A construction of hyperkähler metrics through Riemann-Hilbert problems I
In 2009 Gaiotto, Moore and Neitzke presented a new construction of hyperkähler metrics on the total spaces of certain complex integrable systems, represented as a torus fibration over a base space , except for a divisor in , in which the torus fiber degenerates into a nodal torus. The hyperkähler metric is obtained via solutions of a Riemann-Hilbert problem. We interpret the Kontsevich-Soibelman Wall Crossing Formula as an isomonodromic deformation of a family of RH problems, therefore guaranteeing continuity of at the walls of marginal stability. The technical details about solving the different classes of Riemann-Hilbert problems that arise here are left to a second article. To extend this construction to singular fibers, we use the Ooguri-Vafa case as our model and choose a suitable gauge transformation that allow us to define an integral equation defined at the degenerate fiber, whose solutions are the desired Darboux coordinates . We show that these functions yield a holomorphic symplectic form , which, by Hitchin’s twistor construction, constructs the desired hyperkähler metric.
2010 Mathematics Subject Classification:Primary
Hyperkähler manifolds first appeared within the framework of differential geometry as Riemannian manifolds with holonomy group of special restricted group. Nowadays, hyperkähler geometry forms a separate research subject fusing traditional areas of mathematics such as differential and algebraic geometry of complex manifolds, holomorphic symplectic geometry, Hodge theory and many others.
One of the latest links can be found in theoretical physics: In 2009, Gaiotto, Moore and Neitzke  proposed a new construction of hyperkähler metrics on target spaces of quantum field theories with superysmmetry. Such manifolds were already known to be hyperkähler (see ), but no known explicit hyperkähler metrics have been constructed.
The manifold is a total space of a complex integrable system and it can be expressed as follows. There exists a complex manifold , a divisor and a subset such that is a torus fibration over . On the divisor , the torus fibers of degenerate, as Figure 1 shows.
Moduli spaces of Higgs bundles on Riemann surfaces with prescribed singularities at finitely many points are one of the prime examples of this construction. Hyperkähler geometry is useful since we can use Hitchin’s twistor space construction  and consider all -worth of complex structures at once. In the case of moduli spaces of Higgs bundles, this allows us to consider from three distinct viewpoints:
(Dolbeault) is the moduli space of Higgs bundles, i.e. pairs , a rank degree zero holomorphic vector bundle and a Higgs field.
(De Rham) is the moduli space of flat connections on rank holomorphic vector bundles, consisting of pairs with a holomorphic connection and
(Betti) of conjugacy classes of representations of the fundamental group of .
All these algebraic structures form part of the family of complex structures making into a hyperkähler manifold.
To prove that the manifolds from the integrable systems are indeed hyperkähler, we start with the existence of a simple, explicit hyperkähler metric on . Unfortunately, does not extend to . To construct a complete metric , it is necessary to do “quantum corrections” to . These are obtained by solving a certain explicit integral equation (see (2.12) below). The novelty is that the solutions, acting as Darboux coordinates for the hyperkähler metric , have discontinuities at a specific locus in . Such discontinuities cancel the global monodromy around and is thus feasible to expect that extends to the entire .
We start by defining a Riemann-Hilbert problem on the -slice of the twistor space . That is, we look for functions with prescribed discontinuities and asymptotics. In the language of Riemann-Hilbert theory, this is known as monodromy data. Rather than a single Riemann-Hilbert problem, we have a whole family of them parametrized by the manifold. We show that this family constitutes an isomonodromic deformation since by the Kontsevich-Soibelman Wall-Crossing Formula, the monodromy data remains invariant.
Although solving Riemann-Hilbert problems in general is not always possible, in this case it can be reduced to an integral equation solved by standard Banach contraction principles. We will focus on a particular case known as the “Pentagon” (a case of Hitchin systems with gauge group ). The family of Riemann-Hilbert problems and their methods of solutions is a topic of independent study so we leave this construction to a second article that can be of interest in the study of boundary-value problems.
The extension of the manifold is obtained by gluing a circle bundle with an appropriate gauge transformation eliminating any monodromy problems near the divisor . The circle bundle constructs the degenerate tori at the discriminant locus (see Figure 2).
