$[\tilde{\Sigma}]\in\mathcal{B}^{\circ}_{\Sigma_{0}}$, possibly with different
consistent choice of $A,B$ symplectic basis cycles. The transition function

$\mathcal{B}_{\Sigma}\cap\mathcal{B}_{\tilde{\Sigma}}\times
G_{\Sigma}\xrightarrow{\cong}\left(G|_{\mathcal{B}_{\Sigma}}\right)|_{\mathcal{B}_{\tilde{\Sigma}}}=\left(G|_{\mathcal{B}_{\tilde{\Sigma}}}\right)|_{\mathcal{B}_{\Sigma}}\xrightarrow{\cong}\mathcal{B}_{\Sigma}\cap\mathcal{B}_{\tilde{\Sigma}}\times
G_{\tilde{\Sigma}}$

induced from the identity on
$G|_{\mathcal{B}_{\Sigma}\cap\mathcal{B}_{\tilde{\Sigma}}}$ can be seen to be
continuous. It follows that
$G:=\bigcup_{[\Sigma]\in\mathcal{B}^{\circ}_{\Sigma_{0}}}G|_{\mathcal{B}_{\Sigma}}$
with the canonical projection
$\pi:G\rightarrow\mathcal{B}^{\circ}_{\Sigma_{0}}$ is a vector bundle with
fiber over each point $[\Sigma]\in\mathcal{B}^{\circ}_{\Sigma_{0}}$ given by
$\pi^{-1}([\Sigma])=G_{\Sigma}$ and
$G|_{\mathcal{B}_{\Sigma}}\cong\mathcal{B}_{\Sigma}\times G_{\Sigma}$ giving a
local trivialization over each $\mathcal{B}_{\Sigma}$.

######  Definition 2.5.22.

Given a choice of $\\{\epsilon_{\alpha\in Ram}\\},\\{M_{\alpha\in Ram}\\}$,
$\mathcal{U}_{Ram}$ and let $\mathcal{B}^{\circ}_{\Sigma_{0}}$ be the maximal
open neighbourhood of $[\Sigma_{0}]$ for which C2.5.2.2 is satisfied. We
define $(G\rightarrow\mathcal{B}^{\circ}_{\Sigma_{0}},\nabla_{\mathcal{F}})$
to be the vector bundle
$G:=\bigcup_{[\Sigma]\in\mathcal{B}^{\circ}_{\Sigma_{0}}}G|_{\mathcal{B}_{\Sigma}}$
with fiber $G_{\Sigma}$ over each point
$[\Sigma]\in\mathcal{B}^{\circ}_{\Sigma_{0}}$ and equipped with the connection
$\nabla_{\mathcal{F}}:\Gamma(\mathcal{B}^{\circ}_{\Sigma_{0}},G)\rightarrow\Gamma(\mathcal{B}^{\circ}_{\Sigma_{0}},T^{*}\mathcal{B}\otimes
G)$ given by the differentiation along a leaf of the foliation $\mathcal{F}$.

For any open subset
$\mathcal{B}^{\prime}\subseteq\mathcal{B}^{\circ}_{\Sigma_{0}}$ we denote by
$\Gamma(\mathcal{B}^{\prime},G)$ the set of holomorphic sections of $G$ on
$\mathcal{B}^{\prime}$.

We will mostly be interested in the small deformations of the curve
$\Sigma_{0}$, therefore, it is sufficient to consider the restriction of $G$
to the open neighbourhood
$\mathcal{B}_{\Sigma_{0}}\subseteq\mathcal{B}^{\circ}_{\Sigma_{0}}$ of
$[\Sigma_{0}]$, where $\mathcal{B}_{\Sigma_{0}}$ along with
$\\{\epsilon_{\alpha\in Ram}\\},\\{M_{\alpha\in Ram}\\}$ and
$\mathcal{U}_{Ram}$ are chosen to satisfy Condition 2.5.2. Since
$\mathcal{B}_{\Sigma_{0}}$ is contractible by C2.5.2.1, the restriction of $G$
to $\mathcal{B}_{\Sigma_{0}}$ is a trivial vector bundle which we denote by
$(G\rightarrow\mathcal{B}_{\Sigma_{0}},\nabla_{\mathcal{F}})$. Similarly, the
trivial vector sub-bundles of $G\rightarrow\mathcal{B}_{\Sigma_{0}}$:
$G^{\perp}\rightarrow\mathcal{B}_{\Sigma_{0}}$ with fiber $G^{\perp}_{\Sigma}$
and $V\rightarrow\mathcal{B}_{\Sigma_{0}}$ with fiber $V_{\Sigma}$ over
$[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ may be constructed.

Let us recast some definitions and results in this section as follows. We have
an injective vector bundle morphism $i:G\hookrightarrow W^{Ram}$ covering the
map $\gamma:\mathcal{B}_{\Sigma_{0}}\hookrightarrow Discs^{Ram}$ given by
(59), such that $i$ restricted to the fiber over any point
$[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ is $i:G_{\Sigma}\hookrightarrow
W^{Ram}_{t(\Sigma)}$ given by (60) (see Figure 2.5.6). Therefore, $G$ is a
vector sub-bundle of $W^{Ram}$, in fact by Proposition 2.5.19 it is a
coisotropic vector sub-bundle. Proposition 2.5.20 can be restated as an
isomorphism of vector bundles $G/G^{\perp}\cong\mathcal{H}$. We define
$[.]:G\rightarrow\mathcal{H}$ to be the morphism of vector bundle given fiber-
wise by the natural projection $G_{\Sigma}\rightarrow
G_{\Sigma}/G^{\perp}_{\Sigma}\cong\mathcal{H}_{\Sigma}\cong
H^{1}(\Sigma,\mathbb{C})$ which sends any differential form $\xi\in
G_{\Sigma}$ to its cohomology class $[\xi]\in H^{1}(\Sigma,\mathbb{C})$ (see
Figure 2.5.6).

####  2.5.4. The $Ram$ product residue constraints

So far, the residue constraints were only introduced on a disc with only a
single ramification point. Since there are more than one ramification points
on $\Sigma$, we will need to consider a $|Ram|$ copies of the residue
constraints. The purpose of this section is to fix the needed notations
regarding the $Ram$ product residue constraints Airy structure.

Consider the Airy structure obtained as a product of the residue constraints
Airy structures introduced in Section 2.3. Set
$\mathbb{I}:=\mathbb{Z}_{>0}\times Ram$ and let
$V_{Airy}^{|Ram|}:=\prod_{\alpha\in Ram}V_{Airy}=\sum_{\alpha\in
Ram}[\alpha]\otimes z^{-1}\mathbb{C}[z^{-1}]\frac{dz}{z}$ be a discrete vector
space with a basis $\\{e^{k=1,2,3,\cdots,\alpha\in Ram}:=[\alpha]\otimes
z^{-k}\frac{dz}{z}\\}$. The continuous dual space is given by
$(V^{|Ram|}_{Airy})^{*}:=\prod_{\alpha\in Ram}V^{*}_{Airy}=\sum_{\alpha\in
Ram}[\alpha]\otimes z\mathbb{C}[[z]]\frac{dz}{z}$ with a basis
$\\{f_{k=1,2,3,\cdots,\alpha\in Ram}:=[\alpha]\otimes kz^{k}\frac{dz}{z}\\}$.
Define the Tate space

$W_{Airy}^{|Ram|}:=\left\\{\eta=\sum_{\alpha\in
Ram}[\alpha]\otimes\eta_{\alpha}\ |\
\eta_{\alpha}\in\mathbb{C}((z))dz,Res_{z=0}\eta_{\alpha}=0,\alpha\in
Ram\right\\}=V^{|Ram|}_{Airy}\oplus(V^{|Ram|}_{Airy})^{*}$

with the symplectic form
$\Omega_{Airy}([\alpha_{1}]\otimes\xi_{1},[\alpha_{2}]\otimes\xi_{2})=\delta_{\alpha_{1},\alpha_{2}}Res_{z=0}(\xi_{1}\int\xi_{2})$.
We define the $|Ram|$ product residue constraints Airy structure
$\\{(H_{Airy})_{k=1,2,3,\cdots,\alpha\in Ram}\\}$ on $V^{|Ram|}_{Airy}$ by

$\displaystyle(H_{Airy})_{2n,\alpha}(w)$
$\displaystyle=Res_{z=0}\left(\left(z-\frac{w(z)}{2zdz}\right)[\alpha]\otimes
z^{2n}d(z^{2})\right),$ (68) $\displaystyle(H_{Airy})_{2n-1,\alpha}(w)$
$\displaystyle=\frac{1}{2}Res_{z=0}\left(\left(z-\frac{w(z)}{2zdz}\right)^{2}[\alpha]\otimes
z^{2n-2}d(z^{2})\right),\qquad n=1,2,3,...$

where $w:=\sum_{\alpha\in
Ram}\sum_{k=1}^{\infty}(x^{k,\alpha}f_{k,\alpha}+y_{k,\alpha}e^{k,\alpha})\in
W^{|Ram|}_{Airy}$. We may write (2.5.4) in the standard form (25):

$\displaystyle(H_{Airy})_{i,\alpha}$
$\displaystyle=-y_{i,\alpha}+\sum_{\beta,\gamma\in
Ram}\sum_{j,k=1}^{\infty}(a_{Airy})_{(i,\alpha)(j,\beta)(k,\gamma)}x^{j,\beta}x^{k,\gamma}$
$\displaystyle\qquad+\sum_{\beta,\gamma\in
Ram}\sum_{j,k=1}^{\infty}2(b_{Airy})_{(i,\alpha)(j,\beta)}^{k,\gamma}x^{j,\beta}y_{k,\gamma}+\sum_{\beta,\gamma\in
Ram}\sum_{j,k=1}^{\infty}(c_{Airy})_{i,\alpha}^{(j,\beta)(k,\gamma)}y_{j,\beta}y_{k,\gamma}.$

and express the residue constraints Airy structure as
$(V_{Airy}^{|Ram|},A_{Airy},B_{Airy},C_{Airy})$ where

$A_{Airy}:=((a_{Airy})_{(i,\alpha)(j,\beta)(k,\gamma)}),\qquad
B_{Airy}:=((b_{Airy})^{k,\gamma}_{(i,\alpha)(j,\beta)}),\qquad
C_{Airy}:=((c_{Airy})^{(j,\beta)(k,\gamma)}_{i,\alpha}).$

The corresponding quantum Airy structure is
$(V^{|Ram|}_{Airy},A_{Airy},B_{Airy},C_{Airy},\epsilon_{Airy})$ where
$(\epsilon_{Airy})_{i,\alpha}:=\frac{1}{16}\delta_{i,3}$ and
$\epsilon_{Airy}=((\epsilon_{Airy})_{i,\alpha})$. Define the quadratic
Lagrangian submanifold
$L^{|Ram|}_{Airy}:=\\{(H_{Airy})_{i=1,2,3,\cdots,\alpha\in Ram}=0\\}\subset
W^{|Ram|}_{Airy}$ with $T_{0}L^{|Ram|}_{Airy}:=\sum_{\alpha\in
Ram}[\alpha]\otimes z\mathbb{C}[[z]]\frac{dz}{z}=(V^{|Ram|}_{Airy})^{*}$. It
is clear that $T_{0}L^{|Ram|}_{Airy}$ is a Lagrangian subspace of
$W^{|Ram|}_{Airy}$ and that $V^{|Ram|}_{Airy}$ is a Lagrangian complement of
$T_{0}L^{|Ram|}_{Airy}$.

######  Remark 2.5.23.

Our definitions of $W^{|Ram|}_{Airy}$ and $L^{|Ram|}_{Airy}$ are exactly the
same as how $W^{Ram}_{Airy}$ and $L^{Ram}_{Airy}$ were defined in [21, Section
7.1]. Our choice of symbols came from the fact that $W^{|Ram|}_{Airy}$ and
$L^{|Ram|}_{Airy}$ are the products of $|Ram|$ copies of $W_{Airy}$ and
$L_{Airy}$ (as given in Section 2.3 or [21, Section 3.3]) respectively, where
$|Ram|$ denotes the number of ramification points. Finally, note that by
taking Laurent expansion of any $\xi\in G^{mer}_{\Sigma}$ we obtain the
inclusion $i:G^{mer}_{\Sigma}\hookrightarrow W^{|Ram|}_{Airy}$ can also be
defined exactly as we did in (60).

Let us consider a new Lagrangian complement of $T_{0}L^{|Ram|}_{Airy}$:

$V^{mer}_{\Sigma}:=\left\\{\xi\in G^{mer}_{\Sigma}\ |\
\oint_{A_{i}}\xi=0,i=1,\cdots,g\right\\}$

where $G^{mer}_{\Sigma}$ was given in Remark 2.5.6. Essentially,
$V^{mer}_{\Sigma}\subseteq V_{\Sigma}$ is the subspace of meromorphic
differential forms in $V_{\Sigma}$ with basis
$\\{\bar{e}^{i=1,2,3,\cdots,\alpha\in Ram}\\}$ where $\bar{e}^{i,\alpha}$ is
defined in Definition 2.5.11 and note that (63) can be written as

$\bar{e}^{k,\alpha}=e^{k,\alpha}+\sum_{\alpha^{\prime}\in
Ram}\sum_{k^{\prime}=1}^{\infty}\frac{1}{kk^{\prime}}P^{(k,\alpha)(k^{\prime},\alpha^{\prime})}f_{k^{\prime},\alpha^{\prime}}.$

The gauge transformation
$(V^{|Ram|}_{Airy},A_{Airy},B_{Airy},C_{Airy},\epsilon_{Airy})\mapsto(V^{mer}_{\Sigma},\bar{A}_{Airy},\bar{B}_{Airy},\bar{C}_{Airy},\bar{\epsilon}_{Airy})$
associating with the choice of basis $\\{\eta^{i=1,2,3,\cdots}:=\sum_{\beta\in
Ram}\sum_{j=1}^{\infty}d^{i}_{j,\beta}\bar{e}^{j,\beta}\\}$ for
$V^{mer}_{\Sigma}$ and $\\{\omega_{i=1,2,3,\cdots}:=\sum_{\alpha\in
Ram}\sum_{i=1}^{\infty}c^{i,\alpha}_{j}f_{i,\alpha}\\}$ for
$T_{0}L^{|Ram|}_{Airy}$ for some
$(c^{i,\alpha}_{j}):V^{|Ram|}_{Airy}\rightarrow V^{|Ram|}_{Airy}$ and its
inverse $(d^{i}_{j,\beta}):V^{|Ram|}_{Airy}\rightarrow V^{|Ram|}_{Airy}$, is
given by

(69) $\displaystyle(\bar{a}_{Airy})_{i_{1}i_{2}i_{3}}$
$\displaystyle=(a_{Airy})_{(j_{1},\alpha_{1})(j_{2},\alpha_{2})(j_{3},\alpha_{3})}c^{j_{1},\alpha_{1}}_{i_{1}}c^{j_{2},\alpha_{2}}_{i_{2}}c^{j_{3},\alpha_{3}}_{i_{3}}$
$\displaystyle(\bar{b}_{Airy})_{i_{1}i_{2}}^{i_{3}}$
$\displaystyle=\left((b_{Airy})_{(j_{1},\alpha_{1})(j_{2},\alpha_{2})}^{j_{3},\alpha_{3}}+(a_{Airy})_{(j_{1},\alpha_{1})(j_{2},\alpha_{2})(k,\gamma)}s^{(k,\gamma)(j_{3},\alpha_{3})}\right)c^{j_{1},\alpha_{1}}_{i_{1}}c^{j_{2},\alpha_{2}}_{i_{2}}d^{i_{3}}_{j_{3},\alpha_{3}}$
$\displaystyle(\bar{c}_{Airy})_{i_{1}}^{i_{2}i_{3}}$
$\displaystyle=\Big{(}(c_{Airy})_{(j_{1},\alpha_{1})}^{(j_{2},\alpha_{2})(j_{3},\alpha_{3})}+(b_{Airy})_{(j_{1},\alpha_{1})(k_{1},\gamma_{1})}^{(j_{3},\alpha_{3})}s^{(k_{1},\gamma_{1})(j_{2},\alpha_{2})}$
$\displaystyle\qquad+(b_{Airy})_{(j_{1},\alpha_{1})(k_{2},\gamma_{2})}^{(j_{2},\alpha_{2})}s^{(k_{2},\gamma_{2})(j_{3},\alpha_{3})}+(a_{TR})_{(j_{1},\alpha_{1})(k_{1},\gamma_{1})(k_{2},\gamma_{2})}s^{(k_{1},\gamma_{1})(j_{2},\gamma_{2})}s^{(k_{2},\gamma_{2})(j_{3},\gamma_{3})}\Big{)}$
$\displaystyle\qquad\qquad\qquad\times
c^{j_{1},\alpha_{1}}_{i_{1}}d_{j_{2},\alpha_{2}}^{i_{2}}d_{j_{3},\alpha_{3}}^{i_{3}}$
$\displaystyle(\bar{\epsilon}_{Airy})_{i}$
$\displaystyle=\left((\epsilon_{Airy})_{j_{1},\alpha_{1}}+(a_{Airy})_{(j_{1},\alpha_{1})(k_{1},\gamma_{1})(k_{2},\gamma_{2})}s^{(k_{1},\gamma_{1})(k_{2},\gamma_{2})}\right)c^{j_{1},\alpha_{1}}_{i_{1}}.$

where $s^{(i,\alpha)(j,\beta)}=\frac{1}{ij}P^{(i,\alpha)(j,\beta)}$,
$(s^{(i,\alpha)(j,\beta)}):V^{|Ram|}_{Airy}\times
V^{|Ram|}_{Airy}\rightarrow\mathbb{C}$.

######  Remark 2.5.24.

It is easy to see that
$((V_{Airy})^{|Ram|}_{odd},A_{Airy},B_{Airy},C_{Airy},\epsilon_{Airy})$ is a
quantum Airy sub-structure of
$(V_{Airy}^{|Ram|},A_{Airy},B_{Airy},C_{Airy},\epsilon_{Airy})$ where
$(V_{Airy})^{|Ram|}_{odd}:=\prod_{\alpha\in Ram}(V_{Airy})_{odd}\subset
V_{Airy}^{|Ram|}$. Likewise, let $(V^{mer}_{\Sigma})_{odd}\subset
V^{mer}_{\Sigma}$ be the vector space spanned by finite linear combinations of
vectors $\\{\bar{e}^{k=odd,\alpha\in Ram}\\}$ then
$((V^{mer}_{\Sigma})_{odd},\bar{A}_{Airy},\bar{B}_{Airy},\bar{C}_{Airy},\bar{\epsilon}_{Airy})$
is a quantum Airy sub-structure of
$(V^{mer}_{\Sigma},\bar{A}_{Airy},\bar{B}_{Airy},\bar{C}_{Airy},\bar{\epsilon}_{Airy})$
according to Remark 2.2.30.

The consideration of the analytic residue constraints Airy structure in
Section 2.4.2 straightforwardly generalizes to the case of the $Ram$ product
residue constraints Airy structure because $Ram$ is a finite set. In
particular, we treat (2.5.4) analytically as functions
$(H_{Airy})_{i,\alpha}:W^{Ram}_{Airy}\rightarrow\mathbb{C}$ and we also have
that $L^{Ram}_{Airy}=\\{(H_{Airy})_{i=1,2,3,\cdots,\alpha\in Ram}=0\\}\subset
W^{Ram}_{Airy}$. The analytic counter-part of $V^{|Ram|}_{Airy}$ is obviously
given by $V_{Airy}^{Ram}:=\prod_{\alpha\in Ram}V^{\epsilon_{\alpha}}_{Airy}$.
It is clear that $V^{Ram}_{Airy}$ is a Lagrangian complement of
$T_{0}L^{Ram}_{Airy}$ in $W^{Ram}_{Airy}=V^{Ram}_{Airy}\oplus
T_{0}L^{Ram}_{Airy}$.

According to Corollary 2.5.18, $V_{\Sigma}$ is another Lagrangian complement
of $T_{0}L^{Ram}_{Airy}$. Let us use $\\{\eta^{i=1,2,3,\cdots}:=\sum_{\beta\in
Ram}\sum_{j=1}^{\infty}d^{i}_{j,\beta}\bar{e}^{j,\beta}\\}$ as a basis of
$V_{\Sigma}$ and $\\{\omega_{i=1,2,3,\cdots}:=\sum_{\alpha\in
Ram}\sum_{i=1}^{\infty}c^{i,\alpha}_{j}f_{i,\alpha}\\}$ as a basis of
$T_{0}L^{Ram}_{Airy}$ for some $(c^{i,\alpha}_{j}),(d^{i}_{j,\beta})$
satisfying Condition 2.4.17. Note that $(s^{(i,\alpha)(j,\beta)})$ where
$s^{(i,\alpha)(j,\beta)}=\frac{1}{ij}P^{(i,\alpha)(j,\beta)}$ also satisfies
Condition 2.4.17 since it came from the series expansion of a Bergman kernel.
Therefore, the change of the canonical basis
$\\{e^{k,\alpha},f_{k,\alpha}\\}\mapsto\\{\eta^{k},\omega_{k}\\}$ of
$W^{Ram}_{Airy}$ is of the same type as considered in Section 2.4.2. Suppose
that $w=\sum_{\alpha\in
Ram}\sum_{k=1}^{\infty}(x^{k,\alpha}f_{k,\alpha}+y_{k,\alpha}e^{k,\alpha})\in
W^{Ram}_{Airy}$, then by Proposition 2.4.20 we have
$w=\sum_{k=1}^{\infty}(\alpha^{k}\omega_{k}+\beta_{k}\eta^{k})$ where

(70) $\alpha^{i}:=\sum_{\beta\in
Ram}\sum_{j=1}^{\infty}d^{i}_{j,\beta}\left(x^{j,\beta}-\sum_{\gamma\in
Ram}\sum_{k=1}^{\infty}s^{(j,\beta),(k,\gamma)}y_{k,\gamma}\right),\qquad\beta_{i}:=\sum_{\beta\in
Ram}\sum_{j=1}^{\infty}c^{j,\beta}_{i}y_{j,\beta}$

are the new canonical coordinates of $W^{Ram}_{Airy}$. Also, by Proposition
2.4.20 we have $(\bar{H}_{Airy})_{i}:=\sum_{\beta\in
Ram}\sum_{j=1}^{\infty}c^{j,\beta}_{i}(H_{Airy})_{j,\beta}$ given by

(71)
$(\bar{H}_{Airy})_{i}:=-\beta_{i}+\sum_{j,k=1}^{\infty}(\bar{a}_{Airy})_{ijk}\alpha^{j}\alpha^{k}+2\sum_{j,k=1}^{\infty}(\bar{b}_{Airy})_{ij}^{k}\alpha^{i}\beta_{k}+\sum_{j,k=1}^{\infty}(\bar{c}_{Airy})_{i}^{jk}\beta_{j}\beta_{k}$

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 (69).

######  Remark 2.5.25.

Moreover, it is straightforward to generalize Proposition 2.4.21 to the case
of the $Ram$ product residue constraints Airy structure. We conclude that if
$i_{\alpha}\left(\sum_{i=1}^{\infty}\boldsymbol{\beta}_{i}(\alpha^{1},\cdots,\alpha^{K})\eta^{i}\right)(z_{\alpha})\in
V^{\epsilon_{\alpha}}_{Airy}$, where
$\boldsymbol{\beta}_{i}(\alpha^{1},\cdots,\alpha^{K}):=\beta_{i}(\\{\alpha^{1},\cdots,\alpha^{K},\alpha^{k>K}=0\\})$,
is holomorphic in $\\{\alpha^{1},\cdots,\alpha^{K},z^{-1}_{\alpha}\\}$ on some
open neighbourhood of $(\alpha^{1},\cdots,\alpha^{K})=(0,\cdots,0)$,
$|z_{\alpha}|>\epsilon_{\alpha}$ and

$(\bar{H}_{Airy})_{i}\left(\\{\alpha^{1},\cdots,\alpha^{K},\alpha^{k>K}=0\\},\\{\boldsymbol{\beta}_{k=1,2,3,\cdots}(\alpha^{1},\cdots,\alpha^{K})\\}\right)=0$

then $\boldsymbol{\beta}_{i}=(\partial_{i}S_{0})|_{\alpha^{k>K}=0}$. Where
$S_{0}=\sum_{n=1}^{\infty}S_{0,n}\in\prod_{n=1}^{\infty}Sym_{n}(V^{mer}_{\Sigma})$
is the genus zero part of the ATR output using the quantum Airy structure
$(V^{mer}_{\Sigma},\bar{A}_{Airy},\bar{B}_{Airy},\bar{C}_{Airy},\bar{\epsilon}_{Airy})$.

