S_{1}^{\prime\prime}(x_{\alpha})+2S_{2}^{\prime}(x_{\alpha})S_{0}^{\prime}(x_{\alpha})+\left(S_{1}^{\prime}(x_{\alpha})\right)^{2}=0.$
(6.9) $\displaystyle\hbar^{m+1}{\text{-terms}}:\quad
S_{m}^{\prime\prime}(x_{\alpha})+\sum_{a+b=m+1}S_{a}^{\prime}(x_{\alpha})S_{b}^{\prime}(x_{\alpha})=0,\quad
m\geq 2.$

The symbol ′ denotes the $x_{\alpha}$-derivative.

  * •

Solve (6.6), (6.7), and (6.8) to find $S_{0}(x_{\alpha}),S_{1}(x_{\alpha})$,
and $S_{2}(x_{\alpha})$.

  * •

Construct the normalization
$\tilde{\pi}:\widetilde{\Sigma}\longrightarrow\Sigma$ as in (4.14), and define

(6.10) $W_{0,1}:=\nu^{*}i^{*}\eta.$

  * •

Define

(6.11)
$F_{1,1}(z_{1})=-\int^{z_{1}}\frac{W_{0,2}\left(z_{1},\tilde{\sigma}(z_{1})\right)}{\Omega(z_{1})},$

where integration means a primitive of the meromorphic $1$-form
$\frac{W_{0,2}\left(z_{1},\tilde{\sigma}(z_{1})\right)}{\Omega(z_{1})}$ on
$\tilde{\pi}^{-1}U_{\alpha}$.

  * •

Define

(6.12)
$F_{0,3}(z_{1},z_{2},z_{3})=\iiint\left(-W(z_{1},z_{2},z_{3})+2\big{(}f(z_{1})+f(z_{2})+f(z_{3})\big{)}\right),$

where

(6.13)
$W(z_{1},z_{2},z_{3})=\frac{1}{\Omega(z_{1})}\bigg{(}W_{0,2}(z_{1},z_{2})W_{0,2}\big{(}z_{1},\tilde{\sigma}(z_{3})\big{)}+W_{0,2}(z_{1},z_{3})W_{0,2}\big{(}z_{1},\tilde{\sigma}(z_{2})\big{)}\bigg{)}\\\
+d_{2}\left(\frac{\omega^{\tilde{\sigma}(z_{2})-z_{2}}(z_{1})W_{0,2}(z_{2},\tilde{\sigma}(z_{3}))}{\Omega(z_{2})}\right)+d_{3}\left(\frac{\omega^{\tilde{\sigma}(z_{3})-z_{3}}(z_{1})W_{0,2}(z_{2},\tilde{\sigma}(z_{3}))}{\Omega(z_{3})}\right)$

and

(6.14)
$f(z):=\widetilde{S}_{2}(z)-\left(F_{1,1}(z)-\frac{1}{6}\iiint^{z}W(z_{1},z_{2},z_{3})\right).$

Here, $\widetilde{S}_{2}(z)=S_{2}\big{(}x_{\alpha}(z)\big{)}$ is the lift of
$S_{2}(x_{a})$ to $\tilde{\pi}^{-1}(U_{\alpha})\subset\widetilde{\Sigma}$.

  * •

Note that we have

(6.15)
$S_{2}(x_{\alpha})=F_{1,1}\big{(}z(x_{\alpha})\big{)}+\frac{1}{6}F_{0,3}\big{(}z(x_{\alpha}),z(x_{\alpha}),z(x_{\alpha})\big{)}$

for a local section $z:U_{\alpha}\longrightarrow\tilde{\pi}^{-1}(U_{\alpha})$.

  * •

Solve the PDE recursion (5.6) using the initial data (6.11) and (6.12), and
determine $F_{g,n}(z_{1},\dots,z_{n})$ for $2g-2+n\geq 2$.

  * •

Define

(6.16)
$S_{m}(x_{\alpha})=\sum_{2g-2+n=m-1}\frac{1}{n!}F_{g,n}\big{(}z(x_{\alpha})\big{)},\qquad
m\geq 3,$

where $F_{g,n}\big{(}z(x_{\alpha})\big{)}$ is the principal specialization of
$F_{g,n}(z_{1},\dots,z_{n})$ evaluated at a local section $z=z(x_{\alpha})$ of
$\tilde{\pi}:\widetilde{\Sigma}\longrightarrow C$ on $U_{\alpha}$.

Then (6.4) gives the WKB expansion of the solution to the generator of the
Rees ${\mathcal{D}}_{C}$-module.

######  Remark 6.2.

The WKB method is to solve (6.5) iteratively and find $S_{m}(x_{\alpha})$ for
all $m\geq 0$. Here, (6.6) is the semi-classical limit of (6.3), and (6.7) is
the consistency condition we need for solving the WKB expansion. Since the
$1$-form $dS_{0}(x)$ is a local section of $T^{*}C$, we identify
$y=S_{0}^{\prime}(x)$. Then (6.6) is the local expression of the spectral
curve equation (6.2). This expression is the same everywhere on
$C\setminus{\text{\rm{supp}}}(\Delta)$. We note $q$ is globally defined.
Therefore, we recover the spectral curve $\Sigma$ from the differential
operator (6.1).

######  Remark 6.3.

$W_{1,1}:=dF_{1,1}$ and $W_{0,3}:=d_{1}d_{2}d_{3}F_{0,3}$ are solutions of
(5.5) for $2g-2+n=1$ with respect to the contour of integration along the
diagonal divisors mentioned in Remark 5.7.

The key idea of [23, 24] is the principal specialization of symmetric
functions, which in our case means _restriction_ of a PDE on a symmetric
function to the main diagonal of the variables. Differential forms pull back,
but PDEs do not. The essence of Theorem 6.1 is that the principal
specialization of PDE recursion (5.6) is exactly the quantum curve equation
(6.3).

######  Lemma 6.4 ([58, Lemma A.1]).

Let $f(z_{1},\dots,z_{n})$ be a symmetric function in $n$ variables. Then

(6.17) $\displaystyle\frac{d}{dz}f(z,z,\dots,z)$
$\displaystyle=n\left.\left[\frac{\partial}{\partial
u}f(u,z,\dots,z)\right]\right|_{u=z};$
$\displaystyle\frac{d^{2}}{dz^{2}}f(z,z,\dots,z)$
$\displaystyle=n\left.\left[\frac{\partial^{2}}{\partial
u^{2}}f(u,z,\dots,z)\right]\right|_{u=z}$
$\displaystyle\qquad+n(n-1)\left.\left[\frac{\partial^{2}}{\partial
u_{1}\partial u_{2}}f(u_{1},u_{2},z,\dots,z)\right]\right|_{u_{1}=u_{2}=z}.$

For a function in one variable $f(z)$, we have

(6.18) $\lim_{z_{2}\rightarrow
z_{1}}\left[\omega^{z_{2}-b}(z_{1})\big{(}f(z_{1})-f(z_{2})\big{)}\right]=d_{1}f(z_{1}),$

where $\omega^{z_{2}-b}(z_{1})$ is the $1$-form of (5.4).

