$\\{e^{i\in\mathbb{I}},f_{i\in\mathbb{I}}\\}\mapsto\\{\bar{e}^{i\in\mathbb{I}},\bar{f}_{i\in\mathbb{I}}\\}$,
$\bar{e}^{i}:=e^{i}+\sum_{j\in\mathbb{I}}s^{ij}f_{j},\bar{f}_{i}:=f_{i}$ of
$W$, where $(s^{ij}):V\otimes V\rightarrow\mathbb{C}$ is an arbitrary
symmetric tensor. Note that $\\{\bar{e}^{i\in\mathbb{I}}\\}$ is a new basis
spanning the new Lagrangian complement $\bar{V}$ of $T_{0}L$ which remains
fixed. The corresponding change of canonical coordinates on $W$ is given by
$\bar{x}^{i}=x^{i}-\sum_{j\in\mathbb{I}}s^{ij}y_{j},\bar{y}_{i}=y_{i}$. The
symplectic form retains its canonical form
$\Omega=\sum_{i\in\mathbb{I}}d\bar{x}^{i}\wedge d\bar{y}_{i}$ in this new
coordinates and $T_{0}L\cong W/\bar{V}\cong\bar{V}^{*}$, therefore
$W\cong\bar{V}\oplus\bar{V}^{*}$. Expressing the Airy structure
$\\{H_{i\in\mathbb{I}}\\}$ using the new coordinates
$\\{\bar{x}^{i\in\mathbb{I}},y_{i\in\mathbb{I}}\\}$ in the usual form (25) we
obtain the new tensors
$\bar{A}=(\bar{a}_{ijk}),\bar{B}=(\bar{b}_{ij}^{k}),\bar{C}=(\bar{c}_{i}^{jk})$.

Both Airy structures $(V,A,B,C)$ and $(\bar{V},\bar{A},\bar{B},\bar{C})$ are
the same in the sense that they define the same Lagrangian submanifold
$L\subset W$, but we have chosen to described them differently using the
different choices of Lagrangian complement $\bar{V}$. Hence we call
$(V,A,B,C)\mapsto(\bar{V},\bar{A},\bar{B},\bar{C})$ a _gauge transformation_
of Airy structures.

We can also explore gauge transformation associate with freedom in choosing a
basis $\bar{e}^{i}:=\sum_{j\in\mathbb{I}}d^{i}_{j}e^{j}$ of $V$ and
$\bar{f}_{i}:=\sum_{j\in\mathbb{I}}c^{j}_{i}f_{j}$ of $V^{*}$. The
corresponding change of coordinates given by
$\bar{x}^{i}=\sum_{j\in\mathbb{I}}d^{i}_{j}x^{j}$ and
$\bar{y}_{i}=\sum_{j\in\mathbb{I}}c^{j}_{i}y_{j}$ respectively. We require
$(c^{i}_{j}):V\rightarrow V$ and $(d^{i}_{j}):V\rightarrow V$ to be the
inverse of each other, i.e.
$\sum_{k\in\mathbb{I}}d^{k}_{i}c_{k}^{j}=\delta_{i}^{j}$ and
$\sum_{k\in\mathbb{I}}c^{k}_{i}d^{j}_{k}=\delta^{j}_{i}$, so that the
coordinates $\\{\bar{x}^{i\in\mathbb{I}},\bar{y}_{i\in\mathbb{I}}\\}$ remain
canonical for $W=V\oplus V^{*}$. Let us also redefine the basis for
$\mathfrak{g}=\bigoplus_{i\in\mathbb{I}}\mathbb{C}\cdot H_{i}$ by
$H_{i}\mapsto\bar{H}_{i}:=\sum_{j\in\mathbb{I}}c^{j}_{i}H_{j}$. The
commutation relations become

$\\{\bar{H}_{i},\bar{H}_{j}\\}=\sum_{k\in\mathbb{I}}\bar{g}^{k}_{ij}\bar{H}_{k},\qquad\text{where}\qquad\bar{g}_{ij}^{k}:=g_{pq}^{r}c^{p}_{i}c^{q}_{j}d^{k}_{r}.$

Here and the remaining of this section, Einstein summation notations will be
used where appropriate. Note that
$\bar{g}^{k}_{ij}:=g_{pq}^{r}c^{p}_{i}c^{q}_{j}d^{k}_{r}$ only involve finite
summations since $g^{i}_{jk}$ are components of the tensor
$(g^{i}_{jk}):V\rightarrow V\otimes V$ as explained in Remark 2.2.16 (see also
Remark 2.2.12). Hence $\bar{g}^{i}_{jk}$ is well-defined and we have a tensor
$(\bar{g}^{i}_{jk}):V\rightarrow V\otimes V$.

######  Remark 2.2.28.

For a fixed $i\in\mathbb{I}$, $c^{i}_{j}$ will be non-zero for only finitely
many $j\in\mathbb{I}$. It follows that the elements of the inverse matrix must
satisfy the same properties. Namely, given an $i\in\mathbb{I}$, $d^{i}_{j}$
will be non-zero for only finitely many $j\in\mathbb{I}$. On the other hand,
$s^{ij}$ can be non-zero for infinitely many pairs of
$(i,j)\in\mathbb{I}^{2}$.

As a summary, let us give an explicit formula for the gauge transformation
$(V,A,B,C)\mapsto(\bar{V},\bar{A},\bar{B},\bar{C})$ associating with the
change of a canonical basis
$\\{e^{i\in\mathbb{I}},f_{i\in\mathbb{I}}\\}\mapsto\\{\bar{e}^{i\in\mathbb{I}},\bar{f}_{i\in\mathbb{I}}\\}$
of $W$:

(33)
$\bar{e}^{i}:=\sum_{j\in\mathbb{I}}d_{j}^{i}\left(e^{j}+\sum_{k\in\mathbb{I}}s^{jk}f_{k}\right),\qquad\bar{f}_{i}:=\sum_{j\in\mathbb{I}}c_{i}^{j}f_{j},$

where $(c^{i}_{j}):V\rightarrow V$ and $(d^{i}_{j}):V\rightarrow V$ are the
inverse of each other and $(s^{ij}):V\times V\rightarrow\mathbb{C}$. It is
easy to check that

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

The corresponding change of a canonical coordinates is given by:

(34)
$\bar{x}^{i}:=\sum_{j=1}^{\infty}d^{i}_{j}\left(x^{j}-\sum_{k=1}^{\infty}s^{jk}y_{k}\right),\qquad\bar{y}_{i}:=\sum_{j=1}^{\infty}c^{j}_{i}y_{j},$

and it is easy to check that

$\\{\bar{x}^{i},\bar{x}^{j}\\}=\\{\bar{y}_{i},\bar{y}_{j}\\}=0,\qquad\\{\bar{y}_{i},\bar{x}^{j}\\}=\delta^{j}_{i}.$

Denote the transformation (33) by $T:W\rightarrow W$. Let $\bar{V}\subset W$
be the vector subspace spanned by $\\{\bar{e}^{i\in\mathbb{I}}\\}$:

(35) $\bar{V}:=T(V).$

Substituting the inverse coordinates transformation
$x^{i}=\sum_{j\in\mathbb{I}}\left(c^{i}_{j}\bar{x}^{j}+\sum_{k\in\mathbb{I}}\bar{y}_{j}d^{j}_{k}s^{ki}\right),y_{i}=\sum_{j\in\mathbb{I}}d^{j}_{i}\bar{y}_{j}$,
into (25) and collecting terms so that
$\bar{H}_{i}:=\sum_{j\in\mathbb{I}}c^{j}_{i}H_{j}$ takes the form

$\bar{H}_{i}:=-\bar{y}_{i}+\sum_{j,k\in\mathbb{I}}\bar{a}_{ijk}\bar{x}^{j}\bar{x}^{k}+2\sum_{j,k\in\mathbb{I}}\bar{b}_{ij}^{k}\bar{x}^{j}\bar{y}_{k}+\sum_{j,k\in\mathbb{I}}\bar{c}^{jk}_{i}\bar{y}_{j}\bar{y}_{k},$

we find that $(V,A,B,C)\mapsto(\bar{V},\bar{A},\bar{B},\bar{C})$ is given by
[21, Section 2.1]

(36) $\displaystyle\bar{a}_{i_{1}i_{2}i_{3}}$
$\displaystyle:=a_{j_{1}j_{2}j_{3}}c^{j_{1}}_{i_{1}}c^{j_{2}}_{i_{2}}c^{j_{3}}_{i_{3}}$
(37) $\displaystyle\bar{b}^{i_{3}}_{i_{1}i_{2}}$
$\displaystyle:=b^{j_{3}}_{j_{1}j_{2}}c^{j_{1}}_{i_{1}}c^{j_{2}}_{i_{2}}d^{i_{3}}_{j_{3}}+a_{j_{1}j_{2}p}s^{pj_{3}}c^{j_{1}}_{i_{1}}c^{j_{2}}_{i_{2}}d^{i_{3}}_{j_{3}}$
(38) $\displaystyle\bar{c}^{i_{2}i_{3}}_{i_{1}}$
$\displaystyle:=c^{j_{2}j_{3}}_{j_{1}}c^{j_{1}}_{i_{1}}d^{i_{2}}_{j_{2}}d^{i_{3}}_{j_{3}}+b^{j_{2}}_{j_{1}p}s^{pj_{3}}c^{j_{1}}_{i_{1}}d^{i_{2}}_{j_{2}}d^{i_{3}}_{j_{3}}+b^{j_{3}}_{j_{1}q}s^{qj_{2}}c^{j_{1}}_{i_{1}}d^{i_{2}}_{j_{2}}d^{i_{3}}_{j_{3}}+a_{j_{1}pq}s^{pj_{2}}s^{qj_{3}}c^{j_{1}}_{i_{1}}d^{i_{2}}_{j_{2}}d^{i_{3}}_{j_{3}}.$

Similarly, we study _gauge transformations_ of quantum Airy structures
$(V,A,B,C,\epsilon)\mapsto(\bar{V},\bar{A},\bar{B},\bar{C},\bar{\epsilon})$ by
making the following substitutions:

$x^{i}\mapsto\hat{x}^{i}:=\sum_{j\in\mathbb{I}}\left(c^{i}_{j}\bar{x}^{j}+\sum_{k\in\mathbb{I}}s^{ik}d^{j}_{k}\hbar\bar{\partial}_{j}\right),\qquad\hbar\partial_{i}=\hbar\frac{\partial}{\partial
x^{i}}\mapsto\hat{y}_{i}:=\sum_{j\in\mathbb{I}}d^{j}_{i}\hbar\bar{\partial}_{j}=\sum_{j\in\mathbb{I}}d^{j}_{i}\hbar\frac{\partial}{\partial\bar{x}^{j}}$

in (26) and collecting terms so that
$\bar{\hat{H}}:=\sum_{j\in\mathbb{I}}c^{j}_{i}\hat{H}_{j}$ takes the form:

$\bar{\hat{H}}_{i}:=-\hbar\bar{\partial}_{i}+\sum_{j,k\in\mathbb{I}}\bar{a}_{ijk}\bar{x}^{j}\bar{x}^{k}+2\hbar\sum_{j,k\in\mathbb{I}}\bar{b}^{k}_{ij}\bar{x}^{j}\bar{\partial}_{k}+\hbar^{2}\sum_{j,k\in\mathbb{I}}\bar{c}^{jk}_{i}\bar{\partial}_{j}\bar{\partial}_{k}+\hbar\bar{\epsilon}_{i}.$

We find that the transformation
$A\mapsto\bar{A},B\mapsto\bar{B},C\mapsto\bar{C}$ are given by (36)-(38) and
$\epsilon\mapsto\bar{\epsilon}$ is given by [21, Section 2.2]

(39) $\bar{\epsilon}_{i}:=\epsilon_{l}c^{l}_{i}+a_{ljk}c^{l}_{i}s^{jk}.$

Because the commutation relations of $\hat{x}^{i}$ and $\hat{y}_{j}$ in
$\mathcal{D}^{\hbar}(W)$ are the same as those of $x^{i}$ and
$\hbar\partial_{j}$ we conclude that
$(\bar{V},\bar{A},\bar{B},\bar{C},\bar{\epsilon})$ is a quantization of
$(\bar{V},\bar{A},\bar{B},\bar{C})$.

######  Remark 2.2.29.

Suppose that $(V,A,B,C,\epsilon)$ is a quantum pre-Airy structure. Given
$(c^{i}_{j}):V\rightarrow V$, $(d^{i}_{j})=(c^{i}_{j})^{-1}:V\rightarrow V$
and $(s^{ij}):V\times V\rightarrow\mathbb{C}$, then
$(V,A,B,C,\epsilon)\mapsto(\bar{V},\bar{A},\bar{B},\bar{C},\bar{\epsilon})$
given by (35)-(39) is also a quantum pre-Airy structure transformation. We
call this a _gauge transformation_ of quantum pre-Airy structure.

######  Remark 2.2.30.

The gauge transformation respects sub-structures in the following sense [21,
Section 2.6]. We write $\bar{V}:=T(V)$. Suppose that $(U,A,B,C,\epsilon)$ is a
quantum (pre-)Airy sub-structure of $(V,A,B,C,\epsilon)$ then
$(\bar{U},\bar{A},\bar{B},\bar{C},\bar{\epsilon})$ is a quantum (pre-)Airy
sub-structure of $(\bar{V},\bar{A},\bar{B},\bar{C},\bar{\epsilon})$ where
$\bar{U}:=T(U)$ and $T:W\rightarrow W$ denotes the transformation (33).

###  2.3. The Residue Constraints Airy Structure

In this section, we are going to review the residue constraints Airy structure
[21, Section 3.3]. It is the Airy structure that will be most interesting to
us due to its connection with topological recursion as we will explain in
Section 2.6. Consider a countably infinite dimensional (let
$\mathbb{I}:=\mathbb{Z}_{>0}$) vector space

$V_{Airy}:=z^{-1}\mathbb{C}[z^{-1}]\frac{dz}{z}$

of meromorphic differential forms in the neighborhood of $z=0$ and equipped
with the discrete topology. The set
$\left\\{e^{k\in\mathbb{I}}:=z^{-k}\frac{dz}{z}\right\\}$ is a basis of
$V_{Airy}$. The continuous dual space is given by

$V^{*}_{Airy}:=z\mathbb{C}[[z]]\frac{dz}{z}$

where a vector $f\in V^{*}_{Airy}$ can be thought of as a linear functional
$v\mapsto Res_{z=0}\left(f\int v\right)\in\mathbb{C}$ on $V_{Airy}$. The set
$\left\\{f_{k\in\mathbb{I}}:=kz^{k}\frac{dz}{z}\right\\}$ is a basis of
$V^{*}_{Airy}$, we can check that $f_{i}(e^{j})=\delta_{i}^{j}$. The
corresponding Tate space is

$W_{Airy}:=V_{Airy}\oplus V^{*}_{Airy}=\left\\{\eta\in\mathbb{C}((z))dz\ |\
Res_{z=0}\eta=0\right\\}$

with the symplectic form

$\Omega_{Airy}(f,g):=Res_{z=0}\left(f\int g\right).$

Any typical element $w$ of $W_{Airy}$ can be expressed as
$w:=\sum_{k\in\mathbb{I}}\left(x^{k}f_{k}+y_{k}e^{k}\right)=\sum_{k=1}^{\infty}\left(y_{k}z^{-k}\frac{dz}{z}+kx^{k}z^{k}\frac{dz}{z}\right)$,
where $y_{k}\in\mathbb{C}$ is non-zero for an only finite number of $k$. The
symbol $\int g$ in the definition of $\Omega_{Airy}(f,g)$ denotes a formal
indefinite integral of $g$. For instance, let us write
$g:=\sum_{k=1}^{\infty}\left(y_{k}z^{-k}\frac{dz}{z}+kx^{k}z^{k}\frac{dz}{z}\right)$
then $\int
g:=\sum_{k=1}^{\infty}\left(-\frac{1}{k}y_{k}z^{-k}+x^{k}z^{k}\right)+const$.
Note that the integration constant does not introduce any ambiguity in the
expression of $\Omega_{Airy}(f,g)$ since the residue of $f$ is zero. It is
easy to check that $f\mapsto\Omega_{Airy}(f,.)$ gives an isomorphism $W\cong
W^{*}$ and in particular, $e^{k}\mapsto\Omega_{Airy}(e^{k},.)$ gives an
isomorphism $V_{Airy}\cong V^{**}_{Airy}$.

Define the _residue constraints Airy structure_ $\\{H_{i\in\mathbb{I}}\\}$ on
$V_{Airy}$ by

$\displaystyle(H_{Airy})_{2n}(w)$
$\displaystyle=Res_{z=0}\left(\left(z-\frac{w(z)}{2zdz}\right)z^{2n}d(z^{2})\right),$
(40) $\displaystyle(H_{Airy})_{2n-1}(w)$
$\displaystyle=\frac{1}{2}Res_{z=0}\left(\left(z-\frac{w(z)}{2zdz}\right)^{2}z^{2n-2}d(z^{2})\right),\qquad
n=1,2,3,...$

where
$w(z):=\sum_{k\in\mathbb{I}}\left(x^{k}f_{k}+y_{k}e^{k}\right)=\sum_{k=1}^{\infty}\left(y_{k}z^{-k}\frac{dz}{z}+kx^{k}z^{k}\frac{dz}{z}\right)\in
W_{Airy}$. By expanding (2.3) into formal polynomials in $\widehat{Sym}_{\leq
2}(W_{Airy})$ of coordinates $\\{x^{i\in\mathbb{I}},y_{i\in\mathbb{I}}\\}$ of
$W_{Airy}$, we find that $\\{(H_{Airy})_{i\in\mathbb{I}}\\}$ can be expressed
in the standard form (25) as [21, Section 3.3]:

(41)
$(H_{Airy})_{2n}=-J_{2n},\qquad(H_{Airy})_{2n-1}=-J_{2n-1}+\frac{1}{4}\sum_{k=-\infty}^{\infty}J_{2k-1}J_{2n-2k-3},\qquad
n=1,2,3,...$

where $J_{+k}:=y_{k},J_{-k}:=kx^{k}$. The quantization of Airy structure
$\\{(H_{Airy})_{i\in\mathbb{I}}\\}$ exists and it is given by [21, Section
3.3]

(42)
$(\hat{H}_{Airy})_{2n}=-\hat{J}_{2n},\qquad(\hat{H}_{Airy})_{2n-1}=-\hat{J}_{2n-1}+\frac{1}{4}\sum_{k=-\infty}^{\infty}\hat{J}_{2k-1}\hat{J}_{2n-2k-3}+\frac{\hbar}{16}\delta_{n,2},\qquad
n=1,2,3,...$

where $\hat{J}_{+k}:=\hbar\frac{\partial}{\partial
x^{k}},\hat{J}_{-k}:=kx^{k}$. By reading the coefficients of (42), we found
that the tensors $A_{Airy},B_{Airy},C_{Airy}$ and $\epsilon_{Airy}$ are given
by [21, Section 3.3]

(43) $\displaystyle A_{Airy}=\frac{1}{4}e^{1}\otimes e^{1}\otimes e^{1},$
$\displaystyle\qquad\text{or}\qquad(a_{Airy})_{ijk}=\frac{1}{4}\delta_{1i}\delta_{1j}\delta_{1k}$
(44) $\displaystyle
B_{Airy}:e^{n}\mapsto\frac{1}{4}\sum_{\begin{subarray}{c}n_{1},n_{2}\geq 1\\\
n_{1}+n_{2}=n+3\\\ n_{1}\ odd\end{subarray}}n_{2}e^{n_{1}}\otimes e^{n_{2}},$
$\displaystyle\qquad\text{or}\qquad(b_{Airy})_{ij}^{k}=\frac{1}{4}\sum_{\begin{subarray}{c}k_{1},k_{2}\geq
1\\\ k_{1}+k_{2}=k+3\\\ k_{1}\
odd\end{subarray}}k_{2}\delta_{k_{1}i}\delta_{k_{2}j}$ (45) $\displaystyle
C_{Airy}:e^{n_{1}}\otimes e^{n_{2}}\mapsto\begin{cases}0,&n_{1}+n_{2}\
\text{odd}\\\ \frac{1}{4}e^{n_{1}+n_{2}+3},&n_{1}+n_{2}\
\text{even}\end{cases}$
$\displaystyle\qquad\text{or}\qquad(c_{Airy})_{i}^{jk}=\begin{cases}0,&j+k\
\text{odd}\\\ \frac{1}{4}\delta_{i,j+k+3},&j+k\ \text{even}\end{cases}$ (46)
$\displaystyle\epsilon_{Airy}=\frac{1}{16}e^{3},$
$\displaystyle\qquad\text{or}\qquad(\epsilon_{Airy})_{i}=\frac{1}{16}\delta_{i,3}.$

######  Remark 2.3.1.

We can also express $A_{Airy},B_{Airy},C_{Airy}$ tensors in term of a residue
of meromorphic forms by substituting
$w(z):=\sum_{k\in\mathbb{I}}(x^{k}f_{k}(z)+y_{k}e^{k}(z))$ into (2.3):

$\displaystyle(a_{Airy})_{ijk}$
$\displaystyle=Res_{z=0}\left(-\frac{1}{4i}\frac{1}{z(dz)^{2}}f_{i}(z)f_{j}(z)f_{k}(z)\right)\delta_{i,odd}$
$\displaystyle(b_{Airy})_{ij}^{k}$
$\displaystyle=Res_{z=0}\left(-\frac{1}{4i}\frac{1}{z(dz)^{2}}f_{i}(z)f_{j}(z)e^{k}(z)\right)\delta_{i,odd}$
$\displaystyle(c_{Airy})_{i}^{jk}$
$\displaystyle=Res_{z=0}\left(-\frac{1}{4i}\frac{1}{z(dz)^{2}}f_{i}(z)e^{j}(z)e^{k}(z)\right)\delta_{i,odd},$

where $\delta_{i,odd}=\sum_{n=1}^{\infty}\delta_{i,2n-1}$ is $1$ if $i$ is an
odd integer and $0$ if $i$ is an even integer. Sometime it is easier to work
with $A_{Airy},B_{Airy},C_{Airy}$ in this form. It is less straightforward to
express $\epsilon_{Airy}$ as a residue.

One last important point to note is that
$(V_{Airy},A_{Airy},B_{Airy},C_{Airy},\epsilon_{Airy})$ contains a primitive
quantum Airy sub-structure
$((V_{Airy})_{odd},A_{Airy},B_{Airy},C_{Airy},\epsilon_{Airy})$ where

$(V_{Airy})_{odd}:=z^{-1}\mathbb{C}[z^{-2}]\frac{dz}{z}.$

####  2.3.1. The Lagrangian ‘submanifold’ $L_{Airy}\subset W_{Airy}$

As we have mentioned in the last section, the interpretation of an Airy
structure as a quadratic Lagrangian submanifold is less obvious when dimension
is infinite. Let us show how
$L_{Airy}:=\\{(H_{Airy})_{i\in\mathbb{I}}=0\\}\subset W_{Airy}$ highlights
this difficulty.

######  Lemma 2.3.2.

$\\{w\in W_{Airy}\ |\
(H_{Airy})_{i\in\mathbb{I}}(w)=0\\}=\\{x^{1}=y_{1}=y_{2}=y_{3}=\cdots=0\\}\subset
W_{Airy}.$

In other words, the solution to $(H_{Airy})_{i\in\mathbb{I}}=0$ in $W_{Airy}$
are the trivial ones. Therefore, by interpreting $L_{Airy}$ as a subset of
$W_{Airy}$ we will have $L_{Airy}$ as a linear subspace of $W_{Airy}$ instead
of a non-linear submanifold as the name _quadratic_ suggests.

