Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles
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 𝒟-module on a smooth projective algebraic curve, whose semi-classical limit produces the Hitchin spectral cur...
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| description | 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 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 variant of the topological recursion of Eynard-Orantin and Mirzakhani for the context of singular Hitchin spectral curves. We show that a PDE version of topological recursion provides all-order WKB analysis for the Rees 𝒟-modules, defined as the quantization of Hitchin spectral curves associated with meromorphic SL(2, ℂ)-Higgs bundles. Topological recursion can be considered 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 recursion of topological type, agree for holomorphic and meromorphic SL(2, 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.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 036, 53 pages
Interplay between Opers, Quantum Curves,
WKB Analysis, and Higgs Bundles
Olivia DUMITRESCU ab and Motohico MULASE cd
a) Department of Mathematics, University of North Carolina at Chapel Hill,
340 Phillips Hall, CB 3250, Chapel Hill, NC 27599–3250 USA
E-mail: dolivia@unc.edu
b) Simion Stoilow Institute of Mathematics, Romanian Academy,
21 Calea Grivitei Street, 010702 Bucharest, Romania
c) Department of Mathematics, University of California, Davis, CA 95616–8633, USA
E-mail: mulase@math.ucdavis.edu
d) Kavli Institute for Physics and Mathematics of the Universe, The University of Tokyo,
Kashiwa, Japan
Received December 31, 2019, in final form March 12, 2021; Published online April 09, 2021
https://doi.org/10.3842/SIGMA.2021.036
Abstract. Quantum curves were introduced in the physics literature. We develop a mathe-
matical framework for the case associated with Hitchin spectral curves. In this context,
a quantum curve is a Rees 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 defor-
mation families of opers. We propose a variant of the topological recursion of Eynard–
Orantin and Mirzakhani for the context of singular Hitchin spectral curves. We show that
a PDE version of topological recursion provides all-order WKB analysis for the Rees D-
modules, defined as the quantization of Hitchin spectral curves associated with meromorphic
SL(2,C)-Higgs bundles. Topological recursion can be considered as a process of quantization
of Hitchin spectral curves. We prove that these two quantizations, one via the construc-
tion of families of opers, and the other via the PDE recursion of topological type, agree
for holomorphic and meromorphic SL(2,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: 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
2020 Mathematics Subject Classification: 14H15; 14N35; 81T45; 14F10; 14J26; 33C05;
33C10; 33C15; 34M60; 53D37
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, 19, 20, 21,
43, 44, 49, 61, 70, 72, 74]). 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
This paper is a contribution to the Special Issue on Integrability, Geometry, Moduli in honor of Motohico Mu-
lase for his 65th birthday. The full collection is available at https://www.emis.de/journals/SIGMA/Mulase.html
mailto:dolivia@unc.edu
mailto:mulase@math.ucdavis.edu
https://doi.org/10.3842/SIGMA.2021.036
https://www.emis.de/journals/SIGMA/Mulase.html
2 O. Dumitrescu and M. Mulase
geometry, when it is encoded in a holomorphic curve, that gives a generating function of A-
model theory of genus g for all g ≥ 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 [46, 47]. A concise overview of our theory is available in [25]. In Definitions 2.10
and 2.11, we introduce a quantum curve as a Rees 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 D-module to every
Hitchin spectral curve.
The Planck constant ~ is a deformation parameter that appears in the definition of Rees
D-modules. For us, it has a geometric meaning, and is naturally identified with an element
~ ∈ H1(C,KC), (1.1)
where KC is the canonical sheaf over C. The cohomology group H1(C,KC) controls the defor-
mation 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,C)-
Higgs bundle through a concrete construction of an ~-family of SL(r,C)-opers on C, as proved
in Theorem 3.10 for holomorphic case, and in Theorem 3.15 for meromorphic case. The ~-family
interpolates opers and Higgs fields. We then prove, in Theorem 3.11, that the Rees 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 ≥ 2 coming from the Fuchsian uniformization, our construction of opers
is the same as those opers predicted by a conjecture of Gaiotto [41], as explained in Section 3.3.
This conjecture has been solved in [26] (see [18] for a subsequent development.)
It has been noticed that topological recursion of Eynard–Orantin [36, 37] and others [17, 33,
34], and also its more recent generalizations (such as [2, 12, 57] and many papers cited in these
articles), provide another aspect of quantization. A notable one is the remodeling conjecture
of Mariño [60] and his collaborators [14, 15], and its complete solution by mathematicians [39, 40].
(For many earlier contributions to the remodeling conjecture and physics oriented discussions,
we refer to the references cited in [38, 39].) 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 D-modules from Hitchin spectral curves?
Topological recursion was originally developed as a computational mechanism to calculate
the multi-resolvent correlation functions of random matrices (see [17, 33, 36] and references cited
there). As mentioned above, it generates a mirror symmetric B-model counterpart of genus g
A-model for all g ≥ 0. This correspondence has been rigorously established for many examples
(see for example, [13, 23, 28, 31, 32, 35, 39, 40, 63, 64, 65, 66, 70, 72], and others). Yet so far
still no clear geometric relation between topological recursion and quantum curves (in particular,
when they appear as difference operators) has been established.
Another tantalizing subject is the relation between the theory of τ -functions and topologi-
cal recursion/quantum curves. The present paper does not attempt to address this relation.
The subject presented in Section 3 is closely related to the work of [7, 50, 72], in terms of for-
malism and mathematical structures. No deep understanding is offered in this paper.
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 3
Among the earliest striking applications of topological recursion in algebraic geometry, there
are new proofs obtained in [66] for the Witten conjecture on cotangent class intersection numbers
and the λg-conjecture. Indeed, these celebrated formulas are straightforward consequences of the
Laplace transform of a combinatorial formula known as the cut-and-join equations of [42, 75].
Applications of topological recursion to enumerative problems are effective when the spectral
curve in the theory is of genus 0. In this case, the residue calculations required in the formalism
of [36] can be explicitly performed. The computational aspect of the formalism as a tool is
not effective in a more general context, such as when the spectral curve is a high genus non-
hyperelliptic curve, or has singularities.
A novel approach proposed in [27] is the implementation of PDE recursions of topological
type, 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
main 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 similar. 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 difficulties of residue calculations
of higher genus curves, still we do not have a full understanding in this direction. The PDE
recursion of topological type [27, 30], 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,C)-Higgs bundles.
The issue of singular spectral curves is addressed in [30], 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 that for the case of SL(2,C), the nor-
malization process of [30] and the PDE recursion of [27] produce an all-order WKB expansion
for the meromorphic Rees D-modules obtained by quantizing singular Hitchin spectral curves
through the construction of ~-families of opers. In this sense, our result shows that quantization
of Hitchin spectral curves, singular or non-singular, through the PDE recursion of topological
type and construction of ~-family of opers are equivalent, for the case of SL(2,C)-Higgs bundles.
We note a relation between meromorphic Higgs bundles over P1 and Painlevé equations [11].
An application of topological recursion to establishing new results in Painlevé theory and con-
struction of associated quantum curves are presented in [52, 53].
The interplay between Rees D-modules, ~-families of opers, Hitchin spectral curves as semi-
classical limit, Gaiotto’s correspondence, and WKB analysis through PDE recursion of topo-
logical type, 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 [57], in which Kontsevich and Soibel-
man 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 spec-
tral curve, in parallel to the philosophy of [57]. As we show in (3.42), the family of spectral
curves is (re)constructed from our deformation family of Rees 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 D-modules, without studying individual spectral curves. Investigating
a possible connection between the Airy structures of [57] and this paper’s results is a future
4 O. Dumitrescu and M. Mulase
subject. A relation between quantum curves and deformation quantization was first discussed
in [71].
Let us briefly describe our quantization process of this paper now. Our geometric setting
is a smooth projective algebraic curve C over C of an arbitrary genus g = g(C) with a choice
of a spin structure, or a theta characteristic, K
1
2
C . There are 22g choices of such spin structures.
We choose any one of them. Let (E, φ) be an SL(r,C)-Higgs bundle on C with a meromorphic
Higgs field φ. Denote by
T ∗C := P(KC ⊕OC)
π−→ C
the compactified cotangent bundle of C (see [5, 56]), which is a ruled surface on the base C.
The Hitchin spectral curve
Σ
π
!!
i // T ∗C
π
��
C
for a meromorphic Higgs bundle is defined as the divisor of zeros on T ∗C of the characteristic
polynomial of φ:
Σ := Σ(φ) = (det(η − π∗φ))0 , (1.2)
where η ∈ H0(T ∗C, π∗KC) is the tautological 1-form on T ∗C extended as a meromorphic 1-form
on the compactification T ∗C. The morphism π : Σ −→ C is a degree r map.
We denote by MDol the moduli space of holomorphic stable SL(r,C)-Higgs bundles on C
for g ≥ 2. The assignment of the coefficients of the characteristic polynomial (1.2) to (E, φ) ∈
MDol defines the Hitchin fibration
µH : MDol −→ B :=
r⊕
i=2
H0(C,K⊗iC ). (1.3)
With the choice of a spin structure K
1
2
C and Kostant’s principal three-dimensional subgroup TDS
of [58], one constructs a cross-section κ : B −→MDol. We denote by 〈H,X+, X−〉 ⊂ sl(r,C) the
Lie algebra of a principal TDS, where we use the standard representation as traceless matrices
acting on Cr. Thus H is diagonal, X− is lower triangular, X+ = Xt
−, and their relations are
[H,X±] = ±2X±, [X+, X−] = H. (1.4)
The map κ is defined by
B 3 q = (q2, . . . , qr) 7−→ κ(q) ∈
(
E0, φ(q)
)
∈MDol,
where
E0 :=
(
K
1
2
C
)H
, φ(q) := X− +
r∑
`=2
q`X
`−1
+ .
Clearly κ is not a section of the fibration µH in a strict sense, because µH ◦ κ is not the
identity map of B for r ≥ 3. But it is a section in a more general sense that the image of κ
always intersects with every fiber of µH exactly at one point. Note that B is the moduli space
of Hitchin spectral curves associated with holomorphic SL(r,C)-Higgs bundles on C. We use
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 5
an unconventional way of defining the universal family S of spectral curves over B, instead of the
natural family associated with (1.3), rather appealing to the Hitchin section κ, as
S
p
$$
// B × T ∗C
pr1
��
B
, p−1(q) = Σ
(
φ(q)
)
. (1.5)
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 ≥ 2, the Fuchsian projective structure, that appears in our solution [26]
to a conjecture of Gaiotto [41], is a natural choice for our purpose. As we show in Section 3,
there is a unique filtered extension E~ for every ~ ∈ H1(C,KC). For r = 2, E~ is the canonical
extension
0 −→ K
1
2
C −→ E~ −→ K
− 1
2
C −→ 0
associated with
~ ∈ H1(C,KC) = Ext1
(
K
− 1
2
C ,K
1
2
C
)
.
With respect to the projective coordinate system, we can define a one-parameter family of opers(
E~,∇~(q)
)
∈MdeR
for ~ 6= 0, where
∇~(q) := d +
1
~
φ(q), (1.6)
d is the exterior differentiation on C, and MdeR is the moduli space of holomorphic irreducible
SL(r,C)-connections on C. The sum of the exterior differentiation and a Higgs field is not a con-
nection in general. Here, the point is that the original vector bundle E0 is deformed to E~, and
we have chosen a projective coordinate system on C. Therefore, (1.6) makes sense as a global
connection on C in E~.
Note that ~∇~(q) is Deligne’s ~-connection interpolating a connection d + φ(q) and a Higgs
field φ(q). We also note that
(
E~, ~∇~(q)
)
defines a global Rees D-module on C. Its generator
is a globally defined differential operator P on C that acts on K
− r−1
2
C , which is what we call
the quantum curve of the Hitchin spectral curve Σ
(
φ(q)
)
corresponding to q ∈ B. The actual
shape (3.40) of P is quite involved due to non-commutativity of the coordinate of C and dif-
ferentiation. 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 σ∗Σ
(
φ(q)
)
of the Higgs field −φ(q), where
σ : T ∗C −→ T ∗C, σ2 = 1, (1.7)
is the involution defined by the fiber-wise action of −1. This extra sign comes from the difference
of conventions in the characteristic polynomial (1.2) and the connection (1.6).
The above process can be generalized in a straightforward way to meromorphic spectral
data q for a curve C of arbitrary genus. The corresponding connections ∇~(q), and hence the
Rees D-modules, then have regular and irregular singularities.
We note that when we use the Fuchsian projective coordinate system of a curve C of ge-
nus g ≥ 2 and holomorphic SL(r,C)-Higgs bundles, our quantization process is exactly the
6 O. Dumitrescu and M. Mulase
same as the construction of SL(r,C)-opers of [26] that was established by solving a conjecture
of Gaiotto [41].
In Section 6, we perform a PDE variant of topological recursion for the case of meromorphic
SL(2,C)-Higgs bundles. For this purpose, we use a normalization method of [30] for singular
Hitchin spectral curves. We then show that the PDE recursion provides the WKB analysis
for the quantum curve constructed through (1.6). When we deal with a singular spectral curve
Σ ⊂ 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 Σ locally to a non-
singular curve, and the other is to blow up T ∗C and obtain a resolution of singularities Σ̃ of Σ.
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 (1.4).
For example, it selects a particular fixed point of C∗-action on the Hitchin section, which cor-
responds to the ~ → ∞ limit of (1.6). It is counterintuitive, but this limit is the connection
d + X− acting on E~=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 D-modules quan-
tizing Hitchin spectral curves is presented in Section 2. Then in Section 3, we quantize Hitchin
spectral curves as Rees D-modules through a concrete construction of ~-families of holomorphic
and meromorphic SL(r,C)-opers. The semi-classical limit of these resulting opers is calculated.
Since our PDE 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 poly-
nomial of the Higgs field φ. 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,C)-Hitchin spectral curves is established through PDE recursion of topo-
logical type. We thus show that two quantizations procedures, one through ~-family of opers
and the other through PDE recursion, agree for SL(2,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 [61].
2 Rees 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 C. The sheaf DC of differential
operators on C is the subalgebra of the C-linear endomorphism algebra EndC(OC) generated by
the anti-canonical sheaf K−1
C and the structure sheaf OC . Here, K−1
C acts on OC as holomorphic
vector fields, and OC acts on itself by multiplication. Locally every element of DC is written as
DC 3 P (x) =
r∑
`=0
a`(x)
(
d
dx
)r−`
, a`(x) ∈ OC
for some r ≥ 0. For a fixed r, we introduce the filtration by order of differential operators
into DC as follows:
FrDC =
{
P (x) =
r∑
`=0
a`(x)
(
d
dx
)r−` ∣∣∣∣ a`(x) ∈ OC
}
.
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 7
The Rees ring D̃C is defined by
D̃C =
∞⊕
r=0
~rFrDC ⊂ C[[~]]⊗C DC .
An element of D̃C on a coordinate neighborhood U ⊂ C can be written as
P (x, ~) =
r∑
`=0
a`(x, ~)
(
~
d
dx
)r−`
. (2.1)
Definition 2.1 (Rees D-module, cf. [59]). The Rees construction
M̃ =
∞⊕
r=0
~rFrM
associated with a filtered DC-module (F•,M) is a Rees D-module if the compatibility condition
FaDC · FbM⊂ Fa+bM holds.
Let
D =
n∑
j=1
mjpj , mj > 0
be an effective divisor on C. The point set {p1, . . . , pn} ⊂ C is the support of D. A meromorphic
Higgs bundle with poles at D is a pair (E, φ) consisting of an algebraic vector bundle E on C
and a Higgs field
φ : E −→ KC(D)⊗OC E.
Since the cotangent bundle
T ∗C = Spec
(
Sym
(
K−1
C
))
is the total space of KC , we have the tautological 1-form η ∈ H0(T ∗C, π∗KC) on T ∗C coming
from the projection
T ∗C ←−−−− π∗KC
π
y
C ←−−−− KC .
The natural holomorphic symplectic form of T ∗C is given by −dη. The compactified cotangent
bundle of C is a ruled surface defined by
T ∗C := P(KC ⊕OC) = Proj
( ∞⊕
n=0
(
K−nC · I0 ⊕K−n+1
C · I ⊕ · · · ⊕K0
C · In
))
,
where I represents 1 ∈ OC being considered as a degree 1 element. The divisor at infinity
C∞ := P(KC ⊕ {0})
is reduced in the ruled surface and supported on the subset P(KC⊕OC)\T ∗C. The tautological
1-form η extends on T ∗C as a meromorphic 1-form with simple poles along C∞. Thus the divisor
of η in T ∗C is given by
(η) = C0 − C∞,
where C0 is the zero section of T ∗C.
