restatethis[1]

######  Theorem LABEL:#1.

# Seiberg-Witten Theory and Topological Recursion

Wee Chaimanowong  School of Mathematics and Statistics, University of
Melbourne, Royal Parade, Parkville, Victoria 3010, Australia
<EMAIL_ADDRESS>
(Date: August 2020)

###### Abstract.

Kontsevich-Soibelman (2017) reformulated Eynard-Orantin topological recursion
(2007) in terms of Airy structure which provides some geometrical insights
into the relationship between the moduli space of curves and topological
recursion. In this work, we investigate the analytical approach to this
relationship using the Seiberg-Witten family of curves as the main example. In
particular, we are going to show that the formula computing the Hitchin
systems’ Special Kahler’s prepotential from the genus zero part of topological
recursion as obtained by Baraglia-Huang (2017) can be generalized for a more
general family of curves embedded inside a foliated symplectic surface,
including the Seiberg-Witten family. Consequently, we obtain a similar formula
relating the Seiberg-Witten prepotential to the genus zero part of topological
recursion on a Seiberg-Witten curve.

###### Contents

  1. 1 Introduction
    1. 1.1 Seiberg-Witten Family of Curves
  2. 2 Airy Structures and Deformation of Curves
    1. 2.1 Deformation of Curves in Foliated Symplectic Surfaces
    2. 2.2 Airy Structures
    3. 2.3 The Residue Constraints Airy Structure
    4. 2.4 The Embedding of Discs and The Analytic Residue Constraints
    5. 2.5 From Local To Global
      1. 2.6 From Airy Structures to Topological Recursion
        1. 2.7 Topological Recursion and Prepotentials
          1. 3 Seiberg-Witten Prepotential and Topological Recursion
            1. 3.1 Embedding of Seiberg-Witten Family of Curves into a Foliated Symplectic Surface
              1. 3.2 From the Seiberg-Witten Family of Curves to Airy Structures
                1. 3.3 Seiberg-Witten Prepotential and Topological Recursion

##  1\. Introduction

Since its introduction in 1994, _Seiberg-Witten theory_ [30] has been an
active research topic in both the physics and mathematics communities. From
the physics point of view, it is an exact low-energy effective field theory of
an $\mathcal{N}=2$ supersymmetric $SU(N+1)$ gauge theory. From the
mathematical point of view, all the physical information can be encoded into
an $N$-dimensional family $\mathcal{B}$ of genus $N$ connected compact Riemann
surfaces $\Sigma$ known as _Seiberg-Witten curves_. Each Seiberg-Witten curve
is equipped with a residueless meromorphic differential form $dS_{SW}$ called
the _Seiberg-Witten differential_. The homology group
$H_{1}(\Sigma,\mathbb{C})$ is naturally equipped with a symplectic structure,
given by the oriented intersection product. By choosing symplectic basis
cycles $A$ and $B$ for the homology group, the corresponding $A$ and
$B$-periods of $dS_{SW}$ recover the _electric_ $(a^{1},\cdots,a^{N})$ and
_magnetic_ $(b_{1},\cdots,b_{N})$ charges of the theory respectively.

The supersymmetric origin of Seiberg-Witten theory enforces that the effective
action of the theory is completely determined by a holomorphic function
$\mathfrak{F}^{SW}=\mathfrak{F}^{SW}(\\{a^{i=1,\cdots,N}\\},\Lambda)$ known as
the _Seiberg-Witten prepotential_. The parameter $\Lambda$ is the energy scale
of the theory where gauge couplings become strong. Therefore, technically we
should have written $\mathcal{B}$ as $\mathcal{B}(\Lambda)$ to emphasize the
dependency of the family on the energy scale $\Lambda$. In the original work
[30], the electric-magnetic duality was employed to constrain the form of the
prepotential. A more mathematical approach to the computation of the
prepotential has been studied via Nekrasov’s partition function a few years
later by various authors [27, 28, 26, 25].

Nekrasov’s partition function
$Z(\\{a^{i=1,\cdots,N}\\},\epsilon_{1},\epsilon_{2},\Lambda)$ is the path-
integral over the non-perturbative instanton contributions of a certain
$(\epsilon_{1},\epsilon_{2})$-deformed $\mathcal{N}=2$ supersymmetric
$SU(N+1)$ gauge theory. It can be written as an integral over the moduli space
of instantons which can be evaluated via Atiyah-Bott equivariant localization
formula [2]. In the end, we find that
$Z(\\{a^{i}\\},\epsilon_{1},\epsilon_{2},\Lambda)$ is a sum of terms indexed
by $N$-tuples of Young’s diagrams over all possible $N$-tuples of Young’s
diagrams with boxes of width $-\epsilon_{1}$ and height $+\epsilon_{2}$. In
the limit $\epsilon_{1},\epsilon_{2}\rightarrow 0$, Young diagrams become
graphs of continuous functions $f$ and there exists the function $f_{0}$ in
which the corresponding term in the summation:

$\mathfrak{F}_{0}(\\{a^{i}\\},\Lambda)=\lim_{\epsilon_{1},\epsilon_{2}\rightarrow
0}\epsilon_{1}\epsilon_{2}\log
Z(\\{a^{i}\\},\epsilon_{1},\epsilon_{2},\Lambda)$

will dominate the contribution. It was proven by Nekrasov-Okounkov [28, 27]
that $\mathfrak{F}_{0}$ is the Seiberg-Witten prepotential
$\mathfrak{F}_{SW}$. The same result was independently proven using techniques
in algebraic geometry by Nakajima-Yoshioka [25, 26].

Mathematically, the Seiberg-Witten prepotential $\mathfrak{F}_{SW}$ contains
information about the deformation of Seiberg-Witten curves $\Sigma$ in the
family $\mathcal{B}$. A Seiberg-Witten curve $\Sigma$ is represented by a
point $[\Sigma]\in\mathcal{B}$. Let us define the _regular locus_
$\mathcal{B}^{reg}$ of $\mathcal{B}$ to be the dense subset of points
$[\Sigma]\in\mathcal{B}$ such that $\Sigma$ is non-singular. The set
$\\{a^{1}:=a^{1}(\Sigma),\cdots,a^{N}:=a^{N}(\Sigma)\\}$ where
$a^{i}(\Sigma):=\oint_{A_{i}\subset\Sigma}dS_{SW}$ are $A$-periods of the
Seiberg-Witten differential on $\Sigma$ is a system of local holomorphic
coordinates on some neighbourhoods
$\mathcal{B}_{\Sigma}\subseteq\mathcal{B}^{reg}$ of an arbitrary point
$[\Sigma]\in\mathcal{B}^{reg}$ giving $\mathcal{B}^{reg}$ the structure of an
$N$-dimensional complex manifold. Let us fix a point
$[\Sigma_{0}]\in\mathcal{B}^{reg}$. The prepotential
$\mathfrak{F}^{SW}_{\Sigma_{0}}:\mathcal{B}_{\Sigma_{0}}\rightarrow\mathbb{C}$
is a holomorphic function locally defined on some open neighbourhood
$\mathcal{B}_{\Sigma_{0}}\subseteq\mathcal{B}^{reg}$ of $[\Sigma_{0}]$ which
relates the $B$-periods on $\Sigma$ of the Seiberg-Witten differential,
$b_{i}(\Sigma):=\oint_{B_{i}\subset\Sigma}dS_{SW}$, to its $A$-periods as
follows:

$b_{i}=b_{i}(\Sigma)=\frac{\partial\mathfrak{F}^{SW}_{\Sigma_{0}}(\\{a^{k}\\},\Lambda)}{\partial
a^{i}},\qquad i=1,\cdots,N,\qquad[\Sigma]\in\mathcal{B}_{\Sigma_{0}}.$

Let us explore the prepotential from another important viewpoint. Similarly to
how we equipped the homology group $H_{1}(\Sigma,\mathbb{C})$ with a
symplectic structure, the cohomology group $H^{1}(\Sigma,\mathbb{C})\cong
H_{1}(\Sigma,\mathbb{C})^{*}$ may be equipped with a symplectic structure
given by the oriented intersection of cohomological classes via Poincaré
duality. We can say that the existence of prepotential
$\mathfrak{F}^{SW}_{\Sigma_{0}}$ is also equivalent to the fact that

$\mathcal{L}_{\Sigma_{0}}:=\left\\{\left(a^{1}(\Sigma),\cdots,a^{N}(\Sigma),b_{1}(\Sigma),\cdots,b_{N}(\Sigma)\right)\
|\ [\Sigma]\in\mathcal{B}_{\Sigma_{0}}\right\\}\subset\mathbb{C}^{2g}\cong
H^{1}(\Sigma_{0},\mathbb{C})$

is a Lagrangian submanifold. From this point of view, the prepotential is the
generating function of $\mathcal{L}_{\Sigma_{0}}$. Here we have identified
$H^{1}(\Sigma,\mathbb{C})$ for every $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$
with $\mathbb{C}^{2g}$. Consider the symplectic vector bundle
$\mathcal{H}\rightarrow\mathcal{B}_{\Sigma_{0}}$ equipped with the _Gauss-
Manin_ connection $\nabla_{GM}$ and with the fiber over
$[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ given by
$\mathcal{H}_{\Sigma}:=H^{1}(\Sigma,\mathbb{C})$. Then
$\mathcal{L}_{\Sigma_{0}}\subset\mathcal{H}_{\Sigma_{0}}=H^{1}(\Sigma_{0},\mathbb{C})$
is the image of the embedding of $\mathcal{B}_{\Sigma_{0}}$ into the fiber
$\mathcal{H}_{\Sigma_{0}}$ given by sending the cohomology class of $dS_{SW}$
over $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ to $[\Sigma_{0}]$ via the parallel
transport.

In fact, the existence of the Seiberg-Witten prepotential
$\mathfrak{F}^{SW}_{\Sigma_{0}}$ is a consequence of an important property of
the Seiberg-Witten differential, namely, any infinitesimal variations of
$dS_{SW}$ in the family $\mathcal{B}$ using a certain choice of connection is
a global holomorphic form on $\Sigma$. Based on this key property, we may
consider a generalized notion of ‘Seiberg-Witten theory’ as follows. Let
$\mathcal{B}$ be an arbitrary $g$ dimensional family of genus $g$ curves
$\Sigma$ and each $\Sigma$ is equipped with a residueless meromorphic
differential $dS$ such that

(1) $\frac{\partial dS}{\partial u^{i}}=\text{holomorphic form on
$\Sigma$},\qquad i=1,\cdots,g$

with respect to some choices of connections, where $\\{u^{1},\cdots,u^{g}\\}$
are local coordinates of $\mathcal{B}$. Let
$\mathcal{B}_{\Sigma_{0}}\subset\mathcal{B}^{reg}$ be some small open
neighbourhood of $[\Sigma_{0}]\in\mathcal{B}^{reg}$. The image of the
embedding

(2)
$[\Sigma]\mapsto(\mathbf{a}^{1}(\Sigma),\cdots,\mathbf{a}^{g}(\Sigma),\mathbf{b}_{1}(\Sigma),\cdots,\mathbf{b}_{g}(\Sigma))\in\mathcal{H}_{\Sigma_{0}}=H^{1}(\Sigma_{0},\mathbb{C}),$

given by sending the cohomology class of $dS$ over
$[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ to $[\Sigma_{0}]$ via parallel
transport, is a Lagrangian submanifold $\mathcal{L}_{\Sigma_{0}}$. Here,
$\mathbf{a}^{i}=\mathbf{a}^{i}(\Sigma),\mathbf{b}_{i}=\mathbf{b}_{i}(\Sigma)$
are $A,B$-periods of $dS$ and the prepotential
$\mathfrak{F}_{\Sigma_{0}}=\mathfrak{F}_{\Sigma_{0}}(\\{\mathbf{a}^{k}\\})$
exists on $\mathcal{B}_{\Sigma_{0}}$ as the generating function of
$\mathcal{L}_{\Sigma_{0}}$:
$\mathbf{b}_{i}=\frac{\partial\mathfrak{F}_{\Sigma_{0}}(\\{\mathbf{a}^{k}\\})}{\partial\mathbf{a}^{i}}$.
In the more general context of _Whitham hierarchy_ [19, 24], instead of just a
single energy scale parameter $\Lambda$, we can consider an infinite number of
_time_ variables $\\{T^{k=0,1,2,\cdots}\\}$ such that $\frac{\partial
dS}{\partial T^{k}}=d\Omega_{k}$ for some meromorphic differentials
$d\Omega_{k}$ on $\Sigma$. In this context, the main problem to consider is
the following: given a set of meromorphic differentials
$\\{d\Omega_{k=0,1,2,\cdots}\\}$ on $\Sigma$ in the family of curves
$\mathcal{B}(\\{T^{k=0,1,2,\cdots}\\})$, find (if possible) the meromorphic
differential $dS$ on $\Sigma$ satisfying (1) and $\frac{\partial dS}{\partial
T^{k}}=d\Omega_{k}$.

In this work, we aim to study some mathematical aspects of the Seiberg-Witten
family of curves and the prepotential from the point of view of _topological
recursion_. In the bigger picture, we will consider the complex analytic
moduli space $\mathcal{B}$ of smooth compact complex genus $g$ curves $\Sigma$
embedded inside a foliated symplectic surface $S$ tangent to the leaves of the
foliation $\mathcal{F}$ at finitely many points with tangency order $1$. Let
us call any curve $[\Sigma]\in\mathcal{B}$ a $\mathcal{F}$-_transveral curve_
and call points where $\Sigma$ is tangent to $\mathcal{F}$ the _ramification
points_ of $\Sigma$. We will examine the deformation of curves in
$\mathcal{B}$, using the Kontsevich-Soibelman approach to topological
recursion [21] with Seiberg-Witten family of curves as the main example.

A _spectral curve_ is a Riemann surface $\Sigma$ with some distinct points
$\\{r_{\alpha\in Ram}\\}$ called _ramification points_ together with some
additional data. Topological recursion is a recursive procedure which takes a
spectral curve as input and produces symmetric meromorphic multi-differentials
$\omega_{g,n}$ which are sections of $(\Omega^{1}_{\Sigma})^{\boxtimes n}$ on
$(\Sigma\setminus\\{r_{\alpha\in Ram}\\})^{n}$. For instance, $\omega_{g,1}$
is a meromorphic differential on $\Sigma$. Topological recursion was
introduced by Eynard-Orantin [12, 13] around 2007-2008 with a wide variety of
applications in areas ranging from mathematical physics to enumerative
geometry. For example, it has been shown [12, 13] by comparing the topological
recursion relations with a particular choice of the spectral curve to loop
equations that the corresponding output $\omega_{g,n}$ are given by the
correlation functions in Matrix models. Similarly, let the spectral curve be
given by an Airy curve with some additional data. It has been shown [10, 11,
9] that coefficients of the Laurent expansion of $\omega_{g,n}$ centered at
the only ramification point is related to the intersection number of divisor
classes of universal cotangent line bundles $\\{\psi_{1},\cdots,\psi_{n}\\}$
(known as $\psi$-classes) on $\bar{\mathcal{M}}_{g,n}$. The generating
function of the intersection numbers was conjectured and later proven to be
the tau function of KdV hierarchy by Witten and Kontsevich respectively [33,
20].

In 2017, Kontsevich-Soibelman [21] introduced the notion of _Airy structures_
which is given by a Lie algebra $\mathfrak{g}$ generated by a set of quadratic
polynomials $\\{H_{k\in\mathbb{I}}\\}$ of a certain form on a symplectic
vector space $W$ where the Lie bracket is naturally given by the Poisson
bracket on $W$. They showed that any Airy structure corresponds to a recursion
called _abstract topological recursion_. This reduces to the usual Eynard-
Orantin topological recursion when considering an example of Airy structure
$\\{H_{i\in\mathbb{I}}\\}:=\\{(H_{Airy})_{k=1,2,3,\cdots}\\}$ and
$W:=W^{Ram}_{Airy}$ arises from the so-called _residue constraints_. The zero
set of the generators of $\mathfrak{g}$ is a Lagrangian submanifold $L\subset
W$ by the fact that $\mathfrak{g}$ is closed under the Lie bracket. We call
$L$ a _quadratic Lagrangian submanifold_ since it is defined by a set of
quadratic polynomials. In particular, we let
$L^{Ram}_{Airy}:=\\{(H_{Airy})_{k=1,2,3,\cdots}=0\\}\subset W^{Ram}_{Airy}$.

Let us return to the discussion of the particular Airy structure
$\\{(H_{Airy})_{k=1,2,3,\cdots}\\}$ corresponding to topological recursion.
Let $Discs^{Ram}$ be a space of $|Ram|$-disjoint union of discs
$\mathbb{D}_{\alpha},\alpha\in Ram$ embedded inside the foliated symplectic
surface $S$. Each $\mathbb{D}_{\alpha}$ are deformed Airy curves which can be
parameterized by

(3)
$z_{\alpha}\mapsto\left(x_{\alpha}=z_{\alpha}^{2}+a,y_{\alpha}=\sum_{k=1}^{\infty}b_{k\alpha}z_{\alpha}^{k}\right)$

on the local coordinates chart, $(U_{\alpha},x_{\alpha},y_{\alpha})$ of $S$
with the ramification point
$r_{\alpha}=(x_{\alpha},y_{\alpha})=(a,b_{0\alpha})\in U_{\alpha}$. Given an
annulus on a disc $\mathbb{D}_{\alpha}$ centered at $r_{\alpha}$, the set of
all Laurent series which converge on the annulus naturally has a structure of
an infinite dimensional symplectic vector space. The infinite-dimensional
symplectic vector space $W^{Ram}_{Airy}$ is given by the direct sum these
vector spaces over $\alpha\in Ram$. The space $Discs^{Ram}$ can be thought of
as the local version of the moduli space of curves $\mathcal{B}$. Introduce a
trivial vector bundle $W^{Ram}\rightarrow Discs^{Ram}$ with fiber over any
point given by $W^{Ram}_{Airy}$ equipped with a certain connection $\nabla$.
Let us fix a reference point $t_{0}\in Discs^{Ram}$ and let
$Discs^{Ram}_{t_{0}}\ni t_{0}$ be a certain subset of $Discs^{Ram}$ analogous
to $\mathcal{B}_{\Sigma_{0}}$. The important observation is that there exists
a local section $\theta$ of $W^{Ram}$ on $Discs^{Ram}_{t_{0}}$ such that the
parallel transport of $\theta_{t}\in W^{Ram}_{t}$ from any point $t\in
Discs^{Ram}_{t_{0}}$ to $t_{0}$ gives an embedding of $Discs^{Ram}_{t_{0}}$
into the fiber $W^{Ram}_{t_{0}}$ with the image contained in the quadratic
Lagrangian submanifold $L^{Ram}_{Airy}$. The section $\theta$ can be thought
of like a local version of $dS$ or a Seiberg-Witten differential.

At first glance, this on-going discussion seems to have little to do with the
deformation of curves or Seiberg-Witten theory. But we note that an
infinitesimal deformation of a smooth curve $\Sigma$ will produce a global
holomorphic section of the normal bundle on $\Sigma$. If $\Sigma$ is embedded
inside a symplectic surface, then by applying the symplectic form to the
global section of the normal bundle, we produce a global holomorphic form on
$\Sigma$ which naturally arises from the infinitesimal deformation of
$\Sigma$. In particular, let $\mathcal{B}$ be the moduli space of
$\mathcal{F}$-transversal genus $g$ curves, then we have the map
$\boldsymbol{\phi}_{\Sigma}:T_{[\Sigma]}\mathcal{B}\rightarrow\Gamma(\Sigma,\Omega^{1}_{\Sigma})\hookrightarrow\mathcal{H}_{\Sigma}$
for each $[\Sigma]\in\mathcal{B}$. It is possible to check that
$d_{\nabla_{GM}}\boldsymbol{\phi}=0$ where $d_{\nabla_{GM}}$ denotes the
exterior covariant derivative using the Gauss-Manin connection. Let us fix a
reference point $[\Sigma_{0}]\in\mathcal{B}$. By the Pointcaré Lemma, there
exists an open neighbourhood $\mathcal{B}_{\Sigma_{0}}$ of $[\Sigma_{0}]$ with
a local section $\boldsymbol{\theta}$ of $\mathcal{H}$ on
$\mathcal{B}_{\Sigma_{0}}$ such that
$d_{\nabla_{GM}}\boldsymbol{\theta}=\boldsymbol{\phi}$. Then the parallel
transport of $\boldsymbol{\theta}_{\Sigma}\in\mathcal{H}_{\Sigma}$ from any
point $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ to $[\Sigma_{0}]$ defines an
embedding of $\mathcal{B}_{\Sigma_{0}}$ into $\mathcal{H}_{\Sigma_{0}}$ as a
Lagrangian submanifold $\mathcal{L}_{\Sigma_{0}}$. This is exactly the same
embedding given in (2) with

$\mathbf{a}^{i}:=\mathbf{a}^{i}(\Sigma)=\oint_{A_{i}}\boldsymbol{\theta}_{\Sigma},\qquad\mathbf{b}_{i}:=\mathbf{b}_{i}(\Sigma)=\oint_{B_{i}}\boldsymbol{\theta}_{\Sigma}=\frac{\partial\mathfrak{F}_{\Sigma_{0}}(\\{\mathbf{a}^{k}\\})}{\partial\mathbf{a}^{i}},\qquad
i=1,\cdots,g,$

where the holomorphic function
$\mathfrak{F}_{\Sigma_{0}}=\mathfrak{F}_{\Sigma_{0}}(\\{\mathbf{a}^{k}\\})$
defined on $\mathcal{B}_{\Sigma_{0}}$ is the generating function of
$\mathcal{L}_{\Sigma_{0}}$ called the prepotential. For each
$[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$, we may consider a lift of the
cohomology class $\boldsymbol{\theta}_{\Sigma}$ to a differential form on
$\Sigma$. Such a lift is not unique since an addition by $df$ for any
meromorphic function $f$ on $\Sigma$ will not change its cohomology class.
However, the key idea of Kontsevich-Soibelman is there exists exactly one lift
$\theta_{\Sigma}$ of $\boldsymbol{\theta}_{\Sigma}$ such that the direct sum
of Laurent series expansions of $\theta_{\Sigma}$ at each ramification point
coincides with $\theta_{t}\in W^{Ram}_{t}$. Here, $t\in Discs^{Ram}_{t_{0}}$
is given in term of $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ by the disjoint
union of some open neighbourhood
$\mathbb{D}_{\alpha}(\Sigma)\subset\Sigma\subset S$ which we shall denote by
$t=\gamma(\Sigma)$. The image of $\theta_{t}$ under the parallel transport
from $t$ to $t_{0}$ is contained inside $L^{Ram}_{Airy}\subset
W^{Ram}_{t_{0}}$.

