A construction of hyperkähler metrics through Riemann-Hilbert problems I
César Garza










Department of Mathematics, IUPUI, Indianapolis, USA

cegarza@iu.edu









[2010]Primary 








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 ℳ over a base space ℬ, except for a divisor D in ℬ, in which the torus fiber degenerates into a nodal torus. The hyperkähler metric g is obtained via solutions 𝒳_γ 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 𝒳_γ 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 𝒳_γ. We show that these functions yield a holomorphic symplectic form ϖ(ζ), which, by Hitchin's twistor construction, constructs the desired hyperkähler metric.


A construction of hyperkähler metrics through Riemann-Hilbert problems I
    C. Garza
    
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§ INTRODUCTION


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 <cit.> proposed  a new construction of hyperkähler metrics g on target spaces ℳ of quantum field theories with d = 4, 𝒩 = 2 superysmmetry. Such manifolds were already known to be hyperkähler (see <cit.>), but no known explicit hyperkähler metrics have been constructed.

The manifold ℳ is a total space of a complex integrable system and it can be expressed as follows. There exists a complex manifold ℬ, a divisor D ⊂ℬ and a subset ℳ' ⊂ℳ such that ℳ' is a torus fibration over ℬ' := ℬ\ D. On the divisor D, the torus fibers of ℳ degenerate, as Figure <ref> shows.



Moduli spaces ℳ 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 <cit.> and consider all -worth of complex structures at once. In the case of moduli spaces of Higgs bundles, this allows us to consider ℳ from three distinct viewpoints:


	
  * (Dolbeault) ℳ_Dol is the moduli space of Higgs bundles, i.e. pairs (E, Φ), E → C a rank n degree zero holomorphic vector bundle and Φ∈Γ(End(E) ⊗Ω^1) a Higgs field.
	
	
  * (De Rham) ℳ_DR  is the moduli space of flat connections on rank n holomorphic vector bundles, consisting of pairs (E, ∇) with ∇ : E →Ω^1 ⊗ E a holomorphic connection and
	
	
  * (Betti) ℳ_B = Hom(π_1(C) →GL_n())/GL_n() of conjugacy classes of representations of the fundamental group of C. 


All these algebraic structures form part of the family of complex structures making ℳ into a hyperkähler manifold. 


To prove that the manifolds ℳ from the integrable systems are indeed hyperkähler, we start with the existence of a simple, explicit hyperkähler metric g^sf on ℳ'. Unfortunately, g^sf does not extend to ℳ.  To construct a complete metric g, it is necessary to do “quantum corrections” to g^sf. These are obtained by solving a certain explicit integral equation (see (<ref>) below). The novelty is that the solutions, acting as Darboux coordinates for the hyperkähler metric g, have discontinuities at a specific locus in ℬ. Such discontinuities cancel the global monodromy around D and is thus feasible to expect that g extends to the entire ℳ.


We start by defining a Riemann-Hilbert problem on the -slice of the twistor space 𝒵 = ℳ' ×. That is, we look for functions 𝒳_γ 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 ℳ' 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 SU(2)). 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 ℳ' is obtained by gluing a circle bundle with an appropriate gauge transformation eliminating any monodromy problems near the divisor D. The circle bundle constructs the degenerate tori at the discriminant locus D  (see Figure <ref>).



On the extended manifold ℳ we prove that the solutions 𝒳_γ of the Riemann-Hilbert problem on ℳ' extend and the resulting holomorphic symplectic form ϖ(ζ) gives the desired hyperkähler metric g. 

Although for the most basic examples of this construction such as the moduli space of Higgs bundles it was already known that ℳ' extends to a hyperkähler manifold ℳ with degenerate torus fibers, the construction here works for the general case of _ℬ = 1. Moreover, the functions 𝒳_γ here are special coordinates arising in moduli spaces of flat connections, Teichmüller theory and Mirror Symmetry. In particular, these functions are used in <cit.> 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 <ref> 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 ℳ considered and hence equip the latter with a hyperkähler structure. In Section <ref> 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 <ref>, we explicitly show that this metric is a smooth deformation of the well-known Taub-NUT metric near the singular fiber of ℳ thus proving its extension to the entire manifold. In Section <ref> 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 <ref> deals with the extension of these solutions 𝒳_γ to singular fibers of ℳ thought as a torus fibration. What we do is to actually complete the manifold ℳ from a regular torus fibration ℳ' by gluing circle bundles near a discriminant locus D. This involves a change of the torus coordinates for the fibers of ℳ'. In terms of the new coordinates, the 𝒳_γ functions extend to the new patch and parametrize the complete manifold ℳ. We finish the paper by showing that, near the singular fibers of ℳ, the hyperkähler metric g 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. 



§ INTEGRABLE SYSTEMS DATA



We start by presenting the complex integrable systems introduced in <cit.>. As motivation, consider the moduli space ℳ of Higgs bundles on a complex curve C with Higgs field Φ having prescribed singularities at finitely many points. In <cit.>, it is shown that the space of quadratic differentials u on C with fixed poles and residues is a complex affine space ℬ and the map det : ℳ→ℬ is proper with generic fiber Jac(Σ_u), a compact torus obtained from the spectral curve Σ_u : = {(z, ϕ) ∈ T^*C : ϕ^2 = u},  a double-branched cover of C over the zeroes of the quadratic differential u. Σ_u has an involution that flips ϕ↦ -ϕ. If we take Γ_u to be the subgroup of H_1(Σ_u, ℤ) odd under this involution, Γ forms a lattice of rank 2 over ℬ', the space of quadratic differentials with only simple zeroes. This lattice comes with a non-degenerate anti-symmetric pairing ⟨ , ⟩ from the intersection pairing in H_1. It is also proved in <cit.> that the fiber Jac(Σ_u) can be identified with the set of characters Hom(Γ_u, /2πℤ). If λ denotes the tautological 1-form in T^* C, then for any γ∈Γ,

    Z_γ = 1/π∮_γλ

defines a holomorphic function Z_γ in ℬ'. Let {γ_1, γ_2} be a local basis of Γ with {γ^1, γ^2} the dual basis of Γ^*. Without loss of generality, we also denote by ⟨ ,  ⟩ the pairing in Γ^*. Let ⟨ dZ ∧ dZ ⟩ be short notation for ⟨γ^1, γ^2 ⟩ dZ_γ_1∧ dZ_γ_2. Since _ℬ' = 1, ⟨ dZ ∧ dZ ⟩ = 0.

This type of data arises in the construction of hyperkähler manifolds as in <cit.> and <cit.>, so we summarize the conditions required:
 

We start with a complex manifold ℬ (later shown to be affine) of dimension n and a divisor D ⊂ℬ. Let ℬ' = ℬ\ D. Over ℬ' there is a local system Γ with fiber a rank 2n lattice, equipped with a non-degenerate anti-symmetric integer valued pairing ⟨  , ⟩. 

We will denote by Γ^* the dual of Γ and, by abuse of notation, we'll also use ⟨  , ⟩ for the dual pairing (not necessarily integer-valued) in Γ^*. Let u denote a general point of ℬ'. We want to obtain a torus fibration over ℬ', so let TChar_u(Γ) be the set of twisted unitary characters of Γ_u[Although we can also work with the set of unitary characters (no twisting involved) by shifting the θ-coordinates, we choose not to do so, as that results in more complex calculations], i.e. maps θ : Γ_u →ℝ/2πℤ satisfying

    θ_γ + θ_γ' = θ_γ + γ' + π⟨γ, γ' ⟩.

Topologically, TChar_u(Γ) is a torus (S^1)^2n. Letting u vary, the TChar_u(Γ) form a torus bundle ℳ' over ℬ'. Any local section γ gives a local angular coordinate of ℳ' by “evaluation on γ”, θ_γ : ℳ' →ℝ/2πℤ.

We also assume there exists a homomorphism Z : Γ→ such that the vector Z(u) ∈Γ^*_u ⊗ varies holomorphically with u. If we pick a patch U ⊂ℬ' on which Γ admits a basis {γ_1, …, γ_2n} of local sections in which ⟨ , ⟩ is the standard symplectic pairing, then (after possibly shrinking U) the functions

    f_i = Re(Z_γ_i)

are real local coordinates. The transition functions on overlaps U ∩ U' are valued on Sp(2n, ℤ), as different choices of basis in Γ must fix the symplectic pairing. This gives an affine structure on ℬ'.

 By differentiating and evaluating in γ, we get 1-forms dθ_γ, d Z_γ on ℳ' which are linear on Γ. For a local basis {γ_1, …, γ_2n} as in the previous paragraph, let {γ^1, …, γ^2n} denote its dual basis on Γ^*. We write ⟨ dZ ∧ dZ ⟩ as short notation for
 
    ⟨γ^i , γ^j ⟩ dZ_γ_i∧ dZ_γ_j,

 where we sum over repeated indices. Observe that the anti-symmetric pairing ⟨  , ⟩ and the anti-symmetric wedge product of 1-forms makes (<ref>) symmetric.  We require that:
 

    ⟨ dZ ∧ dZ ⟩ = 0,


By (<ref>), near u, ℬ' can be locally identified with a complex Lagrangian submanifold of Γ^* ⊗_ℤ.

