the
exponential $e^{\frac{\mu_{1}}{\hbar}S(\mu_{1})(z_{1}+\frac{1}{z_{1}})}$ is
expanded. Furthermore, since the last contour integral is $z_{n}$, we have to
expand also $e^{\frac{\mu_{n}}{\hbar}S(\mu_{n})(z_{n}+\frac{1}{z_{n}})}$
otherwise the expression vanishes. This means that fixing the order of
integration and taking the $[\hbar^{-2}]$ coefficient yields just nontrivial
contribution of the form

$\displaystyle[\mu_{1}^{b_{1}+1}...\mu_{n}^{b_{n}+1}]$
$\displaystyle\operatorname{Res}_{z_{n}=0}dz_{n}...\operatorname{Res}_{z_{1}=0}dz_{1}\mu_{1}S(\mu_{1})(z_{1}+\frac{1}{z_{1}})\mu_{n}S(\mu_{n})(z_{n}+\frac{1}{z_{n}})$
$\displaystyle\times\sum_{\begin{subarray}{c}\sigma\in S_{n}\\\
\sigma=n\text{-cycle}\end{subarray}}\prod_{i=1}^{n}\frac{1}{z_{i}e^{\mu_{i}/2}-z_{\sigma(i)}e^{-\mu_{\sigma(i)}/2}}.$

Now, computing the residues for all $(n-1)!$ permutations $\sigma\in S_{n}$
with $\sigma$ an $n$-cycle, a lot of permutations have vanishing contributions
except of $2^{n-2}$ permutations $\sigma\in S_{n}$. This $2^{n-2}$
permutations can be characterised nicely in the cycle notation as follows. Put
1 on the first place, than we have to put 2 either at the first or last empty
space

$\displaystyle(1,2,....)\qquad\text{or}\qquad(1,....,2)$

otherwise the integral vanishes. Adding 3 again either at the first or last
empty space:

$\displaystyle(1,2,3....)\qquad\text{or}\qquad(1,2,....,3)\qquad\text{or}\qquad(1,3,....,2)\qquad\text{or}\qquad(1,....,3,2)$

etc. Constructing the subset of $2^{n-2}$ permutations $\sigma\in S_{n}$ in
this way, these are the only permutations which do not vanish if we take the
order of integration to be the consecutive order. Performing the integration,
each permutation gives a term of the form
$(\pm)^{\epsilon_{2}+...+\epsilon_{n-1}}e^{\epsilon_{2}\frac{\hbar\mu_{2}}{2}+\epsilon_{3}\frac{\hbar\mu_{3}}{2}...+\epsilon_{n-1}\frac{\hbar\mu_{n-1}}{2}}$,
where $\epsilon_{i}=\pm$ depends on if we have put $i$ at the first or last
empty space by constructing the permutation. Therefore, we find

$\displaystyle\int_{[\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{1},1)]^{vir}}ev^{*}_{i}(\gamma)\prod_{i}\psi_{i}^{b_{i}}$
$\displaystyle=$
$\displaystyle[\mu_{1}^{b_{1}+1}...\mu_{n}^{b_{n}+1}]\operatorname{Res}_{z_{n}=0}dz_{n}...\operatorname{Res}_{z_{1}=0}dz_{1}\mu_{1}S(\mu_{1})(z_{1}+\frac{1}{z_{1}})\mu_{n}S(\mu_{n})(z_{n}+\frac{1}{z_{n}})$
$\displaystyle\times\sum_{\begin{subarray}{c}\sigma\in S_{n}\\\
\sigma=n\text{-cycle}\end{subarray}}\prod_{i=1}^{n}\frac{1}{z_{i}e^{\mu_{i}/2}-z_{\sigma(i)}e^{-\mu_{\sigma(i)}/2}}$
$\displaystyle=$
$\displaystyle[\mu_{1}^{b_{1}+1}...\mu_{n}^{b_{n}+1}]\mu_{1}S(\mu_{1})\mu_{n}S(\mu_{n})\sum_{(\epsilon_{2},...,\epsilon_{n-1})\in\\{+1,-1\\}^{n-1}}\prod_{i=2}^{n-1}(-1)^{\epsilon_{i}}e^{\epsilon_{i}\hbar\mu_{i}/2}$
$\displaystyle=$
$\displaystyle[\mu_{1}^{b_{1}+1}...\mu_{n}^{b_{n}+1}]\prod_{i=1}^{n}(e^{\hbar\mu_{i}/2}-e^{-\hbar\mu_{i}/2})$
$\displaystyle=$
$\displaystyle\prod_{i=1}^{n}\frac{1}{2^{2b_{i}}(2b_{i}+1)!},$

where $\sum_{i}b_{i}=2g$. This result obviously coincides with [Pan00, Theorem
2] as claimed before.

##  5\. Outlook

This article presents new ideas on how the universal $x-y$ duality of the
theory of TR can be applied to a simple class of spectral curves of the form
$e^{x}=F(e^{y})$ (or $e^{x}=F(y)e^{ay}$) with exponential variables. These
types of curves find applications in topological string theory, where the
spectral curve corresponds to the mirror of the toric Calabi-Yau 3-fold, and
the generated correlators are the $B$-model correlators. The main idea was to
define the (a priori trivial family)
$\omega_{g,n}^{\vee}:=\delta_{n,1}\frac{B_{2g}(1/2)}{(2g)!}\partial_{y}^{2g}\omega^{\vee}_{0,1}$.
The same coefficients appear in the asymptotic expansion of the quantum
dilogarithm, providing a primary example. Further examples discussed in this
article include the framed topological vertex and stationary Gromov-Witten
invariants on $\mathbb{P}^{1}$. Additionally, in the context of polynomial
spectral curves $P(x,y)=0$, we provide further examples of the $x-y$ duality
formula in Section 2.5, relating it structurally to the determinantal formula
in the case where $y$ is unramified.

From here, several further directions can be pursued:

  * •

We have seen in Sec. 2.5 that the $x-y$ duality formula can be brought into
the same form as the determinantal formula if $y$ is unramified via the Cauchy
determinant. However, bringing the general $x-y$ formula into such a form,
where the sum over all graphs $\mathcal{G}_{n}$ is turned into a sum over
$n$-cycle permutations, is absolutely nontrivial but might be possible. The
advantage is that the sum over graphs $\mathcal{G}_{n}$ includes many more
terms since the number of these graphs grows much faster than the number of
permutations. Furthermore, having such an $x-y$ duality formula will provide a
new way to express the kernel $K(x_{i},x_{j})$ through the family
$\tilde{\omega}_{g,n}^{\vee}$ rather than through the family
$\tilde{\omega}_{g,n}$.

  * •

The $\hbar$ series of the family $\tilde{\omega}_{g,n}$ is factorially
growing. The determinantal formula can be used to compute the Borel transform.
It is worth investigating whether the $x-y$ duality formula can also be used
to derive the Borel transform or make some other asymptotic predictions.

  * •

The two wave functions, $\tilde{\Psi}(x)$ and $\tilde{\Psi}^{\vee}(y)$,
related by $x-y$ duality, are heuristically Fourier/Laplace transformations of
each other. The $x-y$ duality between the two families, $\omega_{g,n}$ and
$\omega_{g,n}^{\vee}$, should recover this Fourier/Laplace transformation at
each order in a formal $\hbar$ expansion. Making this observation more
rigorous will interplay with the resurgence since the Fourier/Laplace
transformation is in principle defined for a non-vanishing parameter $\hbar$.

  * •

Extending the $x-y$ duality formula to general spectral curves with
exponential variables of the form $P(e^{x},e^{y})=0$, especially with higher
genus.

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