In the end, we get a family of differential forms $\theta_{\Sigma}$ which
relates the Lagrangian submanifold
$\mathcal{L}_{\Sigma_{0}}\subset\mathcal{H}_{\Sigma_{0}}$ with the quadratic
Lagrangian submanifold $L^{Ram}_{Airy}\subset W^{Ram}_{t_{0}}$ and each
$\theta_{\Sigma}$ can be thought of as a generalized Seiberg-Witten
differential $dS$ satisfying (1). We can understand the relationship between
$\mathcal{L}_{\Sigma_{0}}$ and $L^{Ram}_{Airy}$ more geometrically. The map
$[\Sigma]\mapsto\gamma(\Sigma)$ embeds $\mathcal{B}_{\Sigma_{0}}$ as a subset
of $Discs^{Ram}_{t_{0}}$ and $\mathcal{L}_{\Sigma_{0}}$ is a cohomological
image in $\mathcal{H}_{\Sigma_{0}}$ of the cross-section of
$L^{Ram}_{Airy}\subset W^{Ram}_{Airy}$ over $\mathcal{B}_{\Sigma_{0}}\subset
Discs^{Ram}_{t_{0}}$. The quadratic Lagrangian submanifold $L^{Ram}_{Airy}$
may be regarded as _universal_ in the following sense. Any two Langrangian
submanifolds
$\mathcal{L}_{\Sigma_{0}},\mathcal{L}^{\prime}_{\Sigma^{\prime}_{0}}$ arise
from two different moduli spaces $\mathcal{B},\mathcal{B}^{\prime}$ of
$\mathcal{F}$-transversal curves with the same number of genus and
ramification points will correspond to two different cross-sections of
$L^{Ram}_{Airy}$.

From the idea described above, we obtain the main result as follows:

prepotentialvsomega0ntheorem Let $\mathcal{B}$ be the moduli space of
$\mathcal{F}$-transversal genus $g$ curves in a foliated symplectic surface
$S$. Then for some small open neighbourhood
$\mathcal{B}_{\Sigma_{0}}\subset\mathcal{B}$ of the reference point
$[\Sigma_{0}]\in\mathcal{B}$, the prepotential
$\mathfrak{F}_{\Sigma_{0}}=\mathfrak{F}_{\Sigma_{0}}(\mathbf{a}^{1},\cdots,\mathbf{a}^{g})$
can be expressed as

$\displaystyle\mathfrak{F}_{\Sigma_{0}}$
$\displaystyle(\mathbf{a}^{1},\cdots,\mathbf{a}^{g})=\mathfrak{F}_{\Sigma_{0}}(\mathbf{a}^{1}_{0},\cdots,\mathbf{a}^{g}_{0})$
$\displaystyle+\sum_{i=1}^{g}(\mathbf{a}^{i}-\mathbf{a}^{i}_{0})\frac{\partial\mathfrak{F}_{\Sigma_{0}}}{\partial\mathbf{a}^{i}}(\mathbf{a}^{1}_{0},\cdots,\mathbf{a}^{g}_{0})+\frac{1}{2}\sum_{i,j=1}^{g}(\mathbf{a}^{i}-\mathbf{a}^{i}_{0})(\mathbf{a}^{j}-\mathbf{a}^{j}_{0})\frac{\partial^{2}\mathfrak{F}_{\Sigma_{0}}}{\partial\mathbf{a}^{i}\partial\mathbf{a}^{j}}(\mathbf{a}^{1}_{0},\cdots,\mathbf{a}^{g}_{0})$
(4) $\displaystyle+\sum_{n=3}^{\infty}\frac{1}{n!}\left(\frac{1}{2\pi
i}\right)^{n-1}\sum_{i_{1},\cdots,i_{n}=1}^{g}\left(\oint_{p_{1}\in
B_{i_{1}}}\cdots\oint_{p_{n}\in
B_{i_{n}}}\omega_{0,n}(p_{1},\cdots,p_{n})\right)(\mathbf{a}^{i_{1}}-\mathbf{a}^{i_{1}}_{0})\cdots(\mathbf{a}^{i_{n}}-\mathbf{a}^{i_{n}}_{0})$

for all
$(\mathbf{a}^{1},\cdots,\mathbf{a}^{g})=[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$.
Where $\mathbf{a}_{0}^{k}:=\mathbf{a}^{k}(\Sigma_{0})$ and the multi-
differentials
$\left\\{\omega_{g,n}\in\Gamma\left((\Sigma_{0}\setminus\cup_{\alpha\in
Ram}r_{\alpha})^{n},(\Omega_{\Sigma_{0}}^{1})^{\boxtimes n}\right)\right\\}$
are produced from topological recursion on $\Sigma_{0}$.

This theorem is a joint work with Paul Norbury, Michael Swaddle and Mehdi
Tavakol which has been proved previously in [6], however in this paper we are
going to approach it from an analytical point of view as oppose to the formal
approach taken by [6]. The interesting point about Theorem 2.7.3 is that the
prepotential $\mathfrak{F}_{\Sigma_{0}}$ contains information about the
deformation of curves $\Sigma$ within the moduli space $\mathcal{B}$ whereas
$\omega_{0,n}$ is produced from topological recursion, which is done on a
single curve $\Sigma_{0}$ without any knowledge on the moduli space it belongs
to.

We are going to revisit in detail all the discussions about Airy structures,
deformations of curves, and topological recursion later in Section 2 along
with the proof of Theorem 2.7.3. To apply this result to the Seiberg-Witten
family of curves we need to find a foliated symplectic surface $S$ such that
each Seiberg-Witten curve can be embedded as a $\mathcal{F}$-transversal
curve. It turns out that such a foliated symplectic surface exists as we are
going to show in Section 3, hence there is a nice relationship between the
Seiberg-Witten prepotential
$\mathfrak{F}^{SW}_{\Sigma_{0}}=\mathfrak{F}^{SW}_{\Sigma_{0}}(\\{a^{k}\\},\Lambda)$
and the genus zero part of topological recursion on Seiberg-Witten curve with
$\omega_{0,1}=dS_{SW}$.

The formula (4) can also be obtained via the Rauch variational formula [14,
12, 3]. For example, Baraglia-Huang [3] studied the moduli space $\mathcal{B}$
of curves $\Sigma$ arising from the _Hitchin systems_ , where each $\Sigma$ is
embedded inside the total space of the cotangent bundle $T^{*}\mathcal{C}$ of
some compact Riemann surface $\mathcal{C}$. They showed that the Rauch
variational formula holds in this case and that the formula (4) follows.
Theorem 2.7.3 generalizes the result by Baraglia-Huang to the case where $S$
is an arbitrary foliated symplectic surface instead of $T^{*}\mathcal{C}$.
Moreover, even in the $S=T^{*}\mathcal{C}$ case, our approach of proving the
formula via the idea of Kontsevich-Soibelman should shed light on some
interesting underlying geometries between the Lagrangian submanifold in
$\mathcal{H}_{\Sigma_{0}}=H^{1}(\Sigma_{0},\mathbb{C})$ given by the
cohomology of the tautological 1-form and the quadratic Lagrangian
submanifolds with $\omega_{0,n}$ as the generating function.

###  1.1. Seiberg-Witten Family of Curves

In this section, we give a quick summary of the needed mathematical details of
the Seiberg-Witten family of curves. More comprehensive reviews of this
material can be found abundantly in the literature such as [30, 19, 24, 26].
The information of $SU(N+1)$ Seiberg-Witten theory can be encoded into a
family $\mathcal{B}$ of genus $g=N$ _Seiberg-Witten curves_ $\Sigma$ given by

(5)
$\Lambda^{g+1}\left(w+\frac{1}{w}\right)=z^{g+1}+u^{g}z^{g-1}+...+u^{1}=:P(z;u)$

where $\\{u^{1},...,u^{g}\\}$ are the coordinates of $\mathcal{B}$.
Occasionally, we will write the curve $\Sigma$ as $\Sigma(u)$ to emphasize its
dependence on $u=(u^{1},\cdots,u^{g})$. We may regard (5) as a plane curve
equation, in that case, we have $w,z\in\mathbb{C}$ and
$\Sigma(u)\subset\mathbb{C}^{2}$ as a non-compact curve. The compactification
of $\Sigma(u)$ is obtained by adding two points
$\infty_{+}=(w,z)=(\infty,\infty)$ and $\infty_{-}=(w,z)=(0,\infty)$. In this
way, we regard $\Sigma(u)$ defined in (5) as a compact smooth curve. The map
$\pi:\Sigma(u)\rightarrow\mathbb{P}^{1},(w,z)\mapsto z$ is a double cover that
shows that $\Sigma(u)$ is a hyperelliptic curve with the hyperelliptic
involution given by $(w,z)\leftrightarrow(1/w,z)$. An alternative way to
express (5) is

$y^{2}=P^{2}(z;u)-4\Lambda^{2g+2}\qquad\text{where}\qquad
y:=\Lambda^{g+1}\left(w-\frac{1}{w}\right)$

which makes it even more obvious that $\Sigma(u)$ is a hyperelliptic curve
with the hyperelliptic involution $(y,z)\leftrightarrow(-y,z)$. The $2g+2$
ramification points of $\pi$ are given by $(y,z)=(0,z^{i\pm})$ where
$z^{i\pm},i=1,\cdots,g+1$ are the roots of the polynomial $P(z;u)\mp
2\Lambda^{g+1}=0$. We can check that the curve $\Sigma(u)$ is smooth unless
$P^{\prime}(z^{i+};u)=0$ or $P^{\prime}(z^{i-};u)=0$. In other words, we may
define the family of smooth Seiberg-Witten curves to be

$\mathcal{B}^{reg}:=\left\\{u=(u^{1},\cdots,u^{g})\in\mathcal{B}\ |\
\text{$P^{2}(z;u)-4\Lambda^{2g+2}=0$ has $2g+2$ distinct
roots}\right\\}\subset\mathcal{B}.$

We note that $\mathcal{B}^{reg}$ is dense inside $\mathcal{B}$ and since we
will mainly be interested in smooth Seiberg-Witten curves in our works, we
will relabel $\mathcal{B}^{reg}$ as $\mathcal{B}$ for simplicity.

We define the _Seiberg-Witten differential_ to be the meromorphic differential
form

$dS_{SW}:=z\frac{dw}{w}=\frac{zP^{\prime}(z;u)dz}{y}$

defined on each Seiberg-Witten curve $\Sigma(u)$ with residueless poles of
order $2$ at $\infty_{\pm}\in\Sigma(u)$. Occasionally we will write
$dS_{SW}(\Sigma(u))$ if we want to emphasize the specific $dS_{SW}$ defined on
$\Sigma(u)$. Let us check that the variations of $dS_{SW}(\Sigma(u))$ within
the family $\mathcal{B}$ produce global homolomorphic forms on $\Sigma(u)$,
hence $dS_{SW}$ satisfies (1). As we have mentioned, a choice of connection
must be specified in order for the derivatives $\frac{\partial}{\partial
u^{i}}$ to makes sense, in this case let us keep $w=const$, we have

(6) $\frac{\partial dS}{\partial u^{i}}\Big{|}_{w=const}=\frac{\partial
z}{\partial
u^{i}}\Big{|}_{w=const}\frac{dw}{w}=-\frac{z^{i-1}}{P^{\prime}(z;u)}\frac{dw}{w}=\frac{z^{i-1}dz}{y},\qquad
i=1,\cdots,g.$

which are holomorphic on $\Sigma(u)$. We define

$a^{i}:=\oint_{A_{i}}dS_{SW},\qquad b_{i}:=\oint_{B_{i}}dS_{SW},\qquad
i=1,\cdots,g.$

We note that $\\{a^{1},\cdots,a^{g}\\}$ gives another set of holomorphic
coordinates for $\mathcal{B}$. We can see this by noting that
$\left(\frac{\partial a^{i}}{\partial u^{j}}\right)$ is invertible because if
$\sum_{i=1}^{g}c_{i}\frac{\partial a^{i}}{\partial
u^{j}}=\oint_{A_{k}}\sum_{i=1}^{g}\frac{c_{i}z^{i-1}dz}{y}=0$ for all
$j,k=1,\cdots,g$ and for some constants $c_{i}\in\mathbb{C}$ then
$\sum_{i=1}^{g}\frac{c_{i}z^{i-1}dz}{y}=0$ because it is a holomorphic form
with zero $A$-periods. It follows that $\frac{\partial dS_{SW}}{\partial
a^{i}}=\omega_{i}$ are normalized holomorphic differentials:

$\oint_{A_{j}}\omega_{i}=\delta_{j}^{i},\qquad\oint_{B_{j}}\omega_{i}=\frac{\partial
b_{i}}{\partial a^{j}}=:\tau_{ij},$

where $b_{i}=b_{i}(a^{1},\cdots,a^{g})$ is treated as a holomorphic function
of $\\{a^{i=1,\cdots,g}\\}$. The standard argument involving Riemann bilinear
identity shows that $\tau_{ij}=\tau_{ji}$:

$0=\frac{1}{2\pi
i}\sum_{k=1}^{g}\left(\oint_{A_{k}}\omega_{i}\oint_{B_{k}}\omega_{j}-\oint_{B_{k}}\omega_{i}\oint_{A_{k}}\omega_{j}\right)=\frac{1}{2\pi
i}\left(\tau_{ji}-\tau_{ij}\right).$

In other words, $\frac{\partial b_{i}}{\partial a^{j}}=\frac{\partial
b_{j}}{\partial a^{i}}$, therefore, by Poincaré lemma given any
$[\Sigma_{0}]\in\mathcal{B}$ there exists an open neighbourhood
$\mathcal{B}_{\Sigma_{0}}\ni[\Sigma_{0}]$ with a holomorphic function
$\mathfrak{F}^{SW}_{\Sigma_{0}}=\mathfrak{F}^{SW}_{\Sigma_{0}}(\\{a^{k}\\},\Lambda)$
such that

$b_{i}=\frac{\partial\mathfrak{F}^{SW}_{\Sigma_{0}}(\\{a^{k}\\},\Lambda)}{\partial
a^{i}}.$

We call
$\mathfrak{F}^{SW}_{\Sigma_{0}}=\mathfrak{F}^{SW}_{\Sigma_{0}}(\\{a^{k}\\},\Lambda)$
the _Seiberg-Witten prepotential_. The expansion of
$\mathfrak{F}_{\Sigma_{0}}^{SW}$ can be computed via Nekrasov’s partition
function [28, 29] or Nakajima’s blow-up formula [26, 25] or by computing the
odd periods of $M_{A_{g}}\otimes QH^{*}(\mathbb{P}^{1})$ Frobenius manifold
[8].

## Acknowledgements

I would like to express my deepest gratitude to Paul Norbury and Mehdi Tavakol
for supervising me throughout this project. I also wish to thank Michael
Swaddle and Gaëtan Borot for many useful discussions. Part of this work was
done during my visit to the Max Planck Insitute for Mathematics, Bonn, I would
like to thank them for their hospitality and for partially funded this
project. This work is primarily funded by the Melbourne Research Scholarship.

##  2\. Airy Structures and Deformation of Curves

In this section, we summarize the work of Kontsevich-Soibelman [21] on Airy
structures, topological recursion, and the moduli space of curves in a
foliated symplectic surface. In particular, there are 3 important ideas from
[21] which we are going to explore:

  1. (1)

The quantization of Airy structures leads to a generalized version of
topological recursion. The usual Eynard-Orantin topological recursion
corresponds to an example of Airy structure given by the so-called _residue
constraints_.

  2. (2)

A deformation of an Airy curve $y^{2}=x$, i.e. a _disc_ , inside a foliated
symplectic surface produces a differential form satisfying the residue
constraints.

  3. (3)

The cohomology class, arising from the deformation of a curve in a foliated
symplectic surface, can be lifted to the differential form on the curve, which
locally agrees near each ramification point with the differential form
produced from the deformation of a disc.

Connecting these ideas leads to a nice relationship between the prepotential
and the genus zero part of topological recursion which is the content of
Theorem 2.7.3, the main result of this section. We are going to start in
Section 2.1 by reviewing some basic deformation theory of curves $\Sigma$ in a
foliated symplectic surface $(S,\Omega_{S},\mathcal{F})$, following [21,
Section 4]. This eventually leads to an embedding of a Lagrangian submanifold
in $H^{1}(\Sigma,\mathbb{C})$ and establishes the notion of _prepotential_ in
this general context. Section 2.2 reviews definitions and basic results
concerning Airy structures, including the corresponding quadratic Lagrangian
submanifold, quantization of Airy structures, and abstract topological
recursion, all of which can be found in Section 1-3 of [21]. An important
example of an Airy structure arises from residue constraints as introduced in
[21, Section 3.3] will be discussed in Section 2.3.

We have been following the material in [21] closely so far, and we will be for
the rest of this section. However, for the remaining sections, with an
exception of Section 2.6, we are going to approach the subject from an
analytical perspective rather than a more algebraic treatment of [21]. In
particular, we will replace all formal power series with power series
converges on some annulus. We believe that this should be a more natural way
to arrive at our main result involving the prepotential which is a holomorphic
function.

We study the analytic version of residue constraints Airy structure in Section
2.4. In particular, we look at the deformation of analytic discs and show that
an embedding of a neighbourhood in the space of all discs gives an analytic
solution to the residue constraints, as explained in [21, Section 6]. The
connection between the ‘global’ deformations, as studied in Section 2.1, with
the ‘local’ deformations, studied in terms of discs in Section 2.4, will be
discussed in Section 2.5 analytically, with a summary given by Proposition
2.5.38 [21, Proposition 7.1.2]. In Section 2.6 we review the well-known result
from [21, Section 3] how Eynard-Orantin topological recursion arises from the
residue constraints Airy structure. At last, we prove our main result, Theorem
2.7.3, in Section 2.7. We argue that when specializing to an example of a
family of curves arises from the _Hitchin systems_ [18, 3], our result agrees
with the result previously obtained by Baraglia-Huang [3, Theorem 7.4] using a
different technique. An application of Theorem 2.7.3 to the Seiberg-Witten
family of curves will also be briefly mentioned before a more extensive
discussion in Section 3.

###  2.1. Deformation of Curves in Foliated Symplectic Surfaces

####  2.1.1. Foliated symplectic surfaces

Let us start by fixing some frequently used definitions which should be
standard in complex geometry:

######  Definition 2.1.1.

A $2n$-dimensional _holomorphic sympletic manifold_ $(X,\Omega_{X})$ is a
$2n$-dimensional complex manifold $X$ equipped with a holomorphic closed non-
degenerates $2$-form $\Omega_{X}$, i.e. at any point $x\in X$,
$\Omega_{X,x}:T_{x}X\times T_{x}X\rightarrow\mathbb{C}$ is a non-degenerate
anti-symmetric bilinear form on the holomorphic tangent space $T_{x}X$.

######  Definition 2.1.2.

Let $(X,\Omega_{X})$ be a holomorphic symplectic manifold. A complex
submanifold $L\subset X$ is a _holomorphic Lagrangian submanifold_ if for
every $x\in L$, $T_{x}L\subset T_{x}X$ is a maximal isotropic subspace, i.e.
$T_{x}L$ is a maximal subspace of $T_{x}X$ such that
$\Omega_{X,x}|_{T_{x}L}=0$.

Since no other types of symplectic manifolds or Lagrangian submanifolds will
be considered in this work, we are going to refer to $(X,\Omega)$ and
$L\subset X$ in the above definitions simply as _symplectic manifold_ and
_Lagrangian submanifold_ respectively, omitting the term ‘holomorphic’.
Definition 2.1.1 and Definition 2.1.2 also make sense when $X$ is infinite-
dimensional, in that case any Lagrangian submanifold $L\subset X$ will also be
infinite-dimensional. On the other hand, when $X$ is finite-dimensional, it is
common to replace the maximal isotropic subspace condition in Definition 2.1.2
with $\dim L=\frac{1}{2}\dim X$. If $(X,\Omega_{X})$ and $L\subset X$ are
complex vector spaces, then we also call them a _symplectic vector space_ and
a _Lagrangian subspace_ respectively.

######  Definition 2.1.3.

An $m$-dimensional holomorphic foliation $\mathcal{F}$ on an $n$-dimensional
complex manifold $X$ is a decomposition of $X$ into a disjoint union of
connected $(n-m)$-dimensional complex submanifolds $\\{L_{\alpha}\\}$ called
the _leaves_ of the foliation satisfying the following property: At every
point, $x\in X$ there exists an open neighbourhood $U\ni x$ and a system of
local holomorphic coordinates $(x^{1},\cdots,x^{n})$ such that for each leaf
$L_{\alpha}$, each component of $L_{\alpha}\cap U$ is given by
$\\{x^{m+1}=const,\cdots,x^{n}=const\\}$.

In the case where $X$ is a surface, $n=2$ we choose $m=1$ and we will refer to
a $1$-dimensional foliation $\mathcal{F}$ of $X$ simply as a foliation
$\mathcal{F}$ of $X$. The leaves of a $1$-dimensional foliation $\mathcal{F}$
of $X$ are Lagrangian for dimensional reasons.

######  Definition 2.1.4.

Let $(S,\Omega_{S},\mathcal{F})$ be a foliated symplectic surface $S$ with a
symplectic form $\Omega_{S}$ and a foliation $\mathcal{F}$. Let $U\subseteq S$
be an open subset with a coordinates system $(x,y)$ such that
$\Omega_{S}|_{U}=dx\wedge dy$ and the foliation $\mathcal{F}$ is given locally
on $U$ by the equations $x=const$. We call $(x,y)$ a
$(\mathcal{F},\Omega_{S})$-local coordinate system and call $(U,x,y)$ a
$(\mathcal{F},\Omega_{S})$-chart.

######  Lemma 2.1.5.

([21, Section 4.1]) Given any point $p\in S$ there exists a
$(\mathcal{F},\Omega_{S})$-chart $(U_{p},x_{p},y_{p}),U_{p}\subseteq S$
containing $p$.

