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 $\mathcal{D}$-module on a smooth projective algebraic curve, whose semi-classical limit produces the Hitchin spectral curve of a Higgs bundle. We give a method of quantization of Hitchin spectral curves by concretely constructing one-parameter deformation families of opers. We propose a 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 $\mathcal{D}$-modules, defined as the quantization of Hitchin spectral curves associated with meromorphic $SL(2,\mathbb{C})$-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 $SL(2,\mathbb{C})$-Higgs bundles. Classical differential equations such as the Airy differential equation provides a typical example. Through these classical examples, we see that quantum curves relate Higgs bundles, opers, a conjecture of Gaiotto, and quantum invariants, such as Gromov--Witten invariants


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 geometry, when it is encoded in a holomorphic curve, that gives a generating function of Amodel 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 ∈ H 1 (C, K C ), (1.1) where K C is the canonical sheaf over C. The cohomology group H 1 (C, K C ) controls the deformation of a classical object, i.e., a geometric object such as a Higgs bundle in our case, into a quantum object, i.e., a non-commutative quantity such as a differential operator. In our case, the result of quantization is an oper. Using a fixed choice of a theta characteristic and a projective structure on C, we determine a unique quantization of the Hitchin spectral curve of a holomorphic or meromorphic SL(r, 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: 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 2 2g 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 the compactified cotangent bundle of C (see [5,56]), which is a ruled surface on the base C. The Hitchin spectral curve for a meromorphic Higgs bundle is defined as the divisor of zeros on T * C of the characteristic polynomial of φ: where η ∈ H 0 (T * C, π * K C ) 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 M Dol 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) ∈ M Dol defines the Hitchin fibration 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 −→ M Dol . 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 C r . Thus H is diagonal, X − is lower triangular, X + = X t − , and their relations are The map κ is defined by Clearly κ is not a section of the fibration µ H in a strict sense, because µ H • 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 µ H 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 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 (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 ∈ H 1 (C, K C ). For r = 2, E is the canonical extension With respect to the projective coordinate system, we can define a one-parameter family of opers E , ∇ (q) ∈ M deR for = 0, where ∇ (q) := d + 1 φ(q), (1.6) d is the exterior differentiation on C, and M deR 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 connection in general. Here, the point is that the original vector bundle E 0 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 , 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 differentiation. It is determined in the proof of Theorem 3.11. In Example 3.1 we list quantum curves P for r = 2, 3, 4. No matter how complicated its form is, the semi-classical limit of P recovers the spectral curve σ * Σ φ(q) of the Higgs field −φ(q), where σ : T * C −→ T * C, σ 2 = 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 genus g ≥ 2 and holomorphic SL(r, C)-Higgs bundles, our quantization process is exactly the 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 nonsingular curve, and the other is to blow up T * C and obtain a resolution of singularities Σ 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 corresponds 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 quantizing 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 polynomial 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 topological 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].

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 D C of differential operators on C is the subalgebra of the C-linear endomorphism algebra End C (O C ) generated by the anti-canonical sheaf K −1 C and the structure sheaf O C . Here, K −1 C acts on O C as holomorphic vector fields, and O C acts on itself by multiplication. Locally every element of D C is written as for some r ≥ 0. For a fixed r, we introduce the filtration by order of differential operators into D C as follows: The Rees ring D C is defined by An element of D C on a coordinate neighborhood U ⊂ C can be written as Definition 2.1 (Rees D-module, cf. [59]). The Rees construction be an effective divisor on C. The point set {p 1 , . . . , p n } ⊂ 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 Since the cotangent bundle is the total space of K C , we have the tautological 1-form η ∈ H 0 (T * C, π * K C ) on T * C coming from the projection The natural holomorphic symplectic form of T * C is given by −dη. The compactified cotangent bundle of C is a ruled surface defined by where I represents 1 ∈ O C being considered as a degree 1 element. The divisor at infinity is reduced in the ruled surface and supported on the subset P(K C ⊕ O C ) \ 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 where C 0 is the zero section of T * C.
The relation between the sheaf D C and the geometry of the compactified cotangent bundle T * C is the following. First we have Let us denote by gr m D C = F m D C F m−1 D C . By writing I = 1 ∈ H 0 (C, D C ), we then have 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 is a finite morphism of degree r. The spectral curve of a Higgs bundle (E, φ) is the divisor of zeros 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 The construction goes as follows: is the connection defined by the C-linear left-multiplication operation of K −1 C on D C , which satisfies the derivation property for f ∈ O C and P ∈ D C , and • β is the canonical right D C -module structure in K C 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 The connection ∇ of (2.2) is integrable because d 2 = 0. Actually, the statement is true for any dimensions. We note that there is no reason for E to be coherent as an O C -module. Conversely, if an algebraic vector bundle E on C of rank r admits a holomorphic connection ∇ : E −→ K C ⊗ E, then E acquires the structure of a D C -module. This is because ∇ is automatically flat, and the covariant derivative ∇ X for X ∈ K −1 C satisfies A repeated application of (2.4) makes E a D C -module. The fact that every D C -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| 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 When we consider holomorphic dependence of a quantum curve with respect to the quantization 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 quantization.