####  2.5.5. Connection $\nabla_{\mathcal{F}}$ and parallel transport on
$G\rightarrow\mathcal{B}_{\Sigma_{0}}$

In this section, we are going to more carefully examine the connection
$\nabla_{\mathcal{F}}$ and discuss the parallel transport on
$G\rightarrow\mathcal{B}_{\Sigma_{0}}$. For now, let $\\{\epsilon_{\alpha\in
Ram}\\}$, $\\{M_{\alpha\in Ram}\\}$, $\mathcal{U}_{Ram}$ and an open
neighbourhood $\mathcal{B}_{\Sigma_{0}}\subseteq\mathcal{B}$ of
$[\Sigma_{0}]\in\mathcal{B}$ be chosen such that C2.5.2.1 and C2.5.2.2 are
satisfied. Recall from Definition 2.5.22 that $\nabla_{\mathcal{F}}$ is given
by the differentiation along a leaf of the foliation $\mathcal{F}$. Let us
verify that this indeed defines a connection on
$G\rightarrow\mathcal{B}_{\Sigma_{0}}$ which is flat.

######  Lemma 2.5.26.

For any section $\xi\in\Gamma(\mathcal{B}_{\Sigma_{0}},G)$, we have
$\nabla_{\mathcal{F}}\xi\in\Gamma(\mathcal{B}_{\Sigma_{0}},T^{*}\mathcal{B}\otimes
G)$.

###### Proof.

Consider $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$. Let
$\mathcal{U}:=\\{(U_{p\in\sigma},x_{p\in\sigma},y_{p\in\sigma})\\}$ be a
collection of $(\mathcal{F},\Omega_{S})$-charts covering
$\Sigma\setminus\cup_{\alpha\in
Ram}\mathbb{D}_{\alpha,\epsilon_{\alpha}}(\Sigma)$ where $\Sigma\cap U_{p}$
are bi-holomorphic to a simply-connected open subset of $\mathbb{C}$ with
$x_{p}$ as a local coordinate and $\sigma$ is some finite index set. It
follows that $dx_{p}|_{\Sigma\cap U_{p}}\neq 0$ and so
$r_{\alpha}(\Sigma)\notin\cup_{p\in\sigma}\Sigma\cap U_{p}$ for all $\alpha\in
Ram$.

Let $\\{u^{1},...,u^{g}\\}$ be coordinates of $\mathcal{B}_{\Sigma_{0}}$ (this
exists due to the condition C2.5.2.1). Given a section
$\xi\in\Gamma(\mathcal{B}_{\Sigma_{0}},G)$, $\xi_{\Sigma}\in G_{\Sigma}$ and a
tangent vector $v=\sum_{k=1}^{g}v^{k}\frac{\partial}{\partial u^{k}}\in
T_{[\Sigma]}\mathcal{B}_{\Sigma_{0}}$ then for each $p\in\sigma$ we have
$\iota_{v}\nabla_{\mathcal{F}}\xi|_{\Sigma\cap
U_{p}}=\sum_{k=1}^{g}v^{k}\frac{\partial}{\partial
u_{k}}|_{x_{p}=const}(\xi|_{\Sigma\cap U_{p}})$. Using $x_{p}$ as a local
coordinate on $\Sigma\cap U_{p}$ and write $\xi|_{\Sigma\cap
U_{p}}=\xi_{p}(x_{p},u)dx$. It follows that
$\iota_{v}\nabla_{\mathcal{F}}\xi|_{\Sigma\cap
U_{p}}=\sum_{k=1}^{g}v^{k}\frac{\partial}{\partial
u^{k}}(\xi_{p}(x_{p},u))dx_{p}$. We patch
$\iota_{v}\nabla_{\mathcal{F}}\xi|_{\Sigma\cap U_{p}}$ for each $p\in\sigma$
together to obtain $\iota_{v}\nabla_{\mathcal{F}}\xi$ which is holomorphic on
$\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}(\Sigma)\subset\cup_{p\in\sigma}\Sigma\cap
U_{p}$ for some $\bar{\epsilon}_{\alpha}<\epsilon_{\alpha},\forall\alpha\in
Ram$.

Let us show that
$\oint_{\partial\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}(\Sigma)}\nabla_{\mathcal{F}}\xi=0$.
Write $\xi$ on $\mathbb{A}_{\alpha,\epsilon_{\alpha},M_{\alpha}}(\Sigma)$ in
the standard local coordinate as $\xi(z_{\alpha})=\sum_{k\neq
0}\xi_{\alpha,k}z^{k}_{\alpha}\frac{dz_{\alpha}}{z_{\alpha}}$. Since
$x_{\alpha}=a_{\alpha}(\Sigma)+z^{2}_{\alpha}$, we have
$\nabla_{\mathcal{F}}(z_{\alpha})=\sum_{k=1}^{g}du^{k}\frac{\partial
a_{\alpha}(\Sigma)}{\partial u^{k}}\frac{\partial}{\partial
a_{\alpha}}|_{x_{\alpha}=const}z_{\alpha}=-\frac{1}{2z_{\alpha}}\sum_{k=1}^{g}du^{k}\frac{\partial
a_{\alpha}(\Sigma)}{\partial u^{k}}$. It follows that
$\nabla_{\mathcal{F}}(z_{\alpha}^{k}dz_{\alpha})=\frac{1}{2}\sum_{k=1}^{g}du^{k}\frac{\partial
a_{\alpha}(\Sigma)}{\partial u^{k}}(1-k)z^{k-2}_{\alpha}dz_{\alpha}$,
therefore $\iota_{v}\nabla_{\mathcal{F}}\xi$ is free of
$\frac{dz_{\alpha}}{z_{\alpha}}$ term. Hence
$\iota_{v}\nabla_{\mathcal{F}}\xi_{\Sigma}\in G_{\Sigma}$, in other words,
$\nabla_{\mathcal{F}}\xi\in\Gamma(\mathcal{B}_{\Sigma_{0}},T^{*}\mathcal{B}\otimes
G)$, as required. ∎

######  Lemma 2.5.27.

The vector bundle $G\rightarrow\mathcal{B}_{\Sigma_{0}}$ with the connection
$\nabla_{\mathcal{F}}$ is flat.

###### Proof.

This is mostly identical to the proof of Lemma 2.1.18, but let us write it
more concretely this time. Let
$d_{\nabla_{\mathcal{F}}}:\Gamma(\mathcal{B}_{\Sigma_{0}},\Omega^{r}_{\mathcal{B}}\otimes
G)\rightarrow\Gamma(\mathcal{B}_{\Sigma_{0}},\Omega^{r+1}_{\mathcal{B}}\otimes
G)$ be the exterior covariant derivative. Suppose that
$\xi\in\Gamma(\mathcal{B}_{\Sigma_{0}},\Omega^{r}_{\mathcal{B}}\otimes G)$. On
each $\Sigma\cap U_{p}$ we have $\xi|_{\Sigma\cap
U_{p}}=\sum_{i_{1},\cdots,i_{r}=1}^{g}(\xi_{i_{1}\cdots
i_{r};p}(x_{p};u)dx_{p})du^{i_{1}}\wedge\cdots du^{i_{r}}$, therefore

$d_{\nabla_{\mathcal{F}}}\xi|_{\Sigma\cap
U_{p}}=\nabla_{\mathcal{F}}\xi|_{\Sigma\cap
U_{p}}=\sum_{k=1}^{g}\sum_{i_{1},\cdots,i_{r}=1}^{g}\left(\frac{\partial}{\partial
u^{k}}\Big{|}_{x_{p}=const}(\xi_{i_{1}\cdots
i_{r};p}(x_{p},u))dx_{p}\right)du^{k}\wedge du^{i_{1}}\wedge\cdots\wedge
du^{i_{r}}$

and

$d^{2}_{\nabla_{\mathcal{F}}}\xi|_{\Sigma\cap
U_{p}}=\sum_{k,l=1}^{g}\sum_{i_{1},\cdots,i_{r}=1}^{g}\left(\frac{\partial^{2}}{\partial
u^{l}\partial u^{k}}\Big{|}_{x_{p}=const}(\xi_{i_{1}\cdots
i_{r};p}(x_{p},u))dx_{p}\right)du^{l}\wedge du^{k}\wedge
du^{i_{1}}\wedge\cdots\wedge du^{i_{r}}=0.$

Therefore, $d^{2}_{\nabla_{\mathcal{F}}}=0$, so $\nabla_{\mathcal{F}}$ is
flat. ∎

Next, we turn our attention to the parallel transport on $G$ using the
connection $\nabla_{\mathcal{F}}$. Since the connection is flat, parallel
transport is path-independent. The parallel transport of $\xi\in G_{\Sigma}$
to the fiber $G_{\Sigma^{\prime}}$ only exists if $[\Sigma]\in\mathcal{B}$ is
close enough to $[\Sigma^{\prime}]\in\mathcal{B}$. This is much like the
situation in Section 2.4: given $\xi\in W^{\epsilon,M}_{Airy}$ then
$\exp(a\mathcal{L}_{\frac{1}{2z}\partial_{z}})\xi\in W^{\epsilon,M}_{Airy}$
only if $|a|$ is sufficiently small. The idea is to first construct a map
$s_{\Sigma^{\prime},\Sigma}$ sending points on $\Sigma^{\prime}$ to $\Sigma$
along the foliation leaves, then the parallel transport of $\xi\in G_{\Sigma}$
to $G_{\Sigma^{\prime}}$ is given by the pull-back
$s^{*}_{\Sigma^{\prime},\Sigma}\xi$.

Let us discuss this in more detail. Suppose that
$[\Sigma],[\Sigma^{\prime}]\in\mathcal{B}_{\Sigma_{0}}$, then they are path-
connected since $\mathcal{B}_{\Sigma_{0}}$ is contractible as required by
C2.5.2. Let $\boldsymbol{\Sigma}:[0,1]\rightarrow\mathcal{B}_{\Sigma_{0}}$ be
one such path, $\boldsymbol{\Sigma}(0)=[\Sigma^{\prime}]$, and
$\boldsymbol{\Sigma}(1)=[\Sigma]$. For some
$\bar{\epsilon}^{\prime}_{\alpha}<\epsilon_{\alpha}$ let us send a point
$q^{\prime}\in\Sigma^{\prime}\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}^{\prime}_{\alpha}}(\Sigma^{\prime})$
traveling along a leaf of the foliation $\mathcal{F}$. If $[\Sigma^{\prime}]$
is sufficiently close to $\boldsymbol{\Sigma}(t)\in\mathcal{B}_{\Sigma_{0}}$
for any $t\in[0,1]$, then the point $q^{\prime}$ will intersect
$\boldsymbol{\Sigma}(t)$, potentially multiple times. As we move continuously
along the path $\boldsymbol{\Sigma}$, these intersection points will also move
continuously in $S$. Therefore, for all $t\in[0,1]$ sufficiently small, we can
define $q(t)\in\boldsymbol{\Sigma}(t)$ to be the intersection point which
moves continuously from $q(0)=q^{\prime}$. Now, suppose that $[\Sigma]$,
$[\Sigma^{\prime}]$ and every points along the path $\boldsymbol{\Sigma}$ are
sufficiently close to $[\Sigma^{\prime}]$, then we define the following map

######  Definition 2.5.28.

The map $s_{\Sigma^{\prime},\Sigma}:\Sigma^{\prime}\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}^{\prime}_{\alpha}}(\Sigma^{\prime})\rightarrow\Sigma$
is given by sending any point
$q^{\prime}\in\Sigma^{\prime}\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}^{\prime}_{\alpha}}(\Sigma^{\prime})$
along a leaf of the foliation to $q(1)\in\Sigma$.

######  Remark 2.5.29.

Technically, the map $s_{\Sigma^{\prime},\Sigma}$ should be denoted by
$s_{\Sigma^{\prime},\Sigma,\\{\bar{\epsilon}^{\prime}_{\alpha}\\}}$ since the
set of parameters $\\{\bar{\epsilon}^{\prime}_{\alpha\in Ram}\\}$ is a part of
the definition. We are going to stick with the notation
$s_{\Sigma^{\prime},\Sigma}$ for simplicity, keeping in mind that
$s_{\Sigma^{\prime},\Sigma}$ is defined on some open subset in
$\Sigma^{\prime}$ containing $\Sigma^{\prime}\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}(\Sigma^{\prime})$.

Suppose further that $[\Sigma],[\Sigma^{\prime}]\in\mathcal{B}_{\Sigma_{0}}$
are sufficiently close such that the image of $s_{\Sigma^{\prime},\Sigma}$ is
contained in $\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}(\Sigma)$ for some
$\bar{\epsilon}_{\alpha}<\epsilon_{\alpha}$. Then given any holomorphic
differential form $\xi$ on $\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}(\Sigma)$, the pull-back
$s^{*}_{\Sigma^{\prime},\Sigma}\xi$ is a holomorphic differential form on
$\Sigma^{\prime}\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}^{\prime}_{\alpha}}(\Sigma^{\prime})$.
The following lemma gives a more concrete sufficient condition for
$[\Sigma],[\Sigma^{\prime}]\in\mathcal{B}$ to be considered ‘sufficiently
close’ and shows that if $\xi\in G_{\Sigma}$ then
$s^{*}_{\Sigma^{\prime},\Sigma}\xi\in G_{\Sigma^{\prime}}$.

######  Lemma 2.5.30.

Let $[\Sigma],[\Sigma^{\prime}]\in\mathcal{B}$. Given $\xi\in G_{\Sigma}$,
where $\xi$ is holomorphic on $\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}(\Sigma)$ for some
$\bar{\epsilon}_{\alpha}<\epsilon_{\alpha},\forall\alpha\in Ram$. If we could
cover $\Sigma\setminus\cup_{\alpha\in
Ram}\mathbb{D}_{\alpha,\epsilon_{\alpha}}(\Sigma)$ with a collection of
$(\mathcal{F},\Omega_{S})$-charts
$\mathcal{U}=\\{(U_{p\in\sigma},x_{p\in\sigma},y_{p\in\sigma})\\}$ such that
$\Sigma\setminus\cup_{\alpha\in
Ram}\mathbb{D}_{\alpha,\epsilon_{\alpha}}(\Sigma)\subset\cup_{p\in\sigma}\Sigma\cap
U_{p}\subset\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}(\Sigma)$ and the
following properties are satisfied:

  1. (1)

For each $p\in\sigma,\Sigma\cap U_{p}$ is bi-holomorphic to a simply-connected
open subset of $\mathbb{C}$ with $x_{p}$ as a local coordinate.

  2. (2)

Any $q^{\prime}\in U_{p}$ is path-connected in $U_{p}$ via a leaf of the
foliation $\mathcal{F}$ to a unique point $q$ on $\Sigma\cap U_{p}$.

  3. (3)

For each $p\in\sigma$, $\Sigma^{\prime}\cap U_{p}$ is bi-holomorphic to a
simply-connected open subset of $\mathbb{C}$ with $x_{p}$ as a local
coordinate and $\Sigma^{\prime}\setminus\cup_{\alpha\in
Ram}\mathbb{D}_{\alpha,\epsilon_{\alpha}}(\Sigma^{\prime})\subset\cup_{p\in\sigma}\Sigma^{\prime}\cap
U_{p}$.

then $s^{*}_{\Sigma^{\prime},\Sigma}\xi\in G_{\Sigma^{\prime}}$.

###### Proof.

Let $\xi\in G_{\Sigma}$ be a holomorphic form defined on
$\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}(\Sigma)$ for some
$\bar{\epsilon}_{\alpha}<\epsilon_{\alpha}$ such that
$\cup_{p\in\sigma}\Sigma\cap U_{p}\subset\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}(\Sigma)$. By the
properties of the covering $\mathcal{U}$, Definition 2.5.28 gives a well-
defined map

$s_{\Sigma^{\prime},\Sigma}:\Sigma^{\prime}\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}^{\prime}_{\alpha}}(\Sigma^{\prime})\subseteq\cup_{p\in\sigma}\Sigma^{\prime}\cap
U_{p}\rightarrow\cup_{p\in\sigma}\Sigma\cap
U_{p}\subseteq\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}(\Sigma)$

for some $\bar{\epsilon}^{\prime}_{\alpha}<\epsilon_{\alpha}$. Let us verify
that the pull-back $s_{\Sigma^{\prime},\Sigma}^{*}\xi$ is in
$G_{\Sigma^{\prime}}$. We note that $x_{p}$ is a local coordinate for both
$\Sigma\cap U_{p}$ and $\Sigma^{\prime}\cap U_{p}$. In $x_{p}$ local
coordinate, we can express $s_{\Sigma^{\prime},\Sigma}$ as

$x_{p}\circ s_{\Sigma^{\prime},\Sigma}|_{\Sigma^{\prime}\cap U_{p}}\circ
x_{p}^{-1}=id_{\mathbb{C}}|_{x_{p}(\Sigma^{\prime}\cap
U_{p})}:x_{p}(\Sigma^{\prime}\cap U_{p})\rightarrow x_{p}(\Sigma^{\prime}\cap
U_{p})\subset x_{p}(\Sigma\cap U_{p})\subset\mathbb{C}.$

In particular, $s_{\Sigma^{\prime},\Sigma}$ is locally bi-holomorphic from
each $\Sigma^{\prime}\cap U_{p}$ to its image. It follows that
$s^{*}_{\Sigma^{\prime},\Sigma}\xi|_{\Sigma^{\prime}\cap U_{p}}$ is
holomorphic on $\Sigma^{\prime}\cap U_{p}$ for all $p\in\sigma$, hence
$s^{*}_{\Sigma^{\prime},\Sigma}\xi$ is holomorphic on
$\Sigma^{\prime}\setminus\cup_{\alpha\in
Ram}\mathbb{D}_{\alpha,\bar{\epsilon}^{\prime}_{\alpha}}(\Sigma^{\prime})\subset\cup_{p\in\sigma}\Sigma^{\prime}\cap
U_{p}$. Lastly, we need to check that
$\oint_{\partial\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}(\Sigma^{\prime})}s^{*}_{\Sigma^{\prime},\Sigma}\xi=0$,
but this follows from the lemma below:

######  Lemma 2.5.31.

For any $\xi\in G_{\Sigma}$ holomorphic on $\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}(\Sigma)\subset\cup_{p\in\sigma}\Sigma\cap
U_{p}$ and a path $C\subset\Sigma^{\prime}\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}(\Sigma^{\prime})$, we have

$\int_{C}s^{*}_{\Sigma^{\prime},\Sigma}\xi=\int_{(s_{\Sigma^{\prime},\Sigma})_{*}C}\xi,$

where $(s_{\Sigma^{\prime},\Sigma})_{*}C\subset\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}(\Sigma)$ is the image of
$C$ under the map $s_{\Sigma^{\prime},\Sigma}$.

###### Proof.

We dissect $C$ into closed segments $C_{i},i=1,...,N$ such that
$C_{i}\subset\Sigma^{\prime}\cap U_{p_{i}}$ for some $p_{i}\in\sigma$. As we
have seen, $s_{\Sigma^{\prime},\Sigma}|_{\Sigma^{\prime}\cap U_{p_{i}}}$ is a
restriction of the identity, hence it is biholomorphic. So the Lemma holds for
each segment $C_{i}$. Summing the integral over all segments $C_{i}$ proving
the lemma. ∎

It follows from the construction of the map $s_{\Sigma^{\prime},\Sigma}$ that
$(s_{\Sigma^{\prime},\Sigma})_{*}(\partial\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}(\Sigma^{\prime}))$
is homotopic to $\partial\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}(\Sigma)$
because we can deform one to another by moving along a path
$\boldsymbol{\Sigma}$ linking $[\Sigma]$ and $[\Sigma^{\prime}]$. We conclude
that
$\oint_{\partial\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}(\Sigma^{\prime})}s^{*}_{\Sigma^{\prime},\Sigma}\xi=\oint_{\partial\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}(\Sigma)}\xi=0$
because $\xi\in G_{\Sigma}$ which shows $s^{*}_{\Sigma^{\prime},\Sigma}\xi\in
G_{\Sigma^{\prime}}$ as claimed. ∎

Next, we show that the pull-back via $s_{\Sigma^{\prime},\Sigma}$ is the
parallel transport on $G$. If Lemma 2.5.30 can be compared to Lemma 2.4.9 then
the following is analogous to Lemma 2.4.13:

######  Lemma 2.5.32.

Let $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ and $\xi\in G_{\Sigma}$ be given,
then
$[\Sigma^{\prime}]\mapsto\xi_{\Sigma^{\prime}}:=s^{*}_{\Sigma^{\prime},\Sigma}\xi$
is a parallel vector field, defined over the subset of all
$[\Sigma^{\prime}]\in\mathcal{B}_{\Sigma_{0}}$ such that the conditions of
Lemma 2.5.30 are satisfied. In other words, $\xi_{\Sigma^{\prime}}\in
G_{\Sigma^{\prime}}$ is the parallel transport of $\xi\in G_{\Sigma}$ from
$[\Sigma]$ to $[\Sigma^{\prime}]$ on $G\rightarrow\mathcal{B}_{\Sigma_{0}}$
using the connection $\nabla_{\mathcal{F}}$.

###### Proof.

Since $x_{p}$ is a local coordinate of $\Sigma$ on $\Sigma\cap U_{p}$ we can
write $\xi|_{\Sigma\cap U_{p}}=f(x_{p})dx_{p}$ for some holomorphic function
$f$ which is constant with respect to the coordinates of
$\mathcal{B}_{\Sigma_{0}}$. Because
$s^{*}_{\Sigma^{\prime},\Sigma}x_{p}=x_{p}$ by definition, we have
$s^{*}_{\Sigma^{\prime},\Sigma}\xi|_{\Sigma^{\prime}\cap
U_{p}}=f(x_{p})dx_{p}$. Therefore,
$\nabla_{\mathcal{F}}s^{*}_{\Sigma^{\prime},\Sigma}\xi|_{\Sigma^{\prime}\cap
U_{p}}=0$ for all $p\in\sigma$ and so
$\nabla_{\mathcal{F}}s^{*}_{\Sigma^{\prime},\Sigma}\xi=0$. ∎

Now that all the relevant results have been established, let us assume that
$\\{\epsilon_{\alpha\in Ram}\\},\\{M_{\alpha\in Ram}\\},\mathcal{U}_{Ram}$ and
$\mathcal{B}_{\Sigma_{0}}$ also satisfy C2.5.2.3 and C2.5.2.4 in addition to
C2.5.2.1 and C2.5.2.2. Let us show in the following that the image of
$\mathcal{B}_{\Sigma_{0}}$ under $\gamma$ defined in (59) is contained in
$Discs^{Ram}_{t_{0}}$. This fact will be useful in comparing parallel
transport on $W^{Ram}\rightarrow Discs^{Ram}_{t_{0}}$ with parallel transport
on $G\rightarrow\mathcal{B}_{\Sigma_{0}}$.

######  Lemma 2.5.33.

Suppose that $\\{\epsilon_{\alpha\in Ram}\\},\\{M_{\alpha\in
Ram}\\},\mathcal{U}_{Ram}$ and $\mathcal{B}_{\Sigma_{0}}$ satisfies the
Condition 2.5.2 then the image of
$\gamma:\mathcal{B}_{\Sigma_{0}}\hookrightarrow Discs^{Ram}$ is contained in
$Discs^{Ram}_{t_{0}}$. In other words, we have

(72) $\gamma:\mathcal{B}_{\Sigma_{0}}\hookrightarrow
Discs^{Ram}_{t_{0}}\hookrightarrow Discs^{Ram}.$

###### Proof.

C2.5.2.1 and C2.5.2.2 ensure that the map
$\gamma:\mathcal{B}_{\Sigma_{0}}\rightarrow Discs^{Ram}$ is well-defined. We
are going to show that C2.5.2.3 implies that if
$\sum_{k=0}^{\infty}b_{k\alpha}(\Sigma)z_{\alpha}^{k}$ has a radius of
convergence $\bar{M}_{\alpha}$, then
$\min(\bar{M}_{\alpha}^{2}-M_{\alpha}^{2},\epsilon_{\alpha}^{2})>|a_{\alpha}(\Sigma)-a_{\alpha}(\Sigma_{0})|$
for all $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$.