In the example of moduli spaces of Higgs bundles, as u approaches a quadratic differential with non-simple zeros, one homology cycles vanishes (see Figure <ref>). This cycle γ_0 is primitive in H_1 and its monodromy around the critical quadratic differential is governed by the Picard-Lefschetz formula. In the general case, let D_0 be a component of the divisor D ⊂ℬ. We also assume the following:


	
  * Z_γ_0(u) → 0 as u → u_0 ∈ D_0 for some γ_0 ∈Γ.
	
	
	
	
  * γ_0 is primitive (i.e. there exists some γ' with ⟨γ_0, γ'⟩ = 1).
	
	
  * The monodromy of Γ around D_0 is of “Picard-Lefschetz type”, i.e.
	
    γ↦γ + ⟨γ, γ_0 ⟩γ_0




We assign a complex structure and a holomorphic symplectic form on ℳ' as follows (see <cit.> and the references therein for proofs). Take a local basis {γ_1, …, γ_2n} of Γ. If ϵ^ij = ⟨γ_i , γ_j⟩ and ϵ_ij is its dual, let

    ω_+ = ⟨ dZ ∧ dθ⟩ = ϵ_ij  dZ_γ_i∧ dθ_γ_j.

By linearity on γ of the 1-forms, ω_+ is independent of the choice of basis. There is a unique complex structure J on ℳ' for which ω_+ is of type (2,0). The 2-form ω_+ gives a holomorphic symplectic structure on (ℳ', J). With respect to this structure, the projection π: ℳ' →ℬ' is holomorphic, and the torus fibers ℳ'_u = π^-1(u) are compact complex Lagrangian submanifolds. 

Recall that a positive 2-form ω on a complex manifold is a real 2-form for which ω(v,Jv) >0 for all real tangent vectors v. From now on, we assume that ⟨ dZ ∧ dZ⟩ is a positive 2-form on ℬ'. Now fix R > 0. Then we can define a 2-form on ℳ' by

    ω_3^sf = R/4⟨ dZ ∧ dZ⟩ - 1/8π^2 R⟨ dθ∧ dθ⟩.

This is a positive form of type (1,1) in the J complex structure. Thus, the triple (ℳ', J, ω_3^sf) determines a Kähler metric g^sf on ℳ'. This metric is in fact hyperkähler (see <cit.>), so we have a whole -worth of complex structures for ℳ', parametrized by ζ∈. The above complex structure J represents J(ζ = 0), the complex structure at ζ = 0 in . The superscript ^sf stands for “semiflat”. This is because g^sf is flat on the torus fibers ℳ'_u.

Alternatively, it is shown in <cit.> that if

    𝒳_γ^sf(ζ) = exp( π R Z_γ/ζ + iθ_γ + π R ζZ_γ)

Then the 2-form

    ϖ(ζ) = 1/8π^2 R⟨ dlog𝒳^sf(ζ) ∧ dlog𝒳^sf(ζ) ⟩

(where the DeRham operator d is applied to the ℳ' part only) can be expressed as

    -i/2ζω_+ + ω^sf_3 -i ζ/2ω_-,

for ω_- = ω_+ = ⟨ dZ∧ dθ⟩, that is, in the twistor space 𝒵 = ℳ' × of <cit.>, ϖ(ζ) is a holomorphic section of Ω_𝒵/⊗𝒪(2) (the twisting by 𝒪(2) is due to the poles at ζ = 0 and ζ = ∞ in ). This is the key step in Hitchin's twistor space construction. By <cit.>, ℳ' is hyperkähler.

We want to reproduce the same construction of a hyperkähler metric now with corrected Darboux coordinates 𝒳_γ(ζ). For that, we need another piece of data. Namely, a function Ω : Γ→ℤ such that Ω(γ;u) = Ω(-γ;u). Furthermore, we impose a condition on the nonzero Ω(γ;u). Introduce a positive definite norm on Γ. Then we require the existence of K > 0 such that

    |Z_γ|/γ > K

for those γ such that Ω(γ; u) ≠ 0. This is called the Support Property, as in <cit.>.

For a component of the singular locus D_0 and for γ_0 the primitive element in Γ for which Z_γ_0→ 0 as u → u_0 ∈ D_0, we also require

    Ω(γ_0; u) = 1  for all u in a neighborhood of D_0


To see where these invariants arise from, consider the example of moduli spaces of Higgs bundles again. A quadratic differential u ∈ℬ' determines a metric h on C. Namely, if u = P(z)dz^2, h = |P(z)| dz dz. Let C' be the curve obtained after removing the poles and zeroes of u. Consider the finite length inextensible geodesics on C' in the metric h. These come in two types:


	
  * Saddle connections: geodesics running between two zeroes of u. See Figure <ref>.
	
	


  * Closed geodesics: When they exist, they come in 1-parameter families sweeping out annuli in C'. See Figure <ref>.

	



On the branched cover Σ_u → C, each geodesic can be lifted to a union of closed curves in Σ_u, representing some homology class γ∈ H_1(Σ_u, ℤ). See Figure <ref>.

	


 In this case, Ω(γ,u) counts these finite length geodesics: every saddle connection with lift γ contributes +1 and every closed geodesic with lift γ contributes -2.


Back to the general case, we're ready to formulate a Riemann-Hilbert problem on the -slice of the twistor space 𝒵 = ℳ' ×. Recall that in a RH problem we have a contour Σ 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 Σ and fixed asymptotics at the points where Σ 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 γ∈Γ_u as:

    ℓ_γ(u) = Z_γℝ_-.

We also define a transformation of the functions 𝒳_γ' given by each γ∈Γ_u:

    𝒦_γ𝒳_γ' = 𝒳_γ' (1- 𝒳_γ)^⟨γ', γ⟩

Let T_u denote the space of twisted complex characters of Γ_u, i.e. maps 𝒳 : Γ_u →^× satisfying

    𝒳_γ𝒳_γ' = (-1)^⟨γ, γ'⟩𝒳_γ + γ'

T_u has a canonical Poisson structure given by

    {𝒳_γ,  𝒳_γ'} = ⟨γ, γ' ⟩𝒳_γ + γ'

The T_u glue together into a bundle over ℬ' with fiber a complex Poisson torus. Let T be the pullback of this system to ℳ'. We can interpret the transformations 𝒦_γ as birational automorphisms of T.
To each ray ℓ going from 0 to ∞ in the ζ-plane, we can define a transformation

    S_ℓ = ∏_γ : ℓ_γ(u) = ℓ𝒦_γ^Ω(γ;u)

Note that all the γ's involved in this product are multiples of each other, so the 𝒦_γ commute and it is not necessary to specify an order for the product.

To obtain the corrected 𝒳_γ, we can formulate a Riemann-Hilbert problem for which the former functions are solutions to it.  We seek a map 𝒳 : ℳ'_u ×^×→ T_u with the following properties:

	
  * 𝒳 depends piecewise holomorphically on ζ, with discontinuities only at the rays ℓ_γ(u) for which Ω(γ;u) ≠ 0.
	
  * The limits 𝒳^± as ζ approaches any ray ℓ from both sides exist and are related by
	
    𝒳^+ = S_ℓ^-1∘𝒳^-

	
  * 𝒳 obeys the reality condition
	
    𝒳_-γ(-1/ζ) = 𝒳_γ(ζ)

	
  * For any γ∈Γ_u, lim_ζ→ 0𝒳_γ(ζ) / 𝒳^sf_γ(ζ) exists and is real. 


In <cit.>, this RH problem is formulated as an integral equation:

    𝒳_γ(u,ζ) = 𝒳^sf_γ(u,ζ)exp[ -1/4π i∑_γ'Ω(γ';u) ⟨γ, γ' ⟩∫_ℓ_γ'(u)dζ'/ζ'ζ'+ζ/ζ'-ζlog( 1 - 𝒳_γ'(u,ζ'))],

One can define recursively, setting 𝒳^(0) = 𝒳^sf:

    𝒳^(ν+1)_γ(u,ζ) = 𝒳^sf_γ(u,ζ)exp[ -1/4π i∑_γ'Ω(γ';u) ⟨γ, γ' ⟩∫_ℓ_γ'(u)dζ'/ζ'ζ'+ζ/ζ'-ζlog( 1 - 𝒳^(ν)_γ'(u,ζ'))],


More precisely, we have a family of RH problems, parametrized by u ∈ℬ', as this defines the rays ℓ_γ(u), the complex torus T_u where the symplectomorphisms are defined and the invariants Ω(γ;u) involved in the definition of the problem.

