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 -module on a smooth projective algebraic curve, whose semi-classical limit produces the Hitchin spectral curve of a Higgs bundle. We give a method of quantization of Hitchin spectral curves by concretely constructing one-parameter deformation families of opers.
We propose a generalization of the topological recursion of Eynard-Orantin and Mirzakhani for the context of singular Hitchin spectral curves. We show a surprising result that a PDE version of the topological recursion provides all-order WKB analysis for the Rees -modules, defined as the quantization of Hitchin spectral curves associated with meromorphic -Higgs bundles. Topological recursion is thus identified as a process of quantization of Hitchin spectral curves. We prove that these two quantizations, one via the construction of families of opers, and the other via the PDE topological recursion, agree for holomorphic and meromorphic -Higgs bundles.
Classical differential equations such as the Airy differential equation provides a typical example. Through these classical examples, we see that quantum curves relate Higgs bundles, opers, a conjecture of Gaiotto, and quantum invariants, such as Gromov-Witten invariants.
Key words and phrases:Quantum curve; Hitchin spectral curve; Higgs field; Rees D-module; opers; non-Abelian Hodge correspondence; mirror symmetry; Airy function; quantum invariants; WKB approximation; topological recursion.
2010 Mathematics Subject Classification:Primary: 14H15, 14N35, 81T45; Secondary: 14F10, 14J26, 33C05, 33C10, 33C15, 34M60, 53D37
- 1 Introduction
- 2 Rees -modules as quantum curves for Higgs bundles
- 3 Opers
- 4 Geometry of singular spectral curves
- 5 A differential version of topological recursion
- 6 WKB analysis of quantum curves
- 7 A simple classical example
The purpose of this paper is to construct a geometric theory of quantum curves. The notion of quantum curves was introduced in the physics literature (see for example, [1, 16, 17, 37, 38, 43, 54, 65]). A quantum curve is supposed to compactly capture topological invariants, such as certain Gromov-Witten invariants, Seiberg-Witten invariants, and quantum knot polynomials. Geometrically, a quantum curve is a unique quantization of the B-model geometry, when it is encoded in a holomorphic curve, that gives a generating function of A-model theory of genus for all . In a broad setting, a quantum curve can be a differential operator, a difference operator, a mixture of them, or a linear operator defined by a trace-class kernel function.
The geometric theory we present here is focused on the process of quantization of Hitchin spectral curves. A concise overview of our theory is available in . In Definition 2.10 and Definition 2.11, we introduce a quantum curve as a Rees -module on a smooth projective algebraic curve whose semi-classical limit is the Hitchin spectral curve associated with a Higgs bundle on . The process of quantization is therefore an assignment of a Rees -module to every Hitchin spectral curve.
The Planck constant is a deformation parameter that appears in the definition of Rees -modules. For us, it has a geometric meaning, and is naturally identified with an element
where is the canonical sheaf over . The cohomology group 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 , we determine a unique quantization of the Hitchin spectral curve of a holomorphic or meromorphic -Higgs bundle through a concrete construction of an -family of -opers on , 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 -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 of genus coming from the Fuchsian uniformization, our construction of opers is the same as those opers predicted by a conjecture of Gaiotto , as explained in Subsection 3.3. This conjecture has been solved in .
It has been noticed that topological recursion of  and its generalizations provide another aspect of quantization, most notably through the remodeling conjecture of Mariño  and his collaborators [11, 12], and its complete solution by mathematicians [33, 34]. From this point of view, a quantum curve is a quantization of B-model geometry that is obtained as an application of topological recursion. It then becomes a natural question:
What is the relation between quantization via topological recursion and the quantization through our construction of Rees -modules from Hitchin spectral curves?
The topological recursion was originally developed as a computational mechanism to calculate the multi-resolvent correlation functions of random matrices [14, 29]. As mentioned above, it generates a mirror symmetric B-model counterpart of genus A-model for all . This correspondence has been rigorously established for many examples [10, 19, 25, 27, 28, 31, 33, 34, 56, 57, 58, 59]. Yet so far still no clear geometric relation between topological recursion and quantum curves has been established.