###### Proof.

Since $(H_{Airy})_{2n}=0$ implies $y_{2n}=0$, it remains for us to show that
$x^{1}=y_{2n-1}=0$ for all $n=1,2,3,...$. Since
$w=\sum_{k=1}^{\infty}\left(y_{k}z^{-k}\frac{dz}{z}+kx^{k}z^{k}\frac{dz}{z}\right)\in
W_{Airy}$ means there are finitely many non-zero $y_{i\in\mathbb{I}}$, let us
assume that $y_{2n+1}=0$ for all $n\geq N$ for some $N\in\mathbb{Z}_{>0}$.
Then it is easy to check that $(H_{Airy})_{4n+1}=\frac{1}{4}y^{2}_{2n-1}=0$,
which implies $y_{2n-1}=0$. Therefore $y_{2n-1}=0$ for all $n=1,2,3,...$.
Finally, $y_{1}=0$ implies $(H_{Airy})_{1}=\frac{1}{4}(x^{1})^{2}=0$.
Therefore $x^{1}=0$ and the lemma is proven. ∎

This Lemma tells us that any solution
$w=\sum_{k=1}^{\infty}\left(y_{k}z^{-k}\frac{dz}{z}+kx^{k}z^{k}\frac{dz}{z}\right)$
to $(H_{Airy})_{i}(w)=0$ with $x^{1}\neq 0$ must consist of infinitely many
non-zero $y_{i}$ and so $w$ does not belong to $W_{Airy}$. According to
Theorem 2.2.24 or Corollary 2.2.25, there exists
$\psi_{L_{Airy}}:=\exp\left(\sum_{g=0}^{\infty}\hbar^{g-1}(S_{Airy})_{g}\right)$
such that $(\psi_{L_{Airy}})^{-1}(\hat{H}_{Airy})_{i}(\psi_{L_{Airy}})=0$. In
particular, we can still find the generating function $(S_{Airy})_{0}$ such
that

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

although we will have $\frac{\partial S_{0}}{\partial x^{i}}\neq 0$ for
infinitely many $i\in\mathbb{I}$. In this sense, $L_{Airy}$ can be thought of
formally as a Lagrangian submanifold of $W_{Airy}$.

Nevertheless, we define $T_{0}L_{Airy}:=z\mathbb{C}[[z]]\frac{dz}{z}\subset
W_{Airy}$. It is clear that $T_{0}L_{Airy}$ is a Lagrangian subspace of
$W_{Airy}$ and $V_{Airy}$ is a Lagrangian complement of $T_{0}L_{Airy}$.

###  2.4. The Embedding of Discs and The Analytic Residue Constraints

Essentially, our work in this section is an analytic approach to [21, Section
6]. As we have seen, the quadratic Lagrangian submanifold $L_{Airy}\subset
W_{Airy}$ defined by the residue constraints can only be interpreted formally
and not as an actual submanifold. In this section, we will study the residue
constraints from an analytic perspective. We show that there exists non-
trivial solution $L^{M}_{Airy}:=\\{(H_{Airy})_{i=1,2,3,\cdots}=0\\}$ of
residue constraints in a modified version of $W_{Airy}$, which will be
introduced as $W^{\epsilon_{,}M}_{Airy}$. We also will see how the residue
constraints are related to the deformation of _discs_ embedded in the foliated
symplectic surface $(\mathbb{C}^{2},dx\wedge dy,\\{x=const\\})$. Another
motivation for us to take the analytic approach instead of the formal one is
because it should make the connection between Airy structures and the
prepotential $\mathfrak{F}_{\Sigma_{0}}$ more transparent as
$\mathfrak{F}_{\Sigma_{0}}$ is a holomorphic function.

We organize the content of this section as follows. In Section 2.4.1, we
introduce the space of analytic discs $Discs^{M}$ (similar to the space of
formal discs $Discs$ in [21, Section 6]) and the vector bundle
$W^{\epsilon,M}\rightarrow Discs^{M}$. Some interesting properties of
$W^{\epsilon,M}$ will be noted. In Section 2.4.2, we show how the residue
constraints Airy structure can be considered analytically. For example, when
writing $(H_{Airy})_{i}$ in the standard form (25), all summations converge
absolutely and there is no ambiguity in summation order. On the other hand,
not all gauge transformations considered in Section 2.2.4 are allowed in the
analytic context, and we propose a more restricted class of gauge
transformations. We also explore how some of the analytic solutions in
$L^{M}_{Airy}$ can be expressed in terms of the genus zero part of the ATR
with quantum Airy structure $(V_{Airy},A_{Airy},B_{Airy},C_{Airy})$, which we
have discussed previously as a formal power series. Finally, we define in
Section 2.4.3 an embedding map
$\Phi_{t_{0}}:Discs^{\epsilon,M}_{t_{0}}\rightarrow W^{\epsilon,M}_{t_{0}}$
analogous to
$\boldsymbol{\Phi}_{\Sigma_{0}}:\mathcal{B}_{\Sigma_{0}}\rightarrow\mathcal{H}_{\Sigma_{0}}$
defined in Section 2.1 and show that the image of $\Phi_{t_{0}}$ is given by
convergence Laurent series satisfying the residue constraints.

####  2.4.1. The space of discs

We begin by introducing the notion of analytic discs and the trivial weak
symplectic vector bundle $(W^{\epsilon,M}\rightarrow
Discs^{M},\Omega,\nabla)$. Unlike the formal discs and symplectic vector
bundle $W\rightarrow Discs$ introduced in [21, Section 6], the convergence
condition of series on some annulus will be enforced in our version. The
upshot is that $(W^{\epsilon,M}\rightarrow Discs^{M},\Omega,\nabla)$ is flat
(Lemma 2.4.12), covariantly constant (Lemma 2.4.14) and the parallel transport
of $\xi_{1}\in W^{\epsilon,M}_{t_{1}}$ to $W^{\epsilon,M}_{t_{2}}$, for any
two points $t_{1},t_{2}\in Discs^{M}$ sufficiently close, is given by (Lemma
2.4.13)

$\xi_{2}:=\exp\left((a(t_{2})-a(t_{1}))\mathcal{L}_{\frac{1}{2z}\partial_{z}}\right)(\xi_{1})\in
W^{\epsilon,M}_{t_{2}},$

where $\exp(a\mathcal{L}_{\frac{1}{2z}\partial_{z}})$ is an operator
exponential of the Lie derivative $\frac{1}{2z}\partial_{z}$. If
$|a(t_{2})-a(t_{1})|$ is too large then $\xi_{2}$ may not have the desired
convergence properties to be in $W^{\epsilon,M}_{t_{2}}$. This is why it is
important for $t_{1}$ and $t_{2}$ to be sufficiently close. In fact, most of
the contents from Lemma 2.4.9 until Lemma 2.4.12 are dedicated to discussing
the properties of $\exp(a\mathcal{L}_{\frac{1}{2z}\partial_{z}})$ and in
particular, the precise notion for $t_{1}$ and $t_{2}$ to be considered
‘sufficiently close’. However, these details are technical and can be skipped
in the first reading. On the other hand, Lemma 2.4.13 is the key result that
will be used in the construction of
$\Phi_{t_{0}}:Discs^{\epsilon,M}_{t_{0}}\rightarrow W^{\epsilon,M}_{t_{0}}$
later in Section 2.4.3.

In the simplest terms, a disc is an embedding
$t:\mathbb{D}\cong\mathbb{D}_{t}\hookrightarrow\mathbb{C}^{2}$ such that
$\mathbb{D}_{t}$ has a tangency of order $1$ to a leaf of the foliation at
$r:=t(0)$, where $\mathbb{D}\subset\mathbb{C}$ is an open neighbourhood of
$0\in\mathbb{C}$. Using the standard local coordinate $z:=\sqrt{x-x(r)}$, in
general, the embedding $t$ is a parameterization of
$\mathbb{D}_{t}\subset\mathbb{C}^{2}$ which can be written in the form

$t=(x=a+z^{2},y=b_{0}+b_{1}z+b_{2}z^{2}+b_{3}z^{3}+...),\qquad
a,b_{k}\in\mathbb{C},$

where $b_{1}\neq 0$. The study of the embedding of discs helps us understand
at the local level the deformation of $\mathcal{F}$-transversal curves
$\Sigma$ embedded inside a foliated symplectic surface
$(S,\Omega_{S},\mathcal{F})$. As explained in Section 2.1, given a curve
$[\Sigma]\in\mathcal{B}$ then we can find a $(\mathcal{F},\Omega_{S})$-chart
$U_{\alpha}$ around any ramification point $r_{\alpha},\alpha\in Ram$ of
$\Sigma$. Because $U_{\alpha}$ is symplectomorphic to an open subset of the
foliated symplectic surface $(\mathbb{C}^{2},dx\wedge dy,\\{x=const\\})$,
therefore $\Sigma\cap U_{\alpha}$ can be treated as a disc embedded in
$(\mathbb{C}^{2},dx\wedge dy,\\{x=const\\})$. We will compare the global
deformation of $\Sigma\subset S$ to the deformation of discs embedded around
each ramification point of $\Sigma$ later in Section 2.5.

Because discs are only defined locally, it follows that their deformation
space is much larger than that of a $\mathcal{F}$-transversal curve $\Sigma$.
Let us now define the space of discs:

######  Definition 2.4.1.

Let $\mathbb{D}_{M}:=\\{z\in\mathbb{C}\ |\ |z|<M\\}$, then

$Discs^{M}:=\left\\{t:\mathbb{D}_{M}\xrightarrow{\cong}\mathbb{D}_{t,M}\hookrightarrow\mathbb{C}^{2}\
|\
t=\left(x=a(t)+z^{2},y=\sum_{k=0}^{\infty}b_{k}(t)z^{k}\right),\begin{array}[]{c}b_{1}(t)\neq
0\text{ and }\\\ \sum_{k=0}^{\infty}b_{k}(t)z^{k}\text{ converges }\\\ \text{
for $|z|<\bar{M}$ for some $M<\bar{M}$.}\end{array}\right\\}.$

We can think of $Discs^{M}$ as a subset of the infinite dimensional vector
space of infinite tuple of numbers $(a(t),b_{0}(t),b_{1}(t),\cdots)$. It is
almost a vector subspace except for the condition $b_{1}(t)\neq 0$.
Nevertheless, for $t_{1},t_{2}\in Discs^{M}$ and
$\lambda_{1},\lambda_{2}\in\mathbb{C}$ such that
$\lambda_{1}b_{1}(t_{1})+\lambda_{2}b_{1}(t_{2})\neq 0$ we define

$\lambda_{1}t_{1}+\lambda_{2}t_{2}:=\left(x=\lambda_{1}a(t_{1})+\lambda_{2}a(t_{2})+z^{2},y=\sum_{k=0}^{\infty}(\lambda_{1}b_{k}(t_{1})+\lambda_{2}b_{k}(t_{2}))z^{k}\right)\in
Discs^{M}.$

We define the _kinematic tangent space_ $T_{t}Discs^{M}$ as guided by [22]:

$T_{t}Discs^{M}:=\left\\{v=A\frac{\partial}{\partial
a}+\sum_{k=0}^{\infty}B_{k}\frac{\partial}{\partial b_{k}}\ |\
A,B_{k}\in\mathbb{C},\begin{array}[]{c}\sum_{k=0}^{\infty}B_{k}z^{k}\text{
converges }\\\ \text{ for $|z|<\bar{M}$ for some
$M<\bar{M}$.}\end{array}\right\\}.$

More precisely, $T_{t}Discs^{M}$ is the set of $v\in Discs^{M}$ such that
$\gamma(\tau):=t+\tau v$ defines a path $\gamma:(-\delta,\delta)\rightarrow
Discs^{M}$ for some $\delta>0$. The tangent space $T_{t}Discs^{M}$ tells us
information about the infinitesimal deformations of $\mathbb{D}_{t,M}$.

We are going to introduce the vector bundle $W^{\epsilon,M}\rightarrow
Discs^{M}$ which can be thought of as a local counter-part of the vector
bundle $\mathcal{H}\rightarrow\mathcal{B}$ defined in Section 2.1. A fiber of
$W^{\epsilon,M}$ over each point $t\in Discs^{M}$ is given by
$W^{\epsilon,M}_{Airy}$, the analytic version of $W_{Airy}$. Unlike $W_{Airy}$
which is a symplectic topological vector space, $W^{\epsilon,M}_{Airy}$ will
be a _weak symplectic vector space_.

######  Definition 2.4.2.

A weak symplectic (topological) vector space $(W,\Omega)$ is a pair of a
(topological) vector space $W$ and a (continuous) antisymmetric bilinear form
$\Omega:W\times W\rightarrow\mathbb{C}$ such that $w\mapsto\Omega(w,.)$ is
injective $W\hookrightarrow W^{*}$. We call $\Omega$ the _weak symplectic
form_ of $W$.

######  Definition 2.4.3.

For any $\epsilon,M\in\mathbb{R}$ such that $0<\epsilon<M$, the weak
symplectic vector space $(W^{\epsilon,M}_{Airy},\Omega_{Airy})$ is given by

$W_{Airy}^{\epsilon,M}:=\left\\{\xi\in
z^{-1}\mathbb{C}[[z^{-1}]]\frac{dz}{z}\oplus z\mathbb{C}[[z]]\frac{dz}{z}\ |\
\begin{array}[]{c}\text{$\xi(z)$ converges when
$\bar{\epsilon}<|z|<\bar{M}$}\\\ \text{for some
$\bar{\epsilon}<\epsilon<M<\bar{M}$}\end{array}\right\\}.$

and the symplectic form $\Omega_{Airy}$ is given by:

$\Omega_{Airy}(\xi_{1},\xi_{2}):=Res_{z=0}\left(\xi_{1}\int\xi_{2}\right).$

The definition of the symplectic form $\Omega_{Airy}$ is exactly the same as
the one on $W_{Airy}$, hence we are denoting them using the same notation. Let
$(W^{\epsilon,M}\rightarrow Discs^{M},\Omega,\nabla)$ be the trivial weak
symplectic vector bundle, $W^{\epsilon,M}:=Discs^{M}\times
W^{\epsilon,M}_{Airy}$. The fiber over each point $t\in Discs^{M}$ of
$W^{\epsilon,M}$ is given by the weak symplectic vector space
$(W^{\epsilon,M}_{t},\Omega_{t}):=(W^{\epsilon,M}_{Airy},\Omega_{Airy})$. The
connection $\nabla$ is given by:

$\displaystyle\nabla$
$\displaystyle:=d-(da)\mathcal{L}_{\frac{1}{2z}\partial_{z}}=da\frac{\partial}{\partial
a}+\sum_{k=0}^{\infty}db_{k}\frac{\partial}{\partial
b_{k}}-(da)\mathcal{L}_{\frac{1}{2z}\partial_{z}}$ (47)
$\displaystyle:\Gamma(Discs^{M},W^{\epsilon,M})\rightarrow\Gamma(Discs^{M},Hom(TDiscs^{M},W^{\epsilon,M}))$

where $\mathcal{L}_{\frac{1}{2z}\partial_{z}}\in End(W^{\epsilon,M}_{t})$
denotes the Lie derivative operator.

######  Remark 2.4.4.

The trivial vector bundle $W^{\epsilon,M}\rightarrow Discs^{M}$ should be
compared to the trivial vector bundle $W\rightarrow Discs$ introduced in [21,
Section 6] where $Discs$ is the space of formal discs and each fiber of $W$ is
given by $W_{t}:=W_{Airy}=\\{\eta\in\mathbb{C}((z))dz\ |\ Res_{z=0}\eta=0\\}$.

######  Remark 2.4.5.

The following are useful elementary facts when working with any $\xi\in
W^{\epsilon,M}_{Airy}$. If a Laurent series
$\sum_{k=-\infty}^{+\infty}a_{k}z^{k}$ converges for
$\bar{\epsilon}<|z|<\bar{M}$ then it converges absolutely and uniformly for
$\epsilon<|z|<M$ for any $\epsilon>\bar{\epsilon}$ and $M<\bar{M}$. From the
uniform convergence, it follows that the integration of a Laurent series can
be done term-by-term. Similarly, using the Cauchy integral formula, we find
that the differentiation of a Laurent series can also be done term-by-term.

######  Remark 2.4.6.

Let us explain why the symplectic form
$\Omega_{Airy}:W^{\epsilon,M}_{Airy}\times
W^{\epsilon,M}_{Airy}\rightarrow\mathbb{C}$ is well-defined. If
$\xi_{2}=\sum_{k=1}^{\infty}\left((y_{2})_{k}z^{-k}\frac{dz}{z}+(x_{2})^{k}kz^{k}\frac{dz}{z}\right)$
converges to a holomorphic differential on an annulus
$\bar{\epsilon}<|z|<\bar{M}$ with $\bar{\epsilon}<\epsilon<M<\bar{M}$ then the
indefinite integral $\int\xi_{2}$ converges to a holomorphic function on the
same annulus. It follows that $\xi_{1}\int\xi_{2}$ is holomorphic on some
annulus containing $\epsilon<|z|<M$ and hence its residue is well-defined as a
coefficient of $\frac{dz}{z}$ in the Laurent expansion of
$\xi_{1}\int\xi_{2}$. The integration constant arising from $\int\xi_{2}$ will
not introduce any ambiguity in $Res_{z=0}(\xi_{1}\int\xi_{2})$ because the
residue of $\xi_{1}$ is zero.

######  Remark 2.4.7.

It is possible to assign a topology to $W^{\epsilon,M}$. For example, we can
equip $Discs^{M}$ and $W^{\epsilon,M}$ with the topology induced from the
following norms:

$\|t\|_{Discs^{M}}:=\sqrt{|a(t)|^{2}+\left(\sup_{\epsilon<|z|<M}\left|\sum_{k=0}^{\infty}b_{k}(t)z^{k}\right|\right)^{2}},\qquad\|\xi\|_{W^{\epsilon,M}}:=\sup_{\epsilon<|z|<M}\left|\xi(z)\frac{z}{dz}\right|$

and equip $W^{\epsilon,M}:=Discs^{M}\times W^{\epsilon,M}_{Airy}$ with the
product topology. The topology of $W^{\epsilon,M}$ will play no role later.
Note that for $W^{\epsilon,M}_{Airy}$ to be a weak symplectic vector space, a
topology is not needed. The stronger condition of symplectic requires a
topology because an infinite-dimensional vector space is always strictly
smaller than its algebraic dual, hence a topology is needed for the condition
$W\cong W^{*}$ to be satisfied. The fact that $W^{\epsilon,M}_{Airy}$ is weak
symplectic instead of symplectic will not lead to any major consequences.

######  Remark 2.4.8.

We follow [22, 23] in defining a connection $\nabla$ (as given in (2.4.3)) on
an infinite-dimensional vector bundle over an infinite-dimensional manifold.
Let us explain more precisely the meaning of
$\Gamma(Discs^{M},W^{\epsilon,M})$ and
$\Gamma(Discs^{M},Hom(TDiscs^{M},W^{\epsilon,M}))$ as follows.

We call a section $\xi$ of $W^{\epsilon,M}$ differentiable if
$\iota_{v}d\xi=A\frac{\partial\xi}{\partial
a}+\sum_{k=0}^{\infty}B_{k}\frac{\partial\xi}{\partial b_{k}}\in
W^{\epsilon,M}$ for all $v:=A\frac{\partial}{\partial
a}+\sum_{k=0}^{\infty}B_{k}\frac{\partial}{\partial b_{k}}\in T_{t}Discs^{M}$
and call $\xi$ smooth if it is infinitely differentiable. We define
$\Gamma(Discs^{M},W^{\epsilon,M})$ to be the set of smooth global sections of
$W^{\epsilon,M}$. Let $\Gamma(Discs^{M},TDiscs^{M})$ be the set of sections
$t\mapsto A(t)\frac{\partial}{\partial
a}+\sum_{k=0}^{\infty}B_{k}(t)\frac{\partial}{\partial b_{k}}\in
T_{t}Discs^{M}$ such that
$\sum_{k=0}^{\infty}B_{k}(t)z^{k}dz\in\Gamma(Discs^{M},W^{\epsilon,M})$.
Finally, we let $\Gamma(Discs^{M},Hom(TDiscs^{M},W^{\epsilon,M}))$ be a set of
linear maps
$\Gamma(Discs^{M},TDiscs^{M})\rightarrow\Gamma(Discs^{M},W^{\epsilon,M})$.

The term $\mathcal{L}_{\frac{1}{2z}\partial_{z}}$ in the definition of
$\nabla$ naturally arises from the deformation of discs via the variation of
$a$ keeping the coordinate $x$ of $\mathbb{C}^{2}$ constant. This is because
$x=z^{2}+a$. Therefore, the connection $\nabla$ of $W^{\epsilon,M}$ is indeed
the local version of $\nabla_{GM}$ on $\mathcal{H}$.

Let us examine the operator $\mathcal{L}_{\frac{1}{2z}\partial_{z}}$ more
closely. In general, we have that $\mathcal{L}_{h(z)\partial_{z}}\in
End(W^{\epsilon,M}_{t})=End(W^{\epsilon,M}_{Airy})$ for any $h(z)\in
z^{-1}\mathbb{C}[[z^{-1}]]\oplus\mathbb{C}[[z]]$ which converges absolutely
for $\bar{\epsilon}<|z|<\bar{M}$ for some $\bar{\epsilon}<\epsilon<M<\bar{M}$.
To see this, let $\xi=f(z)\frac{dz}{z}\in W^{\epsilon,M}_{Airy}$ be given such
that $f(z)$ converges absolutely for $\bar{\bar{\epsilon}}<|z|<\bar{\bar{M}}$
for some $\bar{\bar{\epsilon}}<\epsilon<M<\bar{\bar{M}}$. Then the Lie
derivative

(48)
$\mathcal{L}_{h(z)\partial_{z}}\xi(z)=(d\iota_{h(z)\partial_{z}}+\iota_{h(z)\partial_{z}}d)(f(z)dz)=d(h(z)f(z))$

is holomorphic for
$\max(\bar{\epsilon},\bar{\bar{\epsilon}})<|z|<\min(\bar{M},\bar{\bar{M}})$.
The Laurent series expansion of $\mathcal{L}_{h(z)\partial_{z}}\xi(z)$ also
will not have the $\frac{dz}{z}$ term because
$\mathcal{L}_{h(z)\partial_{z}}\xi(z)=d(h(z)f(z))$ is a total differential and
we have $\oint_{|z|=\epsilon}\mathcal{L}_{h(z)\partial_{z}}\xi(z)=0$. This
shows us that $\mathcal{L}_{h(z)\partial_{z}}\xi(z)\in W^{\epsilon,M}_{Airy}$.