We still need one more piece of the puzzle, since the latter function Ω may not be continuous. In fact, Ω jumps along a real codimension-1 loci in ℬ' called the “wall of marginal stability”. This is the locus where 2 or more functions Z_γ coincide in phase, so two or more rays ℓ_γ(u) become one. More precisely:

    W = {u ∈ℬ': ∃γ_1, γ_2  with Ω(γ_1;u) ≠ 0, Ω(γ_2;u) ≠ 0, ⟨γ_1, γ_2⟩≠ 0, Z_γ_1/Z_γ_2∈_+}

 The jumps of Ω are not arbitrary; they are governed by the Kontsevich-Soibelman wall-crossing formula.

To describe this, let V be a strictly convex cone in the ζ-plane with apex at the origin. Then for any u ∉ W define

    A_V(u) = ∏^_γ : Z_γ(u) ∈ V𝒦_γ^Ω(γ;u) = ∏^_ℓ⊂ V S_ℓ[This product may be infinite. One should more precisely think of A_V(u) as living in a certain prounipotent completion of the group generated by {𝒦_γ}_γ : Z_γ(u) ∈ V as explained in <cit.>]


The arrow indicates the order of the rational maps 𝒦_γ. A_V(u) is a birational Poisson automorphism of T_u. Define a V-good path to be a path p in ℬ' along which there is no point u with Z_γ(u) ∈∂ V and Ω(γ;u) ≠ 0. (So as we travel along a V-good path, no ℓ_γ rays enter or exit V.) If u, u' are the endpoints of a V-good path p, the wall-crossing formula is the condition that A_V(u), A_V(u') are related by parallel transport in T along p. See Figure <ref>.


	






 §.§ Statement of Results



We will restrict in this paper to the case _ℬ = 1, so n = Γ = 2. We want to extend the torus fibration ℳ' to a manifold ℳ with degenerate torus fibers.  To give an example, in the case of Hitchin systems, the torus bundle ℳ' 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 ℳ' to a manifold ℳ with an extended fibration ℳ→ℬ such that the torus fibers ℳ'_u degenerate to nodal torus (i.e. “singular” or “bad” fibers) for u ∈ D.

We start by fully working out the simplest example known as Ooguri-Vafa <cit.>. Here we have a fibration over the open unit disk ℬ := {u ∈ : |u| < 1 }. At the discriminant locus D : = { u = 0 }, the fibers degenerate into a nodal torus. The local rank-2 lattice Γ has a basis (γ_m, γ_e) and the skew-symmetric pairing is defined by ⟨γ_m, γ_e ⟩ = 1. The monodromy of Γ around u = 0 is γ_e ↦γ_e, γ_m ↦γ_m + γ_e. We also have functions Z_γ_e(u) = u, Z_γ_m(u) = u/2π i ( log u - 1) + f(u), for f holomorphic and admitting an extension to ℬ. Finally, the integer-valued function Ω in Γ is here: Ω(±γ_e; u) = 1 and Ω(γ; u) = 0 for any other γ∈Γ_u. There is no wall of marginal stability in this case. The integral equation (<ref>) can be solved after just 1 iteration.

For all other nontrivial cases, in order to give a satisfactory extension of the 𝒳_γ coordinates, it was necessary to develop the theory of Riemann-Hilbert-Birkhoff problems to suit these infinite-dimensional systems (as the transformations S_ℓ defining the problem can be thought as operators on C^∞(T_u), 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 <cit.> to reformulate the Riemann-Hilbert problem in <cit.> independent of the regions determined by the wall.

Having redefined the problem, we want our 𝒳_γ to be smooth on the parameters θ_γ_1,  θ_γ_2 and u,
away from where the prescribed jumps are. Even at ℳ', there was no mathematical proof that such condition must be true. In the companion paper <cit.>, we combine classical Banach contraction methods and Arzela-Ascoli results on uniform convergence in compact sets to obtain:

If the collection J of nonzero Ω(u; γ) satisfies the support property (<ref>) and if the parameter R of (<ref>) is large enough (determined by the values |Z_γ(u)|, γ∈ J), there exists a unique collection of functions 𝒳_γ with the prescribed asymptotics and jumps as in <cit.>. These functions are smooth on u and the torus coordinates θ_1, θ_2 (even for u at the wall of marginal stability), and piecewise holomorphic on ζ.


Since we're considering only the case n=1, Γ is a rank-1 lattice over the Riemann surface ℬ' and the discriminant locus D where the torus fibers degenerate is a discrete subset of ℬ'. 

From this point on, we restrict our attention to the next nontrivial system, known as the Pentagon case <cit.>. Here ℬ = with 2 bad fibers which we can assume are at u = -2, u = 2 and ℬ' is the twice-punctured plane. There is a wall of marginal stability where all Z_γ are contained in the same line. This separates ℬ in two domains ℬ_out and a simply-connected ℬ_in. See Figure <ref>.





On ℬ_in we can trivialize Γ and choose a basis {γ_1, γ_2} with pairing ⟨γ_1, γ_2⟩ = 1. This basis does not extend to a global basis for Γ since it is not invariant under monodromy. However, the set {γ_1, γ_2, -γ_1, -γ_2, γ_1 + γ_2, -γ_1 - γ_2} is indeed invariant so the following definition of Ω makes global sense:
  
    For u ∈ℬ_in, Ω(γ; u) =    {[                            1 for γ∈{γ_1, γ_2, -γ_1, -γ_2};                            0                    otherwise ]. 
    For u ∈ℬ_out , Ω(γ; u) =    {[                                                   1 for γ∈{γ_1, γ_2, -γ_1, -γ_2, γ_1 + γ_2, -γ_1 - γ_2};                                                   0                                           otherwise ].

  
  
  
  


  
   The Pentagon case appears in the study of Hitchin systems with gauge group SU(2). The extension of ℳ' was previously obtained by hyperkähler quotient methods in <cit.>, but no explicit hyperkähler metric was constructed. 

Once the {𝒳_γ_i} are obtained by Theorem <ref>, 
it is necessary to do an analytic continuation along ℬ' for the particular 𝒳_γ_i for which Z_γ_i→ 0 as u → u_0 ∈ D. Without loss of generality, we can assume there
is a local basis {γ_1, γ_2} of Γ such that Z_γ_2→ 0 in D. After that, an analysis of the possible divergence of 𝒳_γ as u → u_0 shows the necessity of performing a gauge transformation on the torus coordinates of the fibers ℳ_u that allows us to define an integral equation even at u_0 ∈ D. This series of transformations are defined in (<ref>), (<ref>), (<ref>) and (<ref>), and constitute a new result that was not expected in <cit.>. 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 u → u_0. As this is of independent interest, we leave the relevant results to <cit.> and we show that our solutions contain at worst branch singularities at 0 or ∞ in the ζ-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 {γ_1, γ_2} be a local basis for Γ in a small sector centered at u_0 ∈ D such that Z_γ_2→ 0 as u → u_0 ∈ D. For the Pentagon integrable system, the local function 𝒳_γ_1 admits an analytic continuation 𝒳_γ_1 to a punctured disk centered at u_0 in ℬ. There exists a gauge transformation θ_1 ↦θ_1 that extends the torus fibration ℳ' to a manifold ℳ that is locally, for each point in D, a (trivial) fibration over ℬ× S^1 with fiber S^1 coordinatized by θ_1 and with one fiber collapsed into a point. For R > 0 big enough, it is possible to extend 𝒳_γ_1 and 𝒳_γ_2 to ℳ, still preserving the smooth properties as in Theorem <ref>.






After we have the smooth extension of the {𝒳_γ_i} by Theorem <ref>, we can extend the holomorphic symplectic form ϖ(ζ) labeled by ζ∈ as in <cit.> for all points except possibly one at the singular fiber. From ϖ(ζ) we can obtain the hyperkähler metric g and, in the case of the Pentagon, after a change of coordinates, we realize g locally as the Taub-NUT metric plus smooth corrections, finishing the construction of ℳ and its hyperkähler metric. The following is the main theorem of the paper.


For the Pentagon case, the extension ℳ of the manifold ℳ' constructed in Theorem <ref> admits, for R large enough, a hyperkähler metric g obtained by extending the hyperkähler metric on ℳ' determined by the Darboux coordinates {𝒳_γ_i}.






   


§ THE OOGURI-VAFA CASE





 §.§ Classical Case



We start with one of the simplest cases, known as the Ooguri-Vafa case, first treated in <cit.>. To see where this case comes from, recall that by the SYZ picture of K3 surfaces <cit.>, any K3 surface ℳ is a hyperkähler manifold. In one of its complex structures (say J^(ζ = 0)) is elliptically fibered, with base manifold ℬ = and generic fiber a compact complex torus. There are a total of 24 singular fibers, although the total space is smooth. See Figure <ref>.