It is well-known that there is no point on a compact Riemann surface where all
holomorphic differential forms vanish simultaneously [17]. In particular, for
any ramification point, $r_{\alpha},\alpha\in Ram$ there exists a holomorphic
differential form $\omega\in\Gamma(\Sigma_{0},\Omega^{1}_{\Sigma_{0}})\subset
G_{\Sigma_{0}}$ such that $\omega(r_{\alpha})\neq 0$. For clarity, let us
denote the standard local coordinate on $\Sigma_{0}\cap U_{\alpha}$ by
$z_{\alpha,\Sigma_{0}}$ and the standard local coordinate on $\Sigma\cap
U_{\alpha}$ by $z_{\alpha}$. Let us write
$i_{\alpha}(\omega)=\sum_{k=1}^{\infty}\omega_{\alpha,k}z^{k}_{\alpha,\Sigma_{0}}\frac{dz_{\alpha,\Sigma_{0}}}{z_{\alpha,\Sigma_{0}}}\in
T_{0}L^{M_{\alpha}}_{Airy}\subset
W^{\epsilon_{\alpha},M_{\alpha}}_{t_{\alpha}(\Sigma_{0})}$ where
$\sum_{k=1}^{\infty}$ converges for $|z_{\alpha,\Sigma_{0}}|<\bar{M}_{\alpha}$
for some $\bar{M}_{\alpha}>M_{\alpha}$. Using the fact that
$x_{\alpha}=a_{\alpha}(\Sigma)+z^{2}_{\alpha}$ on $\Sigma\cap U_{\alpha}$ and
$x_{\alpha}=a_{\alpha}(\Sigma_{0})+z^{2}_{\alpha,\Sigma_{0}}$ on
$\Sigma_{0}\cap U_{\alpha}$, it follows that

$\omega(z_{\alpha,\Sigma_{0}})|_{\mathbb{A}_{\alpha,\epsilon_{\alpha},M_{\alpha}}(\Sigma_{0})}=\sum_{k=1}^{\infty}\omega_{\alpha,k}\left(\sqrt{x_{\alpha}-a_{\alpha}(\Sigma_{0})}\right)^{k-1}d\sqrt{x_{\alpha}-a_{\alpha}(\Sigma_{0})}$

implies

(73)
$s^{*}_{\Sigma,\Sigma_{0}}\omega(z_{\alpha})|_{\mathbb{A}_{\alpha,\epsilon_{\alpha},M_{\alpha}}(\Sigma)}=\sum_{k=1}^{\infty}\omega_{\alpha,k}\left(\sqrt{z^{2}_{\alpha}+a_{\alpha}(\Sigma)-a_{\alpha}(\Sigma_{0})}\right)^{k-1}d\sqrt{z^{2}_{\alpha}+a_{\alpha}(\Sigma)-a_{\alpha}(\Sigma_{0})}.$

By C2.5.2.3 we have that $s^{*}_{\Sigma,\Sigma_{0}}\omega$ is holomorphic on
$\mathbb{A}_{\alpha,\epsilon_{\alpha},M_{\alpha}}(\Sigma)$. For the expression
in (73) to be holomorphic on
$\mathbb{A}_{\alpha,\epsilon_{\alpha},M_{\alpha}}(\Sigma)$ we must have
$\bar{M}^{2}_{\alpha}>M_{\alpha}^{2}+|a_{\alpha}(\Sigma)-a_{\alpha}(\Sigma_{0})|$
and either $\epsilon_{\alpha}^{2}>|a_{\alpha}(\Sigma)-a_{\alpha}(\Sigma_{0})|$
or that $\omega_{\alpha,k}=0$ for all $k$ odd and all $\alpha\in Ram$. But the
latter option is not possible since $\omega(r_{\alpha})\neq 0$. ∎

####  2.5.6. Comparison of vector bundles
$G\rightarrow\mathcal{B}_{\Sigma_{0}}$, $\mathcal{H}\rightarrow\mathcal{B}$,
and $W^{Ram}\rightarrow Discs^{Ram}$

So far we have introduced covariantly constant flat (weak) symplectic vector
bundles
$(\mathcal{H}\rightarrow\mathcal{B},\Omega_{\mathcal{H}},\nabla_{GM})$,
$(W^{Ram}\rightarrow Discs^{Ram},\Omega,\nabla)$, and a flat coisotropic
vector bundle $(G\rightarrow\mathcal{B}_{\Sigma_{0}},\nabla_{\mathcal{F}})$.
We are now in a position to summarize how they are related.

######  Proposition 2.5.34.

The morphisms of vector bundles $[.]:G\rightarrow\mathcal{H}$ given by taking
the cohomology class and $i:G\rightarrow W^{Ram}$ given by Laurent series
expansions are compatible with the connections in the sense that

  1. (1)

$i(\nabla_{\mathcal{F}}\xi)=\gamma^{*}\nabla i(\xi)$

  2. (2)

$[\nabla_{\mathcal{F}}\xi]=\nabla_{GM}[\xi]$

for any sections $\xi\in\Gamma(\mathcal{B}_{\Sigma_{0}},G)$. Moreover, they
are compatible with parallel transport in the sense that

  1. (3)

$i(s^{*}_{\Sigma^{\prime},\Sigma}\xi_{\Sigma})=\sum_{\alpha\in
Ram}[\alpha]\otimes\exp\left((a(\Sigma^{\prime})-a(\Sigma))\mathcal{L}_{\frac{1}{2z_{\alpha}}\partial_{z_{\alpha}}}\right)i_{\alpha}(\xi_{\Sigma})$

  2. (4)

$[s^{*}_{\Sigma^{\prime},\Sigma}\xi_{\Sigma}]=\Gamma_{[\Sigma]}^{[\Sigma^{\prime}]}[\xi_{\Sigma}]$

for any vectors in the fiber $\xi_{\Sigma}\in G_{\Sigma}$ given that
$[\Sigma],[\Sigma^{\prime}]\in\mathcal{B}_{\Sigma_{0}}$ and $\xi_{\Sigma}$
satisfies the conditions of Lemma 2.5.30.

###### Proof.

  1. (1)

Let $\xi\in\Gamma(\mathcal{B}_{\Sigma_{0}},G)$ and let us examine
$\nabla_{\mathcal{F}}\xi$ on $\Sigma\cap U_{\alpha}\subset\Sigma$. Working in
the standard local coordinates we can write
$\xi_{\Sigma}=\xi_{\alpha}(z_{\alpha},u)dz_{\alpha}$. Using the fact that
$x_{\alpha}=a_{\alpha}(\Sigma)+z^{2}_{\alpha}$, we have $\frac{\partial
z_{\alpha}}{\partial
u^{k}}|_{x_{\alpha}=const}=-\frac{1}{2z_{\alpha}}\frac{\partial
a_{\alpha}(\Sigma)}{\partial u^{k}}$ and so

$\displaystyle i_{\alpha}(\nabla_{\mathcal{F}}\xi_{\Sigma})(z_{\alpha})$
$\displaystyle=i_{\alpha}\left(\sum_{k=1}^{g}du^{k}\left(\left(\frac{\partial}{\partial
u^{k}}\Big{|}_{z_{\alpha}-const}\xi_{\alpha}-\frac{1}{2z_{\alpha}}\frac{\partial
a_{\alpha}(\Sigma)}{\partial u^{k}}\frac{\partial}{\partial
z_{\alpha}}\xi_{\alpha}\right)dz_{\alpha}+\xi_{\alpha}d\left(\frac{1}{2z_{\alpha}}\frac{\partial
a_{\alpha}(\Sigma)}{\partial u^{k}}\right)\right)\right)$
$\displaystyle=\sum_{k=1}^{g}du^{k}\left(\frac{\partial}{\partial
u^{k}}-\frac{\partial a_{\alpha}(\Sigma)}{\partial
u^{k}}\mathcal{L}_{\frac{1}{2z_{\alpha}}\partial_{z_{\alpha}}}\right)i_{\alpha}(\xi)_{t_{\alpha}(\Sigma)}=\gamma^{*}\nabla
i_{\alpha}(\xi)_{t_{\alpha}(\Sigma)}(z_{\alpha})$

for every $\alpha\in Ram$. It follows that
$i(\nabla_{\mathcal{F}}\xi)=\gamma^{*}\nabla i(\xi)$ as claimed.

  2. (2)

This immediately follows from Lemma 2.1.17 with
$D_{\Sigma}:=\Sigma\setminus\cup_{p\in\sigma}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}(\Sigma)$
for some $\bar{\epsilon}_{\alpha}<\epsilon_{\alpha},\alpha\in Ram$.

  3. (3)

Let $\xi_{\Sigma}\in G_{\Sigma}$ and let us write $\xi_{\Sigma}$ on
$\Sigma\cap U_{\alpha}\subset\Sigma$ in the standard local coordinates as
$\xi_{\Sigma}=\xi_{\alpha}(z_{\alpha})dz_{\alpha}$. Recall that the local
coordinates of $S$ on $U_{\alpha}$ are $(x_{\alpha},y_{\alpha})$ and the
foliation $\mathcal{F}$ is given by $x_{\alpha}=const$, so we have
$s^{*}_{\Sigma^{\prime},\Sigma}x_{\alpha}=x_{\alpha}$. For clarity, let us
denote the standard local coordinate on $\Sigma^{\prime}\cap U_{\alpha}$ by
$z_{\alpha,\Sigma^{\prime}}$ whereas $z_{\alpha}$ denotes the standard local
coordinate on $\Sigma\cap U_{\alpha}$. Using the fact that
$x_{\alpha}=a_{\alpha}(\Sigma)+z^{2}_{\alpha}$ on $\Sigma\cap U_{\alpha}$ and
$x_{\alpha}=a_{\alpha}(\Sigma^{\prime})+z^{2}_{\alpha,\Sigma^{\prime}}$ on
$\Sigma^{\prime}\cap U_{\alpha}$, we have

$\xi_{\Sigma}(z_{\alpha})=\xi_{\alpha}(z_{\alpha})dz_{\alpha}=\xi_{\alpha}\left(\sqrt{x_{\alpha}-a_{\alpha}(\Sigma)}\right)d\sqrt{x_{\alpha}-a_{\alpha}(\Sigma)}$

and

$\displaystyle s^{*}_{\Sigma^{\prime},\Sigma}$
$\displaystyle\xi_{\Sigma}(z_{\alpha})=\xi_{\alpha}\left(\sqrt{x_{\alpha}-a_{\alpha}(\Sigma)}\right)d\sqrt{x_{\alpha}-a_{\alpha}(\Sigma)}$
$\displaystyle=\xi_{\alpha}\left(\sqrt{z^{2}_{\alpha,\Sigma^{\prime}}+a_{\alpha}(\Sigma^{\prime})-a_{\alpha}(\Sigma)}\right)d\sqrt{z^{2}_{\alpha,\Sigma^{\prime}}+a_{\alpha}(\Sigma^{\prime})-a_{\alpha}(\Sigma)}=\exp\left((a_{\alpha}(\Sigma^{\prime})-a_{\alpha}(\Sigma))\mathcal{L}_{\frac{1}{2z_{\alpha}}\partial_{z_{\alpha}}}\right)(\xi_{\Sigma})$

By construction, $s^{*}_{\Sigma^{\prime},\Sigma}\xi_{\Sigma}$ is holomorphic
and it can be represented by a Laurent series converging on
$\mathbb{A}_{\alpha,\bar{\epsilon}_{\alpha},\bar{M}_{\alpha}}(\Sigma^{\prime})$
for some
$\bar{\epsilon}_{\alpha}<\epsilon_{\alpha}<M_{\alpha}<\bar{M}_{\alpha}$.
Repeating the above for each ramification point $\alpha\in Ram$ and perform a
Laurent series expansion, the claimed formula follows.

  4. (4)

Since
$(s_{\Sigma^{\prime},\Sigma})_{*}:H_{1}(\Sigma^{\prime},\mathbb{C})\rightarrow
H_{1}(\Sigma,\mathbb{C})$ is an isomorphism, mapping $A,B$-periods on
$\Sigma^{\prime}$ to $A,B$-periods on $\Sigma$, then
$[s^{*}_{\Sigma^{\prime},\Sigma}\xi_{\Sigma}]=\Gamma^{[\Sigma^{\prime}]}_{[\Sigma]}[\xi_{\Sigma}]$
follows straightforwardly from Lemma 2.5.31.

∎

$\displaystyle\mathcal{L}_{\Sigma_{0}}$$\displaystyle\mathcal{H}_{\Sigma_{0}}$$\displaystyle\mathcal{B}_{\Sigma_{0}}$$\displaystyle\mathcal{B}$$\displaystyle\mathcal{H}$$\displaystyle\mathbf{\Phi}_{\Sigma_{0}}$$\displaystyle[\Sigma_{0}]$$\displaystyle[\Sigma]$$\displaystyle
G_{\Sigma_{0}}$$\displaystyle L^{Ram}_{Airy}$$\displaystyle
W^{Ram}_{t_{0}}$$\displaystyle Discs^{Ram}$$\displaystyle
W^{Ram}$$\displaystyle\Phi_{t_{0}}$$\displaystyle[\Sigma_{0}]$$\displaystyle[\Sigma]$$\displaystyle
Discs^{Ram}_{t_{0}}$$[.]$$i$$\gamma$$\displaystyle\mathcal{B}_{\Sigma_{0}}$$\displaystyle\Phi_{\Sigma_{0}}$$\displaystyle[\Sigma_{0}]$$\displaystyle[\Sigma]$$\displaystyle
G$

Figure 1.

Comparison of vector bundles $G\rightarrow\mathcal{B}_{\Sigma_{0}}$,
$\mathcal{H}\rightarrow\mathcal{B}$, and $W^{Ram}\rightarrow Discs^{Ram}$. The
vector bundle morphism $[.]:G\rightarrow\mathcal{H}$ covering the inclusion
map $\mathcal{B}_{\Sigma_{0}}\hookrightarrow\mathcal{B}$ is given fiber-wise
by the natural projection $G_{\Sigma}\rightarrow
G_{\Sigma}/G^{\perp}_{\Sigma}\cong\mathcal{H}_{\Sigma}$, which is the same as
taking a cohomology class. The vector bundle morphism $i:G\rightarrow W^{Ram}$
covering the map $\gamma:\mathcal{B}_{\Sigma_{0}}\hookrightarrow Discs^{Ram}$
is given by taking Laurent expansions around ramification points. In this
sense, the vector bundle $G$ bridges the local point of view given by
$W^{Ram}$ with the global point of view given by $\mathcal{H}$. This Figure
also illustrates the parallel construction of the map
$\boldsymbol{\Phi}_{\Sigma_{0}}:\mathcal{B}_{\Sigma_{0}}\rightarrow\mathcal{H}_{\Sigma_{0}}$,
$\Phi_{\Sigma_{0}}:\mathcal{B}_{\Sigma_{0}}\rightarrow G_{\Sigma_{0}}$, and
$\Phi_{t_{0}}:Discs^{Ram}_{t_{0}}\rightarrow W^{Ram}_{t_{0}}$. Each map embeds
some neighbourhood $\mathcal{B}_{\Sigma_{0}}\ni[\Sigma_{0}]$ or
$Discs^{Ram}_{t_{0}}\ni t_{0}$ into the fiber $\mathcal{H}_{\Sigma_{0}}$,
$G_{\Sigma_{0}}$ or $W^{Ram}_{t_{0}}$ over the reference point by sending the
zero vector at an arbitrary point to the reference point via parallel
transport using the affine connection $\nabla_{GM}+\boldsymbol{\phi}$,
$\nabla_{\mathcal{F}}+\phi$ or $\nabla+\phi$.

####  2.5.7. The embedding of $\mathcal{B}_{\Sigma_{0}}$ into $G_{\Sigma_{0}}$
and the residue constraints

In this section, we will construct 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. The main outcome of this section is Proposition
2.5.38 comparing $\Phi_{\Sigma_{0}}$ to $\Phi_{t_{0}}$ and
$\boldsymbol{\Phi}_{\Sigma_{0}}$, which is the key result to be used in
proving Theorem 2.7.3 later. To construct $\Phi_{\Sigma_{0}}$, we are going to
need $\phi\in\Gamma(\mathcal{B}_{\Sigma_{0}},T^{*}\mathcal{B}\otimes G)$ and
find $\theta\in\Gamma(\mathcal{B}_{\Sigma_{0}},G)$ such that
$\nabla_{\mathcal{F}}\theta=\phi$. Therefore, most of this section leading up
to Proposition 2.5.38 will be allocated to discuss the construction and
properties of $\phi$ and $\theta$. We are also going to assume
$\\{\epsilon_{\alpha\in Ram}\\},\\{M_{\alpha\in Ram}\\}$, $\mathcal{U}_{Ram}$
and $\mathcal{B}_{\Sigma_{0}}$ are chosen such that Condition 2.5.2 is
satisfied.

Let us begin by discussing
$\phi\in\Gamma(\mathcal{B}_{\Sigma_{0}},T^{*}\mathcal{B}\otimes G)$ as
follows. At any point $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$, the map
$\phi_{\Sigma}:T_{[\Sigma]}\mathcal{B}_{\Sigma_{0}}\xrightarrow{\cong}\Gamma(\Sigma,\Omega^{1}_{\Sigma})\hookrightarrow
G_{\Sigma}$ has already been defined previously in Section 2.1, but let us
recall it here. Any given vector $v\in T_{[\Sigma]}\mathcal{B}_{\Sigma_{0}}$
can be mapped into a global holomorphic section of normal bundle
$n_{v}\in\Gamma(\Sigma,\nu_{\Sigma})$. Then
$\phi_{\Sigma}(v):=\Omega_{S}(n_{v},.)|_{\Sigma}\in\Gamma(\Sigma,\Omega^{1}_{\Sigma})$.
Let us cover $\Sigma$ with a collection of $(\mathcal{F},\Omega_{S})$-charts
$\mathcal{U}=\\{(U_{p\in\sigma},x_{p\in\sigma},y_{p\in\sigma})\\}$ then
$\phi_{\Sigma}(v)$ is written locally on each $\Sigma\cap U_{p}$ as

(74) $\phi_{\Sigma}(v)|_{\Sigma\cap
U_{p}}=-\iota_{v}\nabla_{\mathcal{F}}(y_{p}dx_{p})|_{\Sigma\cap U_{p}}.$

Now, let us compare this to
$\boldsymbol{\phi}\in\Gamma(\mathcal{B}_{\Sigma_{0}},T^{*}\mathcal{B}\otimes\mathcal{H})$
from Section 2.1.2 and

$\phi:=\sum_{\alpha\in
Ram}[\alpha]\otimes\phi_{\alpha}\in\Gamma(Discs^{Ram}_{t_{0}},Hom(TDiscs^{Ram}_{t_{0}},W^{Ram})),$

where
$\phi_{\alpha}\in\Gamma(Discs^{\epsilon_{\alpha},M_{\alpha}}_{t_{0\alpha}},Hom(TDiscs^{\epsilon_{\alpha},M_{\alpha}}_{t_{0\alpha}},W^{\epsilon_{\alpha},M_{\alpha}}))$
are as given in Section 2.4. For example, let $t=(t_{\alpha\in Ram})\in
Discs^{Ram}_{t_{0}},t_{\alpha}\in
Discs^{\epsilon_{\alpha},M_{\alpha}}_{t_{0\alpha}}$ and $v=(v_{\alpha\in
Ram})\in T_{t}Discs^{Ram}_{t_{0}},v_{\alpha}\in
T_{t_{\alpha}}Discs^{\epsilon_{\alpha},M_{\alpha}}_{t_{0\alpha}}$, we define
$\phi_{t}(v):=\sum_{\alpha\in
Ram}[\alpha]\otimes\phi_{t_{\alpha}}(v_{\alpha})$. We have the following lemma
compare $\phi_{\Sigma}$ to $\phi_{t(\Sigma)}$ and
$\boldsymbol{\phi}_{\Sigma}$:

######  Lemma 2.5.35.

Let $\phi\in\Gamma(\mathcal{B}_{\Sigma_{0}},T^{*}\mathcal{B}\otimes G)$ be
given as above, then

  1. (1)

$[\phi_{\Sigma}]=\boldsymbol{\phi}_{\Sigma}$

  2. (2)

$i(\phi_{\Sigma})=\phi_{t(\Sigma)}\circ d\gamma$.

We summarize these relationships by the following commutative diagram

${G_{\Sigma}/G^{\perp}_{\Sigma}}$${W^{Ram}_{Airy}}$${\mathcal{H}_{\Sigma}}$${G_{\Sigma}}$${W^{Ram}_{t(\Sigma)}}$${\Gamma(\Sigma,\Omega^{1}_{\Sigma})}$${\Gamma(\Sigma,\Omega^{1}_{\Sigma})}$${T_{0}L^{Ram}_{Airy}}$${T_{[\Sigma]}\mathcal{B}_{\Sigma_{0}}}$${T_{[\Sigma]}\mathcal{B}_{\Sigma_{0}}}$${T_{t(\Sigma)}Discs^{Ram}_{t_{0}}}$$\scriptstyle{\cong}$$\scriptstyle{[.]}$$\scriptstyle{i}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\boldsymbol{\phi}_{\Sigma}}$$\scriptstyle{\cong}$$\scriptstyle{d\gamma}$$\scriptstyle{\phi_{\Sigma}}$$\scriptstyle{\phi_{t(\Sigma)}}$

###### Proof.

  1. (1)

This is obvious since
$\boldsymbol{\phi}_{\Sigma}=\sum_{k=1}^{g}[\Omega_{S}(n_{v_{k}},.)|_{\Sigma}]du^{k}$
by definition.

  2. (2)

Given $t_{\alpha}\in
Discs^{\epsilon_{\alpha},M_{\alpha}}_{t_{0}\alpha},t_{\alpha}=(x_{\alpha}=z^{2}_{\alpha}+a_{\alpha},y_{\alpha}=\sum_{k=0}^{\infty}b_{k\alpha}z_{\alpha}^{k})$
and vector $v_{\alpha}=A_{\alpha}\frac{\partial}{\partial
a_{\alpha}}+\sum_{k=0}^{\infty}B_{k\alpha}\frac{\partial}{\partial
b_{k\alpha}}\in
T_{t_{\alpha}}Discs^{\epsilon_{\alpha},M_{\alpha}}_{t_{0\alpha}}$ we can also
compute $\phi_{\alpha,t_{\alpha}}$ defined in Section 2.4 by keeping
$x_{\alpha}=const$ when taking derivatives and use $\frac{\partial
z_{\alpha}}{\partial a_{\alpha}}|_{x_{\alpha}=const}=-\frac{1}{2z_{\alpha}}$.
We have

$\displaystyle\phi_{\alpha,t_{\alpha}}(v_{\alpha})$
$\displaystyle=\Omega_{S}\left(\left(A_{\alpha}\frac{\partial
y_{\alpha}}{\partial
a_{\alpha}}\Big{|}_{x_{\alpha}=const}+\sum_{k=0}^{\infty}B_{k\alpha}\frac{\partial
y_{\alpha}}{\partial
b_{k\alpha}}\Big{|}_{x_{\alpha}=const}\right)\frac{\partial}{\partial
y_{\alpha}},.\right)\Big{|}_{\mathbb{D}_{t,M_{\alpha}}}$
$\displaystyle=-\left(\sum_{k=0}^{\infty}B_{k\alpha}z^{k}_{\alpha}-\frac{A_{\alpha}}{2z_{\alpha}}\sum_{k=1}^{\infty}kb_{k\alpha}z^{k-1}_{\alpha}\right)dx_{\alpha}\Big{|}_{\mathbb{D}_{t,M_{\alpha}}}=\left(A_{\alpha}\sum_{k=1}^{\infty}kb_{k}z^{k-1}_{\alpha}-2\sum_{k=0}^{\infty}B_{k\alpha}z^{k+1}_{\alpha}\right)dz_{\alpha}$

which should be compared to (2.4.3). Setting
$t_{\alpha}=t_{\alpha}(\Sigma)=\gamma_{\alpha}(\Sigma)$ and

$v_{\alpha}=d\gamma_{\alpha}(v)=\sum_{k=1}^{g}v^{k}\left(\frac{\partial
a_{\alpha}(\Sigma)}{\partial u^{k}}\frac{\partial}{\partial
a_{\alpha}}+\sum_{l=0}^{\infty}\frac{\partial b_{l\alpha}(\Sigma)}{\partial
u^{k}}\frac{\partial}{\partial b_{l\alpha}}\right)$

shows that
$i_{\alpha}\phi_{\Sigma}=\gamma^{*}_{\alpha}\phi_{\alpha}=\phi_{\alpha}\circ
d\gamma_{\alpha}$.