Remark 2.5. The classical limit of a formal -connection is the evaluation = 0 of ∇ , which is simply an O C -module homomorphism i.e., a holomorphic Higgs field in the vector bundle E.
-coherent D C -module is equivalent to a vector bundle on C equipped with an -connection.
In analysis, the semi-classical limit of a differential operator P (x, ) of the form (2.1) is a function defined by then determines the first term of the singular perturbation expansion, or the WKB asymptotic expansion, of a solution ψ(x, ) to the differential equation We note that the expression (2.7) is never meant to be a convergent series in .
Since dS 0 (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 * x C a fiber coordinate. Then (2.5) and (2.6) give an equation a (x, 0)y r− = 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 of the Rees ring D C of the sheaf of differential operators D C on U is the holomorphic function Definition 2.8 (semi-classical limit of a Rees D-module). Suppose a Rees D C -module M globally defined on C is written as on every coordinate neighborhood U ⊂ C with a differential operator P U of the form (2.1). Using this expression (2.1) for P U , we construct a meromorphic function a (x, 0)y r− (2.8) on T * C| U , where y is the fiber coordinate of T * C, which is trivialized on U . Define as the divisor of zero of the function p U (x, y). If Σ U 's glue together to a spectral curve Σ ⊂ T * C, then we call Σ the semi-classical limit of the Rees D C -module M.
Remark 2.9. For the local equation (2.8) to be consistent globally on C, the coefficients of (2.1) have to satisfy 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 D C -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 P 1 , whereas many important classical examples of differential equations are naturally defined over P 1 with regular and irregular singularities.
is said to be a meromorphic connection with poles along an effective divisor D if Then ∇ extends to Since ∇ is holomorphic on C \supp(D), it induces a D C\supp(D) -module structure in E| C\supp (D) .
as a D C -module. Equation (2.9) is called the meromorphic extension of the D C\supp(D) -module E| C\supp(D) . Let us take a local coordinate x of C, this time around a pole p j ∈ supp(D). If a generator P of E near x = 0 has a local expression around p j with locally defined holomorphic functions b (x), b 0 (0) = 0, and an integer k ∈ Z, then P has a regular singular point at p j . Otherwise, p j is an irregular singular point of P .
Definition 2.11 (quantum curve for a meromorphic Higgs bundle). Let (E, φ) be a meromorphic 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 D C -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 P 1 , 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.

Opers
There is a simple mechanism to construct a quantization of a Hitchin spectral curve, using a particular choice of isomorphism between a Hitchin section and the moduli of opers. The quantum 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 holomorphically 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.

Holomorphic SL(r, C)-opers and quantization of Higgs bundles
We first recall projective structures on C following Gunning [45]. Recall that every compact Riemann surface has a projective structure subordinating the given complex structure. A complex projective coordinate system is a coordinate neighborhood covering with a local coordinate x α of U α such that for every U α ∩ U β , we have a Möbius coordinate transformation Since we solve differential equations on C, we always assume that each coordinate neighborhood U α is simply connected. In what follows, we choose and fix a projective coordinate system on C. Since the transition function for the canonical line bundle K C of C is given by the cocycle We choose and fix, once and for all, a theta characteristic, or a spin structure, K 1 2 (3.1) The choice of ± here is an element of H 1 (C, Z/2Z) = (Z/2Z) 2g , indicating that there are 2 2g choices for spin structures in C.
The significance of the projective coordinate system lies in the fact that ∂ 2 β ξ αβ = 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 : 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 det E = O C and trφ = 0. We denote by M Dol the moduli space of stable holomorphic SL(r, C)-Higgs bundles on C. Hitchin [46] defines a holomorphic fibration that induces the structure of an algebraically completely integrable Hamiltonian system in M Dol .
With the choice of a spin structure K 1 2 C , we have a natural section κ : B → M Dol defined by utilizing Kostant's principal three-dimensional subgroup (TDS) [58] as follows.
First, let be an arbitrary point of the Hitchin base B. Define where s i := i(r − i). H is a diagonal matrix whose (i, i)-entry is s i − s i−1 = r − 2i + 1, with s 0 = s r = 0. The Lie algebra X + , X − , H ∼ = sl(2, C) is the Lie algebra of the principal TDS in SL(r, C).
Lemma 3.1. Define a Higgs bundle (E 0 , φ(q)) consisting of a vector bundle and a Higgs field Then it is a stable SL(r, C)-Higgs bundle. The Hitchin section is defined by which gives a biholomorphic map between B and κ(B) ⊂ M Dol .
Proof . We first note that X − : Similarly, since X −1 + is an upper-diagonal matrix with non-zero entries along the ( − 1)-th upper diagonal, we have is globally defined as a Higgs field in E 0 . The Higgs pair is stable because no subbundle of E 0 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 M Dol .