###### Proof of Theorem 6.1.

First let $P\in\Sigma\cap C_{\infty}$ be an odd cusp singularity on the fiber
$\pi^{-1}(p)$ of a point $p\in C=C_{0}\subset\overline{T^{*}C}$. The quadratic
differential $q$ has a pole of odd order at $p$, and the normalization
$\tilde{\pi}:\widetilde{\Sigma}\longrightarrow C$ is simply ramified at a
point $Q\in\widetilde{\Sigma}$ over $p$. We choose a local projective
coordinate $x$ on $C$ centered at $p$. The Galois action of $\tilde{\sigma}$
on $\widetilde{\Sigma}$ fixes $Q$. As we have shown in Case 1 of the proof of
Theorem 4.2, locally over $p$, the spectral curve $\Sigma$ has the shape

$z_{0}^{2}=c(x)x^{2\mu+1},$

where $z_{0}=1/y$, $y$ is the fiber coordinate on $T_{p}^{*}C$, and $c(x)$ is
a unit $c(x)\in{\mathcal{O}}_{C,p}^{*}$. The quadratic differential $q$ has a
local expression

$q=\frac{1}{c(x)x^{2\mu+1}}.$

Define $z_{1}=z_{0}/x$. The proper transform of $\widetilde{\Sigma}$ after the
first blow up at $P$ is locally written by

$z_{1}^{2}=c(x)x^{2\mu-1}.$

Note that $z_{1}$ is an affine coordinate of the first exceptional divisor.
Repeating this process $\mu$-times, we end up with a coordinate
$z_{\mu-1}=z_{\mu}x$ and an equation

$z_{\mu}^{2}=c(x)x.$

Here again, $z_{\mu}$ is an affine coordinate of the exceptional divisor
created by the $\mu$-th blow-up. Write $z=z_{\mu}$ so that the proper
transform of the $\mu$-times blow-ups is given by

(6.19) $z^{2}=c(x)x.$

Note that the Galois action of $\tilde{\sigma}$ at $Q$ is simply
$z\longmapsto-z$. Solving (6.19) as a functional equation, we obtain a Galois
invariant local expression

(6.20) $x=x(z)=c_{Q}(z^{2})z^{2},$

where $c_{Q}\in{\mathcal{O}}_{\widetilde{\Sigma},Q}^{*}$ is a unit element.
This function is precisely the local expression of the normalization
$\tilde{\pi}:{\widetilde{\Sigma}}\longrightarrow C$ at
$Q\in{\widetilde{\Sigma}}$. Since

$z=z_{\mu}=\frac{z_{0}}{x^{\mu}}=\frac{1}{yx^{\mu}},$

we have thus obtained the normalization coordinate $z$ on the desingularized
curve ${\widetilde{\Sigma}}$ near $Q$:

(6.21) $\begin{cases}x=x(z)=c_{Q}(z^{2})z^{2}\\\
y=y(z)=\frac{1}{zx^{\mu}}=c_{Q}(z^{2})^{-\mu}\cdot z^{-2\mu-1}\end{cases}.$

It gives a parametric equation for the singular spectral curve $\Sigma$:

$y^{2}=\frac{1}{c(x)x^{2\mu+1}}.$

As we have shown in the proof of Theorem 4.2, the situation is the same for a
zero of $q$ of odd order.

For the purpose of local calculation near $Q\in{\widetilde{\Sigma}}$, we use
the following local expressions:

(6.22) $\displaystyle\omega^{{\tilde{\sigma}}(z)-z}(z_{1})$
$\displaystyle=\left(\frac{1}{z_{1}-{\tilde{\sigma}}(z)}-\frac{1}{z_{1}-z}+O(1)\right)dz_{1}$
(6.23) $\displaystyle\eta$
$\displaystyle=ydx=h(z)dz:=\frac{1}{z^{2\mu}}\big{(}1+O(z)\big{)}dz.$

Here, we adjust the normalization coordinate $z$ by a constant factor to make
(6.23) simple.

Using the notation $\partial_{z}=\partial/\partial z$, we have a local formula
equivalent to (5.6) that is valid for $2g-2+n\geq 2$:

(6.24) $\partial_{z_{1}}F_{g,n}(z_{1},\dots,z_{n})\\\
=-\sum_{j=2}^{n}\left[\frac{\omega^{z_{j}-{\tilde{\sigma}}(z_{j})}(z_{1})}{2h(z_{1})dz_{1}}\cdot\partial_{z_{1}}F_{g,n-1}\big{(}z_{[\hat{j}]}\big{)}-\frac{\omega^{z_{j}-{\tilde{\sigma}}(z_{j})}(z_{1})}{dz_{1}\cdot
2h(z_{j})}\cdot\partial_{z_{j}}F_{g,n-1}\big{(}z_{[\hat{1}]}\big{)}\right]\\\
-\frac{1}{2h(z_{1})}\frac{\partial^{2}}{\partial u_{1}\partial
u_{2}}\left.\left[F_{g-1,n+1}\big{(}u_{1},u_{2},z_{[\hat{1}]}\big{)}+\sum_{\begin{subarray}{c}g_{1}+g_{2}=g\\\
I\sqcup
J=[\hat{1}]\end{subarray}}^{\text{stable}}F_{g_{1},|I|+1}(u_{1},z_{I})F_{g_{2},|J|+1}(u_{2},z_{J})\right]\right|_{\begin{subarray}{c}u_{1}=z_{1}\\\
u_{2}=z_{1}\end{subarray}}.$

Let us apply principal specialization. The left-hand side becomes
$\frac{1}{n}\partial_{z}F_{g,n}(z,\dots,z)$. To calculate the contributions
from the first line of the right-hand side of (6.24), we choose $j>1$ and set
$z_{i}=z$ for all $i$ except for $i=1,j$. Then take the limit
$z_{j}\rightarrow z_{1}$. In this procedure, we note that the contributions
from the simple pole of $\omega^{z_{j}-{\tilde{\sigma}}(z_{j})}(z_{1})$ at
$z_{1}={\tilde{\sigma}}(z_{j})$ cancel at $z_{1}=z_{j}$. Thus we obtain

$\left.-\sum_{j=2}^{n}\frac{1}{z_{1}-z_{j}}\left(\frac{1}{2h(z_{1})}\partial_{z_{1}}F_{g,n-1}(z_{1},z,\dots,z)-\frac{1}{2h(z_{j})}\partial_{z_{j}}F_{g,n-1}(z_{j},z,\dots,z)\right)\right|_{z_{1}=z_{j}}\\\
=-\sum_{j=2}^{n}\partial_{z_{1}}\left(\frac{1}{2h(z_{1})}\partial_{z_{1}}F_{g,n-1}(z_{1},z,\dots,z)\right)\\\
=-(n-1)\partial_{z_{1}}\left(\frac{1}{2h(z_{1})}\partial_{z_{1}}F_{g,n-1}(z_{1},z,\dots,z)\right)\\\
=-(n-1)\partial_{z_{1}}\left(\frac{1}{2h(z_{1})}\right)\partial_{z_{1}}F_{g,n-1}(z_{1},z,\dots,z)-\frac{n-1}{2h(z_{1})}\partial_{z_{1}}^{2}F_{g,n-1}(z_{1},z,\dots,z).$