∎

Since $\mathcal{B}_{\Sigma_{0}}$ is chosen to be contractible (C2.5.2.1),
$G\rightarrow\mathcal{B}_{\Sigma_{0}}$ is a flat vector bundle and
$d_{\nabla_{\mathcal{F}}}\phi=0$, therefore Poincaré Lemma suggests the
existence of $\theta\in\Gamma(\mathcal{B}_{\Sigma_{0}},G)$ such that
$\nabla_{\mathcal{F}}\theta=\phi$. $\theta_{\Sigma}$ is a holomorphic
differential form on $\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}(\Sigma)$ for some
$\bar{\epsilon}_{\alpha}<\epsilon_{\alpha}$,
$\oint_{\partial\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}}\theta_{\Sigma}=0$
and any infinitesimal variation of $\theta_{\Sigma}$ in the moduli space
directions produce a global holomorphic form on $\Sigma$, i.e. a holomorphic
form on $\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}(\Sigma)$ which can be
analytically extended to the entire curve $\Sigma$.

######  Lemma 2.5.36.

Let $\theta\in\Gamma(\mathcal{B}_{\Sigma_{0}},G)$ be a section satisfying
$\nabla_{\mathcal{F}}\theta=\phi$, then

  1. (1)

$[\theta_{\Sigma}]=\boldsymbol{\theta}_{\Sigma}+\Gamma_{[\Sigma_{0}]}^{[\Sigma]}[\xi_{0}]\in\mathcal{H}_{\Sigma}$
for some $[\xi_{0}]\in\mathcal{H}_{\Sigma_{0}}$.

  2. (2)

$i(\theta_{\Sigma})=\sum_{\alpha\in
Ram}[\alpha]\otimes\left(\theta_{t_{\alpha}(\Sigma)}+\exp\left((a_{\alpha}(\Sigma)-a_{\alpha}(\Sigma_{0}))\mathcal{L}_{\frac{1}{2z_{\alpha}}\partial_{z_{\alpha}}}\right)(\xi_{0\alpha})\right)\in
W^{Ram}_{t(\Sigma)}$ for some $\xi_{0\alpha}\in
W^{\bar{\epsilon}_{\alpha},\bar{M}_{\alpha}}_{t_{0\alpha}}\subset
W^{\epsilon_{\alpha},M_{\alpha}}_{t_{0\alpha}}$ where

$\bar{\epsilon}_{\alpha}:=\sqrt{\epsilon^{2}_{\alpha}-\sup_{[\Sigma]\in\mathcal{B}_{\Sigma_{0}}}|a_{\alpha}(\Sigma)-a_{\alpha}(\Sigma_{0})|},\qquad\bar{M}_{\alpha}:=\sqrt{M^{2}_{\alpha}+\sup_{[\Sigma]\in\mathcal{B}_{\Sigma_{0}}}|a_{\alpha}(\Sigma)-a_{\alpha}(\Sigma_{0})|}.$

where $\boldsymbol{\theta}_{\Sigma}\in\mathcal{H}_{\Sigma}$ is given by (17)
and $\theta_{t_{\alpha}(\Sigma)}\in
W^{\epsilon_{\alpha},M_{\alpha}}_{t_{\alpha}(\Sigma)}$ is given by (58).

###### Proof.

  1. (1)

By taking cohomology class, apply Proposition 2.5.34 and Lemma 2.5.35, we find
that
$\nabla_{GM}[\theta_{\Sigma}]=[\nabla_{\mathcal{F}}\theta_{\Sigma}]=[\phi_{\Sigma}]=\boldsymbol{\phi}_{\Sigma}$.
Therefore we have $[\theta]=\boldsymbol{\theta}+\boldsymbol{c}$ for some
parallel section
$\boldsymbol{c}\in\Gamma(\mathcal{B}_{\Sigma_{0}},\mathcal{H})$,
$\nabla_{GM}\boldsymbol{c}=0$. But then we we can write
$\boldsymbol{c}_{\Sigma}=\Gamma_{[\Sigma_{0}]}^{[\Sigma]}[\xi_{0}]$ where
$[\xi_{0}]:=\boldsymbol{c}_{\Sigma_{0}}\in\mathcal{H}_{\Sigma_{0}}$.

  2. (2)

By applying the map $i:G_{\Sigma}\rightarrow W^{Ram}_{t(\Sigma)}$ follows by
Proposition 2.5.34 and Lemma 2.5.35 we have $\gamma^{*}\nabla
i(\theta_{\Sigma})=i(\nabla_{\mathcal{F}}\theta_{\Sigma})=i(\phi_{\Sigma})=\phi_{t(\Sigma)}\circ
d\gamma$. Therefore, we have $i(\theta_{\Sigma})=\sum_{\alpha\in
Ram}[\alpha]\otimes\theta_{t_{\alpha}(\Sigma)}+c$ for some parallel section
$c\in\Gamma(\mathcal{B}_{\Sigma_{0}},\gamma^{*}W^{Ram})$, $\gamma^{*}\nabla
c=0$. It is not hard to check that

$c_{t(\Sigma)}=\sum_{\alpha\in
Ram}[\alpha]\otimes\exp\left((a_{\alpha}(\Sigma)-a_{\alpha}(\Sigma_{0}))\mathcal{L}_{\frac{1}{2z_{\alpha}}\partial_{z_{\alpha}}}\right)(\xi_{0\alpha})$

where $\sum_{\alpha\in Ram}[\alpha]\otimes\xi_{0\alpha}=c_{t_{0}}\in
W^{Ram}_{t_{0}}$.

∎

Let us show explicitly how $\theta\in\Gamma(\mathcal{B}_{\Sigma_{0}},G)$ such
that $\nabla_{\mathcal{F}}\theta=\phi$ can be constructed. We choose a
collection of $(\mathcal{F},\Omega_{S})$-charts
$\mathcal{U}_{0}=\\{(U_{0,p\in\sigma_{0}},x_{p\in\sigma_{0}},y_{p\in\sigma_{0}})\\}$
which allows the parallel transport of any holomorphic form
$\xi\in\Gamma(\Sigma_{0},\Omega^{1}_{\Sigma_{0}})\subset G_{\Sigma_{0}}$ to
$G_{\Sigma}$ for any $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ according to Lemma
2.5.32. Such a collection $\mathcal{U}_{0}$ exists due to C2.5.2.3. From (74)
we know that $\theta_{\Sigma}|_{\Sigma\cap U_{0,p}}$ must take the form
$-(y_{p}+f_{p}(x_{p}))dx_{p}|_{\Sigma\cap U_{0,p}}$ for us to have
$\nabla_{\mathcal{F}}\theta_{\Sigma}|_{\Sigma\cap
U_{0,p}}=\phi_{\Sigma}|_{\Sigma\cap U_{0,p}}$, where $f_{p}(x_{p})dx_{p}$ is
an integration constant. Write
$y_{p}(x_{p};\Sigma)dx_{p}:=y_{p}dx_{p}|_{\Sigma\cap U_{0,p}}$ for each
$p\in\sigma$. In other words, $\Sigma\cap
U_{0,p}=\\{(x_{p},y_{p}(x_{p};\Sigma))\in U_{p}\subset S\ |\ x_{p}\in
x_{p}(\Sigma\cap U_{0,p})\\}$ is the graph of $y_{p}=y_{p}(x_{p};\Sigma)$.
Then, on each open subset $\Sigma\cap U_{0,p}$, let us define

$\theta_{\Sigma}|_{\Sigma\cap
U_{0,p}}:=-y_{p}(x_{p};\Sigma)dx_{p}+y_{p}(x_{p};\Sigma_{0})dx_{p}.$

If each $\theta_{\Sigma}|_{\Sigma\cap U_{0,p}}$ can be patched together to get
$\theta_{\Sigma}\in G_{\Sigma}$, then it will automatically follows that
$\nabla_{\mathcal{F}}\theta_{\Sigma}=\phi_{\Sigma}$ as this relation holds on
every $\Sigma\cap U_{0,p}$ and we are done. The following guarantees that this
is the case:

######  Lemma 2.5.37.

Each $\theta_{\Sigma}|_{\Sigma\cap U_{0,p}},p\in\sigma_{0}$ can be patched
together as $\theta_{\Sigma}\in G_{\Sigma}$.

###### Proof.

First, let us show that each $\theta_{\Sigma}|_{\Sigma\cap
U_{0,p}},p\in\sigma_{0}$ can be patched together to form a global holomorphic
form $\theta_{\Sigma}$ on $\Sigma\setminus\cup_{\alpha\in
Ram}\bar{\mathbb{D}}_{\alpha,\bar{\epsilon}_{\alpha}}(\Sigma)$ for some
$\bar{\epsilon}_{\alpha}<\epsilon_{\alpha}$. Using the coordinates
transformation on $U_{p,0}\cap U_{0,p^{\prime}}$ between $(x_{p},y_{p})$ and
$(x_{p^{\prime}},y_{p^{\prime}})$ as given in (7), we find that
$x_{p^{\prime}}=F_{p^{\prime}}(x_{p})$ and

$y_{p^{\prime}}(x_{p^{\prime}};\Sigma)=\frac{y_{p}(F^{-1}_{p^{\prime}}(x_{p^{\prime}});\Sigma)}{F^{\prime}_{p^{\prime}}(F^{-1}(x_{p^{\prime}}))}-G^{\prime}_{p^{\prime}}(F^{-1}(x_{p^{\prime}})),\qquad
y_{p^{\prime}}(x_{p^{\prime}};\Sigma_{0})=\frac{y_{p}(F^{-1}_{p^{\prime}}(x_{p^{\prime}});\Sigma_{0})}{F^{\prime}_{p^{\prime}}(F^{-1}(x_{p^{\prime}}))}-G^{\prime}_{p^{\prime}}(F^{-1}(x_{p^{\prime}})).$

This means

$\displaystyle\left(\theta_{\Sigma}|_{\Sigma\cap U_{0,p^{\prime}}}\right)$
$\displaystyle|_{\Sigma\cap U_{0,p}\cap
U_{0,p^{\prime}}}=-y_{p^{\prime}}(x_{p^{\prime}};\Sigma)dx_{p^{\prime}}+y_{p^{\prime}}(x_{p^{\prime}};\Sigma_{0})dx_{p^{\prime}}$
$\displaystyle=-\left(\frac{y_{p}(F^{-1}_{p^{\prime}}(x_{p^{\prime}});\Sigma)}{F^{\prime}_{p^{\prime}}(F^{-1}(x_{p^{\prime}}))}-G^{\prime}_{p^{\prime}}(F^{-1}(x_{p^{\prime}}))\right)dx_{p^{\prime}}+\left(\frac{y_{p}(F^{-1}_{p^{\prime}}(x_{p^{\prime}});\Sigma_{0})}{F^{\prime}_{p^{\prime}}(F^{-1}(x_{p^{\prime}}))}-G^{\prime}_{p^{\prime}}(F^{-1}(x_{p^{\prime}}))\right)dx_{p^{\prime}}$
$\displaystyle=-\left(\frac{y_{p}(x_{p};\Sigma)}{F^{\prime}_{p^{\prime}}(x_{p})}-G^{\prime}_{p^{\prime}}(x_{p})\right)dF(x_{p})+\left(\frac{y_{p}(x_{p};\Sigma_{0})}{F^{\prime}_{p^{\prime}}(x_{p})}-G^{\prime}_{p^{\prime}}(x_{p})\right)dF(x_{p})$
$\displaystyle=-y_{p}(x_{p};\Sigma)dx_{p}+y_{p}(x_{p};\Sigma_{0})dx_{p}$
$\displaystyle=\left(\theta_{\Sigma}|_{\Sigma\cap U_{0,p}}\right)|_{\Sigma\cap
U_{0,p}\cap U_{0,p^{\prime}}}.$

Therefore, we can patch together each $\theta_{\Sigma}|_{\Sigma\cap U_{0,p}}$
for $p\in\sigma_{0}$ to get a well-defined holomorphic form $\theta_{\Sigma}$
on $\cup_{p\in\sigma_{0}}\Sigma\cap
U_{0,p}\supset\Sigma\setminus\cup_{\alpha\in
Ram}\mathbb{D}_{\alpha,\epsilon_{\alpha}}(\Sigma)$.

To finish showing that $\theta_{\Sigma}\in G_{\Sigma}$ we need to check that
$\oint_{\partial\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}}\theta_{\Sigma}=0$.
Since $\theta_{\Sigma}$ we have so far is well-defined on the annulus
$\mathbb{A}_{\alpha,\epsilon_{\alpha},M_{\alpha}}(\Sigma)$, it can be written
in the standard local coordinate $z_{\alpha}$ of $\Sigma\cap U_{\alpha}$ as

$\displaystyle i_{\alpha}\theta_{\Sigma}(z_{\alpha})$
$\displaystyle=-y_{\alpha}(z_{\alpha};\Sigma)dx_{\alpha}(z_{\alpha})+y_{\alpha}(z_{\alpha,\Sigma_{0}};\Sigma_{0})dx_{\alpha}(z_{\alpha,\Sigma_{0}})$
$\displaystyle=\theta_{t_{\alpha}(\Sigma)}(z_{\alpha})-\exp\left((a_{\alpha}(\Sigma)-a_{\alpha}(\Sigma_{0}))\mathcal{L}_{\frac{1}{2z_{\alpha}}\partial_{z_{\alpha}}}\right)\theta_{t_{\alpha}(\Sigma_{0})}(z_{\alpha}).$

For clarity, we have denoted by $z_{\alpha,\Sigma_{0}}$ the standard local
coordinate on $\Sigma_{0}\cap U_{\alpha}$ which is related to $z_{\alpha}$ by
$a_{\alpha}(\Sigma)+z_{\alpha}^{2}=x_{\alpha}=a_{\alpha}(\Sigma_{0})+z_{\alpha,\Sigma_{0}}^{2}$.
Let us give more explanation to the second equality. By (58) we have
$\theta_{t_{\alpha}(\Sigma)}=-y_{\alpha}(z_{\alpha},\Sigma)dx_{\alpha}(z_{\alpha})\in
W^{\epsilon_{\alpha},M_{\alpha}}_{t_{\alpha}(\Sigma)}$ and
$\theta_{t_{\alpha}(\Sigma_{0})}=-y_{\alpha}(z_{\alpha,\Sigma_{0}};\Sigma_{0})dx_{\alpha}(z_{\alpha,\Sigma_{0}})\in
W^{\epsilon_{\alpha},M_{\alpha}}_{t_{\alpha}(\Sigma_{0})}$. In particular,

$y_{\alpha}(z_{\alpha,\Sigma_{0}};\Sigma_{0})dx_{\alpha}(z_{\alpha,\Sigma_{0}})=\exp\left((a_{\alpha}(\Sigma)-a_{\alpha}(\Sigma_{0}))\mathcal{L}_{\frac{1}{2z_{\alpha,\Sigma_{0}}}\partial_{z_{\alpha,\Sigma_{0}}}}\right)\left(y_{\alpha}(z_{\alpha,\Sigma_{0}};\Sigma_{0})dx_{\alpha}(z_{\alpha,\Sigma_{0}})\right)\Big{|}_{z_{\alpha,\Sigma_{0}}\mapsto
z_{\alpha}}.$

Clearly, $i_{\alpha}\theta_{\Sigma}$ contains no
$\frac{dz_{\alpha}}{z_{\alpha}}$ term according to Lemma 2.4.9 and so
$\oint_{\partial\bar{\mathbb{D}}_{\alpha,\epsilon_{\alpha}}}\theta_{\Sigma}=0$.
∎

Let us embed the neighbourhood $\mathcal{B}_{\Sigma_{0}}$ of
$[\Sigma_{0}]\in\mathcal{B}$ into the fiber $G_{\Sigma_{0}}$, using the same
procedure we did to embed $\mathcal{B}_{\Sigma_{0}}$ into
$\mathcal{H}_{\Sigma_{0}}$ in Section 2.1 and $Discs^{\epsilon,M}_{t_{0}}$
into $W^{\epsilon,M}_{t_{0}}$ in Section 2.4 but this time with the vector
bundle $(G\rightarrow\mathcal{B}_{\Sigma_{0}},\nabla_{\mathcal{F}})$. The
embedding map $\Phi_{\Sigma_{0}}:\mathcal{B}_{\Sigma_{0}}\rightarrow
G_{\Sigma_{0}}$ is given by sending the zero vector $0_{\Sigma}\in G_{\Sigma}$
at $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$ into $G_{\Sigma_{0}}$ at
$[\Sigma_{0}]\in\mathcal{B}_{\Sigma_{0}}$ via parallel transport using the
connection $\nabla_{\mathcal{F}}+\phi$. Since
$\nabla_{\mathcal{F}}\theta=\phi$, the parallel transport of $0_{\Sigma}$ from
$[\Sigma]\in\mathcal{B}_{\Sigma^{\prime}}$ to
$[\Sigma^{\prime}]\in\mathcal{B}$ using $\nabla_{\mathcal{F}}+\phi$ is path-
independent and it is uniquely given by

$v_{\Sigma}(\Sigma^{\prime})=-\theta_{\Sigma^{\prime}}+s^{*}_{\Sigma^{\prime},\Sigma}\theta_{\Sigma}.$

Therefore,

$\Phi_{\Sigma_{0}}(\Sigma):=v_{\Sigma}(\Sigma_{0})=-\theta_{\Sigma_{0}}+s^{*}_{\Sigma_{0},\Sigma}\theta_{\Sigma}\in
G_{\Sigma_{0}},$

where the existence of parallel transport
$s^{*}_{\Sigma_{0},\Sigma}\theta_{\Sigma}\in G_{\Sigma_{0}}$ is due to
C2.5.2.4 as $\theta_{\Sigma}$ is holomorphic on
$\cup_{p\in\sigma_{0}}\Sigma\cap U_{0,p}\supset\Sigma\setminus\cup_{\alpha\in
Ram}\mathbb{D}_{\alpha,\epsilon_{\alpha}}(\Sigma)$.

Given $t_{0}=(t_{0\alpha})\in Discs^{Ram}$, let us define the $Ram$ product
version of $\Phi_{t_{0}}:Discs^{Ram}_{t_{0}}\rightarrow W^{Ram}_{t_{0}}$ by
$\Phi_{t_{0}}(t):=\sum_{\alpha\in
Ram}[\alpha]\otimes\Phi_{\alpha,t_{0\alpha}}(t_{\alpha})$ for any
$t=(t_{\alpha})\in Discs^{Ram}_{t_{0}}$ where
$\Phi_{\alpha}:Discs^{\epsilon_{\alpha},M_{\alpha}}_{t_{0\alpha}}\rightarrow
W^{\epsilon_{\alpha},M_{\alpha}}_{t_{0\alpha}}$ is as given in Section 2.4:

$\Phi_{\alpha,t_{0\alpha}}(t_{\alpha}):=-\theta_{t_{0\alpha}}+\exp((a_{\alpha}(t_{0\alpha})-a_{\alpha}(t_{\alpha}))\mathcal{L}_{\frac{1}{2z_{\alpha}}\partial_{z_{\alpha}}})\theta_{t_{\alpha}}\in
W^{\epsilon_{\alpha},M_{\alpha}}_{t_{0\alpha}}.$

Finally, we have the following proposition:

######  Proposition 2.5.38.

([21, Proposition 7.1.2]) The embedding
$\Phi_{\Sigma_{0}}:\mathcal{B}_{\Sigma_{0}}\rightarrow G_{\Sigma_{0}}$
satisfies $[\Phi_{\Sigma_{0}}(\Sigma)]=\boldsymbol{\Phi}_{\Sigma_{0}}(\Sigma)$
and
$i(\Phi_{\Sigma_{0}}(\Sigma))=\Phi_{t_{0}}\circ\gamma(\Sigma)=\Phi_{t_{0}}(t(\Sigma))$,
so $(\mathbf{a})$ and $(\mathbf{b})$ squares in the diagram are commutative.
Moreover, if each $(\mathcal{F},\Omega_{S})$-charts
$(U_{\alpha},x_{\alpha},y_{\alpha})$ in $\mathcal{U}_{Ram}$ are chosen such
that
$\gamma_{\alpha}(\Sigma_{0})=(x_{\alpha}=z_{\alpha}^{2},y_{\alpha}=z_{\alpha})$,
then $\Phi_{\Sigma_{0}}$ factors through $L^{Ram}_{Airy}$ and the triangle
$(\mathbf{c})$ is also commutative.

${G_{\Sigma_{0}}/G^{\perp}_{\Sigma_{0}}}$${W^{Ram}_{Airy}}$${\mathcal{H}_{\Sigma_{0}}}$${G_{\Sigma_{0}}}$${W^{Ram}_{t_{0}}}$${\mathcal{L}_{\Sigma_{0}}}$${(\mathbf{a})}$${(\mathbf{b})}$${(\mathbf{c})}$${L^{Ram}_{Airy}}$${\mathcal{B}_{\Sigma_{0}}}$${\mathcal{B}_{\Sigma_{0}}}$${Discs^{Ram}_{t_{0}}}$$\scriptstyle{\cong}$$\scriptstyle{[.]}$$\scriptstyle{i}$$\scriptstyle{\cong}$$\scriptstyle{\boldsymbol{\Phi}_{\Sigma_{0}}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\Phi_{\Sigma_{0}}}$$\scriptstyle{\gamma}$$\scriptstyle{\Phi_{t_{0}}}$

We note that the left-most triangle is merely a restatement of the definition
$\mathcal{L}_{\Sigma_{0}}:=\text{im}\boldsymbol{\Phi}_{\Sigma_{0}}\subset\mathcal{H}_{\Sigma_{0}}$
as given in Section 2.1.3 and we include it here for completeness.