	

Gross and Wilson <cit.> constructed a hyperkähler metric g on a K3 surface by gluing in the Ooguri-Vafa metric constructed in <cit.> with a standard metric g^sf 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 ℬ := {a ∈ : |a| < 1 }. At the locus D : = { a = 0 } (in the literature this is also called the discriminant locus), the fibers degenerate into a nodal torus. Define ℬ' as ℬ\ D, the punctured unit disk. On ℬ' there exists a local system Γ of rank-2 lattices with basis (γ_m, γ_e) and skew-symmetric pairing defined by ⟨γ_m, γ_e ⟩ = 1. The monodromy of Γ around a = 0 is γ_e ↦γ_e, γ_m ↦γ_m + γ_e. We also have functions Z_γ_e(a) = a, Z_γ_m(a) = a/2π i ( log a - 1). On ℬ' we have local coordinates (θ_m, θ_e) for the torus fibers with monodromy θ_e ↦θ_e, θ_m ↦θ_m + θ_e - π. Finally, the integer-valued function Ω in Γ is here: Ω(±γ_e, a) = 1 and Ω(γ, a) = 0 for any other γ∈Γ_a. 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 <cit.> already mentioned at the beginning of this section. In the next section, we'll generalize this case by adding a function f(a) to the definition of Z_γ_m.

Let

    𝒳^sf_γ(ζ, a) := exp( π R ζ^-1 Z_γ(a) + iθ_γ + π R ζZ_γ(a))

These functions receive corrections defined as in <cit.>. We are only interested in the pair (𝒳_m, 𝒳_e) which will constitute our desired Darboux coordinates for the holomorphic symplectic form ϖ. The fact that Ω(γ_m, a) = 0 gives that 𝒳_e = 𝒳^sf_e. As a → 0, Z_γ_e and Z_γ_m approach 0. Thus 𝒳_e|_a = 0 = e^iθ_e. Since 𝒳_e = 𝒳^sf_e the actual 𝒳_m is obtained after only 1 iteration of (<ref>). For each a ∈ℬ', let ℓ_+ be the ray in the ζ-plane defined by {ζ : a/ζ∈_- }. Similarly, ℓ_- : = {ζ : a/ζ∈_+}.

Let

    𝒳_m = 𝒳^sf_m exp[ i/4π∫_ℓ_+dζ'/ζ'ζ' +
     ζ/ζ' - ζlog[1 - 𝒳_e(ζ')] -  i/4π∫_ℓ_-dζ'/ζ'ζ' +
     ζ/ζ' - ζlog[1 - 𝒳_e(ζ')^-1] ].

For convenience, from this point on we assume a is of the form  sb, where s is a positive number, b is fixed and |b| = 1. Moreover, in ℓ_+, ζ' = -tb, for t ∈ (0, ∞), and a similar parametrization holds in ℓ_-.

For fixed b, 𝒳_m as in (<ref>) has a limit as |a| → 0.


Writing ζ' + ζζ'(ζ' - ζ) = -1ζ' + 2ζ' - ζ, we want to find the limit as a → 0 of

    ∫_ℓ_+{-1ζ' + 2ζ' - ζ}log[1 - exp(π Ra/ζ' + iθ_e + π Rζ' a̅)] dζ' 
        - ∫_ℓ_-{-1ζ' + 2ζ' - ζ}log[1 - exp(-π Ra/ζ' - iθ_e - π Rζ' a̅)] dζ' .

For simplicity, we'll focus in the first integral only, the second one can be handled similarly. Rewrite:

    ∫_ℓ_+{-1ζ' + 2ζ' - ζ}log[1 - exp(π Ra/ζ' + iθ_e + π Rζ' a̅)] dζ' 
        = ∫_0^-b{-1ζ' + 2ζ' - ζ}log[1 - exp(π Ra/ζ' + iθ_e + π Rζ' a̅)] dζ'  
        + ∫_-b^-b∞{-1ζ' + 2ζ' - ζ}log[1 - exp(π Ra/ζ' + iθ_e + π Rζ' a̅)] dζ' 
        = ∫_0^-b{-1ζ' + 2ζ' - ζ}log[1 - exp(π Ra/ζ' + iθ_e + π Rζ' a̅)] dζ' 
        + ∫_-b^-b∞{-1ζ' + 2/ζ' + 2ζ' - ζ - 2/ζ'}log[1 - exp(π Ra/ζ' + iθ_e + π Rζ' a̅)] dζ' 
        = ∫_0^-b-1ζ'log[1 - exp(π Ra/ζ' + iθ_e + π Rζ' a̅)] dζ' 
        + ∫_-b^-b∞1ζ'log[1 - exp(π Ra/ζ' + iθ_e + π Rζ' a̅)] dζ' 
        + ∫_0^-b2ζ' - ζlog[1 - exp(π Ra/ζ' + iθ_e + π Rζ' a̅)] dζ' 
        + ∫_-b^-b∞{2ζ' - ζ - 2/ζ'}log[1 - exp(π Ra/ζ' + iθ_e + π Rζ' a̅)] dζ'


Observe that

    ∫_0^-b-1ζ'log[1 - exp(π Ra/ζ' + iθ_e + π Rζ' a̅)] dζ'
        = -∫_0^1 1/tlog[1 - exp(-π Rs(t + 1/t))] dt
    and after a change of variables t̃ = 1/t, we get    = -∫_1^∞1/t̃log[1 - exp(-π Rs(t̃ + 1/t̃))] dt̃
        = -∫_-b^-b∞1ζ'log[1 - exp(π Ra/ζ' + iθ_e + π Rζ' a̅)] dζ'.

Thus, (<ref>) reduces to

    ∫_0^-b2ζ' - ζlog[1 - exp(π Ra/ζ' + iθ_e + π Rζ' a̅)] dζ' 
        + ∫_-b^-b∞{2ζ' - ζ - 2/ζ'}log[1 - exp(π Ra/ζ' + iθ_e + π Rζ' a̅)] dζ' .

If θ_e = 0, (<ref>) diverges to -∞, in which case 𝒳_m = 0. Otherwise, log[1 - exp(π Ra/ζ' + iθ_e + π Rζ' a̅)] is bounded away from 0. Consequently,
|log[1 - exp(π Ra/ζ' + iθ_e + π Rζ' a̅)]| < C < ∞ in ℓ_+.
As a → 0, the integrals are dominated by

    ∫_0^-b2C|ζ' - ζ| |dζ'| + ∫_-b^-b∞C|ζ/b|/|ζ'(ζ' - ζ)| |dζ'| < ∞

if θ_e ≠ 0. Hence we can interchange the limit and the integral in (<ref>) and obtain that, as a → 0, this reduces to


    2log(1 - e^i θ_e)[∫_0^-bdζ'/ζ' - ζ + ∫_-b^-b∞ dζ' {1/ζ' - ζ - 1/ζ'}] 
        = 2log(1 - e^i θ_e)[F(-b) + G(-b)],

where

    F(z) := log( 1 - zζ), G(z) := log( 1 - ζz)

 are the (unique) holomorphic solutions in the simply connected domain U :=  - {z : z/ζ∈_+} to the ODEs

    F'(z) = 1/z - ζ, F(0) = 0  G'(z) = 1/z - ζ - 1/z, lim_z →∞ G(z) = 0.

This forces us to rewrite (<ref>) uniquely as

    2log(1 - e^i θ_e)[log(1 + b/ζ) - log(1 + ζ/b)]

Here log denotes the principal branch of the log in both cases, and the equation makes sense for {b ∈ : b ∉ℓ_+ } (recall that by construction, we have the additional datum |b| = 1). We want to conclude that

    log(1 + b/ζ) - log(1 + ζ/b) = log(b/ζ),

still using the principal branch of the log. To see this, define H(z) as F(z) - G(z) - log(-z/ζ). This is an analytic function on U and clearly H'(z) ≡ 0. Thus H is constant in U. It is easy to show that the identity holds for a suitable choice of z (for example, if ζ is not real, choose z = 1) and by the above, it holds on all of U; in particular, for z = -b.

All the arguments so far can be repeated to the ray ℓ_- to get the final form of (<ref>):
 
    2{log[b/ζ]log(1 - e^iθ_e)
       -log[- b/ζ]log(1 - e^-iθ_e) }, θ_e ≠ 0.

   This yields that (<ref>) simplifies to:

    𝒳_m     = 𝒳^sf_m exp( i/2π{log[b/ζ]log(1 - e^iθ_e)
       -log[- b/ζ]log(1 - e^-iθ_e)}) 
        = 𝒳^sf_m exp( i/2π{log[a/|a|ζ]log(1 - e^iθ_e)
       -log[- a/|a|ζ]log(1 - e^-iθ_e)})

in the limiting case a → 0.


To obtain a function that is continuous everywhere and independent of a, define regions I, II and III in the a-plane as follows: 𝒳^sf_m has a fixed cut in the negative real axis, both in the ζ-plane and the a-plane. Assuming for the moment that ζ∈ (0,π), define region I as the half plane {a ∈ : Im( a/ζ) < 0 }. Region II is that enclosed by the ℓ_- ray and the cut in the negative real axis, and region III is the remaining domain so that as we travel counterclockwise we traverse regions I, II and III in this order (see Figure <ref>).