On the extended manifold we prove that the solutions of the Riemann-Hilbert problem on extend and the resulting holomorphic symplectic form gives the desired hyperkähler metric .
Although for the most basic examples of this construction such as the moduli space of Higgs bundles it was already known that extends to a hyperkähler manifold with degenerate torus fibers, the construction here works for the general case of . Moreover, the functions here are special coordinates arising in moduli spaces of flat connections, Teichmüller theory and Mirror Symmetry. In particular, these functions are used in  for the construction of holomorphic discs with boundary on special Lagrangian torus fibers of mirror manifolds.
The organization of the paper is as follows. In Section 2 we introduce the complex integrable systems to be considered in this paper. These systems arose first in the study of moduli spaces of Higgs bundles and they can be written in terms of initial data and studied abstractly. This leads to a formulation of a family of Riemann-Hilbert problems, whose solutions provide Darboux coordinates for the moduli spaces considered and hence equip the latter with a hyperkähler structure. In Section 3 we fully work the simplest example of these integrable systems: the Ooguri-Vafa case. Although the existence of this hyperkähler metric was already known, this is the first time it is obtained via Riemann-Hilbert methods. In Section 4, we explicitly show that this metric is a smooth deformation of the well-known Taub-NUT metric near the singular fiber of thus proving its extension to the entire manifold. In Section 5 we introduce our main object of study, the Pentagon case. This is the first nontrivial example of the integrable systems considered and here the Wall Crossing phenomenon is present. We use the KS wall-crossing formula to apply an isomonodromic deformation of the Riemann-Hilbert problems leading to solutions continuous at the wall of marginal stability. Finally, Section 6 deals with the extension of these solutions to singular fibers of thought as a torus fibration. What we do is to actually complete the manifold from a regular torus fibration by gluing circle bundles near a discriminant locus . This involves a change of the torus coordinates for the fibers of . In terms of the new coordinates, the functions extend to the new patch and parametrize the complete manifold . We finish the paper by showing that, near the singular fibers of , the hyperkähler metric looks like the metric for the Ooguri-Vafa case plus some smooth corrections, thus proving that this metric is complete.
Acknowledgment: The author likes to thank Andrew Neitzke for his guidance, support and incredibly helpful conversations.
2. Integrable Systems Data
We start by presenting the complex integrable systems introduced in . As motivation, consider the moduli space of Higgs bundles on a complex curve with Higgs field having prescribed singularities at finitely many points. In , it is shown that the space of quadratic differentials on with fixed poles and residues is a complex affine space and the map is proper with generic fiber , a compact torus obtained from the spectral curve , a double-branched cover of over the zeroes of the quadratic differential . has an involution that flips . If we take to be the subgroup of odd under this involution, forms a lattice of rank 2 over , the space of quadratic differentials with only simple zeroes. This lattice comes with a non-degenerate anti-symmetric pairing from the intersection pairing in . It is also proved in  that the fiber can be identified with the set of characters . If denotes the tautological 1-form in , then for any ,
defines a holomorphic function in . Let be a local basis of with the dual basis of . Without loss of generality, we also denote by the pairing in . Let be short notation for . Since , .
We start with a complex manifold (later shown to be affine) of dimension and a divisor . Let . Over there is a local system with fiber a rank lattice, equipped with a non-degenerate anti-symmetric integer valued pairing .
We will denote by the dual of and, by abuse of notation, we’ll also use for the dual pairing (not necessarily integer-valued) in . Let denote a general point of . We want to obtain a torus fibration over , so let be the set of twisted unitary characters of 111Although we can also work with the set of unitary characters (no twisting involved) by shifting the -coordinates, we choose not to do so, as that results in more complex calculations, i.e. maps satisfying
Topologically, is a torus . Letting vary, the form a torus bundle over . Any local section gives a local angular coordinate of by “evaluation on ”, .
We also assume there exists a homomorphism such that the vector varies holomorphically with . If we pick a patch on which admits a basis of local sections in which is the standard symplectic pairing, then (after possibly shrinking ) the functions
are real local coordinates. The transition functions on overlaps are valued on , as different choices of basis in must fix the symplectic pairing. This gives an affine structure on .