###### Proof.

The proof is mostly an application of Proposition 2.5.34 and Lemma 2.5.36.
From Lemma 2.5.36 we know that
$[\theta_{\Sigma}]=\boldsymbol{\theta}_{\Sigma}+\Gamma^{[\Sigma]}_{[\Sigma_{0}]}[\xi_{0}]$
for some $[\xi_{0}]\in\mathcal{H}_{\Sigma_{0}}$ and

$i_{\alpha}(\theta_{\Sigma})=\theta_{t_{\alpha}(\Sigma)}+\exp\left((a_{\alpha}(\Sigma)-a_{\alpha}(\Sigma_{0}))\mathcal{L}_{\frac{1}{2z_{\alpha}}\partial_{z_{\alpha}}}\right)(\xi_{0\alpha})$

for some $\xi_{0\alpha}\in W^{\epsilon_{\alpha},M_{\alpha}}_{t_{0\alpha}}$.
Therefore, we have

$\displaystyle[\Phi_{\Sigma_{0}}(\Sigma)]$
$\displaystyle=-[\theta_{\Sigma_{0}}]+[s^{*}_{\Sigma_{0},\Sigma}\theta_{\Sigma}]=-[\theta_{\Sigma_{0}}]+\Gamma^{[\Sigma_{0}]}_{[\Sigma]}[\theta_{\Sigma}]$
$\displaystyle=-(\boldsymbol{\theta}_{\Sigma_{0}}+[\xi_{0}])+\Gamma^{[\Sigma_{0}]}_{[\Sigma]}(\boldsymbol{\theta}_{\Sigma}+\Gamma_{[\Sigma_{0}]}^{[\Sigma]}[\xi_{0}])=-\boldsymbol{\theta}_{\Sigma_{0}}-[\xi_{0}]+\Gamma_{[\Sigma]}^{[\Sigma_{0}]}\boldsymbol{\theta}_{\Sigma}+[\xi_{0}]=-\boldsymbol{\theta}_{\Sigma_{0}}+\Gamma_{[\Sigma]}^{[\Sigma_{0}]}\boldsymbol{\theta}_{\Sigma}=\boldsymbol{\Phi}_{\Sigma_{0}}(\Sigma)$

and so $(\mathbf{a})$ is commutative. Similarly,

$\displaystyle i(\Phi_{\Sigma_{0}}(\Sigma))$
$\displaystyle=-i(\theta_{\Sigma_{0}})+i(s^{*}_{\Sigma_{0},\Sigma}\theta_{\Sigma})$
$\displaystyle=-i(\theta_{\Sigma_{0}})+\sum_{\alpha\in
Ram}[\alpha]\otimes\exp\left((a_{\alpha}(\Sigma_{0})-a_{\alpha}(\Sigma))\mathcal{L}_{\frac{1}{2z_{\alpha}}\partial_{z_{\alpha}}}\right)i_{\alpha}(\theta_{\Sigma})$
$\displaystyle=-\sum_{\alpha\in
Ram}[\alpha]\otimes\left(\theta_{t_{\alpha}(\Sigma_{0})}+\xi_{0\alpha}\right)$
$\displaystyle\qquad+\sum_{\alpha\in
Ram}[\alpha]\otimes\exp\left((a_{\alpha}(\Sigma_{0})-a_{\alpha}(\Sigma))\mathcal{L}_{\frac{1}{2z_{\alpha}}\partial_{z_{\alpha}}}\right)\left(\theta_{t_{\alpha}(\Sigma)}+\exp\left((a_{\alpha}(\Sigma)-a_{\alpha}(\Sigma_{0}))\mathcal{L}_{\frac{1}{2z_{\alpha}}\partial_{z_{\alpha}}}\right)(\xi_{0\alpha})\right)$
$\displaystyle=\sum_{\alpha\in
Ram}[\alpha]\otimes\left(-\theta_{t_{\alpha}(\Sigma_{0})}-\xi_{0\alpha}+\exp\left((a_{\alpha}(\Sigma_{0})-a_{\alpha}(\Sigma))\mathcal{L}_{\frac{1}{2z_{\alpha}}\partial_{z_{\alpha}}}\right)\theta_{t_{\alpha}(\Sigma)}+\xi_{0\alpha}\right)$
$\displaystyle=\sum_{\alpha\in
Ram}[\alpha]\otimes\left(-\theta_{t_{\alpha}(\Sigma_{0})}+\exp\left((a_{\alpha}(\Sigma_{0})-a_{\alpha}(\Sigma))\mathcal{L}_{\frac{1}{2z_{\alpha}}\partial_{z_{\alpha}}}\right)\theta_{t_{\alpha}(\Sigma)}\right)$
$\displaystyle=\sum_{\alpha\in
Ram}[\alpha]\otimes\Phi_{\alpha,t_{\alpha}(\Sigma_{0})}(t_{\alpha}(\Sigma))=\Phi_{t_{0}}(t(\Sigma))=\Phi_{t_{0}}\circ\gamma(\Sigma)$

and so $(\mathbf{b})$ is commutative. Lastly, if
$\gamma_{\alpha}(\Sigma_{0})=(x_{\alpha}=z^{2}_{\alpha},y_{\alpha}=z_{\alpha})$
then it follows from Proposition 2.4.24 that
$\text{im}\Phi_{\alpha,t_{0\alpha}}\subseteq L^{M_{\alpha}}_{Airy}$ and
therefore $\text{im}\Phi_{t_{0}}\subseteq L^{Ram}_{Airy}$ which shows that
$(\mathbf{c})$ is commutative. ∎

###  2.6. From Airy Structures to Topological Recursion

As we have mentioned earlier in Section 2.3, the main reason for our interest
in the residue constraints Airy structure is due to its connection to
topological recursion. This connection is well-known and has been studied in
various places such as [4, 1, 21], however we include this review here for
completeness.

####  2.6.1. Review of Topological Recursion

We recall the following basic facts and setup of the original Eynard-Orantin
topological recursion [12, 13]. A _spectral curve_ $(\Sigma_{0},x,y,B)$ is
given by a smooth compact Riemann surface $\Sigma_{0}$, global functions
$x,y:\Sigma_{0}\rightarrow\mathbb{P}^{1}$ and a choice of Bergman kernel
$B=B(p,q)$ on $\Sigma_{0}\times\Sigma_{0}$. We assume that $dx(p)$ has simple
zeros at a finite number of points $p\in\\{r_{\alpha\in Ram}\in\Sigma_{0}\\}$
called _ramification points_ and that $y\sim
y(r_{\alpha})+C\sqrt{x-x(r_{\alpha})}$ near each $r_{\alpha}$ (it follows that
$dy(r_{\alpha})\neq 0$). We equip to each ramification point the _involution_
map $\sigma_{\alpha}$ defined locally in some neighbourhood of each
$r_{\alpha}$ satisfying $\sigma_{\alpha}(r_{\alpha})=r_{\alpha}$,
$x\circ\sigma_{\alpha}=x$ and
$d\sigma_{\alpha}|_{r_{\alpha}}=-id:T_{r_{\alpha}}\Sigma_{0}\rightarrow
T_{r_{\alpha}}\Sigma_{0}$. We set the initial condition of the recursion to be

$\omega_{0,1}(p):=ydx,\qquad\omega_{0,2}(p,q)=B(p,q).$

For $g\geq 0,n\geq 3$ or $g\geq 1,n\geq 1$ we compute the multi-differential
$\omega_{g,n}\in\Gamma\left((\Sigma_{0}\setminus\cup_{\alpha\in
Ram}r_{\alpha})^{n},(\Omega_{\Sigma_{0}}^{1})^{\boxtimes n}\right)$ using the
following recursion formula

$\displaystyle\omega_{g,n}(p_{1},...,p_{n})$ $\displaystyle=\sum_{\alpha\in
Ram}Res_{p=r_{\alpha}}K(p_{1},p)\omega_{g-1,n+1}(p,\sigma_{\alpha}(p),p_{2},...,p_{n})$
(75) $\displaystyle\qquad+\sum_{\alpha\in
Ram}\sum^{*}_{\begin{subarray}{c}g_{1}+g_{2}=g\\\ I_{1}\coprod
I_{2}=\\{2,...,n\\}\end{subarray}}Res_{p=r_{\alpha}}K(p_{1},p)\omega_{g_{1},1+|I_{1}|}(p,p_{I_{1}})\omega_{g_{2},1+|I_{2}|}(\sigma_{\alpha}(p),p_{I_{2}})$

where $\sum^{*}$ indicates that we exclude all terms involving $\omega_{0,1}$
from the summation. We also define the _recursion kernel_ $K=K(p_{1},p)$ for
$p_{1}\in\Sigma$ and $p$ in the vicinity of a ramification point by

$K(p_{1},p):=-\frac{1}{2}\frac{\int_{p^{\prime}=\sigma_{\alpha}(p)}^{p^{\prime}=p}\omega_{0,2}(p_{1},p^{\prime})}{\omega_{0,1}(p)-\omega_{0,1}(\sigma_{\alpha}(p))}.$

We note that $K(p_{1},p)$ is a global meromorphic differential on $\Sigma_{0}$
in $p_{1}$ with simple poles at $p_{1}=p,\sigma_{\alpha}(p)$. On the other
hand, $K(p_{1},p)$ is only defined as an inverse of differential locally for
$p$ close to $r_{\alpha}$ with simple poles at $p=r_{\alpha}$ for any
$\alpha\in Ram$. The formula (2.6.1) expresses $\omega_{g,n}$ only in terms of
$\omega_{g^{\prime},n^{\prime}}$ for $2g^{\prime}-2+n^{\prime}<2g-2+n$ and
therefore the recursion is guaranteed to terminate.

######  Remark 2.6.1.

It is known that $\omega_{g,n}(p_{1},\cdots,p_{n})$ is a meromorphic
differential on $\Sigma_{0}$ symmetric in each of its variable $p_{i}$ with
poles only at ramification points of $\Sigma_{0}$ [12, 13] and zero residues
$\oint_{p_{i}=r_{\alpha}}\omega_{g,n}(p_{1},\cdots,p_{n})=0$ [12, Corollary
4.1]. Moreover, if we have chosen $B=B(p,q)$ to be the normalized Bergman
kernel, then $\oint_{p_{i}\in A_{k}}\omega_{g,n}(p_{1},\cdots,p_{n})=0$ [12,
Theorem 4.3].

We observe that we have only used local information of $\omega_{0,1}$ in the
vicinity of each $r_{\alpha}$. This suggests a slight generalization of the
original version of topological recursion. We consider a smooth compact
Riemann surface $\Sigma_{0}$ equipped with a choice of Bergman kernel
$B=B(p,q)$ and a set of distinct points $\\{r_{\alpha\in Ram}\\}$ indexed by
the set $Ram$. We let $x$ and $y$ be locally defined functions on some open
neighbourhood $D_{\alpha}$ of each $r_{\alpha}\in Ram$ such that
$dx(r_{\alpha})=0$ and $\tau_{\alpha}:=\sqrt{x-x(r_{\alpha})}$ is a local
coordinate on $D_{\alpha}$ of $r_{\alpha}$. Then in general, we have

$x|_{D_{\alpha}}=\tau_{\alpha}^{2}+a_{\alpha},\qquad
y|_{D_{\alpha}}=b_{0\alpha}+b_{1\alpha}\tau_{\alpha}+b_{2\alpha}\tau_{\alpha}^{2}+\cdots$

and the involution is given by
$\sigma_{\alpha}(\tau_{\alpha})=-\tau_{\alpha}$. As before, we set the initial
condition of the recursion to be $\omega_{0,1}(p)=ydx$ and
$\omega_{0,2}(p,q)=B(p,q)$, notice that $\omega_{0,1}$ is now only a locally
defined differential form in the vicinity of each $r_{\alpha}$. The rest of
the recursion is the same as above. This variance of topological recursion
sometimes called _local topological recursion_ [9] and it is the version we
are going to be interested in.

####  2.6.2. Relations to Airy structures

Let us introduce the following pre-Airy structure
$\\{(H_{TR})_{i=1,2,3,\cdots,\alpha\in Ram}\\}$ on $V^{|Ram|}_{Airy}$ which is
a slight modification of $\\{(H_{Airy})_{n,\alpha}\\}$ given in (2.5.4):

$\displaystyle(H_{TR})_{2n,\alpha}(w)$
$\displaystyle=Res_{z=0}\left(\left(z-\frac{w(z)}{2zdz}\right)[\alpha]\otimes
z^{2n-2}d(z^{2})\right),$ (76) $\displaystyle(H_{TR})_{2n-1,\alpha}(w)$
$\displaystyle=\frac{1}{2}Res_{z=0}\left(\left(z-\frac{w(z)}{2zdz}\right)\left(z+\frac{w(-z)}{2zdz}\right)[\alpha]\otimes
z^{2n-2}d(z^{2})\right)$

for $n\geq 1$ where $w:=\sum_{\alpha\in
Ram}\sum_{k=1}^{\infty}(x^{k,\alpha}f_{k,\alpha}+y_{k,\alpha}e^{k,\alpha})\in
W^{|Ram|}_{Airy}$. We note the only change from $\\{(H_{n,\alpha})_{Airy}\\}$
is that $w(z)$ in one of the factor of $(H_{TR})_{2n-1,\alpha}$ has been
replaced by $-w(-z)$. Expanding (2.6.2) we get

(77)
$(H_{TR})_{i,\alpha}=-y_{i,\alpha}+(a_{TR})_{(i,\alpha)(j,\beta)(k,\gamma)}x^{j,\beta}x^{k,\gamma}+2(b_{TR})_{(i,\alpha)(j,\beta)}^{k,\gamma}x^{j,\beta}y_{k,\gamma}+(c_{TR})_{i,\alpha}^{(j,\beta)(k,\gamma)}y_{j,\beta}y_{k,\gamma}$

where

$\displaystyle(a_{TR})_{(i,\alpha)(j,\beta)(k,\gamma)}$
$\displaystyle=Res_{z=0}\left(-\frac{1}{4i}\frac{1}{z(dz)^{2}}f_{i,\alpha}(z)f_{j,\beta}(z)f_{k,\gamma}(-z)\right)\delta_{i,odd},\qquad
A_{TR}:=((a_{TR})_{(i,\alpha)(j,\beta)(k,\gamma)})$
$\displaystyle(b_{TR})_{(i,\alpha)(j,\beta)}^{k,\gamma}$
$\displaystyle=Res_{z=0}\left(-\frac{1}{4i}\frac{1}{z(dz)^{2}}f_{i,\alpha}(z)f_{j,\beta}(z)e^{k,\gamma}(-z)\right)\delta_{i,odd},\qquad
B_{TR}:=((b_{TR})_{(i,\alpha)(j,\beta)}^{k,\gamma})$
$\displaystyle(c_{TR})_{i,\alpha}^{(j,\beta)(k,\gamma)}$
$\displaystyle=Res_{z=0}\left(-\frac{1}{4i}\frac{1}{z(dz)^{2}}f_{i,\alpha}(z)e^{j,\beta}(z)e^{k,\gamma}(-z)\right)\delta_{i,odd},\qquad
C_{TR}:=((c_{TR})_{i,\alpha}^{(j,\beta)(k,\gamma)}).$

The expression (77) should be compared to (2.3). Additionally, we define
$(\epsilon_{TR})_{i,\alpha}=\frac{1}{16}\delta_{i,3},\epsilon_{TR}=((\epsilon_{TR})_{i,\alpha})=\epsilon_{Airy}$.
The key point is that: $(V_{Airy}^{|Ram|},A_{TR},B_{TR},C_{TR},\epsilon_{TR})$
is a quantum pre-Airy structure. Additionally, since $-w(-z)=w(z)$ if
$x^{i=even,\alpha\in Ram}=y_{i=even,\alpha\in Ram}=0$, it is clear from
(2.6.2) and Remark 2.5.24 that both
$(V^{|Ram|}_{Airy},A_{TR},B_{TR},C_{TR},\epsilon_{TR})$ and
$(V^{|Ram|}_{Airy},A_{Airy},B_{Airy},C_{Airy},\epsilon_{Airy})$ contain the
same quantum Airy sub-structure
$((V_{Airy})^{|Ram|}_{odd},A_{Airy},B_{Airy},C_{Airy},\epsilon_{Airy})$.

We will now present the main result of this section.

######  Proposition 2.6.2.

([4, Theorem G],[1, Lemma 9.1],[21, Section 3.1]) Consider a smooth compact
Riemann surface $\Sigma_{0}$ with a set of distinct points $\\{r_{\alpha\in
Ram}\\}$ and a neighbourhood $D_{\alpha}\ni r_{\alpha}$ for each $\alpha\in
Ram$ with local coordinate $z_{\alpha}$. Let
$\left\\{\omega_{g,n}\in\Gamma\left((\Sigma_{0}\setminus\cup_{\alpha\in
Ram}r_{\alpha})^{n},(\Omega_{\Sigma_{0}}^{1})^{\boxtimes n}\right)\right\\}$
be the multi-differentials produced from topological recursion on $\Sigma_{0}$
using:

  1. (1)

The involution $\sigma_{\alpha}(z_{\alpha}):=-z_{\alpha}$.

  2. (2)

$\omega_{0,1}(p)$ locally defined on each $D_{\alpha}$ for $\alpha\in Ram$
such that
$\omega_{0,1}(z_{\alpha})-\omega_{0,1}(\sigma_{\alpha}(z_{\alpha}))=4z_{\alpha}^{2}dz_{\alpha}$.

  3. (3)

$\omega_{0,2}(p,q)=B(p,q)$ the normalized Bergman kernel.

Suppose that $V_{\Sigma_{0}}^{mer}\subset W^{|Ram|}_{Airy}$ is a Lagrangian
complement of $T_{0}L_{Airy}^{|Ram|}$ with basis
$\\{\eta^{i=1,2,3,\cdots}:=\sum_{\beta\in
Ram}\sum_{j=1}^{\infty}d^{i}_{j,\beta}\bar{e}^{j,\beta}\\}$, for some
$(d^{i}_{j,\beta}):V^{|Ram|}_{Airy}\rightarrow V^{|Ram|}_{Airy}$ and
$\\{\bar{e}^{k=1,2,3,\cdots,\alpha\in
Ram}=e^{k,\alpha}+\sum_{j,\beta}s^{(k,\alpha)(j,\beta)}f_{j,\beta}\\}$ are
given by Definition 2.5.11. Let
$(V^{mer}_{\Sigma_{0}},\bar{A}_{Airy},\bar{B}_{Airy},\bar{C}_{Airy},\bar{\epsilon}_{Airy})$
be the quantum Airy structure obtained from the gauge transformation of the
quantum Airy structure
$(V^{|Ram|}_{Airy},A_{Airy},B_{Airy},C_{Airy},\epsilon_{Airy})$ (see Section
2.5.4) corresponding to the change of the canonical basis
$\\{e^{k,\alpha},f_{k,\alpha}\\}\mapsto\\{\eta^{k},\omega_{k}\\}$ of
$W^{|Ram|}_{Airy}$. Also, let $\\{S_{g,n}\in Sym_{n}(V^{mer}_{\Sigma_{0}})\\}$
be the output of the ATR using
$(V^{mer}_{\Sigma_{0}},\bar{A}_{Airy},\bar{B}_{Airy},\bar{C}_{Airy},\bar{\epsilon}_{Airy})$.
Then for $g\geq 0,n\geq 3$ or $g\geq 1,n\geq 1$, we have:

(78)
$\omega_{g,n}(p_{1},...,p_{n})=\sum_{i_{1},\cdots,i_{n}=1}^{\infty}S_{g,n;i_{1},\cdots,i_{n}}\eta^{i_{1}}(p_{1})\cdots\eta^{i_{n}}(p_{n}),$

where $\eta^{i_{1}}(p_{1})\cdots\eta^{i_{n}}(p_{n})$ denotes symmetric tensor
product $\sum_{\sigma\in
S_{n}}\frac{1}{n!}\eta^{i_{\sigma(1)}}(p_{\sigma(1)})\otimes...\otimes\eta^{i_{\sigma(n)}}(p_{\sigma(n)})$.

###### Proof.

We let the basis of $T_{0}L_{Airy}^{|Ram|}$ be given by
$\\{\omega_{i=1,2,3,\cdots}:=\sum_{\beta\in
Ram}\sum_{j=1}^{\infty}c^{j,\beta}_{i}f_{j,\beta}\\}$, where
$(c^{j,\beta}_{i})=(d^{i}_{j,\beta})^{-1}$. From Remark 2.6.1, we may write
for $g\geq 0,n\geq 3$ or $g\geq 1,n\geq 1$:

$\omega_{g,n}(p_{1},...,p_{n})=\sum_{i_{1},\cdots,i_{n}=1}^{\infty}(S_{TR})_{g,n;i_{1},...,i_{n}}\eta^{i_{1}}(p_{1})\cdots\eta^{i_{n}}(p_{n}).$

In the following let us assume that $p$ is close to one of the ramification
point but $p_{1}$ is further away in the sense that if $p,p_{1}\in
U_{\alpha}\cap\Sigma_{0}$ were in the neighbourhood of $r_{\alpha},\alpha\in
Ram$ then in standard local coordinates we have
$|z_{\alpha}(p)|<<|z_{\alpha}(p_{1})|$. The Laurent expansion of
$\omega_{0,2}(p_{1},p)$ when $p$ is close to a ramification point in the
standard local coordinates can be written as

$\omega_{0,2}(p_{1},p)=\sum_{\alpha\in
Ram,k>0}\bar{e}^{k,\alpha}(p_{1})f_{k,\alpha}(z_{\alpha})=\sum_{k>0}\eta^{k}(p_{1})\omega_{k}(z_{\alpha}),$

where $z$ is the standard local coordinate corresponding to $p$. Similarly,
for $K(p_{1},p)$ we have

$K(p_{1},p)=-\frac{1}{4}\frac{1}{z_{\alpha}(dz_{\alpha})^{2}}\sum_{\begin{subarray}{c}\alpha\in
Ram,k>0\\\ k\
odd\end{subarray}}\frac{1}{k}\bar{e}^{k,\alpha}(p_{1})f_{k,\alpha}(z_{\alpha}).$

Now we have everything we need to evaluate the recursion formula. For
$g=0,n=3$ we have

$\displaystyle\omega_{0,3}(p_{1},p_{2},p_{3})=2\sum_{\alpha\in
Ram}Res_{p=r_{\alpha}}\left(K(p_{1},p)\omega_{0,2}(p,p_{2})\omega_{0,2}(\sigma_{\alpha}(p),p_{3})\right)$

or,

$\displaystyle\sum_{i_{1},i_{2},i_{3}>0}$
$\displaystyle(S_{TR})_{0,3;i_{1}i_{2}i_{3}}\eta_{0}^{i_{1}}(p_{1})\eta_{0}^{i_{2}}(p_{2})\eta_{0}^{i_{3}}(p_{3})$
$\displaystyle=2\sum_{\begin{subarray}{c}\alpha_{1}\in
Ram,i_{1},i_{2},i_{3}>0\\\ i_{1}\
odd\end{subarray}}Res_{z_{\alpha}=0}\left(-\frac{1}{4i_{1}}\frac{1}{z_{\alpha}(dz_{\alpha})^{2}}f_{i_{1},\alpha_{1}}(z_{\alpha})\omega_{i_{2}}(z_{\alpha})\omega_{i_{3}}(-z_{\alpha})\right)\bar{e}^{i_{1},\alpha_{1}}(p_{1})\eta^{i_{2}}(p_{2})\eta^{i_{3}}(p_{3}).$

Comparing the coefficients, we have

$\displaystyle(S_{TR})_{0,3;i_{1}i_{2}i_{3}}$
$\displaystyle=2\sum_{\begin{subarray}{c}\alpha_{1},\alpha_{2},\alpha_{3}\in
Ram,j_{1},j_{2},j_{3}>0\\\ j_{1}\
odd\end{subarray}}Res_{z_{\alpha}=0}\left(-\frac{1}{4j_{1}}f_{j_{1},\alpha_{1}}(z_{\alpha})f_{j_{2},\alpha_{2}}(z_{\alpha})f_{j_{3},\alpha_{3}}(-z_{\alpha})\right)c^{j_{1},\alpha_{1}}_{i_{1}}c^{j_{2},\alpha_{2}}_{i_{2}}c^{j_{3},\alpha_{3}}_{i_{3}}$
$\displaystyle=2\sum_{\begin{subarray}{c}\alpha_{1},\alpha_{2},\alpha_{3}\in
Ram,j_{1},j_{2},j_{3}>0\\\ j_{1}\
odd\end{subarray}}(a_{TR})_{(j_{1},\alpha_{1})(j_{2},\alpha_{2})(j_{3},\alpha_{3})}c^{j_{1},\alpha_{1}}_{i_{1}}c^{j_{2},\alpha_{2}}_{i_{2}}c^{j_{3},\alpha_{3}}_{i_{3}}.$