8 O. Dumitrescu and M. Mulase
The relation between the sheaf DC and the geometry of the compactified cotangent bundle
T ∗C is the following. First we have
Spec
( ∞⊕
m=0
FmDC
/
Fm−1DC
)
= Spec
( ∞⊕
m=0
K−mC
)
= T ∗C.
Let us denote by grmDC = FmDC
/
Fm−1DC . By writing I = 1 ∈ H0(C,DC), we then have
T ∗C = Proj
( ∞⊕
m=0
(
grmDC · I0 ⊕ grm−1DC · I ⊕ grm−2DC · I⊗2 ⊕ · · · ⊕ gr0DC · I⊗m
))
.
Definition 2.2 (spectral curve). A spectral curve of degree r is a divisor Σ in T ∗C such that
the projection π : Σ −→ C defined by the restriction
Σ
π
!!
i // T ∗C
π
��
C
is a finite morphism of degree r. The spectral curve of a Higgs bundle (E, φ) is the divisor
of zeros
Σ = (det(η − π∗φ))0
on T ∗C of the characteristic polynomial det(η − π∗φ). Here,
π∗φ : π∗E −→ π∗(KC(D))⊗OP(KC⊕OC )
π∗E.
Remark 2.3. The Higgs field φ is holomorphic on C \ supp(D). Thus we can define the divisor
of zeros
Σ◦ =
(
det(η − π∗(φ|C\supp(D)))
)
0
of the characteristic polynomial on T ∗(C \ supp(D)). The spectral curve Σ is the complex
topology closure of Σ◦ with respect to the compactification
T ∗(C \ supp(D)) ⊂ T ∗C.
A left DC-module E on C is naturally an OC-module with a C-linear integrable (i.e., flat)
connection ∇ : E −→ KC ⊗OC E . The construction goes as follows:
∇ : E α−−−−→ DC ⊗OC E
∇D⊗id−−−−→ (KC ⊗OC DC)⊗OC E
β⊗id−−−−→ KC ⊗OC E , (2.2)
where
� α is the natural inclusion E 3 v 7−→ 1⊗ v ∈ DC ⊗OC E ,
� ∇D : DC −→ KC ⊗OC DC is the connection defined by the C-linear left-multiplication
operation of K−1
C on DC , which satisfies the derivation property
∇D(f · P ) = f · ∇D(P ) + df · P ∈ KC ⊗OC DC (2.3)
for f ∈ OC and P ∈ DC , and
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 9
� β is the canonical right DC-module structure in KC defined by the Lie derivative of vector
fields.
If we choose a local coordinate neighborhood U ⊂ C with a coordinate x, then (2.3) takes the
following form. Let us denote by P ′ = [d/dx, P ], and define
∇d
dx
(P ) := P · d
dx
+ P ′.
Then we have
∇d
dx
(f · P ) = f · ∇d
dx
(P ) +
df
dx
· P.
The connection ∇ of (2.2) is integrable because d2 = 0. Actually, the statement is true for any
dimensions. We note that there is no reason for E to be coherent as an OC-module.
Conversely, if an algebraic vector bundle E on C of rank r admits a holomorphic connection
∇ : E −→ KC ⊗ E, then E acquires the structure of a DC-module. This is because ∇ is
automatically flat, and the covariant derivative ∇X for X ∈ K−1
C satisfies
∇X(fv) = f∇X(v) +X(f)v (2.4)
for f ∈ OC and v ∈ E. A repeated application of (2.4) makes E a DC-module. The fact that
every DC-module on a curve is principal implies that for every point p ∈ C, there is an open
neighborhood p ∈ U ⊂ C and a linear differential operator P of order r on U , called a generator,
such that E|U ∼= DU/DUP . 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 DU/DUP
is OU -coherent.
Definition 2.4 (formal ~-connection, cf. [3]). A formal ~-connection on a vector bundle E −→ C
is a C[[~]]-linear homomorphism
∇~ : C[[~]]⊗ E −→ C[[~]]⊗KC ⊗OC E
subject to the derivation condition
∇~(f · v) = f∇~(v) + ~df ⊗ v,
where f ∈ OC ⊗ C[[~]] and v ∈ C[[~]]⊗ E.
When we consider holomorphic dependence of a quantum curve with respect to the quanti-
zation parameter ~, we need to use a particular ~-deformation family of vector bundles. We will
discuss the holomorphic case in Section 3, where we explain how (1.1) appears in our quanti-
zation.
Remark 2.5. The classical limit of a formal ~-connection is the evaluation ~ = 0 of ∇~, which
is simply an OC-module homomorphism
∇0 : E −→ KC ⊗OC E,
i.e., a holomorphic Higgs field in the vector bundle E.
Remark 2.6. An OC⊗C[[~]]-coherent D̃C-module is equivalent to a vector bundle on C equipped
with an ~-connection.
10 O. Dumitrescu and M. Mulase
In analysis, the semi-classical limit of a differential operator P (x, ~) of the form (2.1) is
a function defined by
lim
~→0
(
e−
1
~S0(x)P (x, ~)e
1
~S0(x)
)
=
r∑
`=0
a`(x, 0)(S′0(x))r−`, (2.5)
where S0(x) ∈ OC(U). The equation
lim
~→0
(
e−
1
~S0(x)P (x, ~)e
1
~S0(x)
)
= 0 (2.6)
then determines the first term of the singular perturbation expansion, or the WKB asymptotic
expansion,
ψ(x, ~) = exp
( ∞∑
m=0
~m−1Sm(x)
)
(2.7)
of a solution ψ(x, ~) to the differential equation
P (x, ~)ψ(x, ~) = 0
on U . We note that the expression (2.7) is never meant to be a convergent series in ~.
Since dS0(x) is a local section of T ∗C on U ⊂ C, y = S′0(x) gives a local trivialization
of T ∗C|U , with y ∈ T ∗xC a fiber coordinate. Then (2.5) and (2.6) give an equation
r∑
`=0
a`(x, 0)yr−` = 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 ⊂ 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, ~) =
r∑
`=0
a`(x, ~)
(
~
d
dx
)r−`
of the Rees ring D̃C of the sheaf of differential operators DC on U is the holomorphic function
r∑
`=0
a`(x, 0)yr−`
defined on T ∗C|U .
Definition 2.8 (semi-classical limit of a Rees D-module). Suppose a Rees D̃C-module M̃ glob-
ally defined on C is written as
M̃(U) = D̃C(U)
/
D̃C(U)PU
on every coordinate neighborhood U ⊂ C with a differential operator PU of the form (2.1).
Using this expression (2.1) for PU , we construct a meromorphic function
pU (x, y) =
r∑
`=0
a`(x, 0)yr−` (2.8)
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 11
on T ∗C|U , where y is the fiber coordinate of T ∗C, which is trivialized on U . Define
ΣU = (pU (x, y))0
as the divisor of zero of the function pU (x, y). If ΣU ’s glue together to a spectral curve Σ ⊂ T ∗C,
then we call Σ the semi-classical limit of the Rees D̃C-module M̃.
Remark 2.9. For the local equation (2.8) to be consistent globally on C, the coefficients of (2.1)
have to satisfy
a`(x, 0) ∈ Γ
(
U,K⊗`C
)
.
Definition 2.10 (quantum curve for holomorphic Higgs bundle). A quantum curve associated
with the spectral curve Σ ⊂ T ∗C of a holomorphic Higgs bundle on a projective algebraic
curve C is a Rees D̃C-module E whose semi-classical limit is Σ.
The main reason we wish to extend our framework to meromorphic connections is that there
are no non-trivial holomorphic connections on P1, whereas many important classical examples
of differential equations are naturally defined over P1 with regular and irregular singularities.
A C-linear homomorphism
∇ : E −→ KC(D)⊗OC E
is said to be a meromorphic connection with poles along an effective divisor D if
∇(f · v) = f∇(v) + df ⊗ v
for every f ∈ OC and v ∈ E. Let us denote by
OC(∗D) := lim
−→
OC(mD), E(∗D) := E ⊗OC OC(∗D).
Then ∇ extends to
∇ : E(∗D) −→ KC(∗D)⊗OC(∗D) E(∗D).
Since ∇ is holomorphic on C \supp(D), it induces a DC\supp(D)-module structure in E|C\supp(D).
The DC-module direct image Ẽ = j∗
(
E|C\supp(D)
)
associated with the open inclusion map
j : C \ supp(D) −→ C is then naturally isomorphic to
Ẽ = j∗
(
E|C\supp(D)
) ∼= E(∗D) (2.9)
as a DC-module. Equation (2.9) is called the meromorphic extension of the DC\supp(D)-module
E|C\supp(D).
Let us take a local coordinate x of C, this time around a pole pj ∈ supp(D). If a generator P̃
of Ẽ near x = 0 has a local expression
P̃
(
x,
d
dx
)
= xk
r∑
`=0
b`(x)
(
x
d
dx
)r−`
around pj with locally defined holomorphic functions b`(x), b0(0) 6= 0, and an integer k ∈ Z,
then P̃ has a regular singular point at pj . Otherwise, pj is an irregular singular point of P̃ .
12 O. Dumitrescu and M. Mulase
Definition 2.11 (quantum curve for a meromorphic Higgs bundle). Let (E, φ) be a meromor-
phic Higgs bundle defined over a projective algebraic curve C of any genus with poles along
an effective divisor D, and Σ ⊂ T ∗C its spectral curve. A quantum curve associated with Σ
is the meromorphic extension of a Rees D̃C-module E on C \ supp(D) such that the complex
topology closure of its semi-classical limit Σ◦ ⊂ T ∗C|C\supp(D) in the compactified cotangent
bundle T ∗C agrees with Σ.
In Section 3, we prove that every Hitchin spectral curve associated with a holomorphic
or a meromorphic SL(r,C)-Higgs bundle has a quantum curve.
Remark 2.12. We remark that several examples of quantum curves that are constructed in [13,
32, 64], for various Hurwitz numbers and Gromov–Witten theory of P1, do not fall into our
definition in terms of Rees 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 a Hitchin spectral curve, using a par-
ticular choice of isomorphism between a Hitchin section and the moduli of opers. The quan-
tum deformation parameter ~, originated in physics as the Planck constant, is a purely formal
parameter in WKB analysis. Since we will be using the PDE recursion (5.3) for the analysis
of quantum curves, ~ 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 holomorphi-
cally on ~. 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 ~-connections
as quantum curves, and the geometric interpretation of ~ given in (1.1). The key concept is
opers of Beilinson–Drinfeld [6]. Although a vast generalization of the current paper is possible,
we restrict our attention to SL(r,C)-opers for an arbitrary r ≥ 2 in this paper.
In this section, most of the time C is a smooth projective algebraic curve of genus g ≥ 2
defined over C, unless otherwise specified.
3.1 Holomorphic SL(r,C)-opers and quantization of Higgs bundles
We first recall projective structures on C following Gunning [45]. Recall that every compact Rie-
mann surface has a projective structure subordinating the given complex structure. A complex
projective coordinate system is a coordinate neighborhood covering
C =
⋃
α
Uα
with a local coordinate xα of Uα such that for every Uα ∩ Uβ, we have a Möbius coordinate
transformation
xα =
aαβxβ + bαβ
cαβxβ + dαβ
,
[
aαβ bαβ
cαβ dαβ
]
∈ SL(2,C).
Since we solve differential equations on C, we always assume that each coordinate neighbor-
hood Uα is simply connected. In what follows, we choose and fix a projective coordinate system
on C. Since
dxα =
1
(cαβxβ + dαβ)2
dxβ,
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 13
the transition function for the canonical line bundle KC of C is given by the cocycle{
(cαβxβ + dαβ)2
}
on Uα ∩ Uβ.
We choose and fix, once and for all, a theta characteristic, or a spin structure, K
1
2
C such that(
K
1
2
C
)⊗2 ∼= KC . Let {ξαβ} denote the 1-cocycle corresponding to K
1
2
C . Then we have
ξαβ = ±(cαβxβ + dαβ). (3.1)
The choice of ± here is an element of H1(C,Z/2Z) = (Z/2Z)2g, indicating that there are 22g
choices for spin structures in C.
The significance of the projective coordinate system lies in the fact that ∂2
βξαβ = 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 the projective coordinate system is the vanishing
of Schwarzian derivatives, as explained in [25]. 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, φ) is stable if for every vector subbundle F ⊂ E that is
invariant with respect to φ, i.e., φ : F −→ F ⊗KC , the slope condition
degF
rankF
<
degE
rankE
holds. The moduli space of stable Higgs bundles is constructed [73]. An SL(r,C)-Higgs bundle is
a pair (E, φ) with a fixed isomorphism detE = OC and trφ = 0. We denote byMDol the moduli
space of stable holomorphic SL(r,C)-Higgs bundles on C. Hitchin [46] defines a holomorphic
fibration
µH : MDol 3 (E, φ) 7−→ det(η − π∗φ) ∈ B, B :=
r⊕
`=2
H0
(
C,K⊗`C
)
,
that induces the structure of an algebraically completely integrable Hamiltonian system inMDol.
With the choice of a spin structure K
1
2
C , we have a natural section κ : B ↪→ MDol defined by
utilizing Kostant’s principal three-dimensional subgroup (TDS) [58] as follows.
First, let
q = (q2, q3, . . . , qr) ∈ B =
r⊕
`=2
H0
(
C,K⊗`C
)
be an arbitrary point of the Hitchin base B. Define
X− :=
[√
si−1δi−1,j
]
ij
=
0 0 · · · 0 0√
s1 0
0
√
s2 0
...
. . .
...
0 0 · · · √sr−1 0
, X+ := Xt
−,
H := [X+, X−],
where si := i(r − i). H is a diagonal matrix whose (i, i)-entry is si − si−1 = r − 2i + 1, with
s0 = sr = 0. The Lie algebra 〈X+, X−, H〉 ∼= sl(2,C) is the Lie algebra of the principal TDS
in SL(r,C).
14 O. Dumitrescu and M. Mulase
Lemma 3.1. Define a Higgs bundle (E0, φ(q)) consisting of a vector bundle
E0 :=
(
K
1
2
C
)⊗H
=
r⊕
i=1
(
K
1
2
C
)⊗(r−2i+1)
(3.2)
and a Higgs field
φ(q) := X− +
r∑
`=2
q`X
`−1
+ . (3.3)
Then it is a stable SL(r,C)-Higgs bundle. The Hitchin section is defined by
κ : B 3 q 7−→ (E0, φ(q)) ∈MDol, (3.4)
which gives a biholomorphic map between B and κ(B) ⊂MDol.
Proof. We first note that X− : E0 −→ E0 ⊗KC is a globally defined End0(E0)-valued 1-form,
since it is a collection of constant maps
√
si :
(
K
1
2
C
)⊗(r−2i+1) =−→
(
K
1
2
C
)⊗(r−2(i+1)+1)
⊗KC . (3.5)
Similarly, since X`−1
+ is an upper-diagonal matrix with non-zero entries along the (` − 1)-th
upper diagonal, we have
q` :
(
K
1
2
C
)⊗(r−2i+1)
−→
(
K
1
2
C
)⊗(r−2i+1+2`)
=
(
K
1
2
C
)⊗(r−2(i−`+1)+1)
⊗KC .
Thus φ(q) : E0 −→ E0 ⊗KC is globally defined as a Higgs field in E0. The Higgs pair is stable
because no subbundle of E0 is invariant under φ(q), unless q = 0. And when q = 0, the invariant
subbundles all have positive degrees, since g ≥ 2. �
The image κ(B) is a holomorphic Lagrangian submanifold of a holomorphic symplectic
space MDol.
To define ~-connections holomorphically depending on ~, we need to construct a one-para-
meter holomorphic family of deformations of vector bundles
E~ −−−−→ Ey y
C × {~} −−−−→ C ×H1(C,KC)
and a C-linear first-order differential operator
~∇~ : E~ −→ E~ ⊗KC
depending holomorphically on ~ ∈ H1(C,KC) ∼= C for ~ 6= 0. Let us introduce the notion
of filtered extensions.
Definition 3.2 (filtered extension). A one-parameter family of filtered holomorphic vector
bundles
(
F •~ , E~
)
on C with a trivialized determinant det(E~) ∼= OC is a filtered extension
of the vector bundle E0 of (3.2) parametrized by ~ ∈ H1(C,KC) if the following conditions
hold:
� E~ has a filtration
0 = F r~ ⊂ F r−1
~ ⊂ F r−2
~ ⊂ · · · ⊂ F 0
~ = E~.