For any $a\in\mathbb{C}$, we also define an exponential operator
$\exp(a\mathcal{L}_{h(z)\partial_{z}})$ by

(49)
$\exp(a\mathcal{L}_{h(z)\partial_{z}})(\xi):=\sum_{k=0}^{\infty}\frac{a^{k}}{k!}(\mathcal{L}_{h(z)\partial_{z}})^{k}(\xi),\qquad\xi\in
W^{\epsilon,M}_{Airy}.$

The operator $\exp(a\mathcal{L}_{h(z)\partial_{z}})$ can be interpreted as a
flow after ‘time’ $a\in\mathbb{C}$ generated by
$\mathcal{L}_{h(z)\partial_{z}}$. Note that although
$\mathcal{L}_{h(z)\partial_{z}}\in End(W^{\epsilon,M}_{Airy})$, its
exponential $\exp(a\mathcal{L}_{h(z)\partial_{z}})$ will not be in
$End(W^{\epsilon,M}_{Airy})$ in general. For instance, the flow
$\exp(a\mathcal{L}_{h(z)\partial_{z}})\xi$ of $\xi\in W^{\epsilon,M}_{Airy}$
can fail to have the desired convergence properties and ‘fall out of’
$W^{\epsilon,M}_{Airy}$. Let us consider the case where
$h(z):=z^{k},k\in\mathbb{Z}$ in the following

######  Lemma 2.4.9.

Given $\xi=f(z)dg(z)\in W^{\epsilon,M}_{Airy}$ converges absolutely for
$\bar{\epsilon}<|z|<\bar{M}$ where $\bar{\epsilon}<\epsilon<M<\bar{M}$. Then
for $a\in\mathbb{C}$ with a sufficiently small $|a|$:

$|a|<\begin{cases}\displaystyle\min\left(\frac{\epsilon^{1-k}-\bar{\epsilon}^{1-k}}{1-k},\frac{\bar{M}^{1-k}-M^{1-k}}{1-k}\right),&\qquad
k\neq 1,\\\
\displaystyle\min\left(\log\frac{\epsilon}{\bar{\epsilon}},\log\frac{\bar{M}}{M}\right),&\qquad
k=1\end{cases}$

we have

$\exp(a\mathcal{L}_{z^{k}\partial_{z}})\xi(z)=\left\\{\begin{aligned}
&f\left((z^{1-k}+(1-k)a)^{\frac{1}{1-k}}\right)dg\left((z^{1-k}+(1-k)a)^{\frac{1}{1-k}}\right)&,k\neq
1\\\ &f(e^{a}z)dg(e^{a}z)&,k=1\end{aligned}\right\\}\in
W^{\epsilon,M}_{Airy}.$

In fact, $\exp(a\mathcal{L}_{z^{k}\partial_{z}})\xi(z)$ converges absolutely
for

$\left\\{\begin{aligned}
(\bar{\epsilon}^{1-k}+(1-k)|a|)^{\frac{1}{1-k}}&<|z|<(\bar{M}^{1-k}-(1-k)|a|)^{\frac{1}{1-k}}&,k\neq
1\\\
\bar{\epsilon}|e^{a}|^{-1}&<|z|<\bar{M}|e^{a}|^{-1}&,k=1\end{aligned}\right\\}.$

###### Proof.

For $k\neq 1$ we consider a $z$-plane as a $|k-1|$-fold cover of a $w$-plane
where $w:=z^{1-k}$. We have $z^{k}\partial_{z}=(1-k)\partial_{w}$ and
$\mathcal{L}_{z^{k}\partial_{z}}=\mathcal{L}_{(k-1)\partial_{w}}=(k-1)\mathcal{L}_{\partial_{w}}$.
For any holomorphic functions $f_{1}=f_{1}(w),f_{2}=f_{2}(w)$ we have
$\mathcal{L}_{\partial_{w}}(f_{1}(w)df_{2}(w))=(\mathcal{L}_{\partial_{w}}f_{1}(w))df_{2}(w)+f_{1}(w)d(\mathcal{L}_{\partial_{w}}f_{2}(w))=(\partial_{w}f_{1}(w))df_{2}(w)+f_{1}(w)d(\partial_{w}f_{2}(w))$.
We recognize $\exp((k-1)a\mathcal{L}_{\partial_{w}})$ as the shift operator
$\exp(\mathcal{L}_{(1-k)a\partial_{w}})(f_{1}(w)df_{2}(w))=f_{1}(w+(1-k)a)df_{2}(w+(1-k)a)$.
Choose a branch for $z$ and write $\xi=f(z)dg(z)$ as a differential form on
$w$-plane $\xi=f(w^{\frac{1}{1-k}})dg(w^{\frac{1}{1-k}})$, then

$\displaystyle\exp(a\mathcal{L}_{z^{k}\partial_{z}})(f(z)dg(z))$
$\displaystyle=f\left((w+(1-k)a)^{\frac{1}{1-k}}\right)dg\left((w+(1-k)a)^{\frac{1}{1-k}}\right)$
$\displaystyle=f\left((z^{1-k}+(1-k)a)^{\frac{1}{1-k}}\right)dg\left((z^{1-k}+(1-k)a)^{\frac{1}{1-k}}\right)=:f_{a}(z)dg_{a}(z).$

We have that $f_{a}(z)dg_{a}(z)$ would be holomorphic on the chosen branch-cut
whenever $\bar{\epsilon}<|(z^{1-k}+(1-k)a)^{\frac{1}{1-k}}|<\bar{M}$ or
equivalently
$(\bar{\epsilon}^{1-k}+(1-k)|a|)^{\frac{1}{1-k}}<|z|<(\bar{M}^{1-k}-(1-k)|a|)^{\frac{1}{1-k}}$.
The condition
$|a|<\min\left(\frac{\epsilon^{1-k}-\bar{\epsilon}^{1-k}}{1-k},\frac{\bar{M}^{1-k}-M^{1-k}}{1-k}\right)$
ensures that $(\bar{\epsilon}^{1-k}+(1-k)|a|)^{\frac{1}{1-k}}<\epsilon$ and
$(\bar{M}^{1-k}-(1-k)|a|)^{\frac{1}{1-k}}>M$. On the other hand,
$(z^{1-k}+(1-k)a)^{\frac{1}{1-k}}=z(1+(1-k)az^{k-1})^{\frac{1}{1-k}}$ is a
well-defined function and can be expanded as a Laurent series converges in $z$
for $(|1-k||a|)^{\frac{1}{1-k}}<|z|$ if $1-k>0$ and for
$|z|<(|1-k||a|)^{\frac{1}{1-k}}$ if $1-k<0$. These conditions automatically
follow if
$(\bar{\epsilon}^{1-k}+(1-k)|a|)^{\frac{1}{1-k}}<|z|<(\bar{M}^{1-k}-(1-k)|a|)^{\frac{1}{1-k}}$
since
$(|1-k||a|)^{\frac{1}{1-k}}<(\bar{\epsilon}^{1-k}+(1-k)|a|)^{\frac{1}{1-k}}$
when $1-k>0$ and
$(|1-k||a|)^{\frac{1}{1-k}}>(\bar{M}^{1-k}-(1-k)|a|)^{\frac{1}{1-k}}$ when
$1-k<0$.

Since the series $\xi\in W^{\epsilon,M}_{Airy}$ doesn’t contain the term
$\frac{dz}{z}$, by substituting $z=w^{\frac{1}{1-k}}$ we also do not have the
term $\frac{1}{1-k}\frac{dw}{w}$ in
$f(w^{\frac{1}{1-k}})dg(w^{\frac{1}{1-k}})$. The operator
$\exp(a\mathcal{L}_{z^{k}\partial_{z}})=\exp((k-1)a\mathcal{L}_{\partial_{w}})$
shifts $w$ to $w+(k-1)a$ and will not create the term $\frac{dw}{w}$
therefore, we do not have the term $\frac{dz}{z}$ in the Laurent series of
$\exp(a\mathcal{L}_{z^{k}\partial_{z}})\xi(z)$.

For $k=1$, we note that $(\mathcal{L}_{z\partial_{z}})^{k}(z)=z$ and
$\exp(a\mathcal{L}_{z\partial_{z}})(\xi(z))=f(e^{a}z)dg(e^{a}z)$ follows
directly for (49). Then we can see that if
$|a|<\min\left(\log\frac{\epsilon}{\bar{\epsilon}}.\log\frac{\bar{M}}{M}\right)$
we have $\bar{\epsilon}<|e^{a}z|<\bar{M}$ whenever
$\bar{\epsilon}|e^{a}|^{-1}<\epsilon<|z|<M<\bar{M}|e^{a}|^{-1}$ ∎

As an immediate consequence of Lemma 2.4.9 we have the following useful
statement:

######  Corollary 2.4.10.

Let $k\neq 1$, $a\in\mathbb{C}$ and define

$\epsilon(a):=(\epsilon^{1-k}-(1-k)|a|)^{\frac{1}{1-k}},\qquad
M(a):=(M^{1-k}+(1-k)|a|)^{\frac{1}{1-k}}.$

Then we have $\exp\left(a\mathcal{L}_{z^{k}\partial_{z}}\right)\in
Hom_{\mathbb{C}}\left(W^{\epsilon(a),M(a)}_{Airy},W^{\epsilon,M}_{Airy}\right)$.

######  Lemma 2.4.11.

Let $k\in\mathbb{Z}$ then $\mathcal{L}_{z^{k}\partial_{z}}\in
sp(W^{\epsilon,M}_{Airy})$. Moreover, if $\xi_{1},\xi_{2}\in
W^{\epsilon,M}_{Airy}$ and $a\in\mathbb{C}$ are given such that
$\exp(a\mathcal{L}_{z^{k}\partial_{z}})\xi_{1},\exp(a\mathcal{L}_{z^{k}\partial_{z}})\xi_{2}\in
W^{\epsilon,M}_{Airy}$ then

$\Omega_{Airy}\left(\exp(a\mathcal{L}_{z^{k}\partial_{z}})\xi_{1},\exp(a\mathcal{L}_{z^{k}\partial_{z}})\xi_{2}\right)=\Omega_{Airy}(\xi_{1},\xi_{2}).$

###### Proof.

Using (48) it is straightforward to check that

$\displaystyle\Omega_{Airy}\left(\mathcal{L}_{z^{k}\partial_{z}}z^{m}\frac{dz}{z},z^{n}\frac{dz}{z}\right)$
$\displaystyle+\Omega_{Airy}\left(z^{m}\frac{dz}{z},\mathcal{L}_{z^{k}\partial_{z}}z^{n}\frac{dz}{z}\right)$
$\displaystyle=\Omega_{Airy}\left((m+k-1)z^{m+k-1}\frac{dz}{z},z^{n}\frac{dz}{z}\right)+\Omega_{Airy}\left(z^{m}\frac{dz}{z},(n+k-1)z^{n+k-1}\frac{dz}{z}\right)$
$\displaystyle=\frac{m+k-1}{n}\delta_{-m-k+1,n}+\delta_{-m,n+k-1}=0.$

Since differentiation of a power series can be done term-by-term (Remark
2.4.5), it follows that

$\Omega_{Airy}\left(\mathcal{L}_{z^{k}\partial_{z}}\xi_{1},\xi_{2}\right)+\Omega_{Airy}\left(\xi_{1},\mathcal{L}_{z^{k}\partial_{z}}\xi_{2}\right)=0$

for any $\xi_{1},\xi_{2}\in W^{\epsilon,M}_{Airy}$. Therefore
$\mathcal{L}_{z^{k}\partial_{z}}\in sp(W^{\epsilon,M}_{Airy})$, proving the
first part of the Lemma. Using Lemma 2.4.9 we check that
$\frac{d}{da}\exp(a\mathcal{L}_{z^{k}\partial_{z}})\xi_{i}=\mathcal{L}_{z^{k}\partial_{z}}\exp(a\mathcal{L}_{z^{k}\partial_{z}})\xi_{i},i=1,2$.
It follows that

$\displaystyle\frac{d}{da}\Omega_{Airy}$
$\displaystyle\left(\exp(a\mathcal{L}_{z^{k}\partial_{z}})\xi_{1},\exp(a\mathcal{L}_{z^{k}\partial_{z}})\xi_{2}\right)$
$\displaystyle=\Omega_{Airy}\left(\mathcal{L}_{z^{k}\partial_{z}}\exp(a\mathcal{L}_{z^{k}\partial_{z}})\xi_{1},\exp(a\mathcal{L}_{z^{k}\partial_{z}})\xi_{2}\right)+\Omega_{Airy}\left(\exp(a\mathcal{L}_{z^{k}\partial_{z}})\xi_{1},\mathcal{L}_{z^{k}\partial_{z}}\exp(a\mathcal{L}_{z^{k}\partial_{z}})\xi_{2}\right)$
$\displaystyle=0.$

Which means
$\Omega_{Airy}\left(\exp(a\mathcal{L}_{z^{k}\partial_{z}})\xi_{1},\exp(a\mathcal{L}_{z^{k}\partial_{z}})\xi_{2}\right)$
is constant and must equal to its value at $a=0$ which is
$\Omega_{Airy}(\xi_{1},\xi_{2})$. ∎

For this section, we will only need $\mathcal{L}_{z^{k}\partial_{z}}$, and its
exponentiation when $k=-1$. According to Lemma 2.4.9, given $\xi\in
W^{\epsilon,M}_{Airy}$ converges absolutely for
$\bar{\epsilon}<|z|<\bar{M},\bar{\epsilon}<\epsilon<M<\bar{M}$ and given
$a\in\mathbb{C},|a|<\min(\epsilon^{2}-\bar{\epsilon}^{2},\bar{M}^{2}-M^{2},\bar{\epsilon}^{2},\bar{M}^{2})$
we have
$\exp(a\mathcal{L}_{\frac{1}{2z}\partial_{z}})\xi(z)=f(\sqrt{z^{2}+a})dg(\sqrt{z^{2}+a})\in
W^{\epsilon,M}_{Airy}$ since it is holomorphic for
$\sqrt{\bar{\epsilon}^{2}+|a|}<|z|<\sqrt{\bar{M}^{2}-|a|}$. In particular, it
is straightforward to check that

(50)
$\exp(a\mathcal{L}_{\frac{1}{2z}\partial_{z}})(z)=z+\frac{a}{2z}+\sum_{k=2}^{\infty}\frac{a^{k}}{k!}\frac{(-1)^{k-1}(2k-3)!!}{2^{k}z^{2k-1}},$

which can be recognized as a Laurent series expansion of
$\sqrt{z^{2}+a}=z\sqrt{1+\frac{a}{z^{2}}}$ for $|z|>\epsilon>\sqrt{|a|}$.

We will now show that the vector bundle $(W^{\epsilon,M}\rightarrow
Discs^{M},\Omega,\nabla)$ is flat and the symplectic form $\Omega$ is
$\nabla$-covariantly constant.

######  Lemma 2.4.12.

The vector bundle $W^{\epsilon,M}\rightarrow Discs^{M}$ with the connection
$\nabla$ is flat.

###### Proof.

We define the exterior covariant derivative

$d_{\nabla}:\Gamma\left(Discs^{M},Hom\left(\bigwedge^{r}TDiscs^{M},W^{\epsilon,M}\right)\right)\rightarrow\Gamma\left(Discs^{M},Hom\left(\bigwedge^{r+1}TDiscs^{M},W^{\epsilon,M}\right)\right)$

by

$d_{\nabla}\psi:=(ds)\xi+(-1)^{r}s\wedge\nabla(\xi)=(ds)\xi-(-1)^{r}s\wedge
da\mathcal{L}_{\frac{1}{2z}\partial_{z}}(\xi)$

where $\psi=s\xi,s\in\Gamma(Discs^{M},\Omega^{r}_{Discs^{M}})$ and $\xi\in
W^{\epsilon,M}_{Airy}$, then extend by linearity. We note that
$d_{\nabla}=\nabla$ for $r=0$. The condition that $\nabla$ is flat is the same
as the vanishing of curvature tensor $R_{\nabla}=d_{\nabla}\circ d_{\nabla}=0$
and it is easy to check that

$d^{2}_{\nabla}\psi=(d^{2}s)\xi-(-1)^{r+1}(ds)\wedge
da\mathcal{L}_{\frac{1}{2z}\partial_{z}}(\xi)-(-1)^{r}ds\wedge
da\mathcal{L}_{\frac{1}{2z}\partial_{z}}(\xi)+s\wedge da\wedge
da(\mathcal{L}_{\frac{1}{2z}\partial_{z}})^{2}(\xi)=0.$

This concludes that $\nabla$ is flat. ∎

######  Lemma 2.4.13.

Pick $t_{1},t_{2}\in Discs^{M}$ such that $|a(t_{1})-a(t_{2})|$ is
sufficiently small, then the parallel transport of $\xi\in
W^{\epsilon,M}_{t_{1}}$ to the fiber $W^{\epsilon,M}_{t_{2}}$ exists, path-
independent and given by
$\xi_{t_{2}}=\exp((a(t_{2})-a(t_{1}))\mathcal{L}_{\frac{1}{2z}\partial_{z}})\xi$.
In fact,
$\xi_{t}=\exp((a(t)-a(t_{1}))\mathcal{L}_{\frac{1}{2z}\partial_{z}})(\xi)$ is
a parallel vector field.

###### Proof.

The path-independence of the parallel transport follows from the fact that
$\nabla$ is flat. The condition that $|a(t_{1})-a(t_{2})|$ being sufficiently
small is needed to ensure that
$\exp((a(t_{2})-a(t_{1}))\mathcal{L}_{\frac{1}{2z}\partial_{z}})(\xi)\in
W^{\epsilon,M}_{t_{2}}$ according to Lemma 2.4.9. Finally, we can check that

$\nabla\xi_{t}=da\frac{\partial}{\partial
a}\exp((a(t)-a(t_{1}))\mathcal{L}_{\frac{1}{2z}\partial_{z}})\xi-
da\mathcal{L}_{\frac{1}{2z}\partial_{z}}\exp((a(t)-a(t_{1}))\mathcal{L}_{\frac{1}{2z}\partial_{z}})(\xi)=0.$

∎

######  Lemma 2.4.14.

The symplectic form $\Omega$ is $\nabla$-covariantly constant.

###### Proof.

For any $\xi_{1},\eta_{1}\in W^{\epsilon,M}_{t_{1}}$ we have from Lemma 2.4.13
that their parallel transport to $W^{\epsilon,M}_{t_{2}}$ are
$\xi_{2}=\exp((a(t_{2})-a(t_{1}))\mathcal{L}_{\frac{1}{2z}\partial_{z}})(\xi_{1}),\eta_{2}=\exp((a(t_{2})-a(t_{1}))\mathcal{L}_{\frac{1}{2z}\partial_{z}})(\eta_{1})\in
W^{\epsilon,M}_{t_{2}}$. Therefore,
$\Omega_{t_{2}}(\xi_{2},\eta_{2})=\Omega_{t_{1}}(\xi_{1},\eta_{1})$ by Lemma
2.4.11. ∎

####  2.4.2. The analytic residue constraints

Observe that the residue constraints Hamiltonians:

$\displaystyle(H_{Airy})_{2n}(w)$
$\displaystyle=Res_{z=0}\left(\left(z-\frac{w(z)}{2zdz}\right)z^{2n}d(z^{2})\right),$
$\displaystyle(H_{Airy})_{2n-1}(w)$
$\displaystyle=\frac{1}{2}Res_{z=0}\left(\left(z-\frac{w(z)}{2zdz}\right)^{2}z^{2n-2}d(z^{2})\right),\qquad
n=1,2,3,...$

as given in Section 2.3 is not only well-defined when $w(z)\in W_{Airy}$, but
also when $w(z)\in W^{\epsilon,M}_{Airy}$ is a Laurent series converging on
some annulus containing $\epsilon<|z|<M$. Define
$L_{Airy}^{M}:=\\{(H_{Airy})_{i=1,2,3,\cdots}=0\\}\subset
W_{Airy}^{\epsilon,M}$. As we will see later, unlike its counter-part
$L_{Airy}\subset W_{Airy}$ (see Lemma 2.3.2), the submanifold $L_{Airy}^{M}$
contains many non-trivial solutions of $(H_{Airy})_{i=1,2,3,\cdots}=0$ because
an infinite number of terms with negative power in $z$ is allowed. Define the
tangent space to $L_{Airy}^{M}$ at any point $p\in L_{Airy}^{M}$ by

$T_{p}L_{Airy}^{M}:=\left\\{w\in W^{\epsilon,M}_{Airy}\ |\
\iota_{w}d(H_{Airy})_{i}=0,i=1,2,3,\cdots\right\\}\subset
W^{\epsilon,M}_{Airy}.$

In particular,

$T_{0}L_{Airy}^{M}:=\left\\{\xi\in z\mathbb{C}[[z]]\frac{dz}{z}\ |\
\begin{array}[]{c}\text{$\xi(z)$ converges for $|z|<\bar{M}$}\\\ \text{for
some $\bar{M}>M$}\end{array}\right\\}\subset W^{\epsilon,M}_{Airy},\qquad
T_{0}L^{M}_{Airy}\subset T_{0}L_{Airy}\cong V^{*}_{Airy}.$

is a Lagrangian subspace of $W^{\epsilon,M}_{Airy}$. Let us also define

$V_{Airy}^{\epsilon}:=\left\\{\xi\in z^{-1}\mathbb{C}[[z^{-1}]]\frac{dz}{z}\
|\ \begin{array}[]{c}\text{$\xi(z)$ converges for $|z|>\bar{\epsilon}$}\\\
\text{for some $\bar{\epsilon}<\epsilon$}\end{array}\right\\}\subset
W^{\epsilon,M}_{Airy}\qquad V_{Airy}\subset V^{\epsilon}_{Airy}.$

$V^{\epsilon}_{Airy}$ is also a Lagrangian subspace of
$W^{\epsilon,M}_{Airy}$, in fact, it is a Lagrangian complement of
$T_{0}L^{M}_{Airy}$ : $W^{\epsilon,M}_{Airy}=V^{\epsilon}_{Airy}\oplus
T_{0}L^{M}_{Airy}$. This is similar to $W_{Airy}=V_{Airy}\oplus V_{Airy}^{*}$,
but in this case, $T_{0}L_{Airy}^{M}\subsetneq(V^{\epsilon}_{Airy})^{*}$. We
let $\\{e^{k=1,2,3,\cdots},f_{k=1,2,3,\cdots}\\}$ to be a canonical basis of
$W^{\epsilon,M}_{Airy}$:

$\Omega_{Airy}(e^{i},e^{j})=\Omega_{Airy}(f_{i},f_{j})=0,\qquad\Omega_{Airy}(e^{i},f_{j})=\delta^{i}_{j},$

where $\\{e^{k=1,2,3,\cdots}:=z^{-k}\frac{dz}{z}\\}$ is a basis of
$V_{Airy}^{\epsilon}$ and $\\{f_{k=1,2,3,\cdots}:=kz^{k}\frac{dz}{z}\\}$ is a
basis of $T_{0}L_{Airy}^{M}$.

Among many topics we have discussed regarding Airy structures in Section 2.2,
we will mostly be interested in the genus zero ATR output and the gauge
transformation in the context of the quantum residue constraints Airy
structure introduced in Section 2.3. Since we only have considered these
topics from the formal perspective in Section 2.2 and 2.3, the goal of this
section is to revisit these topics from an analytical perspective within the
context of $\\{(H_{Airy})_{i=1,2,3,\cdots}\\}$. We will show that most things
we have learned previously continue to hold, possibly with some minor
adjustments. Most difficulties came from the following: In $W_{Airy}$, the
Tate topology dictates that the contraction between vectors in $V_{Airy}$ and
$V^{*}_{Airy}$ always result in finite sums since vectors in $V_{Airy}$ are
finite linear combinations of $\\{e^{i\in\mathbb{I}}\\}$. On the other hand,
$W^{\epsilon,M}_{Airy}$ is not a Tate space and $w(z)\in
W^{\epsilon,M}_{Airy}$ in general can be written as
$w(z)=\sum_{k=1}^{\infty}\left(y_{k}z^{-k}\frac{dz}{z}+kx^{k}z^{k}\frac{dz}{z}\right)$
with infinitely many non-zero $x^{k}$ and $y_{k}$. It turns out that the lack
of finiteness can be replaced with absolute uniform convergence. Here are
summaries of the key points:

  1. (1)

By expressing $(H_{Airy})_{i}$ in the form (25), we get multiple terms with
double infinite summation which could lead to some issues such as divergences
and ambiguity of summation order. It turns out that there is no problem
because each $(H_{Airy})_{i}$ is given by the residue of some convergence
Laurent series, hence each double infinite summation actually converges
absolutely (see Lemma 2.4.15).