To define -connections holomorphically depending on , we need to construct a one-parameter holomorphic family of deformations of vector bundles and a C-linear first-order differential operator 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 ) ∼ = O C is a filtered extension of the vector bundle E 0 of (3.2) parametrized by ∈ H 1 (C, K C ) if the following conditions hold: • E has a filtration • The second term is given by .
Remark 3.3. Since we need to identify a deformation parameter and extension classes, we make the natural identification for every pair of vector bundles E and F . We also identify Ext 1 (E, F ) as the class of extensions of E by a vector bundle V . These identifications are done by a choice of a projective coordinate system on C as explained below. C and a non-zero element ∈ H 1 (C, K C ), there is a unique non-trivial filtered exten- Proof . First let us examine the case of r = 2 to see how things work. Since we have a unique extension corresponding to . Obviously which proves (3.7). We also note that as a vector bundle of rank 2, we have the isomorphism 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 .6) and (3.7), F r−2 is determined by the class in Assume that for a given i + 1, we have 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 Then F i−1 as an extension is determined by a class in The exact sequence for every m ≥ 1. Taking the cohomology long exact sequence, we obtain for m = 1, which proves (3.9) for n = i. Similarly, 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.
Filtration. There is a filtration F • by vector subbundles Griffiths transversality. The connection respects the filtration: Grading condition. The connection induces O C -module isomorphisms 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 ∈ H 1 (C, K C )-dependence in the filtered extensions. For example, the extension E of (3.8) is given by a system of transition functions The cocycle condition for the transition functions translates into a condition The application of the exterior differentiation d to the cocycle condition ξ αγ = ξ αβ ξ βγ yields Noticing that we see that solves (3.15). We note that Therefore, in the multiplicative sense, the extension class is determined by ∂ β log ξ αβ .
Proof . The cohomology long exact sequence From the exact sequence, we see that if d log{ξ αβ } = 0 ∈ H 1 (C, K C ), then it comes from a class in H 1 (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 c 1 K Remark 3.7. The same exact sequences in the above proof were used in [9] for constructing Bloch regulators of the algebraic K 2 -group. The torsion property of the Steinberg symbol of generators of K 2 (Σ) and quantizability of a spectral curve Σ was first discussed in [44] for the case of difference equations.
The class {σ αβ } of (3.16) gives a natural isomorphism H 1 (C, K C ) ∼ = C. We identify the deformation parameter ∈ C with the cohomology class { σ αβ } ∈ H 1 (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 U α , and write each q as {(q ) α } that satisfies the transition relation The transition function of the vector bundle E 0 is given by Since X − : E 0 −→ E 0 ⊗ K C is a global Higgs field, its local expressions {X − dx α } with respect to the projective coordinate system satisfies the transition relation on every U α ∩ U β . The same relation holds for the Higgs field φ(q) as well: We have the following: Then • The collection f αβ satisfies the cocycle condition hence it defines a holomorphic bundle on C. It is exactly the filtered extension E of Proposition 3.4.
• The locally defined differential operator for every = 0 forms a global holomorphic connection in E , i.e., gives rise to a oneparameter family of SL(r, C)-opers E , ∇ (q) ∈ M deR . In other words, the locally defined differential operator for every = 0 determines a global holomorphic connection in E satisfying the oper conditions.
• After a suitable gauge transformation depending on , the → ∞ limit of the oper ∇ (q) exists and is equal to ∇ =1 (0).
Hence the cocycle condition f αβ = f αγ f γβ follows. We note that the factor exp c in (3.24) produces exactly the cocycle ξ 2 αβ corresponding to the canonical sheaf K C . To prove (3.20), we use the power series expansion of the adjoint action We then find and Here, we have used the formula which follows from (3.1). Therefore, noticing (3.17), we obtain To prove (3.22), we need, in addition to (3.20), the following relation: 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, They all satisfy the cocycle condition with respect to the projective structure, and define a sequence 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 coming 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 Finally, the gauge transformation of ∇ (q) by a bundle automorphism which follows from the adjoint formula (3.25). Therefore, 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 ξ αβ .
The above construction theorem yields the following.
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 −→ λφ. The oper corresponding to 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 µ H , we need the gauge transformation. The action φ −→ λφ on the Hitchin section induces a weighted action through µ H . Then we have the gauge equivalence via the gauge transformation λ H 2 : 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 (E 0 , 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 Dmodules 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 U α with coordinate x α of the chosen projective coordinate system. A flat section Ψ α of E over U α is a solution of with an appropriate unknown function ψ.