For $g=1,n=1$ we have

$\displaystyle\omega_{1,1}$ $\displaystyle(p_{1})=\sum_{\alpha\in
Ram}Res_{p=r_{\alpha}}(K(p_{1},p)\omega_{0,2}(p,\sigma_{\alpha}(p)))$
$\displaystyle\sum_{i_{1}>0}(S_{TR})_{1,1;i_{1}}$
$\displaystyle\eta^{i_{1}}_{0}(p_{1})=\lim_{z^{\prime}_{\alpha}\rightarrow
z_{\alpha}}\sum_{\alpha\in
Ram}Res_{z_{\alpha}=r_{\alpha}}(K(p_{1},p)\omega_{0,2}(z_{\alpha},-z^{\prime}_{\alpha}))$
$\displaystyle=$ $\displaystyle\lim_{z^{\prime}_{\alpha}\rightarrow
z_{\alpha}}\sum_{\begin{subarray}{c}\alpha_{1},\alpha_{2}\in
Ram,i_{1},i_{2}>0\\\ i_{1}\
odd\end{subarray}}Res_{z_{\alpha}=0}\left(-\frac{1}{4i_{1}}\frac{1}{z_{\alpha}(dz_{\alpha})^{2}}f_{i_{1},\alpha_{1}}(z_{\alpha})f_{i_{2},\alpha_{2}}(z_{\alpha})\bar{e}^{i_{2},\alpha_{2}}(-z^{\prime}_{\alpha})\right)\bar{e}^{i_{1},\alpha_{1}}(p_{1})$
$\displaystyle=$ $\displaystyle\lim_{z^{\prime}_{\alpha}\rightarrow
z_{\alpha}}\sum_{\begin{subarray}{c}\alpha_{1},\alpha_{2}\in
Ram,i_{1},i_{2}>0\\\ i_{1}\
odd\end{subarray}}Res_{z_{\alpha}=0}\left(-\frac{1}{4i_{1}}\frac{1}{z_{\alpha}(dz_{\alpha})^{2}}f_{i_{1},\alpha_{1}}(z_{\alpha})f_{i_{2},\alpha_{2}}(z_{\alpha})e^{i_{2},\alpha_{2}}(-z^{\prime}_{\alpha})\right)\bar{e}^{i_{1},\alpha_{1}}(p_{1})$
$\displaystyle+\lim_{z^{\prime}_{\alpha}\rightarrow z_{\alpha}}$
$\displaystyle\sum_{\begin{subarray}{c}\alpha_{1},\alpha_{2}\in
Ram,i_{1},i_{2}>0\\\ i_{1}\
odd\end{subarray}}s^{(i_{2},\alpha_{2})(j_{2},\beta_{2})}Res_{z_{\alpha}=0}\left(-\frac{1}{4i_{1}}\frac{1}{z_{\alpha}(dz_{\alpha})^{2}}f_{i_{1},\alpha_{1}}(z_{\alpha})f_{i_{2},\alpha_{2}}(z_{\alpha})f_{j_{2},\beta_{2}}(-z^{\prime}_{\alpha})\right)\bar{e}^{i_{1},\alpha_{1}}(p_{1})$
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}\alpha_{1}\in
Ram,i_{1}>0\\\ i_{1}\
odd\end{subarray}}Res_{z_{\alpha}=0}\left(\frac{1}{4z_{\alpha}}\left(z^{i_{1}}_{\alpha}\frac{dz_{\alpha}}{z_{\alpha}}\right)\frac{1}{4z^{2}_{\alpha}}\right)\bar{e}^{i_{1},\alpha_{1}}(p_{1})$
$\displaystyle+\sum_{\begin{subarray}{c}\alpha_{1},\alpha_{2}\in
Ram,i_{1},i_{2}>0\\\ i_{1}\
odd\end{subarray}}s^{(i_{2},\alpha_{2})(j_{2},\beta_{2})}Res_{z=0}\left(-\frac{1}{4i_{1}}\frac{1}{z_{\alpha}(dz_{\alpha})^{2}}f_{i_{1},\alpha_{1}}(z_{\alpha})f_{i_{2},\alpha_{2}}(z_{\alpha})f_{j_{2},\beta_{2}}(-z_{\alpha})\right)\bar{e}^{i_{1},\alpha_{1}}(p_{1})$

Where the limit $\lim_{z^{\prime}_{\alpha}\rightarrow z_{\alpha}}$ is taken
from the direction $|z^{\prime}_{\alpha}|>|z_{\alpha}|$. We can easily see
that the first term of the last line is simply $\sum_{\alpha\in
Ram}\frac{1}{16}\bar{e}^{3,\alpha}(p_{1})=\epsilon_{TR}$ and the residue in
the second term can be recognized as $A_{TR}$. Comparing the coefficients, we
have

$\displaystyle(S_{TR})_{1,1;i_{1}}$
$\displaystyle=\sum_{\begin{subarray}{c}\alpha_{1}\in Ram,j_{1}>0\\\ j_{1}\
odd\end{subarray}}\left((\epsilon_{TR})_{j_{1},\alpha_{1}}+\sum_{\alpha_{2},\alpha_{3}\in
Ram,j_{2},j_{3}>0}(a_{TR})_{(j_{1},\alpha_{1})(j_{2},\alpha_{2})(j_{3},\alpha_{3})}s^{(j_{2},\alpha_{2})(j_{3},\alpha_{3})}\right)c^{j_{1},\alpha_{1}}_{i_{1}}.$

For $g\geq 0,n\geq 4$ or $g\geq 1,n\geq 2$ or $g\geq 2,n\geq 1$ we have from
the recursion formula

$\displaystyle\sum_{i_{1},\cdots,i_{n}>0}(S_{TR})_{g,n;i_{1},...,i_{n}}\eta^{i_{1}}(p_{1})\cdots\eta^{i_{n}}(p_{n})$
$\displaystyle=\sum_{\begin{subarray}{c}\alpha_{1}\in
Ram,i_{1},\cdots,i_{n}>0\\\ i_{1}\
odd\end{subarray}}\Bigg{[}\sum_{j_{1},j_{2}>0}Res_{z_{\alpha}=0}\left(-\frac{1}{4i_{1}}\frac{1}{z_{\alpha}(dz_{\alpha})^{2}}f_{i_{1},\alpha_{1}}(z_{\alpha})\eta^{j_{1}}(z_{\alpha})\eta^{j_{2}}(-z_{\alpha})\right)(S_{TR})_{g-1,n+1;j_{1}j_{2}i_{2}...i_{n}}$
$\displaystyle\qquad+2\sum_{k=2}^{n}\sum_{j_{1}>0}Res_{z_{\alpha}=0}\left(-\frac{1}{4i_{1}}\frac{1}{z_{\alpha}(dz_{\alpha})^{2}}f_{i_{1},\alpha_{1}}(z_{\alpha})\omega_{i_{k}}(z_{\alpha})\eta^{j_{1}}(-z_{\alpha})\right)(S_{TR})_{g,n-1;j_{1}i_{\\{2,..,n\\}\setminus\\{k\\}}}$
$\displaystyle+\sum_{\begin{subarray}{c}g_{1}+g_{2}=g\\\ I_{1}\coprod
I_{2}=\\{2,...,n\\}\end{subarray}}\sum_{j_{1},j_{2}>0}Res_{z_{\alpha}=0}\left(-\frac{1}{4i_{1}}\frac{1}{z_{\alpha}(dz_{\alpha})^{2}}f_{i_{1},\alpha_{1}}(z_{\alpha})\eta^{j_{1}}(z_{\alpha})\eta^{j_{2}}(z_{\alpha})\right)(S_{TR})_{g_{1},1+|I_{1}|;j_{1}i_{I_{1}}}(S_{TR})_{g_{2},1+|I_{2}|;j_{2}i_{I_{2}}}\Bigg{]}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times\bar{e}^{i_{1},\alpha_{1}}(p_{1})\eta^{i_{2}}(p_{2})...\eta^{i_{n}}(p_{n}).$

Comparing the coefficients, we find that the recursion relations (2.6.1)
becomes

(79) $\displaystyle(S_{TR})_{g,n;i_{1},...,i_{n}}$
$\displaystyle=2\sum_{k=2}^{n}\sum_{j>0}(\bar{b}_{TR})_{i_{1}i_{k}}^{j}(S_{TR})_{g,n-1;ji_{\\{2,...,n\\}\setminus\\{k\\}}}+$
$\displaystyle\qquad+\sum_{\begin{subarray}{c}g_{1}+g_{2}=g\\\ I_{1}\coprod
I_{2}=\\{2,...,n\\}\end{subarray}}\sum_{j_{1},j_{2}>0}(\bar{c}_{TR})_{i_{1}}^{j_{1}j_{2}}(S_{TR})_{g_{1},1+|I_{1}|;j_{1}i_{I_{1}}}(S_{TR})_{g_{2},1+|I_{2}|;j_{2}i_{I_{2}}}$
$\displaystyle\qquad+\sum_{j_{1},j_{2}>0}(\bar{c}_{TR})_{i_{1}}^{j_{1}j_{2}}(S_{TR})_{g-1,n+1;j_{1}j_{2}i_{2}...i_{n}},$

for $g\geq 0,n\geq 4$ or $g\geq 2,n\geq 1$ or $g\geq 1,n\geq 2$ with the
initial condition

(80)
$(S_{TR})_{0,3;i_{1}i_{2}i_{3}}=2(\bar{a}_{TR})_{i_{1}i_{2}i_{3}},\qquad(S_{TR})_{1,1;i}=(\bar{\epsilon}_{TR})_{i},\qquad(S_{TR})_{0,n\leq
2}=(S_{TR})_{g\geq 0,0}=0$

where

(81) $\displaystyle(\bar{a}_{TR})_{i_{1}i_{2}i_{3}}$
$\displaystyle=(a_{TR})_{(j_{1},\alpha_{1})(j_{2},\alpha_{2})(j_{3},\alpha_{3})}c^{j_{1},\alpha_{1}}_{i_{1}}c^{j_{2},\alpha_{2}}_{i_{2}}c^{j_{3},\alpha_{3}}_{i_{3}}$
$\displaystyle(\bar{b}_{TR})_{i_{1}i_{2}}^{i_{3}}$
$\displaystyle=\left((b_{TR})_{(j_{1},\alpha_{1})(j_{2},\alpha_{2})}^{j_{3},\alpha_{3}}+(a_{TR})_{(j_{1},\alpha_{1})(j_{2},\alpha_{2})(k,\gamma)}s^{(k,\gamma)(j_{3},\alpha_{3})}\right)c^{j_{1},\alpha_{1}}_{i_{1}}c^{j_{2},\alpha_{2}}_{i_{2}}d^{i_{3}}_{j_{3},\alpha_{3}}$
$\displaystyle(\bar{c}_{TR})_{i_{1}}^{i_{2}i_{3}}$
$\displaystyle=\Big{(}(c_{TR})_{(j_{1},\alpha_{1})}^{(j_{2},\alpha_{2})(j_{3},\alpha_{3})}+(b_{TR})_{(j_{1},\alpha_{1})(k_{1},\gamma_{1})}^{(j_{3},\alpha_{3})}s^{(k_{1},\gamma_{1})(j_{2},\alpha_{2})}$
$\displaystyle\qquad+(b_{TR})_{(j_{1},\alpha_{1})(k_{2},\gamma_{2})}^{(j_{2},\alpha_{2})}s^{(k_{2},\gamma_{2})(j_{3},\alpha_{3})}+(a_{TR})_{(j_{1},\alpha_{1})(k_{1},\gamma_{1})(k_{2},\gamma_{2})}s^{(k_{1},\gamma_{1})(j_{2},\gamma_{2})}s^{(k_{2},\gamma_{2})(j_{3},\gamma_{3})}\Big{)}$
$\displaystyle\qquad\qquad\qquad\times
c^{j_{1},\alpha_{1}}_{i_{1}}d_{j_{2},\alpha_{2}}^{i_{2}}d_{j_{3},\alpha_{3}}^{i_{3}}$
$\displaystyle(\bar{\epsilon}_{TR})_{i}$
$\displaystyle=\left((\epsilon_{TR})_{j_{1},\alpha_{1}}+(a_{TR})_{(j_{1},\alpha_{1})(k_{1},\gamma_{1})(k_{2},\gamma_{2})}s^{(k_{1},\gamma_{1})(k_{2},\gamma_{2})}\right)c^{j_{1},\alpha_{1}}_{i_{1}}.$

In (81) the summation is over positive odd integers $j_{1}$ and over all
positive integers for all other indices. Equivalently, we may take all
summations in (81) to be over all positive integers since components of
$A_{TR},B_{TR},C_{TR},\epsilon_{TR}$ with $j_{1}=even$ are zero. Let
$\bar{A}_{TR}=((\bar{a}_{TR})_{i_{1}i_{2}i_{3}}),\bar{B}_{TR}=((\bar{b}_{TR})_{i_{1}i_{2}}^{i_{3}}),\bar{C}_{TR}=((\bar{c}_{TR})_{i_{1}}^{i_{2}i_{3}})$
and $\bar{\epsilon}_{TR}=((\bar{\epsilon}_{TR})_{i})$. We recognize (2.2.2)
together with (80) as the ATR of quantum pre-Airy structure
$(V^{mer}_{\Sigma_{0}},\bar{A}_{TR},\bar{B}_{TR},\bar{C}_{TR},\bar{\epsilon}_{TR})$
which is the gauge transform of the quantum pre-Airy structure
$(V^{|Ram|}_{Airy},A_{TR},B_{TR},C_{TR},\epsilon_{TR})$ corresponding to the
change of the canonical basis
$\\{e^{k,\alpha},f_{k,\alpha}\\}\mapsto\\{\eta^{k},\omega_{k}\\}$ of
$W^{|Ram|}_{Airy}$. Using Remark 2.2.30 and Remark 2.5.24 we have that
$(V^{mer}_{\Sigma_{0}},\bar{A}_{TR},\bar{B}_{TR},\bar{C}_{TR},\bar{\epsilon}_{TR})$
and
$(V^{mer}_{\Sigma_{0}},\bar{A}_{Airy},\bar{B}_{Airy},\bar{C}_{Airy},\bar{\epsilon}_{Airy})$
both contain the quantum Airy sub-structure
$((V^{mer}_{\Sigma_{0}})_{odd},\bar{A}_{Airy},\bar{B}_{Airy},\bar{C}_{Airy},\bar{\epsilon}_{Airy})$.
It follows that $(S_{TR})_{g,n}=S_{g,n}\in
Sym_{n}((V^{mer}_{\Sigma_{0}})_{odd})$ (see Remark 2.2.14), which completes
the proof. ∎

###  2.7. Topological Recursion and Prepotentials

In this section, we are going to prove the main theorem which relates the
prepotential $\mathfrak{F}_{\Sigma_{0}}$ (see Definition 2.1.22), defined on
some open neighbourhood $\mathcal{B}_{\Sigma_{0}}\ni[\Sigma_{0}]$ of the
moduli space $\mathcal{B}$ of $\mathcal{F}$-transversal genus $g$ curves in
$(S,\Omega_{S},\mathcal{F})$, to the genus zero part of topological recursion
on $\Sigma_{0}$. The basic idea is as follows. The map
$\boldsymbol{\Phi}_{\Sigma_{0}}:\mathcal{B}_{\Sigma_{0}}\rightarrow\mathcal{H}_{\Sigma_{0}}$
embeds $\mathcal{B}_{\Sigma_{0}}$ into the fiber $\mathcal{H}_{\Sigma_{0}}$ as
the Lagrangian submanifold $\mathcal{L}_{\Sigma_{0}}$ with
$\mathfrak{F}_{\Sigma_{0}}$ as the generating function. The image of
$\Phi_{t_{0}}:Discs^{Ram}_{t_{0}}\rightarrow W^{Ram}_{t_{0}}$ is contained
inside the Lagrangian submanifold $L^{Ram}_{Airy}$ with $S_{0}$, the $g=0$
part of the ATR output of the $Ram$ product residue constraints Airy structure
$\\{(H_{Airy})_{i=1,2,3,\cdots,\alpha\in Ram}\\}$, as the generating function.
In Section 2.5 we constructed the map
$\Phi_{\Sigma_{0}}:\mathcal{B}_{\Sigma_{0}}\rightarrow G_{\Sigma_{0}}$
relating $\boldsymbol{\Phi}_{\Sigma_{0}}$ to $\Phi_{t_{0}}$, therefore also
relating $\mathfrak{F}_{\Sigma_{0}}$ to $S_{0}$. Finally, we use Proposition
2.6.2 to relate $S_{0}$ to $\omega_{0,n}$.

The important step in simplifying the proof of this result is to use the right
gauge when working with the Airy structure
$\\{(H_{Airy})_{i=1,2,3,\cdots,\alpha\in Ram}\\}$. For example, the canonical
basis

$\left\\{e^{k=1,2,3,\cdots,\alpha\in Ram}:=[\alpha]\otimes
z^{-k}_{\alpha}\frac{dz_{\alpha}}{z_{\alpha}},f_{k=1,2,3,\cdots,\alpha\in
Ram}:=[\alpha]\otimes kz^{k}_{\alpha}\frac{dz_{\alpha}}{z_{\alpha}}\right\\}$

for $W^{Ram}_{Airy}$ will not make the comparison between
$S_{0}=S_{0}(\\{x^{k=1,2,3,\cdots,\alpha\in Ram}\\})$ and
$\mathfrak{F}_{\Sigma_{0}}=\mathfrak{F}_{\Sigma_{0}}(\mathbf{a}^{1},\cdots,\mathbf{a}^{g})$
easy. Each $x^{k,\alpha}$ is a coefficient of the Laurent series expansion of
$\Phi_{\Sigma_{0}}(\Sigma)$ whereas each $\mathbf{a}^{i}$ relates to
$A$-periods of $\Phi_{\Sigma_{0}}(\Sigma)$. Therefore, we start by choosing
the new canonical basis for $W^{Ram}_{Airy}$ which makes the cohomological
information of $\Phi_{\Sigma_{0}}(\Sigma)$ becomes more visible from the point
of view of $W^{Ram}_{Airy}$.

######  Lemma 2.7.1.

Let $[\Sigma_{0}]\in\mathcal{B}$ and let $\\{\bar{e}^{k=1,2,3,\cdots,\alpha\in
Ram}\\}$ be the basis of the Lagrangian complement $V_{\Sigma_{0}}$ as given
in Definition 2.5.11. There exists a canonical basis

$\left\\{\eta^{i=1,2,3,\cdots}:=\sum_{\beta\in
Ram}\sum_{j=1}^{\infty}d^{i}_{j,\beta}\bar{e}^{j,\beta},\omega_{i=1,2,3,\cdots}:=\sum_{\beta\in
Ram}\sum_{j=1}^{\infty}c^{j,\beta}_{i}f_{j,\beta}\right\\}$

of $W^{Ram}_{t(\Sigma_{0})}\cong W^{Ram}_{Airy}$,

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

corresponding to the Lagrangian complement $V_{\Sigma_{0}}$ such that
$\omega_{1},\cdots,\omega_{g}\in\Gamma(\Sigma_{0},\Omega^{1}_{\Sigma_{0}})$,
$\eta^{i=1,2,3,\cdots}\in V_{\Sigma_{0}}$ and $\omega_{i>g}$ are locally
defined holomorphic forms on $\sqcup_{\alpha\in
Ram}\mathbb{D}_{\alpha,\bar{M}_{\alpha}}(\Sigma_{0})\subset\Sigma_{0}$ for
some $\bar{M}_{\alpha}>M_{\alpha}$. Furthermore, we have that for
$i,j=1,\cdots,g$:

(82)
$\oint_{A_{i}}\omega_{j}=\delta_{j}^{i},\qquad\oint_{B_{i}}\omega_{j}=\tau_{ij}(\Sigma_{0}),\qquad\oint_{A_{i}}\eta^{j}=0,\qquad\oint_{B_{i}}\eta^{j}=2\pi
i\delta^{j}_{i},$

and all $A,B$-periods of $\eta^{i>g}$ vanish. Lastly,
$(c^{i,\alpha}_{j}):V^{|Ram|}_{Airy}\rightarrow V^{|Ram|}_{Airy}$,
$(d^{i}_{j,\beta}):V^{|Ram|}_{Airy}\rightarrow V^{|Ram|}_{Airy}$ both satisfy
Condition 2.4.17 and $\\{\bar{e}^{i=1,2,3,\cdots,\alpha\in Ram}\\}$ is given
by Definition 2.5.11.

######  Remark 2.7.2.

Equivalently, we describe the canonical basis in 2.7.1 as follows. Choose
$\\{\omega_{1},\cdots,\omega_{g}\\}$ to be the set of normalized holomorphic
differentials which is a basis for
$\Gamma(\Sigma_{0},\Omega^{1}_{\Sigma_{0}})$. The set
$\\{\omega_{1},\cdots,\omega_{g},\omega_{i>g}\\}$ is a basis for
$T_{0}L^{Ram}_{Airy}$, where $\omega_{i>g}$ are locally defined. The set
$\\{\eta^{1},\cdots,\eta^{g},\eta^{i>g}\\}$ is a basis for $V_{\Sigma_{0}}$
because $\oint_{A_{i}}\eta^{j}=0$ for all $j=1,2,3,\cdots$ and
$\\{\eta^{i>g}\\}$ is a basis for $G^{\perp}_{\Sigma_{0}}$ because all
$A,B$-periods of $\eta^{i>g}$ vanish.

It follows that
$\\{\omega_{1},\cdots,\omega_{g},\eta^{1},\cdots,\eta^{g},\eta^{i>g}\\}$ is a
basis of $G_{\Sigma_{0}}$ and
$\\{\omega_{1},\cdots,\omega_{g},\eta^{1},\cdots,\eta^{g}\\}$ is a basis for
$\mathcal{H}_{\Sigma_{0}}=H^{1}(\Sigma_{0},\mathbb{C})=G_{\Sigma_{0}}/G^{\perp}_{\Sigma_{0}}$.
Finally, $\oint_{B_{i}}\eta^{j}=2\pi i\delta_{i}^{j}$ for $j=1,\cdots,g$ can
be deduced from $\Omega_{Airy}(\eta^{j},\omega_{i})=\delta^{i}_{j}$.

###### Proof.

Let $\\{\omega_{1},\cdots,\omega_{g}\\}$ be the normalized holomorphic
differentials on $\Sigma_{0}$,
$\oint_{A_{i}}\omega_{j}=\delta^{i}_{j},\oint_{B_{i}}\omega_{j}=\tau_{ij}(\Sigma_{0})$.
It is clear that $\\{\omega_{1},\cdots,\omega_{g}\\}$ is linearly independent
in $T_{0}L^{Ram}_{Airy}\subset W^{Ram}_{Airy}$. Define

$(T_{0}L_{Airy}^{Ram})_{>n}:=\left\\{\xi=\sum_{\alpha\in
Ram}[\alpha]\otimes\xi_{\alpha}\ |\ \xi_{\alpha}(z)\in
z^{n+1}_{\alpha}\mathbb{C}[[z_{\alpha}]]\frac{dz_{\alpha}}{z_{\alpha}},\begin{array}[]{c}\text{$\xi_{\alpha}(z)$
converges for $|z_{\alpha}|<\bar{M}_{\alpha}$}\\\ \text{for some
$\bar{M}_{\alpha}>M_{\alpha}$}\end{array}\right\\}\subset
T_{0}L^{Ram}_{Airy}.$

We take $\\{f^{i>n,\alpha\in Ram}\\}$ to be the basis of
$(T_{0}L^{Ram}_{Airy})_{>n}$. It follows that
$T_{0}L^{Ram}_{Airy}/(T_{0}L^{Ram}_{Airy})_{>n}$ is an $n|Ram|$-dimensional
vector space. By choosing a large enough $n$, the image of
$\\{\omega_{1},\cdots,\omega_{g}\\}$ in
$T_{0}L^{Ram}_{Airy}/(T_{0}L^{Ram}_{Airy})_{>n}$ is linearly independent and
we can complement it with $\\{\omega_{g+1},\cdots,\omega_{n|Ram|}\\}$ such
that the image of $\\{\omega_{1},\cdots,\omega_{n|Ram|}\\}$ in
$T_{0}L^{Ram}_{Airy}/(T_{0}L^{Ram}_{Airy})_{>n}$ is a basis for
$T_{0}L^{Ram}_{Airy}/(T_{0}L^{Ram}_{Airy})_{>n}$. Let
$\\{\omega_{i>n|Ram|}\\}$ be given by $\\{f_{i>n,\alpha\in Ram}\\}$ rearranged
in some order, for example, let $r:\\{0,\cdots,|Ram|-1\\}\rightarrow Ram$ be
an arbitrary bijection then we define $\omega_{i}:=f_{\lfloor
i/|Ram|\rfloor,r(i\bmod{|Ram|})}$. Then $\\{\omega_{i=1,2,3,\cdots}\\}$ is a
basis of
$(T_{0}L^{Ram}_{Airy}/(T_{0}L^{Ram}_{Airy})_{>n})\oplus((T_{0}L^{Ram}_{Airy})_{>n}\cong
T_{0}L^{Ram}_{Airy}$ and $i(\omega_{k})=\sum_{\beta\in
Ram}\sum_{j=1}^{\infty}c^{j,\beta}_{k}f_{j,\beta}$ for some
$(c^{j,\beta}_{i})$ satisfying Condition 2.4.17. In particular,
$(c^{j,\beta}_{i})$ satisfies C2.4.17.3.

It follows that its inverse $(d^{i}_{j,\beta})$ also satisfies Condition
2.4.17 and that $\\{\eta^{i=1,2,3,\cdots},\omega_{i=1,2,3,\cdots}\\}$ gives a
canonical basis for $W^{Ram}_{Airy}$. Since $\eta^{i}=\sum_{\beta\in
Ram}\sum_{j=1}^{\infty}d^{i}_{j,\beta}\bar{e}^{j,\beta}$ is a finite summation
for each $i$, we have $\eta^{i}\in V_{\Sigma_{0}}$ and in particular that
$\oint_{A_{i}}\eta^{j}=0$. Suppose that $i>g$ and $j=1,\cdots,g$ then by using
(2.5.3) we have $\oint_{B_{j}}\eta^{i}=2\pi
i\Omega_{Airy}(\eta^{i},\omega_{j})=0$. Similarly, by using (2.5.3) for
$i,j=1,\cdots,g$ we have $\oint_{B_{j}}\eta^{i}=2\pi
i\Omega_{Airy}(\eta^{i},\omega_{j})=2\pi i\delta^{i}_{j}$. ∎

The main theorem is a joint work with Paul Norbury, Michael Swaddle and Mehdi
Tavakol which was proven in [6] using a different approach. Here we are
stating the theorem and prove it using an analytical framework:

######  Theorem 2.7.3.

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

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

can be expressed as

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

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

  1. (1)

The involution $\sigma_{\alpha}(z_{\alpha}):=-z_{\alpha}$.