For a ≠ 0, Gaiotto, Moore and Neitzke <cit.> proved that 𝒳_m has a continuous extension to the punctured disk of the form:


    𝒳_m = {[                                   𝒳_m                           in region I;                      (1 - 𝒳^-1_e) 𝒳_m                          in region II; - 𝒳_e (1 - 𝒳^-1_e) 𝒳_m = (1 - 𝒳_e)𝒳_m                         in region III ].


If we regard ℳ' as a S^1-bundle over ℬ' × S^1, with the fiber parametrized by θ_m, then we seek to extend ℳ' to a manifold ℳ by gluing to ℳ' another S^1-bundle over D × (0,2π), for D a small open disk around a = 0, and θ_e ∈ (0,2π). The S^1-fiber is parametrized by a different coordinate θ'_m where the Darboux coordinate 𝒳_m can be extended to ℳ. This is the content of the next theorem. 


ℳ' can be extended to a manifold ℳ where the torus fibers over ℬ' degenerate at D = {a = 0} and 𝒳_m can be extended to D, independent of the value of a.


We'll use the following identities:

    log(1 - e^iθ_e)     = log(1 - e^-iθ_e) +i(θ_e - π), for θ_e ∈ (0, 2π) 
    log[-a/|a|ζ]     = {[      log[a/|a|ζ] + iπ           in region I;      log[a/|a|ζ] - iπ in regions II and III ]. 
    log [a/ζ]     = {[        log a - logζ in regions I and II; log a - logζ + 2π i       in region III ].

to obtain a formula for 𝒳_m at a = 0 independent of the region. Formula (<ref>) can be proved with an argument analogous to that used for the proof of (<ref>). Starting with region I, by (<ref>), (<ref>), (<ref>) and (<ref>):

    𝒳_m     = exp[ iθ_m - 1/2π  (θ_e - π) log[a/|a|ζ] + 1/2log(1 - e^-iθ_e)  ] in region I.
    By (<ref>),    = exp[ iθ_m - 1/2π  (θ_e - π) log[a/|a|] + θ_e - π/2πlogζ + 1/2log(1 - e^-iθ_e) ]

In region II, by our formulas above, we get

    𝒳_m     = exp[iθ_m - 1/2π  (θ_e - π) log[a/|a|ζ] - 1/2log(1 - e^-iθ_e)  ](1 - e^-iθ_e) 
        =  exp[iθ_m -  1/2π  (θ_e - π) log[a/|a|ζ] - 1/2log(1 - e^-iθ_e) + log(1 - e^-iθ_e)  ]
        = exp[ iθ_m - 1/2π  (θ_e - π) log[a/|a|] + θ_e - π/2πlogζ + 1/2log(1 - e^-iθ_e) ]  in region II.

Finally, in region III, and making use of (<ref>), (<ref>), (<ref>):

    𝒳_m     = exp[iθ_m - 1/2π  (θ_e - π) log[a/|a|ζ]  - 1/2log(1 - e^-iθ_e)  ](1 - e^iθ_e) 
        = exp[ iθ_m - 1/2π  (θ_e - π) log[a/|a|] + θ_e - π/2πlogζ - i(θ_e - π) . 
       .  - 1/2log(1 - e^-iθ_e) + log(1 - e^-iθ_e) + i(θ_e - π)   ] 
        = exp[ iθ_m - 1/2π  (θ_e - π) log[a/|a|] + θ_e - π/2πlogζ + 1/2log(1 - e^-iθ_e)  ] .


 Observe that, throughout all these calculations, we only had to use the natural branch of the complex logarithm. In summary, (<ref>) works for any region in the a-plane, with a cut in the negative real axis.

This also suggest the following coordinate transformation

    θ'_m = θ_m + i(θ_e - π)/4π( loga/Λ - loga̅/Λ)

Here Λ is the same cutoff constant as in <cit.>. Let φ parametrize the phase of a/|a|. Then (<ref>) simplifies to

    θ'_m = θ_m - (θ_e - π)φ/2π


On a coordinate patch around the singular fiber, θ'_m is single-valued. 
Thus, the above shows that we can glue to ℳ' another S^1-bundle over D × (0,2π), for D a small open disk around a = 0, and θ_e ∈ (0,2π). The S^1-fiber is parametrized by θ'_m and the transition function is given by (<ref>), yielding a manifold ℳ. In this patch, we can extend 𝒳_m to a = 0 as:


    . 𝒳_m|_a = 0 = e^iθ'_mζ^θ_e - π/2π (1 - e^-iθ_e)^1/2

where the branch of ζ^θ_e - π/2π is determined by the natural branch of the logarithm in the ζ plane. Note that when θ_e = 0, 𝒳_m ≡ 0 in (<ref>) and by definition, 𝒳_e ≡ 1. Since these two functions are Darboux coordinates for ℳ, the S^1 fibration over D × (0, 2π) we glued to ℳ' to get ℳ degenerates into a point when θ_e = 0. 

Now consider the case that ζ∈ (-π, 0). Label the regions as one travels counterclockwise, starting with the region bounded by the cut and the ℓ_- (See Figure <ref>). We can do an analytic continuation similar to (<ref>) starting in region I, but formulas (<ref>), (<ref>) become now:




    log[-a/|a|ζ]     = {[     log[a/|a|ζ] - iπ         in region II;     log[a/|a|ζ] + iπ in regions I and III ]. 
    log [a/ζ]     = {[        log a - logζ in regions I and II; log a - logζ - 2π i       in region III ].


By an argument entirely analogous to the case ζ > 0, we get again:

    . 𝒳_m|_a = 0 = e^iθ'_mζ^θ_e - π/2π (1 - e^-iθ_e)^1/2


The case ζ real and positive is even simpler, as Figure <ref> shows. Here we have only two regions, and the jumps at the cut and the ℓ_+ ray are combined, since these two lines are the same. Label the lower half-plane as region I and the upper half-plane as region II. Start an analytic continuation of 𝒳_m in region I as before, using the formulas:




    log[-a/|a|ζ]     = {[ log[a/|a|ζ] - iπ     in region II; log[a/|a|ζ] + iπ      in region I ]. 
    log [a/ζ]     = log a - logζin both regions


The result is equation (<ref>) again. The case ζ = π is entirely analogous to this and it yields the same formula, thus proving that (<ref>) holds for all ζ and is independent of a.





 §.§ Alternative Riemann-Hilbert problem


We may obtain the function 𝒳_m (and consequently, the analytic extension 𝒳_m) at a = 0 through a slightly different formulation of the Riemann-Hilbert problem stated in (<ref>). Namely, instead of defining a jump of 𝒳_m at two opposite rays ℓ_+, ℓ_-, we combine these into a single jump at the line ℓ defined by ℓ_+ and ℓ_-, as in Figure <ref>. Note that because of the orientation of ℓ one of the previous jumps has to be reversed.



For all values a ≠ 0, 𝒳_e = 𝒳_e^sf approaches 0 as ζ→ 0 or ζ→∞ along the ℓ ray due to the exponential decay in formula (<ref>). Thus, the jump function

    G(ζ) := {[               1-𝒳^-1_e  for ζ = t a, 0 ≤ t ≤∞;                 1- 𝒳_e for ζ = t a, -∞≤ t ≤ 0 ].
                                     
is continuous on ℓ regarded as a closed contour on , and it approaches the identity transformation exponentially fast at the points 0 and ∞.









The advantage of this reformulation of the Riemann-Hilbert problem is that it can be extended to the case a = 0 and we can obtain estimates on the solutions 𝒳_m even without an explicit formulation. If we fix a and let |a| → 0 as before, the jump function G(ζ) approaches the constant jumps

    . G(ζ) |_|a|=0 := {[               1-e^-iθ_e  for ζ = t a, 0 < t < ∞;               1- e^iθ_e for ζ = t a, -∞ < t < 0 ].

Thus, . G(ζ) |_|a|=0 has two discontinuities at 0 and ∞. If we denote by 

    Δ_0 = lim_t → 0^+ G(ζ) - lim_t → 0^- G(ζ),     Δ_∞ = lim_t →∞^+ G(ζ) - lim_t →∞^- G(ζ),

then, by (<ref>),

    Δ_0 = -Δ_∞










Let D^+ be the region in  bounded by ℓ with the positive, counterclockwise orientation. Denote by D^- the region where ℓ as a boundary has the negative orientation. We look for solutions of the homogeneous boundary problem

    X_m^+(ζ) = G(ζ) X_m^-(ζ)

with G(ζ) as in (<ref>). This is Lemma 4.1 in <cit.>.



