There are r linearly independent solutions of (3.30), because q 2 , . . . , q r are represented by holomorphic functions on U α . The entries of X + are given by the formula for k = 0, 1, . . . , r − 1, where we use s k = s r−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 By solving (3.31) for k = 0, 1, . . . , r − 2 recursively, we obtain an expression of ψ k as a linear combination of with coefficients in differential polynomials of q 2 , q 3 , . . . , q k . Moreover, the coefficients of these differential polynomials are in Q[ ]. For example, Since ψ 1 is proportional to ψ , inductively we can show that the linear combination expression of ψ k 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 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 U α . 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 characteristic 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 U α with a projective coordinate x α . Let From the choice of local trivialization (3.30) we have an expression where ∆ is a constant diagonal matrix ∆ = δ ij (−1) r−i 1 √ s r−1 s r−2 · · · s i 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: Condition 3.13.
• A(q, ) is a nilpotent upper triangular matrix.
• Each entry a(q, ) i,j is a differential polynomial in q 2 , . . . , q r with coefficients in Q[ ].
• All diagonal and the first upper diagonal entries of A(q, ) are 0: The last property is because ψ k does not contain the (k − 1)-th derivative of ψ, as we have noted above. The diagonal matrix ∆ is designed so that Let us calculate the gauge transformation of ∇ (q) by ∆ + A(q, ): 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 By definition, B(q, ) satisfies the exact same properties listed in Condition 3.13 above. We then have Thus we define 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∆ −1 A(q, ) is an upper triangular 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 The basis vector Ψ 0 defined on a simply connected coordinate neighborhood U α is designed so that the following equation holds for any solution ψ of (3.30): Let ψ [1] , . . . , ψ [r] be r linearly independent solutions of (3.30). The Wronskian matrix is defined by . . . . . . . . . . . .
Thus (3.30) is equivalent to a single linear ordinary differential equation of order r In other words, P α (x α , ; q) is a generator of the Rees D-module E , ∇ (q) on U α . This expression of differential equation of order r is what is commonly referred to as a quantum curve. The coefficients ω i (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 P α (x α , ; 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 particular, 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 P α (x α , ; q) is exactly the characteristic polynomial.
As defined in Definition 2.8, the semi-classical limit of (3.40) is the limit where S 0 (x α ) is a holomorphic function on U α so that dS 0 = ydx α gives a local trivialization of T * C over U α . 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 S 0 (xα) I r×r , and then take the limit as goes to 0: A(q, 0)).
The determinant of the above matrix is the characteristic polynomial det (y + φ(q)). Note that from (3.35), we have Taking the determinant of the above, we conclude that This completes the proof of Theorem 3.11.
For every ∈ H 1 (C, K C ), the -connection E , ∇ (q) of (3.23) defines a global Rees D Cmodule structure in E . Thus we have constructed a universal family E C of Rees D C -modules on a given C with a fixed spin and a projective structures: The universal family S C of spectral curves is defined over C × B.
The semi-classical limit is thus a map of families (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 = ∪ α U α a projective coordinate system subordinating the complex structure of C. on C. Then the semi-classical limit of E(q) is the spectral curve Proof . The proof is exactly the same as Theorem 3.11 on each simply connected coordinate neighborhood U α ⊂ 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 P α (x α , ; q) of (3.40) for r = 2, 3, 4. These formulas show that our quantization procedure is quite nontrivial.

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 M Dol and M deR 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 E top the topologically trivial complex vector bundle of rank r on C. The prototype of the correspondence between stable holomorphic vector bundles on C and differential geometric data goes back to Narasimhan-Seshadri [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 E top , a unitary connection ∇ in E top 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: 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 E top , 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 Then the flatness of ∇(ζ) for all ζ is equivalent to (3.44).
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 (E 0 , φ(q)) on the SL(r, C)-Hitchin section (3.4). Consider the following two-parameter family of connections Then the scaling limit 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.  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 Conjecture 3.16, the limit oper of [26] is given by

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 is biholomorphic, unlike the non-Abelian Hodge correspondence.
Proof . The key point is that since E 0 is made out of K C , the fiber metric h naturally comes from the metric of C itself. Hitchin's equations (3.44) 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].

Geometry of singular spectral curves
The quantization mechanism of Section 3 applies to all Hitchin spectral curves, and it is not sensitive to whether the spectral curve is smooth or not. However, the local quantization mechanism of [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 Σ of the spectral curve is realized as a smooth divisor. This geometric information determines part of the initial data for topological recursion, i.e., the spectral curve of Eynard-Orantin, and the differential form W 0,1 . Geometry of spectral curve also gives us information of singularities of the quantization. For example, when we have a component of a spectral curve tangent to the divisor C ∞ , 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 determine the genus of the normalization Σ. This genus is necessary to identify the Riemann prime form on it, which determines another input datum W 0,2 for the topological recursion.
There are two different ways of defining the spectral curve for Higgs bundles with meromorphic Higgs field. Our definition uses the compactified cotangent bundle. This idea also appears in [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 ∞ ∈ P 1 . 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 E 0 , φ(q) defined on a smooth projective algebraic curve C of genus g ≥ 0 with a fixed projective structure. Here, and D is an effective divisor of C. The spectral curve is the zero-locus in T * C of the characteristic equation 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 E 0 , φ(q) .