By differentiating and evaluating in , we get 1-forms on which are linear on . For a local basis as in the previous paragraph, let denote its dual basis on . We write as short notation for
where we sum over repeated indices. Observe that the anti-symmetric pairing and the anti-symmetric wedge product of 1-forms makes (2.1) symmetric. We require that:
By (2.2), near , can be locally identified with a complex Lagrangian submanifold of .
In the example of moduli spaces of Higgs bundles, as approaches a quadratic differential with non-simple zeros, one homology cycles vanishes (see Figure 1). This cycle is primitive in and its monodromy around the critical quadratic differential is governed by the Picard-Lefschetz formula. In the general case, let be a component of the divisor . We also assume the following:
as for some .
is primitive (i.e. there exists some with ).
The monodromy of around is of “Picard-Lefschetz type”, i.e.
We assign a complex structure and a holomorphic symplectic form on as follows (see  and the references therein for proofs). Take a local basis of . If and is its dual, let
By linearity on of the 1-forms, is independent of the choice of basis. There is a unique complex structure on for which is of type (2,0). The 2-form gives a holomorphic symplectic structure on . With respect to this structure, the projection is holomorphic, and the torus fibers are compact complex Lagrangian submanifolds.
Recall that a positive 2-form on a complex manifold is a real 2-form for which for all real tangent vectors . From now on, we assume that is a positive 2-form on . Now fix . Then we can define a 2-form on by
This is a positive form of type (1,1) in the complex structure. Thus, the triple determines a Kähler metric on . This metric is in fact hyperkähler (see ), so we have a whole -worth of complex structures for , parametrized by . The above complex structure represents , the complex structure at in . The superscript stands for “semiflat”. This is because is flat on the torus fibers .
Alternatively, it is shown in  that if
Then the 2-form
(where the DeRham operator is applied to the part only) can be expressed as
for , that is, in the twistor space of , is a holomorphic section of (the twisting by is due to the poles at and in ). This is the key step in Hitchin’s twistor space construction. By [6, §3], is hyperkähler.
We want to reproduce the same construction of a hyperkähler metric now with corrected Darboux coordinates . For that, we need another piece of data. Namely, a function such that . Furthermore, we impose a condition on the nonzero . Introduce a positive definite norm on . Then we require the existence of such that
for those such that . This is called the Support Property, as in .
For a component of the singular locus and for the primitive element in for which as , we also require
To see where these invariants arise from, consider the example of moduli spaces of Higgs bundles again. A quadratic differential determines a metric on . Namely, if , . Let be the curve obtained after removing the poles and zeroes of . Consider the finite length inextensible geodesics on in the metric . These come in two types:
On the branched cover , each geodesic can be lifted to a union of closed curves in , representing some homology class . See Figure 5.
In this case, counts these finite length geodesics: every saddle connection with lift contributes and every closed geodesic with lift contributes .
Back to the general case, we’re ready to formulate a Riemann-Hilbert problem on the -slice of the twistor space . Recall that in a RH problem we have a contour dividing a complex plane (or its compactification) and one tries to obtain functions which are analytic in the regions defined by the contour, with continuous extensions along the boundary and with prescribed discontinuities along and fixed asymptotics at the points where is non-smooth. In our case, the contour is a collection of rays at the origin and the discontinuities can be expressed as symplectomorphisms of a complex torus:
Define a ray associated to each as:
We also define a transformation of the functions given by each :
Let denote the space of twisted complex characters of , i.e. maps satisfying
has a canonical Poisson structure given by
The glue together into a bundle over with fiber a complex Poisson torus. Let be the pullback of this system to . We can interpret the transformations as birational automorphisms of . To each ray going from 0 to in the -plane, we can define a transformation
Note that all the ’s involved in this product are multiples of each other, so the commute and it is not necessary to specify an order for the product.
To obtain the corrected , we can formulate a Riemann-Hilbert problem for which the former functions are solutions to it. We seek a map with the following properties:
depends piecewise holomorphically on , with discontinuities only at the rays for which .