The limit $z_{1}\rightarrow z$ then produces

(6.25)
$-\partial_{z}\frac{1}{2h(z)}\cdot\partial_{z}F_{g,n-1}(z,\dots,z)-\frac{1}{2h(z)}\partial_{z}^{2}F_{g,n-1}(z\dots,z)\\\
+\left.\frac{(n-1)(n-2)}{2h(z)}\frac{\partial^{2}}{\partial u_{1}\partial
u_{2}}F_{g,n-1}(u_{1},u_{2},z\dots,z)\right|_{u_{1}=u_{2}=z}.$

To calculate the principal specialization of the second line of the right-hand
side of (6.24), we note that since all points $z_{i}$’s for $i\geq 2$ are set
to be equal, a set partition by index sets $I$ and $J$ becomes a partition of
$n-1$ with a combinatorial factor that counts the redundancy. The result is

(6.26) $-\left.\frac{1}{2h(z)}\frac{\partial^{2}}{\partial u_{1}\partial
u_{2}}F_{g-1,n+1}(u_{1},u_{2},z\dots,z)\right|_{u_{1}=u_{2}=z}\\\
-\frac{1}{2h(z)}\sum_{\begin{subarray}{c}g_{1}+g_{2}=g\\\
n_{1}+n_{2}=n-1\end{subarray}}^{\text{stable}}\partial_{z}F_{g_{1},n_{1}+1}(z,\dots,z)\cdot\partial_{z}F_{g_{2},n_{2}+1}(z_{,}\dots,z).$

Assembling (6.25) and (6.26) together, we obtain

(6.27)
$\frac{1}{2h(z)}\left[\partial_{z}^{2}F_{g,n-1}(z\dots,z)+\sum_{\begin{subarray}{c}g_{1}+g_{2}=g\\\
n_{1}+n_{2}=n-1\end{subarray}}^{\text{stable}}\partial_{z}F_{g_{1},n_{1}+1}(z,\dots,z)\cdot\partial_{z}F_{g_{2},n_{2}+1}(z_{,}\dots,z)\right]\\\
+\frac{1}{n}\partial_{z}F_{g,n}(z,\dots,z)+\partial_{z}\frac{1}{2h(z)}\cdot\partial_{z}F_{g,n-1}(z,\dots,z)\\\
=\left.\frac{(n-1)(n-2)}{2h(z)}\frac{\partial^{2}}{\partial u_{1}\partial
u_{2}}F_{g,n-1}(u_{1},u_{2},z\dots,z)\right|_{u_{1}=u_{2}=z}\\\
-\left.\frac{1}{2h(z)}\frac{\partial^{2}}{\partial u_{1}\partial
u_{2}}F_{g-1,n+1}(u_{1},u_{2},z\dots,z)\right|_{u_{1}=u_{2}=z}.$

We now apply the operation $\sum_{2g-2+n=m}\frac{1}{(n-1)!}$ to (6.27) above,
and write the result in terms of

$S_{m}(z):=S_{m}\big{(}x(z)\big{)}=\sum_{2g-2+n=m-1}\frac{1}{n!}F_{g,n}(z,\dots,z)$

of (6.16) to fit into the WKB formalism (6.4). We observe that summing over
all possibilities of $(g,n)$ with the fixed value of $2g-2+n$, the right-hand
side of (6.27) exactly cancels out. Thus we have established that the
functions $S_{m}(z)$ of (6.16) for $m\geq 2$ satisfy the recursion formula

(6.28)
$\frac{1}{2h(z)}\left(\frac{d^{2}S_{m}}{dz^{2}}+\sum_{\begin{subarray}{c}a+b=m+1\\\
a,b\geq
2\end{subarray}}\frac{dS_{a}}{dz}\;\frac{dS_{b}}{dz}\right)+\frac{dS_{m+1}}{dz}+\frac{d}{dz}\left(\frac{1}{2h(z)}\right)\frac{dS_{m}}{dz}=0.$

Using (6.23) we identify the derivation

(6.29) $\frac{d}{dx}=\frac{y}{h(z)}\;\frac{d}{dz},$

which is the push-forward ${\tilde{\pi}}_{*}(d/dz)$ of the vector field
$d/dz$. The transformation (6.29) is singular at $z=0$. If we allow terms
$a=0$ or $b=0$ in (6.28), then what we have in addition is

$\frac{1}{2h(z)}2\frac{dS_{0}}{dz}\frac{dS_{m+1}}{dz}=\frac{1}{h(z)}\frac{h(z)}{y}\frac{dS_{0}}{dx}\frac{dS_{m+1}}{dz}=\frac{dS_{m+1}}{dz},$

since $dS_{0}=ydx$. In other words, the $\frac{dS_{m+1}}{dz}$ term already
there in (6.28) is absorbed in the split differentiation for $a=0$ and $b=0$.

From (6.29) and (6.6), we find that the second derivative with respect to $x$
is given by

$\frac{d^{2}}{dx^{2}}=\frac{d}{dx}\left(\frac{S_{0}^{\prime}}{h(z)}\frac{d}{dz}\right)=\frac{(S_{0}^{\prime})^{2}}{h(z)^{2}}\frac{d^{2}}{dz^{2}}+\frac{S_{0}^{\prime}}{h(z)}\frac{d}{dz}\left(\frac{S_{0}^{\prime}}{h(z)}\right)\cdot\frac{d}{dz},$

denoting by $S_{0}^{\prime}=dS_{0}/dx$. Then (6.9) yields

(6.30)
$\frac{(S_{0}^{\prime})^{2}}{h(z)^{2}}\left(\frac{d^{2}}{dz^{2}}S_{m}+\sum_{a+b=m+1}\frac{dS_{a}}{dz}\frac{dS_{b}}{dz}\right)+\frac{S_{0}^{\prime}}{h(z)}\frac{d}{dz}\left(\frac{S_{0}^{\prime}}{h(z)}\right)\cdot\frac{dS_{m}}{dz}=0.$

The coefficients of $dS_{m}/dz$ in (6.30) are

$2\frac{(S_{0}^{\prime})^{2}}{h(z)^{2}}\frac{dS_{1}}{dz}+\frac{S_{0}^{\prime}}{h(z)}\frac{d}{dz}\left(\frac{S_{0}^{\prime}}{h(z)}\right)=2\frac{(S_{0}^{\prime})^{2}}{h(z)^{2}}\frac{h(z)}{S_{0}^{\prime}}S_{1}^{\prime}+\frac{d}{dx}\left(\frac{S_{0}^{\prime}}{h(z)}\right)\\\
=\frac{1}{h(z)}\big{(}2S_{0}^{\prime}S_{1}^{\prime}+S_{0}^{\prime\prime}\big{)}+S_{0}^{\prime}\frac{d}{dx}\left(\frac{1}{h(z)}\right)=S_{0}^{\prime}\frac{d}{dx}\left(\frac{1}{h(z)}\right)=\frac{(S_{0}^{\prime})^{2}}{h(z)^{2}}\cdot
2h(z)\frac{d}{dz}\left(\frac{1}{2h(z)}\right)\\\
=\frac{2(S_{0}^{\prime})^{2}}{h(z)}\frac{d}{dz}\left(\frac{1}{2h(z)}\right).$

This is exactly what the last term of (6.28) has, after adjusting overall
multiplication by $\frac{2(S_{0}^{\prime})^{2}}{h(z)}$. Therefore, we have
established that (6.6), (6.7), and (6.8) make (6.9) equivalent to (6.28). This
competed the proof of Theorem 6.1. ∎

######  Remark 6.5.

As noted in [10, 58], when the spectral curve is an analytic curve, the
procedure of Theorem 6.1 still produces a second order linear differential
equation as a quantum curve. But it is _not_ the same as the difference
equation one expects as a quantum curve. In the examples of various Hurwitz
numbers considered in [10, 58], these two totally different quantizations are
indeed compatible, in the sense that they have the same asymptotic solution
(6.4).