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 15
� The second term is given by
F r−1
~ =
(
K
1
2
C
)⊗(r−1)
. (3.6)
� For every i = 1, 2, . . . , r − 1, there is an OC-module isomorphism
F i~
/
F i+1
~
∼−→
(
F i−1
~
/
F i~
)
⊗KC . (3.7)
Remark 3.3. Since we need to identify a deformation parameter ~ and extension classes,
we make the natural identification
Ext1(E,F ) = H1(C,E∗ ⊗ F )
for every pair of vector bundles E and F . We also identify Ext1(E,F ) as the class of extensions
0 −→ F −→ V −→ E −→ 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 characte-
ristic K
1
2
C and a non-zero element ~ ∈ H1(C,KC), there is a unique non-trivial filtered exten-
sion
(
F •~ , E~
)
of E0.
Proof. First let us examine the case of r = 2 to see how things work. Since
~ ∈ H1(C,KC) = Ext1
(
K
− 1
2
C ,K
1
2
C
)
∼= C,
we have a unique extension
0 −→ K
1
2
C −→ E~ −→ K
− 1
2
C −→ 0 (3.8)
corresponding to ~. Obviously
K
1
2
C
∼−→
(
E~
/
K
1
2
C
)
⊗KC ,
which proves (3.7). We also note that as a vector bundle of rank 2, we have the isomorphism
E~ ∼=
{
E1, ~ 6= 0,
E0, ~ = 0.
Now consider the general case. We use the induction on i = r − 1, r − 2, r − 3, . . . , 0, in the
reverse direction to construct each term of the filtration F i~ subject to (3.6) and (3.7). The base
case i = r − 1 is the following. Since
0 −→ F r−1
~ −→ F r−2
~ −→ F r−2
~ /F r−1
~ −→ 0
and
F r−2
~ /F r−1
~
∼= F r−1
~ ⊗K−1
C
∼=
(
K
1
2
C
)⊗(r−1)
⊗K−1
C
16 O. Dumitrescu and M. Mulase
from (3.6) and (3.7), F r−2
~ is determined by the class ~ in
Ext1
(
F r−2
~ /F r−1
~ , F r−1
~
)
= H1
(
C,F r−1
~ ⊗
(
F r−2
~ /F r−1
~
)−1) ∼= H1(C,KC).
Assume that for a given i+ 1, we have
H1
(
C,Fn~ ⊗
(
F r−1
~
)−1 ⊗K⊗(−n+r)
C
) ∼= H1(C,KC), (3.9)
H1
(
C,Fn~ ⊗
(
F r−1
~
)−1 ⊗K⊗(−n+m+r)
C
)
= 0, m ≥ 1, (3.10)
for every n in the range i + 1 ≤ n ≤ r − 1. We wish to prove that the same relation holds
for n = i.
The sequence of isomorphisms (3.7) implies that
F i−1
~ /F i~
∼=
(
F i~/F
i+1
~
)
⊗K−1
C
∼= F r−1
~ ⊗K⊗(i−r)
C =
(
K
1
2
C
)⊗(r−1)
⊗K⊗(i−r)
C
∼= K
⊗
(
i− r+1
2
)
C .
Then F i−1
~ as an extension
0 −→ F i~ −→ F i−1
~ −→ F r−1
~ ⊗K⊗(i−r)
C −→ 0 (3.11)
is determined by a class in
Ext1
(
F r−1
~ ⊗K⊗(i−r)
C , F i~
)
= H1
(
C,F i~ ⊗
(
F r−1
~
)−1 ⊗K⊗(−i+r)
C
)
.
The exact sequence
0 −→ Fn~ −→ Fn−1
~ −→ F r−1
~ ⊗K⊗(n−r)
C −→ 0
implies that
0 −→ Fn~ ⊗
(
F r−1
~
)−1⊗K⊗(−n+m+r)
C −→ Fn−1
~ ⊗
(
F r−1
~
)−1⊗K⊗(−n+m+r)
C −→ K⊗mC −→ 0
for every m ≥ 1. Taking the cohomology long exact sequence, we obtain
H1
(
C,Fn−1
~ ⊗
(
F r−1
~
)−1 ⊗K⊗(−n+1+r)
C
) ∼= H1(C,KC)
for m = 1, which proves (3.9) for n = i. Similarly,
H1
(
C,Fn−1
~ ⊗
(
F r−1
~
)−1 ⊗K⊗(−(n−1)+(m−1)+r)
C
)
= H1
(
C,Fn−1
~ ⊗
(
F r−1
~
)−1 ⊗K⊗(−n+m+r)
C
) ∼= H1(C,Km
C ) = 0
for m ≥ 2, which is (3.10) for n = i. By induction on i in the decreasing direction, we have
established that the class ~ determines the unique extension (3.11) for every i. �
Definition 3.5 (SL(r,C)-opers). A point (E,∇) ∈ MdeR, i.e., an irreducible holomorphic
SL(r,C)-connection ∇ : E −→ E ⊗KC acting on a vector bundle E, is an SL(r,C)-oper if the
following conditions are satisfied.
Filtration. There is a filtration F • by vector subbundles
0 = F r ⊂ F r−1 ⊂ F r−2 ⊂ · · · ⊂ F 0 = E. (3.12)
Griffiths transversality. The connection respects the filtration:
∇|F i : F i −→ F i−1 ⊗KC , i = 1, . . . , r. (3.13)
Grading condition. The connection induces OC-module isomorphisms
∇ : F i/F i+1 ∼−→
(
F i−1/F i
)
⊗KC , i = 1, . . . , r − 1. (3.14)
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 17
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 ~ ∈ H1(C,KC)-dependence in the filtered extensions. For example,
the extension E~ of (3.8) is given by a system of transition functions
E~ ←→
{[
ξαβ ~σαβ
0 ξ−1
αβ
]}
on each Uα ∩Uβ. The cocycle condition for the transition functions translates into a condition
σαγ = ξαβσβγ + σαβξ
−1
βγ . (3.15)
The application of the exterior differentiation d to the cocycle condition ξαγ = ξαβξβγ yields
dξαγ
dxγ
dxγ =
dξαβ
dxβ
dxβξβγ + ξαβ
dξβγ
dxγ
dxγ .
Noticing that
ξ2
αβ =
dxβ
dxα
,
we see that
σαβ :=
dξαβ
dxβ
= ∂βξαβ (3.16)
solves (3.15). We note that[
ξαβ ~σαβ
ξ−1
αβ
]
= exp
(
log ξαβ
[
1 0
0 −1
])
exp
(
~∂β log ξαβ
[
0 1
0 0
])
.
Therefore, in the multiplicative sense, the extension class is determined by ∂β log ξαβ.
Lemma 3.6. The extension class σαβ of (3.16) defines a non-trivial extension (3.8).
Proof. The cohomology long exact sequence
H1(C,C) −−−−→ H1(C,OC) −−−−→ H1(C,KC)
∼−−−−→ H2(C,C)y y ∥∥∥
H1(C,C∗) −−−−→ H1(C,O∗C)
d log−−−−→ H1(C,KC)y0
yc1 y
H2(C,Z) H2(C,Z) −−−−→ 0
associated with
0 0y y
0 −−−−→ Z Z −−−−→ 0y y y
0 −−−−→ C −−−−→ OC
d−−−−→ KC −−−−→ 0y y y
0 −−−−→ C∗ −−−−→ O∗C
d log−−−−→ KC −−−−→ 0y y y
0 0 0
18 O. Dumitrescu and M. Mulase
tells us that {σαβ} corresponds to the image of {ξab} via the map
H1(C,O∗C)
d log−−−−→ H1(C,KC).
From the exact sequence, we see that if d log{ξαβ} = 0 ∈ H1(C,KC), then it comes from a class
in H1(C,C∗), which is the moduli space of line bundles with holomorphic connections (see, for
example, [9]). Hence the first Chern class of the theta characteristic c1
(
K
1
2
C
)
should be 0. But
it is g − 1, not 0, since g(C) ≥ 2. �
Remark 3.7. The same exact sequences in the above proof were used in [9] for constructing
Bloch regulators of the algebraic K2-group. The torsion property of the Steinberg symbol
of generators of K2(Σ) and quantizability of a spectral curve Σ was first discussed in [44] for the
case of difference equations.
The class {σαβ} of (3.16) gives a natural isomorphism H1 (C,KC) ∼= C. We identify the
deformation parameter ~ ∈ C with the cohomology class {~σαβ} ∈ H1 (C,KC) = C. Let
q = (q2, q3, . . . , qr) ∈ B =
r⊕
`=2
H0
(
C,K⊗`C
)
be an arbitrary point of the Hitchin base B. We trivialize the line bundle K⊗`C with respect
to our projective coordinate chart C =
⋃
α Uα, and write each q` as {(q`)α} that satisfies the
transition relation
(q`)α = (q`)βξ
2`
αβ.
The transition function of the vector bundle E0 is given by
ξHαβ = exp(H log ξαβ).
Since X− : E0 −→ E0 ⊗KC is a global Higgs field, its local expressions {X−dxα} with respect
to the projective coordinate system satisfies the transition relation
X−dxα = exp(H log ξαβ)X−dxβ exp(−H log ξαβ) (3.17)
on every Uα ∩ Uβ. The same relation holds for the Higgs field φ(q) as well:
φα(q)dxα = exp(H log ξαβ)φβ(q)dxβ exp(−H log ξαβ). (3.18)
We have the following:
Theorem 3.8 (construction of SL(r,C)-opers). On each Uα ∩ Uβ define a transition function
f~αβ := exp(H log ξαβ) exp
(
~∂β log ξαβX+
)
, (3.19)
where ∂β = d
dxβ
, and ~∂β log ξαβ ∈ H1(C,KC). Then
� The collection
{
f~αβ
}
satisfies the cocycle condition
f~αβf
~
βγ = f~αγ ,
hence it defines a holomorphic bundle on C. It is exactly the filtered extension E~ of Pro-
position 3.4.
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 19
� The locally defined differential operator
∇~
α(0) := d +
1
~
X−dxa
for every ~ 6= 0 forms a global holomorphic connection in E~, i.e.,
1
~
X−dxα =
1
~
f~αβX−dxβ
(
f~αβ
)−1 − df~αβ ·
(
f~αβ
)−1
. (3.20)
� Every point (E0, φ(q)) ∈ κ(B) ⊂ MDol of the Hitchin section (3.4) gives rise to a one-
parameter family of SL(r,C)-opers
(
E~,∇~(q)
)
∈MdeR. In other words, the locally defined
differential operator
∇~
α(q) := d +
1
~
φα(q)dxα (3.21)
for every ~ 6= 0 determines a global holomorphic connection
∇~
α(q) = f~αβ∇~
β(q)
(
f~αβ
)−1
(3.22)
in E~ satisfying the oper conditions.
� Deligne’s ~-connection(
E~, ~∇~(q)
)
(3.23)
interpolates the Higgs pair and the oper, i.e., at ~ = 0, the family (3.23) gives the Higgs
pair (E, φ(q)) ∈MDol, and at ~ = 1 it gives an SL(r,C)-oper
(
E1,∇1(q)
)
∈MdeR.
� After a suitable gauge transformation depending on ~, the ~→∞ limit of the oper ∇~(q)
exists and is equal to ∇~=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 ∈ C. Then
eAeBe−A = eB exp c. (3.24)
From this formula,
dxβ
dxα
= ξ2
αβ, and [H,X+] = 2X+, we calculate
f~αβ
(
f~γβ
)−1
= exp(H log ξαβ) exp(~∂β log ξαβX+) exp(−~∂β log ξγβX+) exp(−H log ξγβ)
= exp(H log ξαβ) exp(~∂β log ξαγX+) exp(−H log ξγβ)
= exp(H log ξαβ) exp(~ξ2
βγ∂γ log ξαγX+) exp(−H log ξγβ)
= exp(H log ξαγ) exp(H log ξγβ) exp(~ξ2
βγ∂γ log ξαγX+) exp(−H log ξγβ)
= exp(H log ξαγ) exp(~ξ2
γβξ
2
βγ∂γ log ξαγX+) = f~αγ .
Hence the cocycle condition f~αβ = f~αγf
~
γβ follows. We note that the factor exp c in (3.24)
produces exactly the cocycle ξ2
αβ corresponding to the canonical sheaf KC .
To prove (3.20), we use the power series expansion of the adjoint action
e~ABe−~A =
∞∑
n=0
1
n!
~n(adA)n(B) :=
∞∑
n=0
1
n!
~n
n︷ ︸︸ ︷
[A, [A, [· · · , [A,B] · · · ]]]. (3.25)
20 O. Dumitrescu and M. Mulase
We then find
f~αβX−
(
f~αβ
)−1
= exp(H log ξαβ) exp(~∂β log ξαβX+)X− exp(−~∂β log ξαβX+) exp(−H log ξαβ)
= exp(H log ξαβ)X− exp(−H log ξαβ) + ~∂β log ξαβH
− ~2(∂β log ξαβ)2 exp(H log ξαβ)X+ exp(−H log ξαβ),
and
∂βf
~
αβ
(
f~αβ
)−1
= ∂β log ξαβH − ~(∂β log ξαβ)2 exp(H log ξαβ)X+ exp(−H log ξαβ).
Here, we have used the formula
∂β∂β log ξαβ = ∂β
(
ξ−1
αβ∂βξαβ
)
= −ξ−2
αβ (∂βξαβ)2 = −(∂β log ξαβ)2,
which follows from (3.1). Therefore, noticing (3.17), we obtain(
1
~
f~αβX−
(
f~αβ
)−1 − ∂βf~αβ
(
f~αβ
)−1
)
dxβ =
1
~
exp(H log ξαβ)X−dxβ exp(−H log ξαβ)
=
1
~
X−dxα.
To prove (3.22), we need, in addition to (3.20), the following relation:
r∑
`=2
(q`)αX
`−1
+ dxα = f~αβ
r∑
`=2
(q`)βX
`−1
+ dxβ
(
f~αβ
)−1
. (3.26)
But (3.26) is obvious from (3.18) and (3.19).
Noticing that f~αβ is an upper-triangular matrix, we denote by
(
f~αβ
)
i
the principal truncation
of f~αβ to the first (r − i+ 1)× (r − i+ 1) upper-left corner. For example,(
f~αβ
)
r
=
[
ξr−1
αβ
]
,(
f~αβ
)
r−1
=
[
ξr−1
αβ
ξr−3
αβ
][
1 ~√s1ξ
−1
αβσαβ
1
]
=
[
ξr−1
αβ ~√s1ξ
r−2
αβ σαβ
ξr−3
αβ
]
,
(
f~αβ
)
r−2
=
ξ
r−1
αβ ~√s1ξ
r−2
αβ σαβ
1
2~
2√s1s2ξ
r−3
αβ σ2
αβ
ξr−3
αβ ~√s2ξ
r−4
αβ σαβ
ξr−5
αβ
, etc.
They all satisfy the cocycle condition with respect to the projective structure, and define a se-
quence of vector bundles F i~ on C. From the shape of matrices we see that these vector bundles
give the filtered extension (F •~ , E~) of Proposition 3.4, and hence satisfies the requirement (3.12).
Since the connection ∇~(q) has non-zero lower-diagonal entries in its matrix representation com-
ing from X−, F i~ is not invariant under ∇~(q). But since X− has non-zero entries exactly along
the first lower-diagonal, (3.13) holds. The isomorphism (3.14) is a consequence of (3.5), since
si = i(r − i) 6= 0 for i = 1, . . . , r − 1.
Finally, the gauge transformation of ∇~(q) by a bundle automorphism
~−
H
2 =
~
− r−1
2
. . .
~
r−1
2
(3.27)
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 21
on each coordinate neighborhood Uα gives
d +
1
~
φ(q) 7−→ ~−
H
2
(
d +
1
~
φ(q)
)
~
H
2 = d +X− +
r∑
`=2
q`
~`
X`−1
+ . (3.28)
This is because
~−
H
2 X−~
H
2 = ~X− and ~−
H
2 X`
+~
H
2 = ~−`X`
+,
which follows from the adjoint formula (3.25). Therefore,
lim
~→∞
∇~(q) ∼ d +X− = ∇~=1(0),
where the symbol ∼ 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 (3.1). Only in such a coordinate, the definition of the global connection (3.22) makes
sense. This is due to the vanishing of the second derivative of ξαβ.
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 ≥ 2 with a chosen projective coordinate system subordinating its
complex structure. We denote by MDol the moduli space of stable holomorphic SL(r,C)-Higgs
bundles over C, and by MdeR the moduli space of irreducible holomorphic SL(r,C)-connections
on C. For a fixed theta characteristic K
1
2
C , we have a Hitchin section κ(B) ⊂ MDol of (3.4).
We denote by Op ⊂MdeR the moduli space of SL(r,C)-opers with the condition that the second
term of the filtration is given by F r−1 = K
r−1
2
C . Then the map
MDol ⊃ κ(B) 3 (E0, φ(q))
γ7−→
(
E~,∇~(q)
)
∈ Op ⊂MdeR (3.29)
evaluated at ~ = 1 is a biholomorphic map with respect to the natural complex structures induced
from the ambient spaces.