###### Proof.

By Definition 2.1.3, we choose a local coordinate system,
$(\bar{x}_{p},\bar{y}_{p})$ on some open set $\bar{U}_{p}\ni p$ such that
leaves of the foliation $\mathcal{F}$ are given by $\bar{x}_{p}=const$. Using
this coordinate system, the symplectic form can be written as
$\Omega_{S}=\Omega_{S}(\bar{x}_{p},\bar{y}_{p})d\bar{x}_{p}\wedge
d\bar{y}_{p}$. Consider a coordinates transformation of the form

$\bar{x}_{p}=x_{p},\qquad\bar{y}_{p}=G(x_{p},y_{p}),$

so that the leaves of foliation are still given by $x_{p}=const$. Without loss
of generality, let us suppose that $(x_{p},y_{p})=(0,0)$ is the point $p\in
S$. The symplectic form becomes

$\Omega_{S}=\Omega_{S}\left(x_{p},G(x_{p},y_{p})\right)dx_{p}\wedge\left(\frac{\partial
G}{\partial x_{p}}dx_{p}+\frac{\partial G}{\partial
y_{p}}dy_{p}\right)=\Omega_{S}\left(x_{p},G(x_{p},y_{p})\right)\frac{\partial
G}{\partial y_{p}}dx_{p}\wedge dy_{p}.$

Therefore, we need to find $G=G(x_{p},y_{p})$ such that
$\Omega_{S}(x_{p},G(x_{p},y_{p}))\frac{\partial G}{\partial y_{p}}=1$.
Integrating this equation, we get that $G$ is implicitly defined by
$\tilde{\Omega}_{S}(x_{p},y_{p},G):=\int_{0}^{G}\Omega_{S}(x_{p},\tau)d\tau-
y_{p}+C=0$ for some constant $C$. Let us pick $C=0$, although this choice is
arbitrary. We have $\tilde{\Omega}_{S}(0,0,0)=0$ and
$\frac{\partial\tilde{\Omega}_{S}}{\partial G}(0,0,0)=\Omega_{S}(0,0)\neq 0$
because the symplectic form is non-degenerates by definition. Therefore, we
can apply the complex implicit function theorem to conclude that there exists
$G=G(x_{p},y_{p})$ holomorphic on some open neighbourhood $U$ of
$p=(x_{p}=0,y_{p}=0)$. ∎

We note that the $(\mathcal{F},\Omega_{S})$-local coordinates in the
neighbourhood of a point $p\in S$ is not unique. In fact, given a
$(\mathcal{F},\Omega_{S})$-local coordinates system $(x,y)$ then the new
coordinates system $(\bar{x},\bar{y})$ is also a
$(\mathcal{F},\Omega_{S})$-local coordinates system if and only if it is
related to $(x,y)$ under the transformation of type

(7) $\bar{x}=F(x),\qquad\bar{y}=\frac{y}{F^{\prime}(x)}+G^{\prime}(x),$

where $F$ and $G$ are locally defined holomorphic functions and
$F^{\prime}(x(p))\neq 0$. In particular, $F$ is any function locally bi-
holomorphic in a neighbourhood of $p$.

####  2.1.2. Deformation of $\mathcal{F}$-transversal curves

######  Definition 2.1.6.

([21, Definition 4.1.2]) A smooth compact complex curve $\Sigma$ embedded
inside a foliated symplectic surface $(S,\Omega_{S},\mathcal{F})$ is called
$\mathcal{F}$-_transversal_ if $\Sigma$ is tangent to the leaves of
$\mathcal{F}$ at only finitely many points, with tangency order $1$. We call
points on a $\mathcal{F}$-transversal curve $\Sigma$ where $\Sigma$ is tangent
to a leaf of foliation the _ramification points_ $\\{r_{\alpha\in
Ram(\Sigma)}\\}$ where $Ram(\Sigma)$ is an index set.

######  Remark 2.1.7.

Given a Riemann surface $\Sigma$ and a holomorphic map
$f:\Sigma\rightarrow\mathbb{P}^{1}$, the term ‘ramification point’
traditionally refers to a point $r\in\Sigma$ such that $f$ can be locally
written as $f(z)=z^{n}$, for some $n\in\mathbb{Z}_{\geq 2}$, using some
holomorphic coordinates charts $(U,z)$ and $(V,w)$ around $r$ and $f(r)$
respectively. For us, instead of a map $f:\Sigma\rightarrow\mathbb{P}^{1}$, we
have the foliation $\mathcal{F}$. By Lemma 2.1.5, a
$(\mathcal{F},\Omega_{S})$-chart $(U_{p},x_{p},y_{p})$ can be chosen around
any point $p\in\Sigma$, hence the foliation locally induces a map
$q\in\Sigma\cap U_{p}\mapsto x_{p}(q)\in\mathbb{C}$. Our usage of the term
‘ramification point’ follows from the fact that each $r_{\alpha},\alpha\in
Ram$ are ramification points of these locally foliation induced maps.
Additionally, the ‘tangency order $1$’ condition means the ramification index
of each ramification point is $2$. Note that a different choice of
$(\mathcal{F},\Omega_{S})$-chart $(U_{p},\bar{x}_{p},\bar{y}_{p})$ will give a
different local map $q\mapsto\bar{x}_{p}(q)$ but ramification points will be
the same. This is because $\bar{x}_{p}=F_{p}(x_{p})$ for some function $F_{p}$
such that $F_{p}^{\prime}(x_{p}(q))\neq 0,q\in U_{p}$ by (7), hence
$d\bar{x}_{p}|_{\Sigma}=F_{p}^{\prime}(x_{p})dx_{p}|_{\Sigma}$ means
$d\bar{x}_{p}|_{\Sigma}=0$ exactly when $dx_{p}|_{\Sigma}=0$.

Let $\mathcal{B}$ be the complex analytic moduli space of
$\mathcal{F}$-transversal genus $g$ curves in $(S,\Omega_{S},\mathcal{F})$. We
are going to show later that the dimension of $\mathcal{B}$ is $g$. We write
$[\Sigma]\in\mathcal{B}$ for the point in $\mathcal{B}$ corresponding to a
$\mathcal{F}$-transversal curve $\Sigma$. Since the number of ramification
points is constant for every $[\Sigma]\in\mathcal{B}$, we will simply write
the index set as $Ram=Ram(\Sigma)$. For any $p\in\Sigma\subseteq S$ we find a
$(\mathcal{F},\Omega_{S})$-chart $(U_{p},x_{p},y_{p})$ containing $p$
according to Lemma 2.1.5. Let us think of $x_{p},y_{p}$ as functions
$\Sigma\cap U_{p}\rightarrow\mathbb{C}$, the ramification points
$\\{r_{\alpha\in Ram}\\}$ are exactly where $dx_{p}|_{\Sigma}$ vanishes with
an order $1$ zero. Consequently, if $r_{\alpha}\notin\Sigma\cap U_{p}$ for all
$\alpha\in Ram$ then $dx_{p}|_{\Sigma}\neq 0$ on $\Sigma\cap U_{p}$ and we may
assume that $U_{p}$ is chosen small enough such that $x_{p}$ is a local
coordinate on $\Sigma\cap U_{p}$ according to the inverse function theorem. On
the other hand, if $p=r_{\alpha}\in\Sigma,\alpha\in Ram$, we denote by
$(U_{\alpha},x_{\alpha},y_{\alpha})$ a $(\mathcal{F},\Omega_{S})$-chart such
that $\Sigma\cap U_{\alpha}$ is bi-holomorphic to a simply-connected open
subset of $\mathbb{C}$ with a local coordinate
$z_{\alpha}:=\sqrt{x_{\alpha}-x_{\alpha}(r_{\alpha})}$.

######  Definition 2.1.8.

Given a $(\mathcal{F},\Omega_{S})$-chart $(U_{\alpha},x_{\alpha},y_{\alpha})$,
we call the local coordinate
$z_{\alpha}:=\sqrt{x_{\alpha}-x_{\alpha}(r_{\alpha})}$ on an open
neighbourhood $\Sigma\cap U_{\alpha}$ of the ramification point $r_{\alpha}$ a
_standard local coordinate_.

The parameterization of $\Sigma\cap U_{\alpha}$ using the standard local
coordinate takes the form

(8)
$(x_{\alpha}=a_{\alpha}(\Sigma)+z_{\alpha}^{2},y_{\alpha}=b_{0\alpha}(\Sigma)+b_{1\alpha}(\Sigma)z_{\alpha}+b_{2\alpha}(\Sigma)z^{2}_{\alpha}+...),\qquad
a_{\alpha}(\Sigma),b_{k\alpha}(\Sigma)\in\mathbb{C},$

where $b_{1\alpha}(\Sigma)\neq 0$ by the tangency order $1$ condition.
Therefore, we have $dy_{p}|_{\Sigma}(r_{\alpha})\neq 0$ and $y_{p}$ is a local
coordinate on some neighbourhoods of $r_{\alpha}$. In particular, since
$\Sigma$ is compact, we can always cover $\Sigma$ with a finite collection of
$(\mathcal{F},\Omega_{S})$-charts
$\mathcal{U}=\\{(U_{p\in\sigma},x_{p\in\sigma},y_{p\in\sigma})\\}$, where
$\sigma$ is a finite index set, such that either $x_{p}$ or $y_{p}$ is a local
coordinate on each $\Sigma\cap U_{p}$.

The following fact will be useful to us later:

######  Lemma 2.1.9.

([21, Section 4.1 and 6.2]) Given $[\Sigma]\in\mathcal{B}$, there exists a
$(\mathcal{F},\Omega_{S})$-chart $(U_{\alpha},x_{\alpha},y_{\alpha})$
containing the ramification point $r_{\alpha},\alpha\in Ram$ of $\Sigma$ such
that the parameterization of $\Sigma\cap U_{\alpha}$ using the the standard
local coordinate $z_{\alpha}$ takes the form
$(x_{\alpha}=z_{\alpha}^{2},y_{\alpha}=z_{\alpha})$. The
$(\mathcal{F},\Omega_{S})$-chart $(U_{\alpha},x_{\alpha},y_{\alpha})$ is
unique up to the $\mathbb{Z}_{3}$ automorphism action
$x\mapsto\xi^{2}x,y\mapsto\xi y$, where $\xi$ is a third-root of unity.

###### Proof.

Let $(U_{\alpha},x_{\alpha},y_{\alpha})$ be a $(\mathcal{F},\Omega_{S})$-chart
around $r_{\alpha}$ which exists due to Lemma 2.1.5 and
$z_{\alpha}:=\sqrt{x_{\alpha}-x_{\alpha}(r_{\alpha})}$ be the standard local
coordinate on $\Sigma\cap U_{\alpha}$. We suppose that the parameterization of
$\Sigma\cap U_{\alpha}$ is given by
$(x_{\alpha}=z^{2}_{\alpha}+a,y_{\alpha}=z_{\alpha}\psi(z^{2}_{\alpha})+\phi(z^{2}_{\alpha}))$
for some $\psi$ and $\phi$ holomorphic on $\Sigma\cap U_{\alpha}$. We wish to
find $F$ and $G$ such that the new $(\mathcal{F},\Omega_{S})$-local
coordinates $(\bar{x}_{\alpha},\bar{y}_{\alpha})$ given by

$\bar{x}_{\alpha}=F(x_{\alpha}),\qquad\bar{y}_{\alpha}=\frac{y_{\alpha}}{F^{\prime}(x_{\alpha})}+G^{\prime}(x_{\alpha})$

with the new standard local coordinate
$\bar{z}_{\alpha}:=\sqrt{\bar{x}_{\alpha}-\bar{x}_{\alpha}(r_{\alpha})}=\sqrt{F(z_{\alpha}^{2}+a)-F(a)}$
satisfies

$\bar{z}^{2}_{\alpha}=F(z_{\alpha}^{2}+a),\qquad\bar{z}_{\alpha}=\frac{z_{\alpha}\psi(z_{\alpha}^{2})+\phi(z_{\alpha}^{2})}{F^{\prime}(z_{\alpha}^{2}+a)}+G^{\prime}(z_{\alpha}^{2}+a).$

This is equivalent to

(9) $F^{-1}(0)=a,\qquad
F^{\prime}\left(z_{\alpha}^{2}+F^{-1}(0)\right)\left(\left(F\left(z_{\alpha}^{2}+F^{-1}(0)\right)\right)^{1/2}-G^{\prime}\left(z_{\alpha}^{2}+F^{-1}(0)\right)\right)=z_{\alpha}\psi(z_{\alpha}^{2})+\phi(z_{\alpha}^{2}).$

The second equation above can be solved by separately equating the odd and
even terms. The odd terms yields:

(10)
$F^{\prime}\left(z^{2}_{\alpha}+F^{-1}(0)\right)F\left(z^{2}_{\alpha}+F^{-1}(0)\right)^{1/2}=z_{\alpha}\psi(z^{2}_{\alpha})\implies\frac{1}{3}\frac{d}{dz_{\alpha}}\left(F\left(z^{2}_{\alpha}+F^{-1}(0)\right)\right)^{3/2}=z^{2}_{\alpha}\psi(z^{2}_{\alpha}).$

Both sides can be expanded as power series in $z^{2}$, starting with the
$z^{2}$ term, so $F(z^{2}+a)$ and hence $F(x)$ can be found by comparing
coefficients. Similarly, we can define
$G(x):=\frac{\phi(x-F^{-1}(0))}{F^{\prime}(x)}$. Note that if
$\left(F(z^{2}+a)\right)^{1/2}$ satisfies (10), then
$\xi\left(F(z^{2}+a)\right)^{1/2}$ also does for any third root of unity
$\xi$. ∎

The tangent space $T_{[\Sigma]}\mathcal{B}$ describes the infinitesimal
deformations of the curve $\Sigma$ inside $S$. It is isomorphic to the space
of global sections of the holomorphic normal bundle $\nu_{\Sigma}\cong
TS|_{\Sigma}/T\Sigma$, where $TS$ and $T\Sigma$ are holomorphic tangent
bundles of $S$ and $\Sigma$ respectively. In general, given any holomorphic
vector bundle $\mathcal{V}$ on a complex manifold $X$, we denote the set of
holomorphic sections of $\mathcal{V}$ by $\Gamma(X,\mathcal{V})$. The
availability of the symplectic form $\Omega_{S}$ gives the following result:

######  Lemma 2.1.10.

([21, Section 4.2]) Let $\Omega^{1}_{\Sigma}$ be the holomorphic canonical
line bundle on $\Sigma$, then the symplectic form $\Omega_{S}$ induces the
isomorphism
$\Gamma(\Sigma,\nu_{\Sigma})\cong\Gamma(\Sigma,\Omega^{1}_{\Sigma})$.
Consequently, $\dim
T_{[\Sigma]}\mathcal{B}=\dim\Gamma(\Sigma,\Omega^{1}_{\Sigma})=g$, hence
$\mathcal{B}$ is $g$-dimensional.

###### Proof.

We establish the isomorphism
$\Gamma(\Sigma,\nu_{\Sigma})\cong\Gamma(\Sigma,\Omega^{1}_{\Sigma})$ by
$v\mapsto\Omega_{S}(v,.)|_{\Sigma}$. Any holomorphic section of the normal
bundle $v\in\Gamma(\Sigma,TS|_{\Sigma}/T\Sigma)$ can be represented as a
collection of sections $\\{v_{p\in\sigma}\in\Gamma(V_{p},TS|_{\Sigma})\\}$
such that $v_{p}-v_{p^{\prime}}\in\Gamma(V_{p}\cap V_{p^{\prime}},T\Sigma)$
for some open cover $\\{V_{p\in\sigma}\\}$ of $\Sigma$. We note that for each
$p\in\sigma$, $\Omega_{S}(v_{p},.)|_{V_{p}}$ is a holomorphic form on
$V_{p}\subset\Sigma$ as it is a pull-back of the contraction between
$\Omega_{S}$ and $v_{p}$ which are both holomorphic. Since
$\Omega_{S}(t,.)|_{\Sigma}$ for all $t\in\Gamma(\Sigma,T\Sigma)$, we can patch
together $\Omega_{S}(v_{p},.)|_{V_{p}}$ for each $p\in\sigma$ to get a global
section $\Omega_{S}(v,.)|_{\Sigma}\in\Gamma(\Sigma,\Omega^{1}_{\Sigma})$.

The map $v\mapsto\Omega_{S}(v,.)|_{\Sigma}$ is injective because if
$\Omega_{S}(v,.)|_{\Sigma}=0$ then by the fact that $\Sigma\subseteq S$ is a
Lagrangian submanifold, we have $v(p)\in T_{p}\Sigma$ for all $p\in\Sigma$.
Therefore, $v=0\in\Gamma(\Sigma,\nu_{\Sigma})$. For surjectivity, given a
holomorphic differential form $\omega\in\Gamma(\Sigma,\Omega^{1}_{\Sigma})$,
we cover $\Sigma$ with a collection of $(\mathcal{F},\Omega_{S})$-charts
$\mathcal{U}=\\{(U_{p\in\sigma},x_{p\in\sigma},y_{p\in\sigma})\\}$ such that
either $x_{p}$ or $y_{p}$ is a local coordinate on $\Sigma\cap U_{p}$. Let us
assume without loss of generality that $x_{p}$ is a local coordinate on
$\Sigma\cap U_{p}$. Then in general $\omega|_{\Sigma\cap
U_{p}}=\omega_{p}(x_{p})dx_{p}$ and by taking
$v_{p}:=-\omega_{p}(x_{p})\frac{\partial}{\partial y_{p}}\in\Gamma(\Sigma\cap
U_{p},TS|_{\Sigma})$ we have $\Omega_{S}(v_{p},.)|_{\Sigma\cap
U_{p}}=\omega|_{\Sigma\cap U_{p}}$. For $p,p^{\prime}\in\sigma$ we have on
$U_{p}\cap U_{p^{\prime}}$ that

$\Omega_{S}(v_{p}-v_{p^{\prime}},.)|_{\Sigma\cap U_{p}\cap
U_{p^{\prime}}}=\left(\omega|_{\Sigma\cap U_{p}}\right)|_{\Sigma\cap U_{p}\cap
U_{p^{\prime}}}-\left(\omega|_{\Sigma\cap U_{p^{\prime}}}\right)|_{\Sigma\cap
U_{p}\cap U_{p^{\prime}}}=0.$

Therefore, $(v_{p}-v_{p^{\prime}})|_{\Sigma\cap U_{p}\cap
U_{p^{\prime}}}\in\Gamma(\Sigma\cap U_{p}\cap U_{p^{\prime}},T\Sigma)$ and
$\\{v_{p\in\sigma}\\}$ represents a global section of a normal bundle
$v\in\Gamma(\Sigma,\nu_{\Sigma})$ such that $\Omega_{S}(v,.)=\omega$. ∎

Alternatively, since the symplectic form $\Omega_{S}$ is a global section of
the holomorphic canonical line bundle $\Omega_{S}^{2}$ of $S$ with no zeros or
poles, we have from the adjunction formula:
$\Omega^{1}_{\Sigma}\cong\Omega^{2}_{S}|_{\Sigma}\otimes\nu_{\Sigma}$ that
$\Omega^{1}_{\Sigma}\cong\nu_{\Sigma}$. It follows that
$\Gamma(\Sigma,\nu_{\Sigma})\cong\Gamma(\Sigma,\Omega^{1}_{\Sigma})$ and the
map $v\mapsto\Omega_{S}(v,.)$ explicitly express this isomorphism.

######  Definition 2.1.11.

Consider the _universal family_ of $\mathcal{F}$-transversal curves:

$Z:=\\{([\Sigma],p)\in\mathcal{B}\times S\ |\
p\in\Sigma\\}\subseteq\mathcal{B}\times S,$

let $\pi:Z\rightarrow\mathcal{B}$ be the canonical projection and let
$\mathbb{C}$ be a constant sheaf on $Z$. Then we define the holomorphic
symplectic vector bundle
$(\mathcal{H}\rightarrow\mathcal{B},\Omega_{\mathcal{H}},\nabla_{GM})$ where
$\mathcal{H}:=R^{1}\pi_{*}\mathbb{C}$ is a holomorphic vector bundle with
fiber over each point $[\Sigma]\in\mathcal{B}$ given by
$\mathcal{H}_{\Sigma}=H^{1}(\Sigma,\mathbb{C})$. $\nabla_{GM}$ is the _Gauss-
Manin_ connection and the symplectic form $\Omega_{\mathcal{H}}$ is given by

(11)
$\Omega_{\mathcal{H}}([\xi_{1}],[\xi_{2}]):=\iint_{\Sigma}\xi_{1}\wedge\xi_{2}$

where $\xi_{1},\xi_{2}$ are smooth differential forms representing
$[\xi_{1}],[\xi_{2}]\in\mathcal{H}_{\Sigma}$.

For any open subset, $\mathcal{B}^{\prime}\subseteq\mathcal{B}$ we denote by
$\Gamma(\mathcal{B}^{\prime},\mathcal{H})$ the set of holomorphic sections of
$\mathcal{H}$ over $U$.

######  Remark 2.1.12.

The natural symplectic form $\Omega^{*}_{\mathcal{H}}$ on
$H_{1}(\Sigma,\mathbb{C})$ can be defined using the oriented intersection
product of homology classes as follows. For any $[C_{1}],[C_{2}]\in
H_{1}(\Sigma,\mathbb{Z})$, the intersection product is given by
$[C_{1}]\cdot[C_{2}]=[C_{1}\cap C_{2}]\in
H_{0}(\Sigma,\mathbb{Z})\cong\mathbb{Z}$ which can be extended linearly to the
intersection product on
$H_{1}(\Sigma,\mathbb{C})=H_{1}(\Sigma,\mathbb{Z})\otimes\mathbb{C}$. Then,
for any $[C_{1}],[C_{2}]\in H_{1}(\Sigma,\mathbb{C})$, we define
$\Omega^{*}_{\mathcal{H}}([C_{1}],[C_{2}]):=[C_{1}]\cdot[C_{2}]$. Typically,
we denote a symplectic basis of $H_{1}(\Sigma,\mathbb{C})$ by $A$ and $B$
cycles: $\\{[A_{1}],\cdots,[A_{g}],[B_{1}],\cdots,[B_{g}]\\}$,

$\Omega_{\mathcal{H}}^{*}([A_{i}],[A_{j}])=\Omega_{\mathcal{H}}^{*}([B_{i}],[B_{j}])=0,\qquad\Omega_{\mathcal{H}}^{*}([A_{i}],[B_{j}])=\delta_{ij}.$

On $H^{1}(\Sigma,\mathbb{C})$, the symplectic form $\Omega_{\mathcal{H}}$ is
defined by the intersection product of cohomology classes via Poincaré duality
$P.D.:H^{1}(\Sigma,\mathbb{C})\cong H_{1}(\Sigma,\mathbb{C})$. In other words,
given $[\xi_{1}],[\xi_{2}]\in H^{1}(\Sigma,\mathbb{C})$ then (11) is
equivalent to
$\Omega_{\mathcal{H}}([\xi_{1}],[\xi_{2}]):=\Omega^{*}_{\mathcal{H}}(P.D.[\xi_{1}],P.D.[\xi_{2}])$.
Note also that the expression for $\Omega_{\mathcal{H}}([\xi_{1}],[\xi_{2}])$
as given in (11) only depends on the cohomology classes of $\xi_{1}$ and
$\xi_{2}$, because $\iint_{\Sigma}df\wedge\xi_{2}=\iint_{\Sigma}d(f\xi_{2})=0$
by Stokes Theorem.

