# Interplay between opers, quantum curves, WKB analysis, and Higgs bundles

Olivia Dumitrescu Olivia Dumitrescu: Department of Mathematics
Pierce Hall 209
Central Michigan University
Mount Pleasant 48859, Michigan and Simion Stoilow Institute of Mathematics
Romanian Academy
21 Calea Grivitei Street
010702 Bucharest, Romania<EMAIL_ADDRESS>and  Motohico Mulase Motohico
Mulase: Department of Mathematics
University of California
Davis, CA 95616–8633, U.S.A. and Kavli Institute for Physics and Mathematics
of the Universe
The University of Tokyo
Kashiwa, Japan<EMAIL_ADDRESS>
###### Abstract.

Quantum curves were introduced in the physics literature. We develop a
mathematical framework for the case associated with Hitchin spectral curves.
In this context, a quantum curve is a Rees ${\mathcal{D}}$-module on a smooth
projective algebraic curve, whose semi-classical limit produces the Hitchin
spectral curve of a Higgs bundle. We give a method of quantization of Hitchin
spectral curves by concretely constructing one-parameter deformation families
of opers.

We propose a generalization of the topological recursion of Eynard-Orantin and
Mirzakhani for the context of singular Hitchin spectral curves. We show a
surprising result that a PDE version of the topological recursion provides
all-order WKB analysis for the Rees ${\mathcal{D}}$-modules, defined as the
quantization of Hitchin spectral curves associated with meromorphic
$SL(2,{\mathbb{C}})$-Higgs bundles. Topological recursion is thus identified
as a process of quantization of Hitchin spectral curves. We prove that these
two quantizations, one via the construction of families of opers, and the
other via the PDE topological recursion, agree for holomorphic and meromorphic
$SL(2,{\mathbb{C}})$-Higgs bundles.

Classical differential equations such as the Airy differential equation
provides a typical example. Through these classical examples, we see that
quantum curves relate Higgs bundles, opers, a conjecture of Gaiotto, and
quantum invariants, such as Gromov-Witten invariants.

###### Key words and phrases:

Quantum curve; Hitchin spectral curve; Higgs field; Rees D-module; opers; non-
Abelian Hodge correspondence; mirror symmetry; Airy function; quantum
invariants; WKB approximation; topological recursion.

###### 2010 Mathematics Subject Classification:

Primary: 14H15, 14N35, 81T45; Secondary: 14F10, 14J26, 33C05, 33C10, 33C15,
34M60, 53D37

###### Contents

  1. 1 Introduction
  2. 2 Rees ${\mathcal{D}}$-modules as quantum curves for Higgs bundles
  3. 3 Opers
    1. 3.1 Holomorphic $SL(r,{\mathbb{C}})$-opers and quantization of Higgs bundles
    2. 3.2 Semi-classical limit of $SL(r,{\mathbb{C}})$-opers
    3. 3.3 Non-Abelian Hodge correspondence and a conjecture of Gaiotto
  4. 4 Geometry of singular spectral curves
  5. 5 A differential version of topological recursion
  6. 6 WKB analysis of quantum curves
  7. 7 A simple classical example
    1. 7.1 The Higgs bundle for the Airy function
    2. 7.2 Blowing up a Hirzebruch surface

##  1\. Introduction

The purpose of this paper is to construct a geometric theory of quantum
curves. The notion of quantum curves was introduced in the physics literature
(see for example, [1, 16, 17, 37, 38, 43, 54, 65]). A quantum curve is
supposed to compactly capture topological invariants, such as certain Gromov-
Witten invariants, Seiberg-Witten invariants, and quantum knot polynomials.
Geometrically, a quantum curve is a unique quantization of the B-model
geometry, when it is encoded in a holomorphic curve, that gives a generating
function of A-model theory of genus $g$ for all $g\geq 0$. In a broad setting,
a quantum curve can be a differential operator, a difference operator, a
mixture of them, or a linear operator defined by a trace-class kernel
function.

The geometric theory we present here is focused on the process of quantization
of Hitchin spectral curves. A concise overview of our theory is available in
[21]. In Definition 2.10 and Definition 2.11, we introduce a quantum curve as
a Rees ${\mathcal{D}}$-module on a smooth projective algebraic curve $C$ whose
_semi-classical limit_ is the Hitchin spectral curve associated with a Higgs
bundle on $C$. The process of quantization is therefore an assignment of a
Rees ${\mathcal{D}}$-module to every Hitchin spectral curve.

The Planck constant $\hbar$ is a deformation parameter that appears in the
definition of Rees ${\mathcal{D}}$-modules. For us, it has a geometric
meaning, and is naturally identified with an element

(1.1) $\hbar\in H^{1}(C,K_{C}),$

where $K_{C}$ is the canonical sheaf over $C$. The cohomology group
$H^{1}(C,K_{C})$ controls the deformation of a classical object, i.e., a
geometric object such as a Higgs bundle in our case, into a quantum object,
i.e., a non-commutative quantity such as a differential operator. In our case,
the result of quantization is an oper.

Using a fixed choice of a theta characteristic and a projective structure on
$C$, we determine a unique quantization of the Hitchin spectral curve of a
holomorphic or meromorphic $SL(r,{\mathbb{C}})$-Higgs bundle through a
concrete construction of an $\hbar$-family of $SL(r,{\mathbb{C}})$-opers on
$C$, as proved in Theorem 3.10 for holomorphic case, and in Theorem 3.15 for
meromorphic case. The $\hbar$-family interpolates opers and Higgs fields. We
then prove, in Theorem 3.11, that the Rees ${\mathcal{D}}$-module as the
quantization result recovers the starting Hitchin spectral curve via semi-
classical limit of WKB analysis. This is our main theorem of the paper. When
we choose the projective structure of $C$ of genus $g\geq 2$ coming from the
Fuchsian uniformization, our construction of opers is the same as those opers
predicted by a conjecture of Gaiotto [35], as explained in Subsection 3.3.
This conjecture has been solved in [22].

It has been noticed that topological recursion of [32] and its generalizations
provide another aspect of quantization, most notably through the _remodeling
conjecture_ of Mariño [53] and his collaborators [11, 12], and its complete
solution by mathematicians [33, 34]. From this point of view, a quantum curve
is a quantization of B-model geometry that is obtained as an application of
topological recursion. It then becomes a natural question:

######  Question 1.1.

What is the relation between quantization via topological recursion and the
quantization through our construction of Rees ${\mathcal{D}}$-modules from
Hitchin spectral curves?

The topological recursion was originally developed as a computational
mechanism to calculate the multi-resolvent correlation functions of random
matrices [14, 29]. As mentioned above, it generates a mirror symmetric B-model
counterpart of genus $g$ A-model for all $g\geq 0$. This correspondence has
been rigorously established for many examples [10, 19, 25, 27, 28, 31, 33, 34,
56, 57, 58, 59]. Yet so far still no clear geometric relation between
topological recursion and quantum curves has been established.

The earliest striking application of topological recursion in algebraic
geometry is the simple proofs obtained in [59] for the Witten conjecture on
cotangent class intersection numbers and the $\lambda_{g}$-conjecture. Indeed,
these celebrated formulas are straightforward consequences of the _Laplace
transform_ of a combinatorial formula known as the _cut-and-join equations_
[36, 66]. This particular success of topological recursion is due to the fact
that the spectral curve of the theory is of genus $0$, and hence the residue
calculations in the formalism of [32] can be explicitly performed. The result
is a PDE topological recursion obtained as the Laplace transform of the
combinatorial equation. The difficulty of topological recursion as a
computational tool in a more general context comes from two fronts: one is the
case of singular spectral curves, and the other is the impossibility of
performing residue calculations appearing in the recursion formula for a high
genus spectral curve.

A novel approach proposed in [23] is the implementation of PDE topological
recursions, which appear naturally in enumerative geometry problems, to the
context of Hitchin spectral curves. It replaces the integral topological
recursion formulated in terms of residue calculations at the ramification
divisor of a spectral curve by a recursive set of partial differential
equations that captures _local_ nature of topological recursion. As we explain
in Section 5, the difference of the two recursion formulas lies in the choice
of contours of integration in the original format of integral topological
recursion. All other ingredients are the same. For a genus $0$ spectral curve,
the two sets of recursions are equivalent. In general, these two recursions
aim at achieving different goals. The original choice of contours should
capture some global nature of _periods_ hidden in the quantum invariants. Due
to the impossibility of calculating the residues mentioned above, still we do
not have a full understanding in this direction. The PDE topological recursion
of [23, 24], on the other hand, captures local nature of the functions
involved, and leads to an all-order WKB analysis of quantum curves for
$SL(2,{\mathbb{C}})$-Higgs bundles. The issue of singular spectral curves is
addressed in [24], in which we have developed a systematic process of
normalization of singular Hitchin spectral curves associated with meromorphic
rank $2$ Higgs bundles.

Theorem 6.1 is our answer to Question 1.1. It states a surprising result that
for the case of $SL(2,{\mathbb{C}})$, the normalization process of [24] and
the PDE topological recursion of [23] produce an all-order WKB expansion for
the meromorphic Rees ${\mathcal{D}}$-modules obtained by quantizing singular
Hitchin spectral curves through the construction of $\hbar$-families of opers.
In this sense, our result shows that quantization of Hitchin spectral curves,
singular or non-singular, through the PDE topological recursion and
construction of $\hbar$-family of opers are equivalent, for the case of
$SL(2,{\mathbb{C}})$-Higgs bundles.

We note the relation between meromorphic Higgs bundles over ${\mathbb{P}}^{1}$
and Painlevé equations. We refer to [45, 46] for the application of
topological recursion and quantum curves to Painlevé theory.

The interplay between Rees ${\mathcal{D}}$-modules, $\hbar$-families of opers,
Hitchin spectral curves as semi-classical limit, Gaiotto’s correspondence, and
WKB analysis through PDE topological recursion, creates a sense of
inevitability of the notion of quantization. Section 7 serves as an overview
to this interplay, where we present the _Airy differential equation_ as a
prototypical example.

A totally new mathematical framework is presented in [50], in which Kontsevich
and Soibelman formulate topological recursion as a special case of deformation
quantization. They call the formalism Airy structures. In their work, spectral
curves no longer serve as input data for topological recursion. Although
construction of quantum curves is not the only purpose of the original
topological recursion, what we present in our current paper is that our
general procedure of quantization of Hitchin spectral curves has nothing to do
with individual spectral curve, in parallel to the philosophy of [50]. As we
show in (3.59), the family of spectral curves is (re)constructed from our
deformation family of Rees ${\mathcal{D}}$-modules, not the other way around.
Yet at this moment we do not have a mechanism to give the WKB expansion
directly for the family of Rees ${\mathcal{D}}$-modules, without studying
individual spectral curves. Investigating a possible connection between the
Airy structures of [50] and this paper’s results is a future subject. A
relation between quantum curves and deformation quantization is first
discussed in [63].

Let us briefly describe our quantization process of this paper now. Our
geometric setting is a smooth projective algebraic curve $C$ over
${\mathbb{C}}$ of an arbitrary genus $g=g(C)$ with a choice of a _spin
structure_ , or a _theta characteristic_ , $K_{C}^{\frac{1}{2}}$. There are
$2^{2g}$ choices of such spin structures. We choose any one of them. Let
$(E,\phi)$ be an $SL(r,{\mathbb{C}})$-Higgs bundle on $C$ with a meromorphic
Higgs field $\phi$. Denote by

(1.2)
$\overline{T^{*}C}:={\mathbb{P}}(K_{C}\oplus{\mathcal{O}}_{C})\overset{\pi}{\longrightarrow}C$

the compactified cotangent bundle of $C$ (see [5, 49]), which is a ruled
surface on the base $C$. The _Hitchin spectral curve_

(1.3)
$\textstyle{\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{i}$$\textstyle{\overline{T^{*}C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{C}$

for a meromorphic Higgs bundle is defined as the divisor of zeros on
$\overline{T^{*}C}$ of the characteristic polynomial of $\phi$:

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

where $\eta\in H^{0}(T^{*}C,\pi^{*}K_{C})$ is the tautological $1$-form on
$T^{*}C$ extended as a meromorphic $1$-form on the compactification
$\overline{T^{*}C}$. The morphism $\pi:\Sigma\longrightarrow C$ is a degree
$r$ map.

We denote by ${\mathcal{M}}_{{\text{\rm{Dol}}}}$ the moduli space of
holomorphic stable $SL(r,{\mathbb{C}})$-Higgs bundles on $C$ for $g\geq 2$.
The assignment of the coefficients of the characteristic polynomial (1.4) to
$(E,\phi)\in{\mathcal{M}}_{{\text{\rm{Dol}}}}$ defines the Hitchin fibration

(1.5) $\mu_{H}:{\mathcal{M}}_{{\text{\rm{Dol}}}}\longrightarrow
B:=\bigoplus_{i=2}^{r}H^{0}(C,K_{C}^{\otimes i}).$

With the choice of a spin structure $K_{C}^{\frac{1}{2}}$ and Kostant’s
_principal three-dimensional subgroup_ TDS of [51], one constructs a cross-
section $\kappa:B\longrightarrow{\mathcal{M}}_{{\text{\rm{Dol}}}}$. We denote
by ${\langle}H,X_{+},X_{-}{\rangle}\subset sl(r,{\mathbb{C}})$ the Lie algebra
of a principal TDS, where we use the standard representation as traceless
matrices acting on ${\mathbb{C}}^{r}$. Thus $H$ is diagonal, $X_{-}$ is lower
triangular, $X_{+}=X_{-}^{t}$, and their relations are

(1.6) $[H,X_{\pm}]=\pm 2X_{\pm},\qquad[X_{+},X_{-}]=H.$

The map $\kappa$ is defined by

$B\owns{\mathbf{q}}=(q_{2},\dots,q_{r})\longmapsto\kappa({\mathbf{q}})\in\big{(}E_{0},\phi({\mathbf{q}})\big{)}\in{\mathcal{M}}_{{\text{\rm{Dol}}}},$

where

$E_{0}:=\big{(}K_{C}^{\frac{1}{2}}\big{)}^{H}\\\
,\qquad\phi({\mathbf{q}}):=X_{-}+\sum_{\ell=2}^{r}q_{\ell}X_{+}^{\ell-1}.$

Clearly $\kappa$ is not a section of the fibration $\mu_{H}$ in general
because $\mu_{H}\circ\kappa$ is not the identity map of $B$ for $r\geq 3$.
Nonetheless, the image of $\kappa$ intersects with every fiber of $\mu_{H}$
exactly at one point. Note that $B$ is the moduli space of Hitchin spectral
curves associated with holomorphic $SL(r,{\mathbb{C}})$-Higgs bundles on $C$.
We use an unconventional way of defining the universal family of spectral
curves over $B$, appealing to the Hitchin section $\kappa$, as

(1.7) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.33334pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
37.1667pt\raise-30.81729pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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30.33334pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{B\times\overline{T^{*}C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt\raise
0.0pt\hbox{$\textstyle{\;B}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad\qquad
p^{-1}({\mathbf{q}})=\Sigma\big{(}\phi({\mathbf{q}})\big{)}.$

Now we choose and fix, once and for all, a _projective coordinate system_ of
$C$ subordinating the complex structure of $C$. This process does not depend
algebraically on the moduli space of $C$. For a curve of genus $g\geq 2$, the
Fuchsian projective structure, that appears in our solution [22] to a
conjecture of Gaiotto [35], is a natural choice for our purpose. As we show in
Section 3, there is a unique _filtered extension_ $E_{\hbar}$ for every
$\hbar\in H^{1}(C,K_{C})$. For $r=2$, $E_{\hbar}$ is the canonical extension

$0\longrightarrow K_{C}^{\frac{1}{2}}\longrightarrow E_{\hbar}\longrightarrow
K_{C}^{-{\frac{1}{2}}}\longrightarrow 0$

associated with

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

With respect to the projective coordinate system, we can define a one-
parameter family of opers

$\left(E_{\hbar},\nabla^{\hbar}({\mathbf{q}})\right)\in{\mathcal{M}}_{{\text{\rm{deR}}}}$

for $\hbar\neq 0$, where

(1.8) $\nabla^{\hbar}({\mathbf{q}}):=d+\frac{1}{\hbar}\phi({\mathbf{q}}),$

$d$ is the exterior differentiation on C, and
${\mathcal{M}}_{{\text{\rm{deR}}}}$ is the moduli space of holomorphic
irreducible $SL(r,{\mathbb{C}})$-connections on $C$. The sum of the exterior
differentiation and a Higgs field is _not_ a connection in general. Here, the
point is that the original vector bundle $E_{0}$ is deformed to $E_{\hbar}$,
and we have chosen a projective coordinate system on $C$. Therefore, (1.8)
makes sense as a global connection on $C$ in $E_{\hbar}$.

Note that $\hbar\nabla^{\hbar}({\mathbf{q}})$ is Deligne’s $\hbar$-connection
interpolating a connection $d+\phi({\mathbf{q}})$ and a Higgs field
$\phi({\mathbf{q}})$. We also note that
$\left(E_{\hbar},\hbar\nabla^{\hbar}({\mathbf{q}})\right)$ defines a global
Rees ${\mathcal{D}}$-module on $C$. Its generator is a globally defined
differential operator $P$ on $C$ that acts on $K_{C}^{-\frac{r-1}{2}}$, which
is what we call the _quantum curve_ of the Hitchin spectral curve
$\Sigma\big{(}\phi({\mathbf{q}})\big{)}$ corresponding to ${\mathbf{q}}\in B$.
The actual shape (3.55) of $P$ is quite involved due to non-commutativity of
the coordinate of $C$ and differentiation. It is determined in the proof of
Theorem 3.11. In Example 3.1 we list quantum curves $P$ for $r=2,3,4$. No
matter how complicated its form is, the semi-classical limit of $P$ recovers
the spectral curve $\sigma^{*}\Sigma\big{(}\phi({\mathbf{q}})\big{)}$ of the
Higgs field $-\phi({\mathbf{q}})$, where

(1.9)
$\sigma:\overline{T^{*}C}\longrightarrow\overline{T^{*}C},\qquad\sigma^{2}=1,$

is the involution defined by the fiber-wise action of $-1$. This extra sign
comes from the difference of conventions in the characteristic polynomial
(1.4) and the connection (1.8).

The above process can be generalized in a straightforward way to _meromorphic_
spectral data ${\mathbf{q}}$ for a curve $C$ of arbitrary genus. The
corresponding connections $\nabla^{\hbar}({\mathbf{q}})$, and hence the Rees
${\mathcal{D}}$-modules, then have regular and irregular singularities.

We note that when we use the Fuchsian projective coordinate system of a curve
$C$ of genus $g\geq 2$ and holomorphic $SL(r,{\mathbb{C}})$-Higgs bundles, our
quantization process is exactly the same as the construction of
$SL(r,{\mathbb{C}})$-opers of [22] that was established by solving a
conjecture of Gaiotto [35].

In Section 6, we perform the generalized topological recursion in terms of the
PDE version for the case of meromorphic $SL(2,{\mathbb{C}})$-Higgs bundles.
For this purpose, we use a normalization method of [24] for singular Hitchin
spectral curves. We then show that the WKB analysis that the PDE recursion
provides is exactly for the quantum curve constructed through (1.8). When we
deal with a singular spectral curve $\Sigma\subset\overline{T^{*}C}$, the key
question is how to relate the singular curve with smooth ones. In terms of the
Hitchin fibration, a singular spectral curve corresponds to a degenerate
Abelian variety in the family. There are two different approaches to this
question: one is to deform $\Sigma$ locally to a non-singular curve, and the
other is to blow up $\overline{T^{*}C}$ and obtain a resolution of
singularities $\widetilde{\Sigma}$ of $\Sigma$. In this paper we will pursue
the second path, and give a WKB analysis of the quantum curve using the
geometric information of the desingularization.

Kostant’s principal TDS plays a crucial role in our quantization through the
relation (1.6). For example, it selects a particular fixed point of
${\mathbb{C}}^{*}$-action on the Hitchin section, which corresponds to the
$\hbar\rightarrow\infty$ limit of (1.8). It is counterintuitive, but this
limit is the connection $d+X_{-}$ acting on $E_{\hbar=1}$, not just $d$ which
looks to be the case from the formula. This limiting connection then defines a
vector space structure in the moduli space of opers.