The earliest striking application of topological recursion in algebraic geometry is the simple proofs obtained in  for the Witten conjecture on cotangent class intersection numbers and the -conjecture. Indeed, these celebrated formulas are straightforward consequences of the Laplace transform of a combinatorial formula known as the cut-and-join equations [36, 66]. This particular success of topological recursion is due to the fact that the spectral curve of the theory is of genus , and hence the residue calculations in the formalism of  can be explicitly performed. The result is a PDE topological recursion obtained as the Laplace transform of the combinatorial equation. The difficulty of topological recursion as a computational tool in a more general context comes from two fronts: one is the case of singular spectral curves, and the other is the impossibility of performing residue calculations appearing in the recursion formula for a high genus spectral curve.
A novel approach proposed in  is the implementation of PDE topological recursions, which appear naturally in enumerative geometry problems, to the context of Hitchin spectral curves. It replaces the integral topological recursion formulated in terms of residue calculations at the ramification divisor of a spectral curve by a recursive set of partial differential equations that captures local nature of topological recursion. As we explain in Section 5, the difference of the two recursion formulas lies in the choice of contours of integration in the original format of integral topological recursion. All other ingredients are the same. For a genus spectral curve, the two sets of recursions are equivalent. In general, these two recursions aim at achieving different goals. The original choice of contours should capture some global nature of periods hidden in the quantum invariants. Due to the impossibility of calculating the residues mentioned above, still we do not have a full understanding in this direction. The PDE topological recursion of [23, 24], on the other hand, captures local nature of the functions involved, and leads to an all-order WKB analysis of quantum curves for -Higgs bundles. The issue of singular spectral curves is addressed in , in which we have developed a systematic process of normalization of singular Hitchin spectral curves associated with meromorphic rank Higgs bundles.
Theorem 6.1 is our answer to Question 1.1. It states a surprising result that for the case of , the normalization process of  and the PDE topological recursion of  produce an all-order WKB expansion for the meromorphic Rees -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 topological recursion and construction of -family of opers are equivalent, for the case of -Higgs bundles.
The interplay between Rees -modules, -families of opers, Hitchin spectral curves as semi-classical limit, Gaiotto’s correspondence, and WKB analysis through PDE topological recursion, creates a sense of inevitability of the notion of quantization. Section 7 serves as an overview to this interplay, where we present the Airy differential equation as a prototypical example.
A totally new mathematical framework is presented in , in which Kontsevich and Soibelman formulate topological recursion as a special case of deformation quantization. They call the formalism Airy structures. In their work, spectral curves no longer serve as input data for topological recursion. Although construction of quantum curves is not the only purpose of the original topological recursion, what we present in our current paper is that our general procedure of quantization of Hitchin spectral curves has nothing to do with individual spectral curve, in parallel to the philosophy of . As we show in (3.59), the family of spectral curves is (re)constructed from our deformation family of Rees -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 -modules, without studying individual spectral curves. Investigating a possible connection between the Airy structures of  and this paper’s results is a future subject. A relation between quantum curves and deformation quantization is first discussed in .
Let us briefly describe our quantization process of this paper now. Our geometric setting is a smooth projective algebraic curve over of an arbitrary genus with a choice of a spin structure, or a theta characteristic, . There are choices of such spin structures. We choose any one of them. Let be an -Higgs bundle on with a meromorphic Higgs field . Denote by
for a meromorphic Higgs bundle is defined as the divisor of zeros on of the characteristic polynomial of :
where is the tautological -form on extended as a meromorphic -form on the compactification . The morphism is a degree map.