######  Remark 6.6.

Geometry of normalization of the singular spectral curve leads us to the
analysis of Stokes phenomena. It is beyond the scope of current paper, and
will be treated elsewhere.

##  7\. A simple classical example

Riemann and Poincaré worked on an interplay between algebraic geometry of
curves in a ruled surface and the asymptotic expansion of an analytic solution
to a differential equation defined on the base curve of the ruled surface. The
theme of the current paper lies exactly on this link, looking at this
classical subject from a modern point of view. The simple examples for
$SL(2,{\mathbb{C}})$-meromorphic Higgs bundles on ${\mathbb{P}}^{1}$
illustrate the relation between a Higgs bundle, the compactified cotangent
bundle of a curve, a quantum curve, a classical differential equation, non-
Abelian Hodge correspondence, and the quantum invariants that the quantum
curve captures.

###  7.1. The Higgs bundle for the Airy function

The Higgs bundle $(E,\phi)$ we consider consists of the base curve
$C={\mathbb{P}}^{1}$ and a particular vector bundle

(7.1) $E_{0}=K_{{\mathbb{P}}^{1}}^{\frac{1}{2}}\oplus
K_{{\mathbb{P}}^{1}}^{-{\frac{1}{2}}}={\mathcal{O}}_{{\mathbb{P}}^{1}}(-1)\oplus{\mathcal{O}}_{{\mathbb{P}}^{1}}(1)$

of rank $2$ on ${{\mathbb{P}}^{1}}$. A meromorphic Higgs field is given by

(7.2) $\phi=\begin{bmatrix}\phi_{11}&\phi_{12}\\\
\phi_{21}&\phi_{22}\end{bmatrix}:E\longrightarrow E\otimes
K_{{\mathbb{P}}^{1}}(m),\qquad m\geq 0.$

Each matrix component is given by

$\displaystyle\phi_{11}:K_{{\mathbb{P}}^{1}}^{\frac{1}{2}}\longrightarrow
K_{{\mathbb{P}}^{1}}^{{\frac{1}{2}}}\otimes
K_{{\mathbb{P}}^{1}}(m)=K_{{\mathbb{P}}^{1}}^{\frac{3}{2}}(m),\qquad\phi_{11}\in
H^{0}\left(C,K_{{\mathbb{P}}^{1}}(m)\right)$
$\displaystyle\phi_{12}:K_{{\mathbb{P}}^{1}}^{-{\frac{1}{2}}}\longrightarrow
K_{{\mathbb{P}}^{1}}^{{\frac{1}{2}}}\otimes
K_{{\mathbb{P}}^{1}}(m)=K_{{\mathbb{P}}^{1}}^{\frac{3}{2}}(m),\qquad\phi_{12}\in
H^{0}\left(C,K_{{\mathbb{P}}^{1}}^{\otimes 2}(m)\right)$
$\displaystyle\phi_{21}:K_{{\mathbb{P}}^{1}}^{\frac{1}{2}}\longrightarrow
K_{{\mathbb{P}}^{1}}^{-{\frac{1}{2}}}\otimes
K_{{\mathbb{P}}^{1}}(m)=K_{{\mathbb{P}}^{1}}^{\frac{1}{2}}(m),\qquad\phi_{21}\in
H^{0}\left(C,{\mathcal{O}}_{{\mathbb{P}}^{1}}(m)\right)$
$\displaystyle\phi_{22}:K_{{\mathbb{P}}^{1}}^{-{\frac{1}{2}}}\longrightarrow
K_{{\mathbb{P}}^{1}}^{-{\frac{1}{2}}}\otimes
K_{{\mathbb{P}}^{1}}(m)=K_{{\mathbb{P}}^{1}}^{\frac{1}{2}}(m),\qquad\phi_{22}\in
H^{0}\left(C,K_{{\mathbb{P}}^{1}}(m)\right).$

Since we are considering a point on a Hitchin section, we take $\phi_{21}=1$
to be the identity map
$K_{{\mathbb{P}}^{1}}^{\frac{1}{2}}\overset{\cong}{\longrightarrow}K_{{\mathbb{P}}^{1}}^{\frac{1}{2}}\hookrightarrow
K_{{\mathbb{P}}^{1}}^{\frac{1}{2}}(m)$, and $\phi_{11}=\phi_{22}=0$. When we
allow singularities, we can make other choices for $\phi_{21}$ as well.

The Planck constant $\hbar$ has a geometric meaning (1.1) as a parameter of
the extension classes of line bundles. For ${\mathbb{P}}^{1}$, it is

$\hbar\in{\text{\rm{Ext}}}^{1}\left(K_{{\mathbb{P}}^{1}}^{-{\frac{1}{2}}},K_{{\mathbb{P}}^{1}}^{{\frac{1}{2}}}\right)\cong
H^{1}\left({\mathbb{P}}^{1},K_{{\mathbb{P}}^{1}}\right)={\mathbb{C}}.$

It determines the unique extension

(7.3) $0\longrightarrow K_{{\mathbb{P}}^{1}}^{\frac{1}{2}}\longrightarrow
E_{\hbar}\longrightarrow K_{{\mathbb{P}}^{1}}^{-{\frac{1}{2}}}\longrightarrow
0,$

where

(7.4)
$E_{\hbar}\cong\begin{cases}{\mathcal{O}}_{{\mathbb{P}}^{1}}(-1)\oplus{\mathcal{O}}_{{\mathbb{P}}^{1}}(1)\qquad\;\;\hbar=0\\\
{\mathcal{O}}_{{\mathbb{P}}^{1}}\oplus{\mathcal{O}}_{{\mathbb{P}}^{1}}\qquad\qquad\qquad\hbar\neq
0\end{cases}$

as a vector bundle, since every vector bundle on ${\mathbb{P}}^{1}$ splits.