 





 
 The solutions X_m^± obtained therein are related to 𝒳_m via 𝒳_m (ζ) = 𝒳^sf_m (ζ) X_m (ζ). Uniqueness of solutions of the homogeneous Riemann-Hilbert problem shows that these are the same functions (up to a constant factor) constructed in the previous section. Observe that the term ζ^θ_e - π/2π appears naturally due to the nature of the discontinuity of the jump function at 0 and ∞. The analytic continuation around the point a = 0 and the gauge transformation θ_m ↦θ'_m are still performed as before.
                                      





 §.§ Generalized Ooguri-Vafa coordinates




We can generalize the previous extension to the case Z_γ_m := 1/2π ia log a + f(a), where f : ℬ' → is holomorphic and admits a holomorphic extension into ℬ. In particular,

    𝒳_m^sf = exp( -iR/2ζa log a + π R f(a)/ζ + i θ_m + i ζ R/2aloga + π R ζf(a))

The value at the singular locus f(0) does not have to be 0. All the other data remains the same. 

The first thing we observe is that 𝒳_e remains the same. Consequently, the corrections for the generalized 𝒳_m are as before. Using the change of coordinates as in (<ref>), we can thus write

    . 𝒳_m|_a = 0 = exp[ π R f(0)/ζ + iθ'_m + π R ζ f(0) ] ζ^θ_e - π/2π (1 - e^-iθ_e)^1/2




§ EXTENSION OF THE OOGURI-VAFA METRIC




 §.§ Classical Case





  §.§.§ A C^1 extension of the coordinates



In section <ref> we extended the fibered manifold ℳ' to a manifold ℳ with a degenerate fiber at a = 0 in ℬ. We also extended 𝒳_m continuously to this bad fiber. Now we extend the metric by enlarging the holomorphic symplectic form ϖ(ζ). Recall that this is of the form

    ϖ(ζ) = -1/4π^2 Rd 𝒳_e/𝒳_e∧d𝒳_m/𝒳_m

Clearly there are no problems extending d log𝒳_e, so it remains only to extend d log𝒳_m.


Let 𝒳_m denote the analytic continuation around a = 0 of the magnetic function, as in the last section. The 1-form

    d log𝒳_m = d 𝒳_m/𝒳_m,

(where d denotes the differential of a function on the torus fibration ℳ' only) has an extension to ℳ


We proceed as in section <ref> and work in different regions in the a-plane (see Figure <ref>), starting with region I, where 𝒳_m = 𝒳_m. Then observe that we can write the corrections on 𝒳_m as a complex number Υ_m(ζ) ∈ (ℳ'_a)^ such that

    𝒳_m = exp( -i R /2ζ(alog a - a) + i Υ_m + iζ R/2 (aloga - a )).

Thus, by (<ref>) and ignoring the i factor, it suffices to obtain an extension of 

    d[ - R /2ζ(alog a - a) +  Υ_m + ζ R/2 (aloga - a ) ] 
        =  -R/2ζlog a   da +  d Υ_m + ζ R/2loga  da.

Using (<ref>),

    d Υ_m  = dθ_m     - 1/4π∫_ℓ_+dζ'ζ'ζ'+ζ/ζ'-ζ𝒳_e/1-𝒳_e( π R/ζ' da +idθ_e+ π R ζ' da) 
        +1/4π∫_ℓ_-dζ'ζ'ζ'+ζ/ζ'-ζ𝒳^-1_e/1-𝒳^-1_e( -π R/ζ' da -idθ_e - π R ζ' da).

We have to change our θ_m coordinate into θ'_m according to (<ref>) and differentiate to obtain:

    d Υ_m      =  
    dθ'_m - i(θ_e - π)/4π( da/a - da/a)+ a/2πdθ_e 
        - 1/4π∫_ℓ_+dζ'ζ'ζ'+ζ/ζ'-ζ𝒳_e/1-𝒳_e( π R/ζ' da +idθ_e+ π R ζ' da) 
        +1/4π∫_ℓ_-dζ'ζ'ζ'+ζ/ζ'-ζ𝒳^-1_e/1-𝒳^-1_e( -π R/ζ' da -idθ_e - π R ζ' da)

Recall that, since we have introduced the change of coordinates θ_m ↦θ'_m, we are working on a patch on ℳ that contains a = 0 with a degenerate fiber here. It then makes sense to ask if (<ref>) extends to a =0. If this is true, then every independent 1-form extends individually. Let's consider the form involving dθ_e first. By (<ref>), this part consists of:

    a/2πdθ_e  - i/4π∫_ℓ_+dζ'ζ'ζ'+ζ/ζ'-ζ𝒳_e/1-𝒳_e dθ_e - i/4π∫_ℓ_-dζ'ζ'ζ'+ζ/ζ'-ζ𝒳^-1_e/1-𝒳^-1_e dθ_e.

We can use the exact same technique in section <ref> to find the limit of (<ref>) as a → 0. Namely, split each integral into four parts, use the symmetry of 𝒳_e1- 𝒳_e between 0 and ∞ to cancel two of these integrals and take the limit in the remaining ones. The result is:

    a/2π - ie^iθ_e/2π(1-e^iθ_e)log[ e^i  a/ζ] - 
    ie^-iθ_e/2π(1-e^-iθ_e)log[ -e^i  a/ζ]  
        =  a/2π - ie^iθ_e/2π(1-e^iθ_e)log[ e^i  a/ζ] + 
    i/2π(1-e^iθ_e)log[ -e^i  a/ζ]

in region I (we omitted the dθ_e factor for simplicity). Making use of formulas (<ref>) and (<ref>), we can simplify the above expression and get rid of the apparent dependence on a until finally getting:

    -ilogζ/2π - 1/2(1-e^iθ_e), θ_e ≠ 0.

In other regions of the a-plane we have to modify 𝒳_m as in (<ref>). Nonetheless, by (<ref>) and (<ref>), the result is the same and we conclude that at least the terms involving dθ_e have an extension to a=0 for θ_e ≠ 0.

Next we extend the terms involving da. By (<ref>) and (<ref>), these are:

    -R/2ζlog a   da  - i(θ_e - π)/4π a da - R/4∫_ℓ_+dζ'(ζ')^2ζ'+ζ/ζ'-ζ𝒳_e/1-𝒳_e da - R/4∫_ℓ_-dζ'(ζ')^2ζ'+ζ/ζ'-ζ𝒳^-1_e/1-𝒳^-1_e da

In what follows, we ignore the da part and focus on the coefficients for the extension. The partial fraction decomposition

    ζ'+ζ/(ζ')^2(ζ'-ζ) = 2/ζ'(ζ'-ζ) - 1/(ζ')^2

splits each integral above into two parts. We will consider first the terms

    - i(θ_e - π)/4π a + R/4∫_ℓ_+dζ'(ζ')^2𝒳_e/1-𝒳_e + R/4∫_ℓ_-dζ'(ζ')^2𝒳^-1_e/1-𝒳^-1_e.

Use the fact that 𝒳_e (resp. 𝒳^-1_e) has norm less than 1 on ℓ_+ (resp. ℓ_-) and the uniform convergence of the geometric series on ζ' to write (<ref>) as:

    - i(θ_e - π)/4π a + R/4∑_n=1^∞{.    ∫_ℓ_+dζ'/(ζ')^2exp(
    π R n a/ζ' +i n θ_e +π R n ζ' a) + 
       .
    ∫_ℓ_-dζ'/(ζ')^2exp(
    -π R n a/ζ' -i n θ_e -π R n ζ' a)},


    = - i(θ_e - π)/4π a + (R/4) ( -2|a|/a)∑_n=1^∞( e^inθ_e - e^-inθ_e)K_1(2π R n |a|)
        = - i(θ_e - π)/4π a - R|a|/2a∑_n=1^∞( e^inθ_e - e^-inθ_e)K_1(2π R n |a|).

Since K_1(x)  1/x, for x real and x → 0, we obtain, letting a → 0:

    - i(θ_e - π)/4π a - R|a|/2a· 2π R |a|∑_n=1^∞( e^inθ_e - e^-inθ_e)/n
        =  - i(θ_e - π)/4π a + 1/4π a[log(1-e^iθ_e)-log(1-e^-iθ_e)]
    and by (<ref>),    =  - i(θ_e - π)/4π a +i(θ_e -π)/4π a = 0.

Therefore this part of the da terms extends trivially to 0 in the singular fiber.

It remains to extend the other terms involving da. Recall that by (<ref>), these terms are (after getting rid of a factor of -R/2):

    log a/ζ + ∫_ℓ_+dζ'/ζ'(ζ'-ζ)𝒳_e/1-𝒳_e +  ∫_ℓ_-dζ'/ζ'(ζ'-ζ)𝒳^-1_e/1-𝒳^-1_e.


We'll focus in the first integral in (<ref>). As a starting point, we'll prove that as a → 0, the limiting value of this integral is the same as the limit of

    ∫_ℓ_+dζ'/ζ'(ζ'-ζ)exp( π R a/ζ' +iθ_e )/1-exp( π R a/ζ' +iθ_e +π R ζ' a).