This paper is organized as follows. The notion of quantum curves as Rees
${\mathcal{D}}$-modules quantizing Hitchin spectral curves is presented in
Section 2. Then in Section 3, we quantize Hitchin spectral curves as Rees
${\mathcal{D}}$-modules through a concrete construction $\hbar$-families of
holomorphic and meromorphic $SL(r,{\mathbb{C}})$-opers. The semi-classical
limit of these resulting opers is calculated. Since our topological recursion
depends solely on the geometry of normalization of singular Hitchin spectral
curves, we provide detailed study of the blow-up process in Sections 4. We
give the genus formula for the normalization of the spectral curve in terms of
the characteristic polynomial of the Higgs field $\phi$. Then in Section 5, we
define topological recursions for the case of degree $2$ coverings. In Section
6, we prove that an all-order WKB analysis for quantization of meromorphic
$SL(2,{\mathbb{C}})$-Hitchin spectral curves is established through PDE
topological recursion. We thus show that two quantizations procedures, one
through $\hbar$-family of opers and the other through PDE topological
recursion, agree for $SL(2,{\mathbb{C}})$. The general structure of the theory
is explained using the Airy differential equation as an example in Section 7.
This example shows how the WKB analysis computes quantum invariants.

The current paper does not address difference equations that appear as quantum
curves in knot theory, nor the mysterious spectral theory of [54].

##  2\. Rees ${\mathcal{D}}$-modules as quantum curves for Higgs bundles

In this section, we give the definition of quantum curves in the context of
Hitchin spectral curves. Let $C$ be a non-singular projective algebraic curve
defined over ${\mathbb{C}}$. The sheaf ${\mathcal{D}}_{C}$ of differential
operators on $C$ is the subalgebra of the ${\mathbb{C}}$-linear endomorphism
algebra ${\mathcal{E}}nd_{{\mathbb{C}}}({\mathcal{O}}_{C})$ generated by the
anti-canonical sheaf $K_{C}^{-1}$ and the structure sheaf ${\mathcal{O}}_{C}$.
Here, $K_{C}^{-1}$ acts on ${\mathcal{O}}_{C}$ as holomorphic vector fields,
and ${\mathcal{O}}_{C}$ acts on itself by multiplication. Locally every
element of ${\mathcal{D}}_{C}$ is written as

${\mathcal{D}}_{C}\owns
P(x)=\sum_{\ell=0}^{r}a_{\ell}(x)\left(\frac{d}{dx}\right)^{r-\ell},\qquad
a_{\ell}(x)\in{\mathcal{O}}_{C}$

for some $r\geq 0$. For a fixed $r$, we introduce the filtration by order of
differential operators into ${\mathcal{D}}_{C}$ as follows:

$F_{r}{\mathcal{D}}_{C}=\left.\left\\{P(x)=\sum_{\ell=0}^{r}a_{\ell}(x)\left(\frac{d}{dx}\right)^{r-\ell}\right|a_{\ell}(x)\in{\mathcal{O}}_{C}\right\\}.$

The _Rees_ ring $\widetilde{{\mathcal{D}}_{C}}$ is defined by

(2.1)
$\widetilde{{\mathcal{D}}_{C}}=\bigoplus_{r=0}^{\infty}\hbar^{r}F_{r}{\mathcal{D}}_{C}\subset{\mathbb{C}}[\\![\hbar]\\!]\otimes_{\mathbb{C}}{\mathcal{D}}_{C}.$

An element of $\widetilde{{\mathcal{D}}_{C}}$ on a coordinate neighborhood
$U\subset C$ can be written as

(2.2)
$P(x,\hbar)=\sum_{\ell=0}^{r}a_{\ell}(x,\hbar)\left(\hbar\frac{d}{dx}\right)^{r-\ell}.$

######  Definition 2.1 (Rees ${\mathcal{D}}$-module).

The Rees construction

(2.3)
$\widetilde{{\mathcal{M}}}=\bigoplus_{r=0}^{\infty}\hbar^{r}F_{r}{\mathcal{M}}$

associated with a filtered ${\mathcal{D}}_{C}$-module
$(F_{\bullet},{\mathcal{M}})$ is a Rees ${\mathcal{D}}$-module if the
compatibility condition $F_{a}{\mathcal{D}}_{C}\cdot F_{b}{\mathcal{M}}\subset
F_{a+b}{\mathcal{M}}$ holds.

Let

$D=\sum_{j=1}^{n}m_{j}p_{j},\qquad m_{j}>0$

be an effective divisor on $C$. The point set $\\{p_{1},\dots,p_{n}\\}\subset
C$ is the support of $D$. A _meromorphic Higgs bundle_ with poles at $D$ is a
pair $(E,\phi)$ consisting of an algebraic vector bundle $E$ on $C$ and a
Higgs field

(2.4) $\phi:E\longrightarrow K_{C}(D)\otimes_{{\mathcal{O}}_{C}}E.$

Since the cotangent bundle

$T^{*}C={\text{\rm{Spec}}}\left({\text{\rm{Sym}}}\left(K_{C}^{-1}\right)\right)$

is the total space of $K_{C}$, we have the tautological $1$-form $\eta\in
H^{0}(T^{*}C,\pi^{*}K_{C})$ on $T^{*}C$ coming from the projection

$\begin{CD}T^{*}C@<{}<{}<\pi^{*}K_{C}\\\ @V{\pi}V{}V\\\
<EMAIL_ADDRESS>
The natural holomorphic symplectic form of $T^{*}C$ is given by $-d\eta$. The
compactified cotangent bundle of $C$ is a ruled surface defined by

(2.5)
$\overline{T^{*}C}:={\mathbb{P}}\left(K_{C}\oplus{\mathcal{O}}_{C}\right)={\text{\rm{Proj}}}\left(\bigoplus_{n=0}^{\infty}\left(K_{C}^{-n}\cdot
I^{0}\oplus K_{C}^{-n+1}\cdot I\oplus\cdots\oplus K_{C}^{0}\cdot
I^{n}\right)\right),$

where $I$ represents $1\in{\mathcal{O}}_{C}$ being considered as a degree $1$
element. The divisor at infinity

(2.6) $C_{\infty}:={\mathbb{P}}\left(K_{C}\oplus\\{0\\}\right)$

is reduced in the ruled surface and supported on the subset
${\mathbb{P}}\left(K_{C}\oplus{\mathcal{O}}_{C}\right)\setminus T^{*}C$. The
tautological $1$-form $\eta$ extends on $\overline{T^{*}C}$ as a meromorphic
$1$-form with simple poles along $C_{\infty}$. Thus the divisor of $\eta$ in
$\overline{T^{*}C}$ is given by

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

where $C_{0}$ is the zero section of $T^{*}C$.

The relation between the sheaf ${\mathcal{D}}_{C}$ and the geometry of the
compactified cotangent bundle $\overline{T^{*}C}$ is the following. First we
have

(2.8)
${\text{\rm{Spec}}}\left(\bigoplus_{m=0}^{\infty}F_{m}{\mathcal{D}}_{C}\big{/}F_{m-1}{\mathcal{D}}_{C}\right)={\text{\rm{Spec}}}\left(\bigoplus_{m=0}^{\infty}K_{C}^{-m}\right)=T^{*}C.$

Let us denote by
$gr_{m}{\mathcal{D}}_{C}=F_{m}{\mathcal{D}}_{C}\big{/}F_{m-1}{\mathcal{D}}_{C}$.
By writing $I=1\in H^{0}(C,{\mathcal{D}}_{C})$, we then have

$\overline{T^{*}C}={\text{\rm{Proj}}}\left(\bigoplus_{m=0}^{\infty}\left(gr_{m}{\mathcal{D}}_{C}\cdot
I^{0}\oplus gr_{m-1}{\mathcal{D}}_{C}\cdot I\oplus
gr_{m-2}{\mathcal{D}}_{C}\cdot I^{\otimes 2}\oplus\cdots\oplus
gr_{0}{\mathcal{D}}_{C}\cdot I^{\otimes m}\right)\right).$

######  Definition 2.2 (Spectral curve).

A spectral curve of degree $r$ is a divisor $\Sigma$ in $\overline{T^{*}C}$
such that the projection $\pi:\Sigma\longrightarrow C$ defined by the
restriction

$\textstyle{\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{i}$$\textstyle{\overline{T^{*}C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{C}$

is a finite morphism of degree $r$. The spectral curve of a Higgs bundle
$(E,\phi)$ is the divisor of zeros

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

on $\overline{T^{*}C}$ of the characteristic polynomial
$\det(\eta-\pi^{*}\phi)$. Here,

$\pi^{*}\phi:\pi^{*}E\longrightarrow\pi^{*}\left(K_{C}(D)\right)\otimes_{{\mathcal{O}}_{{\mathbb{P}}\left(K_{C}\oplus{\mathcal{O}}_{C}\right)}}\pi^{*}E.$

######  Remark 2.3.

The Higgs field $\phi$ is holomorphic on $C\setminus{\text{\rm{supp}}}(D)$.
Thus we can define the divisor of zeros

$\Sigma^{\circ}=\left(\det\left(\eta-\pi^{*}\left(\phi|_{C\setminus{\text{\rm{supp}}}(D)}\right)\right)\right)_{0}$

of the characteristic polynomial on $T^{*}(C\setminus{\text{\rm{supp}}}(D))$.
The spectral curve $\Sigma$ is the complex topology closure of
$\Sigma^{\circ}$ with respect to the compactification

(2.10) $T^{*}(C\setminus{\text{\rm{supp}}}(D))\subset\overline{T^{*}C}.$

A left ${\mathcal{D}}_{C}$-module ${\mathcal{E}}$ on $C$ is naturally an
${\mathcal{O}}_{C}$-module with a ${\mathbb{C}}$-linear integrable (i.e.,
flat) connection $\nabla:{\mathcal{E}}\longrightarrow
K_{C}\otimes_{{\mathcal{O}}_{C}}{\mathcal{E}}$. The construction goes as
follows:

(2.11)
$\begin{CD}\nabla:{\mathcal{E}}@>{\alpha}>{}>{\mathcal{D}}_{C}\otimes_{{\mathcal{O}}_{C}}{\mathcal{E}}@>{\nabla_{\\!{\mathcal{D}}}\otimes
id}>{}>\left(K_{C}\otimes_{{\mathcal{O}}_{C}}{\mathcal{D}}_{C}\right)\otimes_{{\mathcal{O}}_{C}}{\mathcal{E}}@>{\beta\otimes
id}>{}>K_{C}\otimes_{{\mathcal{O}}_{C}}{\mathcal{E}},\end{CD}$

where

  * •

$\alpha$ is the natural inclusion ${\mathcal{E}}\owns v\longmapsto 1\otimes
v\in{\mathcal{D}}_{C}\otimes_{{\mathcal{O}}_{C}}{\mathcal{E}}$;

  * •

$\nabla_{\\!{\mathcal{D}}}:{\mathcal{D}}_{C}\longrightarrow
K_{C}\otimes_{{\mathcal{O}}_{C}}{\mathcal{D}}_{C}$ is the connection defined
by the ${\mathbb{C}}$-linear left-multiplication operation of $K_{C}^{-1}$ on
${\mathcal{D}}_{C}$, which satisfies the derivation property

(2.12) $\nabla_{\\!{\mathcal{D}}}(f\cdot
P)=f\cdot\nabla_{\\!{\mathcal{D}}}(P)+df\cdot P\in
K_{C}\otimes_{{\mathcal{O}}_{C}}{\mathcal{D}}_{C}$

for $f\in{\mathcal{O}}_{C}$ and $P\in{\mathcal{D}}_{C}$; and

  * •

$\beta$ is the canonical right ${\mathcal{D}}_{C}$-module structure in $K_{C}$
defined by the Lie derivative of vector fields.

If we choose a local coordinate neighborhood $U\subset C$ with a coordinate
$x$, then (2.12) takes the following form. Let us denote by
$P^{\prime}=[d/dx,P]+P\cdot d/dx$, and define

$\nabla_{\\!\\!\frac{d}{dx}}(P):=P\cdot\frac{d}{dx}+P^{\prime}.$

Then we have

$\nabla_{\\!\\!\frac{d}{dx}}(f\cdot
P)=f\cdot\nabla_{\\!\\!\frac{d}{dx}}(P)+\frac{df}{dx}\cdot P.$

The connection $\nabla$ of (2.11) is integrable because $d^{2}=1$. Actually,
the statement is true for any dimensions. We note that there is no reason for
${\mathcal{E}}$ to be coherent as an ${\mathcal{O}}_{C}$-module.

Conversely, if an algebraic vector bundle $E$ on $C$ of rank $r$ admits a
holomorphic connection $\nabla:E\longrightarrow K_{C}\otimes E$, then $E$
acquires the structure of a ${\mathcal{D}}_{C}$-module. This is because
$\nabla$ is automatically flat, and the covariant derivative $\nabla_{\\!X}$
for $X\in K_{C}^{-1}$ satisfies

(2.13) $\nabla_{\\!X}(fv)=f\nabla_{\\!X}(v)+X(f)v$

for $f\in{\mathcal{O}}_{C}$ and $v\in E$. A repeated application of (2.13)
makes $E$ a ${\mathcal{D}}_{C}$-module. The fact that every
${\mathcal{D}}_{C}$-module on a curve is principal implies that for every
point $p\in C$, there is an open neighborhood $p\in U\subset C$ and a linear
differential operator $P$ of order $r$ on $U$, called a generator, such that
$E|_{U}\cong{\mathcal{D}}_{U}/{\mathcal{D}}_{U}P.$ Thus on an open curve $U$,
a holomorphic connection in a vector bundle of rank $r$ gives rise to a
differential operator of order $r$. The converse is true if
${\mathcal{D}}_{U}/{\mathcal{D}}_{U}P$ is ${\mathcal{O}}_{U}$-coherent.

######  Definition 2.4 (Formal $\hbar$-connection).

A formal $\hbar$-connection on a vector bundle $E\longrightarrow C$ is a
${\mathbb{C}}[\\![\hbar]\\!]$-linear homomorphism

$\nabla^{\hbar}:{\mathbb{C}}[\\![\hbar]\\!]\otimes
E\longrightarrow{\mathbb{C}}[\\![\hbar]\\!]\otimes
K_{C}\otimes_{{\mathcal{O}}_{C}}E$

subject to the derivation condition

(2.14) $\nabla^{\hbar}(f\cdot v)=f\nabla^{\hbar}(v)+\hbar df\otimes v,$

where $f\in{\mathcal{O}}_{C}\otimes{\mathbb{C}}[\\![\hbar]\\!]$ and
$v\in{\mathbb{C}}[\\![\hbar]\\!]\otimes E$.

When we consider _holomorphic_ dependence of a quantum curve with respect to
the quantization parameter $\hbar$, we need to use a particular
$\hbar$-deformation family of vector bundles. We will discuss the holomorphic
case in Section 3, where we explain how (1.1) appears in our quantization.

######  Remark 2.5.

The classical limit of a formal $\hbar$-connection is the evaluation $\hbar=0$
of $\nabla^{\hbar}$, which is simply an ${\mathcal{O}}_{C}$-module
homomorphism

$\nabla^{0}:E\longrightarrow K_{C}\otimes_{{\mathcal{O}}_{C}}E,$

i.e., a holomorphic Higgs field in the vector bundle $E$.

######  Remark 2.6.

An ${\mathcal{O}}_{C}\otimes{\mathbb{C}}[\\![\hbar]\\!]$-coherent
$\widetilde{{\mathcal{D}}_{C}}$-module is equivalent to a vector bundle on $C$
equipped with an $\hbar$-connection.

In analysis, the semi-classical limit of a differential operator $P(x,\hbar)$
of (2.2) is a function defined by

(2.15) $\lim_{\hbar\rightarrow
0}\left(e^{-\frac{1}{\hbar}S_{0}(x)}P(x,\hbar)e^{\frac{1}{\hbar}S_{0}(x)}\right)=\sum_{\ell=0}^{r}a_{\ell}(x,0)\left(S_{0}^{\prime}(x)\right)^{r-\ell},$

where $S_{0}(x)\in{\mathcal{O}}_{C}(U)$. The equation

(2.16) $\lim_{\hbar\rightarrow
0}\left(e^{-\frac{1}{\hbar}S_{0}(x)}P(x,\hbar)e^{\frac{1}{\hbar}S_{0}(x)}\right)=0$

then determines the first term of the singular perturbation expansion, or the
WKB asymptotic expansion,

(2.17) $\psi(x,\hbar)=\exp\left(\sum_{m=0}^{\infty}\hbar^{m-1}S_{m}(x)\right)$

of a solution $\psi(x,\hbar)$ to the differential equation

$P(x,\hbar)\psi(x,\hbar)=0$

on U. We note that the expression (2.17) is never meant to be a convergent
series in $\hbar$.

Since $dS_{0}(x)$ is a local section of $T^{*}C$ on $U\subset C$,
$y=S_{0}^{\prime}(x)$ gives a local trivialization of $T^{*}C|_{U}$, with
$y\in T_{x}^{*}C$ a fiber coordinate. Then (2.15) and (2.16) give an equation

(2.18) $\sum_{\ell=0}^{r}a_{\ell}(x,0)y^{r-\ell}=0$

of a curve in $T^{*}C|_{U}$. This motivates us to give the following
definition:

######  Definition 2.7 (Semi-classical limit of a Rees differential operator).

Let $U\subset C$ be an open subset of $C$ with a local coordinate $x$ such
that $T^{*}C$ is trivial over $U$ with a fiber coordinate $y$. The semi-
classical limit of a local section

$P(x,\hbar)=\sum_{\ell=0}^{r}a_{\ell}(x,\hbar)\left(\hbar\frac{d}{dx}\right)^{r-\ell}$

of the Rees ring $\widetilde{{\mathcal{D}}_{C}}$ of the sheaf of differential
operators ${\mathcal{D}}_{C}$ on $U$ is the holomorphic function

$\sum_{\ell=0}^{r}a_{\ell}(x,0)y^{r-\ell}$

defined on $T^{*}C|_{U}$.

######  Definition 2.8 (Semi-classical limit of a Rees
${\mathcal{D}}$-module).

Suppose a Rees $\widetilde{{\mathcal{D}}_{C}}$-module
$\widetilde{{\mathcal{M}}}$ globally defined on $C$ is written as

(2.19)
$\widetilde{{\mathcal{M}}}(U)=\widetilde{{\mathcal{D}}_{C}}(U)\big{/}\widetilde{{\mathcal{D}}_{C}}(U)P_{U}$

on every coordinate neighborhood $U\subset C$ with a differential operator
$P_{U}$ of the form (2.2). Using this expression (2.2) for $P_{U}$, we
construct a meromorphic function

(2.20) $p_{U}(x,y)=\sum_{\ell=0}^{r}a_{\ell}(x,0)y^{r-\ell}$

on $\overline{T^{*}C}|_{U}$, where $y$ is the fiber coordinate of $T^{*}C$,
which is trivialized on $U$. Define

(2.21) $\Sigma_{U}=(p_{U}(x,y))_{0}$

as the divisor of zero of the function $p_{U}(x,y)$. If $\Sigma_{U}$’s glue
together to a spectral curve $\Sigma\subset\overline{T^{*}C}$, then we call
$\Sigma$ the semi-classical limit of the Rees
$\widetilde{{\mathcal{D}}_{C}}$-module $\widetilde{{\mathcal{M}}}$.

######  Remark 2.9.

For the local equation (2.20) to be consistent globally on $C$, the
coefficients of (2.2) have to satisfy

(2.22) $a_{\ell}(x,0)\in\Gamma\\!\left(U,K_{C}^{\otimes\ell}\right).$

######  Definition 2.10 (Quantum curve for holomorphic Higgs bundle).

A quantum curve associated with the spectral curve $\Sigma\subset T^{*}C$ of a
holomorphic Higgs bundle on a projective algebraic curve $C$ is a Rees
$\widetilde{{\mathcal{D}}_{C}}$-module ${\mathcal{E}}$ whose semi-classical
limit is $\Sigma$.