The quantization of the Higgs field $\phi$ is an $\hbar$-connection of Deligne
in $E_{\hbar}$ defined on ${\mathbb{P}}^{1}$ and is given by
$\hbar\nabla^{\hbar}$, where

(7.5) $\nabla^{\hbar}=d+\frac{1}{\hbar}\phi:E_{\hbar}\longrightarrow
E_{\hbar}\otimes K_{{\mathbb{P}}^{1}}(m),$

and $d$ is the exterior differentiation operator acting on sections of the
trivial bundle $E_{\hbar}$ for $\hbar\neq 0$. The operator $\nabla^{\hbar}$ is
a meromorphic connection in the vector bundle $E_{\hbar}$. Of course $d+\phi$
is never a connection in general, because $\phi$ is a Higgs field belonging to
a different bundle and satisfying a different transition rule with respect to
coordinate changes. However, as explained in Section 3 (see also [21, 26]), a
Higgs field $\phi$ associated with a complex simple Lie group on a Hitchin
section gives rise to a connection $\nabla^{\hbar}=d+\frac{1}{\hbar}\phi$ in
$E_{\hbar}$ with respect to the coordinate system associated with a
_projective structure_ subordinating the complex structure of the base curve.
Since our examples are constructed on ${\mathbb{P}}^{1}$, the affine
coordinate $x\in{\mathbb{A}}^{1}={\mathbb{P}}^{1}\setminus\\{\infty\\}$ is a
natural coordinate representing the projective structure. Hence the expression
$d+\phi$ makes sense as a connection in $E_{\hbar}$ for every
$SL(2,{\mathbb{C}})$-Higgs bundle $(E_{0},\phi)$. This is due to the vanishing
of the Schwarzian derivatives for the coordinate change in a projective
structure (see [21] for more detail on how the Schwarzian derivative plays a
role here).

To see the effect of quantization, i.e., the passage from the Higgs bundle to
an $\hbar$-family of connections (7.5), let us use the local coordinate and
write everything concretely. The transition function defined on
${\mathbb{C}}^{*}=U_{\infty}\cap U_{0}$ of the vector bundle $E_{0}$ on
${\mathbb{P}}^{1}=U_{\infty}\cup U_{0}$ is given by $\begin{bmatrix}x\\\
&\frac{1}{x}\end{bmatrix},$ where
$U_{0}={\mathbb{P}}^{1}\setminus\\{\infty\\}={\mathbb{A}}^{1}$ and
$U_{\infty}={\mathbb{P}}^{1}\setminus\\{0\\}$. With respect to the same
coordinate, the extension $E_{\hbar}$ is given by $\begin{bmatrix}x&\hbar\\\
&\frac{1}{x}\end{bmatrix}.$ The equality

$\begin{bmatrix}1\\\ -\frac{1}{\hbar
x}&1\end{bmatrix}\begin{bmatrix}x&\hbar\\\
&\frac{1}{x}\end{bmatrix}\begin{bmatrix}&-\hbar\\\
\frac{1}{\hbar}&x\end{bmatrix}=\begin{bmatrix}1\\\ &1\end{bmatrix}$

proves (7.4). The local expressions of the $1$-form $\phi_{11}$ and the
quadratic differential $\phi_{12}$ satisfy

$(\phi_{11})_{u}du=(\phi_{11})_{x}dx,\qquad(\phi_{12})_{u}du^{2}=(\phi_{12})_{x}dx^{2},$

where $u=1/x$ is a coordinate on $U_{\infty}$. Then the local expressions of
the Higgs field (7.2) with $\phi_{22}=0$ satisfy the following transition
relation with respect to $E_{0}$:

(7.6) $\begin{bmatrix}(\phi_{11})_{u}&-(\phi_{12})_{u}\\\
-1&\end{bmatrix}du=\begin{bmatrix}x&\\\
&\frac{1}{x}\end{bmatrix}\begin{bmatrix}(\phi_{11})_{x}&(\phi_{12})_{x}\\\
1&\end{bmatrix}dx\begin{bmatrix}x&\\\ &\frac{1}{x}\end{bmatrix}^{-1}.$

The negative signs are due to $du=-\frac{1}{x^{2}}dx.$ For the case of
$SL(2,{\mathbb{C}})$-Higgs bundles, we further assume
${\text{\rm{tr}}}(\phi)=\phi_{11}=0$. In this case, we note that (7.6) is
equivalent to the gauge transformation rule of connection matrices with
respect to $E_{\hbar}$:

(7.7) $-\frac{1}{\hbar}\begin{bmatrix}&(\phi_{12})_{u}\\\
1&\end{bmatrix}du=\frac{1}{\hbar}\begin{bmatrix}x&\hbar\\\
&\frac{1}{x}\end{bmatrix}\begin{bmatrix}&(\phi_{12})_{x}\\\
1&\end{bmatrix}dx\begin{bmatrix}x&\hbar\\\
&\frac{1}{x}\end{bmatrix}^{-1}-d\begin{bmatrix}x&\hbar\\\
&\frac{1}{x}\end{bmatrix}\begin{bmatrix}x&\hbar\\\
&\frac{1}{x}\end{bmatrix}^{-1}.$

In other words,

$d_{u}-\frac{1}{\hbar}\begin{bmatrix}&(\phi_{12})_{u}\\\
1&\end{bmatrix}du=\begin{bmatrix}x&\hbar\\\
&\frac{1}{x}\end{bmatrix}\left(d_{x}+\frac{1}{\hbar}\begin{bmatrix}&(\phi_{12})_{x}\\\
1&\end{bmatrix}dx\right)\begin{bmatrix}x&\hbar\\\
&\frac{1}{x}\end{bmatrix}^{-1}.$

Therefore, $\nabla^{\hbar}=d+\frac{1}{\hbar}\phi$ is a globally defined
connection in $E_{\hbar}$ for every $\hbar\neq 0$, and

$\left(\left.\hbar\nabla^{\hbar}\right)\right|_{\hbar=0}=\phi$

is the original Higgs field.

Let us start with a particular spectral curve, the algebraic curve

$\Sigma\subset{\mathbb{F}}_{2}={\mathbb{P}}\left(K_{{\mathbb{P}}^{1}}\oplus{\mathcal{O}}_{{\mathbb{P}}^{1}}\right)=\overline{T^{*}{\mathbb{P}}^{1}}$

embedded in the Hirzebruch surface with the defining equation

(7.8) $y^{2}-x=0$

on $U_{0}$. Here, $y$ is a fiber coordinate of the cotangent bundle
$T^{*}{\mathbb{P}}^{1}\subset{\mathbb{F}}^{2}$ over $U_{0}$. The Hirzebruch
surface is the natural compactification of the cotangent bundle
$T^{*}{\mathbb{P}}^{1}$, which is the total space of the canonical bundle
$K_{{\mathbb{P}}^{1}}$. We denote by $\eta\in
H^{0}(T^{*}{\mathbb{P}}^{1},\pi^{*}K_{{\mathbb{P}}^{1}})$ the tautological
$1$-form associated with the projection
$\pi:T^{*}{\mathbb{P}}^{1}\longrightarrow{\mathbb{P}}^{1}$. It is expressed as
$\eta=ydx$ in terms of the affine coordinates. The holomorphic symplectic form
on $T^{*}{\mathbb{P}}^{1}$ is given by $-d\eta=dx\wedge dy$. The $1$-form
$\eta$ extends to ${\mathbb{F}}_{2}$ as a meromorphic differential form and
defines a divisor