It suffices to show that

    ∫_ℓ_+dζ'/ζ'(ζ'-ζ)exp( π R a/ζ')/1-exp( π R a/ζ' +iθ_e +π R ζ' a) [1-exp(π R ζ' a)] → 0, as a → 0, θ_e ≠ 0

To see this, we can assume |a| < 1. Let b = a/|a|. Observe that in the ℓ_+ ray, |exp(π Ra/ζ')| < 1, and since θ_e ≠ 0, we can bound (<ref>) by

    const∫_ℓ_+dζ'/ζ'(ζ'-ζ) [1-exp(π R ζ' b)] < ∞.

Equation (<ref>) now follows from Lebesgue Dominated Convergence and the fact that 1-exp(π R ζ' a) → 0 as a → 0. A similar application of Dominated Convergence allows us to reduce the problem to the extension of

    ∫_ℓ_+dζ'/ζ'(ζ'-ζ)exp( π R a/ζ' +iθ_e )/1-exp( π R a/ζ' +iθ_e ).

Introduce the real variable s = -π R a / ζ'. We can write (<ref>) as:

    e^iθ_e∫_0^∞ds/s[ -π R a/s - ζ]e^-s/1-e^iθ_e-s
        = -1/ζ∫_0^∞ds/s+π R a/ζ·e^-s/e^-iθ_e-e^-s
        = 1/ζ∫_0^∞ds/s+π R a/ζ·1/1-e^s-iθ_e

The integrand of (<ref>) has a double zero at ∞, when a → 0, so the only possible non-convergent part in the limit a=0 is the integral

    1/ζ∫_0^1 ds/s+π R a/ζ·1/1-e^s-iθ_e.

Since

    ∫_0^1 ds/s[ 1/1-e^s-iθ_e - 1/1-e^-iθ_e] < ∞,

we can simplify this analysis even further and focus only on

    1/ζ(1-e^-iθ_e)∫_0^1 ds/s+π R a/ζ
        = -log (π R a /ζ)/ζ(1-e^-iθ_e).

We can apply the same technique to obtain a limit for the second integral in (<ref>). The result is

    -log (-π R a /ζ)/ζ(1-e^iθ_e),

which means that the possibly non-convergent terms in (<ref>) are:

    log a/ζ - log a/ζ(1-e^-iθ_e) - log a/ζ(1-e^iθ_e) = 0.

Note that the corrections of 𝒳_m in other regions of the a-plane as in (<ref>) depend only on 𝒳_e, which clearly has a smooth extension to the singular fiber.


The extension of the da part is performed in exactly the same way as with the da forms. We conclude that the 1-form

    d𝒳_m/𝒳_m

has an extension to ℳ; more explicitly, to the fiber at a=0 in the classical Ooguri-Vafa case. This holds true also in the generalized Ooguri-Vafa case since here we simply add factors of the form f'(a)da and it is assumed that f(a) has a smooth extension to the singular fiber.


In section <ref>, we will reinterpret these extension of the derivatives of 𝒳_m if we regard the gauge transformation (<ref>) as a contour integral between symmetric contours. It will be then easier to see that the extension can be made smooth.



  §.§.§ Extension of the metric




The results of the previous section already show the continuous extension of the holomorphic symplectic form

    ϖ(ζ) = -1/4π^2 Rd 𝒳_e/𝒳_e∧d𝒳_m/𝒳_m

to the limiting case a = 0, but we excluded the special case θ_e = 0. Here we obtain ϖ(ζ) at the singular fiber with a different approach that will allow us to see that such an extension is smooth without testing the extension for each derivative. Although it was already known that ℳ' extends to the hyperkähler manifold ℳ constructed here, this approach is new, as it gives an explicit construction of the metric as we will see. Furthermore, the Ooguri-Vafa model can be thought as an elementary model for which more complex integrable systems are modeled locally (see <ref>).

The holomorphic symplectic form ϖ(ζ) extends smoothly to ℳ. Near a = 0 and θ_e = 0, the hyperkähler metric g looks like a constant multiple of the Taub-NUT metric g_Taub-NUT plus some smooth corrections.


 By <cit.>, near a = 0,

    ϖ(ζ) = -1/4π^2 Rd 𝒳_e/𝒳_e∧[ idθ_m + 2π i A + π i V 
    (1/ζda - ζ da̅)],

where

    A = 1/8π^2( loga/Λ - loga̅/Λ)dθ_e - R/4π( da/a - da̅/a̅)∑_n ≠ 0 (sgn n) e^inθ_e |a|
       K_1(2π R|na|)

should be understood as a U(1) connection over the open subset of × S^1 parametrized by (a,θ_e) and V is given by Poisson re-summation as

    V = R/4π[ 1/√(R^2|a|^2 + θ_e^2/4π^2) + ∑_n = -∞
     n ≠ 0^∞( 1/√(R^2 |a|^2 + (θ_e/2π + n)^2) - κ_n ) ].

Here κ_n is a regularization constant introduced to make the sum convergent, even at a = 0, θ_e ≠ 0. The curvature F of the unitary connection satisfies

    dA = *dV.

Consider now a gauge transformation θ_m ↦θ_m + α and its induced change in the connection A ↦ A' = A - dα/2π (see <cit.>). We have idθ'_m + 2π i A' = idθ_m + idα + 2π i A - idα = idθ_m + 2π i A. Furthermore, for the particular gauge transformation in (<ref>), at a = 0 and for θ_e ≠ 0:

    A'     = A - dα/2π
        = 1/8π^2( loga/Λ - loga̅/Λ)dθ_e - 1/8π^2( da/a - da̅/a̅) [ ∑_n = 1^∞e^inθ_e/n - ∑_n = 1^∞e^-inθ_e/n]
        - 1/8π^2( loga/Λ - loga̅/Λ)dθ_e - i(θ_e - π)/8π^2( da/a - da̅/a̅),
    (here we're using the fact that K_1(x) → 1/x as x → 0)    = i(θ_e - π)/8π^2( da/a - da̅/a̅) - i(θ_e - π)/8π^2( da/a - da̅/a̅) = 0.
    since the above sums converge to -log(1 - e^iθ_e) + log(1 - e^-iθ_e) = -i(θ_e - π) for θ_e ≠ 0.


Writing V_0 (observe that this only depends on θ_e) for the limit of V as a → 0, we get at a = 0

    ϖ(ζ)     = -1/4π^2 R( π R/ζda + idθ_e + π R ζ da̅) ∧(
      idθ'_m + π i V_0 ( da/ζ - ζ da̅) )  
        = 1/4π^2 R dθ_e ∧ dθ'_m + iV_0/2da ∧ da̅ -i/4πζda ∧ dθ'_m - V_0/4π Rζda ∧ dθ_e  
        - iζ/4π da̅∧ dθ'_m + V_0 ζ/4π R da̅∧ dθ_e.


This yields that, at the singular fiber,

    ω_3     = 1/4π^2 R dθ_e ∧ dθ'_m + iV_0/2da ∧ da̅
    ω_+     = 1/2π da ∧( dθ'_m - iV_0/Rdθ_e )
    ω_-     = 1/2π da̅∧( dθ'_m + iV_0/Rdθ_e )


From the last two equations we obtain that dθ'_m - iV_0/R dθ_e and dθ'_m + iV_0/R dθ_e are respectively (1,0) and (0,1) forms under the complex structure J_3. A (1,0) vector field dual to the (1,0) form above is then 12(∂_θ'_m + iR/V_0 ∂_θ_e). In particular,

    J_3(∂_θ'_m) = -R/V_0∂_θ_e,  J_3 (-R/V_0∂_θ_e) = -∂_θ'_m.

With this and (<ref>) we can reconstruct the metric at a = 0. Observe that

    g(∂_θ_e, ∂_θ_e)     = ω_3(∂_θ_e, J_3(∂_θ_e)) = ω_3(∂_θ_e, V_0/R∂_θ'_m) = V_0/4π^2 R^2
    
    g(∂_θ'_m, ∂_θ'_m)     = ω_3(∂_θ'_m, J_3(∂_θ'_m)) = ω_3(∂_θ'_m, -R/V_0∂_θ_e) = 1/4π^2 V_0


Consequently,

    g = 1/V_0( dθ'_m/2π)^2 + V_0 dx⃗^2,

where a = x^1 + ix^2, θ_e = 2π R x^3. Since V_0(θ_e) is undefined for θ_e = 0, we have to check that g extends to this point. Let (r,ϑ, ϕ) denote spherical coordinates for x⃗. The formula above is the natural extension of the metric given in <cit.> for nonzero a:

    g = 1/V(x⃗)( dθ'_m/2π + A'(x⃗))^2 + V(x⃗) dx⃗^2

To see that this extends to r =0, we rewrite

    V     = R/4π[ 1/√(R^2 |a|^2 + θ_e^2/4π^2) + ∑_n ≠ 0(
     1/√(R^2 |a|^2 + (θ_e/2π + n)^2) - κ_n )] 
        = 1/4π[ 1/√( |a|^2 + θ_e^2/4R^2 π^2) + R∑_n ≠ 0(
     1/√(R^2 |a|^2 + (θ_e/2π + n)^2) - κ_n ) ] 
        = 1/4π( 1/r + C(x⃗) ),

where C(x⃗) is smooth and bounded in a neighborhood of the origin.