The main reason we wish to extend our framework to meromorphic connections is
that there are no non-trivial holomorphic connections on ${\mathbb{P}}^{1}$,
whereas many important classical examples of differential equations are
naturally defined over ${\mathbb{P}}^{1}$ with regular and irregular
singularities. A ${\mathbb{C}}$-linear homomorphism

$\nabla:E\longrightarrow K_{C}(D)\otimes_{{\mathcal{O}}_{C}}E$

is said to be a _meromorphic connection_ with poles along an effective divisor
$D$ if

$\nabla(f\cdot v)=f\nabla(v)+df\otimes v$

for every $f\in{\mathcal{O}}_{C}$ and $v\in E$. Let us denote by

${\mathcal{O}}_{C}(*D):=\lim_{\longrightarrow}{\mathcal{O}}_{C}(mD),\qquad
E(*D):=E\otimes_{{\mathcal{O}}_{C}}{\mathcal{O}}_{C}(*D).$

Then $\nabla$ extends to

$\nabla:E(*D)\longrightarrow K_{C}(*D)\otimes_{{\mathcal{O}}_{C}(*D)}E(*D).$

Since $\nabla$ is holomorphic on $C\setminus{\text{\rm{supp}}}(D)$, it induces
a ${\mathcal{D}}_{C\setminus{\text{\rm{supp}}}(D)}$-module structure in
$E|_{C\setminus{\text{\rm{supp}}}(D)}$. The ${\mathcal{D}}_{C}$-module direct
image $\widetilde{E}=j_{*}\left(E|_{C\setminus{\text{\rm{supp}}}(D)}\right)$
associated with the open inclusion map
$j:C\setminus{\text{\rm{supp}}}(D)\longrightarrow C$ is then naturally
isomorphic to

(2.23)
$\widetilde{E}=j_{*}\left(E|_{C\setminus{\text{\rm{supp}}}(D)}\right)\cong
E(*D)$

as a ${\mathcal{D}}_{C}$-module. (2.23) is called the _meromorphic extension_
of the ${\mathcal{D}}_{C\setminus{\text{\rm{supp}}}(D)}$-module
$E|_{C\setminus{\text{\rm{supp}}}(D)}$.

Let us take a local coordinate $x$ of $C$, this time around a pole
$p_{j}\in{\text{\rm{supp}}}(D)$. If a generator $\widetilde{P}$ of
$\widetilde{E}$ near $x=0$ has a local expression

(2.24)
$\widetilde{P}(x,d/dx)=x^{k}\sum_{\ell=0}^{r}b_{\ell}(x)\left(x\frac{d}{dx}\right)^{r-\ell}$

around $p_{j}$ with locally defined holomorphic functions $b_{\ell}(x)$,
$b_{0}(0)\neq 0$, and an integer $k\in{\mathbb{Z}}$, then $\widetilde{P}$ has
a _regular_ singular point at $p_{j}$. Otherwise, $p_{j}$ is an _irregular_
singular point of $\widetilde{P}$.

######  Definition 2.11 (Quantum curve for a meromorphic Higgs bundle).

Let $(E,\phi)$ be a meromorphic Higgs bundle defined over a projective
algebraic curve $C$ of any genus with poles along an effective divisor $D$,
and $\Sigma\subset\overline{T^{*}C}$ its spectral curve. A quantum curve
associated with $\Sigma$ is the meromorphic extension of a Rees
$\widetilde{{\mathcal{D}}_{C}}$-module ${\mathcal{E}}$ on
$C\setminus{\text{\rm{supp}}}(D)$ such that the complex topology closure of
its semi-classical limit $\Sigma^{\circ}\subset
T^{*}C|_{C\setminus{\text{\rm{supp}}}(D)}$ in the compactified cotangent
bundle $\overline{T^{*}C}$ agrees with $\Sigma$.

In Section 3, we prove that every Hitchin spectral curve associated with a
holomorphic or a meromorphic $SL(r,{\mathbb{C}})$-Higgs bundle has a quantum
curve.

######  Remark 2.12.

We remark that several examples of quantum curves that are constructed in [10,
28, 57], for various Hurwitz numbers and Gromov-Witten theory of
${\mathbb{P}}^{1}$, do not fall into our definition in terms of Rees
${\mathcal{D}}$-modules. This is because in the above mentioned examples,
quantum curves involve _infinite_ -order differential operators, or
_difference_ operators, while we consider only differential operators of
finite order in this paper.

##  3\. Opers

There is a simple mechanism to construct a quantization of Hitchin spectral
curve, using a particular choice of isomorphism between a _Hitchin section_
and the moduli of _opers_. The quantum deformation parameter $\hbar$,
originated in physics as the Planck _constant_ , is a purely formal parameter
in WKB analysis. Since we will be using the PDE recursion (5.6) for the
analysis of quantum curves, $\hbar$ plays the role of a formal parameter for
the asymptotic expansion. This point of view motivates our definition of
quantum curves as Rees $D$-modules in the previous section. However, the
quantum curves appearing in the quantization of Hitchin spectral curves
associated with $G$-Higgs bundles for a complex simple Lie group $G$ always
depend _holomorphically_ on $\hbar$. Therefore, we need a more geometric setup
for quantum curves to deal with this holomorphic dependence. The purpose of
this section is to explain holomorphic $\hbar$-connections as quantum curves,
and the geometric interpretation of $\hbar$ given in (1.1). The key concept is
_opers_ of Beilinson-Drinfeld [4]. Although a vast generalization of the
current paper is possible, we restrict our attention to
$SL(r,{\mathbb{C}})$-opers for an arbitrary $r\geq 2$ in this paper.

In this section, most of the time $C$ is a smooth projective algebraic curve
of genus $g\geq 2$ defined over ${\mathbb{C}}$, unless otherwise specified.

###  3.1. Holomorphic $SL(r,{\mathbb{C}})$-opers and quantization of Higgs
bundles

We first recall projective structures on $C$ following Gunning [39]. Recall
that every compact Riemann surface has a projective structure subordinating
the given complex structure. A _complex projective coordinate system_ is a
coordinate neighborhood covering

$C=\bigcup_{\alpha}U_{\alpha}$

with a local coordinate $x_{\alpha}$ of $U_{\alpha}$ such that for every
$U_{\alpha}\cap U_{\beta}$, we have a _Möbius_ coordinate transformation

(3.1)
$x_{\alpha}=\frac{a_{\alpha\beta}x_{\beta}+b_{\alpha\beta}}{c_{\alpha\beta}x_{\beta}+d_{\alpha\beta}},\qquad\begin{bmatrix}a_{\alpha\beta}&b_{\alpha\beta}\\\
c_{\alpha\beta}&d_{\alpha\beta}\end{bmatrix}\in SL(2,{\mathbb{C}}).$

Since we solve differential equations on $C$, we always assume that each
coordinate neighborhood $U_{\alpha}$ is simply connected. In what follows, we
choose and fix a projective coordinate system on $C$. Since

$dx_{\alpha}=\frac{1}{(c_{\alpha\beta}x_{\beta}+d_{\alpha\beta})^{2}}\;dx_{\beta},$

the transition function for the canonical line bundle $K_{C}$ of $C$ is given
by the cocycle

$\left\\{(c_{\alpha\beta}x_{\beta}+d_{\alpha\beta})^{2}\right\\}\quad\text{on}\quad
U_{\alpha}\cap U_{\beta}.$

We choose and fix, once and for all, a theta characteristic, or a spin
structure, $K_{C}^{\frac{1}{2}}$ such that
$\left(K_{C}^{\frac{1}{2}}\right)^{\otimes 2}\cong K_{C}$. Let
$\\{\xi_{\alpha\beta}\\}$ denote the $1$-cocycle corresponding to
$K_{C}^{\frac{1}{2}}$. Then we have

(3.2) $\xi_{\alpha\beta}=\pm(c_{\alpha\beta}x_{\beta}+d_{\alpha\beta}).$

The choice of $\pm$ here is an element of
$H^{1}(C,{\mathbb{Z}}/2{\mathbb{Z}})=({\mathbb{Z}}/2{\mathbb{Z}})^{2g}$,
indicating that there are $2^{2g}$ choices for spin structures in $C$.

The significance of the projective coordinate system lies in the fact that
$\partial^{2}_{\beta}\xi_{\alpha\beta}=0$. This simple property plays an
essential role in our construction of global connections on $C$, as we see in
this section. Another way of appreciating projective coordinate system is the
vanishing of Schwarzian derivatives, as explained in [21]. A scalar valued
single linear ordinary differential equation of any order can be globally
defined in terms of a projective coordinate.

A holomorphic Higgs bundle $(E,\phi)$ is _stable_ if for every vector
subbundle $F\subset E$ that is invariant with respect to $\phi$, i.e.,
$\phi:F\longrightarrow F\otimes K_{C}$, the slope condition

$\frac{\deg F}{{\text{\rm{rank}}}F}<\frac{\deg E}{{\text{\rm{rank}}}E}$

holds. The moduli space of stable Higgs bundles is constructed [64]. An
$SL(r,{\mathbb{C}})$-Higgs bundle is a pair $(E,\phi)$ with a fixed
isomorphism $\det E={\mathcal{O}}_{C}$ and ${\text{\rm{tr}}}\phi=0$. We denote
by ${\mathcal{M}}_{{\text{\rm{Dol}}}}$ the moduli space of stable holomorphic
$SL(r,{\mathbb{C}})$-Higgs bundles on $C$. Hitchin [40] defines a holomorphic
fibration

(3.3)
$\mu_{H}:{\mathcal{M}}_{{\text{\rm{Dol}}}}\owns(E,\phi)\longmapsto\det(\eta-\pi^{*}\phi)\in
B,\qquad B:=\bigoplus_{\ell=2}^{r}H^{0}\left(C,K_{C}^{\otimes\ell}\right),$

that induces the structure of an algebraically completely integrable
Hamiltonian system in ${\mathcal{M}}_{{\text{\rm{Dol}}}}$. With the choice of
a spin structure $K_{C}^{\frac{1}{2}}$, we have a natural section
$\kappa:B\hookrightarrow{\mathcal{M}}_{{\text{\rm{Dol}}}}$ defined by
utilizing Kostant’s _principal three-dimensional subgroup_ (TDS) [51] as
follows.

First, let

$\mathbf{q}=(q_{2},q_{3},\dots,q_{r})\in
B=\bigoplus_{\ell=2}^{r}H^{0}(C,K_{C}^{\otimes\ell})$

be an arbitrary point of the Hitchin base $B$. Define

(3.4) $\displaystyle
X_{-}:=\big{[}\sqrt{s_{i-1}}\delta_{i-1,j}\big{]}_{ij}=\begin{bmatrix}0&0&\cdots&0&0\\\
\sqrt{s_{1}}&&&&0\\\ 0&\sqrt{s_{2}}&&&0\\\ \vdots&&\ddots&&\vdots\\\
0&0&\cdots&\sqrt{s_{r-1}}&0\end{bmatrix},\qquad X_{+}:=X_{-}^{t},$
$\displaystyle H:=[X_{+},X_{-}],$

where $s_{i}:=i(r-i)$. $H$ is a diagonal matrix whose $(i,i)$-entry is
$s_{i}-s_{i-1}=r-2i+1$, with $s_{0}=s_{r}=0$. The Lie algebra
${\langle}X_{+},X_{-},H{\rangle}\cong sl(2,{\mathbb{C}})$ is the Lie algebra
of the principal TDS in $SL(r,{\mathbb{C}})$.

######  Lemma 3.1.

Define a Higgs bundle $\left(E_{0},\phi(\mathbf{q})\right)$ consisting of a
vector bundle

(3.5) $E_{0}:=\left(K_{C}^{\frac{1}{2}}\right)^{\otimes
H}=\bigoplus_{i=1}^{r}\left(K_{C}^{\frac{1}{2}}\right)^{\otimes(r-2i+1)}$

and a Higgs field

(3.6) $\phi(\mathbf{q}):=X_{-}+\sum_{\ell=2}^{r}q_{\ell}X_{+}^{\ell-1}.$

Then it is a stable $SL(r,{\mathbb{C}})$-Higgs bundle. The Hitchin section is
defined by

(3.7)
$\kappa:B\owns{\mathbf{q}}\longmapsto\left(E_{0},\phi(\mathbf{q})\right)\in{\mathcal{M}}_{{\text{\rm{Dol}}}},$

which gives a biholomorphic map between $B$ and
$\kappa(B)\subset{\mathcal{M}}_{{\text{\rm{Dol}}}}$.

###### Proof.

We first note that $X_{-}:E_{0}\longrightarrow E_{0}\otimes K_{C}$ is a
globally defined ${\text{\rm{End}}}_{0}(E_{0})$-valued $1$-form, since it is a
collection of constant maps

(3.8)
$\sqrt{s_{i}}:\left(K_{C}^{\frac{1}{2}}\right)^{\otimes(r-2i+1)}\overset{=}{\longrightarrow}\left(K_{C}^{\frac{1}{2}}\right)^{\otimes(r-2(i+1)+1)}\otimes
K_{C}.$

Similarly, since $X_{+}^{\ell-1}$ is an upper-diagonal matrix with non-zero
entries along the $(\ell-1)$-th upper diagonal, we have

$q_{\ell}:\left(K_{C}^{\frac{1}{2}}\right)^{\otimes(r-2i+1)}{\longrightarrow}\left(K_{C}^{\frac{1}{2}}\right)^{\otimes(r-2i+1+2\ell)}=\left(K_{C}^{\frac{1}{2}}\right)^{\otimes(r-2(i-\ell+1)+1)}\otimes
K_{C}.$

Thus $\phi({\mathbf{q}}):E_{0}\longrightarrow E_{0}\otimes K_{C}$ is globally
defined as a Higgs field in $E_{0}$. The Higgs pair is stable because no
subbundle of $E_{0}$ is invariant under $\phi({\mathbf{q}})$, unless
${\mathbf{q}}=0$. And when ${\mathbf{q}}=0$, the invariant subbundles all have
positive degrees, since $g\geq 2$. ∎

The image $\kappa(B)$ is a holomorphic Lagrangian submanifold of a holomorphic
symplectic space ${\mathcal{M}}_{{\text{\rm{Dol}}}}$.

To define $\hbar$-connections holomorphically depending on $\hbar$, we need to
construct a one-parameter holomorphic family of deformations of vector bundles

$\begin{CD}E_{\hbar}@>{}>{}>{\mathcal{E}}\\\ @V{}V{}V@V{}V{}V\\\
C\times\\{\hbar\\}@>{}>{}>C\times H^{1}(C,K_{C})\end{CD}$

and a ${\mathbb{C}}$-linear first-order differential operator

$\hbar\nabla^{\hbar}:E_{\hbar}\longrightarrow E_{\hbar}\otimes K_{C}$

depending holomorphically on $\hbar\in H^{1}(C,K_{C})\cong{\mathbb{C}}$ for
$\hbar\neq 0$. Let us introduce the notion of _filtered extensions_.

######  Definition 3.2 (Filtered extension).

A one-parameter family of filtered holomorphic vector bundles
$\left(F_{\hbar}^{\bullet},E_{\hbar}\right)$ on $C$ with a trivialized
determinant $\det(E_{\hbar})\cong{\mathcal{O}}_{C}$ is a filtered extension of
the vector bundle $E_{0}$ of (3.5) parametrized by $\hbar\in H^{1}(C,K_{C})$
if the following conditions hold:

  * •

$E_{\hbar}$ has a filtration

(3.9) $0=F_{\hbar}^{r}\subset F_{\hbar}^{r-1}\subset
F_{\hbar}^{r-2}\subset\cdots\subset F_{\hbar}^{0}=E_{\hbar}.$

  * •

The second term is given by

(3.10) $F_{\hbar}^{r-1}=\left(K_{C}^{\frac{1}{2}}\right)^{\otimes(r-1)}.$

  * •

For every $i=1,2,\dots,r-1$, there is an ${\mathcal{O}}_{C}$-module
isomorphism

(3.11)
$F_{\hbar}^{i}\big{/}F_{\hbar}^{i+1}\overset{\sim}{\longrightarrow}\left(F_{\hbar}^{i-1}\big{/}F_{\hbar}^{i}\right)\otimes
K_{C}.$

######  Remark 3.3.

Since we need to identify a deformation parameter $\hbar$ and extension
classes, we make the natural identification

(3.12) ${\text{\rm{Ext}}}^{1}\left(E,F\right)=H^{1}\left(C,E^{*}\otimes
F\right)$

for every pair of vector bundles $E$ and $F$. We also identify
${\text{\rm{Ext}}}^{1}(E,F)$ as the class of extensions

$0\longrightarrow F\longrightarrow V\longrightarrow E\longrightarrow 0$

of $E$ by a vector bundle $V$. These identifications are done by a choice of a
projective coordinate system on $C$ as explained below.

######  Proposition 3.4 (Construction of filtered extension).

For every choice of the theta characteristic $K_{C}^{\frac{1}{2}}$ and a non-
zero element $\hbar\in H^{1}(C,K_{C})$, there is a unique non-trivial filtered
extension $\left(F_{\hbar}^{\bullet},E_{\hbar}\right)$ of $E_{0}$.

###### Proof.

First let us examine the case of $r=2$ to see how things work. Since

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

we have a unique extension

(3.13) $0\longrightarrow K_{C}^{\frac{1}{2}}\longrightarrow
E_{\hbar}\longrightarrow K_{C}^{-{\frac{1}{2}}}\longrightarrow 0$

corresponding to $\hbar$. Obviously

$K_{C}^{\frac{1}{2}}\overset{\sim}{\longrightarrow}\left(E_{\hbar}\big{/}K_{C}^{{\frac{1}{2}}}\right)\otimes
K_{C},$

which proves (3.11). We also note that as a vector bundle of rank $2$, we have
the isomorphism

(3.14) $E_{\hbar}\cong\begin{cases}E_{1}\qquad\hbar\neq 0,\\\
E_{0}\qquad\hbar=0.\end{cases}$

Now consider the general case. We use the induction on
$i=r-1,r-2,r-3,\dots,0,$ in the reverse direction to construct each term of
the filtration $F_{\hbar}^{i}$ subject to (3.10) and (3.11). The base case
$i=r-1$ is the following. Since

(3.15) $0\longrightarrow F_{\hbar}^{r-1}\longrightarrow
F_{\hbar}^{r-2}\longrightarrow F_{\hbar}^{r-2}/F_{\hbar}^{r-1}\longrightarrow
0$

and

$F_{\hbar}^{r-2}/F_{\hbar}^{r-1}\cong F_{\hbar}^{r-1}\otimes
K_{C}^{-1}\cong\left(K_{C}^{\frac{1}{2}}\right)^{\otimes(r-1)}\otimes
K_{C}^{-1}$

from (3.10) and (3.11), $F_{\hbar}^{r-2}$ is determined by the class $\hbar$
in

${\text{\rm{Ext}}}^{1}\left(F_{\hbar}^{r-2}/F_{\hbar}^{r-1},F_{\hbar}^{r-1}\right)=H^{1}\left(C,F_{\hbar}^{r-1}\otimes(F_{\hbar}^{r-2}/F_{\hbar}^{r-1})^{-1}\right)\cong
H^{1}(C,K_{C}).$

Assume that for a given $i+1$, we have

(3.16) $\displaystyle
H^{1}\left(C,F_{\hbar}^{n}\otimes\big{(}F_{\hbar}^{r-1}\big{)}^{-1}\otimes
K_{C}^{\otimes(-n+r)}\right)\cong H^{1}(C,K_{C}),$ (3.17) $\displaystyle
H^{1}\left(C,F_{\hbar}^{n}\otimes\big{(}F_{\hbar}^{r-1}\big{)}^{-1}\otimes
K_{C}^{\otimes(-n+m+r)}\right)=0,\qquad m\geq 1,$

for every $n$ in the range $i+1\leq n\leq r-1$. We wish to prove that the same
relation holds for $n=i$.