(7.9) $(\eta)=C_{0}-C_{\infty},$

where $C_{0}$ is the zero-section of $T^{*}{\mathbb{P}}^{1}$, and $C_{\infty}$
the section at infinity of $\overline{T^{*}{\mathbb{P}}^{1}}$. The Picard
group ${\text{\rm{Pic}}}({\mathbb{F}}_{2})$ of the Hirzebruch surface is
generated by the class $C_{0}$ and a fiber class $F$ of $\pi$. Although (7.8)
is a perfect parabola in the affine plane, it has a quintic cusp singularity
at $x=\infty$. Let $(u,w)$ be a coordinate system on another affine chart of
${\mathbb{F}}_{2}$ defined by

(7.10) $\begin{cases}x={1}/{u}\\\ ydx=vdu,\qquad w=1/v.\end{cases}$

Then $\Sigma$ in the $(u,w)$-plane is given by

(7.11) $w^{2}=u^{5}.$

The expression of $\Sigma\in{\text{\rm{NS}}}({\mathbb{F}}_{2})$ as an element
of the Néron-Severy group of ${\mathbb{F}}_{2}$, in this case the same as
${\text{\rm{Pic}}}({\mathbb{F}}_{2})$, is thus given by $\Sigma=2C_{0}+5F.$

Define a stable Higgs pair $\left(E_{0},\phi(q)\right)$ on ${\mathbb{P}}^{1}$
with $E_{0}=K_{{\mathbb{P}}^{1}}^{\frac{1}{2}}\oplus
K_{{\mathbb{P}}^{1}}^{-{\frac{1}{2}}}={\mathcal{O}}_{{\mathbb{P}}^{1}}(-1)\oplus{\mathcal{O}}_{{\mathbb{P}}^{1}}(1)$
and

(7.12) $\phi(q)=\begin{bmatrix}&q\\\ 1\end{bmatrix}:E_{0}\longrightarrow
E_{0}\otimes K_{{\mathbb{P}}^{1}}(4).$

Here, we choose a meromorphic quadratic differential $q\in
H^{0}\left({\mathbb{P}}^{1},K_{{\mathbb{P}}^{1}}(2)^{\otimes 2}\right)$ that
has a simple zero at $0\in{\mathbb{P}}^{1}$ and a pole of order $5$ at
$\infty\in{\mathbb{P}}^{1}$. Up to a constant factor, there is only one such
differential

$q=x(dx)^{2}=\frac{1}{u^{5}}(du)^{2}\in
H^{0}\left({\mathbb{P}}^{1},K_{{\mathbb{P}}^{1}}(2)^{\otimes
2}\right)={\mathbb{C}}.$

The spectral curve $\Sigma$ of $\left(E_{0},\phi(q)\right)$ is given by the
characteristic equation

(7.13)
$\det(\eta-\pi^{*}\phi)=\eta^{2}-\pi^{*}{\text{\rm{tr}}}(\phi)+\pi^{*}\det(\phi)=\eta^{2}-q=0$

in ${\mathbb{F}}_{2}$. As explained above, $\left(E_{0},\phi(q)\right)$
uniquely determines a meromorphic _oper_

(7.14)
$\nabla^{\hbar}(q)=d+\frac{1}{\hbar}\phi(q)=d+\frac{1}{\hbar}\begin{bmatrix}&q\\\
1\end{bmatrix}$

on the extension $E_{\hbar}$ of (7.3) over ${\mathbb{P}}^{1}$ [21]. Indeed,
the case $q=0$ of (7.14) for $\hbar=1$ is the _non-Abelian Hodge
correspondence_ [20, 40, 64]

$H^{0}\left(C,{\text{\rm{End}}}(E_{0})\otimes
K_{{\mathbb{P}}^{1}}\right)\owns\begin{bmatrix}0&0\\\
1&0\end{bmatrix}\Longleftrightarrow d+\begin{bmatrix}0&0\\\
1&0\end{bmatrix}:E_{1}\longrightarrow E_{1}\otimes K_{{\mathbb{P}}^{1}}.$

The quantization procedure of this paper is the following $\hbar$-deformation

(7.15) $H^{0}\left(C,{\text{\rm{End}}}(E_{0})\otimes
K_{{\mathbb{P}}^{1}}(m)\right)\owns\begin{bmatrix}&q\\\
1&\end{bmatrix}\Longleftrightarrow d+\frac{1}{\hbar}\begin{bmatrix}&q\\\
1&\end{bmatrix}:E_{\hbar}\longrightarrow E_{\hbar}\otimes
K_{{\mathbb{P}}^{1}}(m),$

where $q\in H^{0}\left(C,K_{{\mathbb{P}}^{1}}^{\otimes
2}\otimes{\mathcal{O}}_{{\mathbb{P}}^{1}}(m)\right)$ is a meromorphic
quadratic differential on $C$.

Let $\psi(x,\hbar)$ denote an analytic function in $x$ with a formal parameter
$\hbar$ such that

(7.16) $\hbar\nabla_{q}^{\hbar}\begin{bmatrix}-\hbar\psi^{\prime}\\\
\psi\end{bmatrix}=0,\qquad\hbar\neq 0,$

where ′ denotes the $x$ differentiation. Then it satisfies a Schrödinger
equation

(7.17) $\left(\left(\hbar\frac{d}{dx}\right)^{2}-x\right)\psi(x,\hbar)=0.$

The differential operator

(7.18) $P(x,\hbar):=\left(\hbar\frac{d}{dx}\right)^{2}-x$

quantizing the spectral curve $\Sigma$ of (7.8) is an example of a quantum
curve. Reflecting the fact (7.11) that $\Sigma$ has a quintic cusp singularity
at $x=\infty$, (7.17) has an _irregular_ singular point of _class_
$\frac{3}{2}$ at $x=\infty$.

Let us recall the definition of regular and irregular singular points of a
second order differential equation here.

######  Definition 7.1.

Let

(7.19)
$\left(\frac{d^{2}}{dx^{2}}+a_{1}(x)\frac{d}{dx}+a_{2}(x)\right)\psi(x)=0$

be a second order differential equation defined around a neighborhood of $x=0$
on a small disc $|x|<\epsilon$ with meromorphic coefficients $a_{1}(x)$ and
$a_{2}(x)$ with poles at $x=0$. Denote by $k$ (reps. $\ell$) the order of the
pole of $a_{1}(x)$ (resp. $a_{2}(x)$) at $x=0$. If $k\leq 1$ and $\ell\leq 2$,
then (7.19) has a regular singular point at $x=0$. Otherwise, consider the
_Newton polygon_ of the order of poles of the coefficients of (7.19). It is
the upper part of the convex hull of three points $(0,0),(1,k),(2,\ell)$. As a
convention, if $a_{j}(x)$ is identically $0$, then we assign $-\infty$ as its
pole order. Let $(1,r)$ be the intersection point of the Newton polygon and
the line $x=1$. Thus

(7.20) $r=\begin{cases}k\qquad 2k\geq\ell,\\\ \frac{\ell}{2}\qquad
2k\leq\ell.\end{cases}$

The differential equation (7.19) has an irregular singular point of class
$r-1$ at $x=0$ if $r>1$.