Recall that Pic(T * C) is generated by the zero section C 0 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 where a j=1 p j ∈ Pic a (C) is a divisor on C of degree a. As an element of the Néron-Severi group NS(T * C) = Pic(T * C)/Pic 0 (T * C), it is simply 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 p a (Σ) = 1 2 Σ · (Σ + K T * C ) + 1 and we find that the arithmetic genus of the spectral curve Σ is 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]: The discriminant divisor of the spectral curve (4.1) is a divisor on C defined by Since q is a meromorphic section of K ⊗2 C , Theorem 4.2 (geometric genus formula). Define Then the geometric genus of the spectral curve Σ of (4.1) is given by We note that (4.7) implies δ ≡ 0 mod 2. 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 If it has two components, then it is locally isomorphic to 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 : Σ −→ Σ be the non-singular model of Σ. Thenπ = π • ν : Σ −→ C is a double sheeted covering of C by a smooth curve Σ. If Σ has two components at a singularity P as in (4.11), thenπ −1 (P ) consists of two points andπ is not ramified there. If P is a cusp (4.10), thenπ −1 (P ) is a ramification point of the coveringπ. 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 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 : Σ −→ Σ through a sequence of blow-ups of the ambient space T * C. This is because we need to construct differential forms on Σ 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.
is defined by blowing up T * C in the following way: • At each r i of (4.5), blow up r i ∈ Σ ∩ C 0 ⊂ T * C a total of m i 2 times.
• At each p j of (4.6), blow up at the intersection Σ ∩ π −1 (p j ) ⊂ C ∞ a total of n j 2 times.
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 ∈ H 0 (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 C 0 . 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 p g (Σ) = p a (Σ) = 4g − 3 + n from the Riemann-Hurwitz formula. As a divisor class, we have in agreement of (4.2).
Theorem 4.6. In the blow-up space Bl(T * C), we have the following.
• The Galois action σ : Σ −→ Σ lifts to an involution of Σ, and the morphismπ : Σ −→ C is a Galois covering with the Galois group Gal( Σ/C) (4.14) Proof . We need to consider only when ∆ is non-reduced. Let r i ∈ supp(∆) be a zero of q of degree m i > 1. The curve germ of Σ near r i ∈ Σ ∩ C 0 is given by a formula y 2 = x m i , where x is the base coordinate on C and y a fiber coordinate. We blow up once at (x, y) = (0, 0), using a local parameter y 1 = y/x on the exceptional divisor. The proper transform of the curve germ becomes y 2 1 = x m i −2 . Repeat this process at (x, y 1 ) = (0, 0), until we reach the equation where = 0 or 1. The proper transform of the curve germ is now non-singular. We see that after a sequence of m i 2 blow-ups starting at the point r i , the proper transform of Σ is simply ramified over r i ∈ C = C 0 if m i is odd, and unramified if m i is even. We apply the same sequence of blow-ups at each r i with multiplicity greater than 1.
Let p j ∈ supp(∆) be a pole of q with order n j > 1. The intersection P = Σ ∩ π −1 (p j ) lies on C ∞ , and Σ has a singularity at P . Let z = 1/y be a fiber coordinate of π −1 (p j ) at the infinity. Then the curve germ of Σ at the point P is given by The involution σ in this coordinate is simply z −→ −z. The blow-up process we apply at P is the same as before. After n j 2 blow-ups starting at the point P ∈ Σ ∩ π −1 (p j ), the proper transform of Σ is simply ramified over p j ∈ C if n j is odd, and unramified if n j is even. Again we do this process for all p j with a higher multiplicity.
The blow-up space Bl(T * C) is defined as the application of a total of m i=1 m i 2 + n j=1 n j 2 times blow-ups on T * C as described above. The proper transform Σ is the minimal resolution of Σ. Note that the morphism is a double covering, ramified exactly at δ points. Since p a ( Σ) = p g (Σ), (4.9) follows from the Riemann-Hurwitz formula applied toπ : Σ −→ 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 C 0 and C ∞ are point-wise invariant under the involution σ. Since Bl(T * C) is constructed by blowing up points on C 0 and C ∞ and their proper transforms, we have a natural liftσ : Bl(T * C) −→ Bl(T * C) of σ which induces (4.14).

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 expansions of solutions for quantum curves. Since a linear differential equation is characterized by its solutions, topological recursion can be used as a mechanism of defining the quantization process from a spectral curve to a quantum curve. Then a natural question arises: 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(z 1 , z 2 ) on the product Σ × Σ of a compact Riemann surface Σ 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 We also need to choose a symplectic basis for H 1 ( Σ, 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 projective algebraic curve together with a choice of a symplectic basis for H 1 (C, Z), andπ : Σ −→ C a degree 2 covering by another non-singular curve Σ. We denote by R the ramification divisor ofπ. The coveringπ is a Galois covering with the Galois group Z/2Z = σ , and R is the fixed-point divisor of the involutionσ. We also choose a labeling of points of R, and define the A-cycles of a symplectic bases for H 1 ( Σ, 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 W g,n on the Hilbert scheme Σ [n] of n-points on Σ for all g ≥ 0 and n ≥ 1 in the stable range 2g − 2 + n > 0, from given initial data W 0,1 and W 0,2 . The differential form W g,n is a meromorphic n-linear form, i.e., a 1-form on each factor of Σ [n] for 2g − 2 + n > 0.