The sequence of isomorphisms (3.11) implies that

$F_{\hbar}^{i-1}/F_{\hbar}^{i}\cong\big{(}F_{\hbar}^{i}/F_{\hbar}^{i+1}\big{)}\otimes
K_{C}^{-1}\cong F_{\hbar}^{r-1}\otimes
K_{C}^{\otimes(i-r)}=\left(K_{C}^{\frac{1}{2}}\right)^{\otimes(r-1)}\otimes
K_{C}^{\otimes(i-r)}\cong K_{C}^{\otimes\left(i-\frac{r+1}{2}\right)}.$

Then $F_{\hbar}^{i-1}$ as an extension

(3.18) $0\longrightarrow F_{\hbar}^{i}\longrightarrow
F_{\hbar}^{i-1}\longrightarrow F_{\hbar}^{r-1}\otimes
K_{C}^{\otimes(i-r)}\longrightarrow 0$

is determined by a class in

${\text{\rm{Ext}}}^{1}\left(F_{\hbar}^{r-1}\otimes
K_{C}^{\otimes(i-r)},F_{\hbar}^{i}\right)=H^{1}\left(C,F_{\hbar}^{i}\otimes\big{(}F_{\hbar}^{r-1}\big{)}^{-1}\otimes
K_{C}^{\otimes(-i+r)}\right).$

The exact sequence

$0\longrightarrow F_{\hbar}^{n}\longrightarrow F_{\hbar}^{n-1}\longrightarrow
F_{\hbar}^{r-1}\otimes K_{C}^{\otimes(n-r)}\longrightarrow 0$

implies that

$0\longrightarrow
F_{\hbar}^{n}\otimes\big{(}F_{\hbar}^{r-1}\big{)}^{-1}\otimes
K_{C}^{\otimes(-n+m+r)}\longrightarrow
F_{\hbar}^{n-1}\otimes\big{(}F_{\hbar}^{r-1}\big{)}^{-1}\otimes
K_{C}^{\otimes(-n+m+r)}\longrightarrow K_{C}^{\otimes m}\longrightarrow 0$

for every $m\geq 1$. Taking the cohomology long exact sequence, we obtain

$H^{1}\left(C,F_{\hbar}^{n-1}\otimes\big{(}F_{\hbar}^{r-1}\big{)}^{-1}\otimes
K_{C}^{\otimes(-n+1+r)}\right)\cong H^{1}(C,K_{C})$

for $m=1$, which proves (3.16) for $n=i$. Similarly,

$\displaystyle
H^{1}\left(C,F_{\hbar}^{n-1}\otimes\big{(}F_{\hbar}^{r-1}\big{)}^{-1}\otimes
K_{C}^{\otimes(-(n-1)+(m-1)+r)}\right)$
$\displaystyle=H^{1}\left(C,F_{\hbar}^{n-1}\otimes\big{(}F_{\hbar}^{r-1}\big{)}^{-1}\otimes
K_{C}^{\otimes(-n+m+r)}\right)$ $\displaystyle\cong
H^{1}\left(C,K_{C}^{m}\right)=0$

for $m\geq 2$, which is (3.17) for $n=i$. By induction on $i$ in the
decreasing direction, we have established that the class $\hbar$ determines
the unique extension (3.18) for every $i$. ∎

######  Definition 3.5 ($SL(r,{\mathbb{C}})$-opers).

A point $(E,\nabla)\in{\mathcal{M}}_{{\text{\rm{deR}}}}$, i.e., an irreducible
holomorphic $SL(r,{\mathbb{C}})$-connection $\nabla:E\longrightarrow E\otimes
K_{C}$ acting on a vector bundle $E$, is an $SL(r,{\mathbb{C}})$-oper if the
following conditions are satisfied.

Filtration. There is a filtration $F^{\bullet}$ by vector subbundles

(3.19) $0=F^{r}\subset F^{r-1}\subset F^{r-2}\subset\cdots\subset F^{0}=E.$

Griffiths Transversality. The connection respects the filtration:

(3.20) $\nabla|_{F^{i}}:F^{i}\longrightarrow F^{i-1}\otimes K_{C},\qquad
i=1,\dots,r.$

Grading Condition. The connection induces ${\mathcal{O}}_{C}$-module
isomorphisms

(3.21)
$\overline{\nabla}:F^{i}/F^{i+1}\overset{\sim}{\longrightarrow}\left(F^{i-1}/F^{i}\right)\otimes
K_{C},\qquad i=1,\dots,r-1.$

For the purpose of defining differential operators globally on the Riemann
surface $C$, we need a projective coordinate system on $C$ subordinating its
complex structure. The coordinate also allows us to give a concrete $\hbar\in
H^{1}(C,K_{C})$-dependence in the filtered extensions. For example, the
extension $E_{\hbar}$ of (3.13) is given by a system of transition functions

(3.22)
$E_{\hbar}\longleftrightarrow\left\\{\begin{bmatrix}\xi_{\alpha\beta}&\hbar\sigma_{\alpha\beta}\\\
0&\xi_{\alpha\beta}^{-1}\end{bmatrix}\right\\}$

on each $U_{\alpha}\cap U_{\beta}$. The cocycle condition for the transition
functions translates into a condition

(3.23)
$\sigma_{\alpha\gamma}=\xi_{\alpha\beta}\sigma_{\beta\gamma}+\sigma_{\alpha\beta}\xi_{\beta\gamma}^{-1}.$

The application of the exterior differentiation $d$ to the cocycle condition
$\xi_{\alpha\gamma}=\xi_{\alpha\beta}\xi_{\beta\gamma}$ yields

$\frac{d\xi_{\alpha\gamma}}{dx_{\gamma}}dx_{\gamma}=\frac{d\xi_{\alpha\beta}}{dx_{\beta}}dx_{\beta}\xi_{\beta\gamma}+\xi_{\alpha\beta}\frac{d\xi_{\beta\gamma}}{dx_{\gamma}}dx_{\gamma}.$

Noticing that

(3.24) $\xi_{\alpha\beta}^{2}=\frac{dx_{\beta}}{dx_{\alpha}},$

we see that

(3.25)
$\sigma_{\alpha\beta}:=\frac{d\xi_{\alpha\beta}}{dx_{\beta}}=\partial_{\beta}\xi_{\alpha\beta}$

solves (3.23). We note that

$\begin{bmatrix}\xi_{\alpha\beta}&\hbar\sigma_{\alpha\beta}\\\
&\xi_{\alpha\beta}^{-1}\end{bmatrix}=\exp\left(\log\xi_{\alpha\beta}{\begin{bmatrix}1&0\\\
0&-1\end{bmatrix}}\right)\exp\left(\hbar\partial_{\beta}\log\xi_{\alpha\beta}\begin{bmatrix}0&1\\\
0&0\end{bmatrix}\right).$

Therefore, in the multiplicative sense, the extension class is determined by
$\partial_{\beta}\log\xi_{\alpha\beta}$.

######  Lemma 3.6.

The extension class $\sigma_{\alpha\beta}$ of (3.25) defines a non-trivial
extension (3.13).

###### Proof.

The cohomology long exact sequence

$\begin{CD}H^{1}(C,{\mathbb{C}})@>{}>{}>H^{1}(C,{\mathcal{O}}_{C})@>{}>{}>H^{1}(C,K_{C})@>{\sim}>{}>H^{2}(C,{\mathbb{C}})\\\
@V{}V{}V@V{}V{}V\Big{\|}\\\
H^{1}(C,{\mathbb{C}}^{*})@>{}>{}>H^{1}(C,{\mathcal{O}}^{*}_{C})@>{d\log}>{}>H^{1}(C,K_{C})\\\
@V{}V{0}V@V{}V{c_{1}}V@V{}V{}V\\\
H^{2}(C,{\mathbb{Z}})@=H^{2}(C,{\mathbb{Z}})@>{}>{}>0\end{CD}$

associated with

$\begin{CD}00\\\ &&@V{}V{}V@V{}V{}V\\\
0@>{}>{}>{\mathbb{Z}}@={\mathbb{Z}}@>{}>{}>0\\\ &&@V{}V{}V@V{}V{}V@V{}V{}V\\\
0@>{}>{}>{\mathbb{C}}@>{}>{}>{\mathcal{O}}_{C}@>{d}>{}>K_{C}@>{}>{}>0\\\
&&@V{}V{}V@V{}V{}V@V{}V{}V\\\
0@>{}>{}>{\mathbb{C}}^{*}@>{}>{}>{\mathcal{O}}_{C}^{*}@>{d\log}>{}>K_{C}@>{}>{}>0\\\
&&@V{}V{}V@V{}V{}V@V{}V{}V\\\ 000\end{CD}$

tells us that $\\{\sigma_{\alpha\beta}\\}$ corresponds to the image of
$\\{\xi_{ab}\\}$ via the map

$\begin{CD}H^{1}(C,{\mathbb{C}}^{*}_{C})@>{d\log}>{}>H^{1}(C,K_{C}).\end{CD}$

From the exact sequence, we see that if $d\log\\{\xi_{\alpha\beta}\\}=0\in
H^{1}(C,K_{C})$, then it comes from a class in $H^{1}(C,{\mathbb{C}}^{*})$,
which is the moduli space of line bundles with holomorphic connections (see
for example, [7]). Hence the first Chern class of the theta characteristic
$c_{1}\big{(}K_{C}^{\frac{1}{2}}\big{)}$ should be $0$. But it is $g-1$, not
$0$, since $g(C)\geq 2$. ∎

######  Remark 3.7.

A peculiar relation between quantizability of a spectral curve and the
algebraic K-group $K_{2}$ of the function field of the spectral curve is
discussed in [38] for the case of difference equations, in relation to Bloch
regulators [7].

The class $\\{\sigma_{\alpha\beta}\\}$ of (3.25) gives a natural isomorphism
$H^{1}\left(C,K_{C}\right)\cong{\mathbb{C}}$. We identify the deformation
parameter $\hbar\in{\mathbb{C}}$ with the cohomology class
$\\{\hbar\sigma_{\alpha\beta}\\}\in H^{1}\left(C,K_{C}\right)={\mathbb{C}}$.
Let

${\mathbf{q}}=(q_{2},q_{3},\dots,q_{r})\in
B=\bigoplus_{\ell=2}^{r}H^{0}\big{(}C,K_{C}^{\otimes\ell}\big{)}$

be an arbitrary point of the Hitchin base $B$. We trivialize the line bundle
$K_{C}^{\otimes\ell}$ with respect to our projective coordinate chart
$C=\bigcup_{\alpha}U_{\alpha}$, and write each $q_{\ell}$ as
$\\{(q_{\ell})_{\alpha}\\}$ that satisfies the transition relation

(3.26) $(q_{\ell})_{\alpha}=(q_{\ell})_{\beta}\xi_{\alpha\beta}^{2\ell}.$

The transition function of the vector bundle $E_{0}$ is given by

(3.27) $\xi_{\alpha\beta}^{H}=\exp(H\log\xi_{\alpha\beta}).$

Since $X_{-}:E_{0}\longrightarrow E_{0}\otimes K_{C}$ is a global Higgs filed,
its local expressions $\\{X_{-}dx_{\alpha}\\}$ with respect to the projective
coordinate system satisfies the transition relation

(3.28)
$X_{-}dx_{\alpha}=\exp(H\log\xi_{\alpha\beta})X_{-}dx_{\beta}\exp(-H\log\xi_{\alpha\beta})$

on every $U_{\alpha}\cap U_{\beta}$. The same relation holds for the Higgs
field $\phi({\mathbf{q}})$ as well:

(3.29)
$\phi_{\alpha}({\mathbf{q}})dx_{\alpha}=\exp(H\log\xi_{\alpha\beta})\phi_{\beta}({\mathbf{q}})dx_{\beta}\exp(-H\log\xi_{\alpha\beta}).$

We have the following:

######  Theorem 3.8 (Construction of $SL(r,{\mathbb{C}})$-opers).

On each $U_{\alpha}\cap U_{\beta}$ define a transition function

(3.30)
$f_{\alpha\beta}^{\hbar}:=\exp(H\log\xi_{\alpha\beta})\exp\big{(}\hbar\partial_{\beta}\log\xi_{\alpha\beta}X_{+}\big{)},$

where $\partial_{\beta}=\frac{d}{dx_{\beta}}$, and
$\hbar\partial_{\beta}\log\xi_{\alpha\beta}\in H^{1}(C,K_{C})$. Then

  * •

The collection $\\{f_{\alpha\beta}^{\hbar}\\}$ satisfies the cocycle condition

(3.31)
$f_{\alpha\beta}^{\hbar}f_{\beta\gamma}^{\hbar}=f_{\alpha\gamma}^{\hbar},$

hence it defines a holomorphic bundle on $C$. It is exactly the filtered
extension $E_{\hbar}$ of Proposition 3.4.

  * •

The locally defined differential operator

(3.32) $\nabla_{\alpha}^{\hbar}(0):=d+\frac{1}{\hbar}X_{-}dx_{a}$

for every $\hbar\neq 0$ forms a global holomorphic connection in $E_{\hbar}$,
i.e.,

(3.33)
$\frac{1}{\hbar}X_{-}dx_{\alpha}=\frac{1}{\hbar}f_{\alpha\beta}^{\hbar}X_{-}dx_{\beta}\left(f_{\alpha\beta}^{\hbar}\right)^{-1}-df_{\alpha\beta}^{\hbar}\cdot\left(f_{\alpha\beta}^{\hbar}\right)^{-1}.$

  * •

Every point
$\left(E_{0},\phi({\mathbf{q}})\right)\in\kappa(B)\subset{\mathcal{M}}_{{\text{\rm{Dol}}}}$
of the Hitchin section (3.7) gives rise to a one-parameter family of
$SL(r,{\mathbb{C}})$-opers
$\left(E_{\hbar},\nabla^{\hbar}({\mathbf{q}})\right)\in{\mathcal{M}}_{{\text{\rm{deR}}}}$.
In other words, the locally defined differential operator

(3.34)
$\nabla_{\alpha}^{\hbar}({\mathbf{q}}):=d+\frac{1}{\hbar}\phi_{\alpha}({\mathbf{q}})dx_{\alpha}$

for every $\hbar\neq 0$ determines a global holomorphic connection

(3.35)
$\nabla^{\hbar}_{\alpha}({\mathbf{q}})=f_{\alpha\beta}^{\hbar}\nabla^{\hbar}_{\beta}({\mathbf{q}})\left(f_{\alpha\beta}^{\hbar}\right)^{-1}$

in $E_{\hbar}$ satisfying the oper conditions.

  * •

Deligne’s $\hbar$-connection

(3.36) $\left(E_{\hbar},\hbar\nabla^{\hbar}({\mathbf{q}})\right)$

interpolates the Higgs pair and the oper, i.e., at $\hbar=0$, the family
(3.36) gives the Higgs pair
$\left(E,\phi({\mathbf{q}})\right)\in{\mathcal{M}}_{{\text{\rm{Dol}}}}$, and
at $\hbar=1$ it gives an $SL(r,{\mathbb{C}})$-oper
$\left(E_{1},\nabla^{1}({\mathbf{q}})\right)\in{\mathcal{M}}_{{\text{\rm{deR}}}}$.

  * •

After a suitable gauge transformation depending on $\hbar$, the
$\hbar\rightarrow\infty$ limit of the oper $\nabla^{\hbar}({\mathbf{q}})$
exists and is equal to $\nabla^{\hbar=1}(0)$.

###### Proof.

Recall the Baker-Campbell-Hausdorff formula: Let $A,B$ be elements of a Lie
algebra such that $[A,B]=cB$ for a constant $c\in{\mathbb{C}}$. Then

(3.37) $e^{A}e^{B}e^{-A}=e^{B\exp c}.$

From this formula, $\frac{dx_{\beta}}{dx_{\alpha}}=\xi_{\alpha\beta}^{2}$, and
$[H,X_{+}]=2X_{+}$, we calculate

$\displaystyle
f_{\alpha\beta}^{\hbar}\left(f_{\gamma\beta}^{\hbar}\right)^{-1}$
$\displaystyle=\exp(H\log\xi_{\alpha\beta})\exp\left(\hbar\partial_{\beta}\log\xi_{\alpha\beta}X_{+}\right)\exp\left(-\hbar\partial_{\beta}\log\xi_{\gamma\beta}X_{+}\right)\exp(-H\log\xi_{\gamma\beta})$
$\displaystyle=\exp(H\log\xi_{\alpha\beta})\exp\left(\hbar\partial_{\beta}\log\xi_{\alpha\gamma}X_{+}\right)\exp(-H\log\xi_{\gamma\beta})$
$\displaystyle=\exp(H\log\xi_{\alpha\beta})\exp\left(\hbar\xi_{\beta\gamma}^{2}\partial_{\gamma}\log\xi_{\alpha\gamma}X_{+}\right)\exp(-H\log\xi_{\gamma\beta})$
$\displaystyle=\exp(H\log\xi_{\alpha\gamma})\exp(H\log\xi_{\gamma\beta})\exp\left(\hbar\xi_{\beta\gamma}^{2}\partial_{\gamma}\log\xi_{\alpha\gamma}X_{+}\right)\exp(-H\log\xi_{\gamma\beta})$
$\displaystyle=\exp(H\log\xi_{\alpha\gamma})\exp\left(\hbar\xi_{\gamma\beta}^{2}\xi_{\beta\gamma}^{2}\partial_{\gamma}\log\xi_{\alpha\gamma}X_{+}\right)$
$\displaystyle=f_{\alpha\gamma}^{\hbar}.$

Hence the cocycle condition
$f_{\alpha\beta}^{\hbar}=f_{\alpha\gamma}^{\hbar}f_{\gamma\beta}^{\hbar}$
follows. We note that the factor $\exp c$ in (3.37) produces exactly the
cocycle $\xi_{\alpha\beta}^{2}$ corresponding to the canonical sheaf $K_{C}$.

To prove (3.33), we use the power series expansion of the adjoint action

(3.38) $e^{\hbar A}Be^{-\hbar
A}=\sum_{n=0}^{\infty}\frac{1}{n!}\hbar^{n}(\rm{ad}_{A})^{n}(B):=\sum_{n=0}^{\infty}\frac{1}{n!}\;\hbar^{n}\;\overset{n}{\overbrace{[A,[A,[\cdots,[A}},B]\cdots]]].$

We then find

$\displaystyle
f_{\alpha\beta}^{\hbar}X_{-}\left(f_{\alpha\beta}^{\hbar}\right)^{-1}$
$\displaystyle=\exp(H\log\xi_{\alpha\beta})\exp\left(\hbar\partial_{\beta}\log\xi_{\alpha\beta}X_{+}\right)X_{-}\exp\left(-\hbar\partial_{\beta}\log\xi_{\alpha\beta}X_{+}\right)\exp(-H\log\xi_{\alpha\beta})$
$\displaystyle=\exp(H\log\xi_{\alpha\beta})X_{-}\exp(-H\log\xi_{\alpha\beta})+\hbar\partial_{\beta}\log\xi_{\alpha\beta}H$
$\displaystyle\qquad-\hbar^{2}(\partial_{\beta}\log\xi_{\alpha\beta})^{2}\exp(H\log\xi_{\alpha\beta})X_{+}\exp(-H\log\xi_{\alpha\beta}),$

and

$\partial_{\beta}f_{\alpha\beta}^{\hbar}\left(f_{\alpha\beta}^{\hbar}\right)^{-1}=\partial_{\beta}\log\xi_{\alpha\beta}H-\hbar(\partial_{\beta}\log\xi_{\alpha\beta})^{2}\exp(H\log\xi_{\alpha\beta})X_{+}\exp(-H\log\xi_{\alpha\beta}).$

Here, we have used the formula

$\partial_{\beta}\partial_{\beta}\log\xi_{\alpha\beta}=\partial_{\beta}\left(\xi_{\alpha\beta}^{-1}\partial_{\beta}\xi_{\alpha\beta}\right)=-\xi_{\alpha\beta}^{-2}(\partial_{\beta}\xi_{\alpha\beta})^{2}=-(\partial_{\beta}\log\xi_{\alpha\beta})^{2},$

which follows from (3.2). Therefore, noticing (3.28), we obtain

$\displaystyle\left(\frac{1}{\hbar}f_{\alpha\beta}^{\hbar}X_{-}\left(f_{\alpha\beta}^{\hbar}\right)^{-1}-\partial_{\beta}f_{\alpha\beta}^{\hbar}\left(f_{\alpha\beta}^{\hbar}\right)^{-1}\right)dx_{\beta}$
$\displaystyle=\frac{1}{\hbar}\exp(H\log\xi_{\alpha\beta})X_{-}dx_{\beta}\exp(-H\log\xi_{\alpha\beta})$
$\displaystyle=\frac{1}{\hbar}X_{-}dx_{\alpha}.$

To prove (3.35), we need, in addition to (3.33), the following relation:

(3.39)
$\sum_{\ell=2}^{r}(q_{\ell})_{\alpha}X_{+}^{\ell-1}dx_{\alpha}=f_{\alpha\beta}^{\hbar}\sum_{\ell=2}^{r}(q_{\ell})_{\beta}X_{+}^{\ell-1}dx_{\beta}\left(f_{\alpha\beta}^{\hbar}\right)^{-1}.$

But (3.39) is obvious from (3.29) and (3.30).