The class $\frac{3}{2}$ at $\infty$ indicates how the asymptotic expansion of
the solution $\psi$ looks like. Indeed, any non-trivial solution has an
essential singularity at $\infty$. We note that every solution of (7.17) is an
entire function for any value of $\hbar\neq 0$. Applying our main result of
this paper, we construct a particular _all-order_ asymptotic expansion of this
entire solution

(7.21) $\psi(x,\hbar)=\exp F(x,\hbar),\qquad
F(x,\hbar):=\sum_{m=0}^{\infty}\hbar^{m-1}S_{m}(x),$

valid for $|{\text{\rm{Arg}}}(x)|<\pi$, and $\hbar>0$. Here, the first two
terms of the asymptotic expansion are given by

(7.22) $\displaystyle S_{0}(x)$ $\displaystyle=\pm\frac{2}{3}x^{\frac{3}{2}},$
(7.23) $\displaystyle S_{1}(x)$ $\displaystyle=-\frac{1}{4}\log x.$

Although the _classical limit_ $\hbar\rightarrow 0$ of (7.17) does not make
sense under the expansion (7.21), the _semi-classical limit_ through the _WKB_
analysis

(7.24)
$\left[e^{-S_{1}(x)}e^{-\frac{1}{\hbar}S_{0}(x)}\left(\hbar^{2}\frac{d^{2}}{dx^{2}}-x\right)e^{\frac{1}{\hbar}S_{0}(x)}e^{S_{1}(x)}\right]\exp\left(\sum_{m=2}^{\infty}\hbar^{m-1}S_{m}(x)\right)=0$

has a well-defined limit $\hbar\rightarrow 0$. The result is
$S_{0}^{\prime}(x)^{2}=x,$ which gives (7.22), and also (7.8) by defining
$dS_{0}=\eta$. The vanishing of the $\hbar$-linear terms of (7.24) is
$2S_{0}^{\prime}(x)S_{1}^{\prime}(x)+S_{0}^{\prime\prime}(x)=0$, which gives
(7.23) above.

The entire solution in $x$ for $\hbar\neq 0$ and the choice of
$S_{0}(x)=-\frac{2}{3}x^{\frac{3}{2}}$ is called the _Airy function_

(7.25)
$Ai(x,\hbar)=\frac{1}{2\pi}\hbar^{-\frac{1}{6}}\int_{-\infty}^{\infty}\exp\left({\frac{ipx}{\hbar^{2/3}}}+{i\frac{p^{3}}{3}}\right)dp.$

The surprising discovery of Kontsevich [48] is that $S_{m}(x)$ for $m\geq 2$
has the following _closed_ formula:

(7.26) $S_{m}(x):=\sum_{2g-2+n=m-1}\frac{1}{n!}F_{g,n}^{\text{Airy}}(x),$
(7.27) $F_{g,n}^{\text{Airy}}(x):=\frac{(-1)^{n}}{2^{2g-2+n}}\cdot
x^{-\frac{(6g-6+3n)}{2}}\sum_{\begin{subarray}{c}d_{1}+\dots+d_{n}\\\
=3g-3+n\end{subarray}}{\langle}\tau_{d_{1}}\cdots\tau_{d_{n}}{\rangle}_{g,n}\prod_{i=1}^{n}|(2d_{i}-1)|!!,$

where the coefficients

${\langle}\tau_{d_{1}}\cdots\tau_{d_{n}}{\rangle}_{g,n}=\int_{{\overline{\mathcal{M}}}_{g,n}}\psi_{1}^{d_{1}}\cdots\psi_{n}^{d_{n}}$

are the cotangent class intersection numbers on the moduli space
${\overline{\mathcal{M}}}_{g,n}$ of stable curves of genus $g$ with $n$ non-
singular marked points. The expansion coordinate $x^{\frac{3}{2}}$ of (7.27)
indicates the class of the irregular singularity of the Airy differential
equation.

Although (7.26) is not a generating function of all intersection numbers, the
quantum curve (7.8) alone actually determines every intersection number
${\langle}\tau_{d_{1}}\cdots\tau_{d_{n}}{\rangle}_{g,n}$. This mechanism is
the _PDE topological recursion_ of [23, 24], which computes _free energies_

(7.28)
$F_{g,n}^{\text{Airy}}(t_{1},\cdots,t_{n}):=\frac{(-1)^{n}}{2^{2g-2+n}}\sum_{\begin{subarray}{c}d_{1}+\dots+d_{n}\\\
=3g-3+n\end{subarray}}{\langle}\tau_{d_{1}}\cdots\tau_{d_{n}}{\rangle}_{g,n}\prod_{i=1}^{n}\left(\frac{t_{i}}{2}\right)^{2d_{i}+1}|(2d_{i}-1)|!!$

as a function in $n$ variables from $\Sigma$ through the process of blow-ups
of ${\mathbb{F}}_{2}$.

###  7.2. Blowing up a Hirzebruch surface

Let us now give a detailed algebraic geometry procedure for this example. We
start with the spectral curve $\Sigma$ of (7.8). Our goal is to come up with
(7.17). The first step is to blow up ${\mathbb{F}}_{2}$ and to construct a
normalization of $\Sigma$. The construction of $Bl(\overline{T^{*}C})$ is
given in Definition 4.4. It is the minimal resolution of the divisor

$\Sigma=\left(\det(\eta-\pi^{*}\phi)\right)_{0}$

of the characteristic polynomial. The discriminant of the defining equation
(7.13) of the spectral curve is

$-\det(\phi)=x(dx)^{2}=\frac{1}{u^{5}}(du)^{2}.$

It has a simple zero at $x=0$ and a pole of order $5$ at $x=\infty$. The
_Geometric Genus Formula_ (4.10) for the general base curve $C$ reads

$g(\widetilde{\Sigma})=2g(C)-1+{\frac{1}{2}}\delta,$

where $\delta$ is the sum of the number of cusp singularities of $\Sigma$ and
the ramification points of $\pi:\Sigma\longrightarrow C$ (Theorem 4.2). In our
case, it tells us that $\widetilde{\Sigma}$ is a non-singular curve of genus
$0$, i.e., a ${\mathbb{P}}^{1}$, after blowing up
$\lfloor\frac{5}{2}\rfloor=2$ times.

The center of blow-up is $(u,w)=(0,0)$ for the first time. Put $w=w_{1}u$, and
denote by $E_{1}$ the exceptional divisor of the first blow-up. The proper
transform of $\Sigma$ for this blow-up, $w_{1}^{2}=u^{3}$, has a cubic cusp
singularity, so we blow up again at the singular point. Let $w_{1}=w_{2}u$,
and denote by $E_{2}$ the exceptional divisor created by the second blow-up.
The self-intersection of the proper transform of $E_{1}$ is $-2$. We then
obtain the desingularized curve ${\widetilde{\Sigma}}$, locally given by
$w_{2}^{2}=u$. The proof of Theorem 4.2 also tells us that
${\widetilde{\Sigma}}\longrightarrow{\mathbb{P}}^{1}$ is ramified at two
points. Choose the affine coordinate $t=2w_{2}$ of the exceptional divisor
added at the second blow-up. Our choice of the constant factor is to make the
formula the same as in [27]. We have

(7.29) $\begin{cases}x=\frac{1}{u}=\frac{1}{w_{2}^{2}}=\frac{4}{t^{2}}\\\
y=-\frac{u^{2}}{w}=-\frac{u^{2}}{w_{2}u^{2}}=-\frac{2}{t}.\end{cases}$

In the $(u,w)$-coordinate, we see that the parameter $t$ is a normalization
parameter of the quintic cusp singularity:

$\begin{cases}u=\frac{t^{2}}{4}\\\ w=\frac{t^{5}}{32}.\end{cases}$

Note that ${\widetilde{\Sigma}}$ intersects transversally with the proper
transform of $C_{\infty}$. The blow-up space $Bl({\mathbb{F}}^{2})$ is the
result of the twice blow-ups of the Hirzebruch surface.