  2. (2)

In Section 2.2.4, gauge transformations are given by $(c^{i}_{j})$, its
inverse $(d^{i}_{j})$, and $(s^{ij})$ where $c^{i}_{j}$ is non-zero for only
finitely many $j=1,2,3,\cdots$ for each fixed $i$. In the analytic setting, we
need to check that canonical coordinates transformation (34) follows from the
canonical basis transformation (33). In Section 2.2.4 this is easy because
there are only finitely many terms in each summation, but now we have infinite
sums and getting (34) from (33) requires interchanging the orders of these
infinite sums. Moreover, we also need to make sure that the summation of the
type $\sum_{i=1}^{\infty}c^{i}_{j}y_{i}$ which appears in (34) will converge,
as $y_{i}$ may be non-zero for an infinitely many $i=1,2,3,\cdots$. We will
see that if $(c^{i}_{j}),(d^{i}_{j})$ and $(s^{ij})$ are further restricted to
those satisfying Condition 2.4.17, then all of these concerns will be taken
care of. In particular, expressing $(H_{Airy})_{i}$ in the transformed
canonical coordinates will give us exactly the same gauge transformation of
$A_{Airy},B_{Airy},C_{Airy}$ as given in Section 2.2.4 (see Proposition
2.4.20).

  3. (3)

We discussed in Section 2.2 that
$(H_{Airy})_{k}(\\{x^{i\in\mathbb{I}}\\},\\{y_{i\in\mathbb{I}}:=\partial
S_{0}/\partial x^{i}\\})=0$ where
$S_{0}\in\prod_{n=1}^{\infty}Sym_{n}(V_{Airy})$ is the genus zero output of
the ATR using $(V_{Airy},A_{Airy},B_{Airy},C_{Airy},\epsilon_{Airy})$. Even
though $S_{0}$ was introduced as a formal series, we will see in Proposition
2.4.21 that it can be interpreted analytically and that its derivatives are
relevant to the solution of analytic residue constraints.

For the remainder of this section, we present the technical detail leading to
each point above. It is possible to skip these detail without losing any
continuity to Section 2.4.3 where we will resume to discuss the embedding map
$\Phi_{t_{0}}:Discs^{\epsilon,M}_{t_{0}}\rightarrow W^{\epsilon,M}_{t_{0}}$.

######  Lemma 2.4.15.

Let
$w(z):=\sum_{k=1}^{\infty}(y_{k}e^{k}(z)+x^{k}f_{k}(z))=\sum_{k=1}^{\infty}\left(y_{k}z^{-k}\frac{dz}{z}+kx^{k}z^{k}\frac{dz}{z}\right)\in
W^{\epsilon,M}_{Airy}$. Then

$(H_{Airy})_{i}=-y_{i}+\sum_{j,k=1}^{\infty}(a_{Airy})_{ijk}x^{j}x^{k}+2\sum_{j,k=1}^{\infty}(b_{Airy})_{ij}^{k}x^{j}y_{k}+\sum_{j,k=1}^{\infty}(c_{Airy})^{jk}_{i}y_{j}y_{k},$

where
$A_{Airy}=((a_{Airy})_{ijk}),B_{Airy}=((b_{Airy})_{ij}^{k}),C_{Airy}=((c_{Airy})_{i}^{jk})$
are given by (43)-(45) and all the summations converge absolutely.

###### Proof.

Note that there are possibly infinitely many non-zero $y_{k=1,2,3,\cdots}$ and
$x^{k=1,2,3,\cdots}$ therefore each $\sum_{j,k=1}^{\infty}$ are double
infinite summations. To check that each double infinite summations in
$(H_{Airy})_{i}$ converges absolutely we note that $(H_{Airy})_{2n}=-y_{2n}$
and

(51) $(H_{Airy})_{2n-1}(w)=\frac{1}{4\pi
i}\oint_{|z|=\epsilon}\left(\left(z-\frac{\sum_{k=1}^{\infty}\left(y_{k}e^{k}(z)+x^{k}f_{k}(z)\right)}{2zdz}\right)^{2}z^{2n-2}d(z^{2})\right)$

where
$\sum_{k=1}^{\infty}(y_{k}e^{k}(z)+x^{k}f_{k}(z))=\sum_{k=1}^{\infty}\left(y_{k}z^{-k}\frac{dz}{z}+kx^{k}z^{k}\frac{dz}{z}\right)$
converges absolutely uniformly for $\epsilon<|z|<M$. Adding or multiplying
convergence Laurent series a finite number of times will produce another
convergence Laurent series. We can expand (51) and examine each term more
closely. For example

$\displaystyle\frac{1}{2\pi i}\oint_{|z|=\epsilon}$
$\displaystyle\left(-\frac{1}{4i}\frac{1}{z(dz)^{2}}f_{i}(z)\left(\sum_{j=1}^{\infty}x^{j}f_{j}(z)\right)\left(\sum_{k=1}^{\infty}y_{k}e^{k}(z)\right)\right)\delta_{i,odd}$
$\displaystyle=\sum_{j,k=1}^{\infty}Res_{z=0}\left(-\frac{1}{4i}\frac{1}{z(dz)^{2}}f_{i}(z)f_{j}(z)e^{k}(z)\right)\delta_{i,odd}x^{j}y_{k}=\sum_{j,k=1}^{\infty}(b_{Airy})^{k}_{ij}x^{j}y_{k},$

where we have used Remark 2.3.1 to identify the residue with
$(b_{Airy})^{k}_{ij}$. Note that
$\sum_{j=1}^{\infty}\sum_{k=1}^{\infty}x^{j}y_{k}f_{j}(z)e^{k}(z)$ converges
absolutely uniformly for $\epsilon<|z|<M$ as it is the product of
$\sum_{j=1}^{\infty}x^{j}f_{j}(z)$ and $\sum_{k=1}^{\infty}y_{k}e^{k}(z)$.
Therefore term-by-term integration is allowed and
$\sum_{j,k=1}^{\infty}(b_{Airy})^{k}_{ij}x^{j}y_{k}$ converges absolutely.
Similarly, the sum $\sum_{j,k=1}^{\infty}(a_{Airy})_{ijk}x^{j}x^{k}$ and
$\sum_{j,k=1}^{\infty}(c_{Airy})^{jk}_{i}y_{j}y_{k}$ also converges
absolutely. ∎

######  Remark 2.4.16.

Note that
$-\sum_{i=1}^{\infty}\frac{\delta_{i,odd}}{4i}\frac{1}{z(dz)^{2}}f_{i}(z)e^{i}(z_{1})=\frac{1}{4z(z^{2}-z^{2}_{1})}\frac{dz_{1}}{dz}=:K(z_{1},z)$.
It follows that

$\sum_{i,j,k=1}^{\infty}(b_{Airy})^{k}_{ij}x^{j}y_{k}e^{i}(z_{1})=\left(\frac{1}{2\pi
i}\oint_{|z|=\bar{\epsilon}}\frac{\left(\sum_{j=1}^{\infty}x^{j}f_{j}(z)\right)\left(\sum_{k=1}^{\infty}y_{k}e^{k}(z)\right)}{4z(z^{2}-z_{1}^{2})dz}\right)dz_{1}\in
V^{\epsilon}_{Airy}.$

Similarly, we have
$\sum_{i,j,k=1}^{\infty}(a_{Airy})_{ijk}x^{j}x^{k}e^{i}(z_{1}),\sum_{i,j,k=1}^{\infty}(c_{Airy})_{i}^{jk}y_{j}y_{k}e^{i}(z_{1})\in
V^{\epsilon}_{Airy}$. Therefore, we obtain the analytic analogues to the
definition of tensors $A_{Airy},B_{Airy},C_{Airy}$:

$A_{Airy}:T_{0}L_{Airy}^{M}\otimes T_{0}L_{Airy}^{M}\rightarrow
V_{Airy}^{\epsilon},\qquad B_{Airy}:T_{0}L_{Airy}^{M}\otimes
V_{Airy}^{\epsilon}\rightarrow V_{Airy}^{\epsilon},\qquad
C_{Airy}:V^{\epsilon}_{Airy}\otimes V^{\epsilon}_{Airy}\rightarrow
V^{\epsilon}_{Airy}.$

We can recognise $K(z_{1},z)$ to be the recursive kernel in the topological
recursion of an Airy curve $y^{2}=x$ with
$\omega_{0,2}(z_{1},z_{2})=\frac{dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}$ and
$\omega_{0,1}(z)=ydx$.

Let us examine the analytic version of the gauge transformation introduced in
Section 2.2.4. Consider the change of a canonical basis
$\\{e^{k=1,2,3,\cdots},f_{k=1,2,3,\cdots}\\}\mapsto\\{\bar{e}^{k=1,2,3,\cdots},\bar{f}_{k=1,2,3,\cdots}\\}$
of $W^{\epsilon,M}_{Airy}$ where

(52)
$\bar{e}^{i}:=\sum_{j=1}^{\infty}d^{i}_{j}\left(e^{j}+\sum_{k=1}^{\infty}s^{jk}f_{k}\right),\qquad\bar{f}_{i}:=\sum_{j=1}^{\infty}c^{j}_{i}f_{j},$

where $(c^{j}_{i}),(d^{j}_{i})$ and $(s^{ij})$ are tensors satisfying the
following conditions:

######  Condition 2.4.17.

C2.4.17.1:

$(c^{j}_{i})$ and $(d^{j}_{i})$ are inverse of each other:
$\sum_{k=1}^{\infty}c^{k}_{i}d^{j}_{k}=\delta_{i}^{j}$ and
$\sum_{k=1}^{\infty}d^{k}_{i}c^{j}_{k}=\delta_{i}^{j}$.

C2.4.17.2:

$\sum_{j=1}^{\infty}c^{j}_{i}f_{j}=\sum_{j=1}^{\infty}jc^{j}_{i}z^{j}\frac{dz}{z}\in
T_{0}L_{Airy}^{M}$ and
$\sum_{j=1}^{\infty}d^{j}_{i}f_{j}=\sum_{j=1}^{\infty}jd^{j}_{i}z^{j}\frac{dz}{z}\in
T_{0}L^{M}_{Airy}$.

C2.4.17.3:

There exists $I\in\mathbb{Z}_{>0}$ such that for all $j=1,2,3,\cdots$ we have
$c^{j}_{i}=d^{j}_{i}=\delta^{j}_{i}$ for all $i>I$.

C2.4.17.4:

$s^{ij}$ is symmetric and
$\sum_{i,j=1}^{\infty}s^{ij}f_{i}(z_{1})f_{j}(z_{2})=\sum_{i,j=1}^{\infty}ijs^{ij}\frac{dz_{1}dz_{2}}{z_{1}z_{2}}$
is a power series in two variables $z_{1},z_{2}$ converges for
$|z_{1}|,|z_{2}|<\bar{M}$ for some $\bar{M}>M$ (hence a holomorphic symmetric
bi-differential).

######  Remark 2.4.18.

Compare the conditions on $(c^{i}_{j}),(d^{i}_{j})$ and $(s^{ij})$ defined
here with those in Section 2.2.4, especially in Remark 2.2.28. We find that
the analytic version of gauge transformation is more restrictive. This is
because $V^{\epsilon}_{Airy}$ is a larger space than $V_{Airy}$, so loosely
speaking there are more ways for things to go wrong. In other words, any
analytic gauge transformation will be a gauge transformation as considered in
Section 2.2.4 but not the other way around.

Note that, we do not need to specify $\sum_{j=1}^{\infty}d^{i}_{j}e^{j}\in
V^{\epsilon}_{Airy}$ and $\sum_{j=1}^{\infty}c^{i}_{j}e^{j}\in
V^{\epsilon}_{Airy}$ as another condition, because C2.4.17.3 ensures this. If
one of $(c^{i}_{j})$ or $(d^{i}_{j})$ satisfies C2.4.17.2 or C2.4.17.3 then so
will the other one because they are the inverse of each other. This can be
seen using the formula for the inverse of block matrices.

######  Lemma 2.4.19.

Suppose that $(c^{j}_{i}),(d^{j}_{i})$ and $(s^{ij})$ satisfy the Condition
2.4.17, then we have the following

  1. (1)

Let $\sum_{k=1}^{\infty}x^{k}f_{k}\in T_{0}L^{M}_{Airy}$ and
$\sum_{k=1}^{\infty}y_{k}e^{k}\in V^{\epsilon}_{Airy}$ then
$\sum_{j=1}^{\infty}\left(\sum_{i=1}^{\infty}y_{i}d^{i}_{j}\right)e^{j},\sum_{j=1}^{\infty}\left(\sum_{i=1}^{\infty}y_{i}c^{i}_{j}\right)e^{j}\in
V^{\epsilon}_{Airy}$ and
$\sum_{j=1}^{\infty}\left(\sum_{i=1}^{\infty}x^{i}d^{j}_{i}\right)f_{j},\sum_{j=1}^{\infty}\left(\sum_{i=1}^{\infty}x^{i}c^{j}_{i}\right)f_{j}\in
T_{0}L_{Airy}^{M}$ converges absolutely uniformly.

  2. (2)

Let $\sum_{k=1}^{\infty}y_{k}e^{k}\in V^{\epsilon}_{Airy}$ and Let $(v^{i})$
be an arbitrary vector such that $\sum_{k=1}^{\infty}y_{k}v^{k}$ converges
then
$\sum_{i=1}^{\infty}y_{i}\left(\sum_{j=1}^{\infty}d^{i}_{j}v^{j}\right)=\sum_{j=1}^{\infty}\left(\sum_{i=1}^{\infty}y_{i}d^{i}_{j}\right)v^{j}$
and
$\sum_{i=1}^{\infty}y_{i}\left(\sum_{j=1}^{\infty}c^{i}_{j}v^{j}\right)=\sum_{j=1}^{\infty}\left(\sum_{i=1}^{\infty}y_{i}c^{i}_{j}\right)v^{j}$.

  3. (3)

Let $\sum_{k=1}^{\infty}u_{k}e^{k}\in V^{\epsilon}_{Airy}$ and
$\sum_{k=1}^{\infty}v_{k}e^{k}\in V^{\epsilon}_{Airy}$ then
$\sum_{i,j=1}^{\infty}s^{ij}u_{i}v_{j}$ converges absolutely.

  4. (4)

Let $\sum_{k=1}^{\infty}y_{k}e^{k}\in V^{\epsilon}_{Airy}$ then
$\sum_{i=1}^{\infty}y_{i}\left(\sum_{j=1}^{\infty}s^{ij}f_{j}\right)=\sum_{i,j=1}^{\infty}y_{i}s^{ij}f_{j}\in
T_{0}L^{M}_{Airy}$ converges absolutely uniformly.

###### Proof.

  1. (1)

Using C2.4.17.2, we have that $\sum_{i=1}^{\infty}y_{i}d^{i}_{j}=\frac{1}{2\pi
i}\oint_{|z|=\epsilon}\left(\left(\sum_{i=1}^{\infty}\frac{y_{i}}{z^{i}}\right)\int\left(\sum_{k=1}^{\infty}d^{k}_{j}f_{k}(z)\right)\right)$
because of the uniform convergences of each series in the integral. Therefore
$\sum_{i=1}^{\infty}y_{i}d^{i}_{j}$ converges for all $j=1,2,3,\cdots$. By
C2.4.17.3, there exists $J$ such that $d^{i}_{j}=\delta^{i}_{j}$ for all
$j>J$, therefore
$\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}y_{i}d^{i}_{j}e^{j}=\sum_{j=1}^{J}\left(\sum_{i=1}^{\infty}y_{i}d^{i}_{j}\right)e^{j}+\sum_{j=J+1}^{\infty}y_{j}e^{j}\in
V^{\epsilon}_{Airy}$. The remaining claims follow similarly.

  2. (2)

This is a straightforward consequence of C2.4.17.3:

$\displaystyle\sum_{i=1}^{\infty}y_{i}\left(\sum_{j=1}^{\infty}d^{i}_{j}v^{j}\right)$
$\displaystyle=\sum_{i=1}^{\infty}y_{i}\left(\sum_{j=1}^{J}d^{i}_{j}v^{j}\right)+\sum_{i=1}^{\infty}y_{i}\left(\sum_{j=J+1}^{\infty}\delta^{i}_{j}v^{j}\right)=\sum_{j=1}^{J}\left(\sum_{i=1}^{\infty}y_{i}d^{i}_{j}\right)v^{j}+\sum_{i=J+1}^{\infty}y_{i}v^{i}$
$\displaystyle=\sum_{j=1}^{J}\left(\sum_{i=1}^{\infty}y_{i}d^{i}_{j}\right)v^{j}+\sum_{j=J+1}^{\infty}\left(\sum_{i=1}^{\infty}y_{i}d^{i}_{j}\right)v^{j}=\sum_{j=1}^{\infty}\left(\sum_{i=1}^{\infty}y_{i}d^{i}_{j}\right)v^{j}.$

Similarly,
$\sum_{i=1}^{\infty}y_{i}\left(\sum_{j=1}^{\infty}c^{i}_{j}v^{j}\right)=\sum_{j=1}^{\infty}\left(\sum_{i=1}^{\infty}y_{i}c^{i}_{j}\right)v^{j}$.

  3. (3)

Note that

(53) $\displaystyle\sum_{i,j=1}^{\infty}s^{ij}u_{i}v_{j}=\left(\frac{1}{2\pi
i}\right)^{2}\oint_{|z_{1}|=\epsilon}\oint_{|z_{2}|=\epsilon}\left(\sum_{k_{1}=1}^{\infty}\frac{u_{k_{1}}}{k_{1}z_{1}^{k_{1}}}\right)\left(\sum_{i,j=1}^{\infty}s^{ij}f_{i}(z_{1})f_{j}(z_{2})\right)\left(\sum_{k_{2}=1}^{\infty}\frac{v_{k_{2}}}{k_{2}z_{2}^{k_{2}}}\right).$

To obtain the first equality, we have used the fact that each series in the
integral converges uniformly for $\bar{\epsilon}<|z_{1}|,|z_{2}|<\bar{M}$ (see
C2.4.17.4) for some $\bar{\epsilon}<\epsilon$ and $\bar{M}>M$. By symmetry,
the second equality follows.

  4. (4)

The proof is mostly identical to the part 3.

∎

Using Lemma 2.4.19 to help compute the symplectic pairings, we can verify that
$\\{\bar{e}^{k=1,2,3,\cdots},\bar{f}_{k=1,2,3,\cdots}\\}$ as given in (52) is
a canonical basis of $W^{\epsilon,M}_{Airy}$:

$\Omega_{Airy}(\bar{e}^{i},\bar{e}^{j})=\Omega_{Airy}(\bar{f}_{i},\bar{f}_{j})=0,\qquad\Omega_{Airy}(\bar{e}^{i},\bar{f}_{j})=\delta^{i}_{j}.$

######  Proposition 2.4.20.

Let us consider the change of a canonical basis
$\\{e^{k=1,2,3,\cdots},f_{k=1,2,3,\cdots}\\}\mapsto\\{\bar{e}^{k=1,2,3,\cdots},\bar{f}_{k=1,2,3,\cdots}\\}$
of $W^{\epsilon,M}_{Airy}$ as given in (52) with $(c^{j}_{i}),(d^{j}_{i})$ and
$(s^{ij})$ satisfying Condition 2.4.17. Given
$w:=\sum_{k=1}^{\infty}(y_{k}e^{k}+x^{k}f_{k})\in W^{\epsilon,M}_{Airy}$, then
by setting

(54)
$\bar{x}^{i}:=\sum_{j=1}^{\infty}d^{i}_{j}\left(x^{j}-\sum_{k=1}^{\infty}s^{jk}y_{k}\right),\qquad\bar{y}_{i}:=\sum_{j=1}^{\infty}c^{j}_{i}y_{j},$

we have $\sum_{k=1}^{\infty}(\bar{y}_{k}\bar{e}^{k}+\bar{x}^{k}\bar{f}_{k})$
converges absolutely uniformly to $w$. Let
$(\bar{H}_{Airy})_{i}:=\sum_{j=1}^{\infty}c^{j}_{i}(H_{Airy})_{j}$, then

$(\bar{H}_{Airy})_{i}=-\bar{y}_{i}+\sum_{j,k=1}^{\infty}(\bar{a}_{Airy})_{ijk}\bar{x}^{j}\bar{x}^{k}+2\sum_{j,k=1}^{\infty}(\bar{b}_{Airy})_{ij}^{k}\bar{x}^{j}\bar{y}_{k}+\sum_{j,k=1}^{\infty}(\bar{c}_{Airy})^{jk}_{i}\bar{y}_{j}\bar{y}_{k},$

where
$\bar{A}_{Airy}=((\bar{a}_{Airy})_{ijk}),\bar{B}_{Airy}=((\bar{b}_{Airy})_{ij}^{k}),\bar{C}_{Airy}=((\bar{c}_{Airy})_{i}^{jk})$
are given by the gauge transformation of $A_{Airy},B_{Airy},C_{Airy}$
according to (36)-(38).

###### Proof.

Substitute
$e^{i}=\sum_{j=1}^{\infty}\left(c^{i}_{j}\bar{e}^{j}-\sum_{k=1}^{\infty}\bar{f}_{j}d^{j}_{k}s^{ki}\right)$
and $f_{i}=\sum_{j=1}^{\infty}d^{j}_{i}\bar{f}_{j}$ into
$\sum_{k=1}^{\infty}(y_{k}e^{k}+x^{k}f_{k})$. Then using Lemma 2.4.19 to help
interchange infinite summations when needed and look at coefficients of
$\bar{e}^{i=1,2,3,\cdots}$, $\bar{f}_{i=1,2,3,\cdots}$, we have (54). The fact
that $\sum_{k=1}^{\infty}(\bar{y}_{k}\bar{e}^{k}+\bar{x}^{k}\bar{f}_{k})$
converges absolutely uniformly follows from part 1 and part 4 of Lemma 2.4.19.

Substitute the inverse of (54):
$x^{i}=\sum_{j=1}^{\infty}\left(c^{i}_{j}\bar{x}^{j}+\sum_{k=1}^{\infty}\bar{y}_{j}d^{j}_{k}s^{ki}\right)$
and $y_{i}=\sum_{j=1}^{\infty}d^{j}_{i}\bar{y}_{j}$ into $(H_{Airy})_{i}$.
Using Lemma 2.4.19, we find that
$(H_{Airy})_{2n}=\sum_{j=1}^{\infty}d^{j}_{2n}\bar{y}_{j}$ and

$(H_{Airy})_{2n-1}=\frac{1}{4\pi
i}\oint_{|z|=\epsilon}\left(\left(z-\frac{\sum_{k=1}^{\infty}(\bar{y}_{k}\bar{e}^{k}(z)+\bar{x}^{k}\bar{f}_{k}(z))}{2zdz}\right)^{2}z^{2n-2}d(z^{2})\right).$

From C2.4.17.2,
$\sum_{j=1}^{\infty}c^{j}_{i}\delta_{j,odd}z^{j}dz=\sum_{n=1}^{\infty}c^{2n-1}_{i}z^{2n-1}dz$
converges absolutely uniformly for $|z|<\bar{M}$ for some $\bar{M}>M$.
Therefore, we can commute the summation in
$(\bar{H}_{Airy})_{i}=\sum_{j=1}^{\infty}c^{j}_{i}(H_{Airy})_{j}$ with the
integral $\oint_{|z|=\epsilon}$. Expand $(\bar{H}_{Airy})_{i}$ into the
standard form, the proposition is proven. ∎

Let $\bar{V}^{\epsilon}_{Airy}$ be the vector space containing all
$\sum_{k=1}^{\infty}\bar{y}_{k}\bar{e}^{k}\in W^{\epsilon,M}_{Airy}$ such that
$\sum_{k=1}^{\infty}y_{k}e^{k}\in V^{\epsilon}_{Airy}$, where
$\bar{y}_{i}=\sum_{j=1}^{\infty}c^{j}_{i}y_{j}$. Equivalently by Lemma 2.4.19,
we have that $\sum_{k=1}^{\infty}y_{k}\bar{e}^{k}\in\bar{V}^{\epsilon}_{Airy}$
if and only if $\sum_{k=1}^{\infty}y_{k}e^{k}\in V^{\epsilon}_{Airy}$. It
follows from Proposition 2.4.20 that
$W^{\epsilon,M}_{Airy}=\bar{V}^{\epsilon}_{Airy}\oplus T_{0}L^{M}_{Airy}$. In
particular, $\bar{V}^{\epsilon}_{Airy}$ is the new Lagrangian complement of
$T_{0}L_{Airy}^{M}$ corresponding to the new canonical basis
$\\{\bar{e}^{k=1,2,3,\cdots},\bar{f}_{k=1,2,3,\cdots}\\}$ of
$W^{\epsilon,M}_{Airy}$.