  2. (2)

$\omega_{0,1}(p)$ locally defined on each $\Sigma_{0}\cap U_{\alpha}$ for
$\alpha\in Ram$ such that
$\omega_{0,1}(z_{\alpha})-\omega_{0,1}(\sigma_{\alpha}(z_{\alpha}))=4z_{\alpha}^{2}dz_{\alpha}$.

  3. (3)

$\omega_{0,2}(p,q)=B(p,q)$ the normalized Bergman kernel.

with $z_{\alpha}$ for $\alpha\in Ram$ denotes the standard local coordinates
corresponding to the collection of $(\mathcal{F},\Omega_{S})$-charts
$\mathcal{U}_{Ram}=\\{(U_{\alpha\in Ram},x_{\alpha\in Ram},y_{\alpha\in
Ram})\\}$, $r_{\alpha}(\Sigma_{0})\in\Sigma_{0}\cap U_{\alpha}$ such that
$\gamma_{\alpha}(\Sigma_{0})=(x_{\alpha}=z^{2}_{\alpha},y_{\alpha}=z_{\alpha})$.

###### Proof.

Let $\\{\epsilon_{\alpha\in Ram}\\},\\{M_{\alpha\in Ram}\\}$,
$\mathcal{U}_{Ram}$ and $\mathcal{B}_{\Sigma_{0}}$ be chosen such that
Condition 2.5.2 is satisfied. In particular, the collection of
$(\mathcal{F},\Omega_{S})$-charts $\mathcal{U}_{Ram}$ is chosen such that
$\gamma_{\alpha}(\Sigma_{0})=(x_{\alpha}=z_{\alpha}^{2},y_{\alpha}=z_{\alpha})$.
Consider the embedding map
$\Phi_{\Sigma_{0}}:\mathcal{B}_{\Sigma_{0}}\rightarrow G_{\Sigma_{0}}$ given
by
$\Phi_{\Sigma_{0}}(\Sigma):=-\theta_{\Sigma_{0}}+s^{*}_{\Sigma_{0}}\theta_{\Sigma}$
(see Section 2.5.7). It follows from Proposition 2.5.38 that
$i(\Phi_{\Sigma_{0}}(\Sigma))$ satisfies the $Ram$ product analytic residue
constraints:

(84)
$i(\Phi_{\Sigma_{0}}(\Sigma))=\Phi_{t(\Sigma_{0})}(\gamma(\Sigma))=\sum_{\alpha\in
Ram}\sum_{k=1}^{\infty}(x^{k,\alpha}f_{k,\alpha}+y_{k,\alpha}e^{k,\alpha})\in
L^{Ram}_{Airy}\subset W^{Ram}_{Airy}.$

Let us change the canonical basis of $W^{Ram}_{Airy}$ to
$\\{\eta^{i=1,2,3,\cdots},\omega_{i=1,2,3,\cdots}\\}$ as provided by Lemma
2.7.1. Then,

(85)
$\Phi_{\Sigma_{0}}(\Sigma)=\sum_{k=1}^{\infty}\beta_{k}\eta^{k}+\sum_{k=1}^{g}\alpha^{k}\omega_{k},$

by Proposition 2.4.20 where
$\\{\alpha^{k=1,2,3,\cdots}\\},\\{\beta_{k=1,2,3,\cdots}\\}$ are given by (70)
and $\alpha^{i>g}=0$ since $\omega_{i>g}\notin G_{\Sigma_{0}}$ (alternatively,
we could apply Corollary 2.5.17). Computing the $A$-periods of
$\Phi_{\Sigma_{0}}(\Sigma)$ using (82), we obtain for $i=1,\cdots,g$:

$\displaystyle\alpha^{i}$
$\displaystyle=\oint_{A_{i}}\left(\sum_{k=1}^{\infty}\beta_{k}\eta^{k}+\sum_{k=1}^{g}\alpha^{k}\omega_{k}\right)$
(86)
$\displaystyle=\oint_{A_{i}}\Phi_{\Sigma_{0}}(\Sigma)=\oint_{A_{i}}\boldsymbol{\Phi}_{\Sigma_{0}}(\Sigma)=-\oint_{A_{i}}\boldsymbol{\theta}_{\Sigma_{0}}+\oint_{A_{i}}\boldsymbol{\theta}_{\Sigma}=\mathbf{a}^{i}(\Sigma)-\mathbf{a}^{i}(\Sigma_{0}).$

Similarly, computing $B$-periods of $\Phi_{\Sigma_{0}}(\Sigma)$ using (82), we
obtain for $i=1,\cdots,g$:

$\displaystyle 2\pi
i\beta_{i}+\sum_{k=1}^{\infty}\tau_{ik}(\Sigma_{0})\alpha^{k}$
$\displaystyle=\oint_{B_{i}}\left(\sum_{k=1}^{\infty}\beta_{k}\eta^{k}+\sum_{k=1}^{g}\alpha^{k}\omega_{k}\right)$
(87)
$\displaystyle=\oint_{B_{i}}\Phi_{\Sigma_{0}}(\Sigma)=\oint_{B_{i}}\boldsymbol{\Phi}_{\Sigma_{0}}(\Sigma)=-\oint_{B_{i}}\boldsymbol{\theta}_{\Sigma_{0}}+\oint_{B_{i}}\boldsymbol{\theta}_{\Sigma}=\mathbf{b}_{i}(\Sigma)-\mathbf{b}_{i}(\Sigma_{0}).$

Where we have used Lemma 2.5.14 in conjunction with the fact that
$(d^{i}_{j,\beta})$ satisfies C2.4.17.3 to justify the term-by-term
integration. In particular for each $i=1,2,3,\cdots$, $\eta^{i}$ is a finite
linear combination of $\bar{e}^{i,\alpha}$. We also have used the relation
$[\Phi_{\Sigma_{0}}(\Sigma)]=\boldsymbol{\Phi}_{\Sigma_{0}}(\Sigma)$ from
Proposition 2.5.38. From now on we will write
$\mathbf{a}^{i}:=\mathbf{a}^{i}(\Sigma),\mathbf{b}_{i}:=\mathbf{b}_{i}(\Sigma)$
and
$\mathbf{a}^{i}_{0}:=\mathbf{a}^{i}(\Sigma_{0}),\mathbf{b}^{0}_{i}:=\mathbf{b}_{i}(\Sigma_{0})$.

Let us cover $\mathcal{B}_{\Sigma_{0}}$ with a coordinates chart
$(\mathcal{B}_{\Sigma_{0}},u^{1},\cdots,u^{g})$. Then it is clear from
construction that for each $\alpha\in Ram$,
$i_{\alpha}(\Phi_{\Sigma_{0}}(\Sigma))$ is holomorphic in
$\\{u^{1},\cdots,u^{g},z_{\alpha}\\}$ for all
$(u^{1},\cdots,u^{g},z_{\alpha})\in\mathcal{B}_{\Sigma_{0}}\times\mathbb{A}_{\alpha,\epsilon_{\alpha},M_{\alpha}}$.
Recall from Lemma 2.1.21 that $\\{\mathbf{a}^{1},\cdots,\mathbf{a}^{g}\\}$ is
also a coordinates system of $\mathcal{B}_{\Sigma_{0}}$. From (85) and (86),
it follows that $\beta_{i}$ are holomorphic functions in
$\\{\alpha^{1},\cdots,\alpha^{g}\\}$. It follows that
$i_{\alpha}\left(\sum_{k=1}^{\infty}\beta_{k}(\alpha^{1},\cdots,\alpha^{g})\eta^{k}\right)(z_{\alpha})$
is holomorphic in $\\{\alpha^{1},\cdots,\alpha^{g},z_{\alpha}^{-1}\\}$ on the
open neighbourhood $\mathcal{B}_{\Sigma_{0}}$ of
$(\alpha^{1},\cdots,\alpha^{g})=(0,\cdots,0)$ and
$|z_{\alpha}|>\epsilon_{\alpha}$. Using (84) together with Proposition 2.4.20
implies that

$(\bar{H}_{Airy})_{i}\left(\\{\alpha^{1},\cdots,\alpha^{g},\alpha^{k>g}=0\\},\\{\beta_{k=1,2,3,\cdots}(\alpha^{1},\cdots,\alpha^{g})\\}\right)=0,$

where $(\bar{H}_{Airy})_{i}$ is the gauge transformed $Ram$ product residue
constraints Airy structure as given in (71). Using the obvious generalization
of Proposition 2.4.21 (see also Remark 2.5.25), we conclude that
$\beta_{i}(\alpha^{1},\cdots,\alpha^{g})=(\partial_{i}S_{0})|_{\alpha^{k>g}=0}$
where
$S_{0}=\sum_{n=1}^{\infty}S_{0,n}\in\prod_{n=1}^{\infty}Sym_{n}(V^{mer}_{\Sigma_{0}})$
is the output of the ATR using the quantum Airy structure
$(V^{mer}_{\Sigma_{0}},\bar{A}_{Airy},\bar{B}_{Airy},\bar{C}_{Airy},\bar{\epsilon}_{Airy})$.
Here,
$(V^{mer}_{\Sigma_{0}},\bar{A}_{Airy},\bar{B}_{Airy},\bar{C}_{Airy},\bar{\epsilon}_{Airy})$
is the quantum Airy structure obtained from the gauge transformation of the
quantum Airy structure
$(V^{|Ram|}_{Airy},A_{Airy},B_{Airy},C_{Airy},\epsilon_{Airy})$, corresponding
to the change of a canonical basis of $W^{|Ram|}_{Airy}$ from
$\\{e^{k=1,2,3,\cdots,\alpha\in Ram},f_{k=1,2,3,\cdots,\alpha\in Ram}\\}$ to
$\\{\eta^{k=1,2,3\cdots},\omega_{k=1,2,3,\cdots}\\}$ (see Section 2.5.4). In
particular, let
$\mathcal{S}_{0}(\alpha^{1},\cdots,\alpha^{g}):=S_{0}(\\{\alpha^{1},\cdots,\alpha^{g},\alpha^{k>g}=0\\})$
then $\mathcal{S}_{0}$ is holomorphic on $\mathcal{B}_{\Sigma_{0}}$ and
$\beta_{i}(\alpha^{1},\cdots,\alpha^{k})=\frac{\partial\mathcal{S}_{0}}{\partial\alpha^{i}}(\alpha^{1},\cdots,\alpha^{k})$
for $i=1,\cdots,g$. Equivalently,
$\beta_{i}=\frac{\partial\mathcal{S}_{0}}{\partial\mathbf{a}^{i}}(\mathbf{a}^{1}-\mathbf{a}^{1}_{0},\cdots,\mathbf{a}^{g}-\mathbf{a}^{g}_{0})$
for $i=1,\cdots,g$. Using the fact that
$\mathbf{b}_{i}=\frac{\partial\mathfrak{F}_{\Sigma_{0}}}{\partial\mathbf{a}^{i}}(\mathbf{a}^{1},\cdots,\mathbf{a}^{g})$
and
$\tau_{ij}(\Sigma)=\frac{\partial^{2}\mathfrak{F}_{\Sigma_{0}}}{\partial\mathbf{a}^{i}\partial\mathbf{a}^{j}}(\mathbf{a}^{1},\cdots,\mathbf{a}^{g})$
then (86) and (87) implies

$2\pi
i\frac{\partial\mathcal{S}_{0}}{\partial\mathbf{a}^{i}}(\mathbf{a}^{1}-\mathbf{a}^{1}_{0},\cdots,\mathbf{a}^{g}-\mathbf{a}^{g}_{0})+\sum_{k=1}^{g}(\mathbf{a}^{k}-\mathbf{a}^{k}_{0})\frac{\partial^{2}\mathfrak{F}_{\Sigma_{0}}}{\partial\mathbf{a}^{k}\partial\mathbf{a}^{i}}(\mathbf{a}^{1}_{0},\cdots,\mathbf{a}^{g}_{0})=\frac{\partial\mathfrak{F}_{\Sigma_{0}}}{\partial\mathbf{a}^{i}}(\mathbf{a}^{1},\cdots,\mathbf{a}^{g})-\frac{\partial\mathfrak{F}_{\Sigma_{0}}}{\partial\mathbf{a}^{i}}(\mathbf{a}^{1}_{0},\cdots,\mathbf{a}^{g}_{0})$

Integrating both sides, we find that for all
$(\mathbf{a}^{1},\cdots,\mathbf{a}^{g})=[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$
we have

$\displaystyle\mathfrak{F}_{\Sigma_{0}}(\mathbf{a}^{1},\cdots,\mathbf{a}^{g})$
$\displaystyle=\mathfrak{F}_{\Sigma_{0}}(\mathbf{a}^{1}_{0},\cdots,\mathbf{a}^{g}_{0})+\sum_{i=1}^{g}(\mathbf{a}^{i}-\mathbf{a}^{i}_{0})\frac{\partial\mathfrak{F}_{\Sigma_{0}}}{\partial\mathbf{a}^{i}}(\mathbf{a}^{1}_{0},\cdots,\mathbf{a}^{g}_{0})$
(88)
$\displaystyle\qquad+\sum_{i,j=1}^{g}(\mathbf{a}^{i}-\mathbf{a}^{i}_{0})(\mathbf{a}^{j}-\mathbf{a}^{j}_{0})\frac{\partial^{2}\mathfrak{F}_{\Sigma_{0}}}{\partial\mathbf{a}^{i}\partial\mathbf{a}^{j}}(\mathbf{a}^{1}_{0},\cdots,\mathbf{a}^{g}_{0})+2\pi
i\mathcal{S}_{0}(\mathbf{a}^{1}-\mathbf{a}_{0}^{1},\cdots,\mathbf{a}^{g}-\mathbf{a}^{g}_{0}).$

From (29),
$S_{0}=\sum_{n=1}^{\infty}\frac{1}{n!}\sum_{i_{1},\cdots,i_{n}=1}^{\infty}S_{0,n;i_{1}\cdots
i_{n}}\alpha^{i_{1}}\cdots\alpha^{i_{n}}$ it follows that

(89)
$\mathcal{S}_{0}(\mathbf{a}^{1}-\mathbf{a}^{1}_{0},\cdots,\mathbf{a}^{g}-\mathbf{a}^{g}_{0})=\sum_{n=1}^{\infty}\frac{1}{n!}\sum_{i_{1},\cdots,i_{n}}^{g}S_{0,n;i_{1},\cdots,i_{n}}(\mathbf{a}^{i_{1}}-\mathbf{a}^{i_{1}}_{0})\cdots(\mathbf{a}^{i_{n}}-\mathbf{a}^{i_{n}}_{0}).$

On the other hand, Proposition 2.6.2 and the fact that
$\oint_{B_{i}}\eta^{j}=0$ if $j>g$ and $\oint_{B_{i}}\eta^{j}=2\pi
i\delta_{i}^{j}$ if $j\leq g$ implies that for $i_{1},\cdots,i_{n}=1,\cdots,g$
we have

(90) $S_{0,n;i_{1},\cdots,i_{n}}=\left(\frac{1}{2\pi
i}\right)^{n}\oint_{p_{1}\in B_{i_{1}}}\cdots\oint_{p_{n}\in
B_{i_{n}}}\omega_{0,n}(p_{1},\cdots,p_{n}).$

Combining (88), (89) and (90) the theorem is proven. ∎

######  Remark 2.7.4.

In Theorem 2.7.3, we can take $\omega_{0,1}(z_{\alpha})=y_{\alpha}dx_{\alpha}$
but other choices for $\omega_{0,1}(p)$ are equally valid as long as
$\omega_{0,1}(z_{\alpha})-\omega_{0,1}(-z_{\alpha})=4z_{\alpha}^{2}dz_{\alpha}$
is satisfied. Note that although
$\omega_{0,1}(z_{\alpha})=y_{\alpha}dx_{\alpha}$ looks very similar to
$-\theta_{\Sigma_{0}}$ which locally takes the form
$\theta_{\Sigma_{0}}|_{\Sigma_{0}\cap U_{p}}=-(y_{p}+f_{p}(x_{p}))dx_{p}$, in
general $\omega_{0,1}$ and $-\theta_{\Sigma_{0}}$ cannot be the same. Firstly,
$\omega_{1,0}$ is usually defined only locally near ramification points
whereas $\theta_{\Sigma_{0}}$ is defined on
$\Sigma_{0}\setminus\cup_{\alpha\in
Ram}\mathbb{D}_{\alpha,\epsilon_{\alpha}}(\Sigma_{0})$. Moreover, it is
generally not true that
$\theta_{\Sigma_{0}}(-z_{\alpha})-\theta_{\Sigma_{0}}(z_{\alpha})=4z_{\alpha}^{2}dz_{\alpha}$
so we cannot choose $\omega_{1,0}=-\theta_{\Sigma_{0}}$. For example, let
$[\Sigma_{1}]\in\mathcal{B}_{\Sigma_{0}}$ be a nearby curve of $\Sigma_{0}$
with the local parameterization:

$\gamma_{\alpha}(\Sigma_{1})=\left(x_{\alpha}=z_{\alpha}^{2}+a_{\alpha}(\Sigma_{1}),y_{\alpha}=b_{0\alpha}(\Sigma_{1})+b_{1\alpha}(\Sigma_{1})z_{\alpha}+b_{2\alpha}(\Sigma_{1})z^{2}_{\alpha}+\cdots\right).$

Let us construct $\theta\in\Gamma(\mathcal{B}_{\Sigma_{0}},G)$ by patching
together $\theta_{\Sigma}|_{\Sigma\cap
U_{p}}:=-y_{p}(x_{p};\Sigma)dx_{p}+y_{p}(x_{p};\Sigma_{1})dx_{p}$ as explained
in Section 2.5.7 (replacing $\mathcal{B}_{\Sigma_{0}}$ by
$\mathcal{B}_{\Sigma_{0}}\cap\mathcal{B}_{\Sigma_{1}}$ if necessary). As in
the proof of Lemma 2.5.37, on the annulus
$\mathbb{A}_{\alpha,\epsilon_{\alpha},M_{\alpha}}(\Sigma_{0})\subset\Sigma_{0}\cap
U_{\alpha}$, we have

$\displaystyle\theta_{\Sigma_{0}}(z_{\alpha})$
$\displaystyle=\theta_{t_{\alpha}(\Sigma_{0})}(z_{\alpha})-\exp\left(-a_{\alpha}(\Sigma_{1})\mathcal{L}_{\frac{1}{2z_{\alpha}}\partial_{z_{\alpha}}}\right)\theta_{t_{\alpha}(\Sigma_{1})}(z_{\alpha})$
$\displaystyle=-z_{\alpha}d(z^{2}_{\alpha})+\sum_{k=0}^{\infty}b_{k\alpha}(\Sigma_{1})z^{k}_{\alpha}\left(\sqrt{1-\frac{a_{\alpha}(\Sigma_{1})}{z_{\alpha}^{2}}}\right)^{k}d(z_{\alpha}^{2}).$

From which it follows that

$\theta_{\Sigma_{0}}(-z_{\alpha})-\theta_{\Sigma_{0}}(z_{\alpha})=4z^{2}_{\alpha}dz_{\alpha}-2\sum_{k=0}^{\infty}b_{2k+1,\alpha}(\Sigma_{1})\left(\sqrt{z_{\alpha}^{2}-a_{\alpha}(\Sigma_{1})}\right)^{2k+1}d(z^{2}_{\alpha})\neq
4z_{\alpha}^{2}dz_{\alpha}.$

However, in some exceptional cases such as those we are going to examine in
the examples below, we can technically set
$\omega_{1,0}:=-\theta_{\Sigma_{0}}$. Suppose that there exists a meromorphic
differential form $dS(\Sigma)$ on each $[\Sigma]\in\mathcal{B}_{\Sigma_{0}}$,
varying holomorphically over $\mathcal{B}_{\Sigma_{0}}$, such that
$\nabla_{\mathcal{F}}dS(\Sigma)=\phi_{\Sigma}$ and $dS(\Sigma)|_{\Sigma\cap
U_{\alpha}}=-(y_{\alpha}+f_{\alpha}(x_{\alpha}))dx_{\alpha}|_{\Sigma\cap
U_{\alpha}}$ for some holomorphic function $f_{\alpha}(x_{\alpha})$ for all
$\alpha\in Ram$. Then we can define the map
$\Phi_{\Sigma_{0}}:\mathcal{B}_{\Sigma_{0}}\rightarrow G_{\Sigma_{0}}$ by
$\Phi_{\Sigma_{0}}(\Sigma):=-dS(\Sigma_{0})+s^{*}_{\Sigma_{0},\Sigma}dS(\Sigma)$.
If $dS(\Sigma)$ has no residues, then it effectively plays the role of
$\theta_{\Sigma}$, even though $dS(\Sigma)$ might not be in $G_{\Sigma}$
because it may have poles at non-ramification points on $\Sigma$. In
particular, we can set
$\boldsymbol{\theta}_{\Sigma_{0}}:=[dS(\Sigma_{0})]\in\mathcal{H}_{\Sigma_{0}}$.
Since
$(dS(\Sigma_{0}))(-z_{\alpha})-dS(\Sigma_{0}))(z_{\alpha})=4z_{\alpha}^{2}dz_{\alpha}$
we can apply Theorem 2.7.3 with $\omega_{1,0}:=dS(\Sigma_{0})$. An example of
this case is the Seiberg-Witten family of curves given in Example 2.7.6. If
$dS(\Sigma)$ turns out to be holomorphic, then we can define
$\theta_{\Sigma}=dS(\Sigma)\in G_{\Sigma}$ and apply Theorem 2.7.3 with
$\omega_{0,1}:=-\theta_{\Sigma_{0}}$. Example 2.7.5 below provides an example
of this scenario.

######  Example 2.7.5.

Let us consider the family of curves arising from the _Hitchin systems_ [18,
3]. Let $\mathcal{M}_{n,d}$ be the moduli space of rank $n$ degree $d$ semi-
stable _Higgs Bundles_ on a compact genus $\bar{g}$ Riemann surface
$\mathcal{C}$. Given a holomorphic vector bundle $E$, we define the _degree_
by $\deg E:=\int_{\mathcal{C}}c_{1}(E)$. A rank $n$ degree $d$ semi-stable
Higgs Bundle is a pair $(E,\Phi)$, where $E$ is a rank $n$ holomorphic vector
bundle on $\mathcal{C}$ and $\Phi:E\rightarrow
E\otimes\Omega_{\mathcal{C}}^{1}$ is a bundle map, such that

$\frac{\deg F}{\text{rank}F}\leq\frac{\deg E}{\text{rank}E}$

for every sub-bundle $F\subset E$ such that $\Phi|_{F}:F\rightarrow
F\otimes\Omega^{1}_{\mathcal{C}}$. We say that $(E,\Phi)$ is stable if the
inequality is strict. Let
$\mathcal{B}:=\bigoplus_{k=1}^{n}\Gamma(\mathcal{C},\Omega^{\otimes
k}_{\mathcal{C}})$ be the base of the _Hitchin map_
$h:\mathcal{M}_{n,d}\rightarrow\mathcal{B}$ given by
$h(E,\Phi)=(p_{1}(\Phi),\cdots,p_{n}(\Phi))$ where
$x^{n}+p_{1}(\Phi)x^{n-1}+\cdots+p_{n}(\Phi)=\det(x\cdot Id-\Phi)$. A fiber of
the Hitchin map over $b=(b_{1},\cdots,b_{n})\in\mathcal{B}$ can be described
by a spectral curve $\Sigma_{b}\subset T^{*}\mathcal{C}$ given by the zero
divisor of the following global section:

$\lambda^{n}+\pi^{*}b_{1}\lambda^{n-1}+\cdots+\pi^{*}b_{n}\in\Gamma(T^{*}\mathcal{C},\pi^{*}(\Omega^{1}_{\mathcal{C}})^{\otimes
n}).$

Where $\pi:T^{*}\mathcal{C}\rightarrow\mathcal{C}$ is the projection morphism
and $\lambda\in\Gamma(T^{*}\mathcal{C},\pi^{*}\Omega^{1}_{\mathcal{C}})$
denotes the tautological section:
$\lambda(p)=p\in(\pi^{*}\Omega_{\mathcal{C}}^{1})_{p}$.