Noticing that $f_{\alpha\beta}^{\hbar}$ is an upper-triangular matrix, we
denote by $\left(f_{\alpha\beta}^{\hbar}\right)_{i}$ the principal truncation
of $f_{\alpha\beta}^{\hbar}$ to the first $(r-i+1)\times(r-i+1)$ upper-left
corner. For example,

$\displaystyle\left(f_{\alpha\beta}^{\hbar}\right)_{r}$
$\displaystyle=\left[\xi_{\alpha\beta}^{r-1}\right],$
$\displaystyle\left(f_{\alpha\beta}^{\hbar}\right)_{r-1}$
$\displaystyle=\begin{bmatrix}\xi_{\alpha\beta}^{r-1}&\\\
&\xi_{\alpha\beta}^{r-3}\end{bmatrix}\begin{bmatrix}1&\hbar\sqrt{s_{1}}\xi_{\alpha\beta}^{-1}\sigma_{\alpha\beta}\\\
&1\end{bmatrix}=\begin{bmatrix}\xi_{\alpha\beta}^{r-1}&\hbar\sqrt{s_{1}}\xi_{\alpha\beta}^{r-2}\sigma_{\alpha\beta}\\\
&\xi_{\alpha\beta}^{r-3}\end{bmatrix},$
$\displaystyle\left(f_{\alpha\beta}^{\hbar}\right)_{r-2}$
$\displaystyle=\begin{bmatrix}\xi_{\alpha\beta}^{r-1}&\hbar\sqrt{s_{1}}\xi_{\alpha\beta}^{r-2}\sigma_{\alpha\beta}&{\frac{1}{2}}\hbar^{2}\sqrt{s_{1}s_{2}}\xi_{\alpha\beta}^{r-3}\sigma_{\alpha\beta}^{2}\\\
&\xi_{\alpha\beta}^{r-3}&\hbar\sqrt{s_{2}}\xi_{\alpha\beta}^{r-4}\sigma_{\alpha\beta}\\\
&&\xi_{\alpha\beta}^{r-5}\end{bmatrix},\qquad\text{etc.}$

They all satisfy the cocycle condition with respect to the projective
structure, and define a sequence of vector bundles $F_{\hbar}^{i}$ on $C$.
From the shape of matrices we see that these vector bundles give the filtered
extension $\left(F_{\hbar}^{\bullet},E_{\hbar}\right)$ of Proposition 3.4, and
hence satisfies the requirement (3.19). Since the connection
$\nabla^{\hbar}({\mathbf{q}})$ has non-zero lower-diagonal entries in its
matrix representation coming from $X_{-}$, $F_{\hbar}^{i}$ is not invariant
under $\nabla^{\hbar}({\mathbf{q}})$. But since $X_{-}$ has non-zero entries
exactly along the first lower-diagonal, (3.20) holds. The isomorphism (3.21)
is a consequence of (3.8), since $s_{i}=i(r-i)\neq 0$ for $i=1,\dots,r-1$.

Finally, the gauge transformation of $\nabla^{\hbar}({\mathbf{q}})$ by a
bundle automorphism

(3.40) $\hbar^{-\frac{H}{2}}=\begin{bmatrix}\hbar^{-\frac{r-1}{2}}&\\\
&\ddots&\\\ &&\hbar^{\frac{r-1}{2}}\end{bmatrix}$

on each coordinate neighborhood $U_{\alpha}$ gives

(3.41)
$d+\frac{1}{\hbar}\phi({\mathbf{q}})\longmapsto\hbar^{-\frac{H}{2}}\left(d+\frac{1}{\hbar}\phi({\mathbf{q}})\right)\hbar^{\frac{H}{2}}=d+X_{-}+\sum_{\ell=2}^{r}\frac{q_{\ell}}{\hbar^{\ell}}X_{+}^{\ell-1}.$

This is because

$\hbar^{-\frac{H}{2}}X_{-}\hbar^{\frac{H}{2}}=\hbar
X_{-}\qquad\text{and}\qquad\hbar^{-\frac{H}{2}}X_{+}^{\ell}\hbar^{\frac{H}{2}}=\hbar^{-\ell}X_{+}^{\ell},$

which follows from the adjoint formula (3.38). Therefore,

$\lim_{\hbar\rightarrow\infty}\nabla^{\hbar}({\mathbf{q}})\sim
d+X_{-}=\nabla^{\hbar=1}(0),$

where the symbol $\sim$ means gauge equivalence.

This completes the proof of the theorem. ∎

######  Remark 3.9.

In the construction theorem, our use of a projective coordinate system is
essential, through (3.2). Only in such a coordinate, the definition of the
global connection (3.35) makes sense. This is due to the vanishing of the
second derivative of $\xi_{\alpha\beta}$.

The above construction theorem yields the following.

######  Theorem 3.10 (Biholomorphic quantization of Hitchin spectral curves).

Let $C$ be a compact Riemann surface of genus $g\geq 2$ with a chosen
projective coordinate system subordinating its complex structure. We denote by
${\mathcal{M}}_{{\text{\rm{Dol}}}}$ the moduli space of stable holomorphic
$SL(r,{\mathbb{C}})$-Higgs bundles over $C$, and by
${\mathcal{M}}_{{\text{\rm{deR}}}}$ the moduli space of irreducible
holomorphic $SL(r,{\mathbb{C}})$-connections on $C$. For a fixed theta
characteristic $K_{C}^{\frac{1}{2}}$, we have a Hitchin section
$\kappa(B)\subset{\mathcal{M}}_{{\text{\rm{Dol}}}}$ of (3.7). We denote by
$Op\subset{\mathcal{M}}_{{\text{\rm{deR}}}}$ the moduli space of
$SL(r,{\mathbb{C}})$-opers with the condition that the second term of the
filtration is given by $F^{r-1}=K_{C}^{\frac{r-1}{2}}$. Then the map

(3.42)
${\mathcal{M}}_{{\text{\rm{Dol}}}}\supset\kappa(B)\owns\left(E_{0},\phi({\mathbf{q}})\right)\overset{\gamma}{\longmapsto}\left(E_{\hbar},\nabla^{\hbar}({\mathbf{q}})\right)\in
Op\subset{\mathcal{M}}_{{\text{\rm{deR}}}}$

evaluated at $\hbar=1$ is a biholomorphic map with respect to the natural
complex structures induced from the ambient spaces.

The biholomorphic quantization (3.42) is also ${\mathbb{C}}^{*}$-equivariant.
The $\lambda\in{\mathbb{C}}^{*}$ action on the Hitchin section is defined by
$\phi\longmapsto\lambda\phi$. The oper corresponding to
$\left(E_{0},\lambda\phi({\mathbf{q}})\right)\in\kappa(B)$ is
$d+\frac{\lambda}{\hbar}\phi({\mathbf{q}})$.

###### Proof.

The ${\mathbb{C}}^{*}$-equivariance follows from the same argument of the
gauge transformation (3.40), (3.41). Since the Hitchin section $\kappa$ is not
the section of the Hitchin fibration $\mu_{H}$, we need the gauge
transformation. The action $\phi\longmapsto\lambda\phi$ on the Hitchin section
induces a weighted action

$B\owns(q_{2},q_{3},\dots,q_{r})\longmapsto\left(\lambda^{2}q_{2},\lambda^{3}q_{3},\dots,\lambda^{r}q_{r}\right)\in
B$

through $\mu_{H}$. Then we have the gauge equivalence via the gauge
transformation $\left(\frac{\lambda}{\hbar}\right)^{\frac{H}{2}}$:

$d+\frac{\lambda}{\hbar}\phi({\mathbf{q}})\sim\left(\frac{\lambda}{\hbar}\right)^{\frac{H}{2}}\left(d+\frac{\lambda}{\hbar}\phi({\mathbf{q}})\right)\left(\frac{\lambda}{\hbar}\right)^{-\frac{H}{2}}=d+X_{-}+\sum_{\ell=2}^{r}\frac{\lambda^{\ell}q_{\ell}}{\hbar^{\ell}}X_{+}^{\ell-1}.$

∎

###  3.2. Semi-classical limit of $SL(r,{\mathbb{C}})$-opers

A holomorphic connection on a compact Riemann surface $C$ is automatically
flat. Therefore, it defines a ${\mathcal{D}}$-module over $C$. Continuing the
last subsection’s conventions, let us fix a projective coordinate system on
$C$, and let $\left(E_{0},\phi({\mathbf{q}})\right)=\kappa({\mathbf{q}})$ be a
point on the Hitchin section of (3.7). It uniquely defines an $\hbar$-family
of opers $\left(E_{\hbar},\nabla^{\hbar}({\mathbf{q}})\right)$.

In this subsection, we establish that the $\hbar$-connection
$\hbar\nabla^{\hbar}({\mathbf{q}})$ defines a family of Rees
${\mathcal{D}}$-modules on $C$ parametrized by $B$ such that the semi-
classical limit of the family agrees with the family of spectral curves (1.7)
over $B$.

To calculate the semi-classical limit, let us trivialize the vector bundle
$E_{\hbar}$ on each simply connected coordinate neighborhood $U_{\alpha}$ with
coordinate $x_{\alpha}$ of the chosen projective coordinate system. A flat
section $\Psi_{\alpha}$ of $E_{\hbar}$ over $U_{\alpha}$ is a solution of

(3.43) $\hbar\nabla^{\hbar}_{\alpha}({\mathbf{q}})\Psi_{\alpha}:=\left(\hbar
d+\phi_{\alpha}({\mathbf{q}})\right)\begin{bmatrix}\psi_{r-1}\\\ \psi_{r-2}\\\
\vdots\\\ \psi_{1}\\\ \psi\end{bmatrix}_{\alpha}=0,$

with an appropriate unknown function $\psi$. Since
$\Psi_{\alpha}=f_{\alpha\beta}^{\hbar}\Psi_{\beta}$, the function $\psi$ on
$U_{\alpha}$ satisfies the transition relation
$(\psi)_{\alpha}=\xi_{\alpha\beta}^{-r+1}(\psi)_{\beta}$. It means that $\psi$
is actually a local section of the line bundle $K_{C}^{-\frac{r-1}{2}}$. There
are $r$ linearly independent solutions of (3.43), because $q_{2},\dots,q_{r}$
are represented by holomorphic functions on $U_{\alpha}$. The entries of
$X_{+}^{\ell}$ are given by the formula

(3.44) $X_{+}^{\ell}=\left[s_{ij}^{(\ell)}\right],\qquad
s_{ij}^{(\ell)}=\delta_{i+\ell,j}\sqrt{s_{i}s_{i+1}\cdots s_{i+\ell-1}}.$

Therefore, (3.43) is equivalent to

(3.45)
$0=\sqrt{s_{r-k-1}}\psi_{k+1}+\hbar\psi_{k}^{\prime}+\sqrt{s_{r-k}}q_{2}\psi_{k-1}+\sqrt{s_{r-k}s_{r-k+1}}q_{3}\psi_{k-2}+\cdots\\\
+\sqrt{s_{r-k}s_{r-k+1}\cdots s_{r-1}}q_{k+1}\psi\\\
=\sqrt{s_{k+1}}\psi_{k+1}+\hbar\psi_{k}^{\prime}+\sqrt{s_{k}}q_{2}\psi_{k-1}+\sqrt{s_{k}s_{k-1}}q_{3}\psi_{k-2}+\cdots+\sqrt{s_{k}s_{k-1}\cdots
s_{1}}q_{k+1}\psi$

for $k=0,1,\dots,r-1$, where we use $s_{k}=s_{r-k}$. Note that the
differentiation is always multiplied by $\hbar$ as $\hbar d$ in (3.43), and
that $\phi({\mathbf{q}})$ is independent of $\hbar$ and takes the form

$\phi({\mathbf{q}})\\\
=\begin{bmatrix}0&\sqrt{s_{r-1}}q_{2}&\sqrt{s_{r-2}s_{r-1}}q_{3}&\cdots&\cdots&\sqrt{s_{2}\cdots
s_{r-1}}q_{r-1}&\sqrt{s_{1}s_{2}\cdots s_{r-1}}q_{r}\\\
\sqrt{s_{r-1}}&0&\sqrt{s_{r-2}}q_{2}&\cdots&\cdots&\sqrt{s_{2}\cdots
s_{r-2}}q_{r-2}&\sqrt{s_{1}\cdots s_{r-2}}q_{r-1}\\\
&\sqrt{s_{r-2}}&0&\ddots&\cdots&\sqrt{s_{2}\cdots
s_{r-3}}q_{r-3}&\sqrt{s_{1}\cdots s_{r-3}}q_{r-2}\\\
&&\ddots&\ddots&\ddots&\vdots&\vdots\\\
&&&\sqrt{s_{3}}&0&\sqrt{s_{2}}q_{2}&\sqrt{s_{1}s_{2}}q_{3}\\\
&&&&\sqrt{s_{2}}&0&\sqrt{s_{1}}q_{2}\\\ &&&&&\sqrt{s_{1}}&0\end{bmatrix}.$

By solving (3.45) for $k=0,1,\dots,r-2$ recursively, we obtain an expression
of $\psi_{k}$ as a linear combination of

$\psi=\psi_{0},\quad\hbar\psi^{\prime}=\hbar\frac{d}{dx_{\alpha}}\psi,\quad\dots\quad\hbar^{k}\psi^{(k)}=\hbar^{k}\frac{d^{k}}{dx_{\alpha}^{k}}\psi,$

with coefficients in differential polynomials of $q_{2},q_{3},\dots,q_{k}$.
Moreover, the coefficients of these differential polynomials are in
$\overline{{\mathbb{Q}}}[\hbar]$. For example,

$\displaystyle\psi_{1}$
$\displaystyle=-\frac{1}{\sqrt{s_{1}}}\hbar\psi^{\prime},$
$\displaystyle\psi_{2}$
$\displaystyle=\frac{1}{\sqrt{s_{1}s_{2}}}\left(\hbar^{2}\psi^{\prime\prime}-s_{1}q_{2}\psi\right),$
$\displaystyle\psi_{3}$
$\displaystyle=\frac{1}{\sqrt{s_{1}s_{2}s_{3}}}\left(-\hbar^{3}\psi^{\prime\prime\prime}+\hbar(s_{1}+s_{2})q_{2}\psi^{\prime}+\left(\hbar
s_{1}q_{2}^{\prime}-s_{1}s_{2}q_{3}\right)\psi\right),$
$\displaystyle\psi_{4}$
$\displaystyle=\frac{1}{\sqrt{s_{1}s_{2}s_{3}s_{4}}}\bigg{(}\hbar^{4}\psi^{\prime\prime\prime\prime}-\hbar^{2}(s_{1}+s_{2}+s_{3})q_{2}\psi^{\prime\prime}+\left(-\hbar^{2}(2s_{1}+s_{2})q_{2}^{\prime}+\hbar(s_{1}s_{2}+s_{2}s_{3})q_{3}\right)\psi^{\prime}$
$\displaystyle\qquad+\left(\hbar^{2}s_{1}q_{2}^{\prime\prime}-\hbar
s_{1}s_{2}q_{3}^{\prime}+s_{1}s_{3}q_{2}^{2}-s_{1}s_{2}s_{3}q_{4}\right)\psi\bigg{)},\qquad\text{etc.}$

Since $\psi_{1}$ is proportional to $\psi^{\prime}$, inductively we can show
that the linear combination expression of $\psi_{k}$ by derivatives of $\psi$
does not contain the $(k-1)$-th order differentiation of $\psi$. Eqn.(3.45)
for $k=r-1$ is a differential equation

(3.46)
$\hbar\psi_{r-1}^{\prime}+\sqrt{s_{1}}q_{2}\psi_{r-2}+\sqrt{s_{1}s_{2}}q_{3}\psi_{r-3}+\cdots+\sqrt{s_{1}s_{2}\cdots
s_{r-1}}q_{r}\psi=0,$

which is an order $r$ differential equation for $\psi\in
K_{C}^{-\frac{r-1}{2}}$. Its actual shape can be calculated using the
procedure of the proof of the next theorem. The equation becomes (3.55). Since
we are using a fixed projective coordinate system, the connection
$\nabla^{\hbar}({\mathbf{q}})$ takes the same form on each coordinate
neighborhood $U_{\alpha}$. Therefore, the shape of the differential equation
of (3.46) as an equation for $\psi\in K_{C}^{-\frac{r-1}{2}}$ is again the
same on every coordinate neighborhood.

We are now ready to calculate the semi-classical limit of the Rees
${\mathcal{D}}$-module corresponding to $\hbar\nabla^{\hbar}({\mathbf{q}})$.
The following is our main theorem of the paper.

######  Theorem 3.11 (Semi-classical limit of an oper).

Under the same setting of Theorem 3.10, let ${\mathcal{E}}({\mathbf{q}})$
denote the Rees ${\mathcal{D}}$-module
$\left(E_{\hbar},\hbar\nabla^{\hbar}({\mathbf{q}})\right)$ associated with the
oper of (3.42). Then the semi-classical limit of ${\mathcal{E}}({\mathbf{q}})$
is the spectral curve $\sigma^{*}\Sigma\subset T^{*}C$ of
$-\phi({\mathbf{q}})$ defined by the characteristic equation
$\det(\eta+\phi({\mathbf{q}}))=0$, where $\sigma$ is the involution of (1.9).

######  Remark 3.12.

The actual shape of the single scalar valued differential equation (3.46) of
order $r$ is given by (3.55) below. This is exactly what we usually refer to
as the quantum curve of the spectral curve $\det(\eta+\phi({\mathbf{q}}))=0$.

###### Proof.

Since the connection and the differential equation (3.46) are globally
defined, we need to analyze the semi-classical limit only at each coordinate
neighborhood $U_{\alpha}$ with a projective coordinate $x_{\alpha}$. Let

$\Psi_{0}=\begin{bmatrix}\hbar^{r-1}\psi^{(r-1)}\\\ \vdots\\\
\hbar\psi^{\prime}\\\ \psi\end{bmatrix}.$

From the choice of local trivialization (3.43) we have an expression

(3.47) $\Psi=\big{(}\Delta+A({\mathbf{q}},\hbar)\big{)}\Psi_{0},$

where $\Delta$ is a constant diagonal matrix

(3.48) $\Delta=\left[\delta_{ij}(-1)^{r-i}\frac{1}{\sqrt{s_{r-1}s_{r-2}\cdots
s_{i}}}\right]_{i,j=1,\dots,r}$

with the understanding that its $(r,r)$-entry is $1$, and
$A({\mathbf{q}},\hbar)=\big{[}a({\mathbf{q}},\hbar)_{i,j}\big{]}$ satisfies
the following properties:

######  Condition 3.13.

  * •

$A({\mathbf{q}},\hbar)$ is a nilpotent upper triangular matrix.

  * •

Each entry $a({\mathbf{q}},\hbar)_{i,j}$ is a differential polynomial in
$q_{2},\dots,q_{r}$ with coefficients in $\overline{{\mathbb{Q}}}[\hbar]$.

  * •

All diagonal and the first upper diagonal entries of $A({\mathbf{q}},\hbar)$
are $0$:

$a({\mathbf{q}},\hbar)_{i,i}=a({\mathbf{q}},\hbar)_{i,i+1}=0,\qquad
i=1,\dots,r.$

The last property is because $\psi_{k}$ does not contain the $(k-1)$-th
derivative of $\psi$, as we have noted above. The diagonal matrix $\Delta$ is
designed so that

(3.49) $\Delta^{-1}X_{-}\Delta=\begin{bmatrix}0\\\ -1&0\\\ &-1&0\\\
&&\ddots\\\ &&&-1&0\end{bmatrix}=:-X.$

Let us calculate the gauge transformation of
$\hbar\nabla^{\hbar}({\mathbf{q}})$ by $\Delta+A({\mathbf{q}},\hbar)$:

(3.50) $\left(\Delta+A({\mathbf{q}},\hbar)\right)^{-1}\left(\hbar
d+\phi({\mathbf{q}})\right)\left(\Delta+A({\mathbf{q}},\hbar)\right)=\left(\hbar
d-Xdx_{\alpha}-\omega({\mathbf{q}},\hbar)dx_{\alpha}\right).$

We wish to determine the matrix $\omega({\mathbf{q}},\hbar)$.