Finally, let us compare the formal solution $L_{Airy}\subset W_{Airy}$ of
$(H_{Airy})_{i=1,2,3,\cdots}=0$ with the analytic solution
$L^{M}_{Airy}\subset W^{\epsilon,M}_{Airy}$. In particular, we would like to
understand how to interpret the generating function $S_{0}$ of $L_{Airy}$
analytically. The following proposition is written for the residue constraints
Airy structure $\\{(H_{Airy})_{i=1,2,3,\cdots}\\}$ to simplify the notations
but its statement and the proof presented are equally valid for any of its
gauge transformations $\\{(\bar{H}_{Airy})_{i=1,2,3,\cdots}\\}$ due to
Proposition 2.4.20.

######  Proposition 2.4.21.

Suppose that we have a set of functions
$\\{y_{i=1,2,3,\cdots}:T_{0}L^{M}_{Airy}\rightarrow\mathbb{C}\\}$ such that

$\sum_{k=1}^{\infty}y_{k}(\\{x^{j=1,2,3,\cdots}\\})e^{k}\in
V^{\epsilon}_{Airy},\qquad\text{and}\qquad(H_{Airy})_{k}(\\{x^{i=1,2,3,\cdots}\\},\\{y_{i=1,2,3,\cdots}(\\{x^{j=1,2,3,\cdots}\\})\\})=0.$

Suppose further that
$\sum_{k=1}^{\infty}\mathbf{y}_{i}(x^{1},...,x^{K})e^{k}(z)$, where
$\mathbf{y}_{i}(x^{1},...,x^{K}):=y_{i}(\\{x^{1},...,x^{K},x^{k>K}=0\\})$, is
holomorphic in $K+1$ variables $\\{x^{1},\cdots,x^{K},z^{-1}\\}$ on some small
open neighbourhoods of $(x^{1},\cdots,x^{K})=(0,\cdots,0)$ and $|z|>\epsilon$,
for any $K\in\mathbb{Z}_{>0}$. Let
$S_{0}=\sum_{n=1}^{\infty}S_{0,n}\in\prod_{n=1}^{\infty}Sym_{n}(V_{Airy})$ be
the formal power series we obtained from the $g=0$ part of the ATR of the
quantum Airy structure
$(V_{Airy},A_{Airy},B_{Airy},C_{Airy},\epsilon_{Airy})$. Then the functions
$(\partial_{i}S_{0})|_{x^{k>K}=0},i=1,2,3,\cdots$ and
$\mathcal{S}_{0}=\mathcal{S}_{0}(x^{1},\cdots,x^{K}):=S_{0}(\\{x^{1},\cdots,x^{K},x^{k>K}=0\\})$
are holomorphic on some small open neighbourhoods of
$(x^{1},\cdots,x^{K})=(0,\cdots,0)$, for any $K\in\mathbb{Z}_{>0}$. In
particular, we have
$(\partial_{i}S_{0})|_{x^{k>K}=0}=\mathbf{y}_{i},i=1,2,3,\cdots$.

###### Proof.

We express $\mathbf{y}_{i}$ as a power series:

$\mathbf{y}_{i}(x^{1},\cdots,x^{K})=\sum_{n=0}^{\infty}\mathbf{y}^{(n)}_{i}(x^{1},\cdots,x^{K}),\qquad\mathbf{y}^{(n)}_{i}(x^{1},\cdots,x^{K})=\sum_{i_{1},\cdots,i_{n}=1}^{K}\frac{1}{n!}\mathbf{y}_{i;i_{1},\cdots,i_{n}}x^{i_{1}}\cdots
x^{i_{n}}.$

By assumption,
$\sum_{k=1}^{\infty}\mathbf{y}_{k}(x^{1},\cdots,x^{K})e^{k}(z)\in
V^{\epsilon}_{Airy}$ is a power series in $\\{x^{1},\cdots,x^{K},z^{-1}\\}$
which converges absolutely uniformly for $|z|>\epsilon$ and
$|x^{i=1,\cdots,K}|<r_{K}$ for some $r_{K}>0$, hence we are allowed to
arbitrarily rearrange its terms. According to (51), the left-hand-side of

(55)
$(H_{Airy})_{i}\left(\\{x^{1},\cdots,x^{K},x^{k>K}=0\\},\\{\mathbf{y}_{j=1,2,3,\cdots}(x^{1},\cdots,x^{K})\\}\right)=0.$

is a residue of a simple expression involving
$\sum_{k=1}^{\infty}\mathbf{y}_{k}(x^{1},\cdots,x^{K})e^{k}$, hence we can
also freely rearrange its terms. Therefore, we can solve for $\mathbf{y}_{i}$
from (55) by substituting in its power series expression and collecting like
terms exactly as we have done in Theorem 2.2.24 for $g_{0}=0$, even though
formal power series has been replaced by an absolutely convergent power
series. The Equation (55) must hold for all $|x^{i=1,\cdots,K}|<r_{K}$, so the
coefficient of each monomial $x^{i_{1}}\cdots x^{i_{n}},i_{k}=1,\cdots,K$ must
vanish independently. In particular, we have
$\mathbf{y}_{i}^{(n_{0})}=(H_{Airy})_{i}^{(0,n_{0})}|_{x^{k>K}=0}=(\partial_{i}S_{0,n_{0}+1})|_{x^{k>K}=0}$
(see (2.2.3)). Therefore,
$(\partial_{i}S_{0})|_{x^{k>K}=0}=\sum_{n=0}^{\infty}(\partial_{i}S_{0,n+1})|_{x^{k>K}=0}=\sum_{n=0}^{\infty}\mathbf{y}_{i}^{(n)}=\mathbf{y}_{i}$
is a holomorphic function in $x^{1},\cdots,x^{K}$ for all $i=1,2,3,\cdots$.
Furthermore, for $i=1,\cdots,K$ we have
$\partial_{i}\mathcal{S}_{0}=(\partial_{i}S_{0})|_{x^{k>K}=0}$. Since
$\partial_{i}\mathcal{S}_{0}=\mathbf{y}_{i}$ are holomorphic for
$i=1,\cdots,K$, we conclude that the function $\mathcal{S}_{0}$ is holomorphic
for all $(x^{1},\cdots,x^{K})$ such that $|x^{i=1,\cdots,K}|<r_{K}$. ∎

####  2.4.3. The embedding of $Discs^{\epsilon,M}_{t_{0}}$ into
$W^{\epsilon,M}_{t_{0}}$

In this section, the procedure we performed in Section 2.1.3 with be repeated
with the covariantly constant weak symplectic flat vector bundle
$(W^{\epsilon,M}\rightarrow Discs^{M},\Omega,\nabla)$ instead of
$(\mathcal{H}\rightarrow\mathcal{B}_{\Sigma_{0}},\Omega_{\mathcal{H}},\nabla_{GM})$.
Following [21, Section 6.1], we will define

$\phi\in\Gamma(Discs^{\epsilon,M}_{t_{0}},Hom(TDiscs_{t_{0}}^{\epsilon,M},W^{\epsilon,M})),$

and use it to construct the map
$\Phi_{t_{0}}:Discs^{\epsilon,M}_{t_{0}}\rightarrow W^{\epsilon,M}_{t_{0}}$
embedding the ‘neighbourhood’ $Discs^{\epsilon,M}_{t_{0}}$ of $t_{0}\in
Discs^{M}$ into the fiber $W^{\epsilon,M}_{t_{0}}$. Finally, we show that the
image of the embedding satisfies the analytic residue constraints. In other
words, $\text{im}\Phi_{t_{0}}$ is contained inside $L^{M}_{Airy}\subset
W^{\epsilon,M}_{t_{0}}\cong W^{\epsilon,M}_{Airy}$ (see Proposition 2.4.24).

Let us fix the reference point $t_{0}:=(x=z^{2},y=z)\in Discs^{M}$ and define
the subspace $Discs^{\epsilon,M}_{t_{0}}\subset Discs^{M}$ containing $t_{0}$
given by

$Discs^{\epsilon,M}_{t_{0}}:=\left\\{t=\left(x=a(t)+z^{2},y=\sum_{k=0}^{\infty}b_{k}(t)z^{k}\right)\in
Discs^{M}\ |\ \begin{array}[]{c}\sum_{k=0}^{\infty}b_{k}(t)z^{k}\text{
converges for }\\\ \text{ $|z|<\bar{M}$ for some $\bar{M}>M$ }\\\ \text{and
$\min(\bar{M}^{2}-M^{2},\epsilon^{2})>|a(t)-a(t_{0})|$}\end{array}\right\\}.$

Conceptually, we think of $Discs^{\epsilon,M}_{t_{0}}$ as a small
neighbourhood of $t_{0}$ inside $Discs^{M}$ (although, it is not a
neighbourhood according to the topology given in Remark 2.4.7). Similar to
before, for any $t\in Discs^{\epsilon,M}_{t_{0}}$, the kinematic tangent space
$T_{t}Discs^{\epsilon,M}_{t_{0}}$ is defined to be the set of $v\in Discs^{M}$
such that $\gamma(\tau):=t+\tau v$ defines a path
$\gamma:(-\delta,\delta)\rightarrow Discs^{\epsilon,M}_{t_{0}}$ for some
$\delta>0$. In other words,

$T_{t}Discs^{\epsilon,M}_{t_{0}}:=\left\\{v=A\frac{\partial}{\partial
a}+\sum_{k=0}^{\infty}B_{k}\frac{\partial}{\partial b_{k}}\in T_{t}Discs^{M}\
|\ \begin{array}[]{c}\sum_{k=0}^{\infty}B_{k}z^{k}\text{ converges for }\\\
\text{ $|z|<\bar{M}$ for some $\bar{M}>M$ }\\\ \text{and
$\min(\bar{M}^{2}-M^{2},\epsilon^{2})>|a(t)-a(t_{0})|$.}\end{array}\right\\}.$

######  Remark 2.4.22.

We will let the definition of
$\Gamma(Discs^{\epsilon,M}_{t_{0}},W^{\epsilon,M})$,
$\Gamma(Discs^{\epsilon,M}_{t_{0}},TDiscs^{\epsilon,M}_{t_{0}})$ and
$\Gamma(Discs^{\epsilon,M}_{t_{0}},Hom(TDiscs_{t_{0}}^{\epsilon,M},W^{\epsilon,M}))$
to be exactly as given in Remark 2.4.8 with $Discs^{M}$ replaced by
$Discs^{\epsilon,M}_{t_{0}}$.

At any point $t\in Discs^{\epsilon,M}_{t_{0}}$, the map $\phi_{t}$ arises from
the consideration of the infinitesimal deformations within the moduli space
$Discs^{\epsilon,M}_{t_{0}}$ of the disc
$\mathbb{D}_{t,M}\hookrightarrow\mathbb{C}^{2}$ embedded inside the foliated
symplectic surface $(\mathbb{C}^{2},\Omega_{\mathbb{C}^{2}}:=dx\wedge
dy,\\{x=const\\})$. It is given by the following diagram:

${T_{t}Discs^{\epsilon,M}_{t_{0}}}$${\displaystyle\varinjlim_{\bar{M}>M(t)}\Gamma(\mathbb{D}_{t,\bar{M}},T\mathbb{C}^{2}|_{\mathbb{D}_{t,\bar{M}}}/T\mathbb{D}_{t,\bar{M}})}$${\displaystyle\varinjlim_{\bar{M}>M(t)}\Gamma(\mathbb{D}_{t,\bar{M}},\Omega^{1}_{\mathbb{D}_{t,\bar{M}}})}$${W_{t}^{\epsilon(t),M(t)}\subset
W_{t}^{\epsilon,M}.}$$\scriptstyle{\cong}$$\scriptstyle{\phi_{t}}$$\scriptstyle{\cong}$

Where $M(t)$ and $\epsilon(t)$ are given as follows

$\epsilon(t):=\sqrt{\epsilon^{2}-|a(t)-a_{0}(t)|}\leq\epsilon<M\leq
M(t):=\sqrt{M^{2}+|a(t)-a_{0}(t)|}.$

Note that for all $t\in Discs^{\epsilon,M}_{t_{0}}$,
$\sum_{k=0}^{\infty}b_{k}(t)z^{k}$ converges for $|z|<\bar{M}$ for some
$\bar{M}$ where $\bar{M}>M(t)>M$. The isomorphism
$T_{t}Discs^{\epsilon,M}_{t_{0}}\xrightarrow{\cong}\varinjlim_{\bar{M}>M(t)}\Gamma(\mathbb{D}_{t,\bar{M}},T\mathbb{C}^{2}|_{\mathbb{D}_{t,\bar{M}}}/T\mathbb{D}_{t,\bar{M}})$
follows because a tangent vector $v\in T_{t}Discs^{\epsilon,M}_{t_{0}}$ gives
an infinitesimal deformation of $\mathbb{D}_{t,\bar{M}}$, $\bar{M}>M(t)$ which
can be thought of as a global section $n_{v}$ of the normal bundle
$\nu_{\mathbb{D}_{t,\bar{M}}/\mathbb{C}^{2}}\cong
T\mathbb{C}^{2}|_{\mathbb{D}_{t,\bar{M}}}/T\mathbb{D}_{t,\bar{M}}$. The map
$\Gamma(\mathbb{D}_{t,\bar{M}},T\mathbb{C}^{2}|_{\mathbb{D}_{t,\bar{M}}}/T\mathbb{D}_{t,\bar{M}})\xrightarrow{\cong}\Gamma(\mathbb{D}_{t,\bar{M}},\Omega^{1}_{\mathbb{D}_{t,\bar{M}}})$
is given by
$n_{v}\mapsto\Omega_{\mathbb{C}^{2}}(n_{v},.)|_{\mathbb{D}_{t,\bar{M}}}$. It
is an isomorphism because $\Omega_{\mathbb{C}^{2}}$ is non-degenerates by
definition and $\mathbb{D}_{t,\bar{M}}\subset\mathbb{C}^{2}$ is obviously a
Lagrangian submanifold. This induces the isomorphism of the direct limit

$\varinjlim_{\bar{M}>M(t)}\Gamma(\mathbb{D}_{t,\bar{M}},T\mathbb{C}^{2}|_{\mathbb{D}_{t,\bar{M}}}/T\mathbb{D}_{t,\bar{M}})\xrightarrow{\cong}\varinjlim_{\bar{M}>M(t)}\Gamma(\mathbb{D}_{t,\bar{M}},\Omega^{1}_{\mathbb{D}_{t,\bar{M}}}).$

Using the standard local coordinates $z$ on $\mathbb{D}_{t,\bar{M}}$, we find
that

$\varinjlim_{\bar{M}>M(t)}\Gamma(\mathbb{D}_{t,\bar{M}},\Omega^{1}_{\mathbb{D}_{t,\bar{M}}})\cong\left\\{\xi\in
z\mathbb{C}[[z]]\frac{dz}{z}\ |\ \begin{array}[]{c}\text{$\xi(z)$ converges
}\\\ \text{ for $|z|<\bar{M}$ for some
$\bar{M}>M(t)$}\end{array}\right\\}\hookrightarrow
W^{\epsilon(t),M(t)}_{t}\subset W^{\epsilon,M}_{t}.$

######  Remark 2.4.23.

Observe that,
$\varinjlim_{\bar{M}>M(t)}\Gamma(\mathbb{D}_{t,\bar{M}},\Omega^{1}_{\mathbb{D}_{t,\bar{M}}})$
is embedded as a Lagrangian subspace of $W^{\epsilon(t),M(t)}_{t}$, i.e. it is
a maximal subspace of $W^{\epsilon(t),M(t)}_{t}$ such that if
$\xi_{1},\xi_{2}\in\varinjlim_{\bar{M}>M(t)}\Gamma(\mathbb{D}_{t,\bar{M}},\Omega^{1}_{\mathbb{D}_{t,\bar{M}}})$
then $\Omega_{Airy}(\xi_{1},\xi_{2})=0$. In other words, $\phi_{t}$ embeds
$T_{t}Discs^{\epsilon,M}_{t_{0}}$ as a Lagrangian subspace of
$W^{\epsilon(t),M(t)}_{t}$. On the other hand, it is not always true that
$\phi_{t}$ embeds $T_{t}Discs^{\epsilon,M}_{t_{0}}$ as a Lagrangian subspace
of $W^{\epsilon,M}_{t}$, in fact
$\varinjlim_{\bar{M}>M(t)}\Gamma(\mathbb{D}_{t,\bar{M}},\Omega^{1}_{\mathbb{D}_{t,\bar{M}}})\subseteq
T_{0}L^{M}_{Airy}$ with an equality if and only if $a(t)=a(t_{0})$.

Let us write down $\phi_{t}$ explicitly. Consider a tangent vector

(56) $v=A\frac{\partial}{\partial
a}+\sum_{k=0}^{\infty}B_{k}\frac{\partial}{\partial b_{k}}\in
T_{t}Discs^{\epsilon,M}_{t_{0}}$

where $\sum_{k=0}^{\infty}B_{k}z^{k}$ converges absolutely for $|z|<\bar{M}$
for some $\bar{M}>M(t)>M$. The vector $v$ maps to

$n_{v}=A\frac{\partial x}{\partial
a}\partial_{x}+\sum_{k=0}^{\infty}B_{k}\frac{\partial y}{\partial
b_{k}}\partial_{y}=A\partial_{x}+\sum_{k=0}^{\infty}B_{k}z^{k}\partial_{y}\in\varinjlim_{\bar{M}>M(t)}\Gamma(\mathbb{D}_{t,\bar{M}},TS|_{\mathbb{D}_{t,\bar{M}}}/T\mathbb{D}_{t,\bar{M}}).$

Therefore, the value of $\phi_{t}$ on $v$ is given by

$\displaystyle\phi_{t}(v)$
$\displaystyle=\Omega_{S}(n_{v},.)|_{\mathbb{D}_{t,\bar{M}}}=\left(Ady-\sum_{k=0}^{\infty}B_{k}z^{k}dx\right)\Big{|}_{\mathbb{D}_{t,\bar{M}}}$
(57)
$\displaystyle=\left(A\sum_{k=1}^{\infty}kb_{k}z^{k-1}-2\sum_{k=0}^{\infty}B_{k}z^{k+1}\right)dz\in\varinjlim_{\bar{M}>M(t)}\Gamma(\mathbb{D}_{t,\bar{M}},\Omega^{1}_{\mathbb{D}_{t,\bar{M}}})\hookrightarrow
W^{\epsilon(t),M(t)}_{t}\subset W^{\epsilon,M}_{t}.$

In other words,

$\phi_{t}=\left(\sum_{k=1}^{\infty}kb_{k}z^{k-1}dz\right)da-2\sum_{k=0}^{\infty}(z^{k+1}dz)db_{k},$

and it is easy to check that $d_{\nabla}\phi=0$.

Let us now construct the map
$\Phi_{t_{0}}:Discs^{\epsilon,M}_{t_{0}}\hookrightarrow
W^{\epsilon,M}_{t_{0}}$ embedding the neighbourhood
$Discs^{\epsilon,M}_{t_{0}}$ of $t_{0}=(x=z^{2},y=z)\in Discs^{M}$ into the
fiber $W_{t_{0}}^{\epsilon,M}$ in the same way we did in Section 2.1.3. We
send the zero vector $0_{t_{1}}\in W^{\epsilon,M}_{t_{1}}$ at any point
$t_{1}\in Discs^{\epsilon,M}_{t_{0}}$ into $W^{\epsilon,M}_{t_{0}}$ at $t_{0}$
via parallel transport using the connection $\nabla+\phi$. Where
$(\nabla+\phi)\xi:=\nabla\xi+\phi$ for
$\xi\in\Gamma(Discs^{M},W^{\epsilon,M})$. Since $d_{\nabla}\phi=0$, there
exists a section $\theta\in\Gamma(Discs^{\epsilon,M}_{t_{0}},W^{\epsilon,M})$
such that $d_{\nabla}\theta=\phi$. We can write down $\theta_{t}$ explicitly
for any $t=\left(x=z^{2}+a(t),y=\sum_{k=0}^{\infty}b_{k}(t)z^{k}\right)\in
Discs^{\epsilon,M}_{t_{0}}$:

(58)
$\theta_{t}=-ydx=-\sum_{k=0}^{\infty}b_{k}(t)z^{k}d(z^{2})=-2\sum_{k=0}^{\infty}b_{k}(t)z^{k+1}dz.$

In general, given $t\in Discs^{M}$, the parallel transport using $\nabla+\phi$
of $0_{t_{1}}\in W^{\epsilon,M}_{t_{1}}$ from $t_{1}\in
Discs^{\epsilon,M}_{t}$ to $W^{\epsilon,M}_{t}$ at $t$ is path-independent and
it is given by

$v_{t_{1}}(t)=-\theta_{t}+\exp((a(t)-a(t_{1}))\mathcal{L}_{\frac{1}{2z}\partial_{z}})\theta_{t_{1}}.$

We note that $v_{t_{1}}(t)$ is well-defined for all $t_{1}\in
Discs^{\epsilon,M}_{t}$ because
$|a(t)-a(t_{1})|<\min(\bar{M}^{2}-M^{2},\epsilon^{2})$ by definition, hence
$\exp((a(t)-a(t_{1}))\mathcal{L}_{\frac{1}{2z}\partial_{z}})\theta_{t_{1}}\in
W^{\epsilon,M}_{t}$. It follows that

$\Phi_{t_{0}}(t):=v_{t}(t_{0})=-\theta_{t_{0}}+\exp((a(t_{0})-a(t))\mathcal{L}_{\frac{1}{2z}\partial_{z}})\theta_{t}\in
W^{\epsilon,M}_{t_{0}},$

for all $t\in Discs^{\epsilon,M}_{t_{0}}$.

######  Proposition 2.4.24.

([21, Section 6.3]) Let $t_{0}=(x=z^{2},y=z)\in Discs^{M}$. The image of the
map $\Phi_{t_{0}}:Discs^{\epsilon,M}_{t_{0}}\hookrightarrow
W^{\epsilon,M}_{t_{0}}$ satisfies the residue constraints (2.3), i.e.
$\text{\emph{im}}\Phi_{t_{0}}\subseteq L_{Airy}^{M}\subset
W^{\epsilon,M}_{t_{0}}\cong W^{\epsilon,M}_{Airy}$.