The biholomorphic quantization (3.29) is also C∗-equivariant. The λ ∈ C∗ action on the
Hitchin section is defined by φ 7−→ λφ. The oper corresponding to (E0, λφ(q)) ∈ κ(B) is
d + λ
~φ(q).
Proof. The C∗-equivariance follows from the same argument of the gauge transformation (3.27),
(3.28). Since the Hitchin section κ is not the section of the Hitchin fibration µH , we need the
gauge transformation. The action φ 7−→ λφ on the Hitchin section induces a weighted action
B 3 (q2, q3, . . . , qr) 7−→
(
λ2q2, λ
3q3, . . . , λ
rqr
)
∈ B
through µH . Then we have the gauge equivalence via the gauge transformation
(
λ
~
)H
2 :
d +
λ
~
φ(q) ∼
(
λ
~
)H
2
(
d +
λ
~
φ(q)
)(
λ
~
)−H
2
= d +X− +
r∑
`=2
λ`q`
~`
X`−1
+ . �
22 O. Dumitrescu and M. Mulase
3.2 Semi-classical limit of SL(r,C)-opers
A holomorphic connection on a compact Riemann surface C is automatically flat. Therefore,
it defines a D-module over C. Continuing the last subsection’s conventions, let us fix a projective
coordinate system on C, and let (E0, φ(q)) = κ(q) be a point on the Hitchin section of (3.4).
It uniquely defines an ~-family of opers
(
E~,∇~(q)
)
.
In this subsection, we establish that the ~-connection ~∇~(q) defines a family of Rees D-
modules on C parametrized by B such that the semi-classical limit of the family agrees with the
family of spectral curves (1.5) over B.
To calculate the semi-classical limit, let us trivialize the vector bundle E~ on each simply
connected coordinate neighborhood Uα with coordinate xα of the chosen projective coordinate
system. A flat section Ψα of E~ over Uα is a solution of
~∇~
α(q)Ψα := (~d + φα(q))
ψr−1
ψr−2
...
ψ1
ψ
α
= 0, (3.30)
with an appropriate unknown function ψ. Since Ψα = f~αβΨβ, the function ψ on Uα satisfies
the transition relation (ψ)α = ξ−r+1
αβ (ψ)β. It means that ψ is actually a local section of the
line bundle K
− r−1
2
C . There are r linearly independent solutions of (3.30), because q2, . . . , qr are
represented by holomorphic functions on Uα. The entries of X`
+ are given by the formula
X`
+ =
[
s
(`)
ij
]
, s
(`)
ij = δi+`,j
√
sisi+1 · · · si+`−1.
Therefore, (3.30) is equivalent to
0 =
√
sr−k−1ψk+1 + ~ψ′k +
√
sr−kq2ψk−1 +
√
sr−ksr−k+1q3ψk−2 + · · ·
+
√
sr−ksr−k+1 · · · sr−1qk+1ψ
=
√
sk+1ψk+1 + ~ψ′k +
√
skq2ψk−1 +
√
sksk−1q3ψk−2 + · · ·+√sksk−1 · · · s1qk+1ψ (3.31)
for k = 0, 1, . . . , r−1, where we use sk = sr−k. Note that the differentiation is always multiplied
by ~ as ~d in (3.30), and that φ(q) is independent of ~ and takes the form
φ(q) =
0
√
sr−1q2
√
sr−2sr−1q3 · · · · · · √s2 · · · sr−1qr−1
√
s1s2 · · · sr−1qr
√
sr−1 0
√
sr−2q2 · · · · · · √s2 · · · sr−2qr−2
√
s1 · · · sr−2qr−1
√
sr−2 0
. . . · · · √s2 · · · sr−3qr−3
√
s1 · · · sr−3qr−2
. . .
. . .
. . .
...
...
√
s3 0
√
s2q2
√
s1s2q3√
s2 0
√
s1q2√
s1 0
.
By solving (3.31) for k = 0, 1, . . . , r − 2 recursively, we obtain an expression of ψk as a linear
combination of
ψ = ψ0, ~ψ′ = ~
d
dxα
ψ, . . . , ~kψ(k) = ~k
dk
dxkα
ψ,
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 23
with coefficients in differential polynomials of q2, q3, . . . , qk. Moreover, the coefficients of these
differential polynomials are in Q[~]. For example,
ψ1 = − 1
√
s1
~ψ′,
ψ2 =
1
√
s1s2
(~2ψ′′ − s1q2ψ),
ψ3 =
1
√
s1s2s3
(−~3ψ′′′ + ~(s1 + s2)q2ψ
′ + (~s1q
′
2 − s1s2q3)ψ),
ψ4 =
1
√
s1s2s3s4
(
~4ψ′′′′ − ~2(s1 + s2 + s3)q2ψ
′′ + (−~2(2s1 + s2)q′2 + ~(s1s2 + s2s3)q3)ψ′
+ (~2s1q
′′
2 − ~s1s2q
′
3 + s1s3q
2
2 − s1s2s3q4)ψ
)
, etc.
Since ψ1 is proportional to ψ′, inductively we can show that the linear combination expression
of ψk by derivatives of ψ does not contain the (k−1)-th order differentiation of ψ. Equation (3.31)
for k = r − 1 is a differential equation
~ψ′r−1 +
√
s1q2ψr−2 +
√
s1s2q3ψr−3 + · · ·+√s1s2 · · · sr−1qrψ = 0, (3.32)
which is an order r differential equation for ψ ∈ K
− r−1
2
C . Its actual shape can be calculated
using the procedure of the proof of the next theorem. The equation becomes (3.40). Since
we are using a fixed projective coordinate system, the connection ∇~(q) takes the same form
on each coordinate neighborhood Uα. Therefore, the shape of the differential equation of (3.32)
as an equation for ψ ∈ K−
r−1
2
C is again the same on every coordinate neighborhood.
We are now ready to calculate the semi-classical limit of the Rees D-module corresponding
to ~∇~(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 E(q) denote the Rees D-module
(
E~, ~∇~(q)
)
associated with the oper of (3.29). Then the
semi-classical limit of E(q) is the spectral curve σ∗Σ ⊂ T ∗C of −φ(q) defined by the characte-
ristic equation det(η + φ(q)) = 0, where σ is the involution of (1.7).
Remark 3.12. The actual shape of the single scalar valued differential equation (3.32) of order r
is given by (3.40) below. This is exactly what we usually refer to as the quantum curve of the
spectral curve det(η + φ(q)) = 0.
Proof. Since the connection and the differential equation (3.32) are globally defined, we need
to analyze the semi-classical limit only at each coordinate neighborhood Uα with a projective
coordinate xα. Let
Ψ0 =
~r−1ψ(r−1)
...
~ψ′
ψ
.
From the choice of local trivialization (3.30) we have an expression
Ψ =
(
∆ +A(q, ~)
)
Ψ0, (3.33)
where ∆ is a constant diagonal matrix
∆ =
[
δij(−1)r−i
1
√
sr−1sr−2 · · · si
]
i,j=1,...,r
(3.34)
with the understanding that its (r, r)-entry is 1, and A(q, ~) =
[
a(q, ~)i,j
]
satisfies the following
properties:
24 O. Dumitrescu and M. Mulase
Condition 3.13.
� A(q, ~) is a nilpotent upper triangular matrix.
� Each entry a(q, ~)i,j is a differential polynomial in q2, . . . , qr with coefficients in Q[~].
� All diagonal and the first upper diagonal entries of A(q, ~) are 0:
a(q, ~)i,i = a(q, ~)i,i+1 = 0, i = 1, . . . , r.
The last property is because ψk does not contain the (k − 1)-th derivative of ψ, as we have
noted above. The diagonal matrix ∆ is designed so that
∆−1X−∆ =
0
−1 0
−1 0
. . .
−1 0
=: −X.
Let us calculate the gauge transformation of ~∇~(q) by ∆ +A(q, ~):
(∆ +A(q, ~))−1(~d+ φ(q))(∆ +A(q, ~)) = (~d−Xdxα − ω(q, ~)dxα). (3.35)
We wish to determine the matrix ω(q, ~).
Lemma 3.14. The matrix ω(q, ~) of (3.35) consists of only the first row, and the matrix
X + ω(q, ~) takes the following canonical form
X + ω(q, ~) =
0 ω2(q, ~) · · · ωr−1(q, ~) ωr(q, ~)
1 0 · · · 0 0
1
. . .
...
...
. . . 0 0
1 0
. (3.36)
Proof of Lemma. First, define
B(q, ~) :=
r−1∑
n=1
(−1)n
(
∆−1A(q, ~)
)n
(3.37)
so that
I +B(q, ~) = (I + ∆−1A(q, ~))−1.
By definition, B(q, ~) satisfies the exact same properties listed in Condition 3.13 above. We then
have
(∆ +A(q, ~))−1
(
~
d
dxα
+ φ(q)
)
(∆ +A(q, ~))
= (I + ∆−1A(q, ~))−1∆−1
(
~
d
dxα
+X− +
r∑
`=2
q`X
`−1
+
)
∆(I + ∆−1A(q, ~))
= (I +B(q, ~))
(
~
d
dxα
−X + ∆−1
r∑
`=2
q`X
`−1
+ ∆
)
(I + ∆−1A(q, ~))
= ~
d
dxα
−X −B(q, ~)X −X∆−1A(q, ~) + ~(I +B(q, ~))∆−1 dA(q, ~)
dxα
−B(q, ~)X∆−1A(q, ~) + (I +B(q, ~))∆−1
( r∑
`=2
q`X
`−1
+ ∆
)
(I + ∆−1A(q, ~)).
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 25
Thus we define
ω(q, ~) = B(q, ~)X +X∆−1A(q, ~)− ~(I +B(q, ~))∆−1 dA(q, ~)
dxα
+B(q, ~)X∆−1A(q, ~)− (∆ +A(q, ~))−1
( r∑
`=2
q`X
`−1
+
)
(∆ +A(q, ~)). (3.38)
Every single matrix in the right-hand side of (3.38) is either diagonal or upper triangular and
nilpotent, except for X. Because of Condition 3.13, B(q, ~)X+X∆−1A(q, ~) is an upper trian-
gular matrix with 0 along the diagonal. Obviously, so are all other terms of (3.38). Thus ω(q, ~)
is upper triangular and nilpotent, and is a polynomial in ~.
Now we note that (3.30) and (3.33) yield(
~
d
dxα
−X − ω(q, ~)
)
Ψ0 = 0. (3.39)
The basis vector Ψ0 defined on a simply connected coordinate neighborhood Uα is designed so
that the following equation holds for any solution ψ of (3.30):
(
~
d
dxα
−X
)
Ψ0 =
(
~
d
dxα
−X
)
~r−1ψ(r−1)
~r−2ψ(r−2)
...
~ψ′
ψ
=
~rψ(r)
0
...
0
0
.
Let ψ[1], . . . , ψ[r] be r linearly independent solutions of (3.30). The Wronskian matrix is defi-
ned by
W =
~r−1ψ
(r−1)
[1] ~r−1ψ
(r−1)
[2] · · · ~r−1ψ
(r−1)
[r]
~r−2ψ
(r−2)
[1] ~r−2ψ
(r−2)
[2] · · · ~r−2ψ
(r−2)
[r]
...
...
. . .
...
~ψ′[1] ~ψ′[2] · · · ~ψ′[r]
ψ[1] ψ[2] · · · ψ[r]
.
Then from (3.39), we have
ω(q, ~) =
[(
~
d
dxα
−X
)
W
]
·W−1 =
~rψ(r)
[1] ~rψ(r)
[2] · · · ~rψ(r)
[r]
0 0 · · · 0
...
...
. . .
...
0 0 · · · 0
W−1.
Clearly we see that ω(q, ~) has non-zero entries only in the first row. The (1, 1)-entry is also 0
because trω(q, ~) = 0. �
Thus (3.30) is equivalent to a single linear ordinary differential equation of order r
Pα(xα, ~; q)ψ :=
[
~r
(
d
dxα
)r
−
r∑
i=2
~r−iωi(q, ~)
(
d
dxα
)r−i]
ψ = 0. (3.40)
In other words, Pα(xα, ~; q) is a generator of the Rees D-module
(
E~, ~∇~(q)
)
on Uα. This
expression of differential equation of order r is what is commonly referred to as a quantum
26 O. Dumitrescu and M. Mulase
curve. The coefficients ωi(q, ~) are calculated by determining the matrix A(q, ~) of (3.33).
Then (3.34), (3.37), and (3.38) give the exact form of Pα(xα, ~; q). As we can see, its shape is
quite involved, and is not obtained by simply replacing y in det
(
y + φ(q)
)
by ~ d
dxα
. In par-
ticular, the coefficients contain derivatives of q`’s, which never appear in the characteristic
polynomial of −φ(q). What we are going to prove now is that nonetheless, the semi-classical
limit of Pα(xα, ~; q) is exactly the characteristic polynomial.
As defined in Definition 2.8, the semi-classical limit of (3.40) is the limit
lim
~→0
e−
1
~S0(xα)
[
~r
(
d
dxα
)r
−
r∑
i=2
~r−iωi(q, ~)
(
d
dxα
)r−i]
e
1
~S0(xa)yr−
r∑
i=2
ωi(q, 0)yr−i, (3.41)
where S0(xα) is a holomorphic function on Uα so that dS0 = ydxα gives a local trivialization
of T ∗C over Uα. Since ω(q, ~) is a polynomial in ~, we can evaluate it at ~ = 0. Notice that (3.41)
is the characteristic polynomial of the matrix X+ω(q, ~) of (3.36) at ~ = 0. The computation of
semi-classical limit is the same as the calculation of the determinant of the connection ~∇~(q),
after taking conjugation by the scalar diagonal matrix e−
1
~S0(xα)Ir×r, and then take the limit
as ~ goes to 0:
e−
1
~S0(xα)I · (∆ +A(q, ~))−1
(
~
d
dxα
+ φ(q)
)
(∆ +A(q, ~)) · e
1
~S0(xα)I
= (∆ +A(q, ~))−1e−
1
~S0(xα)I ·
(
~
d
dxα
+ φ(q)
)
· e
1
~S0(xα)I(∆ +A(q, ~))
= (∆ +A(q, ~))−1
(
dS0(xα)
dxα
+ φ(q)
)
(∆ +A(q, ~)) +O(~)
~→0−→ (∆ +A(q, 0))−1(y + φ(q))(∆ +A(q, 0)).
The determinant of the above matrix is the characteristic polynomial det (y + φ(q)). Note that
from (3.35), we have
e−
1
~S0(xα)I · (∆ +A(q, ~))−1
(
~
d
dxα
+ φ(q)
)
(∆ +A(q, ~)) · e
1
~S0(xα)I
= e−
1
~S0(xα)I ·
(
~
d
dxα
− (X + ω(q, ~))
)
· e
1
~S0(xα)I
= ~
d
dxα
+ y − (X + ω(q, ~))
~→0−→ y − (X + ω(q, 0)).
Taking the determinant of the above, we conclude that
det (y + φ(q)) = yr −
r∑
i=2
ωi(q, 0) yr−i.
This completes the proof of Theorem 3.11. �
For every ~ ∈ H1(C,KC), the ~-connection
(
E~, ~∇~(q)
)
of (3.23) defines a global Rees DC-
module structure in E~. Thus we have constructed a universal family EC of Rees DC-modules
on a given C with a fixed spin and a projective structures:
EC
⊃←−−−−
(
E~,∇~(q)
)y y
C ×B ×H1(C,KC)
⊃←−−−− C × {q} × {~}.
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 27
The universal family SC of spectral curves is defined over C ×B.
P (KC ⊕OC)×B ⊃←−−−− SC
⊃←−−−−
(
det
(
η − φ(q)
))
0y y y
C ×B C ×B ⊃←−−−− C × {q}.
The semi-classical limit is thus a map of families
EC −−−−→ SCy y
C ×B ×H1(C,KC) −−−−→ C ×B.
(3.42)
Our concrete construction (3.21) using the projective coordinate system is not restricted to
the holomorphic Higgs field φ(q). Since the moduli spaces appearing in the correspondence (3.29)
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 = ∪αUα a projec-
tive coordinate system subordinating the complex structure of C.
Theorem 3.15 (quantization of meromorphic data). Let D be an effective divisor of C, and
q ∈ B(D) :=
r⊕
`=2
H0
(
C,KC(D)⊗`
)
.
We define a meromorphic Higgs bundle
(
E0, φ(q)
)
by the same formulae (3.2) and (3.3), as well
as a meromorphic oper
(
E~,∇~(q)
)
by (3.21). The meromorphic oper defines a meromorphic
Rees D-module
E(q) =
(
E~, ~∇~(q))
on C. Then the semi-classical limit of E(q) is the spectral curve(
det(η + φ(q)
)
0
⊂ T ∗(C) (3.43)
of −φ(q).