######  Remark 2.1.13.

Suppose that the $A,B$ symplectic basis cycles
$\\{[A_{1}],\cdots,[A_{g}],[B_{1}],\cdots,[B_{g}]\\}$ of
$H_{1}(\Sigma,\mathbb{C})$ are chosen for $[\Sigma]\in\mathcal{B}$. Since
$H^{1}(\Sigma,\mathbb{C})\cong H_{1}(\Sigma,\mathbb{C})^{*}$, a cohomology
class $[\xi]\in H^{1}(\Sigma,\mathbb{C})$ is determined by its values on $A$
and $B$ cycles. In particular, the cohomology class $[\xi]$ represented by a
closed differential form $\xi$ is given by the $A$ and $B$ periods of $\xi$
and we identify
$\mathcal{H}_{\Sigma}=H^{1}(\Sigma,\mathbb{C})\xrightarrow{\cong}\mathbb{C}^{2g}$
by

$H^{1}(\Sigma,\mathbb{C})\ni[\xi]\mapsto\left(\oint_{A_{1}}\xi,\cdots,\oint_{A_{g}}\xi,\oint_{B_{1}}\xi,\cdots,\oint_{B_{g}}\xi\right)\in\mathbb{C}^{2g}.$

It can happen as we move from $[\Sigma]$ along some closed path in
$\mathcal{B}$, letting $A,B$ cycles deform continuously, that once we return
to the point $[\Sigma]$ the $A,B$ cycles will differ from the original by some
Dehn twist actions. On the other hand, given a sufficiently small open
neighbourhood $\mathcal{B}_{\Sigma}\subseteq\mathcal{B}$, we will be able to
choose $A,B$ symplectic basis cycles consistently throughout
$\mathcal{B}_{\Sigma}$. The Gauss-Manin connection $\nabla_{GM}$ can be
described locally on $\mathcal{B}_{\Sigma}$ as the identification of any two
cohomology classes on two different fibers
$[\xi_{1}]\in\mathcal{H}_{\Sigma_{1}},[\xi_{2}]\in\mathcal{H}_{\Sigma_{2}}$,
if they have the same $A$ and $B$ periods:
$\oint_{A_{k}\subset\Sigma_{1}}\xi_{1}=\oint_{A_{k}\subset\Sigma_{2}}\xi_{2},\oint_{B_{k}\subset\Sigma_{1}}\xi_{1}=\oint_{B_{k}\subset\Sigma_{2}}\xi_{2}$.
This gives a local trivialization
$\mathcal{H}|_{\mathcal{B}_{\Sigma}}\cong\mathbb{C}^{2g}\times\mathcal{B}_{\Sigma}$,
verifying that $\mathcal{H}$ is a vector bundle.

The following gives an alternative expression for $\Omega_{\mathcal{H}}$:

######  Lemma 2.1.14.

(Riemann bilinear identity) Let $[\xi_{1}],[\xi_{2}]\in
H^{1}(\Sigma,\mathbb{C})$ then

$\iint_{\Sigma}\xi_{1}\wedge\xi_{2}=\frac{1}{2\pi
i}\sum_{k=1}^{g}\left(\oint_{A_{k}}\xi_{2}\oint_{B_{k}}\xi_{1}-\oint_{B_{k}}\xi_{2}\oint_{A_{k}}\xi_{1}\right)$

where $\xi_{1},\xi_{2}$ are smooth differential forms representing
$[\xi_{1}],[\xi_{2}]$.

###### Proof.

First, we show that for any $[\xi]\in H^{1}(\Sigma,\mathbb{C})$ has a
representative $\xi=\alpha+\bar{\beta}$ where $\alpha$ is holomorphic
($\bar{\partial}\alpha=0$) and $\bar{\beta}$ is anti-holomorphic
($\partial\bar{\beta}=0$). A smooth differential form $\xi^{\prime}$
representing $[\xi]$ can be decomposed as
$\xi^{\prime}=\alpha^{\prime}+\bar{\beta^{\prime}}$ where $\alpha^{\prime}$ is
a smooth $(1,0)$-form and $\bar{\beta^{\prime}}$ is a smooth $(0,1)$-form.
Then $d\xi^{\prime}=(\partial+\bar{\partial})\xi^{\prime}=0$ implies
$\bar{\partial}\alpha^{\prime}=-\partial\bar{\beta^{\prime}}$. Since Riemann
surfaces are examples of Kähler manifolds, $\partial\bar{\partial}$-Lemma [32,
Proposition 6.17] tells us that there exists a smooth function $f$ such that
$\bar{\partial}\alpha^{\prime}=\partial\bar{\partial}f=-\partial\bar{\beta^{\prime}}$.
We can see that $\xi:=\xi^{\prime}+df=\alpha+\bar{\beta}$ where
$\alpha:=\alpha^{\prime}+\partial f$ and
$\bar{\beta}=\beta^{\prime}+\bar{\partial}f$ is the desired representative.

The rest of the proof follows from Stokes’ theorem and the standard argument
leading to the Riemann bilinear identity. For example, we have

$\iint_{\Sigma}\alpha\wedge\bar{\beta}=\iint_{\Delta}d\phi\wedge\bar{\beta}=\iint_{\Delta}d(\phi\bar{\beta})=\oint_{\partial\bar{\Delta}}\phi\bar{\beta}=\frac{1}{2\pi
i}\sum_{k=1}^{g}\left(\oint_{A_{k}}\bar{\beta}\oint_{B_{k}}\alpha-\oint_{B_{k}}\bar{\beta}\oint_{A_{k}}\alpha\right)$

where $\Delta:=\Sigma\setminus\cup_{k=1}^{g}\left(A_{k}\cup B_{k}\right)$ is a
simply-connected domain and $\alpha=df=\partial f$ for some holomorphic
function $\phi$ on $\Delta$ by Poincaré Lemma. ∎

Let us fix a _reference point_ $[\Sigma_{0}]\in\mathcal{B}$ and an open
neighbourhood $\mathcal{B}_{\Sigma_{0}}\subseteq\mathcal{B}$ of
$[\Sigma_{0}]$. Later we will explain more precisely how
$\mathcal{B}_{\Sigma_{0}}$ is selected (see Condition 2.5.2). But for now, we
simply require $\mathcal{B}_{\Sigma_{0}}$ to be contractible and sufficiently
small such that it can be covered by a single coordinate chart
$(\mathcal{B}_{\Sigma_{0}},u^{1},...,u^{g})$. We will denote by
$\mathcal{H}\rightarrow\mathcal{B}_{\Sigma_{0}}$ the vector bundle given by
the restriction of $\mathcal{H}$ to $\mathcal{B}_{\Sigma_{0}}$, which is
trivial because $\mathcal{B}_{\Sigma_{0}}$ is contractible. Let $T\mathcal{B}$
and $T^{*}\mathcal{B}$ denote the holomorphic tangent and cotangent bundles of
$\mathcal{B}$ respectively. Following [21, Section 5.2], we introduce
$\boldsymbol{\phi}\in\Gamma(\mathcal{B}_{\Sigma_{0}},T^{*}\mathcal{B}\otimes\mathcal{H})$
as follows. At any point $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ we define the
map $\boldsymbol{\phi}_{\Sigma}$ via the following diagram:

${T_{[\Sigma]}\mathcal{B}_{\Sigma_{0}}}$${\Gamma(\Sigma,TS|_{\Sigma}/T\Sigma)}$${\Gamma(\Sigma,\Omega^{1}_{\Sigma})}$${H^{1}(\Sigma,\mathbb{C})\cong\mathcal{H}_{\Sigma}.}$$\scriptstyle{\cong}$$\scriptstyle{\boldsymbol{\phi}_{\Sigma}}$$\scriptstyle{\cong}$$\scriptstyle{[.]}$

Let us discuss the first map
$T_{[\Sigma]}\mathcal{B}_{\Sigma_{0}}\xrightarrow{\cong}\Gamma(\Sigma,TS|_{\Sigma}/T\Sigma)$.
We map $v=\sum_{k=1}^{g}v^{k}\frac{\partial}{\partial u^{k}}$ to a global
normal vector field $n_{v}\in\Gamma(\Sigma,TS|_{\Sigma}/T\Sigma)$ which can be
defined in terms of $TS|_{\Sigma}$-sections on each
$(\mathcal{F},\Omega_{S})$-chart $(U_{p},x_{p},y_{p})$:

(12) $n_{v}|_{\Sigma\cap U_{p}}:=\sum_{k=1}^{g}v^{k}\left(\frac{\partial
x_{p}}{\partial u^{k}}\frac{\partial}{\partial x_{p}}+\frac{\partial
y_{p}}{\partial u^{k}}\frac{\partial}{\partial
y_{p}}\right)\in\Gamma(\Sigma\cap U_{p},TS|_{\Sigma}).$

Taking the quotient of $n_{v}|_{\Sigma\cap U_{p}}$ on each
$(U_{p},x_{p},y_{p})$ and patch each of them together, we get a global section
$n_{v}$. Finally, we define
$\boldsymbol{\phi}_{\Sigma}(v):=[\phi_{\Sigma}(v)]\in\mathcal{H}_{\Sigma}$
where
$\phi_{\Sigma}(v):=\Omega_{S}(n_{v},.)|_{\Sigma}\in\Gamma(\Sigma,\Omega^{1}_{\Sigma})$.
Let $v_{k}:=\frac{\partial}{\partial u^{k}}$, then we can equivalently write
this as

(13) $\phi_{\Sigma}:=\sum_{k=1}^{g}\Omega_{S}(n_{v_{k}},.)|_{\Sigma}du^{k}\in
T^{*}_{[\Sigma]}\mathcal{B}\otimes\Gamma(\Sigma,\Omega^{1}_{\Sigma}),\qquad\boldsymbol{\phi}_{\Sigma}:=[\phi_{\Sigma}]=\sum_{k=1}^{g}[\Omega_{S}(n_{v_{k}},.)|_{\Sigma}]du^{k}\in
T^{*}_{[\Sigma]}\mathcal{B}\otimes\mathcal{H}_{\Sigma}.$

######  Remark 2.1.15.

The image
$\text{im}\boldsymbol{\phi}_{\Sigma}=\Gamma(\Sigma,\Omega^{1}_{\Sigma})$ is a
Lagrangian subspace of $\mathcal{H}_{\Sigma}$ for every
$[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$.

At first glance, the expression (12) seems ambiguous because it depends on the
choice of a connection, in other words, on how the derivatives of $x_{p}$ and
$y_{p}$ are taken with respect to $u^{k}$. Suppose that $\Sigma\cap
U_{p}=\\{(x_{p},y_{p})\in U_{p}\ |\ \mathcal{P}_{p}(x_{p},y_{p};u)=0\\}$ for
some polynomial $\mathcal{P}_{p}$. For any $q\in\Sigma\cap U_{p}$, we have
$t\in T_{q}\Sigma\subset T_{q}S$ if $\iota_{t}d\mathcal{P}_{p}=0$ where
$d\mathcal{P}_{p}=\frac{\partial\mathcal{P}_{p}}{\partial
x_{p}}dx_{p}+\frac{\partial\mathcal{P}_{p}}{\partial y_{p}}dy_{p}$. The
variation of the curve $\Sigma$ in the $u^{k}$-direction in $\mathcal{B}$
gives

(14) $\frac{\partial\mathcal{P}_{p}}{\partial
u^{k}}+\frac{\partial\mathcal{P}_{p}}{\partial x_{p}}\frac{\partial
x_{p}}{\partial u^{k}}+\frac{\partial\mathcal{P}_{p}}{\partial
y_{p}}\frac{\partial y_{p}}{\partial u^{k}}=0.$

Choosing a connection is equivalent to choosing $\frac{\partial
x_{p}}{\partial u^{k}}$ and $\frac{\partial y_{p}}{\partial u^{k}}$ satisfying
this equation. We need to check that if connections are chosen differently on
each $(U_{p},x_{p},y_{p})$ then

$\left(n_{v}|_{\Sigma\cap U_{p}}\right)|_{\Sigma\cap U_{p}\cap
U_{p^{\prime}}}-\left(n_{v}|_{\Sigma\cap U_{p^{\prime}}}\right)|_{\Sigma\cap
U_{p}\cap U_{p^{\prime}}}\in\Gamma(\Sigma\cap U_{p}\cap
U_{p^{\prime}},T\Sigma)$

and therefore we can patch the quotient of each $n_{v}|_{\Sigma\cap U_{p}}$ to
form a global section $n_{v}\in\Gamma(\Sigma,\nu_{\Sigma})$. The following
lemma resolves this concern:

######  Lemma 2.1.16.

Suppose we obtain $(n_{v}|_{\Sigma\cap U_{p}})_{i},i=1,2$ using any two
different choices of connections, then $(n_{v}|_{\Sigma\cap
U_{p}})_{1}-(n_{v}|_{\Sigma\cap U_{p}})_{2}\in\Gamma(\Sigma\cap
U_{p},T\Sigma)$.

###### Proof.

Without loss of generality, let us assume that $v=\frac{\partial}{\partial
u^{k}}$. For $i=1,2$, let

$(n_{v}|_{\Sigma\cap U_{p}})_{i}:=\left(\frac{\partial x_{p}}{\partial
u^{k}}\right)_{i}\frac{\partial}{\partial x_{p}}+\left(\frac{\partial
y_{p}}{\partial u^{k}}\right)_{i}\frac{\partial}{\partial y_{p}}$

where

(15) $\frac{\partial\mathcal{P}_{p}}{\partial
u^{k}}+\frac{\partial\mathcal{P}_{p}}{\partial x_{p}}\left(\frac{\partial
x_{p}}{\partial u^{k}}\right)_{i}+\frac{\partial\mathcal{P}_{p}}{\partial
y_{p}}\left(\frac{\partial y_{p}}{\partial u^{k}}\right)_{i}=0.$

If we subtract (15) with $i=2$ from (15) with $i=1$ then

$\left(\left(\frac{\partial x_{p}}{\partial
u^{k}}\right)_{1}-\left(\frac{\partial x_{p}}{\partial
u^{k}}\right)_{2}\right)\frac{\partial\mathcal{P}_{p}}{\partial
x_{p}}+\left(\left(\frac{\partial y_{p}}{\partial
u^{k}}\right)_{1}-\left(\frac{\partial y_{p}}{\partial
u^{k}}\right)_{2}\right)\frac{\partial\mathcal{P}_{p}}{\partial y_{p}}=0.$

The lemma follows as we recognize the left-hand-side to be
$\iota_{(n_{v}|_{\Sigma\cap U_{p}})_{1}-(n_{v}|_{\Sigma\cap
U_{p}})_{2}}d\mathcal{P}_{p}$. ∎

The next lemma tells us that
$\nabla_{GM}[\xi]=\sum_{k=1}^{g}\left[\frac{\partial\xi}{\partial
u^{k}}\right]du^{k}$ is independent of the choice of the connection and the
differential form $\xi$ we choose to represent $[\xi]$.

######  Lemma 2.1.17.

Let $\xi_{\Sigma}$ be a holomorphic family of meromorphic differential forms
defined for each $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ on an open subsets
$D_{\Sigma}\subseteq\Sigma$ and let
$\left(\frac{\partial\xi_{\Sigma}}{\partial u^{k}}\right)_{i},i=1,2$ be the
derivative $\frac{\partial}{\partial u^{k}}$ taken using two different choices
of connections. Then $\left(\frac{\partial\xi_{\Sigma}}{\partial
u^{k}}\right)_{1}-\left(\frac{\partial\xi_{\Sigma}}{\partial
u^{k}}\right)_{2}$ is an exact meromorphic differential on $D_{\Sigma}$. In
particular, for any closed contour $C\subseteq D_{\Sigma}$ which deforms
homotopically over $\mathcal{B}_{\Sigma_{0}}$ and avoiding all poles of
$\xi_{\Sigma}$ for each $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$, we have

(16) $\frac{\partial}{\partial
u^{k}}\oint_{C}\xi_{\Sigma}=\oint_{C}\left(\frac{\partial}{\partial
u^{k}}\xi_{\Sigma}\right)_{\nabla}$

independent of the the choice of connection $\nabla$ the derivative
$\frac{\partial}{\partial u^{k}}\xi_{\Sigma}$ is taken with respect to.

###### Proof.

First, let us consider the case in which $D_{\Sigma}\subseteq\Sigma\cap U_{p}$
where $(U_{p},x_{p},y_{p})$ is a $(\mathcal{F},\Omega_{S})$-chart with
$dx_{p}|_{\Sigma U_{p}}\neq 0$. We write in general $\xi_{\Sigma}|_{\Sigma\cap
U_{p}}=\xi_{\Sigma,p}(x_{p},u)dx_{p}$. Then we have for $i=1,2$:

$\displaystyle\left(\frac{\partial\xi_{\Sigma}}{\partial u^{k}}\right)_{i}$
$\displaystyle=\frac{\partial\xi_{\Sigma,p}}{\partial
u^{k}}dx_{p}+\left(\frac{\partial x_{p}}{\partial
u^{k}}\right)_{i}\frac{\partial\xi_{\Sigma,p}}{\partial
x_{p}}dx_{p}+\xi_{\Sigma,p}d\left(\frac{\partial x_{p}}{\partial
u^{k}}\right)_{i}$ $\displaystyle=\frac{\partial\xi_{\Sigma,p}}{\partial
u^{k}}dx_{p}+\left(\frac{\partial x_{p}}{\partial
u^{k}}\right)_{i}\frac{\partial\xi_{\Sigma,p}}{\partial
x_{p}}dx_{p}-\left(\frac{\partial x_{p}}{\partial
u^{k}}\right)_{i}\frac{\partial\xi_{\Sigma,p}}{\partial
x_{p}}dx_{p}+d\left(\xi_{p}\left(\frac{\partial x_{p}}{\partial
u^{k}}\right)_{i}\right)=\frac{\partial\xi_{\Sigma,p}}{\partial
u^{k}}dx_{p}+d\left(\xi_{p}\left(\frac{\partial x_{p}}{\partial
u^{k}}\right)_{i}\right).$

Evidently, we have

$\left(\frac{\partial\xi_{\Sigma}}{\partial
u^{k}}\right)_{1}-\left(\frac{\partial\xi_{\Sigma}}{\partial
u^{k}}\right)_{2}=d\left(\xi_{\Sigma,p}\left(\frac{\partial x_{p}}{\partial
u^{k}}\right)_{1}-\xi_{\Sigma,p}\left(\frac{\partial x_{p}}{\partial
u^{k}}\right)_{2}\right)$

which is exact as claimed. In general we cover $D_{\Sigma}$ with a collection
of $(\mathcal{F},\Omega_{S})$-charts
$\\{(U_{p\in\sigma},x_{p\in\sigma},y_{p\in\sigma})\\}$ and check that the
function $\xi_{\Sigma,p}\left(\frac{\partial x_{p}}{\partial
u^{k}}\right)_{1}-\xi_{\Sigma,p}\left(\frac{\partial x_{p}}{\partial
u^{k}}\right)_{2}$ on each $\Sigma\cap U_{p}$ patches together to give us a
function $\Xi_{\Sigma,12}$ defined on $D_{\Sigma}$ such that
$d\Xi_{\Sigma,12}=\left(\frac{\partial\xi_{\Sigma}}{\partial
u^{k}}\right)_{1}-\left(\frac{\partial\xi_{\Sigma}}{\partial
u^{k}}\right)_{2}$.

To see that (16) is true, we dissect $C$ into closed segments
$C_{i},i=1,\cdots,N$ such that $C_{i}\subset\Sigma\cap U_{p}$, then apply the
Leibniz integral rule on each segment $C_{i}$. Since the endpoints of each
$C_{i}$ vary with $u^{k}$ we also get the boundary term contribution, but the
sum of these contributions from each $C_{i}$ will be zero because $C$ is
closed. Note that there is no ambiguity in what connection to use on the
right-hand-side of (16) because the result will only be differed by an exact
differential, which vanishes on a closed contour $C$. ∎

######  Lemma 2.1.18.

([21, Section 5.1]) The symplectic vector bundle
$(\mathcal{H}\rightarrow\mathcal{B},\Omega_{\mathcal{H}},\nabla_{GM})$ is flat
and $\Omega_{\mathcal{H}}$ is $\nabla_{GM}$-covariantly constant.