• W 0,1 is a meromorphic 1-form on Σ to be prescribed according to the geometric setting we have. Then define which satisfiesσ * Ω = −Ω.
• W 0,2 is defined by where E(z 1 , z 2 ) is the A-cycle normalized Riemann prime form on Σ × Σ. W 0,2 is a meromorphic differential 1 ⊗ 1-form on Σ × Σ with 2nd order poles along the diagonal.
• We also define the normalized Cauchy kernel on Σ by 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 Σ. We thus choose a fundamental domain of the universal covering and restrict a and b in that domain for local calculations.
• The integration is taken with respect to z ∈ Γ. Thus we chose a fundamental domain of the universal covering of Σ that contains supp(Ω) 0 ∪ supp(R), and perform the integration locally as residue calculations.
• The expression 1/Ω is a meromorphic section of K −1 C which is multiplied to meromorphic quadratic differentials on Σ in z-variable.
• The sum runs over all partitions of g and set partitions of {2, . . . , n}, other than those containing the (0, 1)-geometry.
• z I = (z i ) i∈I .
Remark 5.3. The integrand of (5.2) is not a well-defined differential form in z ∈ Σ, due to the definition of ωσ (z)−z (z 1 ) 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 ωσ (z)−z (z 1 ) 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 Σ is of genus 0, or hyperelliptic.
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 U n of Σ [n] (or the universal covering, if the global treatment is necessary): 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., 2g 1 − 1 + |I| > 0 and 2g 2 − 1 + |J| > 0. The initial data for the PDE recursion are a function F 1,1 (z 1 ) on U and a symmetric function 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 ωσ (z)−z (z 1 ), W 0,2 (z, z j ), and W 0,2 σ(z), z j at z = z j and z =σ(z j ), 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( Σ) = 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 = z j , z =σ(z j ), j = 1, 2, . . . , n, are equivalent by continuous deformation of the contour Γ.

WKB analysis of quantum curves
We consider in this section a meromorphic SL(2, C)-Higgs bundle (E 0 , φ(q)) associated with a meromorphic quadratic differential q ∈ H 0 C, K C (D) ⊗2 with poles along an effective divisor D on a curve C of arbitrary genus. This includes the case of holomorphic Higgs bundles when g(C) ≥ 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 D C -module is generated by a single differential operator on each projective coordinate neighborhood U α .
Theorem 6.1 (WKB analysis for SL(2, C)-quantum curves). Let q ∈ H 0 C, K C (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 D C -module E on C as the quantization of the possibly singular spectral curve 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 D C -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 U α of C with a projective coordinate x α , so thatπ −1 (U α ) ⊂ Σ in the normalization (4.13) is also simply connected. Let z be a local coordinate ofπ −1 (U α ) and denote by a function x α = x α (z) the projectionπ.
• The formal WKB expansion we wish to construct is a solution to the equation of the specific form Here, F α (x α , ) 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 We interpret the above equation as an infinite sequence of ordinary differential equations for each power of : The symbol denotes the x α -derivative.
• Define where F g,n z(x α ) is the principal specialization of F g,n (z 1 , . . . , z n ) evaluated at a local section z = z(x α ) ofπ : Σ −→ C on U α .
Then (6.4) gives the WKB expansion of the solution to the generator of the Rees D C -module.
Remark 6.2. The WKB method is to solve (6.5) iteratively and find S m (x α ) 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 dS 0 (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).
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). For a function in one variable f (z), we have where ω z 2 −b (z 1 ) 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 = C 0 ⊂ T * C. The quadratic differential q has a pole of odd order at p, and the normalizationπ : Σ −→ C is simply ramified at a point Q ∈ Σ over p. We choose a local projective coordinate x on C centered at p. The Galois action ofσ on Σ 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 where z 0 = 1/y, y is the fiber coordinate on T * p C, and c(x) is a unit c(x) ∈ O * C,p . The quadratic differential q has a local expression Define z 1 = z 0 /x. The proper transform of Σ after the first blow up at P is locally written by Note that z 1 is an affine coordinate of the first exceptional divisor. Repeating this process µ-times, we end up with a coordinate z µ−1 = z µ x and an equation Here again, z µ is an affine coordinate of the exceptional divisor created by the µ-th blow-up. Write z = z µ so that the proper transform of the µ-times blow-ups is given by Note that the Galois action ofσ at Q is simply z −→ −z. Solving (6.13) as a functional equation, we obtain a Galois invariant local expression is a unit element. This function is precisely the local expression of the normalizationπ : Σ −→ C at Q ∈ Σ. Since we have thus obtained the normalization coordinate z on the desingularized curve Σ near Q: It gives a parametric equation for the singular spectral curve Σ: 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 ∈ Σ, we use the following local expressions: 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: [1] (6.15) + stable g 1 +g 2 =g I J= [1] F g 1 ,|I|+1 (u 1 , z I )F g 2 ,|J|+1 (u 2 , z J ) Let us apply principal specialization. The left-hand side becomes 1 n ∂ z F g,n (z, . . . , z). To calculate the contributions from the first line of the right-hand side of (6), we choose j > 1 and set z i = z for all i except for i = 1, j. Then take the limit z j → z 1 . In this procedure, we note that the contributions from the simple pole of ω z j −σ(z j ) (z 1 ) at z 1 =σ(z j ) cancel at z 1 = z j . Thus we obtain . . , z).