We denote by the moduli space of holomorphic stable -Higgs bundles on for . The assignment of the coefficients of the characteristic polynomial (1.4) to defines the Hitchin fibration
With the choice of a spin structure and Kostant’s principal three-dimensional subgroup TDS of , one constructs a cross-section . We denote by the Lie algebra of a principal TDS, where we use the standard representation as traceless matrices acting on . Thus is diagonal, is lower triangular, , and their relations are
The map is defined by
Clearly is not a section of the fibration in general because is not the identity map of for . Nonetheless, the image of intersects with every fiber of exactly at one point. Note that is the moduli space of Hitchin spectral curves associated with holomorphic -Higgs bundles on . We use an unconventional way of defining the universal family of spectral curves over , appealing to the Hitchin section , as
Now we choose and fix, once and for all, a projective coordinate system of subordinating the complex structure of . This process does not depend algebraically on the moduli space of . For a curve of genus , the Fuchsian projective structure, that appears in our solution  to a conjecture of Gaiotto , is a natural choice for our purpose. As we show in Section 3, there is a unique filtered extension for every . For , is the canonical extension
With respect to the projective coordinate system, we can define a one-parameter family of opers
for , where
is the exterior differentiation on C, and is the moduli space of holomorphic irreducible -connections on . 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 is deformed to , and we have chosen a projective coordinate system on . Therefore, (1.8) makes sense as a global connection on in .
Note that is Deligne’s -connection interpolating a connection and a Higgs field . We also note that defines a global Rees -module on . Its generator is a globally defined differential operator on that acts on , which is what we call the quantum curve of the Hitchin spectral curve corresponding to . The actual shape (3.55) of is quite involved due to non-commutativity of the coordinate of and differentiation. It is determined in the proof of Theorem 3.11. In Example 3.1 we list quantum curves for . No matter how complicated its form is, the semi-classical limit of recovers the spectral curve of the Higgs field , where
The above process can be generalized in a straightforward way to meromorphic spectral data for a curve of arbitrary genus. The corresponding connections , and hence the Rees -modules, then have regular and irregular singularities.
We note that when we use the Fuchsian projective coordinate system of a curve of genus and holomorphic -Higgs bundles, our quantization process is exactly the same as the construction of -opers of  that was established by solving a conjecture of Gaiotto .
In Section 6, we perform the generalized topological recursion in terms of the PDE version for the case of meromorphic -Higgs bundles. For this purpose, we use a normalization method of  for singular Hitchin spectral curves. We then show that the WKB analysis that the PDE recursion provides is exactly for the quantum curve constructed through (1.8). When we deal with a singular spectral curve , 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 locally to a non-singular curve, and the other is to blow up and obtain a resolution of singularities of . In this paper we will pursue the second path, and give a WKB analysis of the quantum curve using the geometric information of the desingularization.
Kostant’s principal TDS plays a crucial role in our quantization through the relation (1.6). For example, it selects a particular fixed point of -action on the Hitchin section, which corresponds to the limit of (1.8). It is counterintuitive, but this limit is the connection acting on , not just 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 -modules quantizing Hitchin spectral curves is presented in Section 2. Then in Section 3, we quantize Hitchin spectral curves as Rees -modules through a concrete construction -families of holomorphic and meromorphic -opers. The semi-classical limit of these resulting opers is calculated. Since our topological recursion depends solely on the geometry of normalization of singular Hitchin spectral curves, we provide detailed study of the blow-up process in Sections 4. We give the genus formula for the normalization of the spectral curve in terms of the characteristic polynomial of the Higgs field . Then in Section 5, we define topological recursions for the case of degree coverings. In Section 6, we prove that an all-order WKB analysis for quantization of meromorphic -Hitchin spectral curves is established through PDE topological recursion. We thus show that two quantizations procedures, one through -family of opers and the other through PDE topological recursion, agree for . 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 .
2. Rees -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 be a non-singular projective algebraic curve defined over . The sheaf of differential operators on is the subalgebra of the -linear endomorphism algebra generated by the anti-canonical sheaf and the structure sheaf . Here, acts on as holomorphic vector fields, and acts on itself by multiplication. Locally every element of is written as
for some . For a fixed , we introduce the filtration by order of differential operators into as follows:
The Rees ring is defined by
An element of on a coordinate neighborhood can be written as
Definition 2.1 (Rees -module).