Proof. The proof is exactly the same as Theorem 3.11 on each simply connected coordinate
neighborhood Uα ⊂ C \ supp(D). Thus the semi-classical limit of E(q)|C\supp(D) as a divisor is
defined in T ∗(C) \ π−1(D). The spectral curve (3.43) is its closure in T ∗C with respect to the
complex topology. Thus by Definition 2.11, (3.43) is the semi-classical limit of the meromorphic
extension E(q). �
Example 3.1. Here we list characteristic polynomials and differential operators Pα(xα, ~; q)
of (3.40) for r = 2, 3, 4. These formulas show that our quantization procedure is quite non-
trivial.
� r = 2:
det(y + φ(q)) = y2 − q2,
Pα(xα, ~; q) =
(
~
d
dxα
)2
− q2.
28 O. Dumitrescu and M. Mulase
� r = 3:
det(y + φ(q)) = y3 − 4q2y + 4q3,
Pα(xα, ~; q) =
(
~
d
dxα
)3
− 4q2
(
~
d
dxα
)
+ 4q3 − 2~q′2.
� r = 4:
det(y + φ(q)) = y4 − 10q2 + 24q3y − 36q4 + 9q2
2,
Pα(xα, ~; q) =
(
~
d
dxα
)4
− 10q2
(
~
d
dxα
)2
+ (24q3 − 10~q′2)
(
~
d
dxα
)
− 36q4 + 9q2
2 + 3~2q′′2 − 12~q′3.
3.3 Non-Abelian Hodge correspondence and a conjecture of Gaiotto
The biholomorphic map (3.29) is defined by fixing a projective structure of the base curve C.
Gaiotto [41] conjectured that such a correspondence would be canonically constructed through
a scaling limit of non-Abelian Hodge correspondence. The conjecture has been solved in [26]
for Hitchin moduli spaces MDol and MdeR constructed over an arbitrary complex simple and
simply connected Lie group G. In this subsection, we review the main result of [26] for G =
SL(r,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 ≥ 2. We denote by Etop 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 [69] (see also [4, 68]). Extending
the classical case, the stability condition for an SL(r,C)-Higgs bundle (E, φ) translates into
a differential geometric condition, known as Hitchin’s equations, imposed on a set of geometric
data as follows [24, 46, 73]. The data consist of a Hermitian fiber metric h on Etop, a unitary
connection ∇ in Etop with respect to h, and a differentiable sl(r,C)-valued 1-form φ on C.
The following system of nonlinear equations is called Hitchin’s equations:{
F∇ +
[
φ, φ†
]
= 0,
∇0.1φ = 0.
(3.44)
Here, F∇ is the curvature of ∇, φ† is the Hermitian conjugate of φ with respect to h, and ∇0,1
is the Cauchy–Riemann part of ∇. ∇0,1 gives rise to a natural complex structure in Etop, which
we simply denote by E. Then φ becomes a holomorphic Higgs field in E because of the second
equation of (3.44). The stability condition for the Higgs pair (E, φ) is equivalent to Hitchin’s
equations (3.44). Define a one-parameter family of connections
∇(ζ) :=
1
ζ
· φ+∇+ ζ · φ†, ζ ∈ C∗. (3.45)
Then the flatness of ∇(ζ) for all ζ is equivalent to (3.44).
The non-Abelian Hodge correspondence [24, 46, 62, 73] (see also [8, 10, 77, 78]) is the following
diffeomorphic correspondence
ν : MDol 3 (E, φ) 7−→
(
Ẽ, ∇̃
)
∈MdeR.
First, we construct the solution (∇, φ, h) of Hitchin’s equations corresponding to stable (E, φ).
It induces a family of flat connections ∇(ζ). Then define a complex structure Ẽ in Etop
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 29
by ∇(ζ = 1)0,1. Since ∇(ζ) is flat, ∇̃ := ∇(ζ = 1)1,0 is automatically a holomorphic connection
in Ẽ. Thus (Ẽ, ∇̃) becomes a holomorphic connection. Stability of (E, φ) implies that the resul-
ting connection is irreducible, hence (Ẽ, ∇̃) ∈MdeR. Since this correspondence goes through the
real unitary connection ∇, the assignment E 7−→ Ẽ is not a holomorphic deformation of vector
bundles.
Extending the idea of the one-parameter family (3.45), Gaiotto conjectured the following:
Conjecture 3.16 (Gaiotto [41]). Let (∇, φ, h) be the solution of (3.44) corresponding to a stable
Higgs bundle (E0, φ(q)) on the SL(r,C)-Hitchin section (3.4). Consider the following two-para-
meter family of connections
∇(ζ,R) :=
1
ζ
·Rφ+∇+ ζ ·Rφ†, ζ ∈ C∗, R ∈ R+.
Then the scaling limit
lim
R→0, ζ→0
ζ/R=~
∇(ζ,R)
exists for every ~ ∈ C∗, and forms an ~-family of SL(r,C)-opers.
Remark 3.17. The existence of the limit is non-trivial, because the Hermitian metric h blows
up as R→ 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 ([26]). 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 H. The fun-
damental group π1(C) acts on H through a representation
ρ : π1(C) −→ PSL(2,R) = Aut(H),
and generates an analytic isomorphism
C ∼= H
/
ρ
(
π1(C)
)
.
The representation ρ lifts to SL(2,R) ⊂ SL(2,C), and induces a projective structure in C
subordinating its complex structure coming from H. This projective structure is what we call
the Fuchsian projective structure.
Corollary 3.20 (Gaiotto correspondence and quantization). Under the same setting of Con-
jecture 3.16, the limit oper of [26] is given by
lim
R→0, ζ→0
ζ/R=~
∇(ζ,R) = d +
1
~
φ(q) = ∇~(q), ~ 6= 0, (3.46)
with respect to the Fuchsian projective coordinate system. The operator (3.46) is a connection
in the ~-filtered extension (F •~ , E~) of Definition 3.2. In particular, the correspondence
(E0, φ(q))
γ7−→
(
E~,∇~(q)
)
is biholomorphic, unlike the non-Abelian Hodge correspondence.
Proof. The key point is that since E0 is made out of KC , the fiber metric h naturally comes
from the metric of C itself. Hitchin’s equations (3.44) for 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 [25, 26]. �
30 O. Dumitrescu and M. Mulase
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 mech-
anism of [27, 30] 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 [30]. Then in Section 6, we prove that for the case
of meromorphic SL(2,C)-Higgs bundles, the PDE recursion of topological type 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 Σ ⊂ T ∗C through semi-classical limit,
we first need to identify the choice of the blow-up space Bl(T ∗C) in which the non-singular
model Σ̃ 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 W0,1. 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∞, 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 [30].
In what follows, we give the construction of the canonical blow-up space Bl(T ∗C), and deter-
mine the genus of the normalization Σ̃. This genus is necessary to identify the Riemann prime
form on it, which determines another input datum W0,2 for the topological recursion.
There are two different ways of defining the spectral curve for Higgs bundles with meromor-
phic Higgs field. Our definition uses the compactified cotangent bundle. This idea also appears
in [56]. 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 3
2 at ∞ ∈ P1.
Our point of view of spectral curves is also closely related to considering the stable pairs
of pure dimension 1 on T ∗C. Through Hitchin’s abelianization idea, the moduli space of stable
pairs and the moduli space of Higgs bundles are identified [51].
The geometric setting we start with is a meromorphic SL(2,C)-Higgs bundle
(
E0, φ(q)
)
defi-
ned on a smooth projective algebraic curve C of genus g ≥ 0 with a fixed projective structure.
Here,
q = −det(φ(q)) ∈ H0
(
C,KC(D)⊗2
)
, φ(q) =
[
q
1
]
,
and D is an effective divisor of C. The spectral curve is the zero-locus in T ∗C of the characteristic
equation
Σ :=
(
η2 − π∗(q)
)
0
. (4.1)
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
(
E0, φ(q)
)
.
Recall that Pic(T ∗C) is generated by the zero section C0 of T ∗C and fibers of the projection
map π : T ∗C −→ C. Since the spectral curve Σ is a double covering of C, as a divisor it is
expressed as
Σ = 2C0 +
a∑
j=1
π∗(pj) ∈ Pic(T ∗C), (4.2)
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 31
where
∑a
j=1 pj ∈ Pica(C) is a divisor on C of degree a. As an element of the Néron–Severi group
NS(T ∗C) = Pic(T ∗C)/Pic0(T ∗C),
it is simply
Σ = 2C0 + aF ∈ NS(T ∗C)
for a typical fiber class F . Since the intersection F · C∞ = 1, we have a = Σ · C∞ in NS(T ∗C).
From the genus formula
pa(Σ) =
1
2
Σ · (Σ +KT ∗C) + 1
and
KT ∗C = −2C0 + (4g − 4)F ∈ NS(T ∗C),
we find that the arithmetic genus of the spectral curve Σ is
pa(Σ) = 4g − 3 + a, (4.3)
where a is the number of intersections of Σ and C∞. Now we wish to find the geometric genus
of Σ.
Recall the following from [30]:
Definition 4.1 (discriminant divisor). The discriminant divisor of the spectral curve (4.1) is
a divisor on C defined by
∆ := (q)0 − (q)∞, (4.4)
where
(q)0 =
m∑
i=1
miri, mi > 0, qi ∈ C, (4.5)
(q)∞ =
n∑
j=1
njpj , nj > 0, pj ∈ C. (4.6)
Since q is a meromorphic section of K⊗2
C ,
deg ∆ =
m∑
i=1
mi −
n∑
j=1
nj = 4g − 4. (4.7)
Theorem 4.2 (geometric genus formula). Define
δ = |{i | mi ≡ 1 mod 2}|+ |{j | nj ≡ 1 mod 2}|. (4.8)
Then the geometric genus of the spectral curve Σ of (4.1) is given by
pg(Σ) = 2g − 1 +
1
2
δ. (4.9)
We note that (4.7) implies δ ≡ 0 mod 2.
32 O. Dumitrescu and M. Mulase
Remark 4.3. If φ is a holomorphic Higgs field, then m = δ = 4g − 4 and n = 0. Therefore,
we recover the genus formula g(Σ) = 4g − 3 of [27, equation (2.5)]. In this case, the Hitchin
fibration (1.3) is a family of Prym varieties, which are (3g − 3)-dimensional Abelian varieties
associated with the ramified covering π : Σ −→ C.
Proof. Since Σ ⊂ T ∗C is a double covering of C in a ruled surface, locally at every singular
point p, Σ is either irreducible, or reducible and consisting of two components. When irreducible,
it is locally isomorphic to
t2 − s2m+1 = 0, m ≥ 1. (4.10)
If it has two components, then it is locally isomorphic to
t2 − s2m = (t− sm)(t+ sm) = 0. (4.11)
Note that the local form of Σ at a ramification point of π : Σ −→ C is written as (4.10) with
m = 0. By extending the terminology “singularity” to “critical points” of the morphism π, we
include a ramification point as a cusp with m = 0.
Let ν : Σ̃ −→ Σ be the non-singular model of Σ. Then π̃ = π ◦ ν : Σ̃ −→ C is a double
sheeted covering of C by a smooth curve Σ̃. If Σ has two components at a singularity P as
in (4.11), then π̃−1(P ) consists of two points and π̃ is not ramified there. If P is a cusp (4.10),
then π̃−1(P ) is a ramification point of the covering π̃. Thus the invariant δ of (4.8) counts the
total number of cusp singularities of Σ and the ramification points of π : Σ −→ C. Then the
Riemann–Hurwitz formula gives us
2− 2g
(
Σ̃
)
− δ = 2 (2− 2g(C)− δ) ,
hence
pg(Σ) = 2g(C)− 1 +
1
2
(the number of cusps). (4.12)
The genus formula (4.9) follows from (4.12). �
Our purpose is to apply topological recursion of Section 5 to a singular spectral curve of the
form Σ of (4.1). To this end, we need to construct in a canonical way the normalization
ν : Σ̃ −→ Σ through a sequence of blow-ups of the ambient space T ∗C. This is because we need
to construct differential forms on Σ̃ that reflect geometry of i : Σ −→ T ∗C. We thus proceed
to analyze the local structure of Σ at each singularity using the global equation (4.1) in what
follows.
Definition 4.4 (construction of the blow-up space). The blow-up space Bl(T ∗C)
Σ̃
π̃
��
ĩ //
ν
��
Bl(T ∗C)
ν
$$
Σ
π
��
i // T ∗C
π
ssC
(4.13)
is defined by blowing up T ∗C in the following way:
� At each ri of (4.5), blow up ri ∈ Σ ∩ C0 ⊂ T ∗C a total of
⌊
mi
2
⌋
times.
� At each pj of (4.6), blow up at the intersection Σ ∩ π−1(pj) ⊂ C∞ a total of
⌊nj
2
⌋
times.
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 33
Remark 4.5. Let us consider when both (q)0 and (q)∞ are reduced. From the definition above,
in this case Σ is non-singular, and the two genera (4.3) and (4.9) agree. The spectral curve is
invariant under the involution σ : T ∗C −→ T ∗C of (1.7). If q ∈ H0(C,K⊗2
C ) is holomorphic,
then π : Σ −→ C is simply branched over ∆ = (q)0, and Σ is a smooth curve of genus 4g − 3.
This is in agreement of (4.3) because n = 0 in this case. If q is meromorphic, then its pole
divisor is given by (q)∞ of degree n. Since (q)∞ is reduced, π : Σ −→ C is ramified at the
intersection of C∞ and π∗(q)∞. The spectral curve is also ramified at its intersection with C0.
Note that deg(q)0 = 4g − 4 + n because of (4.7). Thus π : Σ −→ C is simply ramified at a total
of 4g − 4 + 2n points. Therefore, Σ is non-singular, and we deduce that its genus is given by
pg(Σ) = pa(Σ) = 4g − 3 + n
from the Riemann–Hurwitz formula. As a divisor class, we have
Σ = 2C0 + π∗(q)∞ ∈ Pic(T ∗C),
in agreement of (4.2).
Theorem 4.6. In the blow-up space Bl(T ∗C), we have the following.
� The proper transform Σ̃ of the spectral curve Σ ⊂ T ∗C by the birational morphism ν :
Bl(T ∗C) −→ T ∗C is a smooth curve with a holomorphic map π̃ = π ◦ ν : Σ̃ −→ C.
� The Galois action σ : Σ −→ Σ lifts to an involution of Σ̃, and the morphism π̃ : Σ̃ −→ C
is a Galois covering with the Galois group Gal(Σ̃/C) = 〈σ̃〉 ∼= Z/2Z
Σ̃
ν−−−−→ Σ
π−−−−→ C
σ̃
y yσ ∥∥∥
Σ̃ −−−−→
ν
Σ −−−−→
π
C.
(4.14)
Proof. We need to consider only when ∆ is non-reduced. Let ri ∈ supp(∆) be a zero of q
of degree mi > 1. The curve germ of Σ near ri ∈ Σ∩C0 is given by a formula y2 = xmi , 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 y1 = y/x on the exceptional divisor. The proper transform of the curve germ
becomes y2
1 = xmi−2. Repeat this process at (x, y1) = (0, 0), until we reach the equation
y2
` = xε,
where ε = 0 or 1. The proper transform of the curve germ is now non-singular. We see that
after a sequence of
⌊
mi
2
⌋
blow-ups starting at the point ri, the proper transform of Σ is simply
ramified over ri ∈ C = C0 if mi is odd, and unramified if mi is even. We apply the same
sequence of blow-ups at each ri with multiplicity greater than 1.
Let pj ∈ supp(∆) be a pole of q with order nj > 1. The intersection P = Σ ∩ π−1(pj) lies
on C∞, and Σ has a singularity at P . Let z = 1/y be a fiber coordinate of π−1(pj) at the infinity.
Then the curve germ of Σ at the point P is given by
z2 = xnj .
The involution σ in this coordinate is simply z 7−→ −z. The blow-up process we apply at P
is the same as before. After bnj2 c blow-ups starting at the point P ∈ Σ ∩ π−1(pj), the proper
transform of Σ is simply ramified over pj ∈ C if nj is odd, and unramified if nj is even. Again
we do this process for all pj with a higher multiplicity.