###### Proof.

At the point $t=(x=z^{2}+a(t),y=\sum_{k=0}^{\infty}b_{k}(t)z^{k})\in
Discs^{\epsilon,M}_{t_{0}}$, we have

$\Phi_{t_{0}}(t)=zd(z^{2})-\exp((a(t_{0})-a(t))\mathcal{L}_{\frac{1}{2z}\partial_{z}})(yd(z^{2})).$

We will show that $\Phi_{t_{0}}(t)\in W^{\epsilon,M}_{t_{0}}$ satisfies the
residue constraints. In fact, for any $m,n\in\mathbb{Z}_{\geq 0}$ we have

$\displaystyle
Res_{z=0}\left(\left(z-\frac{\Phi_{t_{0}}(t)}{2zdz}\right)^{m}z^{2n}d(z^{2})\right)$
$\displaystyle=Res_{z=0}\left(\left(\frac{\exp(-a(t)\mathcal{L}_{\frac{1}{2z}\partial_{z}})(y)d(z^{2})}{d(z^{2})}\right)^{m}z^{2n}d(z^{2})\right)$
$\displaystyle=Res_{z=0}\left(\exp(-a(t)\mathcal{L}_{\frac{1}{2z}\partial_{z}})(y^{m})z^{2n}d(z^{2})\right)$
$\displaystyle=Res_{z=0}\left(\exp(-a(t)\mathcal{L}_{\frac{1}{2z}\partial_{z}})\left(y^{m}(z^{2}+a(t))^{n}d(z^{2})\right)\right)=0.$

We justify the vanishing of the residue as follows. Since $t\in
Discs^{\epsilon,M}_{t_{0}}$ and $a(t_{0})=0$, we have that
$|a(t)|<\epsilon^{2}$ and $y^{m}(z^{2}+a(t))^{n}d(z^{2})$ is holomorphic for
$|z|<\sqrt{M^{2}+|a(t)|}$. In particular it contains no $\frac{dz}{z}$ and it
belongs to $W^{0,M(t)}_{Airy}\subset W^{\epsilon,M}_{Airy}$. Therefore

$\exp(-a(t)\mathcal{L}_{\frac{1}{2z}\partial_{z}})\left(y^{m}(z^{2}+a(t))^{n}d(z^{2})\right)\in
W^{\epsilon,M}_{Airy}$

by Lemma 2.4.9, which also contains no $\frac{dz}{z}$ term. In the case where
$m=1,2$, we can see that $\Phi_{t_{0}}(t)$ satisfies the residue constraints
for all $t\in Discs^{\epsilon,M}_{t_{0}}$ as claimed. ∎

######  Example 2.4.25.

Let $t_{0}=(x=z^{2},y=z)$ and $t=(x=z^{2}+a,y=z)$, then

$\displaystyle\Phi_{t_{0}}(t)$
$\displaystyle=zd(z^{2})-\exp(-a\mathcal{L}_{\frac{1}{2z}\partial_{z}})(zd(z^{2}))$
$\displaystyle=zd(z^{2})-zd(z^{2})+a\frac{d(z^{2})}{2z}+\sum_{k=2}^{\infty}\frac{a^{k}}{k!}\frac{(2k-3)!!}{2^{k}z^{2k-1}}d(z^{2})=az\frac{dz}{z}+\sum_{k=2}^{\infty}\frac{a^{k}}{k!}\frac{(2k-3)!!}{2^{k-1}z^{2k-3}}\frac{dz}{z}$

In $J$ coordinates we would have

$J_{-1}=a,\qquad J_{2k-3}=\frac{a^{k}(2k-3)!!}{k!2^{k-1}},\qquad k>1$

and the rest of $J_{n}$ are zero. We can check that $\\{J_{n}\\}$ we have
found satisfies the set of equations $\\{(H_{Airy})_{i=1,2,3,\cdots}=0\\}$
defining $L_{Airy}^{M}\subset W^{\epsilon,M}_{Airy}$.

######  Remark 2.4.26.

Since $(W^{\epsilon,M}\rightarrow
Discs^{\epsilon,M}_{t_{0}},\Omega,\nabla,\phi)$ is a weak symplectic flat
vector bundle with $\nabla$-covariantly constant $\Omega$ and
$d_{\nabla}\phi=0$, it would be nice to be able to conclude that
$\text{im}\Phi_{t_{0}}$ is a Lagrangian submanifold using Proposition 2.1.24.
Unfortunately, Proposition 2.1.24 does not apply because $W^{\epsilon,M}$ has
an infinite rank. However, we can show that $\text{im}\Phi_{t_{0}}$ is
isotropic in the sense that

$T_{\Phi_{t_{0}}(t)}\text{\emph{im}}\Phi_{t_{0}}:=\left\\{\iota_{v}d\Phi_{t_{0}}(t)=0\
|\ v\in T_{t}Discs^{\epsilon,M}_{t_{0}}\right\\}\subset
W^{\epsilon,M}_{t_{0}}$

is an isotropic subspace where
$d\Phi_{t_{0}}(t)=\exp((a(t_{0})-a(t))\mathcal{L}_{\frac{1}{2z}\partial_{z}})\phi_{t}$.
For any $v_{1},v_{2}\in T_{t}Discs^{\epsilon,M}_{t_{0}}$ we have by Lemma
2.4.14 and the fact that $\phi_{t}$ embeds $T_{t}Discs^{\epsilon,M}_{t_{0}}$
as a Lagrangian subspace of $W^{\epsilon(t),M(t)}_{t}\subset
W^{\epsilon,M}_{t}$ that
$\Omega_{t_{0}}(\iota_{v_{1}}d\Phi_{t_{0}}(t),\iota_{v_{2}}d\Phi_{t_{0}}(t))=\Omega_{t}(\phi_{t}(v_{1}),\phi_{t}(v_{2}))=0$.

###  2.5. From Local To Global

In Section 2.1 we studied the complex analytic moduli space $\mathcal{B}$ of
$\mathcal{F}$-transversal genus $g$ curves in a foliated symplectic surface
$(S,\Omega_{S},\mathcal{F})$ and the embedding of an open neighbourhood
$\mathcal{B}_{\Sigma_{0}}$ into a fiber
$\mathcal{H}_{\Sigma_{0}}=H^{1}(\Sigma_{0},\mathbb{C})$ of
$\mathcal{H}\rightarrow\mathcal{B}$. All considerations were done at the level
of cohomology which gives a global picture of the deformations. In Section 2.4
we studied what deformations of $\Sigma$ look like near a ramification point
and the embedding of $Discs^{\epsilon,M}_{t_{0}}$ into the fiber
$W^{\epsilon,M}_{t_{0}}$ of $W^{\epsilon,M}\rightarrow Discs^{M}$, which is a
local picture of the deformations. The idea of this section is to connect the
two pictures by lifting each cohomology class in Section 2.1 to a differential
form with Laurent series expansions near each ramification point given by a
series introduced in Section 2.4.

First of all, since there are more than one ramification points on $\Sigma$ in
general, instead of $W^{\epsilon,M}$ we will consider
$W^{Ram}:=\prod_{\alpha\in Ram}W^{\epsilon_{\alpha},M_{\alpha}}$, a vector
bundle of differential forms locally defined on the disjoint union of some
annulus on $\Sigma$, each centered at a ramification point. We are going to
introduce the vector bundle
$G\rightarrow\mathcal{B}_{\Sigma_{0}}\subseteq\mathcal{B}$ where the fibers
$G_{\Sigma}$ are given by the space of elements of $W^{Ram}$ which can be
extended to a holomorphic form on the entire $\Sigma$ with some discs removed
$=\Sigma\setminus\cup_{\alpha\in
Ram}\mathbb{D}_{\alpha,\epsilon_{\alpha}}(\Sigma)$. As we about to see, the
removal of discs is essential in order to make sense of parallel transports on
$G$. There is a map from $G$ to $W^{Ram}$ given by Laurent expansion around
ramification points and a map from $G$ to $\mathcal{H}$ given by taking
cohomology classes. Therefore, $G$ is an ‘intermediate’ vector bundle between
$W^{Ram}$ and $\mathcal{H}$ in this sense.

We equip $G$ with the connection given by differentiation along a foliation
leaf $\nabla_{\mathcal{F}}$. In fact, this is the unique choice of connection
for $G$, since applying $\nabla_{\mathcal{F}}$ on any element of $G_{\Sigma}$
can create poles only at ramification points. It follows that the parallel
transport on $G$ from $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ to
$[\Sigma^{\prime}]\in\mathcal{B}_{\Sigma_{0}}$ is given by pulling-back a
differential form on $\Sigma$ to $\Sigma^{\prime}$ along the foliation leaves.
Clearly, the pull-back will only be well-defined away from ramification
points. For this reason, we are removing a small disc
$\mathbb{D}_{\alpha,\epsilon_{\alpha}}(\Sigma)$ near each ramification point
of each $\Sigma$ to avoid the ambiguity of the pull-back between two near-by
points $[\Sigma],[\Sigma^{\prime}]$. In other words, a parallel transport will
only be possible if $[\Sigma^{\prime}]\in\mathcal{B}$ is sufficiently close to
$[\Sigma]\in\mathcal{B}$ such that each ramification point
$r_{\alpha}(\Sigma^{\prime})$ (or rather, the point on $\Sigma$ connected to
$r_{\alpha}(\Sigma^{\prime})$ via a foliation leaf) does not move too far from
$r_{\alpha}(\Sigma)$ and it remains contained inside the discs
$\mathbb{D}_{\alpha,\epsilon_{\alpha}}(\Sigma)$ to be removed from $\Sigma$.
This adds another reason why we need $\mathcal{B}_{\Sigma_{0}}$ to be
sufficiently small, a condition first mentioned in Section 2.1 and we have
delayed making it precise until this section. We may compare this to the
treatment in [6] where no discs were removed from $\Sigma$, instead, the
_formal neighbourhood_ $\hat{\mathcal{B}}_{\Sigma_{0}}$ of $[\Sigma_{0}]$ was
considered.

Since Section 2.4 is an analytic approach to [21, Section 6], we may think of
this section as an analytic approach to [21, Section 7]. We outline the
content as follows. After fixing terminologies in Section 2.5.1, we discuss
precisely how the open neighbourhood
$\mathcal{B}_{\Sigma_{0}}\subseteq\mathcal{B}$ should be selected. Not
surprisingly, the condition $\mathcal{B}_{\Sigma_{0}}$ needs to satisfy also
depends on the sizes of annulus $\\{\epsilon_{\alpha\in Ram}\\}$,
$\\{M_{\alpha inRam}\\}$, and the choice of $(\mathcal{F},\Omega_{S})$-charts
$U_{\alpha}$ around each ramification point. Then we will introduce the vector
bundle $G$ and discuss some of its properties in Section 2.5.3. Among other
things, we will show that $G$ is a coisotropic sub-bundle of $W^{Ram}$. In
Section 2.5.4, the residue constraints Airy structure on $W^{Ram}$ will be
introduced as the product of residue constraints Airy structure. Then, we will
discuss the connection $\nabla_{\mathcal{F}}$ and parallel transport on $G$ in
detail in Section 2.5.5. The comparison between connection and parallel
transport on $\mathcal{H}\rightarrow\mathcal{B}$, $W^{Ram}\rightarrow
Discs^{Ram}$, and $G\rightarrow\mathcal{B}_{\Sigma_{0}}$ will be studied in
Section 2.5.6. Finally, in Section 2.5.7, we will define the embedding map
$\Phi_{\Sigma_{0}}:\mathcal{B}_{\Sigma_{0}}\rightarrow G_{\Sigma_{0}}$
analogous to
$\boldsymbol{\Phi}_{\Sigma_{0}}:\mathcal{B}_{\Sigma_{0}}\rightarrow\mathcal{H}_{\Sigma_{0}}$
defined in Section 2.1.3 and
$\Phi_{t_{0}}:Discs_{t_{0}}^{\epsilon,M}\rightarrow W^{\epsilon,M}_{t_{0}}$
defined in Section 2.4.3 and argue that the image of $\Phi_{\Sigma_{0}}$
satisfies the residue constraints. The main result of this section,
Proposition 2.5.38 [21, Proposition 7.1.2] summarizes the relationship between
$\Phi_{\Sigma_{0}},\boldsymbol{\Phi}_{\Sigma_{0}}$ and $\Phi_{t_{0}}$. In
fact, since we are working entirely in an analytic framework, we are going to
see the relationship between the Lagrangian submanifold
$\mathcal{L}_{\Sigma_{0}}:=\text{im}\boldsymbol{\Phi}_{\Sigma_{0}}\subset\mathcal{H}_{\Sigma_{0}}$
and $L^{Ram}_{Airy}$, treated as an actual subset of $W^{Ram}_{Airy}$.

####  2.5.1. Basic setup and definitions

Let us fix a reference point $[\Sigma_{0}]\in\mathcal{B}$ and a collection of
$(\mathcal{F},\Omega_{S})$-charts $\mathcal{U}_{Ram}:=\\{(U_{\alpha\in
Ram},x_{\alpha\in Ram},y_{\alpha\in Ram})\\}$ such that each
$U_{\alpha},\alpha\in Ram$ contains the ramification point
$r_{\alpha}(\Sigma_{0})$ of $\Sigma_{0}$ and $\Sigma_{0}\cap U_{\alpha}$ is
bi-holomorphic to a simply-connected open subset of $\mathbb{C}$ with the
standard local coordinate
$z_{\alpha}:=\sqrt{x_{\alpha}-x_{\alpha}(r_{\alpha})}$. We choose
$\mathcal{B}_{\Sigma_{0}}$ to be sufficiently small such that for all
$[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$, $\Sigma\cap U_{\alpha}$ contains the
ramification point $r_{\alpha}(\Sigma)$ of $\Sigma$ and it is bi-holomorphic
to a simply-connected open subset of $\mathbb{C}$ with the standard local
coordinate $z_{\alpha}:=\sqrt{x_{\alpha}-x_{\alpha}(r_{\alpha}(\Sigma))}$. For
each $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ the parameterization of $\Sigma\cap
U_{\alpha}$ takes the form

$t_{\alpha}(\Sigma)=(x_{\alpha}=a_{\alpha}(\Sigma)+z^{2}_{\alpha},\
y_{\alpha}=b_{0\alpha}(\Sigma)+b_{1\alpha}(\Sigma)z_{\alpha}+b_{2\alpha}(\Sigma)z^{2}_{\alpha}+...).$

Let us emphasize that coefficients
$\\{a_{\alpha}(\Sigma),b_{0\alpha}(\Sigma),b_{1\alpha}(\Sigma),b_{2\alpha}(\Sigma),...\\}$
depends on our choice of the $(\mathcal{F},\Omega_{S})$-local coordinates
$(x_{\alpha},y_{\alpha})$. In particular, we often choose
$(x_{\alpha},y_{\alpha})$ such that a reference point
$[\Sigma_{0}]\in\mathcal{B}$ is parameterized by
$t_{\alpha}(\Sigma_{0})=(x_{\alpha}=z^{2}_{\alpha},y_{\alpha}=z_{\alpha})$.

Pick $\\{\epsilon_{\alpha\in Ram}\in\mathbb{R}\\}$ and $\\{M_{\alpha\in
Ram}\in\mathbb{R}\\}$ such that $0<\epsilon_{\alpha}<M_{\alpha}$ and define
the following discs and annulus on $\Sigma$ for
$[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ :

$\displaystyle\mathbb{D}_{\alpha,\epsilon_{\alpha}}(\Sigma)$
$\displaystyle:=\\{p\in\Sigma\cap U_{\alpha}\ |\
|z_{\alpha}(p)|<\epsilon_{\alpha}\\}\subset\Sigma\cap U_{\alpha}$
$\displaystyle\mathbb{D}_{\alpha,M_{\alpha}}(\Sigma)$
$\displaystyle:=\\{p\in\Sigma\cap U_{\alpha}\ |\
|z_{\alpha}(p)|<M_{\alpha}\\}\subset\Sigma\cap U_{\alpha}$
$\displaystyle\mathbb{A}_{\alpha,\epsilon_{\alpha},M_{\alpha}}(\Sigma)$
$\displaystyle:=\mathbb{D}_{\alpha,M_{\alpha}}(\Sigma)-\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}(\Sigma).$

We suppose that $\mathcal{B}_{\Sigma_{0}}$ is chosen to be sufficiently small
such that $\bar{\mathbb{D}}_{\alpha,M_{\alpha}}(\Sigma)\subset\Sigma\cap
U_{\alpha}\subset\Sigma$ for all $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$. In
other words, $\mathbb{D}_{\alpha,M_{\alpha}}(\Sigma)$ is properly contained
inside the coordinate chart $\Sigma\cap U_{\alpha}$ for all
$[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$. Consequently,
$\sum_{k=0}^{\infty}b_{k\alpha}(\Sigma)z^{k}_{\alpha}$ converges in an open
set containing $\mathbb{D}_{\alpha,M_{\alpha}}(\Sigma)$ for all
$[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ and so we have the map
$\gamma_{\alpha}:\mathcal{B}_{\Sigma_{0}}\hookrightarrow Discs^{M_{\alpha}}$
given by $[\Sigma]\mapsto t_{\alpha}(\Sigma)\in Discs^{M_{\alpha}}$.

Over $Discs^{M_{\alpha}}$ for each $\alpha\in Ram$, we have the vector bundle
$W^{\epsilon_{\alpha},M_{\alpha}}\rightarrow Discs^{M_{\alpha}}$ as defined in
Section 2.4. Locally, we can study a curve $\Sigma\subset S$ as an embedding
of $|Ram|$ discs $\mathbb{D}_{\alpha,M_{\alpha}}(\Sigma)\subset\Sigma\cap
U_{\alpha}\cong\mathbb{C}^{2}$ indexed by the ramification index set $Ram$.
Therefore, it makes sense to introduce

$Discs^{Ram}:=\prod_{\alpha\in Ram}Discs^{M_{\alpha}},\qquad
Discs^{Ram}_{t}:=\prod_{\alpha\in
Ram}Discs^{\epsilon_{\alpha},M_{\alpha}}_{t_{\alpha}},\qquad
W^{Ram}:=\prod_{\alpha\in Ram}W^{\epsilon_{\alpha},M_{\alpha}}.$

In other words, $W^{Ram}\rightarrow Discs^{Ram}$ is a trivial vector bundle
where each fiber over $t=(t_{\alpha\in Ram})\in Discs^{Ram}$ is
$W^{Ram}_{t}=\prod_{\alpha\in
Ram}W^{\epsilon_{\alpha},M_{\alpha}}_{t_{\alpha}}=\prod_{\alpha\in
Ram}W^{\epsilon_{\alpha},M_{\alpha}}_{Airy}=:W^{Ram}_{Airy}$. A typical
element $\xi\in W^{Ram}_{t}$ can be written as $\xi=\sum_{\alpha\in
Ram}[\alpha]\otimes\xi_{\alpha}$ where $\xi_{\alpha}\in
W^{\epsilon_{\alpha},M_{\alpha}}_{t_{\alpha}}$. We naturally define the
multiplication $W^{Ram}_{Airy}\times W^{Ram}_{Airy}\rightarrow W^{Ram}_{Airy}$
as follows: let $\xi_{i}=\sum_{\alpha\in
Ram}[\alpha]\otimes\xi_{i\alpha},i=1,2$ then

$\xi_{1}\xi_{2}:=\sum_{\alpha\in
Ram}[\alpha]\otimes\xi_{1\alpha}\xi_{2\alpha}.$

Likewise, for our convenience, if $\xi=\sum_{\alpha\in
Ram}[\alpha]\otimes\xi_{\alpha}$ where $\xi_{\alpha}\in
z^{-1}\mathbb{C}[[z]]dz\oplus\mathbb{C}[[z]]dz$ then
$Res_{z=0}\xi:=\sum_{\alpha\in Ram}Res_{z=0}\xi_{\alpha}$. We define the
symplectic form $\Omega$ and the connection $\nabla$ for $W^{Ram}$ in an
obvious way: $\Omega_{t}:=\Omega_{Airy}$ where

$\Omega_{Airy}(\xi_{1},\xi_{2}):=Res_{z=0}\left(\xi_{1}\int\xi_{2}\right)=\sum_{\alpha\in
Ram}Res_{z=0}\left(\xi_{1\alpha}\int\xi_{2\alpha}\right)$

and $\nabla\xi:=\sum_{\alpha\in
Ram}[\alpha]\otimes\nabla_{\alpha}\xi_{\alpha}$ where $\nabla_{\alpha}$ is a
connection for $W^{\epsilon_{\alpha},M_{\alpha}}$ as defined in Section 2.4.
It is clear that $(W^{Ram}\rightarrow Discs^{Ram},\Omega,\nabla)$ is a weak
symplectic vector bundle and $\Omega_{Airy}$ is $\nabla$-covariantly constant.
The collection of maps
$\gamma_{\alpha}:\mathcal{B}_{\Sigma_{0}}\hookrightarrow Discs^{M_{\alpha}}$
induces

(59) $\gamma:=\prod_{\alpha\in
Ram}\gamma_{\alpha}:\mathcal{B}_{\Sigma_{0}}\hookrightarrow Discs^{Ram}.$

Typically, we denote the point $(t_{\alpha\in
Ram}(\Sigma_{0}))=\gamma([\Sigma_{0}])\in Discs^{Ram}$ simply by
$t_{0}=(t_{0\alpha\in Ram})\in Discs^{Ram}$ where
$t_{0\alpha}:=t_{\alpha}(\Sigma_{0})\in Discs^{M_{\alpha}}$. Moreover, if
$\mathcal{B}_{\Sigma_{0}}$ is sufficiently small then the image of $\gamma$
will even be in $Discs^{Ram}_{t_{0}}$. This fact will be stated more precisely
in Lemma 2.5.33.

Using the map $\gamma$ we obtain the pull-back bundle
$\gamma^{*}W^{Ram}\rightarrow\mathcal{B}_{\Sigma_{0}}$ with the connection
$\gamma^{*}\nabla$. We define the pull-back connection $\gamma^{*}\nabla$ as
follows. For any $\xi\in\Gamma(Discs^{Ram},W^{Ram})$ we have
$\gamma^{*}\xi\in\Gamma(\mathcal{B}_{\Sigma_{0}},\gamma^{*}W^{Ram})$ with
$(\gamma^{*}\xi)_{\Sigma}:=\xi_{\gamma(\Sigma)}=\xi_{t(\Sigma)}=\sum_{\alpha\in
Ram}[\alpha]\otimes(\xi_{\alpha})_{t_{\alpha}(\Sigma)}$ and $\gamma^{*}\nabla$
is defined via the properties
$(\gamma^{*}\nabla)(\gamma^{*}\xi):=\gamma^{*}(\nabla\xi)$. Writing this down
explicitly, we have

$\displaystyle(\gamma^{*}\nabla)(\gamma^{*}\xi)_{\Sigma}$
$\displaystyle:=\sum_{k=1}^{g}\sum_{\alpha\in
Ram,l}du^{k}[\alpha]\otimes\left(\frac{\partial a_{\alpha}(\Sigma)}{\partial
u^{k}}\frac{\partial}{\partial a_{\alpha}}+\frac{\partial
b_{l\alpha}(\Sigma)}{\partial u^{k}}\frac{\partial}{\partial
b_{l\alpha}}-\frac{\partial a_{\alpha}(\Sigma)}{\partial
u^{k}}\mathcal{L}_{\frac{1}{2z_{\alpha}}\partial_{z_{\alpha}}}\right)(\xi_{\alpha})_{t_{\alpha}(\Sigma)}$
$\displaystyle=\sum_{k=1}^{g}du^{k}\left(\frac{\partial}{\partial
u^{k}}-\frac{\partial a_{\alpha}(\Sigma)}{\partial u^{k}}\sum_{\alpha\in
Ram}[\alpha]\otimes\mathcal{L}_{\frac{1}{2z_{\alpha}}\partial_{z_{\alpha}}}\right)(\gamma^{*}\xi)_{\Sigma}.$

####  2.5.2. Choosing $\\{\epsilon_{\alpha\in Ram}\\},\\{M_{\alpha\in
Ram}\\},\mathcal{U}_{Ram}$ and $\mathcal{B}_{\Sigma_{0}}$

We will now write down a precise summary of how $\\{\epsilon_{\alpha\in
Ram}\\},\\{M_{\alpha\in Ram}\\},\mathcal{U}_{Ram}=\\{(U_{\alpha\in
Ram},x_{\alpha\in Ram},y_{\alpha\in Ram})\\}$ and $\mathcal{B}_{\Sigma_{0}}$
can be selected. Once they are selected, they should remain fixed for the
entire calculation. Let $[\Sigma_{0}]\in\mathcal{B}$ be the fixed reference
point. For each $\alpha\in Ram$, we pick
$\epsilon_{\alpha},M_{\alpha}\in\mathbb{R},0<\epsilon_{\alpha}<M_{\alpha}$, a
collection of $(\mathcal{F},\Omega_{S})$-charts
$\mathcal{U}_{Ram}=\\{(U_{\alpha\in Ram},x_{\alpha\in Ram},y_{\alpha\in
Ram})\\}$ with $r_{\alpha}(\Sigma_{0})\in U_{\alpha}$ and an open
neighbourhood $\mathcal{B}_{\Sigma_{0}}$ of $[\Sigma_{0}]$ satisfying
Condition 2.5.2 below:

######  Remark 2.5.1.

These conditions are technical and can be skipped in the first reading. They
are sufficient but not necessary conditions for the results of this section to
hold. It is usually more practical to pick an appropriate $\mathcal{U}_{Ram}$
then select $\\{\epsilon_{\alpha\in Ram}\\},\\{M_{\alpha\in Ram}\\}$ such that
$\bar{\mathbb{D}}_{\alpha,M_{\alpha}}(\Sigma_{0})\subset\Sigma_{0}\cap
U_{\alpha}$ before picking $\mathcal{B}_{\Sigma_{0}}$ and directly check that
it is small enough for all the needed objects, such as
$\Phi:\mathcal{B}_{\Sigma_{0}}\rightarrow G_{\Sigma_{0}}$, to be defined.