######  Lemma 3.14.

The matrix $\omega({\mathbf{q}},\hbar)$ of (3.50) consists of only the first
row, and the matrix $X+\omega({\mathbf{q}},\hbar)$ takes the following
_canonical form_

(3.51)
$X+\omega({\mathbf{q}},\hbar)=\begin{bmatrix}0&\omega_{2}({\mathbf{q}},\hbar)&\cdots&\omega_{r-1}({\mathbf{q}},\hbar)&\omega_{r}({\mathbf{q}},\hbar)\\\
1&0&\cdots&0&0\\\ &1&\ddots&\vdots&\vdots\\\ &&\ddots&0&0\\\
&&&1&0\end{bmatrix}.$

###### Proof of Lemma.

First, define

(3.52)
$B({\mathbf{q}},\hbar):=\sum_{n=1}^{r-1}(-1)^{n}\left(\Delta^{-1}A({\mathbf{q}},\hbar)\right)^{n}$

so that

$I+B({\mathbf{q}},\hbar)=\left(I+\Delta^{-1}A({\mathbf{q}},\hbar)\right)^{-1}.$

By definition, $B({\mathbf{q}},\hbar)$ satisfies the exact same properties
listed in Condition 3.13 above. We then have

$\displaystyle\left(\Delta+A({\mathbf{q}},\hbar)\right)^{-1}\left(\hbar\frac{d}{dx_{\alpha}}+\phi({\mathbf{q}})\right)\left(\Delta+A({\mathbf{q}},\hbar)\right)$
$\displaystyle=\left(I+\Delta^{-1}A({\mathbf{q}},\hbar)\right)^{-1}\Delta^{-1}\left(\hbar\frac{d}{dx_{\alpha}}+X_{-}+\sum_{\ell=2}^{r}q_{\ell}X_{+}^{\ell-1}\right)\Delta\left(I+\Delta^{-1}A({\mathbf{q}},\hbar)\right)$
$\displaystyle=\left(I+B({\mathbf{q}},\hbar)\right)\left(\hbar\frac{d}{dx_{\alpha}}-X+\Delta^{-1}\sum_{\ell=2}^{r}q_{\ell}X_{+}^{\ell-1}\Delta\right)\left(I+\Delta^{-1}A({\mathbf{q}},\hbar)\right)$
$\displaystyle=\hbar\frac{d}{dx_{\alpha}}-X-B({\mathbf{q}},\hbar)X-X\Delta^{-1}A({\mathbf{q}},\hbar)+\hbar\left(I+B({\mathbf{q}},\hbar)\right)\Delta^{-1}\frac{dA({\mathbf{q}},\hbar)}{dx_{\alpha}}$
$\displaystyle\qquad-B({\mathbf{q}},\hbar)X\Delta^{-1}A({\mathbf{q}},\hbar)+\left(I+B({\mathbf{q}},\hbar)\right)\Delta^{-1}\left(\sum_{\ell=2}^{r}q_{\ell}X_{+}^{\ell-1}\Delta\right)\left(I+\Delta^{-1}A({\mathbf{q}},\hbar)\right).$

Thus we define

(3.53) $\displaystyle\omega({\mathbf{q}},\hbar)$
$\displaystyle=B({\mathbf{q}},\hbar)X+X\Delta^{-1}A({\mathbf{q}},\hbar)-\hbar\left(I+B({\mathbf{q}},\hbar)\right)\Delta^{-1}\frac{dA({\mathbf{q}},\hbar)}{dx_{\alpha}}$
$\displaystyle+B({\mathbf{q}},\hbar)X\Delta^{-1}A({\mathbf{q}},\hbar)-\left(\Delta+A({\mathbf{q}},\hbar)\right)^{-1}\left(\sum_{\ell=2}^{r}q_{\ell}X_{+}^{\ell-1}\right)\left(\Delta+A({\mathbf{q}},\hbar)\right).$

Every single matrix in the right-hand side of (3.53) is either diagonal or
upper triangular and nilpotent, except for $X$. Because of Condition 3.13,
$B({\mathbf{q}},\hbar)X+X\Delta^{-1}A({\mathbf{q}},\hbar)$ is an upper
triangular matrix with $0$ along the diagonal. Obviously, so are all other
terms of (3.53). Thus $\omega({\mathbf{q}},\hbar)$ is upper triangular and
nilpotent, and is a polynomial in $\hbar$.

Now we note that (3.43) and (3.47) yield

(3.54)
$\left(\hbar\frac{d}{dx_{\alpha}}-X-\omega({\mathbf{q}},\hbar)\right)\Psi_{0}=0.$

The basis vector $\Psi_{0}$ defined on a simply connected coordinate
neighborhood $U_{\alpha}$ is designed so that the following equation holds for
any solution $\psi$ of (3.43):

$\left(\hbar\frac{d}{dx_{\alpha}}-X\right)\Psi_{0}=\left(\hbar\frac{d}{dx_{\alpha}}-X\right)\begin{bmatrix}\hbar^{r-1}\psi^{(r-1)}\\\
\hbar^{r-2}\psi^{(r-2)}\\\ \vdots\\\ \hbar\psi^{\prime}\\\
\psi\end{bmatrix}=\begin{bmatrix}\hbar^{r}\psi^{(r)}\\\ 0\\\ \vdots\\\ 0\\\
0\end{bmatrix}.$

Let $\psi_{[1]},\dots,\psi_{[r]}$ be $r$ linearly independent solutions of
(3.43). The _Wronskian matrix_ is defined by

$\mathbf{W}=\begin{bmatrix}\hbar^{r-1}\psi_{[1]}^{(r-1)}&\hbar^{r-1}\psi_{[2]}^{(r-1)}&\cdots&\hbar^{r-1}\psi_{[r]}^{(r-1)}\\\
\hbar^{r-2}\psi_{[1]}^{(r-2)}&\hbar^{r-2}\psi_{[2]}^{(r-2)}&\cdots&\hbar^{r-2}\psi_{[r]}^{(r-2)}\\\
\vdots&\vdots&\ddots&\vdots\\\
\hbar\psi_{[1]}^{\prime}&\hbar\psi_{[2]}^{\prime}&\cdots&\hbar\psi_{[r]}^{\prime}\\\
\psi_{[1]}&\psi_{[2]}&\cdots&\psi_{[r]}\end{bmatrix}.$

Then from (3.54), we have

$\omega({\mathbf{q}},\hbar)=\left[\left(\hbar\frac{d}{dx_{\alpha}}-X\right)\mathbf{W}\right]\cdot\mathbf{W}^{-1}=\begin{bmatrix}\hbar^{r}\psi_{[1]}^{(r)}&\hbar^{r}\psi_{[2]}^{(r)}&\cdots&\hbar^{r}\psi_{[r]}^{(r)}\\\
0&0&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&0\end{bmatrix}\mathbf{W}^{-1}.$

Clearly we see that $\omega({\mathbf{q}},\hbar)$ has non-zero entries only in
the first row. The $(1,1)$-entry is also $0$ because
${\text{\rm{tr}}}\;\omega({\mathbf{q}},\hbar)=0$. ∎

Thus (3.43) is equivalent to a single linear ordinary differential equation of
order $r$

(3.55)
$P_{\alpha}(x_{\alpha},\hbar;{\mathbf{q}})\psi:=\left[\hbar^{r}\left(\frac{d}{dx_{\alpha}}\right)^{r}-\sum_{i=2}^{r}\hbar^{r-i}\omega_{i}({\mathbf{q}},\hbar)\left(\frac{d}{dx_{\alpha}}\right)^{r-i}\right]\psi=0.$

In other words, $P_{\alpha}(x_{\alpha},\hbar;{\mathbf{q}})$ is a generator of
the Rees ${\mathcal{D}}$-module
$\left(E_{\hbar},\hbar\nabla^{\hbar}({\mathbf{q}})\right)$ on $U_{\alpha}$.
This expression of differential equation of order $r$ is what is commonly
referred to as a quantum curve. The coefficients
$\omega_{i}({\mathbf{q}},\hbar)$ are calculated by determining the matrix
$A({\mathbf{q}},\hbar)$ of (3.47). Then (3.48), (3.52), and (3.53) give the
exact form of $P_{\alpha}(x_{\alpha},\hbar;{\mathbf{q}})$. As we can see, its
shape is quite involved, and is _not_ obtained by simply replacing $y$ in
$\det\left(y+\phi({\mathbf{q}})\right)$ by $\hbar\frac{d}{dx_{\alpha}}$. In
particular, the coefficients contain derivatives of $q_{\ell}$’s, which never
appear in the characteristic polynomial of $-\phi({\mathbf{q}})$. What we are
going to prove now is that nonetheless, the semi-classical limit of
$P_{\alpha}(x_{\alpha},\hbar;{\mathbf{q}})$ is exactly the characteristic
polynomial.

As defined in Definition 2.8, the semi-classical limit of (3.55) is the limit

(3.56) $\displaystyle\lim_{\hbar\rightarrow
0}e^{-\frac{1}{\hbar}S_{0}(x_{\alpha})}\left[\hbar^{r}\left(\frac{d}{dx_{\alpha}}\right)^{r}-\sum_{i=2}^{r}\hbar^{r-i}\omega_{i}({\mathbf{q}},\hbar)\left(\frac{d}{dx_{\alpha}}\right)^{r-i}\right]e^{\frac{1}{\hbar}S_{0}(x_{a})}$
$\displaystyle=y^{r}-\sum_{i=2}^{r}\omega_{i}({\mathbf{q}},0)\;y^{r-i},$

where $S_{0}(x_{\alpha})$ is a holomorphic function on $U_{\alpha}$ so that
$dS_{0}=ydx_{\alpha}$ gives a local trivialization of $T^{*}C$ over
$U_{\alpha}$. Since $\omega({\mathbf{q}},\hbar)$ is a polynomial in $\hbar$,
we can evaluate it at $\hbar=0$. Notice that (3.56) is the characteristic
polynomial of the matrix $X+\omega({\mathbf{q}},\hbar)$ of (3.51) at
$\hbar=0$. The computation of semi-classical limit is the same as the
calculation of the determinant of the connection
$\hbar\nabla^{\hbar}({\mathbf{q}})$, after taking conjugation by the scalar
diagonal matrix $e^{-\frac{1}{\hbar}S_{0}(x_{\alpha})}I_{r\times r}$, and then
take the limit as $\hbar$ goes to $0$:

$\displaystyle
e^{-\frac{1}{\hbar}S_{0}(x_{\alpha})}I\cdot\left(\Delta+A({\mathbf{q}},\hbar)\right)^{-1}\left(\hbar\frac{d}{dx_{\alpha}}+\phi({\mathbf{q}})\right)\left(\Delta+A({\mathbf{q}},\hbar)\right)\cdot
e^{\frac{1}{\hbar}S_{0}(x_{\alpha})}I$
$\displaystyle=\left(\Delta+A({\mathbf{q}},\hbar)\right)^{-1}e^{-\frac{1}{\hbar}S_{0}(x_{\alpha})}I\cdot\left(\hbar\frac{d}{dx_{\alpha}}+\phi({\mathbf{q}})\right)\cdot
e^{\frac{1}{\hbar}S_{0}(x_{\alpha})}I\left(\Delta+A({\mathbf{q}},\hbar)\right)$
$\displaystyle=\left(\Delta+A({\mathbf{q}},\hbar)\right)^{-1}\left(\frac{dS_{0}(x_{\alpha})}{dx_{\alpha}}+\phi({\mathbf{q}})\right)\left(\Delta+A({\mathbf{q}},\hbar)\right)+O(\hbar)$
$\displaystyle\overset{\hbar\rightarrow
0}{\longrightarrow}\left(\Delta+A({\mathbf{q}},0)\right)^{-1}\left(y+\phi({\mathbf{q}})\right)\left(\Delta+A({\mathbf{q}},0)\right).$

The determinant of the above matrix is the characteristic polynomial
$\det\left(y+\phi({\mathbf{q}})\right)$. Note that from (3.50), we have

$\displaystyle
e^{-\frac{1}{\hbar}S_{0}(x_{\alpha})}I\cdot\left(\Delta+A({\mathbf{q}},\hbar)\right)^{-1}\left(\hbar\frac{d}{dx_{\alpha}}+\phi({\mathbf{q}})\right)\left(\Delta+A({\mathbf{q}},\hbar)\right)\cdot
e^{\frac{1}{\hbar}S_{0}(x_{\alpha})}I$
$\displaystyle=e^{-\frac{1}{\hbar}S_{0}(x_{\alpha})}I\cdot\left(\hbar\frac{d}{dx_{\alpha}}-\left(X+\omega({\mathbf{q}},\hbar)\right)\right)\cdot
e^{\frac{1}{\hbar}S_{0}(x_{\alpha})}I$
$\displaystyle=\hbar\frac{d}{dx_{\alpha}}+y-\left(X+\omega({\mathbf{q}},\hbar)\right)$
$\displaystyle\overset{\hbar\rightarrow
0}{\longrightarrow}y-\left(X+\omega({\mathbf{q}},0)\right).$

Taking the determinant of the above, we conclude that

$\det\left(y+\phi({\mathbf{q}})\right)=y^{r}-\sum_{i=2}^{r}\omega_{i}({\mathbf{q}},0)\;y^{r-i}.$

This completes the proof of Theorem 3.11. ∎

For every $\hbar\in H^{1}(C,K_{C})$, the $\hbar$-connection
$\left(E_{\hbar},\hbar\nabla^{\hbar}({\mathbf{q}})\right)$ of (3.36) defines a
global Rees ${\mathcal{D}}_{C}$-module structure in $E_{\hbar}$. Thus we have
constructed a universal family ${\mathcal{E}}_{C}$ of Rees
${\mathcal{D}}_{C}$-modules on a given $C$ with a fixed spin and a projective
structures:

(3.57)
$\begin{CD}{\mathcal{E}}_{C}@<{\supset}<{}<\left(E_{\hbar},\nabla^{\hbar}({\mathbf{q}})\right)\\\
@V{}V{}V@V{}V{}V\\\ C\times B\times
H^{1}(C,K_{C})@<{\supset}<{}<\;C\times\\{{\mathbf{q}}\\}\times\\{\hbar\\}.\end{CD}$

The universal family ${\mathcal{S}}_{C}$ of spectral curves is defined over
$C\times B$.

(3.58) $\begin{CD}{\mathbb{P}}\left(K_{C}\oplus{\mathcal{O}}_{C}\right)\times
B@<{\supset}<{}<{\mathcal{S}}_{C}@<{\supset}<{}<\left(\det\big{(}\eta-\phi({\mathbf{q}})\big{)}\right)_{0}\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\ C\times B@=C\times
B@<{\supset}<{}<\;C\times\\{{\mathbf{q}}\\}.\end{CD}$

The semi-classical limit is thus a map of families

(3.59) $\begin{CD}{\mathcal{E}}_{C}@>{}>{}>{\mathcal{S}}_{C}\\\
@V{}V{}V@V{}V{}V\\\ C\times B\times H^{1}(C,K_{C})@>{}>{}>\;C\times
B.\end{CD}$

Our concrete construction (3.34) using the projective coordinate system is not
restricted to the holomorphic Higgs field $\phi({\mathbf{q}})$. Since the
moduli spaces appearing in the correspondence (3.42) become more subtle to
define due to the _wildness_ of connections, we avoid the moduli problem, and
state our quantization theorem as a generalization of Theorem 3.8.

Let $C$ be a smooth projective algebraic curve of an arbitrary genus, and
$C=\cup_{\alpha}U_{\alpha}$ a projective coordinate system subordinating the
complex structure of $C$.

######  Theorem 3.15 (Quantization of meromorphic data).

Let $D$ be an effective divisor of $C$, and

(3.60) ${\mathbf{q}}\in
B(D):=\bigoplus_{\ell=2}^{r}H^{0}\big{(}C,K_{C}(D)^{\otimes\ell}\big{)}.$

We define a meromorphic Higgs bundle $\left(E_{0},\phi({\mathbf{q}})\right)$
by the same formulae (3.5) and (3.6), as well as a meromorphic oper
$\left(E_{\hbar},\nabla^{\hbar}({\mathbf{q}})\right)$ by (3.34). The
meromorphic oper defines a meromorphic Rees ${\mathcal{D}}$-module

${\mathcal{E}}({\mathbf{q}})=\left(E_{\hbar},\hbar\nabla^{\hbar}({\mathbf{q}})\right)$

on $C$. Then the semi-classical limit of ${\mathcal{E}}({\mathbf{q}})$ is the
spectral curve

(3.61)
$\big{(}\det(\eta+\phi({\mathbf{q}})\big{)}_{0}\subset\overline{T^{*}(C)}$

of $-\phi({\mathbf{q}})$.

###### Proof.

The proof is exactly the same as Theorem 3.11 on each simply connected
coordinate neighborhood $U_{\alpha}\subset C\setminus{\text{\rm{supp}}}(D)$.
Thus the semi-classical limit of
${\mathcal{E}}({\mathbf{q}})|_{C\setminus{\text{\rm{supp}}}(D)}$ as a divisor
is defined in $\overline{T^{*}(C)}\setminus\pi^{-1}(D)$. The spectral curve
(3.61) is its closure in $\overline{T^{*}C}$ with respect to the complex
topology. Thus by Definition 2.11, (3.61) is the semi-classical limit of the
meromorphic extension ${\mathcal{E}}({\mathbf{q}})$. ∎

######  Example 3.1.

Here we list characteristic polynomials and differential operators
$P_{\alpha}(x_{\alpha},\hbar;{\mathbf{q}})$ of (3.55) for $r=2,3,4$. These
formulas show that our quantization procedure is quite non-trivial.

  * •

$r=2$.

$\displaystyle\det\left(y+\phi({\mathbf{q}})\right)$
$\displaystyle=y^{2}-q_{2}$ $\displaystyle
P_{\alpha}(x_{\alpha},\hbar;{\mathbf{q}})$
$\displaystyle=\left(\hbar\frac{d}{dx_{\alpha}}\right)^{2}-q_{2}.$

  * •

$r=3$.

$\displaystyle\det\left(y+\phi({\mathbf{q}})\right)$
$\displaystyle=y^{3}-4q_{2}y+4q_{3}$ $\displaystyle
P_{\alpha}(x_{\alpha},\hbar;{\mathbf{q}})$
$\displaystyle=\left(\hbar\frac{d}{dx_{\alpha}}\right)^{3}-4q_{2}\left(\hbar\frac{d}{dx_{\alpha}}\right)+4q_{3}-2\hbar
q_{2}^{\prime}.$

  * •

$r=4$.

$\displaystyle\det\left(y+\phi({\mathbf{q}})\right)$
$\displaystyle=y^{4}-10q_{2}+24q_{3}y-36q_{4}+9q_{2}^{2}$ $\displaystyle
P_{\alpha}(x_{\alpha},\hbar;{\mathbf{q}})$
$\displaystyle=\left(\hbar\frac{d}{dx_{\alpha}}\right)^{4}-10q_{2}\left(\hbar\frac{d}{dx_{\alpha}}\right)^{2}+\left(24q_{3}-10\hbar
q_{2}^{\prime}\right)\left(\hbar\frac{d}{dx_{\alpha}}\right)$
$\displaystyle\qquad-36q_{4}+9q_{2}^{2}+3\hbar^{2}q_{2}^{\prime\prime}-12\hbar
q_{3}^{\prime}.$

###  3.3. Non-Abelian Hodge correspondence and a conjecture of Gaiotto

The biholomorphic map (3.42) is defined by fixing a projective structure of
the base curve $C$. Gaiotto [35] conjectured that such a correspondence would
be canonically constructed through a _scaling limit_ of non-Abelian Hodge
correspondence. The conjecture has been solved in [22] for Hitchin moduli
spaces ${\mathcal{M}}_{{\text{\rm{Dol}}}}$ and
${\mathcal{M}}_{{\text{\rm{deR}}}}$ constructed over an arbitrary complex
simple and simply connected Lie group $G$. In this subsection, we review the
main result of [22] for $G=SL(r,{\mathbb{C}})$ and compare it with our
quantization.