###### Proof.

Let
$d_{\nabla_{GM}}:\Gamma(\mathcal{B}_{\Sigma_{0}},\Omega^{r}_{\mathcal{B}}\otimes\mathcal{H})\rightarrow\Gamma(\mathcal{B}_{\Sigma_{0}},\Omega^{r+1}_{\mathcal{B}}\otimes\mathcal{H})$
be the exterior covariant derivative. The condition that $\nabla_{GM}$ is flat
is the same as the vanishing of the curvature tensor
$R_{\nabla_{GM}}=d_{\nabla_{GM}}\circ d_{\nabla_{GM}}=0$. For
$\psi=s[\xi]\in\Gamma(\mathcal{B}_{\Sigma_{0}},\Omega^{r}_{\mathcal{B}}\otimes\mathcal{H}),s\in\Gamma(\mathcal{B}_{\Sigma_{0}},\Omega^{r}_{\mathcal{B}})$
and $[\xi]\in\Gamma(\mathcal{B}_{\Sigma_{0}},\mathcal{H})$, we have

$d_{\nabla_{GM}}\psi=(ds)[\xi]+(-1)^{r}s\wedge\nabla_{GM}[\xi]=(ds)[\xi]+(-1)^{r}s\wedge\sum_{k=1}^{g}\left[\frac{\partial\xi}{\partial
u^{k}}\right]du^{k}$

where we have used Lemma 2.1.17:
$\nabla_{GM}[\xi]=\sum_{k=1}^{g}\left[\frac{\partial\xi}{\partial
u^{k}}\right]du^{k}$. Therefore,

$\displaystyle d^{2}_{\nabla_{GM}}\psi$
$\displaystyle=(d^{2}s)[\xi]+(-1)^{r}(ds)\wedge\sum_{k=1}^{g}\left[\frac{\partial\xi}{\partial
u^{k}}\right]du^{k}$
$\displaystyle\qquad+(-1)^{r+1}(ds)\wedge\sum_{k=1}^{g}\left[\frac{\partial\xi}{\partial
u^{k}}\right]du^{k}+(-1)^{r}s\wedge\sum_{k,l=1}^{g}\left[\frac{\partial\xi}{\partial
u^{k}\partial u^{l}}\right]du^{k}\wedge du^{l}=0$

which means $\nabla_{GM}$ is a flat connection.

Because the vector bundle $\mathcal{H}\rightarrow\mathcal{B}$ is flat,
parallel transport on $\mathcal{H}\rightarrow\mathcal{B}_{\Sigma_{0}}$ using
the Gauss-Manin connection is path-independent. Let us denote by
$\Gamma_{[\Sigma]}^{[\Sigma^{\prime}]}[\xi]\in\mathcal{H}_{\Sigma^{\prime}}$
the parallel transport of $[\xi]\in\mathcal{H}_{\Sigma}$ from the point
$[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ to
$[\Sigma^{\prime}]\in\mathcal{B}_{\Sigma_{0}}$. Since parallel transport
preserves $A$ and $B$-periods, we have
$\Omega_{\mathcal{H}}(\Gamma_{[\Sigma]}^{[\Sigma^{\prime}]}[\xi_{1}],\Gamma_{[\Sigma]}^{[\Sigma^{\prime}]}[\xi_{2}])=\Omega_{\mathcal{H}}([\xi_{1}],[\xi_{2}])$
proving that $\Omega_{\mathcal{H}}$ is $\nabla_{GM}$-covariantly constant. ∎

We will find in Section 2.5 that the connection given by taking derivatives
keeping the direction of foliation leaves constant is particularly important
and we denote it by $\nabla_{\mathcal{F}}$. Let us write down
$\phi_{\Sigma}(v)$ and $\boldsymbol{\phi}_{\Sigma}(v)$ using
$\nabla_{\mathcal{F}}$. Let us cover $\Sigma$ with a collection of
$(\mathcal{F},\Omega_{S})$-charts
$\mathcal{U}=\\{(U_{p\in\sigma},x_{p\in\sigma},y_{p\in\sigma})\\}$. On each
$\Sigma\cap U_{p}$, we write
$\nabla_{\mathcal{F}}=\sum_{k=1}^{g}du^{k}\frac{\partial}{\partial
u^{k}}|_{x_{p}=const}$. Straightforwardly, we have

$\displaystyle\phi_{\Sigma}|_{\Sigma\cap U_{p}}$
$\displaystyle=\sum_{k=1}^{g}\Omega_{S}(n_{v_{k}},.)|_{\Sigma\cap
U_{p}}du^{k}=\sum_{k=1}^{g}\Omega_{S}\left(\frac{\partial x_{p}}{\partial
u^{k}}\Big{|}_{x_{p}=const}\frac{\partial}{\partial x_{p}}+\frac{\partial
y_{p}}{\partial u^{k}}\Big{|}_{x_{p}=const}\frac{\partial}{\partial
y_{p}},.\right)\Big{|}_{\Sigma\cap U_{p}}du^{k}$
$\displaystyle=\sum_{k=1}^{g}\Omega_{S}\left(\frac{\partial y_{p}}{\partial
u^{k}}\Big{|}_{x_{p}=const}\frac{\partial}{\partial
y_{p}},.\right)\Big{|}_{\Sigma\cap
U_{p}}du^{k}=-\sum_{k=1}^{g}\left(\frac{\partial y_{p}}{\partial
u^{k}}\Big{|}_{x_{p}=const}dx_{p}|_{\Sigma\cap U_{p}}\right)du^{k}$
$\displaystyle=-\nabla_{\mathcal{F}}(y_{p}dx_{p}|_{\Sigma\cap U_{p}}),$

and so $\phi_{\Sigma}(v)|_{\Sigma\cap
U_{p}}=-\iota_{v_{k}}\nabla_{\mathcal{F}}(y_{p}dx_{p}|_{\Sigma\cap U_{p}})$.
Lastly, we simply put
$\boldsymbol{\phi}_{\Sigma}(v)=[\phi_{\Sigma}(v)]\in\mathcal{H}_{\Sigma}$.

######  Remark 2.1.19.

Note that, from (14) we have $\frac{\partial y_{p}}{\partial
u^{k}}|_{x_{p}=const}=-\frac{\partial\mathcal{P}_{p}/\partial
u^{k}}{\partial\mathcal{P}_{p}/\partial y_{p}}$. Using

$d\mathcal{P}_{p}|_{\Sigma\cap U_{p}}=\frac{\partial\mathcal{P}_{p}}{\partial
x_{p}}dx_{p}|_{\Sigma\cap U_{p}}+\frac{\partial\mathcal{P}_{p}}{\partial
y_{p}}dy_{p}|_{\Sigma\cap U_{p}}=0,$

we can see that $\frac{\partial\mathcal{P}_{p}}{\partial y_{p}}$ vanishes with
order $1$ zero exactly at ramification points because $dx_{p}|_{\Sigma\cap
U_{p}}$ does. Suppose that $\Sigma\cap U_{p}$ contains a ramification point
$r_{\alpha}$, then $\frac{\partial y_{p}}{\partial u^{k}}|_{x_{p}=const}$ has
a simple pole at $r_{\alpha}$. However, $\frac{\partial y_{p}}{\partial
u^{k}}|_{x_{p}=const}dx_{p}|_{\Sigma\cap U_{p}}$ remains finite at
$r_{\alpha}$ because the simple pole cancels out with the simple zero of
$dx_{p}|_{\Sigma\cap U_{p}}$. From this analysis, we conclude that
$\phi_{\Sigma}(v)$ is holomorphic on $\Sigma$, i.e.
$\phi_{\Sigma}(v)\in\Gamma(\Sigma,\Omega^{1}_{\Sigma})$, as expected since the
definition of $\phi_{\Sigma}$ is independent of our choice of the connection
(see Lemma 2.1.16). We refer to the proof of Lemma 2.5.35 for a version of
this argument repeated using the standard local coordinate $z_{\alpha}$.

Let us compute the exterior covariant derivative of $\phi_{\Sigma}$ using
$\nabla_{\mathcal{F}}$, we have

$\displaystyle d_{\nabla_{\mathcal{F}}}\phi_{\Sigma}|_{\Sigma\cap U_{p}}$
$\displaystyle=d_{\nabla_{\mathcal{F}}}\left(\sum_{k=1}^{g}\Omega_{S}(n_{v_{k}},.)|_{\Sigma}du^{k}\right)\Big{|}_{\Sigma\cap
U_{p}}=d_{\nabla_{\mathcal{F}}}\left(-\nabla_{\mathcal{F}}(y_{p}dx_{p}|_{\Sigma\cap
U_{p}})\right)$
$\displaystyle=\sum_{k,l=1}^{g}\left(\frac{\partial^{2}y_{p}}{\partial
u^{k}\partial u^{l}}\Big{|}_{x_{p}=const}dx_{p}|_{\Sigma\cap
U_{p}}\right)du^{k}\wedge du^{l}=0$

for any $(\mathcal{F},\Omega_{S})$-chart $(U_{p},x_{p},y_{p})\in\mathcal{U}$.
This means $\nabla_{\mathcal{F}}\phi_{\Sigma}=0$ and therefore,

$d_{\nabla_{GM}}\boldsymbol{\phi}_{\Sigma}=d_{\nabla_{GM}}\left[\phi_{\Sigma}\right]=\left[d_{\nabla_{\mathcal{F}}}\phi_{\Sigma}\right]=0,$

where we have applied Lemma 2.1.17 at the second equality. Since
$\mathcal{B}_{\Sigma_{0}}$ is contractible, by the Poincaré Lemma (see Lemma
2.1.23 below), $d_{\nabla_{GM}}\boldsymbol{\phi}=0$ on
$\mathcal{B}_{\Sigma_{0}}$ implies that there exists

(17)
$\text{$\boldsymbol{\theta}\in\Gamma(\mathcal{B}_{\Sigma_{0}},\mathcal{H})$
such that $\nabla_{GM}\boldsymbol{\theta}=\boldsymbol{\phi}$}.$

####  2.1.3. The embedding of $\mathcal{B}_{\Sigma_{0}}$ into
$\mathcal{H}_{\Sigma_{0}}$ as a Lagrangian submanifold

Let us consider the map $\boldsymbol{\Phi}_{\Sigma_{0}}$ embedding the
neighbourhood $\mathcal{B}_{\Sigma_{0}}\subseteq\mathcal{B}$ into a fiber
$\mathcal{H}_{\Sigma_{0}}$ at $[\Sigma_{0}]\in\mathcal{B}_{\Sigma_{0}}$.
Following [21], we first define the affine connection
$\nabla_{GM}+\boldsymbol{\phi}$ on $\mathcal{H}$:

$\left(\nabla_{GM}+\boldsymbol{\phi}\right)[\xi]:=\nabla_{GM}[\xi]+\boldsymbol{\phi}$

for any section $[\xi]\in\Gamma(\mathcal{B}_{\Sigma_{0}},\mathcal{H})$. The
map
$\boldsymbol{\Phi}_{\Sigma_{0}}:\mathcal{B}_{\Sigma_{0}}\rightarrow\mathcal{H}_{\Sigma_{0}}$
is defined by sending a point $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ to the
image of the zero vector $0_{\Sigma}\in\mathcal{H}_{\Sigma}$ under the
parallel transport from point $[\Sigma_{0}]$ to $[\Sigma]$ using the affine
connection $\nabla_{GM}+\boldsymbol{\phi}$. In general, given
$[\Sigma],[\Sigma_{1}]\in\mathcal{B}_{\Sigma_{0}}$ the parallel transport of
$0_{\Sigma_{1}}\in\mathcal{H}_{\Sigma_{1}}$ to $\mathcal{H}_{\Sigma}$ is
unique, path-independent and it is given by

$v_{\Sigma_{1}}(\Sigma):=-\boldsymbol{\theta}_{\Sigma}+\Gamma_{[\Sigma_{1}]}^{[\Sigma]}\boldsymbol{\theta}_{\Sigma_{1}}.$

It follows that

$\boldsymbol{\Phi}_{\Sigma_{0}}(\Sigma):=v_{[\Sigma]}(\Sigma_{0})=-\boldsymbol{\theta}_{\Sigma_{0}}+\Gamma_{[\Sigma]}^{[\Sigma_{0}]}\boldsymbol{\theta}_{\Sigma}.$

We denote the image of $\boldsymbol{\Phi}_{\Sigma_{0}}$ by

$\mathcal{L}_{\Sigma_{0}}:=\text{im}\boldsymbol{\Phi}_{\Sigma_{0}}\subset\mathcal{H}_{\Sigma_{0}}.$

Let $v=\sum_{k=1}^{g}v^{k}\frac{\partial}{\partial u^{k}}\in
T_{[\Sigma]}\mathcal{B}_{\Sigma_{0}}$ be any tangent vector, then

$\iota_{v}d\boldsymbol{\Phi}_{\Sigma_{0}}(\Sigma)=\sum_{k=1}^{g}v^{k}\frac{\partial\boldsymbol{\Phi}_{\Sigma_{0}}(\Sigma)}{\partial
u^{k}}=\Gamma_{[\Sigma]}^{[\Sigma_{0}]}\iota_{v}\nabla_{GM}\boldsymbol{\theta}_{\Sigma}=\Gamma_{[\Sigma]}^{[\Sigma_{0}]}\iota_{v}\boldsymbol{\phi}_{\Sigma}$

is a tangent vector to
$\mathcal{L}_{\Sigma_{0}}\subset\mathcal{H}_{\Sigma_{0}}$. Since
$v\mapsto\iota_{v}d\boldsymbol{\Phi}_{\Sigma_{0}}$ is injective because
$\boldsymbol{\phi}_{\Sigma}$ is injective, by assuming that
$\mathcal{B}_{\Sigma_{0}}$ is small enough we have that
$\boldsymbol{\Phi}_{\Sigma_{0}}:\mathcal{B}_{\Sigma_{0}}\rightarrow\mathcal{H}_{\Sigma_{0}}$
embeds $\mathcal{B}_{\Sigma_{0}}$ as a submanifold in
$\mathcal{H}_{\Sigma_{0}}$.

######  Lemma 2.1.20.

([21, Section 5.1]) $\mathcal{L}_{\Sigma_{0}}$ is a Lagrangian submanifold of
$\mathcal{H}_{\Sigma_{0}}$.

###### Proof.

Since
$\mathcal{H}_{\Sigma_{0}}=H^{1}(\Sigma_{0},\mathbb{C})\cong\mathbb{C}^{2g}$
and
$\dim\mathcal{L}_{\Sigma_{0}}=\dim\mathcal{B}_{\Sigma_{0}}=\frac{1}{2}\dim\mathcal{H}_{\Sigma_{0}}$
it remains for us to check that
$\Omega_{\mathcal{H}}|_{\mathcal{L}_{\Sigma_{0}}}=0$. For $v_{i}\in
T_{[\Sigma]}\mathcal{B}_{\Sigma_{0}},i=1,2$ we have
$\iota_{v_{i}}d\boldsymbol{\Phi}_{\Sigma_{0}}\in
T_{\boldsymbol{\Phi}_{\Sigma_{0}}(\Sigma)}\mathcal{L}_{\Sigma_{0}}$ and

$\Omega_{\mathcal{H}}(\iota_{v_{1}}d\boldsymbol{\Phi}_{\Sigma_{0}},\iota_{v_{2}}d\boldsymbol{\Phi}_{\Sigma_{0}})=\Omega_{\mathcal{H}}(\Gamma_{[\Sigma]}^{[\Sigma_{0}]}\iota_{v_{1}}\boldsymbol{\phi}_{\Sigma},\Gamma_{[\Sigma]}^{[\Sigma_{0}]}\iota_{v_{2}}\boldsymbol{\phi}_{\Sigma})=\Omega_{\mathcal{H}}(\iota_{v_{1}}\boldsymbol{\phi}_{\Sigma},\iota_{v_{2}}\boldsymbol{\phi}_{\Sigma})=0$

because the image
$\text{im}\boldsymbol{\phi}_{\Sigma}=\Gamma(\Sigma,\Omega^{1}_{\Sigma})$ is a
Lagrangian subspace of $\mathcal{H}_{\Sigma}$. ∎

Let us consider a trivialization of
$\mathcal{H}\rightarrow\mathcal{B}_{\Sigma_{0}}$ by choosing $A,B$-symplectic
basis cycles of $H_{1}(\Sigma,\mathbb{C})$ for each
$[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ and identify
$\mathcal{H}_{\Sigma}\cong\mathbb{C}^{2g}$ as in Remark 2.1.13:
$[\xi]\mapsto\left(\oint_{A_{1}}\xi,\cdots,\oint_{A_{g}}\xi,\oint_{B_{1}}\xi,\cdots,\oint_{B_{g}}\xi\right)\in\mathbb{C}^{2g}$.
Let us define

$\mathbf{a}^{i}(\Sigma):=\oint_{A_{i}}\boldsymbol{\theta}_{\Sigma},\qquad\mathbf{b}_{i}(\Sigma):=\oint_{B_{i}}\boldsymbol{\theta}_{\Sigma},\qquad
i=1,...,g.$

We will often write $\mathbf{a}^{i}(\Sigma)$ and $\mathbf{b}_{i}(\Sigma)$
simply as $\mathbf{a}^{i}$ and $\mathbf{b}_{i}$ when they do not cause any
ambiguity.

######  Lemma 2.1.21.

$\\{\mathbf{a}^{1},...,\mathbf{a}^{g}\\}$ is a coordinate system for
$\mathcal{B}_{\Sigma_{0}}$.

###### Proof.

The matrix $\left(\frac{\partial\mathbf{a}^{i}}{\partial u^{j}}\right)$ is
invertible because $\frac{\partial\mathbf{a}^{i}}{\partial
u^{j}}=\frac{\partial}{\partial
u^{j}}\oint_{A_{i}}\boldsymbol{\theta}_{\Sigma}=\oint_{A_{i}}\iota_{\frac{\partial}{\partial
u^{j}}}\nabla_{GM}\boldsymbol{\theta}_{\Sigma}=\oint_{A_{i}}\iota_{\frac{\partial}{\partial
u^{j}}}\boldsymbol{\phi}_{\Sigma}$. So if there exists
$(v^{i})\in\mathbb{C}^{g}$ such that
$\sum_{j=1}^{g}v^{j}\frac{\partial\mathbf{a}^{i}}{\partial u^{j}}=0,i=1,...,g$
then $\phi_{\Sigma}(v)=\sum_{j=1}^{g}v^{j}\iota_{\frac{\partial}{\partial
u^{j}}}\phi_{\Sigma}\in\Gamma(\Sigma,\Omega^{1}_{\Sigma}),v:=\sum_{i=1}^{g}v^{i}\frac{\partial}{\partial
u^{j}}$ is a holomorphic form on $\Sigma$ with zero $A$-periods, therefore
$\phi_{\Sigma}(v)=0$. But we know that $v\mapsto\iota_{v}\phi_{\Sigma}$ is
injective, therefore $v=0$. Hence
$\mathbf{a}^{i}:=\mathbf{a}^{i}(u^{1},...,u^{g}),i=1,...,g$ is a holomorphic
coordinates transformation. ∎

So $\mathbf{b}_{i}$ can be written as a function of $\\{\mathbf{a}^{i}\\}$. We
can write
$\boldsymbol{\Phi}_{\Sigma_{0}}:\mathcal{B}_{\Sigma_{0}}\rightarrow\mathcal{H}_{\Sigma_{0}}$
using this trivialization as

$[\Sigma]:=(\mathbf{a}^{1}(\Sigma),...,\mathbf{a}^{g}(\Sigma))\mapsto\boldsymbol{\Phi}(\Sigma)=(\mathbf{a}^{1}(\Sigma)-\mathbf{a}^{1}(\Sigma_{0}),...,\mathbf{a}^{g}(\Sigma)-\mathbf{a}^{g}(\Sigma_{0}),\mathbf{b}^{1}(\Sigma)-\mathbf{b}^{1}(\Sigma_{0}),...,\mathbf{b}^{g}(\Sigma)-\mathbf{b}^{g}(\Sigma_{0}))$

and

$\frac{\partial\boldsymbol{\Phi}_{\Sigma_{0}}(\Sigma)}{\partial\mathbf{a}^{i}}=\left(0,...,1,...,0,\frac{\partial\mathbf{b}_{1}}{\partial\mathbf{a}^{i}},...,\frac{\partial\mathbf{b}_{g}}{\partial\mathbf{a}^{i}}\right)\in
T_{\boldsymbol{\Phi}_{\Sigma_{0}}(\Sigma)}\text{im}\boldsymbol{\Phi}_{\Sigma_{0}},\qquad
i=1,...,g.$

It follows from Lemma 2.1.20 that for any $i,j=1,...,g$ we have

$0=\Omega_{\mathcal{H}}\left(\frac{\partial\boldsymbol{\Phi}_{\Sigma_{0}}(\Sigma)}{\partial\mathbf{a}^{i}},\frac{\partial\boldsymbol{\Phi}_{\Sigma_{0}}(\Sigma)}{\partial\mathbf{a}^{j}}\right)=\frac{\partial\mathbf{b}_{i}}{\partial\mathbf{a}^{j}}-\frac{\partial\mathbf{b}_{j}}{\partial\mathbf{a}^{i}}.$

Therefore, by an application of Poincaré lemma, there exists a holomorphic
function
$\mathfrak{F}_{\Sigma_{0}}=\mathfrak{F}_{\Sigma_{0}}(\mathbf{a}^{1},...,\mathbf{a}^{g})$
on $\mathcal{B}_{\Sigma_{0}}$ such that

(18)
$\mathbf{b}_{i}=\frac{\partial\mathfrak{F}_{\Sigma_{0}}(\mathbf{a}^{1},...,\mathbf{a}^{g})}{\partial\mathbf{a}^{i}},\qquad
i=1,...,g.$

The section
$\boldsymbol{\theta}\in\Gamma(\mathcal{B}_{\Sigma_{0}},\mathcal{H})$ is
uniquely determined up to an addition of a parallel section
$\mathbf{c}\in\Gamma(\mathcal{B}_{\Sigma_{0}},\mathcal{H}),\nabla_{GM}\mathbf{c}=0$.
Therefore, given the moduli space $\mathcal{B}$ of $\mathcal{F}$-transversal
curves, the function $\mathfrak{F}_{\Sigma_{0}}$ is unique up to an addition
of a linear term.

######  Definition 2.1.22.

Given the moduli space $\mathcal{B}$ of $\mathcal{F}$-transversal curves
inside a foliated symplectic surface $(S,\Omega_{S},\mathcal{F})$, the
_prepotential_
$\mathfrak{F}_{\Sigma_{0}}=\mathfrak{F}_{\Sigma_{0}}(\mathbf{a}^{1},\cdots,\mathbf{a}^{g})$
is a holomorphic function locally defined on the open neighbourhood
$\mathcal{B}_{\Sigma_{0}}\ni[\Sigma_{0}]$ which relates $A$ and $B$-periods of
$\boldsymbol{\theta}_{\Sigma}\in\mathcal{H}_{\Sigma}$ according to (18).

The prepotential is the generating function of the Lagrangian submanifold
$\mathcal{L}_{\Sigma_{0}}:=\text{im}\boldsymbol{\Phi}_{\Sigma_{0}}\subset\mathcal{H}_{\Sigma_{0}}$.
In other words, the prepotential $\mathfrak{F}_{\Sigma_{0}}$ naturally arises
from the study of the deformation of curves $\Sigma$ embedded in a foliated
symplectic surface $(S,\Omega_{S},\mathcal{F})$ around some neighbourhood
$\mathcal{B}_{\Sigma_{0}}$ of $[\Sigma_{0}]$ in the moduli space
$\mathcal{B}$.