######  Condition 2.5.2.

C2.5.2.1:

$\mathcal{B}_{\Sigma_{0}}$ is a contractible open neighbourhood of
$[\Sigma_{0}]$ which can be covered by a single coordinate chart
$(\mathcal{B}_{\Sigma_{0}},u^{1},\cdots,u^{g})$.

C2.5.2.2:

For all $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$, the open subset $\Sigma\cap
U_{\alpha}\subset\Sigma$ is bi-holomorphic to a simply-connected open subset
of $\mathbb{C}$ with the standard local coordinate
$z_{\alpha}:=\sqrt{x_{\alpha}-x_{\alpha}(r_{\alpha}(\Sigma))}$ and
$\bar{\mathbb{D}}_{\alpha,M_{\alpha}}(\Sigma)\subset\Sigma\cap U_{\alpha}$.

C2.5.2.3:

There exists a collection of $(\mathcal{F},\Omega_{S})$-charts
$\mathcal{U}_{0}=\\{(U_{0,p\in\sigma_{0}},x_{p\in\sigma_{0}},y_{p\in\sigma_{0}})\\}$
such that for all $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$, the parallel
transport on the vector bundle
$(G\rightarrow\mathcal{B}_{\Sigma_{0}},\nabla_{\mathcal{F}})$ from the fiber
$G_{\Sigma_{0}}$ to the fiber $G_{\Sigma}$ is possible for any holomorphic
form $\xi\in\Gamma(\Sigma_{0},\Omega^{1}_{\Sigma_{0}})\subset G_{\Sigma_{0}}$
according to Lemma 2.5.32. Additionally, the image of
$\mathbb{A}_{\alpha,\epsilon_{\alpha},M_{\alpha}}(\Sigma)$ via the map
$s_{\Sigma,\Sigma_{0}}$ (see Definition 2.5.28) is contained in
$\Sigma_{0}\cap U_{\alpha}$.

C2.5.2.4:

Let
$\mathcal{U}_{0}=\\{(U_{0,p\in\sigma_{0}},x_{p\in\sigma_{0}},y_{p\in\sigma_{0}})\\}$
be the collection of $(\mathcal{F},\Omega_{S})$-charts from C2.5.2.3. For all
$[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ the parallel transport on the vector
bundle $(G\rightarrow\mathcal{B}_{\Sigma_{0}},\nabla_{\mathcal{F}})$ from the
fiber $G_{\Sigma}$ to the fiber $G_{\Sigma_{0}}$ is possible according to
Lemma 2.5.32 for any $\xi\in G_{\Sigma}$ holomorphic on
$\cup_{p\in\sigma_{0}}\Sigma\cap U_{0,p}\supset\Sigma\setminus\cup_{\alpha\in
Ram}\mathbb{D}_{\alpha,\epsilon_{\alpha}}(\Sigma)$. Additionally, the image of
$\mathbb{A}_{\alpha,\epsilon_{\alpha},M_{\alpha}}(\Sigma_{0})$ via the map
$s_{\Sigma_{0},\Sigma}$ is contained in $\Sigma\cap U_{\alpha}$.

######  Remark 2.5.3.

We can summarize the basic idea behind each part in Condition 2.5.2 as
follows:

C2.5.2.1:

Simplifies our discussions by avoiding any complications arises from
coordinates transformations, transition functions, and possible monodromies of
parallel transport when $\mathcal{B}_{\Sigma_{0}}$ wrap around a singularity
in $\mathcal{B}$.

C2.5.2.2:

Helps us deal with neighbourhoods of ramification points of each
$\Sigma\in\mathcal{B}_{\Sigma_{0}}$. In particular, this helps fix the
definition of the map $\gamma:\mathcal{B}_{\Sigma_{0}}\hookrightarrow
Discs^{Ram}$, the map $i:G\hookrightarrow W^{Ram}$ (see Equation (60)), the
discs
$\mathbb{D}_{\alpha,\epsilon_{\alpha}}(\Sigma),\mathbb{D}_{\alpha,M_{\alpha}}(\Sigma)$
and annulus $\mathbb{A}_{\alpha,\epsilon_{\alpha},M_{\alpha}}(\Sigma)$.

C2.5.2.3:

Helps us define the section $\theta\in\Gamma(\mathcal{B}_{\Sigma_{0}},G)$
which will be important in the construction of
$\Phi:\mathcal{B}_{\Sigma_{0}}\rightarrow G_{\Sigma_{0}}$ embedding the
neighbourhood $\mathcal{B}_{\Sigma_{0}}$ into the fiber $G_{\Sigma_{0}}$
analogous to what we did in Section 2.1 and Section 2.4.

C2.5.2.4:

Makes sure that we are allowed to perform a parallel transport of
$\theta_{\Sigma}$ necessary to define the embedding map
$\Phi:\mathcal{B}_{\Sigma_{0}}\rightarrow G_{\Sigma_{0}}$.

The following Lemma guarantees that given a reference point
$[\Sigma_{0}]\in\mathcal{B}$, it is always possible to find
$\\{\epsilon_{\alpha\in Ram}\\},\\{M_{\alpha\in Ram}\\},\mathcal{U}_{Ram}$ and
$\mathcal{B}_{\Sigma_{0}}$ that satisfy Condition 2.5.2.

######  Lemma 2.5.4.

For any given $\\{\epsilon_{\alpha\in Ram}\\},\\{M_{\alpha\in
Ram}\\},\mathcal{U}_{Ram}$, there exists an open neighbourhood
$\mathcal{B}_{\Sigma_{0}}$ of $[\Sigma_{0}]$ small enough such that
$\\{\epsilon_{\alpha\in Ram}\\},\\{M_{\alpha\in Ram}\\},\mathcal{U}_{Ram}$ and
$\mathcal{B}_{\Sigma_{0}}$ satisfy Condition 2.5.2.

###### Proof.

This follows from the fact that C2.5.2.1-C2.5.2.4 are all open conditions. ∎

####  2.5.3. The vector bundle $G\rightarrow\mathcal{B}_{\Sigma_{0}}$ and its
properties

We are going to define the vector bundle
$(G\rightarrow\mathcal{B}_{\Sigma_{0}},\nabla_{\mathcal{F}})$ in this section
(see Definition 2.5.22). The fiber of $G$ over any point
$[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ is the infinite-dimensional vector space
$G_{\Sigma}$ given by Definition 2.5.5 below. By taking a Laurent expansion in
the vicinity of each ramification point $r_{\alpha},\alpha\in Ram$ we obtain a
vector bundle morphism $i:G\hookrightarrow W^{Ram}$, embedding $G$ as a vector
sub-bundle of $W^{Ram}$. It will follow from Proposition 2.5.19 that $G$ is,
in fact, a coisotropic vector sub-bundle of $W^{Ram}$. Consider
$G^{\perp}_{\Sigma}\subset G_{\Sigma}$ as a group acting on
$W^{Ram}_{t(\Sigma)}$ by vector addition, then we will see in Proposition
2.5.20 that the symplectic reduction of $(W^{Ram}_{t(\Sigma)},\Omega_{Airy})$
is given by $(\mathcal{H}_{\Sigma},\Omega_{\mathcal{H}})$. In particular, we
have the surjective vector bundle morphism $[.]:G\rightarrow\mathcal{H}$ given
by taking the cohomology class. Together, the map $i$ and $[.]$ establish
$G\rightarrow\mathcal{B}_{\Sigma_{0}}$ as an intermediate vector bundle
connecting $W^{Ram}\rightarrow Discs^{Ram}$ and
$\mathcal{H}\rightarrow\mathcal{B}$ as we will discuss further in Section
2.5.6 and Section 2.5.7.

######  Definition 2.5.5.

For any $[\Sigma]\in\mathcal{B}$, the infinite-dimensional vector space
$G_{\Sigma}$ is given by

$G_{\Sigma}:=\left\\{\xi\in\varinjlim_{\bar{\epsilon}_{\alpha}<\epsilon_{\alpha}}\Gamma(\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}},\Omega^{1}_{\Sigma})\ |\
\oint_{\partial\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}}\xi=0,\ \alpha\in
Ram\right\\}.$

What we meant is that if $\xi\in G_{\Sigma}$ then $\xi$ is a holomorphic
differential on some open neighbourhood of $\Sigma\setminus\cup_{\alpha\in
Ram}\mathbb{D}_{\alpha,\epsilon_{\alpha}}(\Sigma)$. In particular, $\xi$ is
holomorphic on some open set containing
$\bar{\mathbb{A}}_{\alpha,\epsilon_{\alpha},M_{\alpha}}(\Sigma)$.

######  Remark 2.5.6.

Definition 2.5.5 of $G_{\Sigma}$ should be compared to the original definition
of $G_{\Sigma}$ in Equation (38) of [21]. We denote the version of
$G_{\Sigma}$ in [21] as $G^{mer}_{\Sigma}$ and recall that it is given by

$G^{mer}_{\Sigma}:=\left\\{\xi\in\Gamma\left(\Sigma,\Omega^{1}_{\Sigma}\left(\sum_{\alpha\in
Ram}nr_{\alpha}\right)\right)\ |\ \text{ for some }n\in\mathbb{Z}_{\geq
0}\text{ and }Res_{r_{\alpha}}\xi=0,\ \alpha\in Ram\right\\}.$

In other words, $G^{mer}_{\Sigma}$ is a vector space of residueless
meromorphic differential forms on $\Sigma$ with poles at ramification points.

For each $\alpha\in Ram$, the Laurent series representation of $\xi\in
G_{\Sigma}$ in the standard local coordinate center at $z_{\alpha}=0$
converges absolutely for
$\bar{\epsilon}_{\alpha}<|z_{\alpha}|<\bar{M}_{\alpha}$ for some
$\bar{\epsilon}_{\alpha}<\epsilon_{\alpha}<M_{\alpha}<\bar{M}_{\alpha}$ and
contains no $\frac{dz_{\alpha}}{z_{\alpha}}$ term because
$\oint_{\partial\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}}\xi=0$. Therefore,
this define the map $i_{\alpha}:G_{\Sigma}\hookrightarrow
W^{\epsilon_{\alpha},M_{\alpha}}_{t_{\alpha}(\Sigma)}$. Performing the Laurent
expansion at every ramification point of $\Sigma$, we obtain the map

(60) $i:=\sum_{\alpha\in Ram}[\alpha]\otimes
i_{\alpha}:G_{\Sigma}\hookrightarrow W^{Ram}_{t(\Sigma)}$

for any $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$. Hence, instead of writing
$i(G_{\Sigma})\subset W^{Ram}_{t(\Sigma)}$ we will sometime write
$G_{\Sigma}\subset W^{Ram}_{t(\Sigma)}$.

######  Remark 2.5.7.

Let us emphasize that the definition of the map $i$ depends on our choice of
the collection of coordinate charts $\mathcal{U}_{Ram}$. Choosing a different
$(\mathcal{F},\Omega_{S})$-local coordinate system $(x_{\alpha},y_{\alpha})$
on $U_{\alpha}$ for any $\alpha\in Ram$ will result in a different map $i$ and
hence, a different embedding $G_{\Sigma}\hookrightarrow W^{Ram}_{t(\Sigma)}$.

We can apply the symplectic form $\Omega_{Airy}$ on vectors in $G_{\Sigma}$.
In fact, using Riemann-bilinear identity we can obtain a nice alternative
expression. For $\xi_{1},\xi_{2}\in G_{\Sigma}$ we have

$\displaystyle\Omega_{Airy}(i(\xi_{1}),i(\xi_{2}))$
$\displaystyle=\sum_{\alpha\in
Ram}Res_{z_{\alpha}=0}\left(i_{\alpha}(\xi_{1})\int
i_{\alpha}(\xi_{2})\right)=\frac{1}{2\pi i}\sum_{\alpha\in
Ram}\oint_{\partial\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}}\left(\xi_{1}\int\xi_{2}\right)$
(61) $\displaystyle=\frac{1}{2\pi
i}\sum_{k=1}^{g}\left(\oint_{A_{k}}\xi_{2}\oint_{B_{k}}\xi_{1}-\oint_{B_{k}}\xi_{2}\oint_{A_{k}}\xi_{1}\right).$

For each $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$, we are going to show that
$G_{\Sigma}\subset W^{Ram}_{t(\Sigma)}$ is a coisotropic subspace and that
$G_{\Sigma}/G^{\perp}_{\Sigma}\cong\mathcal{H}_{\Sigma}$. Then, we will define
the vector bundle $G\rightarrow\mathcal{B}$. But first, we need some
preliminary definitions and results.

######  Definition 2.5.8.

For any $[\Sigma]\in\mathcal{B}$, the infinite-dimensional vector space
$V_{\Sigma}$ is given by

$V_{\Sigma}:=\left\\{\xi\in G_{\Sigma}\ |\
\oint_{A_{i}}\xi=0,i=1,...,g\right\\}\subset G_{\Sigma}.$

Let us quickly recall the following definition of Bergman kernels:

######  Definition 2.5.9.

A _Bergman kernel_ is a meromorphic bi-differential $B=B(p,q)$ on
$\Sigma\times\Sigma$ with zero residue poles of order $2$ along the diagonal
$p=q$ and holomorphic elsewhere. We say that the Bergman kernel $B=B(p,q)$ is
_normalized_ with respect to the chosen $A$-periods if

$B(p,q)=_{p\sim
q}\frac{dz(p)dz(q)}{(z(p)-z(q))^{2}}+\text{holomorphic},\qquad\oint_{p\in
A_{i}}B(p,q)=0,i=1,\cdots,g.$

######  Remark 2.5.10.

Given a curve $\Sigma$ and a choice of $A,B$ symplectic basis cycles, the
normalized Bergman kernel is unique. This is because if $B$ and $B^{\prime}$
are normalized Bergman kernels then $(B-B^{\prime})(p,q)$ is holomorphic in
$p$ with zero $A$-periods for all $q$. By the explicit construction via
Riemann Theta function [13, 12] we find that the normalized Bergman kernels
also satisfy the following properties:

$B(p,q)=B(q,p),\qquad\oint_{p\in B_{i}}B(p,q)=2\pi i\omega_{i}(q)$

where $\omega_{i}$ are the normalized holomorphic forms on $\Sigma$.

######  Definition 2.5.11.

Let $B(p,q)$ be the normalized Bergman kernel on $\Sigma$. For any
$k\in\mathbb{Z}_{\geq 1}$ and $\alpha\in Ram$ we define the meromorphic
differential form on $\Sigma$:

$\bar{e}^{k,\alpha}:=\frac{1}{2\pi
ik}\oint_{z_{\alpha}=0}\frac{B(.,q(z_{\alpha}))}{z_{\alpha}^{k}}.$

Using the standard local coordinates $z_{\alpha}$ and $z_{\beta}$ of
$\Sigma\cap U_{\alpha}$ and $\Sigma\cap U_{\beta}$ respectively,
$\alpha,\beta\in Ram$, the local series expansion of $B(p,q)$ for
$p\in\Sigma\cap U_{\alpha}$ and $q\in\Sigma\cap U_{\beta}$ is given by

(62)
$B(z_{\alpha}(p),z_{\beta}(q))=\frac{\delta_{\alpha\beta}dz_{\alpha}(p)dz_{\beta}(q)}{(z_{\alpha}(p)-z_{\beta}(q))^{2}}+\sum_{i,j=1}^{\infty}P^{(i,\alpha)(j,\beta)}z_{\alpha}^{i-1}(p)z_{\beta}^{j-1}(q)dz_{\alpha}(p)dz_{\beta}(q)$

for some symmetric tensor $P^{(i,\alpha)(j,\beta)}=P^{(j,\beta)(i,\alpha)}$.
We obtain the following easy fact about $\bar{e}^{k,\alpha}$:

######  Lemma 2.5.12.

Let $\bar{e}^{k,\alpha}$ be a meromorphic differential form on $\Sigma$ as
given by Definition 2.5.11. Then $\bar{e}^{k,\alpha}\in V_{\Sigma}$ and

(63) $i(\bar{e}^{k,\alpha})=[\alpha]\otimes
z^{-k}_{\alpha}\frac{dz_{\alpha}}{z_{\alpha}}+\sum_{k^{\prime}\geq
1,\alpha^{\prime}\in
Ram}[\alpha^{\prime}]\otimes\frac{1}{k}P^{(k,\alpha)(k^{\prime},\alpha^{\prime})}z_{\alpha^{\prime}}^{k^{\prime}-1}\frac{dz_{\alpha^{\prime}}}{z_{\alpha^{\prime}}}\in
W^{Ram}_{t(\Sigma)}$

###### Proof.

The expansion (63) is obtained from integrating the series expansion (62).
This shows that $\bar{e}^{k,\alpha}$ is holomorphic everywhere except for a
pole of order $k+1$ at $r_{\alpha}$ with zero residue. Deform $A$-cycles to
avoid intersection with $r_{\alpha}$, then $\oint_{p\in
A_{i}}\bar{e}^{k,\alpha}(p)=\frac{1}{2\pi
ik}\oint_{z_{\alpha}=0}\frac{1}{z_{\alpha}^{k}}\oint_{p\in
A_{i}}B(p,q(z_{\alpha}))=0$ because the Bergman kernel is normalized. ∎

######  Remark 2.5.13.

Note that the definition of $e^{k,\alpha}$ implicitly depends on our choice of
the Bergman kernel (since we are using the normalized Bergman kernel this is
equivalent to our choice of $A,B$ symplectic basis cycles) and the standard
local coordinate $z_{\alpha}$ (which is equivalent to our choice of the
collection of $(\mathcal{F},\Omega_{S})$-charts $\mathcal{U}_{Ram}$).

######  Lemma 2.5.14.

Let $[\Sigma]\in\mathcal{B}$. If
$\xi_{\alpha}=\sum_{k=1}^{\infty}\xi_{\alpha,k}z_{\alpha}^{-k}\frac{dz_{\alpha}}{z_{\alpha}}\in
V^{\epsilon_{\alpha}}_{Airy}$ then
$\bar{\xi}_{\alpha}=\sum_{k=1}^{\infty}\xi_{\alpha,k}\bar{e}^{k,\alpha}\in
V_{\Sigma}$. The principal part of $i_{\alpha^{\prime}}(\bar{\xi}_{\alpha})$
is $\delta_{\alpha\alpha^{\prime}}\xi_{\alpha}$ and for any path
$C\subset\Sigma\setminus\cup_{\alpha\in
Ram}\mathbb{D}_{\alpha,\epsilon_{\alpha}}(\Sigma)$, we have

$\int_{C}\bar{\xi}_{\alpha}=\int_{C}\sum_{k=1}^{\infty}\xi_{\alpha,k}\bar{e}^{k,\alpha}=\sum_{k=1}^{\infty}\xi_{\alpha,k}\int_{C}\bar{e}^{k,\alpha}.$

###### Proof.

We have that
$\xi_{\alpha}=\sum_{k=1}^{\infty}\xi_{\alpha,k}z_{\alpha}^{-k}\frac{dz_{\alpha}}{z_{\alpha}}$
converges absolutely uniformly for $|z_{\alpha}|>\bar{\epsilon}_{\alpha}$ for
some $\bar{\epsilon}_{\alpha}<\epsilon_{\alpha}$. Therefore,

(64)
$\sum_{k=1}^{\infty}\frac{\xi_{\alpha,k}}{kz^{k}_{\alpha}}=-\sum_{k=1}^{\infty}\int\xi_{\alpha,k}z^{-k}_{\alpha}\frac{dz_{\alpha}}{z_{\alpha}}=-\int\sum_{k=1}^{\infty}\xi_{\alpha,k}z_{\alpha}^{-k}\frac{dz_{\alpha}}{z_{\alpha}}$

converges absolutely uniformly for $|z_{\alpha}|>\bar{\epsilon}_{\alpha}$,
because integration of a power series does not change its radius of
convergence. Then by using the Bergman kernel $B(p,q)$ on $\Sigma$ we define

$\displaystyle\bar{\xi}_{\alpha}(p):=\frac{1}{2\pi
i}\oint_{z_{\alpha}\in\partial\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}}\left(\sum_{k=1}^{\infty}\frac{\xi_{\alpha,k}}{kz^{k}_{\alpha}}\right)B(p,q(z_{\alpha}))=\sum_{k=1}^{\infty}\frac{\xi_{\alpha,k}}{2\pi
ik}\oint_{z_{\alpha}=0}\frac{B(p,q(z_{\alpha}))}{z_{\alpha}^{k}}=\sum_{k=1}^{\infty}\xi_{\alpha,k}\bar{e}^{k,\alpha}(p),$

where the term-by-term integration is allowed due to the absolute uniform
convergence. Therefore, $\bar{\xi}_{\alpha}$ is holomorphic for all
$p\in\Sigma\setminus\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}\supset\Sigma\setminus\cup_{\alpha^{\prime}\in
Ram}\mathbb{D}_{\alpha^{\prime},\epsilon_{\alpha^{\prime}}}$ but diverges as
$p$ approaches the integration path
$\partial\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}$. The principal
part of $\bar{e}^{k,\alpha}$ is
$z^{-k}_{\alpha}\frac{dz_{\alpha}}{z_{\alpha}}$ so the principal part of
$i_{\alpha^{\prime}}(\bar{\xi}_{\alpha})$ is $\xi_{\alpha}$ if
$\alpha^{\prime}=\alpha$ and zero otherwise. Given any path
$C\subset\Sigma\setminus\cup_{\alpha\in
Ram}\mathbb{D}_{\alpha,\epsilon_{\alpha}}(\Sigma)$ we have

$\displaystyle\int_{C}\bar{\xi}_{\alpha}(p)$ $\displaystyle=\int_{p\in
C}\left(\frac{1}{2\pi
i}\oint_{z_{\alpha}\in\partial\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}}\left(\sum_{k=1}^{\infty}\frac{\xi_{\alpha,k}}{kz^{k}_{\alpha}}\right)B(p,q(z_{\alpha}))\right)=\frac{1}{2\pi
i}\oint_{z_{\alpha}\in\partial\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}}\left(\sum_{k=1}^{\infty}\frac{\xi_{\alpha,k}}{kz^{k}_{\alpha}}\right)\int_{p\in
C}B(p,q(z_{\alpha}))$
$\displaystyle=\sum_{k=1}^{\infty}\frac{\xi_{\alpha,k}}{2\pi
ik}\oint_{z_{\alpha}\in\partial\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}}\int_{p\in
C}\frac{B(p,q(z_{\alpha}))}{z_{\alpha}^{k}}=\sum_{k=1}^{\infty}\xi_{\alpha,k}\int_{p\in
C}\frac{1}{2\pi
ik}\oint_{z_{\alpha}\in\partial\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}}\frac{B(p,q(z_{\alpha}))}{z_{\alpha}^{k}}$
$\displaystyle=\sum_{k=1}^{\infty}\xi_{\alpha,k}\int_{p\in
C}\bar{e}^{k,\alpha}(p).$

We have used the fact that $C$ do not intersect with
$\partial\bar{\mathbb{D}}_{\epsilon,\bar{\epsilon}_{\alpha}}$, so the order of
integrations can be permuted. The integration can be done term-by-term because
$\sum_{k=1}^{\infty}\xi_{\alpha,k}z^{-k}_{\alpha}\frac{dz_{\alpha}}{z_{\alpha}}$
converges absolutely uniformly for $|z_{\alpha}|>\bar{\epsilon}_{\alpha}$. Let
$C=\partial\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}$ then
$\oint_{\partial\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}}\bar{\xi}_{\alpha}=\sum_{k=1}^{\infty}\xi_{\alpha,k}\oint_{\partial\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}}\bar{e}^{k}=0$.
On the other hand, let $C=A_{i}$ then
$\oint_{A_{i}}\bar{\xi}_{\alpha}=\sum_{k=1}^{\infty}\xi_{\alpha,k}\oint_{A_{i}}\bar{e}^{k,\alpha}=0$.
Therefore, we conclude that $\bar{\xi}_{\alpha}\in V_{\Sigma}$. ∎

Recall the definition of $L^{M}_{Airy}\subset W^{\epsilon,M}_{Airy}$ from
Section 2.4.2 and define $L^{Ram}_{Airy}:=\prod_{\alpha\in
Ram}L^{M_{\alpha}}_{Airy}\subset W^{Ram}_{Airy}$. We note that
$T_{0}L^{Ram}_{Airy}=\prod_{\alpha\in Ram}T_{0}L^{M_{\alpha}}_{Airy}$, or more
explicitly:

$T_{0}L^{Ram}_{Airy}=\left\\{\xi=\sum_{\alpha\in
Ram}[\alpha]\otimes\xi_{\alpha}\ |\ \xi_{\alpha}(z_{\alpha})\in
z_{\alpha}\mathbb{C}[[z_{\alpha}]]\frac{dz_{\alpha}}{z_{\alpha}},\begin{array}[]{c}\text{$\xi_{\alpha}(z_{\alpha})$
converges for $|z_{\alpha}|<\bar{M}_{\alpha}$}\\\ \text{for some
$\bar{M}_{\alpha}>M_{\alpha}$}\end{array}\right\\}\subset W^{Ram}_{Airy},$

is a Lagrangian subspace of $W^{Ram}_{Airy}$.