34 O. Dumitrescu and M. Mulase
The blow-up space Bl(T ∗C) is defined as the application of a total of
m∑
i=1
⌊mi
2
⌋
+
n∑
j=1
⌊nj
2
⌋
times blow-ups on T ∗C as described above. The proper transform Σ̃ is the minimal resolution
of Σ. Note that the morphism
π̃ = π ◦ ν : Σ̃ −→ C
is a double covering, ramified exactly at δ points. Since pa(Σ̃) = pg(Σ), (4.9) follows from the
Riemann–Hurwitz formula applied to π̃ : Σ̃ −→ C. It is also obvious that δ counts the number
of cusp points of Σ, including smooth ramification points of π, in agreement of Theorem 4.2,
and the fact that δ counts the total number of odd cusps on Σ.
Note that C0 and C∞ are point-wise invariant under the involution σ. Since Bl(T ∗C) is
constructed by blowing up points on C0 and C∞ and their proper transforms, we have a natural
lift σ̃ : Bl(T ∗C) −→ Bl(T ∗C) of σ which induces (4.14). �
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 [36] provides an effective tool for calculating asymptotic expan-
sions 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 ~-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 ~-families of opers, for the case of holomorphic and meromorphic
SL(2,C)-Higgs bundles. For this purpose, in this section we review the framework of PDE
recursion developed in [27, 30]. For the case of singular Hitchin spectral curves, our particular
method of normalization of spectral curves of Section 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 (3.42). Thus our quantization procedure in terms of PDE
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,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(z1, z2)
on the product Σ̃× Σ̃ of a compact Riemann surface Σ̃ with values in a certain line bundle [67].
To define the prime form, we have to make a few more extra choices. First, we need to choose
a theta characteristic K
1
2
Σ̂
such that
dimH0
(
Σ̃,K
1
2
Σ̃
)
= 1.
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 35
We also need to choose a symplectic basis for H1(Σ̃,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 [67, p. 3.210]. See also [27, Section 2].
Definition 5.2 (topological recursion for non-singular covering). Let C be a non-singular pro-
jective algebraic curve together with a choice of a symplectic basis for H1(C,Z), and π̃ : Σ̃ −→ C
a degree 2 covering by another non-singular curve Σ̃. We denote by R the ramification divisor
of π̃. The covering π̃ is a Galois covering with the Galois group Z/2Z = 〈σ̃〉, and R is the
fixed-point divisor of the involution σ̃. We also choose a labeling of points of R, and define
the A-cycles of a symplectic bases for H1(Σ̃,Z) as defined in [27, Section 2], which extend the
A-cycles of C. Topological recursion is an inductive mechanism of constructing meromorphic
differential forms Wg,n on the Hilbert scheme Σ̃[n] of n-points on Σ̃ for all g ≥ 0 and n ≥ 1 in the
stable range 2g − 2 + n > 0, from given initial data W0,1 and W0,2. The differential form Wg,n
is a meromorphic n-linear form, i.e., a 1-form on each factor of Σ̃[n] for 2g − 2 + n > 0.
� W0,1 is a meromorphic 1-form on Σ̃ to be prescribed according to the geometric setting we
have. Then define
Ω := σ̃∗W0,1 −W0,1,
which satisfies σ̃∗Ω = −Ω.
� W0,2 is defined by
W0,2(z1, z2) = d1d2 logE(z1, z2),
where E(z1, z2) is the A-cycle normalized Riemann prime form on Σ̃× Σ̃. W0,2 is a mero-
morphic differential 1⊗ 1-form on Σ̃× Σ̃ with 2nd order poles along the diagonal.
� We also define the normalized Cauchy kernel on Σ̃ by
ωa−b(z) := dz log
E(a, z)
E(b, z)
, (5.1)
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 Σ̃. We thus choose a fun-
damental 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.
Wg,n(z1, . . . , zn) =
1
2
1
2π
√
−1
∮
Γ
ωz̃−z(z1)
Ω(z)
×
[
Wg−1,n+1(z, z̃, z2, . . . , zn) +
No(0,1)∑
g1+g2=g
ItJ={2,...,n}
Wg1,|I|+1(z, zI)Wg2,|J |+1(z̃, zJ)
]
. (5.2)
Here,
� The integration contour Γ ⊂ Σ̃ is a collection of positively oriented small loops around
each point p ∈ supp(Ω)0 ∪ supp(R).
36 O. Dumitrescu and M. Mulase
� The integration is taken with respect to z ∈ Γ. Thus we chose a fundamental domain of the
universal covering of Σ̃ that contains supp(Ω)0 ∪ supp(R), and perform the integration
locally as residue calculations.
� z̃ = σ̃(z) is the Galois conjugate of z ∈ Σ̃.
� The expression 1/Ω is a meromorphic section of K−1
C which is multiplied to meromorphic
quadratic differentials on Σ̃ in z-variable.
� “No (0, 1)” means that g1 = 0 and I = ∅, or g2 = 0 and J = ∅, are excluded in the
summation.
� The sum runs over all partitions of g and set partitions of {2, . . . , n}, other than those
containing the (0, 1)-geometry.
� |I| is the cardinality of the subset I ⊂ {2, . . . , n}.
� zI = (zi)i∈I .
Remark 5.3. The integrand of (5.2) is not a well-defined differential form in z ∈ Σ̃, due to
the definition of ωσ̃(z)−z(z1) mentioned above. What the formula defines is a sum of residues at
each point p ∈ supp(Ω)0 ∪ supp(R), which depends on the choice of the domain of ωσ̃(z)−z(z1)
as a function in z.
Remark 5.4. Topological recursion depends on the choice of the integration contour Γ. Since
the integrand of the right-hand side of (5.2) has other poles than the ramification divisor R and
zeros of Ω, other choices of Γ are equally possible.
Remark 5.5. Topological recursion (5.2) can be defined for far more general situations.
The bottle neck of the formalism is difficulty of integration over a high genus non-hyperelliptic
Riemann surface. So the actual calculations have not been done much beyond the cases when
the spectral curve Σ̃ is of genus 0, or hyperelliptic.
When we have a non-singular Hitchin spectral curve i : Σ ↪→ T ∗C associated with a holomor-
phic SL(2,C)-Higgs bundle (E, φ), we apply Definition 5.2 to Σ̃ = Σ, σ̃ = σ, and W0,1 = i∗η,
where σ is the involution of (1.7) and η is the tautological 1-form on T ∗C.
Under the same setting as in topological recursion, we define
Definition 5.6 (PDE recursion for a smooth covering of degree 2). PDE recursion is the
following partial differential equation for all (g, n) subject to 2g− 2 + n ≥ 2 defined on an open
neighborhood Un of Σ̃[n] (or the universal covering, if the global treatment is necessary):
d1Fg,n(z1, . . . , zn) =
n∑
j=2
[
ωzj−σ̃(zj)(z1)
Ω(z1)
d1Fg,n−1
(
z[ĵ]
)
− ωzj−σ̃(zj)(z1)
Ω(zj)
djFg,n−1
(
z[1̂]
)]
+
1
Ω(z1)
du1du2
[
Fg−1,n+1
(
u1, u2, z[1̂]
)
+
stable∑
g1+g2=g
ItJ=[1̂]
Fg1,|I|+1(u1, zI)Fg2,|J |+1(u2, zJ)
]∣∣∣∣u1=z1
u2=z1
. (5.3)
Here, the index subset [ĵ] denotes the complement of j ∈ {1, 2, . . . , n}. The final summation
runs over all indices subject to be in the stable range, i.e., 2g1−1+ |I| > 0 and 2g2−1+ |J | > 0.
The initial data for the PDE recursion are a function F1,1(z1) on U and a symmetric function
F0,3(z1, z2, z3) on U3.
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 37
Remark 5.7. The PDE recursion of topological type (5.3) is obtained from (5.2) by replacing
the original contour Γ about the ramification divisor with the Cauchy kernel integration contour
of [27, 30] about the diagonal poles of ωσ̃(z)−z(z1), W0,2(z, zj), and W0,2
(
σ̃(z), zj
)
at z = zj
and z = σ̃(zj), j = 1, 2, . . . , n. This means the only difference comes from the alternative choice
of the contour of integration Γ. All other ingredients of the formula are the same. Other than
the case of g(Σ̃) = 0, topological recursion and PDE recursion are not equivalent. In many
enumerative problems [13, 16, 28, 31, 35, 63, 65, 66], PDE recursions are established through
the Laplace transform of combinatorial relations. These PDE recursions can then be turned
into the universal form of topological recursion. 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 = zj , z = σ̃(zj), j = 1, 2, . . . , n, are equivalent by continuous deformation
of the contour Γ.
6 WKB analysis of quantum curves
We consider in this section a meromorphic SL(2,C)-Higgs bundle (E0, φ(q)) associated with
a meromorphic quadratic differential q ∈ H0
(
C,KC(D)⊗2
)
with poles along an effective divi-
sor D on a curve C of arbitrary genus. This includes the case of holomorphic Higgs bundles
when g(C) ≥ 2 and D = ∅. As before, we have a fixed spin structure K
1
2
C and a projective
coordinate system on C.
Let
(
E~,∇~(q)
)
be the biholomorphic quantization result of Theorem 3.15. The corresponding
Rees DC-module is generated by a single differential operator
Pα(xa, ~) =
(
~
d
dxα
)2
− qα (6.1)
on each projective coordinate neighborhood Uα.
Theorem 6.1 (WKB analysis for SL(2,C)-quantum curves). Let q ∈ H0
(
C,KC(D)⊗2
)
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 DC-module E on C as the quantization of the possibly singular
spectral curve
Σ = (η2 − q)0 ⊂ T ∗C. (6.2)
Then PDE recursion (5.3) with an appropriate choice of the initial data provides an all-order
WKB analysis for the generator of the Rees DC-module 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 ∈ supp(∆) in the discriminant divisor of (4.4) of odd degree.
Choose a small enough simply connected coordinate neighborhood Uα of C with a projective
coordinate xα, so that π̃−1(Uα) ⊂ Σ̃ in the normalization (4.13) is also simply connected.
Let z be a local coordinate of π̃−1(Uα) and denote by a function xα = xα(z) the projection π̃.
� The formal WKB expansion we wish to construct is a solution to the equation
Pα(xa, ~)ψα(xα, ~) =
[(
~
d
dxα
)2
− qα
]
ψα(xα, ~) = 0 (6.3)
of the specific form
ψα(xα, ~) = exp
( ∞∑
m=0
~m−1Sm(xα)
)
= expFα(xα, ~). (6.4)
38 O. Dumitrescu and M. Mulase
Here, Fα(xα, ~) is a formal Laurent series in ~ starting with the power −1. In WKB
analysis, this series in ~ does not converge.
� Equation (6.3) is equivalent to
~2 d2
dx2
α
Fα + ~2 dFα
dxa
dFα
dxa
− qα = 0. (6.5)
We interpret the above equation as an infinite sequence of ordinary differential equations
for each power of ~:
~0-terms : (S′0(xa))
2 − qα = 0, (6.6)
~1-terms : S′′0 (xα) + 2S′0(xα)S′1(xα) = 0, (6.7)
~2-terms : S′′1 (xα) + 2S′2(xα)S′0(xα) + (S′1(xα))2 = 0, (6.8)
~m+1-terms : S′′m(xα) +
∑
a+b=m+1
S′a(xα)S′b(xα) = 0, m ≥ 2. (6.9)
The symbol ′ denotes the xα-derivative.
� Solve (6.6), (6.7), and (6.8) to find S0(xα), S1(xα), and S2(xα).
� Construct the normalization π̃ : Σ̃ −→ Σ as in (4.13), and define
W0,1 := ν∗i∗η.
� Define
F1,1(z1) = −
∫ z1 W0,2(z1, σ̃(z1))
Ω(z1)
, (6.10)
where integration means a primitive of the meromorphic 1-form
W0,2(z1,σ̃(z1))
Ω(z1) on π̃−1Uα.
� Define
F0,3(z1, z2, z3) =
∫∫∫ (
−W (z1, z2, z3) + 2
(
f(z1) + f(z2) + f(z3)
))
, (6.11)
where
W (z1, z2, z3) =
1
Ω(z1)
(
W0,2(z1, z2)W0,2
(
z1, σ̃(z3)
)
+W0,2(z1, z3)W0,2
(
z1, σ̃(z2)
))
+ d2
(
ωσ̃(z2)−z2(z1)W0,2(z2, σ̃(z3))
Ω(z2)
)
+ d3
(
ωσ̃(z3)−z3(z1)W0,2(z2, σ̃(z3))
Ω(z3)
)
and
f(z) := S̃2(z)−
(
F1,1(z)− 1
6
∫∫∫ z
W (z1, z2, z3)
)
.
Here, S̃2(z) = S2
(
xα(z)
)
is the lift of S2(xa) to π̃−1(Uα) ⊂ Σ̃.
� Note that we have
S2(xα) = F1,1
(
z(xα)
)
+
1
6
F0,3
(
z(xα), z(xα), z(xα)
)
for a local section z : Uα −→ π̃−1(Uα).
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 39
� Solve PDE recursion (5.3) using the initial data (6.10) and (6.11), and determine
Fg,n(z1, . . . , zn) for 2g − 2 + n ≥ 2.
� Define
Sm(xα) =
∑
2g−2+n=m−1
1
n!
Fg,n
(
z(xα)
)
, m ≥ 3, (6.12)
where Fg,n
(
z(xα)
)
is the principal specialization of Fg,n(z1, . . . , zn) evaluated at a local
section z = z(xα) of π̃ : Σ̃ −→ C on Uα.
Then (6.4) gives the WKB expansion of the solution to the generator of the Rees DC-module.
Remark 6.2. The WKB method is to solve (6.5) iteratively and find Sm(xα) for all m ≥ 0.
Here, (6.6) is the semi-classical limit of (6.3), and (6.7) is the consistency condition we need
for solving the WKB expansion. Since the 1-form dS0(x) is a local section of T ∗C, we identify
y = S′0(x). Then (6.6) is the local expression of the spectral curve equation (6.2). This expression
is the same everywhere on C \ supp(∆). We note q is globally defined. Therefore, we recover
the spectral curve Σ from the differential operator (6.1).
Remark 6.3. W1,1 := dF1,1 and W0,3 := d1d2d3F0,3 are solutions of (5.2) for 2g − 2 + n = 1
with respect to the contour of integration along the diagonal divisors mentioned in Remark 5.7.
The key idea of [27, 30] is the principal specialization of symmetric functions, which in our
case means restriction of a PDE on a symmetric function to the main diagonal of the variables.
Differential forms pull back, but PDEs do not. The essence of Theorem 6.1 is that the principal
specialization of PDE recursion (5.3) is exactly the quantum curve equation (6.3).
Lemma 6.4 ([65, Lemma A.1]). Let f(z1, . . . , zn) be a symmetric function in n variables. Then
d
dz
f(z, z, . . . , z) = n
[
∂
∂u
f(u, z, . . . , z)
]∣∣∣∣
u=z
,
d2
dz2
f(z, z, . . . , z) = n
[
∂2
∂u2
f(u, z, . . . , z)
]∣∣∣∣
u=z
+ n(n− 1)
[
∂2
∂u1∂u2
f(u1, u2, z, . . . , z)
]∣∣∣∣
u1=u2=z
.
For a function in one variable f(z), we have
lim
z2→z1
[
ωz2−b(z1)(f(z1)− f(z2))
]
= d1f(z1),
where ωz2−b(z1) is the 1-form of (5.1).
Proof of Theorem 6.1. First let P ∈ Σ ∩ C∞ be an odd cusp singularity on the fiber π−1(p)
of a point p ∈ C = C0 ⊂ T ∗C. The quadratic differential q has a pole of odd order at p, and
the normalization π̃ : Σ̃ −→ C is simply ramified at a point Q ∈ Σ̃ over p. We choose a local
projective coordinate x on C centered at p. The Galois action of σ̃ on Σ̃ fixes Q. As we have
shown in Case 1 of the proof of Theorem 4.2, locally over p, the spectral curve Σ has the shape
z2
0 = c(x)x2µ+1,
where z0 = 1/y, y is the fiber coordinate on T ∗pC, and c(x) is a unit c(x) ∈ O∗C,p. The quadratic
differential q has a local expression
q =
1
c(x)x2µ+1
.
40 O. Dumitrescu and M. Mulase
Define z1 = z0/x. The proper transform of Σ̃ after the first blow up at P is locally written by
z2
1 = c(x)x2µ−1.
Note that z1 is an affine coordinate of the first exceptional divisor. Repeating this process
µ-times, we end up with a coordinate zµ−1 = zµx and an equation
z2
µ = c(x)x.
Here again, zµ is an affine coordinate of the exceptional divisor created by the µ-th blow-up.