The Rees construction
associated with a filtered -module is a Rees -module if the compatibility condition holds.
be an effective divisor on . The point set is the support of . A meromorphic Higgs bundle with poles at is a pair consisting of an algebraic vector bundle on and a Higgs field
Since the cotangent bundle
is the total space of , we have the tautological -form on coming from the projection
The natural holomorphic symplectic form of is given by . The compactified cotangent bundle of is a ruled surface defined by
where represents being considered as a degree element. The divisor at infinity
is reduced in the ruled surface and supported on the subset . The tautological -form extends on as a meromorphic -form with simple poles along . Thus the divisor of in is given by
where is the zero section of .
The relation between the sheaf and the geometry of the compactified cotangent bundle is the following. First we have
Let us denote by . By writing , we then have
Definition 2.2 (Spectral curve).
A spectral curve of degree is a divisor in such that the projection defined by the restriction
is a finite morphism of degree . The spectral curve of a Higgs bundle is the divisor of zeros
on of the characteristic polynomial . Here,
The Higgs field is holomorphic on . Thus we can define the divisor of zeros
of the characteristic polynomial on . The spectral curve is the complex topology closure of with respect to the compactification
A left -module on is naturally an -module with a -linear integrable (i.e., flat) connection . The construction goes as follows:
is the natural inclusion ;
is the connection defined by the -linear left-multiplication operation of on , which satisfies the derivation property
for and ; and
is the canonical right -module structure in defined by the Lie derivative of vector fields.
If we choose a local coordinate neighborhood with a coordinate , then (2.12) takes the following form. Let us denote by , and define
Then we have
The connection of (2.11) is integrable because . Actually, the statement is true for any dimensions. We note that there is no reason for to be coherent as an -module.
Conversely, if an algebraic vector bundle on of rank admits a holomorphic connection , then acquires the structure of a -module. This is because is automatically flat, and the covariant derivative for satisfies
for and . A repeated application of (2.13) makes a -module. The fact that every -module on a curve is principal implies that for every point , there is an open neighborhood and a linear differential operator of order on , called a generator, such that Thus on an open curve , a holomorphic connection in a vector bundle of rank gives rise to a differential operator of order . The converse is true if is -coherent.
Definition 2.4 (Formal -connection).
A formal -connection on a vector bundle is a -linear homomorphism
subject to the derivation condition
where and .
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.
The classical limit of a formal -connection is the evaluation of , which is simply an -module homomorphism
i.e., a holomorphic Higgs field in the vector bundle .
An -coherent -module is equivalent to a vector bundle on equipped with an -connection.
In analysis, the semi-classical limit of a differential operator of (2.2) is a function defined by
where . The equation
then determines the first term of the singular perturbation expansion, or the WKB asymptotic expansion,
of a solution to the differential equation
on U. We note that the expression (2.17) is never meant to be a convergent series in .
of a curve in . This motivates us to give the following definition:
Definition 2.7 (Semi-classical limit of a Rees differential operator).
Let be an open subset of with a local coordinate such that is trivial over with a fiber coordinate . The semi-classical limit of a local section
of the Rees ring of the sheaf of differential operators on is the holomorphic function
defined on .
Definition 2.8 (Semi-classical limit of a Rees -module).
Suppose a Rees -module globally defined on is written as
on , where is the fiber coordinate of , which is trivialized on . Define
as the divisor of zero of the function . If ’s glue together to a spectral curve , then we call the semi-classical limit of the Rees -module .
Definition 2.10 (Quantum curve for holomorphic Higgs bundle).
A quantum curve associated with the spectral curve of a holomorphic Higgs bundle on a projective algebraic curve is a Rees -module whose semi-classical limit is .
The main reason we wish to extend our framework to meromorphic connections is that there are no non-trivial holomorphic connections on , whereas many important classical examples of differential equations are naturally defined over with regular and irregular singularities. A -linear homomorphism
is said to be a meromorphic connection with poles along an effective divisor if
for every and . Let us denote by
Then extends to
Since is holomorphic on , it induces a -module structure in . The -module direct image associated with the open inclusion map is then naturally isomorphic to
as a -module. (2.23) is called the meromorphic extension of the -module .