The limits as approaches any ray from both sides exist and are related by
obeys the reality condition
For any , exists and is real.
In , this RH problem is formulated as an integral equation:
One can define recursively, setting :
More precisely, we have a family of RH problems, parametrized by , as this defines the rays , the complex torus where the symplectomorphisms are defined and the invariants involved in the definition of the problem.
We still need one more piece of the puzzle, since the latter function may not be continuous. In fact, jumps along a real codimension-1 loci in called the “wall of marginal stability”. This is the locus where 2 or more functions coincide in phase, so two or more rays become one. More precisely:
The jumps of are not arbitrary; they are governed by the Kontsevich-Soibelman wall-crossing formula.
To describe this, let be a strictly convex cone in the -plane with apex at the origin. Then for any define
The arrow indicates the order of the rational maps . is a birational Poisson automorphism of . Define a -good path to be a path in along which there is no point with and . (So as we travel along a -good path, no rays enter or exit V.) If are the endpoints of a -good path , the wall-crossing formula is the condition that are related by parallel transport in along . See Figure 6.
2.1. Statement of Results
We will restrict in this paper to the case , so . We want to extend the torus fibration to a manifold with degenerate torus fibers. To give an example, in the case of Hitchin systems, the torus bundle is not the moduli space of Higgs bundles yet, as we have to consider quadratic differentials with non-simple zeroes too. The main results of this paper center on the extension of the manifold to a manifold with an extended fibration such that the torus fibers degenerate to nodal torus (i.e. “singular” or “bad” fibers) for .
We start by fully working out the simplest example known as Ooguri-Vafa . Here we have a fibration over the open unit disk . At the discriminant locus , the fibers degenerate into a nodal torus. The local rank-2 lattice has a basis and the skew-symmetric pairing is defined by . The monodromy of around is . We also have functions , for holomorphic and admitting an extension to . Finally, the integer-valued function in is here: and for any other . There is no wall of marginal stability in this case. The integral equation (2.12) can be solved after just 1 iteration.
For all other nontrivial cases, in order to give a satisfactory extension of the coordinates, it was necessary to develop the theory of Riemann-Hilbert-Birkhoff problems to suit these infinite-dimensional systems (as the transformations defining the problem can be thought as operators on , rather than matrices). It is not clear that such coordinates can be extended, since we may approach the bad fiber from two different sides of the wall of marginal stability and obtain two different extensions. To overcome this first obstacle, we have to use the theory of isomonodromic deformations as in  to reformulate the Riemann-Hilbert problem in  independent of the regions determined by the wall.
Having redefined the problem, we want our to be smooth on the parameters and , away from where the prescribed jumps are. Even at , there was no mathematical proof that such condition must be true. In the companion paper , we combine classical Banach contraction methods and Arzela-Ascoli results on uniform convergence in compact sets to obtain:
If the collection of nonzero satisfies the support property (2.6) and if the parameter of (2.5) is large enough (determined by the values ), there exists a unique collection of functions with the prescribed asymptotics and jumps as in . These functions are smooth on and the torus coordinates (even for at the wall of marginal stability), and piecewise holomorphic on .
Since we’re considering only the case , is a rank-1 lattice over the Riemann surface and the discriminant locus where the torus fibers degenerate is a discrete subset of .
From this point on, we restrict our attention to the next nontrivial system, known as the Pentagon case . Here with 2 bad fibers which we can assume are at and is the twice-punctured plane. There is a wall of marginal stability where all are contained in the same line. This separates in two domains and a simply-connected . See Figure 7.
On we can trivialize and choose a basis with pairing . This basis does not extend to a global basis for since it is not invariant under monodromy. However, the set is indeed invariant so the following definition of makes global sense:
The Pentagon case appears in the study of Hitchin systems with gauge group . The extension of was previously obtained by hyperkähler quotient methods in , but no explicit hyperkähler metric was constructed.