Write z = zµ so that the proper transform of the µ-times blow-ups is given by
z2 = c(x)x. (6.13)
Note that the Galois action of σ̃ at Q is simply z 7−→ −z. Solving (6.13) as a functional equation,
we obtain a Galois invariant local expression
x = x(z) = cQ(z2)z2,
where cQ ∈ O∗Σ̃,Q is a unit element. This function is precisely the local expression of the
normalization π̃ : Σ̃ −→ C at Q ∈ Σ̃. Since
z = zµ =
z0
xµ
=
1
yxµ
,
we have thus obtained the normalization coordinate z on the desingularized curve Σ̃ near Q:x = x(z) = cQ
(
z2
)
z2,
y = y(z) =
1
zxµ
= cQ
(
z2
)−µ
z−2µ−1.
It gives a parametric equation for the singular spectral curve Σ:
y2 =
1
c(x)x2µ+1
.
As we have shown in the proof of Theorem 4.2, the situation is the same for a zero of q of odd
order.
For the purpose of local calculation near Q ∈ Σ̃, we use the following local expressions:
ωσ̃(z)−z(z1) =
(
1
z1 − σ̃(z)
− 1
z1 − z
+O(1)
)
dz1,
η = y dx = h(z) dz :=
1
z2µ
(1 +O(z)) dz. (6.14)
Here, we adjust the normalization coordinate z by a constant factor to make (6.14) simple.
Using the notation ∂z = ∂/∂z, we have a local formula equivalent to (5.3) that is valid for
2g − 2 + n ≥ 2:
∂z1Fg,n(z1, . . . , zn) = −
n∑
j=2
[
ωzj−σ̃(zj)(z1)
2h(z1) dz1
∂z1Fg,n−1(z[ĵ])−
ωzj−σ̃(zj)(z1)
dz1 · 2h(zj)
∂zjFg,n−1
(
z[1̂]
)]
− 1
2h(z1)
∂2
∂u1∂u2
[
Fg−1,n+1
(
u1, u2, z[1̂]
)
(6.15)
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 41
+
stable∑
g1+g2=g
ItJ=[1̂]
Fg1,|I|+1(u1, zI)Fg2,|J |+1(u2, zJ)
]∣∣∣∣u1=z1
u2=z1
.
Let us apply principal specialization. The left-hand side becomes 1
n∂zFg,n(z, . . . , z). To calculate
the contributions from the first line of the right-hand side of (6), we choose j > 1 and set zi = z
for all i except for i = 1, j. Then take the limit zj → z1. In this procedure, we note that
the contributions from the simple pole of ωzj−σ̃(zj)(z1) at z1 = σ̃(zj) cancel at z1 = zj . Thus
we obtain
−
n∑
j=2
1
z1 − zj
(
1
2h(z1)
∂z1Fg,n−1(z1, z, . . . , z)−
1
2h(zj)
∂zjFg,n−1(zj , z, . . . , z)
)∣∣∣∣
z1=zj
= −
n∑
j=2
∂z1
(
1
2h(z1)
∂z1Fg,n−1(z1, z, . . . , z)
)
= −(n− 1)∂z1
(
1
2h(z1)
∂z1Fg,n−1(z1, z, . . . , z)
)
= −(n− 1)∂z1
(
1
2h(z1)
)
∂z1Fg,n−1(z1, z, . . . , z)−
n− 1
2h(z1)
∂2
z1Fg,n−1(z1, z, . . . , z).
The limit z1 → z then produces
−∂z
1
2h(z)
∂zFg,n−1(z, . . . , z)− 1
2h(z)
∂2
zFg,n−1(z, . . . , z)
+
(n− 1)(n− 2)
2h(z)
∂2
∂u1∂u2
Fg,n−1(u1, u2, z, . . . , z)
∣∣∣∣
u1=u2=z
. (6.16)
To calculate the principal specialization of the second line of the right-hand side of (6), we note
that since all points zi’s for i ≥ 2 are set to be equal, a set partition by index sets I and J becomes
a partition of n− 1 with a combinatorial factor that counts the redundancy. The result is
− 1
2h(z)
∂2
∂u1∂u2
Fg−1,n+1(u1, u2, z, . . . , z)
∣∣∣∣
u1=u2=z
− 1
2h(z)
stable∑
g1+g2=g
n1+n2=n−1
(
n− 1
n1
)
∂zFg1,n1+1(z, . . . , z) ∂zFg2,n2+1(z, . . . , z). (6.17)
Assembling (6.16) and (6.17) together, we obtain
1
2h(z)
[
∂2
zFg,n−1(z, . . . , z) +
stable∑
g1+g2=g
n1+n2=n−1
(
n− 1
n1
)
∂zFg1,n1+1(z, . . . , z) ∂zFg2,n2+1(z, . . . , z)
]
+
1
n
∂zFg,n(z, . . . , z) + ∂z
1
2h(z)
∂zFg,n−1(z, . . . , z)
=
(n− 1)(n− 2)
2h(z)
∂2
∂u1∂u2
Fg,n−1(u1, u2, z, . . . , z)
∣∣∣∣
u1=u2=z
− 1
2h(z)
∂2
∂u1∂u2
Fg−1,n+1(u1, u2, z, . . . , z)
∣∣∣∣
u1=u2=z
. (6.18)
42 O. Dumitrescu and M. Mulase
We now apply the operation
∑
2g−2+n=m
1
(n−1)! to (6.18) above, and write the result in
terms of
Sm(z) := Sm(x(z)) =
∑
2g−2+n=m−1
1
n!
Fg,n(z, . . . , z)
of (6.12) to fit into the WKB formalism (6.4). We observe that summing over all possibilities
of (g, n) with the fixed value of 2g − 2 + n, the right-hand side of (6.18) exactly cancels out.
Thus we have established that the functions Sm(z) of (6.12) for m ≥ 2 satisfy the recursion
formula
1
2h(z)
(
d2Sm
dz2
+
∑
a+b=m+1
a,b≥2
dSa
dz
dSb
dz
)
+
dSm+1
dz
+
d
dz
(
1
2h(z)
)
dSm
dz
= 0. (6.19)
Using (6.14) we identify the derivation
d
dx
=
y
h(z)
d
dz
, (6.20)
which is the push-forward π̃∗(d/dz) of the vector field d/dz. The transformation (6.20) is singular
at z = 0. If we allow terms a = 0 or b = 0 in (6.19), then what we have in addition is
1
2h(z)
· 2 dS0
dz
dSm+1
dz
=
1
h(z)
h(z)
y
dS0
dx
dSm+1
dz
=
dSm+1
dz
,
since dS0 = ydx. In other words, the dSm+1
dz term already there in (6.19) is absorbed in the split
differentiation for a = 0 and b = 0.
From (6.20) and (6.6), we find that the second derivative with respect to x is given by
d2
dx2
=
d
dx
(
S′0
h(z)
d
dz
)
=
(S′0)2
h(z)2
d2
dz2
+
S′0
h(z)
d
dz
(
S′0
h(z)
)
d
dz
,
denoting by S′0 = dS0/dx. Then (6.9) yields
(S′0)2
h(z)2
(
d2
dz2
Sm +
∑
a+b=m+1
dSa
dz
dSb
dz
)
+
S′0
h(z)
d
dz
(
S′0
h(z)
)
dSm
dz
= 0. (6.21)
The coefficients of dSm/dz in (6.21) are
2
(S′0)2
h(z)2
dS1
dz
+
S′0
h(z)
d
dz
(
S′0
h(z)
)
= 2
(S′0)2
h(z)2
h(z)
S′0
S′1 +
d
dx
(
S′0
h(z)
)
=
1
h(z)
(
2S′0S
′
1 + S′′0
)
+ S′0
d
dx
(
1
h(z)
)
= S′0
d
dx
(
1
h(z)
)
=
(S′0)2
h(z)2
· 2h(z)
d
dz
(
1
2h(z)
)
=
2(S′0)2
h(z)
d
dz
(
1
2h(z)
)
.
This is exactly what the last term of (6.19) has, after adjusting overall multiplication by
2(S′0)2
h(z) .
Therefore, we have established that (6.6), (6.7), and (6.8) make (6.9) equivalent to (6.19). This
competed the proof of Theorem 6.1. �
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 43
Remark 6.5. In the examples of various Hurwitz numbers considered in [13, 65], PDE recur-
sions can be applied to quantize the spectral curves (generalized Lambert curves) and obtain
second-order linear partial differential equations. Since spectral curves are analytic, the direct
quantization yields difference-differential equations in one variable. These different quantiza-
tion results are compatible in the sense that the same asymptotic solution (6.4) satisfies both
equations. A geometric interpretation is still missing for the analysis of these two different
quantization mechanisms.
Remark 6.6. Geometry of normalization of the singular spectral curve leads us to the analysis
of Stokes phenomena. It is beyond the scope of current paper, and will be treated elsewhere.
7 A simple classical example
Riemann and Poincaré worked on an interplay between algebraic geometry of curves in a ruled
surface and the asymptotic expansion of an analytic solution to a differential equation defined
on the base curve of the ruled surface. The theme of the current paper lies exactly on this
link, looking at this classical subject from a modern point of view. The simple examples for
SL(2,C)-meromorphic Higgs bundles on P1 illustrate the relation between a Higgs bundle, the
compactified cotangent bundle of a curve, a quantum curve, a classical differential equation, non-
Abelian Hodge correspondence, and the quantum invariants that the quantum curve captures.
7.1 The Higgs bundle for the Airy function
The Higgs bundle (E, φ) we consider consists of the base curve C = P1 and a particular vector
bundle
E0 = K
1
2
P1 ⊕K
− 1
2
P1 = OP1(−1)⊕OP1(1)
of rank 2 on P1. A meromorphic Higgs field is given by
φ =
[
φ11 φ12
φ21 φ22
]
: E −→ E ⊗KP1(m), m ≥ 0. (7.1)
Each matrix component is given by
φ11 : K
1
2
P1 −→ K
1
2
P1 ⊗KP1(m) = K
3
2
P1(m), φ11 ∈ H0(C,KP1(m)),
φ12 : K
− 1
2
P1 −→ K
1
2
P1 ⊗KP1(m) = K
3
2
P1(m), φ12 ∈ H0
(
C,K⊗2
P1 (m)
)
,
φ21 : K
1
2
P1 −→ K
− 1
2
P1 ⊗KP1(m) = K
1
2
P1(m), φ21 ∈ H0(C,OP1(m)),
φ22 : K
− 1
2
P1 −→ K
− 1
2
P1 ⊗KP1(m) = K
1
2
P1(m), φ22 ∈ H0(C,KP1(m)).
Since we are considering a point on a Hitchin section, we take φ21 = 1 to be the identity map
K
1
2
P1
∼=−→ K
1
2
P1 ↪→ K
1
2
P1(m), and φ11 = φ22 = 0. When we allow singularities, we can make other
choices for φ21 as well.
The Planck constant ~ has a geometric meaning (1.1) as a parameter of the extension classes
of line bundles. For P1, it is
~ ∈ Ext1
(
K
− 1
2
P1 ,K
1
2
P1
)
∼= H1
(
P1,KP1
)
= C.
It determines the unique extension
0 −→ K
1
2
P1 −→ E~ −→ K
− 1
2
P1 −→ 0, (7.2)
44 O. Dumitrescu and M. Mulase
where
E~ ∼=
{
OP1(−1)⊕OP1(1), ~ = 0,
OP1 ⊕OP1 , ~ 6= 0
(7.3)
as a vector bundle, since every vector bundle on P1 splits.
The quantization of the Higgs field φ is an ~-connection of Deligne in E~ defined on P1 and
is given by ~∇~, where
∇~ = d +
1
~
φ : E~ −→ E~ ⊗KP1(m), (7.4)
and d is the exterior differentiation operator acting on sections of the trivial bundle E~ for ~ 6= 0.
The operator ∇~ is a meromorphic connection in the vector bundle E~. Of course d+φ is never
a connection in general, because φ is a Higgs field belonging to a different bundle and satisfying
a different transition rule with respect to coordinate changes. However, as explained in Section 3
(see also [25, 29]), a Higgs field φ associated with a complex simple Lie group on a Hitchin section
gives rise to a connection ∇~ = d + 1
~φ in E~ with respect to the coordinate system associated
with a projective structure subordinating the complex structure of the base curve. Since our
examples are constructed on P1, the affine coordinate x ∈ A1 = P1 \{∞} is a natural coordinate
representing the projective structure. Hence the expression d + φ makes sense as a connection
in E~ for every SL(2,C)-Higgs bundle (E0, φ). This is due to the vanishing of the Schwarzian
derivatives for the coordinate change in a projective structure (see [25] for more detail on how
the Schwarzian derivative plays a role here).
To see the effect of quantization, i.e., the passage from the Higgs bundle to an ~-family of
connections (7.4), let us use the local coordinate and write everything concretely. The transition
function defined on C∗ = U∞ ∩ U0 of the vector bundle E0 on P1 = U∞ ∪ U0 is given by
[
x
1
x
]
,
where U0 = P1 \ {∞} = A1 and U∞ = P1 \ {0}. With respect to the same coordinate, the
extension E~ is given by
[
x ~
1
x
]
. The equality[
1
− 1
~x 1
] [
x ~
1
x
] [
−~
1
~ x
]
=
[
1
1
]
proves (7.3). The local expressions of the 1-form φ11 and the quadratic differential φ12 satisfy
(φ11)udu = (φ11)xdx, (φ12)udu
2 = (φ12)xdx2,
where u = 1/x is a coordinate on U∞. Then the local expressions of the Higgs field (7.1) with
φ22 = 0 satisfy the following transition relation with respect to E0:[
(φ11)u −(φ12)u
−1
]
du =
[
x
1
x
] [
(φ11)x (φ12)x
1
]
dx
[
x
1
x
]−1
.
The negative signs are due to du = − 1
x2 dx. For the case of SL(2,C)-Higgs bundles, we further
assume tr(φ) = φ11 = 0. In this case, we note that (7.1) is equivalent to the gauge transformation
rule of connection matrices with respect to E~:
−1
~
[
(φ12)u
1
]
du =
1
~
[
x ~
1
x
] [
(φ12)x
1
]
dx
[
x ~
1
x
]−1
− d
[
x ~
1
x
] [
x ~
1
x
]−1
.
In other words,
du −
1
~
[
(φ12)u
1
]
du =
[
x ~
1
x
](
dx +
1
~
[
(φ12)x
1
]
dx
)[
x ~
1
x
]−1
.
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 45
Therefore, ∇~ = d + 1
~φ is a globally defined connection in E~ for every ~ 6= 0, and(
~∇~)∣∣
~=0
= φ
is the original Higgs field.
Let us start with a particular spectral curve, the algebraic curve
Σ ⊂ F2 = P
(
KP1 ⊕OP1
)
= T ∗P1
embedded in the Hirzebruch surface with the defining equation
y2 − x = 0 (7.5)
on U0. Here, y is a fiber coordinate of the cotangent bundle T ∗P1 ⊂ F2 over U0. The Hirzebruch
surface is the natural compactification of the cotangent bundle T ∗P1, which is the total space
of the canonical bundle KP1 . We denote by η ∈ H0
(
T ∗P1, π∗KP1
)
the tautological 1-form asso-
ciated with the projection π : T ∗P1 −→ P1. It is expressed as η = ydx in terms of the affine
coordinates. The holomorphic symplectic form on T ∗P1 is given by −dη = dx∧dy. The 1-form η
extends to F2 as a meromorphic differential form and defines a divisor
(η) = C0 − C∞,
where C0 is the zero-section of T ∗P1, and C∞ the section at infinity of T ∗P1. The Picard
group Pic(F2) of the Hirzebruch surface is generated by the class C0 and a fiber class F of π.
Although (7.5) is a perfect parabola in the affine plane, it has a quintic cusp singularity at x =∞.
Let (u,w) be a coordinate system on another affine chart of F2 defined by{
x = 1/u,
y dx = v du, w = 1/v.
Then Σ in the (u,w)-plane is given by
w2 = u5. (7.6)
The expression of Σ ∈ NS(F2) as an element of the Néron–Severy group of F2, in this case the
same as Pic(F2), is thus given by Σ = 2C0 + 5F .
Define a stable Higgs pair (E0, φ(q)) on P1 with E0 = K
1
2
P1 ⊕K
− 1
2
P1 = OP1(−1)⊕OP1(1) and
φ(q) =
[
q
1
]
: E0 −→ E0 ⊗KP1(4).
Here, we choose a meromorphic quadratic differential q ∈ H0(P1,KP1(2)⊗2) that has a simple
zero at 0 ∈ P1 and a pole of order 5 at ∞ ∈ P1. Up to a constant factor, there is only one such
differential
q = x(dx)2 =
1
u5
(du)2 ∈ H0
(
P1,KP1(2)⊗2
)
= C.