####  2.1.4. Generalization

Previously, we considered a particular vector bundle
$\mathcal{H}\rightarrow\mathcal{B}$ and an embedding of a neighbourhood
$\mathcal{B}_{\Sigma_{0}}$ of $[\Sigma_{0}]\in\mathcal{B}$ into a fiber
$\mathcal{H}_{\Sigma_{0}}$ using the affine connection
$\nabla_{GM}+\boldsymbol{\phi}$. We showed that the image is a Lagrangian
submanifold. This type of embedding can be done with a more general class of
symplectic vector bundles. Let us end this section by giving a discussion of
the general setting. First, let us state and proof a version of Poincaré Lemma
for holomorphic flat vector bundles:

######  Lemma 2.1.23.

Let $(\mathcal{E}\rightarrow X,\nabla)$ be a $n$-dimensional holomorphic
vector bundle over a contractible $m$ dimensional complex manifold $X$ with a
flat connection $\nabla$ and let
$d_{\nabla}:\Omega^{r}_{X}\otimes\mathcal{E}\rightarrow\Omega^{r+1}_{X}\otimes\mathcal{E}$
be the corresponding exterior covariant derivative. Then given a section
$\xi\in\Gamma(X,\Omega^{r}_{X}\otimes\mathcal{E})$ such that
$d_{\nabla}\xi=0$, there exists a section
$\eta\in\Gamma(X,\Omega^{r-1}_{X}\otimes\mathcal{E})$ such that
$\xi=d_{\nabla}\eta$.

###### Proof.

Let us pick a basis $\\{e_{i,x_{0}}\\}$ of $\mathcal{E}_{x_{0}}$ at some point
$x_{0}\in X$ and let the section $e_{i}\in\Gamma(X,\mathcal{E})$ be given at
any $x\in X$ by $e_{i,x}:=\Gamma_{x_{0}}^{x}e_{i,x_{0}}$, where
$\Gamma_{x_{0}}^{x}$ is the parallel transport on $\mathcal{E}$ from $x_{0}$
to $x$ using the flat connection $\nabla$. Since $\mathcal{E}\rightarrow X$ is
flat, the parallel transport with respect to the connection $\nabla$ is path-
independent. On the other hand, the existence and uniqueness of the parallel
transport of $e_{i,x_{0}}$ from $x_{0}$ to $x$ along any path follows from
Picard–Lindelöf theorem. It follows that
$e_{i,x}:=\Gamma_{x_{0}}^{x}e_{i,x_{0}}\in\mathcal{E}_{x}$ is well-defined for
$x\in X$. Therefore, we have parallel sections $\nabla
e_{i}\in\Gamma(X,\mathcal{E})$, $\nabla e_{i}=0$, and $\\{e_{i,x}\\}$ is a
basis of the fiber $\mathcal{E}_{x}$ for all $x\in X$. A section
$\xi\in\Gamma(X,\Omega^{r}_{X}\otimes\mathcal{E})$ can be expressed using the
local coordinates $\\{x^{1},\cdots,x^{m}\\}$ of $X$ as
$\xi:=\sum_{k=1}^{n}\xi_{k}e_{k}$ where
$\xi_{k}:=\sum_{i_{1},\cdots,i_{r}=1}^{m}\xi_{i_{1}\cdots
i_{r};k}(x)dx^{i_{1}}\wedge\cdots\wedge
dx^{i_{r}}\in\Gamma(X,\Omega^{r}_{X})$. Using $\nabla e_{k}=0$, we have that

$d_{\nabla}\xi=\sum_{k=1}^{n}\left((d\xi_{k})e_{k}+(-1)^{r}\xi_{k}\wedge\nabla
e_{k}\right)=\sum_{k=1}^{n}(d\xi_{k})e_{k}.$

Consequently, if $d_{\nabla}\xi=0$ then $d\xi_{k}=0$ for all $k=1,\cdots,n$.
Since $X$ is contractible, we can apply the usual Poincaré Lemma for
differential forms to $\xi_{k}$ and conclude that there exists
$\eta_{k}\in\Gamma(X,\Omega^{r-1}_{X})$ such that $d\eta_{k}=\xi_{k}$. It is
now a matter of simple calculation to verify that $d_{\nabla}\eta=\xi$ where
$\eta=\sum_{k=1}^{n}\eta_{k}e_{k}$. ∎

Let us now present the general version of the embedding
$\mathcal{B}_{\Sigma_{0}}\rightarrow\mathcal{H}_{\Sigma_{0}}$:

######  Proposition 2.1.24.

Let $(\mathcal{E}\rightarrow X,\Omega_{\mathcal{E}},\nabla,\phi)$ be a
$2n$-dimensional holomorphic symplectic vector bundle over a $n$-dimensional
complex manifold $X$ with a flat connection $\nabla$, a $\nabla$-covariantly
constant symplectic form $\Omega_{\mathcal{E}}$ and
$\phi\in\Gamma(X,\Omega^{1}_{X}\otimes\mathcal{E})$, $d_{\nabla}\phi=0$ such
that for each $x\in X$, $\phi_{x}:T_{x}X\rightarrow\mathcal{E}_{x}$ embeds
$T_{x}X$ as a Lagrangian subspace of $\mathcal{E}_{x}$. Fix a reference point
$x_{0}\in X$. There exists an open neighbourhood $X_{x_{0}}\subseteq X$ of
$x_{0}$ such that the map
$\Phi_{x_{0}}:X_{x_{0}}\rightarrow\mathcal{E}_{x_{0}}$, given by sending $x\in
X_{x_{0}}$ to the image of $0_{x}\in\mathcal{E}_{x}$ under the parallel
transport from $x$ to $x_{0}$ using the affine connection $\nabla+\phi$, is a
well-defined embedding of $X_{x_{0}}$ into a Lagrangian submanifold
$\mathcal{L}_{x_{0}}:=\text{im}\Phi_{x_{0}}\subseteq\mathcal{E}_{x_{0}}$.

###### Proof.

Let $X_{x_{0}}\subseteq$ be a contractible open neighbourhood of $x_{0}\in X$.
Since $d_{\nabla}\phi=0$, by Lemma 2.1.23, there exists a section
$\theta\in\Gamma(X_{x_{0}},\mathcal{E})$ such that
$d_{\nabla}\theta=\nabla\theta=\phi$. For $x_{1}\in X_{x_{0}}$, the section
$v_{x_{1}}\in\Gamma(X_{x_{0}},\mathcal{E}),v_{x_{1}}(x):=-\theta_{x}+\Gamma_{x_{1}}^{x}\theta_{x_{1}}$
satisfies $v_{x_{1}}(x_{1})=0$ and $\nabla v_{x_{1}}(x)=\phi_{x}$. For all
$x_{1},x\in X_{x_{0}}$, the parallel transport of
$0_{x_{1}}\in\mathcal{E}_{x_{1}}$ to $\mathcal{E}_{x}$ using the affine
connection $\nabla+\phi$ is path-independent and it is uniquely given by
$v_{x_{1}}(x)$. Therefore, the map
$\Phi_{x_{0}}:X_{x_{0}}\rightarrow\mathcal{E}_{x_{0}}$ is given by

$\Phi_{x_{0}}(x):=v_{x}(x_{0})=-\theta_{x_{0}}+\Gamma_{x}^{x_{0}}\theta_{x}.$

Let us compute the derivatives of $\Phi_{x_{0}}$. Suppose that we can cover
$X_{x_{0}}$ with a single coordinates chart $(X_{x_{0}},x^{1},...,x^{n})$ then

$\displaystyle d\Phi_{x_{0}}(x)$
$\displaystyle=\sum_{i=1}^{n}dx^{i}\frac{\partial\Phi_{x_{0}}(x)}{\partial
x^{i}}=\sum_{i=1}^{n}dx^{i}\frac{\partial}{\partial
x^{i}}\Gamma_{x}^{x_{0}}\left(\sum_{k=1}^{2n}\theta_{k}(x)e_{k,x}\right)=\sum_{i=1}^{n}dx^{i}\frac{\partial}{\partial
x^{i}}\left(\sum_{k=1}^{2n}\theta_{k}(x)e_{k,x_{0}}\right)$
$\displaystyle=\sum_{i=1}^{n}dx^{i}\sum_{k=1}^{2n}\frac{\partial\theta_{k}(x)}{\partial
x^{i}}e_{k,x_{0}}=\sum_{i=1}^{n}dx^{i}\Gamma_{x}^{x_{0}}\left(\sum_{k=1}^{2n}\frac{\partial\theta_{k}(x)}{\partial
x^{i}}e_{k,x}\right)=\Gamma_{x}^{x_{0}}\nabla\left(\sum_{k=1}^{2n}\theta_{k}(x)e_{k,x}\right)=\Gamma_{x}^{x_{0}}\phi_{x}.$

For every $v\in T_{x}X_{x_{0}}$, we have $\iota_{v}d\Phi_{x_{0}}(x)\in
T_{\Phi_{x_{0}}(x)}\mathcal{L}_{x_{0}}$. Furthermore, given $v_{1},v_{2}\in
T_{x}X_{x_{0}}$ we have

$\Omega_{\mathcal{E}}(\iota_{v_{1}}d\Phi_{x_{0}}(x),\iota_{v_{2}}d\Phi_{x_{0}}(x))=\Omega_{\mathcal{E}}(\Gamma_{x}^{x_{0}}\iota_{v_{1}}\phi_{x},\Gamma_{x}^{x_{0}}\iota_{v_{2}}\phi_{x})=\Omega_{\mathcal{E}}(\iota_{v_{1}}\phi_{x},\iota_{v_{2}}\phi_{x})=0$

by the assumption that $\phi_{x}$ embeds $T_{x}X$ as a Lagrangian subspace of
$\mathcal{E}_{x}$. Since $\phi_{x}$ is injective, the rank of
$d\Phi_{x_{0}}(x)$ is $n$ for all $x\in X_{x_{0}}$. It follows that
$\dim\mathcal{L}_{x_{0}}=\frac{1}{2}\dim\mathcal{E}_{x_{0}}$, hence
$\mathcal{L}_{x_{0}}\subseteq\mathcal{E}_{x_{0}}$ is a Lagrangian submanifold.
∎

The proof of Proposition 2.1.24 only works when $\mathcal{E}\rightarrow X$ is
finite-dimensional, hence the result need not hold when the dimension is
infinite. Nevertheless, Proposition 2.1.24 gives us a guideline of what to
expect when we consider the analogous embedding in the infinite-dimensional
case. We will see this in Section 2.4 where we consider the infinite-
dimensional vector bundle $W^{\epsilon,M}\rightarrow Discs^{M}$.

###  2.2. Airy Structures

In this section, we review some basic terminologies and facts about Airy
Structures as introduced in [21] .

####  2.2.1. Topological vector spaces and symplectic structures

######  Definition 2.2.1.

A $k$-_topological vector space_ $V$ is a vector space equipped with a
Hausdorff topology over a topological field $k$ such that

(19) $\text{vector addition : }V\times V\rightarrow V,\qquad\text{scalar
multiplication : }k\times V\rightarrow V$

are continuous.

We need a topology on $V$ to define the dual space $V^{*}:=Hom(V,k)$ of $V$ as
the vector space of continuous linear functionals on $V$. Typically, we equip
$V^{*}$ with the weakest topology such that the map $f\in
V^{*}\mapsto\phi_{v}(f):=f(v)\in k$ is continuous for all $v\in V$, making
$V^{*}$ also a topological vector space. Therefore, we have $\phi_{v}\in
V^{**}:=Hom(V^{*},k)$ by definition, where $V^{**}$ is the vector space of
continuous linear functionals on $V^{*}$. We will not assume vector spaces in
this section to be finite-dimensional. To help with the notation when listing
the basis or coordinates we introduce an index set $\mathbb{I}$. For example,
we write the basis of $V$ as $\\{e_{i\in\mathbb{I}}\\}$ and take
$\mathbb{I}=\\{1,...,n\\}$ if $V$ is finite $n$-dimensional or
$\mathbb{I}=\mathbb{Z}_{>0}$ if $V$ is countably infinite-dimensional.

######  Remark 2.2.2.

The topology of $V$ will only be important to us when $n$ is infinite because
when $n$ is finite, linear functionals on $V$ are automatically continuous.

######  Definition 2.2.3.

A symplectic topological vector space $(W,\Omega)$ is a pair consisting of a
topological vector space $W$ and a continuous antisymmetric bilinear form
$\Omega:W\times W\rightarrow k$ such that $w\mapsto\Omega(w,.)$ is an
isomorphism $W\xrightarrow{\cong}W^{*}$. We call $\Omega$ the _symplectic
form_ of $W$.

Let us introduce the example of symplectic topological vector space
$(W,\Omega)$ which will be important in defining Airy structures. We will
exclusively work with $k:=\mathbb{C}$ equipped with a discrete topology. Let
$V$ be a topological vector space with a discrete topology and a basis
$\\{e^{i\in\mathbb{I}}\\}$ over $\mathbb{C}$. Any vector $v\in V$ can be
written as $v=\sum_{i\in\mathbb{I}}y_{i}e^{i}$ where only a finite number of
$y_{i}\in\mathbb{C}$ are non-zero. Let $\\{f_{i\in\mathbb{I}}\\}\subseteq
V^{*}$ be the set of linearly independent vectors such that
$f_{i}(e^{j})=\delta_{i}^{j}$. Then elements of $V^{*}$ are
$f=\sum_{i\in\mathbb{I}}x^{i}f_{i}$ for $x^{i}\in\mathbb{C}$. We call
$\\{y_{i\in\mathbb{I}}\\}$ and $\\{x^{i\in\mathbb{I}}\\}$ coordinates of $V$
and $V^{*}$ respectively. Equip $V^{*}$ with the weakest topology such that
the map $f\mapsto\phi_{v}(f):=f(v)\in\mathbb{C}$ is continuous for all $v\in
V$. The open sets in $V^{*}$ take the form $f+U_{\mathbb{J}}\subseteq V^{*}$
where $f\in V^{*},\mathbb{J}\subseteq\mathbb{I}$ is a finite set and.

(20)
$U_{\mathbb{J}}:=\left\\{\sum_{i\in\mathbb{I}\setminus\mathbb{J}}x^{i}f_{i}\
|\ x^{i}\in\mathbb{C}\right\\}\subseteq V^{*}.$

######  Remark 2.2.4.

If $V$ is finite dimensional then $\\{f_{i\in\mathbb{I}}\\}$ is a basis of
$V^{*}$. When the dimension is infinite, $\\{f_{i\in\mathbb{I}}\\}$ is a
_Schauder basis_ of $V^{*}$ not an ordinary basis because their finite linear
combination do not span $V^{*}$. However, let us also refer to
$\\{f_{i\in\mathbb{I}}\\}$ as a basis of $V^{*}$ to simplify the language.

Finally, we define $W:=V\oplus V^{*}$ with standard symplectic form
$\Omega:=\sum_{i\in\mathbb{I}}dy_{i}\wedge dx^{i}$ where
$dy_{i}:=f_{i},dx^{i}:=\phi_{e^{i}}$,

(21)
$\Omega(e^{i},e^{j})=\Omega(f_{i},f_{j})=0,\qquad\Omega(e^{i},f_{j})=\delta^{i}_{j}.$

$\\{e^{i\in\mathbb{I}},f_{i\in\mathbb{I}}\\}$ is a basis of $W$. We call
$\\{e^{i\in\mathbb{I}},f_{i\in\mathbb{I}}\\}$, or any basis of $W$ satisfying
(21), a _canonical basis_ of $W$. To show that $(W,\Omega)$ is a symplectic
topological vector space we will need the following:

######  Lemma 2.2.5.

Let $V$ be a discrete topological vector space with basis
$\\{e^{k\in\mathbb{I}}\\}$ over a discrete topological field $\mathbb{C}$,
then $V^{**}\cong V$.

###### Proof.

If $V$ is finite-dimensional, this is trivial. Suppose that $V$ is infinite-
dimensional and let $\phi\in V^{**}=Hom(V^{*},\mathbb{C})$ be a continuous
linear functional. For any sequence $s:\mathbb{Z}_{>0}\rightarrow
V^{*},n\mapsto s_{n}$ converges to $f\in V^{*}$, we have
$\phi(f)=\phi\left(\lim_{n\rightarrow\infty}s_{n}\right)=\lim_{n\rightarrow\infty}\phi(s_{n})$
by the continuity. Let us embed $\mathbb{Z}_{>0}$ into the index set
$\mathbb{I}$. Suppose that
$f:=\sum_{k\in\mathbb{Z}_{>0}\subseteq\mathbb{I}}x^{k}f_{k}$, where
$f_{i}(e^{j})=\delta_{i}^{j}$ and consider the sequence
$s_{n}:=\sum_{k=1}^{n}x^{k}f_{k}+\Lambda_{n+1}f_{n+1}$ converges to $f$ in the
topology of $V^{*}$ for any arbitrary sequence $\\{\Lambda_{n}\\}$.

By the continuity of $\phi$ we have

$\phi(f)=\phi(\lim_{n\rightarrow\infty}s_{n})=\lim_{n\rightarrow\infty}\phi(s_{n})=\lim_{n\rightarrow\infty}\phi\left(\sum_{k=1}^{n}x^{k}f_{k}+\Lambda_{n+1}f_{n+1}\right)=\lim_{n\rightarrow\infty}\left(\sum_{k=1}^{n}x^{k}\phi(f_{k})+\Lambda_{n+1}\phi(f_{n+1})\right).$

But the sequence $\\{\Lambda_{n}\\}$ is arbitrary, therefore, the only way for
the above to be true is if $\phi(f_{n})$ is only non-zero for finitely many
$n\in\mathbb{Z}_{>0}\subseteq\mathbb{I}$. Since the embedding of
$\mathbb{Z}_{>0}$ into $\mathbb{I}$ is also arbitrary, we conclude that
$\phi(f_{k})$ is non-zero for only finite number of $k\in\mathbb{I}$.

The topology on $V^{**}$ is given by the weakest topology such that the map
$\phi\mapsto\phi(f)$ is continuous for all $f\in V^{*}$, which is not hard to
check to be a discrete topology. The isomorphism $V^{**}\cong V$ is given by
$e^{i}\leftrightarrow\phi_{e^{i}},i\in\mathbb{I}$ where
$\phi_{e^{i}}(f_{j}):=f_{j}(e^{i})=\delta_{j}^{i}$, proving the lemma. ∎

Let us finish showing that $(W,\Omega)$ is a symplectic topological vector
space. We have that $e^{i}\mapsto\Omega(e^{i},.)=\phi_{e^{i}}$ is an
isomorphism $V\xrightarrow{\cong}V^{**}$ and
$f_{i}\mapsto\Omega(f_{i},.)=-f_{i}$ is an isomorphism
$V^{*}\xrightarrow{\cong}V^{*}$. It follows that $(W,\Omega)$ is a symplectic
topological vector space because $w\mapsto\Omega(w,.)$ gives an isomorphism
$W=V\oplus V^{*}\xrightarrow{\cong}V^{**}\oplus V^{*}=(V^{*}\oplus
V)^{*}=W^{*}$. The topological symplectic vector space $W$ is an example of a
_Tate space_.

######  Definition 2.2.6.

(Tate space [7]) A Tate space is a topological vector space over a discrete
field $k$ which is isomorphic to $V_{1}\oplus V_{2}^{*}$ for some discrete
topological $k$-vector spaces $V_{1}$ and $V_{2}$.

######  Remark 2.2.7.

The coordinate $x^{i}$ of $V^{*}$ can be thought of as a linear functional
$x^{i}\in Hom(V^{*},\mathbb{C})=V^{**}\cong V$. Equivalently, $x^{i}$ is
identified with the vector $e^{i}\in V$ via the isomorphism
$e^{i}\mapsto\Omega(e^{i},.)$. The reason why we are keeping both notations is
that they carry different conceptual meanings. Similarly, the coordinate
$y_{i}$ can be identified with the vector $-f_{i}\in V^{*}$. Together,
$\\{x^{i\in\mathbb{I}},y_{i\in\mathbb{I}}\\}$ is a coordinate system of $W$.

Given two vector spaces over a field $k$, a _tensor product_ is a frequently
used operation to construct a new vector space. Formally, the tensor product
$V_{1}\otimes V_{2}$ of vector spaces $V_{1}$ and $V_{2}$ is uniquely defined
up to isomorphism as the vector space satisfying a certain universal property.
More concretely, suppose that $V_{1}$ and $V_{2}$ are potentially infinite-
dimensional vector spaces with basis $\\{e^{i\in\mathbb{I}_{1}}_{1}\\}$ and
$\\{e^{i\in\mathbb{I}_{2}}_{2}\\}$ respectively, then $V_{1}\otimes V_{2}$ is
a vector space with a basis $\\{e^{i}_{1}\otimes e^{j}_{2}\ |\
(i,j)\in\mathbb{I}_{1}\times\mathbb{I}_{2}\\}$. In other words,

(22) $V_{1}\otimes
V_{2}=\left\\{\sum_{(i,j)\in\mathbb{I}_{1}\times\mathbb{I}_{2}}v_{ij}e^{i}_{1}\otimes
e^{j}_{2}\ |\ v_{ij}\in k\text{ is non-zero for finitely many pairs of
}(i,j)\in\mathbb{I}_{1}\times\mathbb{I}_{2}\right\\}.$

######  Remark 2.2.8.

The important message of (22) is that only finite linear combinations of
tensor products of vectors are allowed in $V_{1}\otimes V_{2}$ even though
$V_{1}$ or $V_{2}$ are possibly infinite-dimensional.

$V_{1}\otimes V_{2}$ is different from _topological complete tensor product_
$V_{1}\hat{\otimes}V_{2}$ mentioned in [21]. Let us briefly explain the
meaning of $V_{1}\hat{\otimes}V_{2}$. Given a topological vector space $V$,
the _completion_ of $V$ is the unique complete topological vector space
$\hat{V}$ such that $V$ can be embedded densely into $\hat{V}$ [31, Theorem
5.2]. Let $V_{1},V_{2}$ be topological vector spaces. We assign the topology
to $V_{1}\otimes V_{2}$ such that it is the unique object satisfying the
universal properties of the tensor product in the category of $k$-topological
vector spaces [16]. For instance, when $V_{1}$ and $V_{2}$ are locally convex,
then this topology on $V_{1}\otimes V_{2}$ is the _projective topology_ [31,
Definition 43.2].