The limit z 1 → z then produces To calculate the principal specialization of the second line of the right-hand side of (6), we note that since all points z i '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 Assembling (6.16) and (6.17) together, we obtain 1 2h(z) ∂ 2 z F g,n−1 (z, . . . , z) + stable g 1 +g 2 =g n 1 +n 2 =n−1 n − 1 n 1 ∂ z F g 1 ,n 1 +1 (z, . . . , z) ∂ z F g 2 ,n 2 +1 (z, . . . , z) ∂ 2 ∂u 1 ∂u 2 F g,n−1 (u 1 , u 2 , z, . . . , z) We now apply the operation 2g−2+n=m 1 (n−1)! to (6.18) above, and write the result in terms of S m (z) := S m (x(z)) = 2g−2+n=m−1 1 n! F g,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 S m (z) of (6.12) for m ≥ 2 satisfy the recursion formula Using (6.14) we identify the derivation , (6.20) which is the push-forwardπ * (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 since dS 0 = ydx. In other words, the dS m+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 denoting by S 0 = dS 0 /dx. Then (6.9) yields The coefficients of dS m /dz in (6.21) are 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.

Remark 6.5.
In the examples of various Hurwitz numbers considered in [13,65], PDE recursions 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 quantization 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.

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 P 1 illustrate the relation between a Higgs bundle, the compactified cotangent bundle of a curve, a quantum curve, a classical differential equation, non-Abelian Hodge correspondence, and the quantum invariants that the quantum curve captures.

The Higgs bundle for the Airy function
The Higgs bundle (E, φ) we consider consists of the base curve C = P 1 and a particular vector bundle of rank 2 on P 1 . A meromorphic Higgs field is given by Each matrix component is given by Since we are considering a point on a Hitchin section, we take φ 21 = 1 to be the identity map P 1 (m), and φ 11 = φ 22 = 0. When we allow singularities, we can make other choices for φ 21 as well.
The Planck constant has a geometric meaning (1.1) as a parameter of the extension classes of line bundles. For P 1 , it is It determines the unique extension as a vector bundle, since every vector bundle on P 1 splits. The quantization of the Higgs field φ is an -connection of Deligne in E defined on P 1 and is given by ∇ , where and d is the exterior differentiation operator acting on sections of the trivial bundle E for = 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 P 1 , the affine coordinate x ∈ A 1 = P 1 \ {∞} 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 (E 0 , 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 ∞ ∩ U 0 of the vector bundle E 0 on P 1 = U ∞ ∪ U 0 is given by With respect to the same coordinate, the extension E is given by . The local expressions of the 1-form φ 11 and the quadratic differential φ 12 satisfy where u = 1/x is a coordinate on U ∞ . Then the local expressions of the Higgs field (7.1) with φ 22 = 0 satisfy the following transition relation with respect to E 0 : The negative signs are due to du = − 1 x 2 dx. For the case of SL(2, C)-Higgs bundles, we further assume tr(φ) = φ 11 = 0. In this case, we note that (7.1) is equivalent to the gauge transformation rule of connection matrices with respect to E : In other words, Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles 45 Therefore, ∇ = d + 1 φ is a globally defined connection in E for every = 0, and is the original Higgs field. Let us start with a particular spectral curve, the algebraic curve embedded in the Hirzebruch surface with the defining equation on U 0 . Here, y is a fiber coordinate of the cotangent bundle T * P 1 ⊂ F 2 over U 0 . The Hirzebruch surface is the natural compactification of the cotangent bundle T * P 1 , which is the total space of the canonical bundle K P 1 . We denote by η ∈ H 0 T * P 1 , π * K P 1 the tautological 1-form associated with the projection π : T * P 1 −→ P 1 . It is expressed as η = ydx in terms of the affine coordinates. The holomorphic symplectic form on T * P 1 is given by −dη = dx∧dy. The 1-form η extends to F 2 as a meromorphic differential form and defines a divisor where C 0 is the zero-section of T * P 1 , and C ∞ the section at infinity of T * P 1 . The Picard group Pic(F 2 ) of the Hirzebruch surface is generated by the class C 0 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 F 2 defined by Then Σ in the (u, w)-plane is given by The expression of Σ ∈ NS(F 2 ) as an element of the Néron-Severy group of F 2 , in this case the same as Pic(F 2 ), is thus given by Σ = 2C 0 + 5F .