The spectral curve Σ of (E0, φ(q)) is given by the characteristic equation
det(η − π∗φ) = η2 − π∗tr(φ) + π∗ det(φ) = η2 − q = 0 (7.7)
in F2. As explained above, (E0, φ(q)) uniquely determines a meromorphic oper
∇~(q) = d +
1
~
φ(q) = d +
1
~
[
q
1
]
(7.8)
46 O. Dumitrescu and M. Mulase
on the extension E~ of (7.2) over P1 [25]. Indeed, the case q = 0 of (7.8) for ~ = 1 is the
non-Abelian Hodge correspondence [24, 46, 73]
H0(C,End(E0)⊗KP1) 3
[
0 0
1 0
]
⇐⇒ d +
[
0 0
1 0
]
: E1 −→ E1 ⊗KP1 .
The quantization procedure of this paper is the following ~-deformation
H0(C,End(E0)⊗KP1(m)) 3
[
q
1
]
⇐⇒ d +
1
~
[
q
1
]
: E~ −→ E~ ⊗KP1(m),
where q ∈ H0
(
C,K⊗2
P1 ⊗OP1(m)
)
is a meromorphic quadratic differential on C.
Let ψ(x, ~) denote an analytic function in x with a formal parameter ~ such that
~∇~
q
[
−~ψ′
ψ
]
= 0, ~ 6= 0,
where ′ denotes the x differentiation. Then it satisfies a Schrödinger equation((
~
d
dx
)2
− x
)
ψ(x, ~) = 0. (7.9)
The differential operator
P (x, ~) :=
(
~
d
dx
)2
− x
quantizing the spectral curve Σ of (7.5) is an example of a quantum curve. Reflecting the
fact (7.6) that Σ has a quintic cusp singularity at x = ∞, (7.9) has an irregular singular point
of class 3
2 at x =∞.
Let us recall the definition of regular and irregular singular points of a second-order differential
equation here.
Definition 7.1. Let(
d2
dx2
+ a1(x)
d
dx
+ a2(x)
)
ψ(x) = 0 (7.10)
be a second-order differential equation defined around a neighborhood of x = 0 on a small disc
|x| < ε with meromorphic coefficients a1(x) and a2(x) with poles at x = 0. Denote by k (resp. `)
the order of the pole of a1(x) (resp. a2(x)) at x = 0. If k ≤ 1 and ` ≤ 2, then (7.10) has a regular
singular point at x = 0. Otherwise, consider the Newton polygon of the order of poles of the
coefficients of (7.10). It is the upper part of the convex hull of three points (0, 0), (1, k), (2, `).
As a convention, if aj(x) is identically 0, then we assign −∞ as its pole order. Let (1, r) be the
intersection point of the Newton polygon and the line x = 1. Thus
r =
k, 2k ≥ `,
`
2
, 2k ≤ `.
The differential equation (7.10) has an irregular singular point of class r − 1 at x = 0 if r > 1.
The class 3
2 at∞ indicates how the asymptotic expansion of the solution ψ looks like. Indeed,
any non-trivial solution has an essential singularity at ∞. We note that every solution of (7.9)
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 47
is an entire function for any value of ~ 6= 0. Applying our main result of this paper, we construct
a particular all-order asymptotic expansion of this entire solution
ψ(x, ~) = expF (x, ~), F (x, ~) :=
∞∑
m=0
~m−1Sm(x), (7.11)
valid for |Arg(x)| < π, and ~ > 0. Here, the first two terms of the asymptotic expansion are
given by
S0(x) = ±2
3
x
3
2 , (7.12)
S1(x) = −1
4
log x. (7.13)
Although the classical limit ~→ 0 of (7.9) does not make sense under the expansion (7.11), the
semi-classical limit through the WKB analysis[
e−S1(x)e−
1
~S0(x)
(
~2 d2
dx2
− x
)
e
1
~S0(x)eS1(x)
]
exp
( ∞∑
m=2
~m−1Sm(x)
)
= 0 (7.14)
has a well-defined limit ~ → 0. The result is S′0(x)2 = x, which gives (7.12), and also (7.5) by
defining dS0 = η. The vanishing of the ~-linear terms of (7.14) is 2S′0(x)S′1(x) + S′′0 (x) = 0,
which gives (7.13) above.
The entire solution in x for ~ 6= 0 and the choice of S0(x) = −2
3x
3
2 is called the Airy function
Ai(x, ~) =
1
2π
~−
1
6
∫ ∞
−∞
exp
(
ipx
~2/3
+ i
p3
3
)
dp. (7.15)
The surprising discovery of Kontsevich [55] (cf. [22, 76]) is that Sm(x) for m ≥ 2 has the following
closed formula
Sm(x) :=
∑
2g−2+n=m−1
1
n!
FAiry
g,n (x), (7.16)
FAiry
g,n (x) :=
(−1)n
22g−2+n
x−
(6g−6+3n)
2
∑
d1+···+dn
=3g−3+n
〈τd1 · · · τdn〉g,n
n∏
i=1
|(2di − 1)|!!, (7.17)
where the coefficients
〈τd1 · · · τdn〉g,n =
∫
Mg,n
ψd1
1 · · ·ψ
dn
n
are the cotangent class intersection numbers on the moduli spaceMg,n of stable curves of genus g
with n non-singular marked points. The expansion coordinate x
3
2 of (7.17) indicates the class
of the irregular singularity of the Airy differential equation.
Although (7.16) is not a generating function of all intersection numbers, the quantum cur-
ve (7.5) alone actually determines every intersection number 〈τd1 · · · τdn〉g,n. This mechanism is
topological recursion of [36]. PDE recursion computes free energies
FAiry
g,n (t1, . . . , tn) :=
(−1)n
22g−2+n
∑
d1+···+dn
=3g−3+n
〈τd1 · · · τdn〉g,n
n∏
i=1
(
ti
2
)2di+1
|(2di − 1)|!! (7.18)
as a function in n variables from Σ through the process of blow-ups of F2, and the exterior
derivative of free energies are the symplectic invariants of [36].
48 O. Dumitrescu and M. Mulase
7.2 Blowing up a Hirzebruch surface
Let us now give a detailed algebraic geometry procedure for this example. We start with the
spectral curve Σ of (7.5). Our goal is to come up with (7.9). The first step is to blow up F2 and
to construct a normalization of Σ. The construction of Bl(T ∗C) is given in Definition 4.4. It is
the minimal resolution of the divisor
Σ = (det(η − π∗φ))0
of the characteristic polynomial. The discriminant of the defining equation (7.7) of the spectral
curve is
−det(φ) = x(dx)2 =
1
u5
(du)2.
It has a simple zero at x = 0 and a pole of order 5 at x =∞. The geometric genus formula (4.9)
for the general base curve C reads
g(Σ̃) = 2g(C)− 1 +
1
2
δ,
where δ is the sum of the number of cusp singularities of Σ and the ramification points of π:
Σ −→ C (Theorem 4.2). In our case, it tells us that Σ̃ is a non-singular curve of genus 0, i.e.,
a P1, after blowing up b5
2c = 2 times.
The center of blow-up is (u,w) = (0, 0) for the first time. Put w = w1u, and denote by E1 the
exceptional divisor of the first blow-up. The proper transform of Σ for this blow-up, w2
1 = u3, has
a cubic cusp singularity, so we blow up again at the singular point. Let w1 = w2u, and denote
by E2 the exceptional divisor created by the second blow-up. The self-intersection of the proper
transform of E1 is −2. We then obtain the desingularized curve Σ̃, locally given by w2
2 = u.
The proof of Theorem 4.2 also tells us that Σ̃ −→ P1 is ramified at two points. Choose the affine
coordinate t = 2w2 of the exceptional divisor added at the second blow-up. Our choice of the
constant factor is to make the formula the same as in [31]. We have
x =
1
u
=
1
w2
2
=
4
t2
,
y = −u
2
w
= − u2
w2u2
= −2
t
.
(7.19)
In the (u,w)-coordinate, we see that the parameter t is a normalization parameter of the quintic
cusp singularity:
u =
t2
4
,
w =
t5
32
.
Note that Σ̃ intersects transversally with the proper transform of C∞. The blow-up space Bl(F2)
is the result of the twice blow-ups of the Hirzebruch surface:
Σ̃
π̃
��
ĩ //
ν
��
Bl(T ∗P1)
ν
&&
Σ
π
��
i // T ∗P1 = F2.
π
ssP1
(7.20)
Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 49
0
P
Σ
C0
−2
+2 C∞
BlpT ∗C
F
−1
−1
E
Q +1 C∞
C0
−1
−1
−2
E2
E1
F
C0
o
C∞R
BlQ(BlpT ∗C)
Figure 7.1.
Topological recursion (5.2) requires a globally defined meromorphic 1-form W0,1 on Σ̃ and
a symmetric meromorphic 2-form W0,2 on the product Σ̃× Σ̃ as the initial data. We choose{
W0,1 = ĩ∗ν∗η,
W0,2 = d1d2 logE
Σ̃
,
(7.21)
where E
Σ̃
is a normalized Riemann prime form on Σ̃ (see [27, Section 2]). The form W0,2 depends
only on the intrinsic geometry of the smooth curve Σ̃. The geometry of (7.20) is encoded in W0,1.
Now we apply PDE recursion (5.3) to the geometric data (7.20) and (7.21). We claim that
topological recursion of [36] for the geometric data we are considering now is exactly the same
as the recursive equation of [31, equation (6.12)] applied to the curve (7.19) realized as a plane
parabola in C2. This is because topological recursion (5.2) has two residue contributions, one
each from t = 0 and t = ∞. As proved in [31, Section 6], the integrand on the right-hand side
of the recursion formula [31, equation (6.12)] does not have any pole at t = 0. Therefore, the
residue contribution from this point is 0. PDE recursion is obtained by deforming the contour
of integration to enclose only poles of the differential forms Wg,n. Since t = 0 is a regular point,
the two methods have no difference.
The W0,2 of (7.21) is simply dt1·dt2
(t1−t2)2 because Σ̃ ∼= P1. Since t of (7.19) is a normalization
coordinate, we have
W0,1 = ĩ∗ν∗(η) = y(t) dx(t) =
16
t4
,
in agreement of [31, equation (6.8)]. Noticing that the solution to topological recursion is unique
from the initial data, we conclude that
d1 · · · dnFAiry
g,n
(
x(t1), . . . , x(tn)
)
= Wg,n.
By setting the constants of integration by integrating from t = 0 for PDE recursion, we obtain
the expression (7.18). Then its principal specialization gives (7.17). The equivalence of PDE
recursion and the quantum curve equation Theorem 6.1 then proves (7.9) with the expression
of (7.11) and (7.16).
In this process, what is truly amazing is that the single differential equation (7.9), which is
our quantum curve, knows everything about the free energies (7.18). This is because we can
recover the spectral curve Σ from the quantum curve. Then the procedures we need to apply, the
blow-ups and PDE recursion, are canonical. Therefore, we actually recover (7.18) as explained
above.
It is surprising to see that a simple entire function (7.15) contains so much geometric infor-
mation. Our expansion (7.11) is an expression of this entire function viewed from its essential
singularity. We can extract rich information of the solution by restricting the region where the
asymptotic expansion is valid. If we consider (7.11) only as a formal expression in x and ~,
then we cannot see how the coefficients are related to quantum invariants. Topological recur-
sion [36] is a key to connect the two worlds: the world of quantum invariants, and the world
50 O. Dumitrescu and M. Mulase
of holomorphic functions and differentials. This relation is also knows as a mirror symmetry,
or in analysis, simply as the Laplace transform. The intersection numbers 〈τd1 · · · τdn〉g,n belong
to the A-model, while the spectral curve Σ of (7.5) and free energies belong to the B-model.
We consider (7.18) as an example of the Laplace transform, playing the role of mirror sym-
metry [28, 31]. In the context of Hitchin theory, mirror symmetry also plays a different role
through Langland duality (cf. [48, 54, 78, 79]). It is unclear to us how these two different mirror
symmetries are interrelated.
Acknowledgements
The authors wish to thank Philip Boalch for many useful discussions and comments on their work
on quantum curves. In particular, his question proposed at the American Institute of Mathe-
matics Workshop, Spectral data for Higgs bundles in September–October 2015, was critical for
the development of the theory presented in this paper.
This joint research is carried out while the authors have been staying in the following ins-
titutions in the last several years: the American Institute of Mathematics in California, the
Banff International Research Station, Institutul de Matematică “Simion Stoilow” al Academiei
Rom�ane, Institut Henri Poincaré, Institute for Mathematical Sciences at the National University
of Singapore, Kobe University, Leibniz Universität Hannover, Lorentz Center Leiden, Mathema-
tisches Forschungsinstitut Oberwolfach, Max-Planck-Institut für Mathematik-Bonn, and Osaka
City University Advanced Mathematical Institute. Their generous financial support, hospitality,
and stimulating research environments are greatly appreciated.
The authors also thank Jørgen Andersen, Vincent Bouchard, Tom Bridgeland, Bertrand
Eynard, Edward Frenkel, Tamás Hausel, Kohei Iwaki, Maxim Kontsevich, Laura Schaposnik,
Carlos Simpson, Albert Schwarz, Yan Soibelman, Ruifang Song, Jörg Teschner, and Richard
Wentworth, for useful comments, suggestions, and discussions. During the preparation of this
work, the research of O.D. was supported by GRK 1463 Analysis, Geometry, and String Theory
at the Leibniz Universität Hannover, and a grant from MPIM-Bonn. The research of M.M. was
supported by IHÉS, MPIM-Bonn, NSF grants DMS-1104734, DMS-1309298, DMS-1619760,
DMS-1642515, and NSF-RNMS: Geometric Structures And Representation Varieties (GEAR
Network, DMS-1107452, 1107263, 1107367).
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1 Introduction
2 Rees D-modules as quantum curves for Higgs bundles
3 Opers
3.1 Holomorphic SL(r,C)-opers and quantization of Higgs bundles
3.2 Semi-classical limit of SL(r,C)-opers
3.3 Non-Abelian Hodge correspondence and a conjecture of Gaiotto
4 Geometry of singular spectral curves
5 A differential version of topological recursion
6 WKB analysis of quantum curves
7 A simple classical example
7.1 The Higgs bundle for the Airy function
7.2 Blowing up a Hirzebruch surface
References
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| id | nasplib_isofts_kiev_ua-123456789-211313 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T12:09:37Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Dumitrescu, Olivia Mulase, Motohico 2025-12-29T11:09:05Z 2021 Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles. Olivia Dumitrescu and Motohico Mulase. SIGMA 17 (2021), 036, 53 pages 1815-0659 2020 Mathematics Subject Classification: 14H15; 14N35; 81T45; 14F10; 14J26; 33C05; 33C10; 33C15; 34M60; 53D37 arXiv:1702.00511 https://nasplib.isofts.kiev.ua/handle/123456789/211313 https://doi.org/10.3842/SIGMA.2021.036 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 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 variant of the topological recursion of Eynard-Orantin and Mirzakhani for the context of singular Hitchin spectral curves. We show that a PDE version of topological recursion provides all-order WKB analysis for the Rees 𝒟-modules, defined as the quantization of Hitchin spectral curves associated with meromorphic SL(2, ℂ)-Higgs bundles. Topological recursion can be considered 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 recursion of topological type, agree for holomorphic and meromorphic SL(2, 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. The authors wish to thank Philip Boalch for many useful discussions and comments on their work on quantum curves. In particular, his question, proposed at the American Institute of Mathematics Workshop, Spectral data for Higgs bundles in September-October 2015, was critical for the development of the theory presented in this paper. The authors also thank Jürgen Andersen, Vincent Bouchard, Tom Bridgeland, Bertrand Eynard, Edward Frenkel, Tamas Hausel, Kohei Iwaki, Maxim Kontsevich, Laura Schaposnik, Carlos Simpson, Albert Schwarz, Yan Soibelman, Ruifang Song, Jörg Teschner, and Richard Wentworth for useful comments, suggestions, and discussions. During the preparation of this work, the research of O.D. was supported by GRK 1463 Analysis, Geometry, and String Theory at the Leibniz Universität Hannover and a grant from MPIM-Bonn. The research of M.M. was supported by IHES, MPIM-Bonn, NSF grants DMS-1104734, DMS-1309298, DMS-1619760, DMS-1642515, and NSF-RNMS: Geometric Structures And Representation Varieties (GEAR Network, DMS-1107452, 1107263, 1107367). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles Article published earlier |
| spellingShingle | Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles Dumitrescu, Olivia Mulase, Motohico |
| title | Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles |
| title_full | Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles |
| title_fullStr | Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles |
| title_full_unstemmed | Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles |
| title_short | Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles |
| title_sort | interplay between opers, quantum curves, wkb analysis, and higgs bundles |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211313 |
| work_keys_str_mv | AT dumitrescuolivia interplaybetweenopersquantumcurveswkbanalysisandhiggsbundles AT mulasemotohico interplaybetweenopersquantumcurveswkbanalysisandhiggsbundles |