######  Lemma 2.5.15.

For any $[\Sigma]\in\mathcal{B}$ we have

$W^{Ram}_{t(\Sigma)}=W^{Ram}_{Airy}=V_{\Sigma}\oplus T_{0}L^{Ram}_{Airy}.$

Additionally, if $\xi\in V_{\Sigma}$ and the principal part of
$i_{\alpha}(\xi)$ is
$\sum_{k=1}^{\infty}\xi_{\alpha,k}z^{-k}_{\alpha}\frac{dz_{\alpha}}{z_{\alpha}}\in
V^{\epsilon_{\alpha}}_{Airy}$ then it can be written as $\xi=\sum_{\alpha\in
Ram}\sum_{k=1}^{\infty}\xi_{\alpha,k}\bar{e}^{k,\alpha}$.

###### Proof.

If $\xi\in V_{\Sigma}$ is such that $i_{\alpha}(\xi)$ has no principal part
for all $\alpha\in Ram$ then it must be holomorphic on $\Sigma$. Therefore,
$\oint_{A_{i}}\eta=0,i=1,\cdots,g$ which implies that $\eta=0$. This shows
$V_{\Sigma}\cap T_{0}L^{Ram}_{Airy}=\\{0\\}$. To show that
$V_{\Sigma}+T_{0}L^{Ram}_{Airy}=W^{Ram}_{Airy}$ we consider any given
$\xi=(\xi_{\alpha\in Ram})\in W^{Ram}_{Airy}$ where $\xi_{\alpha}\in
z_{\alpha}^{-1}\mathbb{C}[[z_{\alpha}^{-1}]]\frac{dz_{\alpha}}{z_{\alpha}}\oplus
z_{\alpha}\mathbb{C}[[z_{\alpha}]]\frac{dz_{\alpha}}{z_{\alpha}}$ converges
absolutely for $\bar{\epsilon}_{\alpha}<|z_{\alpha}|<\bar{M}_{\alpha}$ for
some $\bar{\epsilon}_{\alpha}<\epsilon_{\alpha}<M_{\alpha}<\bar{M}_{\alpha}$.
Let
$\xi_{\alpha,P}=\sum_{k=1}^{\infty}z^{-k}_{\alpha}\xi_{\alpha,k}\frac{dz_{\alpha}}{z_{\alpha}}\in
V^{\epsilon_{\alpha}}_{Airy}$ be the principal part of $\xi_{\alpha}$. By
Lemma 2.5.14 we have $\bar{\xi}_{P}:=\sum_{\alpha\in
Ram}\bar{\xi}_{\alpha,P}=\sum_{\alpha\in
Ram}\sum_{\alpha,k}\xi_{k,\alpha}\bar{e}^{k,\alpha}\in V_{\Sigma}$ and the
principal part of $i_{\alpha}(\bar{\xi})$ is $\xi_{\alpha,P}$. On the other
hand, because $\Sigma\in\mathcal{B}_{\Sigma_{0}}$ we have that
$i_{\alpha}(\bar{\xi})$ converges absolutely for
$\bar{\epsilon}_{\alpha}<|z|<\bar{\bar{M}}_{\alpha}$ where
$\bar{\bar{M}}_{\alpha}$ is such that
$\mathbb{D}_{\alpha,M_{\alpha}}(\Sigma)\subset\mathbb{D}_{\alpha,\bar{\bar{M}}_{\alpha}}(\Sigma)\subset\Sigma\cap
U_{\alpha}$. It follows that $\xi_{\alpha}-i_{\alpha}(\bar{\xi}_{P})$ has zero
principal part and converges absolutely for
$|z_{\alpha}|<\min(\bar{M}_{\alpha},\bar{\bar{M}}_{\alpha})$. Therefore
$\xi-i(\bar{\xi}_{P})\in T_{0}L^{Ram}_{Airy}$ which proves that
$W^{Ram}_{Airy}=V_{\Sigma}\oplus T_{0}L^{Ram}_{Airy}$. Besides, if $\xi\in
V_{\Sigma}$, then $\xi-\bar{\xi}_{P}$ is holomorphic on $\Sigma$ with
$\oint_{A_{i}}(\xi-\bar{\xi}_{P})=0$ for all $i=1,...,g$ which means
$\xi=\bar{\xi}_{P}=\sum_{\alpha\in
Ram}\sum_{k=1}^{\infty}\xi_{\alpha,k}\bar{e}^{k,\alpha}$. ∎

######  Corollary 2.5.16.

For any $[\Sigma]\in\mathcal{B}$ we have

$G_{\Sigma}+T_{0}L^{Ram}_{Airy}=W^{Ram}_{Airy}.$

###### Proof.

This follows from Lemma 2.5.15 and the fact that $V_{\Sigma}\subset
G_{\Sigma}$. ∎

######  Corollary 2.5.17.

For any $[\Sigma]\in\mathcal{B}$, we have,

$G_{\Sigma}=V_{\Sigma}\oplus\Gamma(\Sigma,\Omega^{1}_{\Sigma}).$

In particular, any $\xi\in G_{\Sigma}$ with the principal part of
$i_{\alpha}(\xi)$ given by
$\sum_{k=1}^{\infty}\xi_{\alpha,k}z_{\alpha}^{-k}\frac{dz_{\alpha}}{z_{\alpha}}$
and $a^{i}:=\oint_{A_{i}}\xi$ can be written as $\xi=\sum_{\alpha\in
Ram}\sum_{k=1}^{\infty}\xi_{\alpha,k}\bar{e}^{k,\alpha}+\sum_{k=1}^{g}a^{k}\omega_{k}$,
where $\omega_{i}$ are the normalized holomorphic forms on $\Sigma$.

###### Proof.

In fact, $G_{\Sigma}=V_{\Sigma}\oplus\Gamma(\Sigma,\Omega^{1}_{\Sigma})$
follows directly from the Definition 2.5.5 and Definition 2.5.8 of
$G_{\Sigma}$ and $V_{\Sigma}$. This is because for any $\xi\in G_{\Sigma}$, we
have $\bar{\xi}:=\xi-\sum_{k=1}^{g}a^{k}\omega_{k}\in V_{\Sigma}$, hence
$V_{\Sigma}+\Gamma(\Sigma,\Omega^{1}_{\Sigma})=G_{\Sigma}$. We also know that
$V_{\Sigma}\cap\Gamma(\Sigma,\Omega^{1}_{\Sigma})=\\{0\\}$ because any global
holomorphic form with zero $A$-periods is zero. The rest of the Corollary
follows from applying Lemma 2.5.15 to $\bar{\xi}\in V_{\Sigma}$. ∎

######  Corollary 2.5.18.

$V_{\Sigma}$ is a Lagrangian complement of $T_{0}L^{Ram}_{Airy}$ in
$W^{Ram}_{Airy}$.

###### Proof.

We already know from Lemma 2.5.15 that $V_{\Sigma}$ is a complement of
$T_{0}L^{Ram}_{Airy}$. Using (2.5.3), it is clear that for any
$\xi_{1},\xi_{2}\in V_{\Sigma}$ we would have
$\Omega_{Airy}(\xi_{1},\xi_{2})=0$ because
$\oint_{A_{i}}\xi_{j}=0,i=1,...,g,j=1,2$. Suppose there exists $\xi_{2}\in
W^{Ram}_{Airy}$ such that $\Omega_{Airy}(\xi_{1},\xi_{2})=0$ for all
$\xi_{1}\in V_{\Sigma}$. Then by Lemma 2.5.15 we can write $\xi_{2}=v+l$ where
$v\in V_{\Sigma},l\in T_{0}L^{Ram}_{Airy}$, but then
$\Omega_{Airy}(\xi_{1},l)=\Omega_{Airy}(\xi_{1},v+l)=\Omega_{Airy}(\xi_{1},\xi_{2})=0,\forall\xi_{1}\in
V_{\Sigma}$ because $\Omega_{Airy}(\xi_{1},v)=0$. Since we also have
$\Omega_{Airy}(l^{\prime},l)=0$ for all $l^{\prime}\in T_{0}L^{Ram}_{Airy}$,
this can only be true if $l=0$, i.e. $\xi_{2}\in V_{\Sigma}$, because
$\Omega_{Airy}$ is non-degenerates. So we have that $V_{\Sigma}$ is a
Lagrangian subspace of $W^{Ram}_{Airy}$ i.e. $V^{\perp}_{\Sigma}=V_{\Sigma}$.
∎

######  Proposition 2.5.19.

([21, Proposition 7.1.3]) For any $[\Sigma]\in\mathcal{B}$, the vector space
$G_{\Sigma}$ is a coisotropic subspace of $W^{Ram}_{t(\Sigma)}$ with the
perpendicular subspace $G^{\perp}_{\Sigma}$ given by

$G^{\perp}_{\Sigma}=\varinjlim_{\bar{\epsilon}_{\alpha}<\epsilon_{\alpha}}d\Gamma(\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}},\mathcal{O}_{\Sigma})\subseteq
G_{\Sigma}.$

###### Proof.

From $V_{\Sigma}\subset G_{\Sigma}$ and Corollary 2.5.18 we have that
$G^{\perp}_{\Sigma}\subset V^{\perp}_{\Sigma}=V_{\Sigma}\subset G_{\Sigma}$.
Let us find $G^{\perp}_{\Sigma}$. If $\xi\in G^{\perp}_{\Sigma}$ then $\xi\in
G_{\Sigma}$ which means
$\oint_{\partial\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}}\xi=0$. Using
(2.5.3) we can see that the only way $\Omega_{Airy}(\xi,\eta)=0,\forall\eta\in
G_{\Sigma}$ can be possible is if
$\oint_{A_{i}}\xi=0,\oint_{B_{i}}\xi=0,i=1,...,g$. We can now define
$f(p):=\int_{p_{0}}^{p}\xi$ which is a holomorphic function in an open
neighbourhood of $\Sigma\setminus\cup_{\alpha\in
Ram}\mathbb{D}_{\alpha,\epsilon_{\alpha}}$. ∎

######  Proposition 2.5.20.

([21, Proposition 7.1.3]) For any $[\Sigma]\in\mathcal{B}$, the sympectic
reduction of $W^{Ram}_{t(\Sigma)}$ is given by
$W^{Ram}_{t(\Sigma)}//G^{\perp}_{\Sigma}:=G_{\Sigma}/G^{\perp}_{\Sigma}\cong\mathcal{H}_{\Sigma}=H^{1}(\Sigma,\mathbb{C})$.
Furthermore, the induced symplectic form $\tilde{\Omega}_{Airy}$ on the
quotient $G_{\Sigma}/G^{\perp}_{\Sigma}$ given by

(65)
$\tilde{\Omega}_{Airy}([\xi_{1}],[\xi_{2}]):=\Omega_{Airy}(\xi_{1},\xi_{2})$

coincides with the symplectic form $\Omega_{\mathcal{H}}$ on
$\mathcal{H}_{\Sigma}$ as given in (11).

###### Proof.

First, we will show that $G_{\Sigma}/G^{\perp}_{\Sigma}$ is isomorphic to the
deRham cohomology group $\mathcal{H}_{\Sigma}=H^{1}(\Sigma,\mathbb{C})$. For
any set $\\{\bar{\epsilon}_{\alpha\in Ram}\\}$ such that
$\bar{\epsilon}_{\alpha}<\epsilon_{\alpha},\alpha\in Ram$ we have the
presentation

(66)
$\mathcal{H}_{\Sigma}=H^{1}(\Sigma,\mathbb{C})\cong\frac{\left\\{\xi\in\Gamma\left(\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}},\Omega^{1}_{\Sigma}\right)\
|\
\oint_{\partial\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}}\xi=0\right\\}}{d\Gamma(\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}},\mathcal{O}_{\Sigma})}.$

To obtain this, we use Mayer–Vietoris sequence to get
$H^{1}(\Sigma,\mathbb{C})\cong\\{\xi\in H^{1}(\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}},\mathbb{C})\ |\
\oint_{\partial\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}}\xi=0\\}$. Then we
use the fact that non-compact Riemann surfaces such as
$\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}$ is a Stein manifold
[15, Behnke and Stein, Section 2.2], hence coherent sheaves are
$\Gamma$-acyclic on $\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}$ by Cartan’s theorem B
[15, Theorem 2.4.1]. Therefore, $H^{1}(\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}},\mathbb{C})\cong
H^{1}\Gamma(\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}},\Omega^{\bullet}_{\Sigma})$,
i.e. deRham cohomology can be computed as a sheaf cohomology of $\mathbb{C}$
using the $\Gamma$-acyclic resolution
$0\rightarrow\mathbb{C}\rightarrow\Omega^{\bullet}_{\Sigma}$ (see, for
example, [5, Corrollary 2.5.3]), which implies (66). Recall that the direct
limit functor preserves short exact sequences. Therefore, taking the direct
limit of (66) and compare the result to the explicit expression of
$G_{\Sigma}$ and $G^{\perp}_{\Sigma}$ in Definition 2.5.5 and Proposition
2.5.19, we have $H^{1}(\Sigma,\mathbb{C})\cong G_{\Sigma}/G^{\perp}_{\Sigma}$
as claimed.

Next, we show that $\tilde{\Omega}_{Airy}$ is a symplectic form on
$\mathcal{H}_{\Sigma}$. If
$\xi_{1}^{\prime}=\xi_{1}+df_{1},\xi_{2}^{\prime}=\xi_{2}+df_{2}$ then
$\tilde{\Omega}_{Airy}([\xi^{\prime}_{1}],[\xi^{\prime}_{2}])=\Omega_{Airy}(\xi_{1}+df_{1},\xi_{2}+df_{2})=\Omega_{Airy}(\xi_{1},\xi_{2})=\tilde{\Omega}_{Airy}([\xi_{1}],[\xi_{2}])$,
so $\tilde{\Omega}_{Airy}$ is well-defined. It is non-degenerates because if
$[\xi]\in\mathcal{H}_{\Sigma}$ is such that
$\tilde{\Omega}_{Airy}([\xi],[\eta])=0$ for all
$[\eta]\in\mathcal{H}_{\Sigma}$ then $\Omega_{Airy}(\xi,\eta)=0$ for all
$\eta\in G_{\Sigma}$ implying that $\xi\in G^{\perp}_{\Sigma}$ and hence
$[\xi]=0$. This proves that $\mathcal{H}_{\Sigma}$ is a symplectic vector
space with the induced symplectic form $\tilde{\Omega}_{Airy}$. Using (2.5.3)
and Lemma 2.1.14 we can see that $\tilde{\Omega}_{Airy}$ coincides with
$\Omega_{\mathcal{H}}$ as claimed. ∎

Given $\xi\in G_{\Sigma}$, the cohomology class $[\xi]\in
H^{1}(\Sigma,\mathbb{C})$ is completely determined by the $A$ and $B$ periods
of $\xi$. In other words, we have a map
$[.]:G_{\Sigma}\rightarrow\mathcal{H}_{\Sigma}$ given by
$\xi\mapsto(H_{1}(\Sigma,\mathbb{C})\ni[C]\mapsto\oint_{C}\xi)$. Proposition
2.5.20 tells us that this map is surjective, in other words, there are enough
differential forms on $G_{\Sigma}$ to represent every cohomology classes in
$H^{1}(\Sigma,\mathbb{C})$.

######  Remark 2.5.21.

Similarly, it has been shown in [21, Proposition 7.1.3] that
$G^{mer}_{\Sigma}$ is a coisotropic subspace of $W^{|Ram|}_{Airy}$ (see
Section 2.5.4 for the definition of $W^{|Ram|}_{Airy}$) with

$(G^{mer}_{\Sigma})^{\perp}=\left\\{df\ |\
f\in\Gamma\left(\Sigma,\mathcal{O}_{\Sigma}\left(\sum_{\alpha\in
Ram}nr_{\alpha}\right)\right),\text{ for some }n\in\mathbb{Z}_{\geq
0}\right\\}\subseteq G^{mer}_{\Sigma}$

and we have
$G^{mer}_{\Sigma}/(G^{mer}_{\Sigma})^{\perp}\cong\mathcal{H}_{\Sigma}$.

At last, let us define the vector bundle
$G\rightarrow\mathcal{B}_{\Sigma_{0}}$ with fiber over each point
$[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ given by $G_{\Sigma}$, where
$\mathcal{B}_{\Sigma_{0}}\subseteq\mathcal{B}$ along with
$\\{\epsilon_{\alpha\in Ram}\\},\\{M_{\alpha\in Ram}\\}$ and
$\mathcal{U}_{Ram}$ are chosen to satisfy Condition 2.5.2. In fact, let
$\mathcal{B}_{\Sigma_{0}}^{\circ}\supseteq\mathcal{B}$ be the maximal open
neighbourhood of $[\Sigma_{0}]$,
$\mathcal{B}_{\Sigma_{0}}\subseteq\mathcal{B}^{\circ}_{\Sigma_{0}}$ such that
$\mathcal{B}^{\circ}_{\Sigma_{0}}$, $\\{\epsilon_{\alpha\in
Ram}\\},\\{M_{\alpha\in Ram}\\}$ and $\mathcal{U}_{Ram}$ satisfy C2.5.2.2 then
we will define the vector bundle
$G\rightarrow\mathcal{B}^{\circ}_{\Sigma_{0}}$. Note that it is not possible
to extend the definition of $G$ as a vector bundle on a larger base than
$\mathcal{B}^{\circ}_{\Sigma_{0}}$ because any different choice of the
collection of $(\mathcal{F},\Omega_{S})$-charts $\mathcal{U}_{Ram}$ will lead
to a different definition for each fiber $G_{\Sigma}$.

For any given point $[\Sigma]\in\mathcal{B}^{\circ}_{\Sigma_{0}}$, there
exists an open neighbourhood
$\mathcal{B}_{\Sigma}\subset\mathcal{B}^{\circ}_{\Sigma_{0}}$ of $[\Sigma]$
such that $A,B$ symplectic basis cycles can be chosen consistently for all
$[\Sigma^{\prime}]\in\mathcal{B}_{\Sigma}$ (see Remark 2.1.13). It follows
that the normalized holomorphic forms $\\{\omega_{i=1,\cdots,g}\\}$ and the
normalized Bergman kernel $B(p,q)$ are defined on every curve
$[\Sigma^{\prime}]\in\mathcal{B}_{\Sigma}$ consistently throughout
$\mathcal{B}_{\Sigma}$. For each $[\Sigma^{\prime}]\in\mathcal{B}_{\Sigma}$,
let us denote by $\\{\omega_{i}^{\Sigma,\Sigma^{\prime}}\\}$,
$\\{\bar{e}^{k,\alpha}_{\Sigma,\Sigma^{\prime}}\\}$ the set of normalized
holomorphic forms and the set of meromorphic differentials on
$\Sigma^{\prime}$ defined by the normalized Bergman kernel (Definition 2.5.11)
corresponding to the given consistent choice of $A,B$ symplectic basis cycles
on $\mathcal{B}_{\Sigma}$. Let the topology on $G_{\Sigma}$ be induced from
the norm

$\|\xi\|_{G_{\Sigma}}:=\sqrt{\sum_{\alpha\in
Ram}\left(\sup_{p\in\mathbb{A}_{\alpha,\epsilon,M_{\alpha}}(\Sigma)}\left|\xi(p)\frac{z_{\alpha}(p)}{dz_{\alpha}(p)}\right|\right)^{2}}.$

and recall from Corollary 2.5.17, any $\xi\in G_{\Sigma^{\prime}}$ can be
written as $\xi=\sum_{\alpha\in
Ram}\sum_{k=1}^{\infty}\xi_{k,\alpha}\bar{e}^{k,\alpha}_{\Sigma,\Sigma^{\prime}}+\sum_{k=1}^{g}a^{k}\omega^{\Sigma,\Sigma^{\prime}}_{k}$.
Then, given $[\Sigma_{1}],[\Sigma_{2}]\in\mathcal{B}_{\Sigma}$ we define a
homeomorphism $G_{\Sigma_{1}}\cong G_{\Sigma_{2}}$ by

(67) $\sum_{\alpha\in
Ram}\sum_{k=1}^{\infty}\xi_{k,\alpha}\bar{e}^{k,\alpha}_{\Sigma,\Sigma_{1}}+\sum_{k=1}^{g}a^{k}\omega^{\Sigma,\Sigma_{1}}_{k}\xleftrightarrow{\hskip
28.45274pt}\sum_{\alpha\in
Ram}\sum_{k=1}^{\infty}\xi_{k,\alpha}\bar{e}^{k,\alpha}_{\Sigma,\Sigma_{2}}+\sum_{k=1}^{g}a^{k}\omega^{\Sigma,\Sigma_{2}}_{k}.$

Equip $\mathcal{B}_{\Sigma}\times G_{\Sigma}$ with the product topology and
assign a topology to the set
$G|_{\mathcal{B}_{\Sigma}}:=\\{([\Sigma^{\prime}],\xi)\ |\
[\Sigma^{\prime}]\in\mathcal{B}_{\Sigma},\xi\in G_{\Sigma^{\prime}}\\}$ such
that $G|_{\mathcal{B}_{\Sigma}}\cong\mathcal{B}_{\Sigma}\times G_{\Sigma}$,
induced from (67), is a homeomorphism over the base $\mathcal{B}_{\Sigma}$. We
can repeat the above construction for any sufficiently small neighbourhood
$\mathcal{B}_{\tilde{\Sigma}}$ of