######  Definition 2.2.9.

The _topological complete tensor product_ $V_{1}\hat{\otimes}V_{2}$ is the
topological completion of the tensor product $V_{1}\otimes V_{2}$.

Let us give an explicit presentation of $W\hat{\otimes}W$ as an example, where
$W=V\oplus V^{*}$ is a Tate space. Assign a topology to $W\otimes W$ by
declaring open sets around $w\in W$ to be
$w+U_{\mathbb{J}_{1}\mathbb{J}_{2}\mathbb{J}_{3}}\subseteq W\otimes W$ where
$\mathbb{J}_{k=1,2}:=(\mathbb{J}_{k}^{i\in\mathbb{I}})$,
$\mathbb{J}^{i}_{1},\mathbb{J}^{i}_{2},\mathbb{J}_{3}\subseteq\mathbb{I}$ are
finite sets and

$U_{\mathbb{J}_{1}\mathbb{J}_{2}\mathbb{J}_{3}}:=\sum_{i\in\mathbb{I}}\left(U_{\mathbb{J}^{i}_{1}}\otimes\mathbb{C}e^{i}+\mathbb{C}e^{i}\otimes
U_{\mathbb{J}^{i}_{2}}\right)+\sum_{\mathbb{J}_{3}\subseteq\mathbb{I}_{1},\mathbb{I}_{2}\subseteq\mathbb{I}}U_{\mathbb{I}_{1}}\otimes
U_{\mathbb{I}_{2}}\subseteq W\otimes W$

where $U_{\mathbb{J}}\subseteq V^{*}$ are vector subspaces given by (20).
Since open neighborhoods of zero
$U_{\mathbb{J}_{1}\mathbb{J}_{2}\mathbb{J}_{3}}$ are subspaces of $W\otimes W$
in this case, the completion can be computed using the inverse limit:

$\displaystyle W\hat{\otimes}W$ $\displaystyle:=\varprojlim W\otimes
W/U_{\mathbb{J}_{1}\mathbb{J}_{2}\mathbb{J}_{3}}$
$\displaystyle=\left\\{\sum_{i,j\in\mathbb{I}}a_{ij}e^{i}\otimes
e^{j}+\sum_{i,j\in\mathbb{I}}(b_{1i}^{j}e^{i}\otimes
f_{j}+b_{2j}^{i}f_{i}\otimes e^{j})+\sum_{i,j\in\mathbb{I}}c^{ij}f_{i}\otimes
f_{j}\ |\
\begin{array}[]{c}a_{ij},b_{1i}^{j},b_{2i}^{j},c^{ij}\in\mathbb{C}\\\
a_{ij}\neq 0\text{ for finitely }\\\ \text{ many pairs of
}(i,j)\in\mathbb{I}\times\mathbb{I},\\\ \text{ and for each
}i\in\mathbb{I},b_{1j}^{i},b_{2j}^{i}\neq 0\\\ \text{ for finitely many
}j\in\mathbb{I}\end{array}\right\\}.$

For $k=1,2$, let us write $\mathbb{J}^{\prime}_{k}\subseteq\mathbb{J}_{k}$ if
$\mathbb{J}^{\prime i}_{k}\subseteq\mathbb{J}^{i}_{k}$ for all
$i\in\mathbb{I}$. The inverse limit is taken over the directed poset of tuples
$(\mathbb{J}_{1},\mathbb{J}_{2},\mathbb{J}_{3})$ with with partial order
$(\mathbb{J}^{\prime}_{1},\mathbb{J}^{\prime}_{2},\mathbb{J}^{\prime}_{3})\leq(\mathbb{J}_{1},\mathbb{J}_{2},\mathbb{J}_{3})$
if $\mathbb{J}^{\prime}_{i}\subseteq\mathbb{J}_{i},\forall i=1,2,3$.

The $n$-th order topological complete tensor product
$W^{\hat{\otimes}n}:=W\hat{\otimes}\cdots\hat{\otimes}W$ can be defined in a
similar way. We denote by $\widehat{Sym}_{n}(W)$ the subspace of symmetric
tensors in $W^{\hat{\otimes}n}$ and let $\widehat{Sym}_{\leq
n}(W):=\bigoplus_{i=0}^{n}\widehat{Sym}_{i}(W)$. Using Remark 2.2.7, we can
think of $\prod_{i=0}^{\infty}\widehat{Sym}_{i}(W)$ as the space of formal
polynomials on $W$ and denote it by
$\mathbb{C}[[W]]=\mathbb{C}[[x^{i\in\mathbb{I}},y_{i\in\mathbb{I}}]]$. In
particular, elements of $\mathbb{C}[[W]]$ are formal series of the form

$\displaystyle\sum_{n=0}^{\infty}\sum_{\begin{subarray}{c}n_{1}+n_{2}=n\\\
n_{1},n_{2}\geq
0\end{subarray}}\sum_{\begin{subarray}{c}i_{1},\cdots,i_{n_{1}}\in\mathbb{I}\\\
j_{1},\cdots,j_{n_{2}}\in\mathbb{I}\end{subarray}}P_{i_{1},\cdots,i_{n_{1}}}^{j_{1},\cdots,j_{n_{2}}}x^{i_{1}}\cdots
x^{i_{n_{1}}}y_{j_{1}}\cdots y_{j_{n_{2}}},\qquad\begin{array}[]{c}\text{Where
$P_{i_{1},\cdots,i_{n_{1}}}^{j_{1},\cdots,j_{n_{2}}}\in\mathbb{C}$ and }\\\
\text{for each $(j_{1},\cdots,j_{n_{2}})\in\mathbb{I}^{n_{2}}$,
$P_{i_{1},\cdots,i_{n_{1}}}^{j_{1},\cdots,j_{n_{2}}}\neq 0$}\\\ \text{for
finitely many $(i_{1},\cdots,i_{n_{1}})\in\mathbb{I}^{n_{1}}$.}\end{array}$

The symplectic form $\Omega$ defines a Poisson bracket on $\mathbb{C}[[W]]$
given by

(23)
$\\{x^{i},x^{j}\\}=\\{y_{i},y_{j}\\}=0,\qquad\\{y_{i},x^{j}\\}=\delta^{j}_{i}$

and extended to arbitrary formal polynomials by the Leibniz rule. We call
coordinates $\\{x^{i\in\mathbb{I}},y_{i\in\mathbb{I}}\\}$ satisfying (23), or
equivalently coordinates such that $\Omega$ takes the canonical form:
$\Omega=\sum_{i\in\mathbb{I}}dy_{i}\wedge dx^{i}$, a _canonical coordinates
system_ of $W$.

####  2.2.2. Pre-Airy structures

Let $V$ be a topological vector space with a discrete topology over the
discrete topological field $\mathbb{C}$.

######  Definition 2.2.10.

([21, Definition 2.6.1]) A _classical pre-Airy structure_ on $V$ is a tuple
$(V,A,B,C)$ where

$A=(a_{ijk})_{i,j,k\in\mathbb{I}}\in V\otimes V\otimes V,\qquad
B=(b^{k}_{ij})_{i,j,k\in\mathbb{I}}:V\rightarrow V\otimes V,\qquad
C=(c_{i}^{jk})_{i,j,k\in\mathbb{I}}:V\otimes V\rightarrow V.$

A _quantum pre-Airy structure_ on $V$ is a tuple $(V,A,B,C,\epsilon)$ where
$(V,A,B,C)$ is a classical pre-Airy structure and $\epsilon=(\epsilon_{i})\in
V$.

######  Remark 2.2.11.

Equivalently, a (quantum) pre-Airy structure is given by any
$A=(a_{ijk})_{i,j,k\in\mathbb{I}},B=(b^{k}_{ij})_{i,j,k\in\mathbb{I}},C=(c_{i}^{jk})_{i,j,k\in\mathbb{I}}$
and $\epsilon=(\epsilon_{i})$ such that: $a_{ijk},\epsilon_{i}\in\mathbb{C}$
are non-zero for only finitely many
$(i,j,k)\in\mathbb{I}\times\mathbb{I}\times\mathbb{I}$,
$b_{ij}^{k}\in\mathbb{C}$ is non-zero for only finitely many
$(i,j)\in\mathbb{I}\times\mathbb{I}$ for each $k\in\mathbb{I}$ and
$c_{i}^{jk}\in\mathbb{C}$ is non-zero for only finitely many $i\in\mathbb{I}$
for each $(j,k)\in\mathbb{I}\times\mathbb{I}$.

Given a quantum pre-Airy structure $(V,A,B,C,\epsilon)$ there exists a
recursive procedure called _Abstract Topological Recursion (ATR)_ [21, Section
2.5] produces a set of numbers $\\{S_{g,n;i_{1},...,i_{n}}\in\mathbb{C}\ |\
i_{k}\in\mathbb{I}\\}$. Let the basis of $V$ be $\\{e^{i\in\mathbb{I}}\\}$. We
package the set of numbers $\\{S_{g,n;i_{1},...,i_{n}}\\}$ into the tensor

$S_{g,n}:=\sum_{i_{1},...,i_{n}\in\mathbb{I}}\frac{1}{n!}S_{g,n;i_{1},...,i_{n}}e^{i_{1}}\otimes...\otimes
e^{i_{n}}\in V^{\otimes n}=V\otimes...\otimes V.$

As the name suggests, ATR is related to topological recursion. We will see
later this section the quantum pre-Airy structure such that the ATR reduces to
the usual topological recursion.

Let us now describe ATR in details. We set the initial conditions to be

$S_{0,3}:=2A\in V\otimes V\otimes V,\qquad S_{1,1}:=\epsilon\in V,\qquad
S_{0,n\leq 2}=S_{g\geq 0,0}=0.$

Then $S_{g,n}$ for $g\geq 0,n\geq 4$ or $g\geq 2,n\geq 1$ or $g\geq 1,n\geq 2$
are computed recursively as follows

$\displaystyle S_{g,n;i_{1},...,i_{n}}$
$\displaystyle=2\sum_{k=2}^{n}\sum_{j\in\mathbb{I}}b_{i_{1}i_{k}}^{j}S_{g,n-1;ji_{\\{2,...,n\\}\setminus\\{k\\}}}+\sum_{\begin{subarray}{c}g_{1}+g_{2}=g\\\
I_{1}\coprod
I_{2}=\\{2,...,n\\}\end{subarray}}\sum_{j_{1},j_{2}\in\mathbb{I}}c_{i_{1}}^{j_{1}j_{2}}S_{g_{1},1+|I_{1}|;j_{1}i_{I_{1}}}S_{g_{2},1+|I_{2}|;j_{2}i_{I_{2}}}$
(24)
$\displaystyle\qquad+\sum_{j_{1},j_{2}\in\mathbb{I}}c_{i_{1}}^{j_{1}j_{2}}S_{g-1,n+1;j_{1}j_{2}i_{2}...i_{n}}.$

Note that the right-hand-side of (2.2.2) only involve
$S_{g^{\prime},n^{\prime}}$ with $2g^{\prime}-2+n^{\prime}<2g-2+n$, therefore
the recursion is guaranteed to terminate.

######  Remark 2.2.12.

The right-hand-side of (2.2.2) will indeed define an element of $V^{\otimes
n}$. Let $v:=\sum_{k\in\mathbb{I}}v_{k}e^{k}\in
V,u:=\sum_{k\in\mathbb{I}}u_{k}e^{k}\in V$, then by definition we have
$\sum_{i,j,k\in\mathbb{I}}b^{k}_{ij}v_{k}e^{i}\otimes e^{j}=B(v)\in V\otimes
V$ and $\sum_{i,j,k\in\mathbb{I}}c^{jk}_{i}v_{j}u_{k}e^{i}=C(v\otimes u)\in
V$. Note that all summations came from contractions of a vectors in $V$ and
$V^{*}$, so they have at most finitely non-zero terms even though the
carnality of the index set $\mathbb{I}$ may be infinite.

Given $g,n$ and suppose that $S_{g^{\prime},n^{\prime}}\in V^{\otimes
n^{\prime}}$ for all $2g^{\prime}-2+n^{\prime}<2g-2+n$ then it follows that
the first and second terms in (2.2.2) are well defined as elements of
$V^{\otimes n}$. Finally, the last term of (2.2.2) is a well-defined element
of $V^{\otimes n}$ because $S_{g-1,n+1}\in V^{\otimes n+1}$, a non-completed
tensor products of $V$, so it contains only finite linear combination of
tensor products of vectors in $V$. More explicitly, we can write
$\sum_{j_{1},j_{2}\in\mathbb{I}}S_{g-1,n+1;j_{1}j_{2}i_{2}...i_{n}}e^{j_{1}}\otimes
e^{j_{2}}$ as a finite sum $\sum_{k=1}^{N}v_{1,k}\otimes v_{2,k}$ for some
$v_{1,k},v_{2,k}\in V$ and $N\in\mathbb{Z}_{\geq 0}$. Then

$\displaystyle\sum_{i_{1}\in\mathbb{I}}\sum_{j_{1},j_{2}\in\mathbb{I}}c^{j_{1},j_{2}}_{i_{1}}S_{g-1,n+1;j_{1}j_{2}i_{2}...i_{n}}e^{i_{1}}$
$\displaystyle=C\left(\sum_{j_{1},j_{2}\in\mathbb{I}}S_{g-1,n+1;j_{1}j_{2}i_{2}...i_{n}}e^{j_{1}}\otimes
e^{j_{2}}\right)$ $\displaystyle=C\left(\sum_{k=1}^{N}v_{1,k}\otimes
v_{2,k}\right)=\sum_{k=1}^{N}C\left(v_{1,k}\otimes v_{2,k}\right)\in V.$

Overall, the output $S_{g,n}$ is a well-defined element of $V^{\otimes n}$.

######  Definition 2.2.13.

([21, Definition 2.6.1]) A classical pre-Airy _sub-structure_ $(U,A,B,C)$ of
$(V,A,B,C)$ is a classical pre-Airy structure given by a vector subspace
$U\subseteq V$ compatible with tensors $A,B,C$ in the sense that

$A\in U\otimes U\otimes U,\qquad B|_{U}:U\rightarrow U\otimes U,\qquad
C|_{U\otimes U}:U\otimes U\rightarrow U.$

A quantum pre-Airy sub-structure $(U,A,B,C,\epsilon)$ of $(V,A,B,C,\epsilon)$
is a quantum pre-Airy structure given by a classical pre-Airy sub-structure
$(U,A,B,C)$ of $(V,A,B,C)$ such that $\epsilon\in U$.

A (quantum or classical) pre-Airy structure on $V$ is called _primitive_ if it
does not contain a non-trivial (quantum or classical) pre-Airy sub-structure
$(U,A,B,C,\epsilon)$ with $U\subsetneq V$.

######  Remark 2.2.14.

Given two quantum pre-Airy structures $(V_{i},A_{i},B_{i},C_{i},\epsilon_{i})$
and their ATR outputs $\\{S^{(i)}_{g,n}\in V^{\otimes n}_{i}\\},i=1,2$ the
important feature is the following. If both quantum pre-Airy structures
$(V_{i},A_{i},B_{i},C_{i},\epsilon_{i})$, $i=1,2$ contain the same quantum
pre-Airy substructure $(V,A,B,C,\epsilon)$, i.e. $V\subseteq V_{1},V_{2}$ and

$A_{1}=A=A_{2},\qquad B_{1}|_{V}=B=B_{2}|_{V},\qquad C_{1}|_{V\otimes
V}=C=C_{2}|_{V\otimes V},\qquad\epsilon_{1}=\epsilon=\epsilon_{2}$

then their ATR outputs will be the same [21, Section 2.6]:

$S^{(1)}_{g,n}=S^{(2)}_{g,n}=S_{g,n}\in V^{\otimes n}\subseteq V^{\otimes
n}_{i},\qquad i=1,2$

where $\\{S_{g,n}\in V^{\otimes n}\\}$ is the output of ATR using the pre-Airy
structure $(V,A,B,C,\epsilon)$.

####  2.2.3. Airy structures

In this section we let $W=V\oplus V^{*}$ be a Tate space and let
$\\{x^{i\in\mathbb{I}},y_{i\in\mathbb{I}}\\}$ be canonical coordinates of $W$.

######  Definition 2.2.15.

([21, Definition 2.1.1]) A _classical Airy structure_ on $V$ is a classical
pre-Airy structure $(V,A,B,C)$ such that the collection of _at-most quadratic
polynomials_ $\\{H_{i\in\mathbb{I}}\in\widehat{Sym}_{\leq 2}(W)\\}$, where

(25)
$H_{i}:=-y_{i}+\sum_{j,k\in\mathbb{I}}a_{ijk}x^{j}x^{k}+2\sum_{j,k\in\mathbb{I}}b_{ij}^{k}x^{j}y_{k}+\sum_{j,k\in\mathbb{I}}c^{jk}_{i}y_{j}y_{k},$

generates the vector space
$\mathfrak{g}:=\bigoplus_{i\in\mathbb{I}}\mathbb{C}\cdot
H_{i}\subset\widehat{Sym}_{\leq 2}(W)$ which is closed under the Poisson
bracket :
$\\{H_{i},H_{j}\\}=\sum_{k\in\mathbb{I}}g^{k}_{ij}H_{k}\in\mathfrak{g}$ for
some _structure constants_ $g^{k}_{ij}\in\mathbb{C}$. In other words, the
structure constants $g^{k}_{ij}$ are the matrix entries of the operator
$\mathfrak{g}$ is a Lie algebra with Lie bracket given by the Poisson bracket.

######  Remark 2.2.16.

We can always assume that the tensors $A=(a_{ijk})$ and $(c_{i}^{jk})$ are
symmetric in the indices $j,k$ because their anti-symmetric part would vanish
in the expression of $H_{i}$ after summing against $x^{j}x^{k}$ and
$y_{j}y_{k}$. On the other hand, the tensor $B=(b_{jk}^{i})$ does not possess
any symmetry in the indices $j,k$.

In fact, given a classical Airy structure
$\\{H_{i\in\mathbb{I}}\in\widehat{Sym}_{\leq 2}(W)\\}$, it can be shown (see
[1, Section 2.2], [21, Section 2.1]) from
$\\{H_{i},H_{j}\\}=\sum_{k\in\mathbb{I}}g^{k}_{ij}H_{k}$ that $a_{ijk}$ is
symmetric in the indices $i,j$ and that $g_{ij}^{k}=2(b^{k}_{ij}-b^{k}_{ji})$.
In other words, $a_{ijk}$ is symmetric in all indices $i,j,k$ and the
structure constants $g_{ij}^{k}$ are components of the tensor
$(g^{k}_{ij})_{i,j,k\in\mathbb{I}}:V\rightarrow V\otimes V$ which is entirely
determined by the tensor $B$.

######  Remark 2.2.17.

Note that each $H_{i}$ may be treated as a function
$H_{i}:W\rightarrow\mathbb{C}$. Denote by $H_{i}(w)$ the evaluation of $H_{i}$
at a point $w:=\sum_{i\in\mathbb{I}}(x^{i}f_{i}+y_{i}e^{i})\in W$ where
$x^{i},y_{i}\in\mathbb{C}$, then $H:=\sum_{i\in\mathbb{I}}H_{i}(w)e^{i}\in V$.

Let us define the Weyl algebra
$\mathcal{D}^{\hbar}(W):=\mathbb{C}\left[\left[x^{i\in\mathbb{I}},\hbar\partial_{i\in\mathbb{I}}\right]\right]\left[\left[\hbar\right]\right]$
where $\hbar\partial_{i}:=\hbar\frac{\partial}{\partial x^{i}}$. We introduce
grading on $\mathcal{D}^{\hbar}(W)$ by setting $\deg\hbar=2,\deg x^{i}=1$ and
$\deg\hbar\partial_{i}=1$. Let $\mathcal{D}^{\hbar}_{n}(W)$ be the vector
subspace of degree $n$ elements in $\mathcal{D}^{\hbar}(W)$ and let
$\mathcal{D}^{\hbar}_{\leq
n}(W):=\bigoplus_{i=0}^{n}\mathcal{D}^{\hbar}_{i}(W)$. We also introduce the
following notations when working with $\mathcal{D}^{\hbar}(W)$:

######  Definition 2.2.18.

Let $\hat{f},\hat{g}\in\mathcal{D}^{\hbar}(W)$. We denote _applying the
operator $\hat{f}$ on $\hat{g}$_ by $\hat{f}(\hat{g})$. We denote _multiplying
the operator $\hat{f}$ with the operator $\hat{g}$_ by $\hat{f}\hat{g}$. For
example, $\hbar\partial_{i}(x^{j})=\hbar\delta^{j}_{i}$, and
$\hbar\partial_{i}x^{j}=\hbar\delta_{i}^{j}+x^{j}\hbar\partial_{i}$.

Conceptually, $\mathcal{D}^{\hbar}(W)$ is the quantization of
$\mathbb{C}[[W]]$ with $y_{i}$ replaced by $\hbar\partial_{i}$ and the Poisson
bracket $\\{.,.\\}$ replaced by the commutator $\frac{1}{\hbar}[.,.]$, where
$[\hat{f},\hat{g}]:=\hat{f}\hat{g}-\hat{g}\hat{f},\hat{f},\hat{g}\in\mathcal{D}^{\hbar}(W)$.
We define the _classical limit_ to be the map
$cl:\mathcal{D}^{\hbar}(W)\rightarrow\mathbb{C}[[W]]$, given by $\hbar\mapsto
0,x^{i}\mapsto x^{i}$ and $\hbar\partial_{i}\mapsto y_{i}$.

######  Definition 2.2.19.

([21, Definition 2.2.1]) A _quantum Airy structure_ on $V$ is a collection
$\\{\hat{H}_{i\in\mathbb{I}}\in\mathcal{D}^{\hbar}_{\leq 2}(W)\\}$ such that
$\hat{\mathfrak{g}}:=\bigoplus_{i\in\mathbb{I}}\mathbb{C}\cdot\hat{H}_{i}\subset\mathcal{D}^{\hbar}_{\leq
2}(W)$ is a Lie algebra with Lie bracket given by the commutator
$\frac{1}{\hbar}[.,.]$. In other words,
$\frac{1}{\hbar}[\hat{H}_{1},\hat{H}_{2}]=\sum_{k\in\mathbb{I}}g^{k}_{ij}\hat{H}_{k}\in\hat{\mathfrak{g}}$
for some constants $g^{k}_{ij}\in\mathbb{C}$.