It follows that $\mathcal{B}$ is a family of genus $g:=2n^{2}(\bar{g}-1)+2$
compact Riemann surfaces $\Sigma$ embedded inside the foliated symplectic
surface $(S,\Omega_{S},\mathcal{F})$ where $S:=T^{*}\mathcal{C}$,
$\mathcal{F}$ is given by the cotangent fibers and $\Omega_{S}:=-d\Theta$
where $\Theta\in\Gamma(T^{*}\mathcal{C},\pi^{*}\Omega^{1}_{\mathcal{C}})$ is
the tautological $1$-form. Denote by $\mathcal{B}^{reg}$, the regular locus of
$\mathcal{B}$ where $\Sigma$ are smooth for all
$[\Sigma]\in\mathcal{B}^{reg}$. For each $[\Sigma]\in\mathcal{B}^{reg}$ we
define
$\theta_{\Sigma}:=-\Theta|_{\Sigma}\in\Gamma(\Sigma,\Omega^{1}_{\Sigma})$. Let
$U\subset\mathcal{C}$ be an open subset with local coordinate $q$, then
$(T^{*}U,q,p),T^{*}U\subset T^{*}\mathcal{C}$ is a
$(\mathcal{F},\Omega_{S})$-chart because $\Theta|_{T^{*}U}=pdq$ means
$\Omega_{S}|_{T^{*}U}=dq\wedge dp$ and the foliation is given by $p=const$.
Moreover, we have $\theta_{\Sigma}|_{\Sigma\cap T^{*}U}=-pdq|_{\Sigma\cap
T^{*}U}$ which implies $\nabla_{\mathcal{F}}\theta_{\Sigma}=\phi_{\Sigma}$.
Let

$z^{i}:=\oint_{A_{i}}\theta_{\Sigma},\qquad
w_{i}:=\oint_{B_{i}}\theta_{\Sigma}=\frac{\partial\mathfrak{F}_{\Sigma_{0}}}{\partial
z^{i}}(z^{1},\cdots,z^{g})$

where $\mathfrak{F}_{\Sigma_{0}}$ is the prepotential. In fact, in this case
the prepotential is simple enough to be written down explicitly:
$\mathfrak{F}_{\Sigma_{0}}(z^{1},\cdots,z^{g})=\sum_{i=1}^{g}z^{i}w_{i}=\frac{1}{2}\sum_{i,j=1}^{g}\tau_{ij}(z^{1},\cdots,z^{g})z^{i}z^{j}$
[3, Proposition 5.11].

Now, consider the topological recursion on $[\Sigma_{0}]\in\mathcal{B}^{reg}$
with $\omega_{0,1}:=-\theta_{\Sigma_{0}}$ and $\omega_{0,2}=B$ the normalized
Bergman kernel. If we picked a $(\mathcal{F},\Omega_{S})$-chart
$(T^{*}U_{\alpha},p_{\alpha},q_{\alpha})$ such that $r_{\alpha}(\Sigma_{0})\in
T^{*}U_{\alpha}$ and
$\gamma_{\alpha}(\Sigma_{0})=(q_{\alpha}=z_{\alpha}^{2},p_{\alpha}=z_{\alpha})$,
then $\theta_{\Sigma_{0}}|_{\Sigma_{0}\cap
T^{*}U_{\alpha}}=-p_{\alpha}dq_{\alpha}|_{\Sigma\cap
T^{*}U_{\alpha}}=-z_{\alpha}d(z_{\alpha}^{2})$ and therefore
$\omega_{0,1}(z_{\alpha})-\omega_{0,1}(-z_{\alpha})=4z_{\alpha}^{2}dz_{\alpha}$.
It follows from Theorem 2.7.3 that formula (83) relating
$\mathfrak{F}_{\Sigma_{0}}=\mathfrak{F}_{\Sigma_{0}}(z^{1},\cdots,z^{g})$ and
$\omega_{0,n}$ holds in this setting over some open neighbourhood
$\mathcal{B}_{\Sigma_{0}}\subset\mathcal{B}^{reg}$ of $[\Sigma_{0}]$. The
exact same result was obtained earlier by Baraglia-Huang [3, Theorem 7.4]
using a different technique. Note that in comparing our (83) with [3, Theorem
7.4], we need to replace $z^{i}$ by $\bar{z}^{i}:=-z^{i}$ and
$\mathfrak{F}_{\Sigma_{0}}=\mathfrak{F}_{\Sigma_{0}}(z^{1},\cdots,z^{g})$ by
$\bar{\mathfrak{F}}_{\Sigma_{0}}=\bar{\mathfrak{F}}_{\Sigma_{0}}(\bar{z}^{1},\cdots,\bar{z}^{g}):=\mathfrak{F}_{\Sigma_{0}}(-\bar{z}^{1},\cdots,-\bar{z}^{g})$
because $\theta_{\Sigma}$ in [3] was defined to be $+\Theta|_{\Sigma}$ instead
of $-\Theta|_{\Sigma}$.

######  Example 2.7.6.

Let $\mathcal{B}$ be the family of smooth Seiberg-Witten curves (see Section
1.1):

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

It can be shown that each Seiberg-Witten curve $\Sigma(u)$ can be embedded
inside a foliated symplectic surface $(S,\Omega_{S},\mathcal{F})$ where $S$ is
a certain compactification of $\mathbb{C}^{2}$. Let us consider the dense
subset $U:=(\mathbb{C}\setminus(-\infty,0])\times\mathbb{C}\subset S$ with
coordinates $(\log w,z)$ and suppose without loss of generality that
$\Sigma_{0}=\Sigma(u_{0})$ is chosen such that all ramification points
$r_{\alpha}(\Sigma_{0})$ are contained in $U$ (we simply pick a different
branch-cut for $\log$ otherwise). On $U$, the symplectic form is
$\Omega_{S}|_{U}=\frac{dw}{w}\wedge dz$ and the foliation is given by
$w=const$, hence $(U,\log w,z)$ is a $(\mathcal{F},\Omega_{S})$-chart. The
Seiberg-Witten differential is given by $dS_{SW}:=z\frac{dw}{w}$, or
$dS_{SW}(\Sigma(u))=z\frac{dw}{w}|_{\Sigma(u)}$ if we would like to emphasize
that we are considering $dS_{SW}$ as a differential form on $\Sigma(u)$. We
define $A,B$-periods and the Seiberg-Witten prepotential
$\mathfrak{F}^{SW}_{\Sigma_{0}}=\mathfrak{F}^{SW}_{\Sigma_{0}}(a^{1},\cdots,a^{g})$
by

$a^{i}:=\oint_{A_{i}}dS_{SW}(\Sigma(u)),\qquad
b_{i}:=\oint_{B_{i}}dS_{SW}(\Sigma(u))=\frac{\partial\mathfrak{F}^{SW}_{\Sigma_{0}}}{\partial
a^{i}}(a^{1},\cdots,a^{g}),$

over some open neighbourhood $\mathcal{B}_{\Sigma_{0}}$ of
$[\Sigma_{0}]\in\mathcal{B}$. By definition of the symplectic form
$\Omega_{S}$, we can see that
$\nabla_{\mathcal{F}}dS_{SW}(\Sigma(u))=-\phi_{\Sigma(u)}$ and hence
$\boldsymbol{\theta}_{\Sigma(u)}:=-[dS_{SW}(\Sigma(u))]\in\mathcal{H}_{\Sigma(u)}$.
The negative sign means that we will need to identify $\mathbf{a}^{i}:=-a^{i}$
and
$\mathfrak{F}^{SW}_{\Sigma_{0}}(a^{1},\cdots,a^{g})=\mathfrak{F}_{\Sigma_{0}}(\mathbf{a}^{1},\cdots,\mathbf{a}^{g})$
after our application of Theorem 2.7.3 later on. Choose a new
$(\mathcal{F},\Omega_{S})$-chart $(U_{\alpha},x_{\alpha},y_{\alpha})$ around
every ramification points $r_{\alpha}$ of $\Sigma_{0}$ such that
$\gamma_{\alpha}(\Sigma_{0})=(x_{\alpha}=z^{2}_{\alpha},y_{\alpha}=z_{\alpha})$.
Then $(\log w,z)$ and $(x_{\alpha},y_{\alpha})$ must be related by the
coordinate transformation of the form (7):

$\log w=F_{\alpha}(x_{\alpha}),\qquad
z=\frac{y_{\alpha}}{F^{\prime}(x_{\alpha})}+G^{\prime}(x_{\alpha}).$

Therefore,

$\displaystyle(dS_{SW}(\Sigma_{0}))(z_{\alpha})-(dS_{SW}(\Sigma_{0}))(-z_{\alpha})$
$\displaystyle=\left(\frac{y_{\alpha}(z_{\alpha})}{F^{\prime}(x_{\alpha}(z_{\alpha}))}+G^{\prime}(x_{\alpha}(z_{\alpha}))\right)F^{\prime}(x_{\alpha}(z_{\alpha}))dx_{\alpha}(z_{\alpha})$
$\displaystyle\qquad-\left(\frac{y_{\alpha}(-z_{\alpha})}{F^{\prime}(x_{\alpha}(-z_{\alpha}))}+G^{\prime}(x_{\alpha}(-z_{\alpha}))\right)F^{\prime}(x_{\alpha}(-z_{\alpha}))dx_{\alpha}(-z_{\alpha})$
$\displaystyle=y_{\alpha}(z_{\alpha})dx_{\alpha}(z_{\alpha})-y_{\alpha}(-z_{\alpha})dx_{\alpha}(-z_{\alpha})=4z_{\alpha}^{2}dz_{\alpha}.$

Using Theorem 2.7.3, we find that on some
$\mathcal{B}_{\Sigma_{0}}\subseteq\mathcal{B}$ the relation (83) holds between
the Seiberg-Witten prepotential $\mathfrak{F}^{SW}_{\Sigma_{0}}$ and
$\omega_{0,n}$ produced from topological recursion on $\Sigma_{0}$ using
$\omega_{0,1}:=dS_{SW}(\Sigma_{0})$ and $\omega_{0,2}=B$, where $B=B(p,q)$ is
the normalized Bergman kernel on $\Sigma_{0}$. In Section 3 we are going to
study this example in more detail. In particular, we will construct the needed
foliated symplectic surface $(S,\Omega_{S},\mathcal{F})$ and explicitly show
the calculations leading to (83) without referring to Theorem 2.7.3.

##  3\. Seiberg-Witten Prepotential and Topological Recursion

In this section we demonstrate the Kontsevich-Soibelman approach to
topological recursion we have studied in Section 2 using an explicit example
of the genus $g\geq 1$ Seiberg-Witten family of curves, parameterized by
$u=(u_{1},\cdots,u_{g})$:

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

In the end, we obtain the formula (117) relating the Seiberg-Witten
prepotential
$\mathfrak{F}^{SW}_{\Sigma_{0}}=\mathfrak{F}^{SW}_{\Sigma_{0}}(\\{a^{i}\\})$
to the genus zero part of topological recursion on a Seiberg-Witten curve with
$\omega_{0,1}=dS_{SW}(\Sigma)$ on some open neighbourhood
$\mathcal{B}_{\Sigma_{0}}\subseteq\mathcal{B}$ of
$[\Sigma_{0}]\in\mathcal{B}$. Note that the energy scale parameter $\Lambda$
plays no role here and we will set $\Lambda=1$. Alternatively, we can simply
absorb $\Lambda$ into $u_{k}$ and $z$ by re-defining
$\frac{u_{k}}{\Lambda^{g+2-k}}\mapsto u_{k}$ and $\frac{z}{\Lambda}\mapsto z$.
First, we will show in Section 3.1 that the family of smooth Seiberg-Witten
curves

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

is the moduli space of $\mathcal{F}$-transversal genus $g$ curves in some
foliated symplectic surface $(S,\Omega_{S},\mathcal{F})$. Once this is done,
using Theorem 2.7.3, the formula (117) immediately follows. However, the
purpose of this section is to demonstrate various calculations in Section 2 in
the light of the Seiberg-Witten family of curves. Therefore, in the remainder
of this section, we will not refer to Theorem 2.7.3, in fact, we will re-prove
the Seiberg-Witten case of Theorem 2.7.3 in Section 3.3. Some other relevant
details will be presented in Section 3.2, such as the choice of
$\\{\epsilon_{\alpha\in Ram}\\},\\{M_{\alpha\in
Ram}\\},\mathcal{U}_{Ram},\mathcal{B}_{\Sigma_{0}}$ and the verification of
Proposition 2.5.38 in the context of Seiberg-Witten family of curves.

###  3.1. Embedding of Seiberg-Witten Family of Curves into a Foliated
Symplectic Surface

####  3.1.1. Outline of the construction of the foliated symplectic surface
$(S,\Omega_{S},\mathcal{F})$

To apply the technique we have discussed in Section 2 we need to find a
foliated symplectic surface $(S,\Omega_{S},\mathcal{F})$, such that any smooth
genus $g\geq 1$ Seiberg-Witten curve $\Sigma(u)$, in the family $\mathcal{B}$,
can be embedded into $S$ as an $\mathcal{F}$-transversal curve. We require
that $S$ contains a dense open subset $U=\mathbb{C}^{*}\times\mathbb{C}$ where
$\mathbb{C}^{*}:=\mathbb{C}\setminus\\{0\\}$, which can be covered using two
$(\mathcal{F},\Omega_{S})$-charts $(U_{i},\log w,z),i=1,2$ given by any two
different branch-cuts of $\log$, $\Omega_{S}|_{U}=\frac{dw}{w}\wedge dz$ and
the foliation given by $w=const$. This is so that the section
$\theta\in\Gamma(\mathcal{B}_{\Sigma_{0}},G)$ introduced in Section 2.5.7 will
be related to the Seiberg-Witten differential $dS_{SW}=z\frac{dw}{w}$. In
other words: if $dS_{SW}$ is treated as a differential form on $U$ then
$d(dS_{SW})|_{U}=\Omega_{S}|_{U}$.

######  Remark 3.1.1.

We cannot simply set $S=\mathbb{C}^{*}\times\mathbb{C}$ or $\mathbb{C}^{2}$
because Seiberg-Witten curves $\Sigma(u)$ are non-compact as curves in
$\mathbb{C}^{*}\times\mathbb{C}$ or $\mathbb{C}^{2}$.

There are many ways to construct $(S,\Omega_{S},\mathcal{F})$ but let us now
describe one of the possible ways. We embed Seiberg-Witten curves $\Sigma(u)$
into $\mathbb{P}^{2}$ as follows

$\Sigma(u):=\left\\{[w:z:x]\in\mathbb{P}^{2}\ |\
w^{2}x^{g}-w(z^{g+1}+u_{g}x^{2}z^{g-1}+\cdots+u_{1}x^{g+1})+x^{g+2}=0\right\\}\subset\mathbb{P}^{2}.$

When $g>1$ the curve $\Sigma(u)$ embeds non-smoothly into $\mathbb{P}^{2}$
with a singular point of multiplicity $g$ at $[w:z:x]=[1:0:0]$. There also
exists no globally defined holomorphic symplectic forms on $\mathbb{P}^{2}$
because the canonical divisor of $\mathbb{P}^{2}$ is
$K_{\mathbb{P}^{2}}\sim-3H$, where $H$ is the hyperplane divisor. In other
words, any symplectic form on $\mathbb{P}^{2}$ is meromorphic with pole
divisor a degree 3 curve. If $\Sigma(u)$ intersects $K_{\mathbb{P}^{2}}$
trivially we could still consider $S$ to be an open subset of $\mathbb{P}^{2}$
containing $\Sigma(u)$ but not $K_{\mathbb{P}^{2}}$, hence it is symplectic.
However, Bézout’s theorem guarantees that $\Sigma(u)$ intersects
$K_{\mathbb{P}^{2}}$ at $3(g+2)$ points counting multiplicity. For these
reasons, we cannot simply take $S$ to be $\mathbb{P}^{2}$ or any open subset
of $\mathbb{P}^{2}$.

In particular, let us consider
$\Omega_{\mathbb{P}^{2}}:=\frac{d(w/x)}{w/x}\wedge d\left(\frac{z}{x}\right)$.
We define divisors $H_{w}:=\\{w=0\\},H_{x}:=\\{x=0\\}$ and $H_{z}:=\\{z=0\\}$
on $\mathbb{P}^{2}$. Then it is clear that $\Omega_{\mathbb{P}^{2}}$ reduces
to $dz\wedge\frac{dw}{w}$ on $\mathbb{C}^{2}=\mathbb{P}^{2}\setminus H_{x}$
where we can set $x=1$. We can see that $\Omega_{\mathbb{P}^{2}}$ has order
$1$ pole along the hyperplane $H_{w}\subset\mathbb{P}^{2}$ and order $2$ pole
along $H_{x}\subset\mathbb{P}^{2}$. Hence the canonical divisor is

(91) $K_{\mathbb{P}^{2}}=(\Omega_{\mathbb{P}^{2}})=-H_{w}-2H_{x}$

and $\Sigma(u)$ intersects $K_{\mathbb{P}^{2}}$ at $[w:z:x]=[0:1:0]$ and
$[w:z:x]=[1:0:0]$.

We are going to show that the symplectic surface $S$ can be obtained by
blowing-up $\mathbb{P}^{2}$ repeatedly starting from points $[0:1:0]$ and
$[1:0:0]$ until $\Sigma(u)$ no longer intersects the canonical divisor.

$H_{x}$$H_{z}$$H_{w}$$2$$1$$0$$\Sigma(u)$

Figure 2.

$\mathbb{P}^{2}$ with divisors $H_{w}:=\\{w=0\\},H_{x}:=\\{x=0\\}$ and
$H_{z}:=\\{z=0\\}$. The numbers are order of poles of
$\Omega_{\mathbb{P}^{2}}$ on each respective divisor. We notice that
$\Sigma(u)$ intersects the point $[w:z:x]=[1:0:0]$ tangentially to $H_{x}$ and
intersects $[w:z:x]=[0:1:0]$ tangentially to $H_{w}$.

####  3.1.2. Some algebraic geometry background and intuition

The adjunction formula states that for a non-singular complex curve $C$
embedded in a complex surface $X$ then

(92) $\Omega_{C}^{1}\cong\Omega_{X}^{2}|_{C}\otimes\nu_{C}$

where $\nu_{C}$ is the holomorphic normal bundle of $C$, $\Omega^{1}_{C}$ is
the holomorphic canonical line bundle of $C$ and $\Omega^{2}_{X}$ is the
holomorphic canonical sheaf of $X$. Therefore, if a genus $g$ Seiberg-Witten
curve $\Sigma(u)\subset\mathbb{P}^{2}$ was smooth and intersects
$K_{\mathbb{P}^{2}}$ trivially, then
$\Omega^{2}_{\mathbb{P}^{2}}|_{\Sigma(u)}\cong\mathcal{O}_{\Sigma(u)}$ and
(92) implies that $\nu_{\Sigma(u)}\cong\Omega_{\Sigma(u)}^{1}$. Instead,
$\Sigma(u)$ is singular and intersects non-trivially with $K_{\mathbb{P}^{2}}$
and we have from Bézout’s theorem that

$\deg\nu_{\Sigma(u)}=\Sigma(u)\cdot\Sigma(u)=(\deg\Sigma(u))^{2}=(g+2)^{2}>2g-2=\deg
K_{\Sigma(u)}$

where $N_{\Sigma(u)}$ denotes the normal bundle divisor, and so
$\nu_{\Sigma(u)}\ncong\Omega^{1}_{\Sigma(u)}$. From Riemann-Roch, the
deformation space of $\Sigma(u)$ inside $\mathbb{P}^{2}$ has dimension

$\displaystyle\dim\Gamma(\Sigma(u),\nu_{\Sigma(u)})$
$\displaystyle=\dim\Gamma(\Sigma(u),\Omega^{1}_{\Sigma(u)}\otimes\nu^{*}_{\Sigma(u)})+\deg\nu_{\Sigma(u)}-g+1$
$\displaystyle=0+(g+2)^{2}-g+1=g^{2}+3g+5>g$

whereas the Seiberg-Witten family has dimension $g$ = number of parameters
$\\{u_{1},...,u_{g}\\}$. So the problem is that the deformation of $\Sigma(u)$
within the Seiberg-Witten family is more restrictive. In particular,
$\Sigma(u)$ will always pass through the point $[w:z:x]=[0:1:0]$ and
$[w:z:x]=[1:0:0]$ for every value of $\\{u_{1},...,u_{g}\\}$, whereas if we
are allowed to deform $\Sigma(u)$ freely it will not be fixed at these two
points.

It suggests that the problem can be solved if we construct $S$ by performing a
series of blow-ups starting from $\mathbb{P}^{2}$ until no such fix points
exists for the Seiberg-Witten family and each curve $\Sigma(u)$ is non-
singular. Conceptually, after each blow-up, the neighbourhood of $\Sigma(u)$
will become more restrictive and the dimension of its deformation space will
be reduced. This can be seen as follows. Given a curve $C$ inside a surface
$X$ and let $p\in X$. Then blowing up $p$ we have $\pi:\tilde{X}\rightarrow X$
and

$\pi^{*}C=\tilde{C}+mE,\qquad\pi^{*}C\cdot E=0.$

Where $\tilde{C}$ is the proper transform of $C$, $E$ is the exceptional
divisor and $m$ is the multiplicity of the point $p$. In particular, $m=0$ if
$p\notin C$, $m=1$ if $p\in C$ is a non-singular point and $m>1$ if $p\in C$
is a singular point of multiplicity $m$. The important thing for us is that

(93)
$\deg\nu_{\tilde{C}}=\tilde{C}\cdot\tilde{C}=(\pi^{*}C-mE)\cdot(\pi^{*}C-mE)=C\cdot
C-m^{2}=\deg\nu_{C}-m^{2}.$

Therefore, we have
$\dim\Gamma(\tilde{C},\nu_{\tilde{C}})\leq\dim\Gamma(C,\nu_{C})$ if
$\deg\Omega^{1}_{\tilde{C}}\leq\deg\nu_{\tilde{C}}$ by the application of
Riemann-Roch.

It is also well-known what happens to the canonical class after the blow-up.
We have

(94) $K_{\tilde{X}}=\pi^{*}K_{X}+E.$

We will apply (94) successively to $K_{\mathbb{P}^{2}}$ as given in (91) and
show that eventually, the canonical class will intersect the proper transform
of the Seiberg-Witten curve $\Sigma(u)$ trivially. Then we define $S$ to be
$\mathbb{P}^{2}$, after a series of blow-ups, excluding the canonical divisor.
We will also check for consistency using (93) that the degree of
$\nu_{\Sigma(u)}$ drops to $2g-2$. In the end, we conclude that the moduli
space $\mathcal{B}$ of $\mathcal{F}$-transversal genus $g$ curves in the
resulting foliated symplectic surface $(S,\Omega_{S},\mathcal{F})$ coincides
with the family of smooth Seiberg-Witten curves.

$H_{x}$$H_{z}$$H_{w}$$E_{w,1}$$E_{w,2}$$E_{w,3}$$E_{w,g}$$E_{w,g+1}$$E_{w,g+2}$$E_{w,g+3}$$E_{z,1}$$E_{z,2}$$E_{z,g+2}$$E_{z,g+1}$$E_{z,g}$$E_{z,g+3}$$E_{z,g+4}$$\Sigma(u)$$2$$2$$2$$2$$2$$2$$1$$1$$0$$0$$1$$2$$2$$2$$2$$1$$0$$...$$...$

Figure 3.

Summary of the construction of the foliated symplectic surface
$(S,\Omega_{S},\mathcal{F})$. We notice that $\Omega_{S}$ is holomorphic on
$E_{z,g+4}$ and $E_{w,g+3}$ which $\Sigma(u)$ intersects with transversely.

####  3.1.3. Constructing $S$. Part 1: Blowing up $[w:z:x]=[1:0:0]$

Let us start by considering the blow-up starting from $[w:z:x]=[1:0:0]$. We
consider the open set $U_{w}:=\\{w\neq
0\\}=\mathbb{A}^{2}\subset\mathbb{P}^{2}$. Set $w=1$, the local coordinates on
$U_{w}$ are $(z,x)$ and we have