Define a stable Higgs pair (E 0 , φ(q)) on P 1 with E 0 = K 1 2 Here, we choose a meromorphic quadratic differential q ∈ H 0 (P 1 , K P 1 (2) ⊗2 ) that has a simple zero at 0 ∈ P 1 and a pole of order 5 at ∞ ∈ P 1 . Up to a constant factor, there is only one such differential The spectral curve Σ of (E 0 , φ(q)) is given by the characteristic equation det(η − π * φ) = η 2 − π * tr(φ) + π * det(φ) = η 2 − q = 0 (7.7) in F 2 . As explained above, (E 0 , φ(q)) uniquely determines a meromorphic oper on the extension E of (7.2) over P 1 [25]. Indeed, the case q = 0 of (7.8) for = 1 is the non-Abelian Hodge correspondence [24,46,73] H 0 (C, End(E 0 ) ⊗ K P 1 ) 0 0 1 0 ⇐⇒ d + 0 0 1 0 : The quantization procedure of this paper is the following -deformation is a meromorphic quadratic differential on C. Let ψ(x, ) denote an analytic function in x with a formal parameter such that where denotes the x differentiation. Then it satisfies a Schrödinger equation The differential operator 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 d 2 dx 2 + a 1 (x) d dx + a 2 (x) ψ(x) = 0 (7.10) be a second-order differential equation defined around a neighborhood of x = 0 on a small disc |x| < with meromorphic coefficients a 1 (x) and a 2 (x) with poles at x = 0. Denote by k (resp. ) the order of the pole of a 1 (x) (resp. a 2 (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 a j (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 , 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) is an entire function for any value of = 0. Applying our main result of this paper, we construct a particular all-order asymptotic expansion of this entire solution ψ(x, ) = exp F (x, ), F (x, ) := ∞ m=0 m−1 S m (x), (7.11) valid for |Arg(x)| < π, and > 0. Here, the first two terms of the asymptotic expansion are given by S 0 (x) = ± 2 3 x 3 2 , (7.12) (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 m−1 S m (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 dS 0 = 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 surprising discovery of Kontsevich [55] (cf. [22,76]) is that S m (x) for m ≥ 2 has the following closed formula τ d 1 · · · τ dn g,n n i=1 |(2d i − 1)|!!, (7.17) where the coefficients τ d 1 · · · τ dn g,n = Mg,n ψ d 1 1 · · · ψ dn n are the cotangent class intersection numbers on the moduli space M g,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 curve (7.5) alone actually determines every intersection number τ d 1 · · · τ dn g,n . This mechanism is topological recursion of [36]. PDE recursion computes free energies

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 F 2 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 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 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 Σ is a non-singular curve of genus 0, i.e., a P 1 , after blowing up 5 2 = 2 times. The center of blow-up is (u, w) = (0, 0) for the first time. Put w = w 1 u, and denote by E 1 the exceptional divisor of the first blow-up. The proper transform of Σ for this blow-up, w 2 1 = u 3 , has a cubic cusp singularity, so we blow up again at the singular point. Let w 1 = w 2 u, and denote by E 2 the exceptional divisor created by the second blow-up. The self-intersection of the proper transform of E 1 is −2. We then obtain the desingularized curve Σ, locally given by w 2 2 = u. The proof of Theorem 4.2 also tells us that Σ −→ P 1 is ramified at two points. Choose the affine coordinate t = 2w 2 of the exceptional divisor added at the second blow-up. Our choice of the constant factor is to make the formula the same as in [31]. We have In the (u, w)-coordinate, we see that the parameter t is a normalization parameter of the quintic cusp singularity: Note that Σ intersects transversally with the proper transform of C ∞ . The blow-up space Bl(F 2 ) is the result of the twice blow-ups of the Hirzebruch surface:  Topological recursion (5.2) requires a globally defined meromorphic 1-form W 0,1 on Σ and a symmetric meromorphic 2-form W 0,2 on the product Σ × Σ as the initial data. We choose W 0,1 =ĩ * ν * η, W 0,2 = d 1 d 2 log E Σ , (7.21) where E Σ is a normalized Riemann prime form on Σ (see [27,Section 2]). The form W 0,2 depends only on the intrinsic geometry of the smooth curve Σ. The geometry of (7.20) is encoded in W 0,1 . 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 C 2 . 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 W g,n . Since t = 0 is a regular point, the two methods have no difference.
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 information. 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 recursion [36] is a key to connect the two worlds: the world of quantum invariants, and the world of holomorphic functions and differentials. This relation is also knows as a mirror symmetry, or in analysis, simply as the Laplace transform. The intersection numbers τ d 1 · · · τ dn 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 symmetry [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.