Once the are obtained by Theorem 2.1, it is necessary to do an analytic continuation along for the particular for which as . Without loss of generality, we can assume there is a local basis of such that in . After that, an analysis of the possible divergence of as shows the necessity of performing a gauge transformation on the torus coordinates of the fibers that allows us to define an integral equation even at . This series of transformations are defined in (6.9), (6.10), (6.11) and (6.27), and constitute a new result that was not expected in . We basically deal with a family of boundary value problems for which the jump function vanishes at certain points and singularities of certain kind appear as . As this is of independent interest, we leave the relevant results to  and we show that our solutions contain at worst branch singularities at 0 or in the -plane. As in the case of normal fibers, we can run a contraction argument to obtain Darboux coordinates even at the singular fibers and conclude:
Let be a local basis for in a small sector centered at such that as . For the Pentagon integrable system, the local function admits an analytic continuation to a punctured disk centered at in . There exists a gauge transformation that extends the torus fibration to a manifold that is locally, for each point in , a (trivial) fibration over with fiber coordinatized by and with one fiber collapsed into a point. For big enough, it is possible to extend and to , still preserving the smooth properties as in Theorem 2.1.
After we have the smooth extension of the by Theorem 2.2, we can extend the holomorphic symplectic form labeled by as in  for all points except possibly one at the singular fiber. From we can obtain the hyperkähler metric and, in the case of the Pentagon, after a change of coordinates, we realize locally as the Taub-NUT metric plus smooth corrections, finishing the construction of and its hyperkähler metric. The following is the main theorem of the paper.
For the Pentagon case, the extension of the manifold constructed in Theorem 2.2 admits, for large enough, a hyperkähler metric obtained by extending the hyperkähler metric on determined by the Darboux coordinates .
3. The Ooguri-Vafa Case
3.1. Classical Case
We start with one of the simplest cases, known as the Ooguri-Vafa case, first treated in . To see where this case comes from, recall that by the SYZ picture of K3 surfaces , any K3 surface is a hyperkähler manifold. In one of its complex structures (say ) is elliptically fibered, with base manifold and generic fiber a compact complex torus. There are a total of 24 singular fibers, although the total space is smooth. See Figure 8.
Gross and Wilson  constructed a hyperkähler metric on a K3 surface by gluing in the Ooguri-Vafa metric constructed in  with a standard metric away from the degenerate fiber. Thus, this simple case can be regarded as a local model for K3 surfaces.
We have a fibration over the open unit disk . At the locus (in the literature this is also called the discriminant locus), the fibers degenerate into a nodal torus. Define as , the punctured unit disk. On there exists a local system of rank-2 lattices with basis and skew-symmetric pairing defined by . The monodromy of around is . We also have functions . On we have local coordinates for the torus fibers with monodromy . Finally, the integer-valued function in is here: and for any other . There is no wall of marginal stability in this case.
We call this the “classical Ooguri-Vafa” case as it is the one appearing in  already mentioned at the beginning of this section. In the next section, we’ll generalize this case by adding a function to the definition of .
These functions receive corrections defined as in . We are only interested in the pair which will constitute our desired Darboux coordinates for the holomorphic symplectic form . The fact that gives that . As , and approach 0. Thus . Since the actual is obtained after only 1 iteration of (2.13). For each , let be the ray in the -plane defined by . Similarly, .
For convenience, from this point on we assume is of the form , where is a positive number, is fixed and . Moreover, in , , for , and a similar parametrization holds in .
For fixed , as in (3.2) has a limit as .
Writing , we want to find the limit as of
For simplicity, we’ll focus in the first integral only, the second one can be handled similarly. Rewrite:
|and after a change of variables , we get|
Thus, (3.4) reduces to
If , (3.3) diverges to , in which case . Otherwise, is bounded away from 0. Consequently, in . As , the integrals are dominated by
if . Hence we can interchange the limit and the integral in (3.5) and obtain that, as , this reduces to
are the (unique) holomorphic solutions in the simply connected domain to the ODEs
This forces us to rewrite (3.6) uniquely as
Here denotes the principal branch of the log in both cases, and the equation makes sense for (recall that by construction, we have the additional datum ). We want to conclude that
still using the principal branch of the log. To see this, define as . This is an analytic function on and clearly . Thus is constant in . It is easy to show that the identity holds for a suitable choice of (for example, if is not real, choose ) and by the above, it holds on all of ; in particular, for .