The setting of this subsection is the following. The base curve $C$ is a
compact Riemann surface of genus $g\geq 2$. We denote by $E^{\text{top}}$ the
topologically trivial complex vector bundle of rank $r$ on $C$. The prototype
of the correspondence between stable holomorphic vector bundles on $C$ and
differential geometric data goes back to Narasimhan-Seshadri [62] (see also
[3, 61]). Extending the classical case, the stability condition for an
$SL(r,{\mathbb{C}})$-Higgs bundle $(E,\phi)$ translates into a differential
geometric condition, known as _Hitchin’s equations_ , imposed on a set of
geometric data as follows [20, 40, 64]. The data consist of a Hermitian fiber
metric $h$ on $E^{\rm{top}}$, a unitary connection $\nabla$ in $E^{\rm{top}}$
with respect to $h$, and a differentiable $sl(r,{\mathbb{C}})$-valued $1$-form
$\phi$ on $C$. The following system of nonlinear equations is called Hitchin’s
equations.

(3.62) $\begin{cases}F_{\nabla}+[\phi,\phi^{\dagger}]=0\\\
\nabla^{0.1}\phi=0.\end{cases}$

Here, $F_{\nabla}$ is the curvature of $\nabla$, $\phi^{\dagger}$ is the
Hermitian conjugate of $\phi$ with respect to $h$, and $\nabla^{0,1}$ is the
Cauchy-Riemann part of $\nabla$. $\nabla^{0,1}$ gives rise to a natural
complex structure in $E^{\rm{top}}$, which we simply denote by $E$. Then
$\phi$ becomes a holomorphic Higgs field in $E$ because of the second equation
of (3.62). The stability condition for the Higgs pair $(E,\phi)$ is equivalent
to Hitchin’s equations (3.62). Define a one-parameter family of connections

(3.63)
$\nabla(\zeta):=\frac{1}{\zeta}\cdot\phi+\nabla+\zeta\cdot\phi^{\dagger},\qquad\zeta\in{\mathbb{C}}^{*}.$

Then the flatness of $\nabla(\zeta)$ for all $\zeta$ is equivalent to (3.62).

The _non-Abelian Hodge correspondence_ [20, 40, 55, 64] is the following
diffeomorphic correspondence

$\nu:{\mathcal{M}}_{{\text{\rm{Dol}}}}\owns(E,\phi)\longmapsto(\widetilde{E},\widetilde{\nabla})\in{\mathcal{M}}_{{\text{\rm{deR}}}}.$

First, we construct the solution $(\nabla,\phi,h)$ of Hitchin’s equations
corresponding to stable $(E,\phi)$. It induces a family of flat connections
$\nabla(\zeta)$. Then define a complex structure $\widetilde{E}$ in
$E^{\rm{top}}$ by $\nabla(\zeta=1)^{0,1}$. Since $\nabla(\zeta)$ is flat,
$\widetilde{\nabla}:=\nabla(\zeta=1)^{1,0}$ is automatically a holomorphic
connection in $\widetilde{E}$. Thus $(\widetilde{E},\widetilde{\nabla})$
becomes a holomorphic connection. Stability of $(E,\phi)$ implies that the
resulting connection is irreducible, hence
$(\widetilde{E},\widetilde{\nabla})\in{\mathcal{M}}_{{\text{\rm{deR}}}}$.
Since this correspondence goes through the real unitary connection $\nabla$,
the assignment $E\longmapsto\widetilde{E}$ is not a holomorphic deformation of
vector bundles.

Extending the idea of the one-parameter family (3.63), Gaiotto conjectured the
following:

######  Conjecture 3.16 (Gaiotto [35]).

Let $(\nabla,\phi,h)$ be the solution of (3.62) corresponding to a sable Higgs
bundle $\left(E_{0},\phi({\mathbf{q}})\right)$ on the
$SL(r,{\mathbb{C}})$-Hitchin section (3.7). Consider the following two-
parameter family of connections

(3.64) $\nabla(\zeta,R):=\frac{1}{\zeta}\cdot R\phi+\nabla+\zeta\cdot
R\phi^{\dagger},\qquad\zeta\in{\mathbb{C}}^{*},R\in{\mathbb{R}}_{+}.$

Then the scaling limit

(3.65) $\lim_{\begin{subarray}{c}R\rightarrow 0,\zeta\rightarrow 0\\\
\zeta/R=\hbar\end{subarray}}\nabla(\zeta,R)$

exists for every $\hbar\in{\mathbb{C}}^{*}$, and forms an $\hbar$-family of
$SL(r,{\mathbb{C}})$-opers.

######  Remark 3.17.

The existence of the limit is non-trivial, because the Hermitian metric $h$
blows up as $R\rightarrow 0$.

######  Remark 3.18.

Unlike the case of non-Abelian Hodge correspondence, the Gaiotto limit works
only for a point in the Hitchin section.

######  Theorem 3.19 ([22]).

Gaiotto’s conjecture holds for an arbitrary simple and simply connected
complex algebraic group $G$.

The universal covering of a compact Riemann surface $C$ is the upper-half
plane ${\mathbb{H}}$. The fundamental group $\pi_{1}(C)$ acts on
${\mathbb{H}}$ through a representation

$\rho:\pi_{1}(C)\longrightarrow
PSL(2,{\mathbb{R}})={\text{\rm{Aut}}}({\mathbb{H}}),$

and generates an analytic isomorphism

$C\cong{\mathbb{H}}\big{/}\rho\big{(}\pi_{1}(C)\big{)}.$

The representation $\rho$ lifts to $SL(2,{\mathbb{R}})\subset
SL(2,{\mathbb{C}})$, and induces a projective structure in $C$ subordinating
its complex structure coming from ${\mathbb{H}}$. This projective structure is
what we call the _Fuchsian_ projective structure.

######  Corollary 3.20 (Gaiotto correspondence and quantization).

Under the same setting of Conjecture 3.16, the limit oper of [22] is given by

(3.66) $\lim_{\begin{subarray}{c}R\rightarrow 0,\zeta\rightarrow 0\\\
\zeta/R=\hbar\end{subarray}}\nabla(\zeta,R)=d+\frac{1}{\hbar}\phi({\mathbf{q}})=\nabla^{\hbar}({\mathbf{q}}),\qquad\hbar\neq
0,$

with respect to the Fuchsian projective coordinate system. The operator (3.66)
is a connection in the $\hbar$-filtered extension
$(F_{\hbar}^{\bullet},E_{\hbar})$ of Definition 3.2. In particular, the
correspondence

$\left(E_{0},\phi({\mathbf{q}})\right)\overset{\gamma}{\longmapsto}\left(E_{\hbar},\nabla^{\hbar}({\mathbf{q}})\right)$

is biholomorphic, unlike the non-Abelian Hodge correspondence.

###### Proof.

The key point is that since $E_{0}$ is made out of $K_{C}$, the fiber metric
$h$ naturally comes from the metric of $C$ itself. Hitchin’s equations (3.62)
for ${\mathbf{q}}=0$ then become a harmonic equation for the metric of $C$,
and its solution is given by the constant curvature hyperbolic metric. This
metric in turn defines the Fuchsian projective structure in $C$. For more
detail, we reefer to [21, 22]. ∎

##  4\. Geometry of singular spectral curves

The quantization mechanism of Section 3 applies to all Hitchin spectral
curves, and it is not sensitive to whether the spectral curve is smooth or
not. However, the local quantization mechanism of [23, 24] using the PDE
version of topological recursion requires a _non-singular_ spectral curve. The
goal of this section is to review the systematic construction of the non-
singular models of singular Hitchin spectral curves of [24]. Then in Section
6, we prove that for the case of meromorphic $SL(2,{\mathbb{C}})$-Higgs
bundles, the PDE topological recursion based on the non-singular model of this
section provides WKB analysis for the quantum curves constructed in Section 3.

Since the quantum curve reflects the geometry of
$\Sigma\subset\overline{T^{*}C}$ through semi-classical limit, we first need
to identify the choice of the blow-up space $Bl(\overline{T^{*}C})$ in which
the non-singular model $\widetilde{\Sigma}$ of the spectral curve is realized
as a smooth divisor. This geometric information determines part of the initial
data for topological recursion, i.e., the spectral curve of Eynard-Orantin,
and the differential form $W_{0,1}$. Geometry of spectral curve also gives us
information of singularities of the quantization. For example, when we have a
component of a spectral curve tangent to the divisor $C_{\infty}$, the quantum
curve has an irregular singular point, and the class of the irregularity is
determined by the degree of tangency. We have given a classification of
singularities of the quantum curves in terms of the geometry of spectral
curves in [24].

In what follows, we give the construction of the canonical blow-up space
$Bl(\overline{T^{*}C})$, and determine the genus of the normalization
$\widetilde{\Sigma}$. This genus is necessary to identify the Riemann prime
form on it, which determines another input datum $W_{0,2}$ for the topological
recursion.

There are two different ways of defining the spectral curve for Higgs bundles
with meromorphic Higgs field. Our definition uses the compactified cotangent
bundle. This idea also appears in [49]. The traditional definition, which
assumes the pole divisor $D$ of the Higgs field to be reduced, is suitable for
the study of moduli spaces of parabolic Higgs bundles. When we deal with non-
reduced effective divisors, parabolic structures do not play any role. Non-
reduced divisors appear naturally when we deal with classical equations such
as the Airy differential equation, which has an irregular singular point of
class $\frac{3}{2}$ at $\infty\in{\mathbb{P}}^{1}$.

Our point of view of spectral curves is also closely related to considering
the _stable pairs_ of pure dimension $1$ on $\overline{T^{*}C}$. Through
Hitchin’s abelianization idea, the moduli space of stable pairs and the moduli
space of Higgs bundles are identified [44].

The geometric setting we start with is a meromorphic
$SL(2,{\mathbb{C}})$-Higgs bundle $\left(E_{0},\phi(q)\right)$ defined on a
smooth projective algebraic curve $C$ of genus $g\geq 0$ with a fixed
projective structure. Here,

(4.1) $q=-\det(\phi(q))\in H^{0}\left(C,K_{C}(D)^{\otimes
2}\right),\qquad\phi(q)=\begin{bmatrix}&q\\\ 1\end{bmatrix},$

and $D$ is an effective divisor of $C$. The spectral curve is the zero-locus
in $\overline{T^{*}C}$ of the characteristic equation

(4.2) $\Sigma:=\left(\eta^{2}-\pi^{*}(q)\right)_{0}.$

The only condition we impose here is that _the spectral curve is irreducible._
In the language of Higgs bundles, this condition corresponds to the stability
of $\left(E_{0},\phi(q)\right)$.

Recall that ${\text{\rm{Pic}}}(\overline{T^{*}C})$ is generated by the zero
section $C_{0}$ of $T^{*}C$ and fibers of the projection map
$\pi:\overline{T^{*}C}\longrightarrow C$. Since the spectral curve $\Sigma$ is
a double covering of $C$, as a divisor it is expressed as

(4.3)
$\Sigma=2C_{0}+\sum_{j=1}^{a}\pi^{*}(p_{j})\in{\text{\rm{Pic}}}(\overline{T^{*}C}),$

where $\sum_{j=1}^{a}p_{j}\in{\text{\rm{Pic}}}^{a}(C)$ is a divisor on $C$ of
degree $a$. As an element of the Néron-Severi group

${\text{\rm{NS}}}(\overline{T^{*}C})={\text{\rm{Pic}}}(\overline{T^{*}C})/{\text{\rm{Pic}}}^{0}(\overline{T^{*}C}),$

it is simply

$\Sigma=2C_{0}+aF\in{\text{\rm{NS}}}(\overline{T^{*}C})$

for a typical fiber class $F$. Since the intersection $F\cdot C_{\infty}=1$,
we have $a=\Sigma\cdot C_{\infty}$ in ${\text{\rm{NS}}}(\overline{T^{*}C})$.
From the genus formula

$p_{a}(\Sigma)={\frac{1}{2}}\Sigma\cdot(\Sigma+K_{\overline{T^{*}C}})+1$

and

$K_{\overline{T^{*}C}}=-2C_{0}+(4g-4)F\in{\text{\rm{NS}}}(\overline{T^{*}C}),$

we find that the arithmetic genus of the spectral curve $\Sigma$ is

(4.4) $p_{a}(\Sigma)=4g-3+a,$

where $a$ is the number of intersections of $\Sigma$ and $C_{\infty}$. Now we
wish to find the geometric genus of $\Sigma$.

Recall the following from [24]:

######  Definition 4.1 (Discriminant divisor).

The discriminant divisor of the spectral curve (4.2) is a divisor on $C$
defined by

(4.5) $\Delta:=(q)_{0}-(q)_{\infty},$

where

(4.6) $\displaystyle(q)_{0}=\sum_{i=1}^{m}m_{i}r_{i},\quad m_{i}>0,\quad
q_{i}\in C,$ (4.7) $\displaystyle(q)_{\infty}=\sum_{j=1}^{n}n_{j}p_{j},\quad
n_{j}>0,\quad p_{j}\in C.$

Since $q$ is a meromorphic section of $K_{C}^{\otimes 2}$,

(4.8) $\deg\Delta=\sum_{i=1}^{m}m_{i}-\sum_{j=1}^{n}n_{j}=4g-4.$

######  Theorem 4.2 (Geometric genus formula).

Define

(4.9) $\delta=|\\{i\;|\;m_{i}\equiv 1\mod 2\\}|+|\\{j\;|\;n_{j}\equiv 1\mod
2\\}|.$

Then the geometric genus of the spectral curve $\Sigma$ of (4.2) is given by

(4.10) $p_{g}(\Sigma)=2g-1+{\frac{1}{2}}\delta.$

We note that (4.8) implies $\delta\equiv 0\mod 2$.

######  Remark 4.3.

If $\phi$ is a holomorphic Higgs field, then $m=\delta=4g-4$ and $n=0$.
Therefore, we recover the genus formula $g(\Sigma)=4g-3$ of [23, Eq.(2.5)].

###### Proof.

Since $\Sigma\subset\overline{T^{*}C}$ is a double covering of $C$ in a ruled
surface, locally at every singular point $p$, $\Sigma$ is either irreducible,
or reducible and consisting of two components. When irreducible, it is locally
isomorphic to

(4.11) $t^{2}-s^{2m+1}=0,\qquad m\geq 1.$

If it has two components, then it is locally isomorphic to

(4.12) $t^{2}-s^{2m}=(t-s^{m})(t+s^{m})=0.$

Note that the local form of $\Sigma$ at a ramification point of
$\pi:\Sigma\longrightarrow C$ is written as (4.11) with $m=0$. By extending
the terminology “singularity” to “critical points” of the morphism $\pi$, we
include a ramification point as a cusp with $m=0$.

Let ${\nu}:{\widetilde{\Sigma}}\longrightarrow\Sigma$ be the non-singular
model of $\Sigma$. Then
${\tilde{\pi}}=\pi\circ{\nu}:{\widetilde{\Sigma}}\longrightarrow C$ is a
double sheeted covering of $C$ by a smooth curve ${\widetilde{\Sigma}}$. If
$\Sigma$ has two components at a singularity $P$ as in (4.12), then
${\tilde{\pi}}^{-1}(P)$ consists of two points and ${\tilde{\pi}}$ is not
ramified there. If $P$ is a cusp (4.11), then ${\tilde{\pi}}^{-1}(P)$ is a
ramification point of the covering ${\tilde{\pi}}$. Thus the invariant
$\delta$ of (4.9) counts the total number of cusp singularities of $\Sigma$
and the ramification points of $\pi:\Sigma\longrightarrow C$. Then the
Riemann-Hurwitz formula gives us

$2-2g\big{(}{\widetilde{\Sigma}}\big{)}-\delta=2\left(2-2g(C)-\delta\right),$

hence

(4.13) $p_{g}(\Sigma)=2g(C)-1+{\frac{1}{2}}\;(\text{\rm{the number of
cusps}}).$

The genus formula (4.10) follows from (4.13). ∎

Our purpose is to apply topological recursion of Section 5 to a singular
spectral curve $\Sigma$ of (4.2). To this end, we need to construct in a
canonical way the normalization
$\nu:{\widetilde{\Sigma}}\longrightarrow\Sigma$ through a sequence of blow-ups
of the ambient space $\overline{T^{*}C}$. This is because we need to construct
differential forms on ${\widetilde{\Sigma}}$ that reflect geometry of
$i:\Sigma\longrightarrow\overline{T^{*}C}$. We thus proceed to analyze the
local structure of $\Sigma$ at each singularity using the global equation
(4.2) in what follows.

######  Definition 4.4 (Construction of the blow-up space).

The blow-up space $Bl(\overline{T^{*}C})$

(4.14) |  |
---|---|---
$\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^{*}C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{i}$$\textstyle{\overline{T^{*}C}.\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{C}$

is defined by blowing up $\overline{T^{*}C}$ in the following way.

  * •

At each $r_{i}$ of (4.6), blow up $r_{i}\in\Sigma\cap
C_{0}\subset\overline{T^{*}C}$ a total of $\lfloor\frac{m_{i}}{2}\rfloor$
times.

  * •

At each $p_{j}$ of (4.7), blow up at the intersection
$\Sigma\cap\pi^{-1}(p_{j})\subset C_{\infty}$ a total of
$\lfloor\frac{n_{j}}{2}\rfloor$ times.

######  Remark 4.5.

Let us consider when both $(q)_{0}$ and $(q)_{\infty}$ are reduced. From the
definition above, in this case $\Sigma$ is non-singular, and the two genera
(4.4) and (4.10) agree. The spectral curve is invariant under the involution
$\sigma:\overline{T^{*}C}\longrightarrow\overline{T^{*}C}$ of (1.9). If $q\in
H^{0}(C,K_{C}^{\otimes 2})$ is holomorphic, then $\pi:\Sigma\longrightarrow C$
is simply branched over $\Delta=(q)_{0}$, and $\Sigma$ is a smooth curve of
genus $4g-3$. This is in agreement of (4.4) because $n=0$ in this case. If $q$
is meromorphic, then its pole divisor is given by $(q)_{\infty}$ of degree
$n$. Since $(q)_{\infty}$ is reduced, $\pi:\Sigma\longrightarrow C$ is
ramified at the intersection of $C_{\infty}$ and $\pi^{*}(q)_{\infty}$. The
spectral curve is also ramified at its intersection with $C_{0}$. Note that
$\deg(q)_{0}=4g-4+n$ because of (4.8). Thus $\pi:\Sigma\longrightarrow C$ is
simply ramified at a total of $4g-4+2n$ points. Therefore, $\Sigma$ is non-
singular, and we deduce that its genus is given by

$p_{g}(\Sigma)=p_{a}(\Sigma)=4g-3+n$

from the Riemann-Hurwitz formula. As a divisor class, we have

$\Sigma=2C_{0}+\pi^{*}(q)_{\infty}\in{\text{\rm{Pic}}}(\overline{T^{*}C}),$

in agreement of (4.3).

######  Theorem 4.6.

In the blow-up space $Bl(\overline{T^{*}C})$, we have the following.

  * •

The proper transform $\widetilde{\Sigma}$ of the spectral curve
$\Sigma\subset\overline{T^{*}C}$ by the birational morphism
$\nu:Bl(\overline{T^{*}C})\longrightarrow\overline{T^{*}C}$ is a smooth curve
with a holomorphic map
$\tilde{\pi}=\pi\circ\nu:\widetilde{\Sigma}\longrightarrow C$.

  * •

The Galois action $\sigma:\Sigma\longrightarrow\Sigma$ lifts to an involution
of $\widetilde{\Sigma}$, and the morphism
$\tilde{\pi}:\widetilde{\Sigma}\longrightarrow C$ is a Galois covering with
the Galois group
${\text{\rm{Gal}}}(\widetilde{\Sigma}/C)={\langle}\tilde{\sigma}{\rangle}\cong{\mathbb{Z}}/2{\mathbb{Z}}$.