(7.30) |  |
---|---|---
$\textstyle{\widetilde{\Sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\pi}}$$\scriptstyle{\tilde{i}}$$\scriptstyle{\nu}$$\textstyle{Bl(\overline{T^{*}{\mathbb{P}}^{1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{i}$$\textstyle{\overline{T^{*}{\mathbb{P}}^{1}}={\mathbb{F}}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{{\mathbb{P}}^{1}}$

Figure 7.1.

The generalized topological recursion (5.5) requires a globally defined
meromorphic $1$-form $W_{0,1}$ on $\widetilde{\Sigma}$ and a symmetric
meromorphic $2$-form $W_{0,2}$ on the product
$\widetilde{\Sigma}\times\widetilde{\Sigma}$ as the initial data. We choose

(7.31) $\begin{cases}W_{0,1}=\tilde{i}^{*}\nu^{*}\eta\\\
W_{0,2}=d_{1}d_{2}\log E_{\widetilde{\Sigma}},\end{cases}$

where $E_{\widetilde{\Sigma}}$ is a normalized Riemann prime form on
${\widetilde{\Sigma}}$ (see [23, Section 2]). The form $W_{0,2}$ depends only
on the intrinsic geometry of the smooth curve $\widetilde{\Sigma}$. The
geometry of (7.30) is encoded in $W_{0,1}$.

Now we apply the differential recursion (5.6) to the geometric data (7.30) and
(7.31). We claim that the integral topological recursion of [32] for the
geometric data we are considering now is exactly the same as the integral
topological recursion of [27, (6.12)] applied to the curve (7.29) _realized as
a plane parabola_ in ${\mathbb{C}}^{2}$. This is because our integral
topological recursion (5.5) has two residue contributions, one each from $t=0$
and $t=\infty$. As proved in [27, Section 6], the integrand on the right-hand
side of the integral recursion formula [27, (6.12)] does not have any pole at
$t=0$. Therefore, the residue contribution from this point is $0$. The
differential recursion is obtained by deforming the contour of integration to
enclose only poles of the differential forms $W_{g,n}$. Since $t=0$ is a
regular point, the two methods have no difference.

The $W_{0,2}$ of (7.31) is simply $\frac{dt_{1}\cdot
dt_{2}}{(t_{1}-t_{2})^{2}}$ because $\widetilde{\Sigma}\cong{\mathbb{P}}^{1}$.
Since $t$ of (7.29) is a normalization coordinate, we have

$W_{0,1}=\tilde{i}^{*}\nu^{*}(\eta)=y(t)dx(t)=\frac{16}{t^{4}},$

in agreement of [27, (6.8)]. Noting that the solution to the integral
topological recursion is unique from the initial data, we conclude that

$d_{1}\cdots
d_{n}F_{g,n}^{\text{\rm{Airy}}}\big{(}x(t_{1}),\dots,x(t_{n})\big{)}=W_{g,n}.$

By setting the constants of integration by integrating from $t=0$ for the
differential recursion equation, we obtain the expression (7.28). Then its
principal specialization gives (7.27). The equivalence of the differential
recursion and the quantum curve equation Theorem 6.1 then proves (7.17) with
the expression of (7.21) and (7.26).

In this process, what is truly amazing is that the single differential
equation (7.17), which is our quantum curve, knows everything about the free
energies (7.28). This is because we can recover the spectral curve $\Sigma$
from the quantum curve. Then the procedures we need to apply, the blow-ups and
the differential recursion equation, are canonical. Therefore, we do recover
(7.28) as explained above.

It is surprising to see that a simple entire function (7.25) contains so much
geometric information. Our expansion (7.21) is an expression of this entire
function viewed from its essential singularity. We can extract rich
information of the solution by restricting the region where the asymptotic
expansion is valid. If we consider (7.21) only as a formal expression in $x$
and $\hbar$, then we cannot see how the coefficients are related to quantum
invariants. The topological recursion [32] is a key to connect the two worlds:
the world of quantum invariants, and the world of holomorphic functions and
differentials. This relation is also knows as a mirror symmetry, or in
analysis, simply as the _Laplace transform_. The intersection numbers
${\langle}\tau_{d_{1}}\cdots\tau_{d_{n}}{\rangle}_{g,n}$ belong to the
$A$-model, while the spectral curve $\Sigma$ of (7.8) and free energies belong
to the $B$-model. We consider (7.28) as an example of the Laplace transform,
playing the role of mirror symmetry [25, 27].

######  Acknowledgement.

The authors wish to thank Philip Boalch for many useful discussions and
comments on their work on quantum curves. In particular, his question proposed
at the American Institute of Mathematics Workshop, _Spectral data for Higgs
bundles_ in September-October 2015, was critical for the development of the
theory presented in this paper.

This joint research is carried out while the authors have been staying in the
following institutions in the last four years: the American Institute of
Mathematics in California, the Banff International Research Station,
Institutul de Matematică “Simion Stoilow” al Academiei Roma͡ne, Institut Henri
Poincaré, Institute for Mathematical Sciences at the National University of
Singapore, Kobe University, Leibniz Universität Hannover, Lorentz Center
Leiden, Mathematisches Forschungsinstitut Oberwolfach, Max-Planck-Institut für
Mathematik-Bonn, and Osaka City University Advanced Mathematical Institute.
Their generous financial support, hospitality, and stimulating research
environments are greatly appreciated.

The authors also thank Jørgen Andersen, Vincent Bouchard, Tom Bridgeland,
Bertrand Eynard, Edward Frenkel, Tamás Hausel, Kohei Iwaki, Maxim Kontsevich,
Laura Schaposnik, Carlos Simpson, Albert Schwarz, Yan Soibelman, Ruifang Song,
Jörg Teschner, and Richard Wentworth, for useful comments, suggestions, and
discussions. During the preparation of this work, the research of O.D. was
supported by GRK 1463 _Analysis, Geometry, and String Theory_ at the Leibniz
Universität Hannover, and a grant from MPIM-Bonn. The research of M.M. was
supported by IHÉS, MPIM-Bonn, NSF grants DMS-1104734, DMS-1309298,
DMS-1619760, DMS-1642515, and NSF-RNMS: Geometric Structures And
Representation Varieties (GEAR Network, DMS-1107452, 1107263, 1107367).

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