A classical Airy structure $\\{H_{i\in\mathbb{I}}\\}$ is called the _classical
limit_ of $\\{\hat{H}_{i\in\mathbb{I}}\\}$ if $H_{i}=cl(\hat{H}_{i})$ in which
case we also have
$\\{H_{i},H_{j}\\}=cl\left(\frac{1}{\hbar}[\hat{H}_{i},\hat{H}_{j}]\right)=\sum_{k\in\mathbb{I}}g^{k}_{ij}H_{k}$.
We say that $\\{\hat{H}_{i\in\mathbb{I}}\\}$ _quantizes_ or it is a
_quantization_ of $\\{H_{i\in\mathbb{I}}\\}$.

######  Remark 2.2.20.

The condition
$\frac{1}{\hbar}[\hat{H}_{i},\hat{H}_{j}]=\sum_{k\in\mathbb{I}}g^{k}_{ij}\hat{H}_{k}$
can be written as relations among tensors $A,B,C$ and $\epsilon$, [1]. If we
ignored all relations involving $\epsilon$, then the remaining are relations
among $A,B$ and $C$ arise from the classical condition
$\\{H_{i},H_{j}\\}=\sum_{k\in\mathbb{I}}g^{k}_{ij}H_{k}$.

######  Remark 2.2.21.

Quantization is more difficult than just replacing $x^{i}$ by $x^{i}$ and
$y_{i}$ by $\hbar\partial_{i}$ because $\mathbb{C}[[W]]$ is a commutative
algebra whereas $\mathcal{D}^{\hbar}(W)$ is non-commutative, hence there is an
ambiguity in ordering $x^{i}$ and $\hbar\partial_{i}$. For instance, it is
unclear whether the term $x^{i}y_{j}$ should be quantized as
$x^{i}\hbar\partial_{j}$ or
$\hbar\partial_{j}x^{i}=\hbar\delta_{j}^{i}+x_{i}\hbar\partial_{j}$. In
general, suppose that $\hat{f},\hat{g},\hat{h}\in\mathcal{D}^{\hbar}_{\leq
2}(W)$ with classical limit
$f=cl(\hat{f}),g=cl(\hat{g}),h=cl(\hat{h})\in\widehat{Sym}_{\leq 2}(W)$ then
$\\{\hat{f},\hat{g}\\}=\hat{h}$ implies
$\\{f,g\\}=cl\left(\frac{1}{\hbar}[\hat{f},\hat{g}]\right)=h$ but on the other
hand, if $\\{f,g\\}=h$ then
$cl\left(\frac{1}{\hbar}[\hat{f},\hat{g}]\right)=\\{f,g\\}+c=h+c$ for some
constant $c\in\mathbb{C}$. Therefore, the quantization of
$\\{H_{i\in\mathbb{I}}\\}$ where $H_{i}$ are given by (25) and
$\\{H_{i},H_{j}\\}=\sum_{k\in\mathbb{I}}g^{k}_{ij}H_{k}$ amount to finding
$\\{\epsilon_{i\in\mathbb{I}}\in\mathbb{C}\\}$ such that

(26)
$\hat{H}_{i}:=-\hbar\partial_{i}+\sum_{j,k\in\mathbb{I}}a_{ijk}x^{j}x^{k}+2\hbar\sum_{j,k\in\mathbb{I}}b^{k}_{ij}x^{j}\partial_{k}+\hbar^{2}\sum_{j,k\in\mathbb{I}}c^{jk}_{i}\partial_{j}\partial_{k}+\hbar\epsilon_{i}$

satisfying
$\frac{1}{\hbar}[\hat{H}_{i},\hat{H}_{j}]=\sum_{k\in\mathbb{I}}g^{k}_{ij}\hat{H}_{k}$.

If follows from (26) we can write a quantum Airy structure on $V$ as a tuple
$(V,A,B,C,\epsilon)$. In other words, a quantum Airy structure is a quantum
pre-Airy structure $(V,A,B,C,\epsilon)$ such that
$\hat{\mathfrak{g}}:=\bigoplus_{i\in\mathbb{I}}\mathbb{C}\cdot\hat{H}_{i}$,
where $\hat{H}_{i}$ given by (26), is a Lie algebra.

Given a classical Airy structure $\\{H_{i\in\mathbb{I}}\\}$, then neither the
uniqueness nor existence of its quantization is guaranteed in general. It
turns out that the sufficient condition for the existence of quantization of
$\\{H_{i\in\mathbb{I}}\\}$ is given by the vanishing of the second cohomology
of Lie algebra $H^{2}(\mathfrak{g},\mathbb{C})$. On the other hand, the
necessary and sufficient condition for the uniqueness of the quantization is
given by the vanishing of $H^{1}(\mathfrak{g},\mathbb{C})$ [21].

We will often refer to classical (pre-)Airy structures simply as (pre-)Airy
structures unless it is important to emphasize their distinction from quantum
(pre-)Airy structures.

There is a correspondence between Airy structures and _quadratic Lagrangian
submanifolds_ $L:=\\{H_{i\in\mathbb{I}}=0\\}$ of $W$, i.e. Lagrangian
submanifolds defined by at-most quadratic polynomials in $W$. When $V$ is
finite-dimensional, this correspondence is easy:

######  Lemma 2.2.22.

([21, Section 2.1]) Let $V$ be an $n$-dimensional vector space and
$\\{H_{1},...,H_{n}\\}$ is a collection of at-most quadratic polynomials on
$W=V\oplus V^{*}$ in the form given in (25). Then
$L:=\\{H_{i=1,\cdots,n}=0\\}\subset W$ is a Lagrangian submanifold if and only
if the collection $\\{H_{1},...,H_{n}\\}$ gives an Airy structure.

###### Proof.

Suppose that $\\{H_{1},...,H_{n}\\}$ is an Airy structure. Let $v_{H_{i}}$ be
a Hamiltonian vector field corresponding to $H_{i}$, i.e.
$\iota_{v_{H_{i}}}\Omega=dH_{i}$. Then $v_{H_{i}}(x)\in T_{x}L$ for all $x\in
L$ because $\iota_{v_{H_{i}}(x)}dH_{j}(x)=\\{H_{i},H_{j}\\}(x)=0$. Because
$\dim L=n$, we conclude that $\\{v_{H_{1}}(x),...,v_{H_{n}}(x)\\}$ is a basis
of $T_{x}L$. Finally, $\Omega(v_{H_{i}},v_{H_{j}})=\\{H_{i},H_{j}\\}=0$ on $L$
and $\dim L=\frac{1}{2}\dim W$, therefore $L\subset W$ is a Lagrangian
submanifold.

Conversely, suppose that $L\subset W$ is a Lagrangian submanifold. Then for
all $v\in T_{x}L$ we have $\Omega(v_{H_{i}}(x),v)=\iota_{v}dH_{i}(x)=0$.
Because $T_{x}L\subset W$ is a Lagrangian subspace, i.e. a maximal isotropic
subspace, it follows that $v_{H_{i}}\in T_{x}L$. Then
$\\{H_{i},H_{j}\\}(x)=\Omega(v_{H_{i}},v_{H_{j}})=0$ for all $x\in L$, which
means $\\{H_{i},H_{j}\\}$ belongs to an ideal generated by
$\\{H_{1},...,H_{n}\\}$. But each $H_{i}$ is at most quadratic, so
$\\{H_{i},H_{j}\\}$ also is at most quadratic, we conclude that
$\\{H_{i},H_{j}\\}=\sum_{k=1}^{n}g^{k}_{ij}H_{k}$ for some constants
$g_{ij}^{k}\in\mathbb{C}$. ∎

######  Remark 2.2.23.

When $V$ is finite dimensional, we define the tangent space to $L$ at $x\in
L\subset W$ to be

$T_{x}L:=\\{v\in W\ |\ \iota_{v}dH_{i}(x)=0,i=1,\cdots,n\\}.$

It is more subtle how to interpret this definition when $V$ is infinite-
dimensional, except at $x=0\in W$. It is easy to see from the form of each
$H_{i}$ in (25) even when $V$ is infinite-dimensional that $H_{i}(0)=0$ and
$dH_{i}(0)=dy_{i}$ for all $i\in\mathbb{I}$, therefore we define
$T_{0}L:=\\{y_{i\in\mathbb{I}}=0\\}=V^{*}\subset W$ which is a Lagrangian
subspace of $W$. Since $V\subset W$ is also a Lagrangian subspace of $W$ and
$W=V\oplus V^{*}\cong V\oplus T_{0}L$ we call $V$ a _Lagrangian complement of
$T_{0}L$_.

When $V$ is infinite-dimensional the arguments used in the proof of Lemma
2.2.22 will no longer hold. Instead, what we mean by quadratic Lagrangian
submanifold is that there exists a generating function
$S_{0}\in\prod_{n=1}^{\infty}Sym_{n}(V)$ such that

(27)
$H_{i}\left(\\{x^{i\in\mathbb{I}}\\},\left\\{y_{i\in\mathbb{I}}:=\frac{\partial
S_{0}}{\partial x^{i}}\right\\}\right)=0,\qquad i\in\mathbb{I}.$

We are going to show this in a moment. Keep in mind that
$\partial_{i}S_{0}=\frac{\partial S_{0}}{\partial
x^{i}}\in\prod_{n=1}^{\infty}Sym_{n}(V)$ is a formal derivative and (27)
should be understood as: the image of $H_{i}$ under the homomorphism
$\widehat{Sym}_{\leq
2}(W)\rightarrow\prod^{\infty}_{n=1}Sym_{n}(V),x^{i}\mapsto
x^{i},y_{i}\mapsto\partial_{i}S_{0}$ is zero.

Note that we can think of a quantum Airy structures
$\\{\hat{H}_{i\in\mathbb{I}}\\}$ as a quantization of a quadratic Lagrangian
submanifold $L:=\\{H_{i\in\mathbb{I}}=0\\}$. Let us consider solving for
$\psi_{L}$ such that $\psi_{L}^{-1}\hat{H}_{i}(\psi_{L})=0$ where $\psi_{L}$
is given by the formal expression

(28) $\psi_{L}:=\exp\left(\sum_{g=0}^{\infty}\hbar^{g-1}S_{g}\right),\qquad
S_{g}=\sum_{n=0}^{\infty}S_{g,n}\in\prod_{n=1}^{\infty}Sym_{n}(V),$

such that
$\psi_{L}^{-1}(\hbar\partial_{i})\psi_{L}:=\hbar\partial_{i}+\partial_{i}\left(\sum_{g=0}^{\infty}\hbar^{g}S_{g}\right)$
and each $S_{g,n}$ are homogeneous polynomial of degree $n$:

(29)
$S_{g,n}:=\sum_{i_{1},...,i_{n}\in\mathbb{I}}\frac{1}{n!}S_{g,n;i_{1},\cdots,i_{n}}x^{i_{1}}\cdots
x^{i_{n}}\in Sym_{n}(V).$

Observe that $S_{g,n}\in Sym_{n}(V)$ without completion of symmetric tensor
product and $S_{g}$ is a formal sum of $S_{g,n}$. The following Proposition
guarantee the existence and uniqueness of the solution $\psi_{L}$:

######  Theorem 2.2.24.

([21, Theorem 2.4.2]) Let $\\{\hat{H}_{i\in\mathbb{I}}\\}$ be a quantum Airy
structure. There exists a unique solution of the form $\psi_{L}$ of the form
(28) such that $\psi^{-1}_{L}\hat{H}_{i}(\psi_{L})=0,\forall i\in\mathbb{I}$.

###### Proof.

First of all, we note that if
$\psi:=\exp\left(\sum_{g=0}^{\infty}\hbar^{g-1}\sum_{n=0}^{\infty}S_{g,n}\right)$
for an arbitrary $S_{g,n}\in Sym_{n}(V)$ then
$\psi^{-1}\hat{H}_{i}\psi\in\mathcal{D}^{\hbar}(W)$. To verify this we need to
check that for each $g$ and $n$, the degree $n$ term of the coefficient of
$\hbar^{g}$ in $\psi^{-1}\hat{H}_{i}\psi(1)$ belongs to $Sym_{n}(V)$. These
type of terms came from

(30)
$H^{(g,n)}_{i}:=-\hbar^{g}\partial_{i}S_{g,n+1}+\hbar^{g}\sum_{j,k\in\mathbb{I}}b^{k}_{ij}x^{j}\partial_{k}S_{g,n}+\hbar^{g}\sum_{\begin{subarray}{c}g_{1}+g_{2}=g\\\
n_{1}+n_{2}=n+2\end{subarray}}\sum_{j,k\in\mathbb{I}}c^{jk}_{i}\partial_{j}S_{g_{1},n_{1}}\partial_{k}S_{g_{2},n_{2}}+\hbar^{g}\sum_{j,k\in\mathbb{I}}c^{jk}_{i}\partial_{j}\partial_{k}S_{g-1,n+2}.$

The first term belongs to $Sym_{n}(V)\hbar^{g}$. Since
$\sum_{k\in\mathbb{I}}b^{k}\partial_{k}S_{g,n}$ is the same as contracting the
tensor $B$ with $S_{g,n}$, by Remark 2.2.12 the second term also belongs to
$Sym_{n}(V)\hbar^{g}$. Similarly, by Remark 2.2.12 we also conclude that the
third and the fourth terms also belong to $Sym_{n}(V)\hbar^{g}$. This verifies
our claim that $\psi^{-1}\hat{H}_{i}\psi\in\mathcal{D}^{\hbar}(W)$. Remark
2.2.12 tells us that $H^{(g,n)}:=\sum_{i\in\mathbb{I}}H^{(g,n)}_{i}e^{i}\in
V^{\otimes(n+1)}$.

The proof is by induction in $g$ and for each fixed $g$ we proceed with
induction in $n$. Suppose that we have found the solution $S_{g}$ for all
$g=0,...,g_{0}-1$ and for $S_{g_{1}}$ up to degree $n_{0}$ term in the sense
that if
$\psi^{(g_{0},n_{0})}_{L}:=\exp\left(\sum_{g=0}^{g_{0}-1}\hbar^{g-1}S_{g}+\hbar^{g_{0}-1}\sum_{n=1}^{n_{0}}S_{g_{0},n}\right)$
then

$\displaystyle\hat{H}^{(g_{0},n_{0})}_{i}$
$\displaystyle:=(\psi^{(g_{0},n_{0})}_{L})^{-1}\hat{H}_{i}\psi^{(g_{0},n_{0})}_{L}$
(31)
$\displaystyle=-\hbar\partial_{i}+H_{i}^{(g_{0},n_{0})}+\hbar^{g_{0}}x^{\geq
n_{0}+1}+\hbar^{\geq g_{0}+1}x^{\geq
0}+2\hbar\sum_{j,k\in\mathbb{I}}b^{k}_{ij}x^{j}\partial_{k}+\hbar^{2}\sum_{j,k\in\mathbb{I}}c^{jk}_{i}\partial_{j}\partial_{k}\in\mathcal{D}^{\hbar}(W),$

where $H^{(g_{0},n_{0})}_{i}:=\hbar^{g_{0}}x^{=n_{0}}\in
Sym_{n}(V)\hbar^{g_{0}}$. Since $\\{\hat{H}_{i\in\mathbb{I}}\\}$ is a quantum
Airy structure we can check that
$\\{\hat{H}_{i\in\mathbb{I}}^{(g_{0},n_{0}-1)}\\}$ satisfies the similar
commutation relations

$\displaystyle\frac{1}{\hbar}[\hat{H}_{i}^{(g_{0},n_{0})},\hat{H}_{j}^{(g_{0},n_{0})}]$
$\displaystyle=\frac{1}{\hbar}\left(\hat{H}_{i}^{(g_{0},n_{0})}\hat{H}_{j}^{(g_{0},n_{0})}-\hat{H}_{j}^{(g_{0},n_{0})}\hat{H}_{i}^{(g_{0},n_{0})}\right)=\frac{1}{\hbar}(\psi_{L}^{(g_{0},n_{0})})^{-1}\left(\hat{H}_{i}\hat{H}_{j}-\hat{H}_{j}\hat{H}_{i}\right)\psi_{L}^{(g_{0},n_{0})}$
(32)
$\displaystyle=(\psi_{L}^{(g_{0},n_{0})})^{-1}\left(\frac{1}{\hbar}[\hat{H}_{i},\hat{H}_{j}]\right)\psi_{L}^{(g_{0},n_{0})}=\sum_{k\in\mathbb{I}}g^{k}_{ij}\hat{H}_{k}^{(g_{0},n_{0})}.$

The left hand side of (2.2.3) contains the term
$\partial_{j}\left(H_{i}^{(g_{0},n_{0})}\right)-\partial_{i}\left(H_{j}^{(g_{0},n_{0})}\right)=\hbar^{g_{0}}x^{=n_{0}-1}$
but the right hand side only contain $\hbar^{\geq g_{0}}x^{\geq n_{0}}$, so it
must be the case that
$\partial_{j}\left(H_{i}^{(g_{0},n_{0})}\right)=\partial_{i}\left(H_{j}^{(g_{0},n_{0})}\right)$.
In other words, there exists a symmetric tensor
$S_{g_{0},n_{0}+1}:=\frac{1}{(n_{0}+1)!}\sum_{i_{1},...,i_{n_{0}}\in\mathbb{I}}S_{g_{0},n_{0};i_{1},...,i_{n_{0}+1}}x^{i_{1}}\cdots
x^{i_{n_{0}+1}}\in Sym_{n_{0}+1}(V)$ such that
$H_{i}^{(g_{0},n_{0})}=\hbar^{g_{0}}\partial_{i}S_{g_{0},n_{0}+1}$.

For the base case of the induction, $(g_{0}=0)$, we can look at the first non-
trivial $S_{0,n_{0}}$ which is $S_{0,3}=2A\in Sym_{3}(V)$. By setting
$\psi^{(0,3)}_{L}=\exp\left(\frac{1}{\hbar}S_{0,3}\right)$ we can see that
$(\psi_{L}^{(0,3)})^{-1}\hat{H}_{i}\psi_{L}^{(0,3)}$ takes the form of (2.2.3)
with $(g_{0},n_{0})=(0,3)$. ∎

Taking the limit $\hbar\rightarrow 0$ of the expression
$\psi^{-1}_{L}\hat{H}_{i}\psi_{L}(1)=0$ is the same as replacing every
$\hbar\partial_{i}$ in $\hat{H}_{i}$ with $\partial_{i}S_{0}$. Therefore, we
obtain (27), showing that
$S_{0}=\sum_{n=0}^{\infty}S_{0,n}\in\prod_{n=1}^{\infty}Sym_{n}(V)$ is the
generating function of $L=\\{H_{i}=0\\}$. It is in this sense that $\psi_{L}$
is considered a quantization of $L$. The following Corollary of Theorem 2.2.24
makes the relationship between quantum Airy structures and ATR becomes clear:

######  Corollary 2.2.25.

If a quantum pre-Airy structure $(V,A,B,C,\epsilon)$ is a quantum Airy
structure then the output $S_{g,n}\in V^{\otimes n}$ of the corresponding ATR
are symmetric tensors. Moreover, if
$\psi_{L}:=\exp\left(\sum_{g=0}^{\infty}\hbar^{g-1}\sum_{n=0}^{\infty}S_{g,n}\right)$
then $\psi^{-1}_{L}\hat{H}_{i}(\psi_{L})=0$.

###### Proof.

Theorem 2.2.24 tells us that the solution to
$\psi_{L}^{-1}\hat{H}_{i}(\psi_{L})=0$ of the form
$\psi_{L}:=\exp\left(\sum_{g=0}^{\infty}\hbar^{g-1}\sum_{n=0}^{\infty}S_{g,n}\right)$
where $S_{g,n}\in Sym_{n}(V)$ exists. The equation
$\psi^{-1}_{L}\hat{H}_{i}\psi_{L}(1)=0$ implies that the right-hand side of
(30) must vanish. In particular, the coefficient of each $x^{i_{1}}\cdots
x^{i_{n}},i_{k}\in\mathbb{I}$ must vanish and this gives us the recursion
relations (2.2.2). ∎

Finally, the notion of sub-structures we introduced for (quantum) pre-Airy
structures translates straightforwardly to the notion of sub-structures for
(quantum) Airy structures:

######  Definition 2.2.26.

([21, Section 2.6]) A classical Airy sub-structure $(U,A,B,C)$ of a classical
Airy structure $(V,A,B,C)$ is a classical pre-Airy sub-structure of a pre-Airy
structure $(V,A,B,C)$. A quantum Airy sub-structure $(U,A,B,C,\epsilon)$ of a
quantum Airy structure $(V,A,B,C,\epsilon)$ is a quantum pre-Airy sub-
structure of a quantum pre-Airy structure $(V,A,B,C,\epsilon)$.

######  Lemma 2.2.27.

([21, Section 2.6]) An Airy sub-structure of an Airy structure is an Airy
structure. A quantum Airy sub-structure of a quantum Airy structure is a
quantum Airy structure

###### Proof.

According to Remark 2.2.20, certain relations among $A,B,C$ and $\epsilon$ are
satisfied if and only if $A,B,C$ and $\epsilon$ define a (quantum) Airy
structure. The fact that we can restrict these tensors to a subspace $U\subset
V$ will not affect these relations. ∎

####  2.2.4. Gauge transformation of (quantum) Airy structures

Given an Airy structure $(V,A,B,C)$, it defines a quadratic Lagrangian
submanifold $L=\\{H_{i\in\mathbb{I}}=0\\}\subset W=V\oplus V^{*}$. Suppose
that $\\{e^{i\in\mathbb{I}},f_{i\in\mathbb{I}}\\}$ and
$\\{x^{i\in\mathbb{I}},y_{i\in\mathbb{I}}\\}$ are the given canonical basis
and canonical coordinates of $W$ respectively. As we have mentioned in Remark
2.2.23, the vector subspace $V\subset W$, with a basis
$\\{e^{i\in\mathbb{I}}\\}$, is a Lagrangian complement of
$T_{0}L=\left\\{\sum_{i\in\mathbb{I}}x^{k}f_{i}\right\\}\subset W$. But $V$ is
not a unique Lagrangian complement of $T_{0}L$. Any other Lagrangian
complement $\bar{V}\subset W$ of $T_{0}L$ corresponds to the change of a
canonical basis