Let us take a local coordinate of , this time around a pole . If a generator of near has a local expression
around with locally defined holomorphic functions , , and an integer , then has a regular singular point at . Otherwise, is an irregular singular point of .
Definition 2.11 (Quantum curve for a meromorphic Higgs bundle).
Let be a meromorphic Higgs bundle defined over a projective algebraic curve of any genus with poles along an effective divisor , and its spectral curve. A quantum curve associated with is the meromorphic extension of a Rees -module on such that the complex topology closure of its semi-classical limit in the compactified cotangent bundle agrees with .
In Section 3, we prove that every Hitchin spectral curve associated with a holomorphic or a meromorphic -Higgs bundle has a quantum curve.
We remark that several examples of quantum curves that are constructed in [10, 28, 57], for various Hurwitz numbers and Gromov-Witten theory of , do not fall into our definition in terms of Rees -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.
There is a simple mechanism to construct a quantization of Hitchin spectral curve, using a particular choice of isomorphism between a Hitchin section and the moduli of opers. The quantum deformation parameter , originated in physics as the Planck constant, is a purely formal parameter in WKB analysis. Since we will be using the PDE recursion (5.6) for the analysis of quantum curves, plays the role of a formal parameter for the asymptotic expansion. This point of view motivates our definition of quantum curves as Rees -modules in the previous section. However, the quantum curves appearing in the quantization of Hitchin spectral curves associated with -Higgs bundles for a complex simple Lie group 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 . Although a vast generalization of the current paper is possible, we restrict our attention to -opers for an arbitrary in this paper.
In this section, most of the time is a smooth projective algebraic curve of genus defined over , unless otherwise specified.
3.1. Holomorphic -opers and quantization of Higgs bundles
We first recall projective structures on following Gunning . 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 of such that for every , we have a Möbius coordinate transformation
Since we solve differential equations on , we always assume that each coordinate neighborhood is simply connected. In what follows, we choose and fix a projective coordinate system on . Since
the transition function for the canonical line bundle of is given by the cocycle
We choose and fix, once and for all, a theta characteristic, or a spin structure, such that . Let denote the -cocycle corresponding to . Then we have
The choice of here is an element of , indicating that there are choices for spin structures in .
The significance of the projective coordinate system lies in the fact that . This simple property plays an essential role in our construction of global connections on , as we see in this section. Another way of appreciating projective coordinate system is the vanishing of Schwarzian derivatives, as explained in . 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 is stable if for every vector subbundle that is invariant with respect to , i.e., , the slope condition
holds. The moduli space of stable Higgs bundles is constructed . An -Higgs bundle is a pair with a fixed isomorphism and . We denote by the moduli space of stable holomorphic -Higgs bundles on . Hitchin  defines a holomorphic fibration
that induces the structure of an algebraically completely integrable Hamiltonian system in . With the choice of a spin structure , we have a natural section defined by utilizing Kostant’s principal three-dimensional subgroup (TDS)  as follows.
be an arbitrary point of the Hitchin base . Define
where . is a diagonal matrix whose -entry is , with . The Lie algebra is the Lie algebra of the principal TDS in .
Define a Higgs bundle consisting of a vector bundle
and a Higgs field
Then it is a stable -Higgs bundle. The Hitchin section is defined by
which gives a biholomorphic map between and .
We first note that is a globally defined -valued -form, since it is a collection of constant maps
Similarly, since is an upper-diagonal matrix with non-zero entries along the -th upper diagonal, we have
Thus is globally defined as a Higgs field in . The Higgs pair is stable because no subbundle of is invariant under , unless . And when , the invariant subbundles all have positive degrees, since . ∎
The image is a holomorphic Lagrangian submanifold of a holomorphic symplectic space .
To define -connections holomorphically depending on , we need to construct a one-parameter holomorphic family of deformations of vector bundles
and a -linear first-order differential operator