(4.15) $\begin{CD}\widetilde{\Sigma}@>{\nu}>{}>\Sigma @>{\pi}>{}>C\\\
@V{\tilde{\sigma}}V{}V@V{}V{\sigma}V\Big{\|}\\\
\widetilde{\Sigma}@>{}>{\nu}>\Sigma @>{}>{\pi}>C\end{CD}$

###### Proof.

We need to consider only when $\Delta$ is non-reduced. Let
$r_{i}\in{\text{\rm{supp}}}(\Delta)$ be a zero of $q$ of degree $m_{i}>1$. The
curve germ of $\Sigma$ near $r_{i}\in\Sigma\cap C_{0}$ is given by a formula
$y^{2}=x^{m_{i}},$ where $x$ is the base coordinate on $C$ and $y$ a fiber
coordinate. We blow up once at $(x,y)=(0,0)$, using a local parameter
$y_{1}=y/x$ on the exceptional divisor. The proper transform of the curve germ
becomes $y_{1}^{2}=x^{m_{i}-2}.$ Repeat this process at $(x,y_{1})=(0,0)$,
until we reach the equation

$y_{\ell}^{2}=x^{\epsilon},$

where $\epsilon=0$ or $1$. The proper transform of the curve germ is now non-
singular. We see that after a sequence of $\lfloor\frac{m_{i}}{2}\rfloor$
blow-ups starting at the point $r_{i}$, the proper transform of $\Sigma$ is
simply ramified over $r_{i}\in C=C_{0}$ if $m_{i}$ is odd, and unramified if
$m_{i}$ is even. We apply the same sequence of blow-ups at each $r_{i}$ with
multiplicity greater than $1$.

Let $p_{j}\in{\text{\rm{supp}}}(\Delta)$ be a pole of $q$ with order
$n_{j}>1$. The intersection $P=\Sigma\cap\pi^{-1}(p_{j})$ lies on
$C_{\infty}$, and $\Sigma$ has a singularity at $P$. Let $z=1/y$ be a fiber
coordinate of $\pi^{-1}(p_{j})$ at the infinity. Then the curve germ of
$\Sigma$ at the point $P$ is given by

$z^{2}=x^{n_{j}}.$

The involution $\sigma$ in this coordinate is simply $z\longmapsto-z$. The
blow-up process we apply at $P$ is the same as before. After
$\lfloor\frac{n_{j}}{2}\rfloor$ blow-ups starting at the point
$P\in\Sigma\cap\pi^{-1}(p_{j})$, the proper transform of $\Sigma$ is simply
ramified over $p_{j}\in C$ if $n_{j}$ is odd, and unramified if $n_{j}$ is
even. Again we do this process for all $p_{j}$ with a higher multiplicity.

The blow-up space $Bl(\overline{T^{*}C})$ is defined as the application of a
total of

(4.16)
$\sum_{i=1}^{m}\left\lfloor\frac{m_{i}}{2}\right\rfloor+\sum_{j=1}^{n}\left\lfloor\frac{n_{j}}{2}\right\rfloor$

times blow-ups on $\overline{T^{*}C}$ as described above. The proper transform
${\widetilde{\Sigma}}$ is the minimal resolution of $\Sigma$. Note that the
morphism

(4.17) ${\tilde{\pi}}=\pi\circ\nu:{\widetilde{\Sigma}}\longrightarrow C$

is a double covering, ramified exactly at $\delta$ points. Since
$p_{a}({\widetilde{\Sigma}})=p_{g}(\Sigma)$, (4.10) follows from the Riemann-
Hurwitz formula applied to
${{\tilde{\pi}}}:{\widetilde{\Sigma}}\longrightarrow C$. It is also obvious
that $\delta$ counts the number of cusp points of $\Sigma$, including smooth
ramification points of $\pi$, in agreement of Theorem 4.2, and the fact that
$\delta$ counts the total number of odd cusps on $\Sigma$.

Note that $C_{0}$ and $C_{\infty}$ are point-wise invariant under the
involution $\sigma$. Since $Bl(\overline{T^{*}C})$ is constructed by blowing
up points on $C_{0}$ and $C_{\infty}$ and their proposer transforms, we have a
natural lift ${\tilde{\sigma}}:Bl(\overline{T^{*}C})\longrightarrow
Bl(\overline{T^{*}C})$ of $\sigma$ which induces (4.15). ∎

##  5\. A differential version of topological recursion

The Airy example of Section 7 suggests that the asymptotic expansion of a
solution to a given quantum curve at its singularity contains information of
quantum invariants. It also suggests that the topological recursion of [32]
provides an effective tool for calculating asymptotic expansions of solutions
for quantum curves. Since a linear differential equation is characterized by
its solutions, topological recursion can be used as a mechanism of defining
the quantization process from a spectral curve to a quantum curve. Then a
natural question arises:

######  Question 5.1.

How are the two quantizations, one with the construction of an $\hbar$-family
of opers, and the other via topological recursion, related?

In Section 6, we prove that topological recursion provides WKB analysis of the
quantum curves constructed through $\hbar$-families of opers, for the case of
holomorphic and meromorphic $SL(2,{\mathbb{C}})$-Higgs bundles. For this
purpose, in this section we review the generalized framework of topological
recursion developed in [23, 24] in the format of PDE recursion. For the case
of singular Hitchin spectral curves, our particular method of normalization of
spectral curves of Sections 4 produces the same result of quantization of
Section 3.

If we consider a family of spectral curves that degenerate to a singular
curve, the necessity of normalization for WKB analysis may sound unnatural. We
emphasize that the semi-classical limit of the quantum curve thus obtained
remains the original _singular_ spectral curve, not the normalization,
consistent with (3.59). Thus our quantization procedure in terms of
topological recursion is also a natural process.

Although many aspects of our current framework can be generalized to arbitrary
complex simple Lie groups, since our calculation mechanism of Section 6 has
been developed so far only for the $SL(2,{\mathbb{C}})$ case, we restrict our
attention to this case in this section.

We start with topological recursion for a degree $2$ covering, not necessarily
restricted to Hitchin spectral curves. The key ingredient of the theory is the
Riemann prime form $E(z_{1},z_{2})$ on the product
$\widetilde{\Sigma}\times\widetilde{\Sigma}$ of a compact Riemann surface
$\widetilde{\Sigma}$ with values in a certain line bundle [60]. To define the
prime form, we have to make a few more extra choices. First, we need to choose
a theta characteristic $K_{\widehat{\Sigma}}^{\frac{1}{2}}$ such that

(5.1) $\dim
H^{0}\left(\widetilde{\Sigma},K_{\widetilde{\Sigma}}^{\frac{1}{2}}\right)=1.$

We also need to choose a symplectic basis for
$H_{1}(\widetilde{\Sigma},{\mathbb{Z}})$, usually referred to as $A$-cycles
and $B$-cycles. We follow Mumford’s convention and use the unique Riemann
prime form as defined in [60, Page 3.210]. See also [23, Section 2].

######  Definition 5.2 (Integral topological recursion for non-singular
covering).

Let $C$ be a non-singular projective algebraic curve together with a choice of
a symplectic basis for $H^{1}(C,{\mathbb{Z}})$, and
$\tilde{\pi}:\widetilde{\Sigma}\longrightarrow C$ a degree $2$ covering by
another non-singular curve $\widetilde{\Sigma}$. We denote by $R$ the
ramification divisor of $\tilde{\pi}$. The covering $\tilde{\pi}$ is a Galois
covering with the Galois group
${\mathbb{Z}}/2{\mathbb{Z}}={\langle}\tilde{\sigma}{\rangle}$, and $R$ is the
fixed-point divisor of the involution $\tilde{\sigma}$. We also choose a
labeling of points of $R$, and define the $A$-cycles of a symplectic bases for
$H_{1}(\widetilde{\Sigma},{\mathbb{Z}})$ as defined in [23, Section 2], which
extend the $A$-cycles of $C$. The integral topological recursion is an
inductive mechanism of constructing meromorphic differential forms $W_{g,n}$
on the Hilbert scheme $\widetilde{\Sigma}^{[n]}$ of $n$-points on
$\widetilde{\Sigma}$ for all $g\geq 0$ and $n\geq 1$ in the _stable range_
$2g-2+n>0$, from given initial data $W_{0,1}$ and $W_{0,2}$. The differential
form $W_{g,n}$ is a meromorphic $n$-linear form, i.e., a $1$-form on each
factor of $\widetilde{\Sigma}^{[n]}$ for $2g-2+n>0$.

  * •

$W_{0,1}$ is a meromorphic $1$-form on $\widetilde{\Sigma}$ to be prescribed
according to the geometric setting we have. Then define

(5.2) $\Omega:=\tilde{\sigma}^{*}W_{0,1}-W_{0,1},$

which satisfies $\tilde{\sigma}^{*}\Omega=-\Omega$.

  * •

$W_{0,2}$ is defined by

(5.3) $W_{0,2}(z_{1},z_{2})=d_{1}d_{2}\log E(z_{1},z_{2}),$

where $E(z_{1},z_{2})$ is the $A$-cycle normalized Riemann prime form on
$\widetilde{\Sigma}\times\widetilde{\Sigma}$. $W_{0,2}$ is a meromorphic
differential $1\otimes 1$-form on
${\widetilde{\Sigma}}\times{\widetilde{\Sigma}}$ with $2$nd order poles along
the diagonal.

  * •

We also define the normalized Cauchy kernel on $\widetilde{\Sigma}$ by

(5.4) $\omega^{a-b}(z):=d_{z}\log\frac{E(a,z)}{E(b,z)},$

which is a meromorphic $1$-form in $z$ with simple poles at $z=a$ of residue
$1$ and at $z=b$ of residue $-1$. We note that the ratio $E(a,z)/E(b,z)$ is a
meromorphic function on the universal covering in $a$ or $b$, but _not_ a
meromorphic function on ${\widetilde{\Sigma}}$. We thus choose a fundamental
domain of the universal covering and restrict $a$ and $b$ in that domain for
local calculations.

The inductive formula of the topological recursion then takes the following
shape for $2g-2+n>0$.

(5.5)
$W_{g,n}(z_{1},\dots,z_{n})={\frac{1}{2}}\frac{1}{2\pi\sqrt{-1}}\oint_{\Gamma}\frac{\omega^{\tilde{z}-z}(z_{1})}{\Omega(z)}\\\
\times\left[W_{g-1,n+1}(z,\tilde{z},z_{2},\dots,z_{n})+\sum_{\begin{subarray}{c}g_{1}+g_{2}=g\\\
I\sqcup
J=\\{2,\dots,n\\}\end{subarray}}^{\rm{No}\;(0,1)}W_{g_{1},|I|+1}(z,z_{I})W_{g_{2},|J|+1}(\tilde{z},z_{J})\right].$

Here,

  * •

The integration contour $\Gamma\subset{\widetilde{\Sigma}}$ is a collection of
positively oriented small loops around each point
$p\in{\text{\rm{supp}}}(\Omega)_{0}\cup{\text{\rm{supp}}}(R)$.

  * •

The integration is taken with respect to $z\in\Gamma$. Thus we chose a
fundamental domain of the universal covering of ${\widetilde{\Sigma}}$ that
contains ${\text{\rm{supp}}}(\Omega)_{0}\cup{\text{\rm{supp}}}(R)$, and
perform the integration locally as residue calculations.

  * •

$\tilde{z}=\tilde{\sigma}(z)$ is the Galois conjugate of
$z\in\widetilde{\Sigma}$.

  * •

The expression $1/\Omega$ is a meromorphic section of $K_{C}^{-1}$ which is
multiplied to meromorphic quadratic differentials on $\widetilde{\Sigma}$ in
$z$-variable.

  * •

“No $(0,1)$” means that $g_{1}=0$ and $I=\emptyset$, or $g_{2}=0$ and
$J=\emptyset$, are excluded in the summation.

  * •

The sum runs over all partitions of $g$ and set partitions of
$\\{2,\dots,n\\}$, other than those containing the $(0,1)$-geometry.

  * •

$|I|$ is the cardinality of the subset $I\subset\\{2,\dots,n\\}$.

  * •

$z_{I}=(z_{i})_{i\in I}$.

######  Remark 5.3.

The integrand of (5.5) is not a well-defined differential form in
$z\in{\widetilde{\Sigma}}$, due to the definition of
$\omega^{{\tilde{\sigma}}(z)-z}(z_{1})$ mentioned above. What the formula
defines is a sum of residues at each point
$p\in{\text{\rm{supp}}}(\Omega)_{0}\cup{\text{\rm{supp}}}(R)$, which depends
on the choice of the domain of $\omega^{{\tilde{\sigma}}(z)-z}(z_{1})$ as a
function in $z$.

######  Remark 5.4.

Topological recursion depends on the choice of the integration contour
$\Gamma$. Since the integrand of the right-hand side of (5.5) has other poles
than the ramification divisor $R$ and zeros of $\Omega$, other choices of
$\Gamma$ are equally possible.

######  Remark 5.5.

Topological recursion (5.5) can be defined for far more general situations.
The bottle neck of the formalism is that integrations over a high genus
Riemann surface is almost always impossible. So the actual calculations have
not been done much, other than the cases when the spectral curve
${\widetilde{\Sigma}}$ is of genus $0$.

When we have a non-singular Hitchin spectral curve $i:\Sigma\hookrightarrow
T^{*}C$ associated with a holomorphic $SL(2,{\mathbb{C}})$-Higgs bundle
$(E,\phi)$, we apply Definition 5.2 to
$\widetilde{\Sigma}=\Sigma,\tilde{\sigma}=\sigma$, and $W_{0,1}=i^{*}\eta$,
where $\sigma$ is the involution of (1.9) and $\eta$ is the tautological
$1$-form on $T^{*}C$.

Under the same setting as in the integral topological recursion, we define

######  Definition 5.6 (PDE topological recursion for a smooth covering of
degree $2$).

The PDE topological recursion is the following partial differential equation
for all $(g,n)$ subject to $2g-2+n\geq 2$ defined on an open neighborhood
$U^{n}$ of $\widetilde{\Sigma}^{[n]}$ (or the universal covering, if the
global treatment is necessary):

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

Here, the index subset $[\hat{j}]$ denotes the complement of
$j\in\\{1,2,\dots,n\\}$. The final summation runs over all indices subject to
be in the stable range, i.e., $2g_{1}-1+|I|>0$ and $2g_{2}-1+|J|>0$. The
initial data for the PDE recursion are a function $F_{1,1}(z_{1})$ on $U$ and
a symmetric function $F_{0,3}(z_{1},z_{2},z_{3})$ on $U^{3}$.

######  Remark 5.7.

The PDE topological recursion (5.6) is obtained from the integral one (5.5) by
replacing the original contour $\Gamma$ about the ramification divisor with
the Cauchy kernel integration contour of [23, 24] about the diagonal poles of
$\omega^{{\tilde{\sigma}}(z)-z}(z_{1})$, $W_{0,2}(z,z_{j})$, and
$W_{0,2}\big{(}{\tilde{\sigma}}(z),z_{j}\big{)}$ at $z=z_{j}$ and
$z={\tilde{\sigma}}(z_{j})$, $j=1,2,\dots,n$. This means the only difference
comes from the alternative choice of the contour of integration $\Gamma$. All
other ingredients of the formula are the same. Other than the case of
$g({\widetilde{\Sigma}})=0$, the integral and PDE topological recursions are
not equivalent. In many enumerative problems [10, 13, 25, 27, 31, 58, 56, 58,
59], however, the PDE topological recursion is more fundamental, and the
integral topological recursion is derived only after the PDE version is first
established through the Laplace transform of combinatorial relations. In these
examples, the residue integral around ramification points of the spectral
curve and Cauchy kernel integrations, i.e., residue calculations around the
diagonals $z=z_{j},z={\tilde{\sigma}}(z_{j})$, $j=1,2,\dots,n$, are indeed
equivalent, by continuous deformation of the contour $\Gamma$. More detail on
the relation between integral and PDE topological recursions will be discussed
elsewhere.

##  6\. WKB analysis of quantum curves

We consider in this section a meromorphic $SL(2,{\mathbb{C}})$-Higgs bundle
$\left(E_{0},\phi(q)\right)$ associated with a meromorphic quadratic
differential $q\in H^{0}\left(C,K_{C}(D)^{\otimes 2}\right)$ with poles along
an effective divisor $D$ on a curve $C$ of arbitrary genus. This includes the
case of holomorphic Higgs bundles when $g(C)\geq 2$ and $D=\emptyset$. As
before, we have a fixed spin structure $K_{C}^{\frac{1}{2}}$ and a projective
coordinate system on $C$.

Let $\left(E_{\hbar},\nabla^{\hbar}(q)\right)$ be the biholomorphic
quantization result of Theorem 3.15. The corresponding Rees
${\mathcal{D}}_{C}$-module is generated by a single differential operator

(6.1)
$P_{\alpha}(x_{a},\hbar)=\left(\hbar\frac{d}{dx_{\alpha}}\right)^{2}-q_{\alpha}$

on each projective coordinate neighborhood $U_{\alpha}$.

######  Theorem 6.1 (WKB analysis for $SL(2,{\mathbb{C}})$-quantum curves).

Let $q\in H^{0}\left(C,K_{C}(D)^{\otimes 2}\right)$ be a quadratic
differential with poles along an effective divisor $D$ on a curve $C$ of
arbitrary genus. We choose and fix a spin structure and a projective
coordinate system on $C$. Theorem 3.15 tells us that we have a unique Rees
${\mathcal{D}}_{C}$-module ${\mathcal{E}}$ on $C$ as the quantization of the
possibly singular spectral curve

(6.2) $\Sigma=(\eta^{2}-q)_{0}\subset\overline{T^{*}C}.$

Then the PDE topological recursion (5.6) with an appropriate choice of the
initial data provides an all-order WKB analysis for the generator of the Rees
${\mathcal{D}}_{C}$-module ${\mathcal{E}}$ on a small neighborhood in $C$ of
each zero or a pole of $q$ of odd order.

More precisely, the WKB analysis is given as follows.

  * •

Take an arbitrary point $p\in{\text{\rm{supp}}}(\Delta)$ in the discriminant
divisor of (4.5) of odd degree. Choose a small enough simply connected
coordinate neighborhood $U_{\alpha}$ of $C$ with a projective coordinate
$x_{\alpha}$, so that $\tilde{\pi}^{-1}(U_{\alpha})\subset\widetilde{\Sigma}$
in the normalization (4.14) is also simply connected. Let $z$ be a local
coordinate of $\tilde{\pi}^{-1}(U_{\alpha})$ and denote by a function
$x_{\alpha}=x_{\alpha}(z)$ the projection $\tilde{\pi}$.

  * •

The formal WKB expansion we wish to construct is a solution to the equation

(6.3)
$P_{\alpha}(x_{a},\hbar)\psi_{\alpha}(x_{\alpha},\hbar)=\left[\left(\hbar\frac{d}{dx_{\alpha}}\right)^{2}-q_{\alpha}\right]\psi_{\alpha}(x_{\alpha},\hbar)=0$

of the specific form

(6.4)
$\psi_{\alpha}(x_{\alpha},\hbar)=\exp\left(\sum_{m=0}^{\infty}\hbar^{m-1}S_{m}(x_{\alpha})\right)=\exp
F_{\alpha}(x_{\alpha},\hbar).$

Here, $F_{\alpha}(x_{\alpha},\hbar)$ is a formal Laurent series in $\hbar$
starting with the power $-1$. In WKB analysis, this series in $\hbar$ does not
converge.

  * •

Eqn.(6.3) is equivalent to

(6.5)
$\hbar^{2}\frac{d^{2}}{dx_{\alpha}^{2}}F_{\alpha}+\hbar^{2}\frac{dF_{\alpha}}{dx_{a}}\frac{dF_{\alpha}}{dx_{a}}-q_{\alpha}=0.$

We interpret the above equation as an infinite sequence of ordinary
differential equations for each power of $\hbar$.

(6.6)
$\displaystyle\hbar^{0}{\text{-terms}}:\quad\left(S_{0}^{\prime}(x_{a})\right)^{2}-q_{\alpha}=0.$
(6.7) $\displaystyle\hbar^{1}{\text{-terms}}:\quad
S_{0}^{\prime\prime}(x_{\alpha})+2S_{0}^{\prime}(x_{\alpha})S_{1}^{\prime}(x_{\alpha})=0.$
(6.8) $\displaystyle\hbar^{2}{\text{-terms}}:\quad