Similarly, we do Poisson re-summation for the unitary connection


    A' = - 1/4π( da/a - da̅/a̅) [ i(θ_e - π)/2π + R ∑_n ≠ 0 (sgn n) e^inθ_e |a| K_1(2π R|na|) ].


Using the fact that the inverse Fourier transform of (sgn ξ)e^iθ_e ξ|a|K_1(2π R|aξ|) is

    i(θ_e/2π + t)/2R√(R^2|a|^2 + ( θ_e/2π + t)^2),

we obtain

    A'     = - i/8π( da/a - da̅/a̅)∑_n = -∞^∞( θ_e/2π + n√(R^2 |a|^2 + (θ_e/2π + n)^2) - κ_n  ) 
        = 1/4π( da/a - da̅/a̅)[-iθ_e/4π√(R^2 |a|^2 + (  
     θ_e/2π)^2) - i/2∑_n ≠ 0( θ_e/2π + n√(R^2 |a|^2 + (θ_e/2π + n)^2) - κ_n  )] 
    since dϕ = d a = -idloga|a| = -i2(daa - da̅a̅) and cosϑ = x^3r, this simplifies to:    = 1/4π(cosϑ + D(x⃗))dϕ.

Here κ_n is a regularization constant that makes the sum converge, and D(x⃗) is smooth and bounded in a neighborhood of r = 0. By (<ref>) and (<ref>), it follows that near r = 0

    g     = V^-1( dθ'_m/2π + A' )^2 + Vdx⃗^2
        = 4π( 1/r + C )^-1( dθ'_m/2π + 1/4πcosϑ dϕ + D dϕ)^2 + 1/4π( 1/r + C ) dx⃗^2
        = 1/4π[ ( 1/r + C )^-1( 2dθ'_m + cosϑ dϕ + D̃ dϕ)^2
     + ( 1/r + C ) dx⃗^2 ]
        = 1/4π g_Taub-NUT + smooth corrections.

This shows that our metric extends to r = 0 and finishes the construction of the singular fiber.




 §.§ General case



Here we work with the assumption in subsection <ref>. To distinguish this case to the previous one, we will denote by ϖ_old, g_old, etc. the forms obtained in the classical case.

Let C := -i/2 + π f'(0) and let

    B_0 = V_0 + R  Im C/π.

We will see that, to extend the holomorphic symplectic form ϖ(ζ)  and consequently the hyperkähler metric g to ℳ, it is necessary to impose a restriction on the class of functions f(a) on ℬ for the generalized Ooguri-Vafa case.


In the General Ooguri-Vafa case, the holomorphic symplectic form ϖ(ζ) and the hyperkähler metric g extend to ℳ, at least for the set of functions f(a) as in <ref> with f'(0) > B_0.


 By formula (<ref>),

    d log𝒳_m^sf = d log𝒳_m, old^sf + R/ζ( -i/2 + π f'(a) )da + Rζ( i/2 + πf'(a))da


Recall that the corrections of 𝒳_m are the same as the classical Ooguri-Vafa case. Thus, using (<ref>), at a = 0

    ϖ(ζ) = ϖ_old(ζ) + iR /2πIm  C da ∧ da + i C/4π^2 ζ
       da ∧ dθ_e + i ζC/4π^2 da∧ dθ_e.


Decomposing ϖ(ζ) = -i/2ζω_+ + ω_3 -iζ /2 ω_-, we obtain:

    ω_3     = ω_3, old + i R/2πIm  C da ∧ da, 
    ω_+     = ω_+, old - C/2π^2 da ∧ dθ_e 
    ω_-     = ω_-, old - C/2π^2 da∧ dθ_e


By (<ref>) and (<ref>),

    dθ'_m - i/R( V_0 - iRC/π)dθ_e and dθ'_m + i/R( V_0 + iRC/π)dθ_e

are, respectively, (1,0) and (0,1) forms. It's not hard to see that

    -V_0 π -iR C/Rπ∂_θ'_m     - i∂_θ_e
    or, rearranging real parts,( -V_0/R -Im C/π) ∂_θ'_m    -i ( Re C/π∂_θ'_m + ∂_θ_e)

is a (1,0) vector field. This allow us to obtain

    J_3[ ( -V_0/R -Im C/π) ∂_θ'_m]     = Re C/π∂_θ'_m + ∂_θ_e
    
    J_3[Re C/π∂_θ'_m + ∂_θ_e]     = ( V_0/R +Im C/π) ∂_θ'_m.

By linearity,

    J_3(∂_θ'_m)     = const·∂_θ'_m - Rπ/V_0 π + RIm C∂_θ_e
    
    J_3(∂_θ_e)     = ( V_0 π + RIm C /π R + (Re C)^2 R/π(V_0 π 
      + RIm C))∂_θ'_m + const·∂_θ_e.

With this we can compute

    g(∂_θ'_m, ∂_θ'_m)     = ω_3(∂_θ'_m, J_3(∂_θ'_m))
        = 1/4π(V_0 π + RIm C)
    
    g(∂_θ_e, ∂_θ_e)     = ω_3(∂_θ_e, J_3(∂_θ_e))
        = V_0 π + RIm C/4π^3 R^2 + (Re C)^2/4π^3(V_0 π + RIm C)
        = B_0/4π^3 R^2 + (Re C)^2/4π^3 B_0


We can see that, if B_0 > 0, the metric at a = 0 is

    g = 1/B_0( dθ'_m/2π)^2 + B_0 dx⃗^2 + (R·Re C/π)^2 dx_3^2/B_0.

This metric can be extended to the point θ_e = 0 (r = 0 in <ref>) exactly as before, by writing g as the Taub-NUT metric plus smooth corrections and observing that, since lim_θ_e → 0 B_0 = ∞,

    lim_θ_e → 0(R·Re C/π)^2 dx_3^2/B_0 = 0.






§ THE PENTAGON CASE






 §.§ Monodromy Data



Now we will extend the results of the Ooguri-Vafa case to the general problem. We will start with the Pentagon example. This  example is presented in detail in <cit.>. By <cit.>, this example represents the moduli space of Higgs bundles with gauge group SU(2) over  with 1 irregular singularity at z = ∞.

Here ℬ = with discriminant locus a 2-point set, which we can assume is {-2,2}  in the complex plane. Thus ℬ' is the twice-punctured plane. ℬ is divided into two domains ℬ_in and ℬ_out by the locus

    W = {u : Z(Γ_u)  is contained in a line in }⊂ℬ

See Figure <ref>. Since ℬ_in is simply connected Γ can be trivialized over ℬ_in by primitive cycles γ_1, γ_2, with Z_γ_1 = 0 at u = -2, Z_γ_2 = 0 at u = 2. We can choose them also so that ⟨γ_1, γ_2 ⟩ = 1. 



Take the set {γ_1, γ_2}. To compute its monodromy around infinity, take cuts at each point of D = {-2,2} (see Figure <ref>) and move counterclockwise. By (<ref>), the jump of γ_2 when you cross the cut at -2 is of the form γ_2 ↦γ_1 + γ_2. As you return to the original place and cross the cut at 2, the jump of γ_1 is of the type γ_1 ↦γ_1 - γ_2.




Thus, around infinity, {γ_1, γ_2} transforms into {-γ_2, γ_1 + γ_2}. The set {γ_1, γ_2, -γ_1, -γ_2, γ_1 + γ_2, -γ_1 - γ_2} is therefore invariant under monodromy at infinity and it makes global sense to define

  
    For u ∈ℬ_in, Ω(γ; u) =    {[                            1 for γ∈{γ_1, γ_2, -γ_1, -γ_2};                            0                    otherwise ]. 
    For u ∈ℬ_out , Ω(γ; u) =    {[                                                   1 for γ∈{γ_1, γ_2, -γ_1, -γ_2, γ_1 + γ_2, -γ_1 - γ_2};                                                   0                                           otherwise ].





Let ℳ' denote the torus fibration over ℬ' constructed in <cit.>. Near u=2, we'll denote γ_1 by γ_m and γ_2 by γ_e (the labels will change for u = - 2). To shorten notation, we'll write ℓ_e, Z_e, etc. instead of ℓ_γ_e, Z_γ_e, etc. Let θ denote the vector of torus coordinates (θ_e, θ_m). With the change of variables a := Z_e(u) we can assume, without loss of generality, that the bad fiber is at a = 0 and

    lim_a → 0 Z_m(a) = c ≠ 0.

Let T denote the complex torus fibration over ℳ' constructed in <cit.>. By the definition of Ω(γ; a), the functions (𝒳_e, 𝒳_m) both receive corrections. Recall that by (<ref>), for each ν∈ℕ, we get a function 𝒳_γ^(ν), which is the ν-th iteration of the function 𝒳_γ. We can write

    𝒳_γ^(ν)(a, ζ, θ) = 𝒳_γ^sf(a, ζ, θ)C_γ^(ν)(a, ζ, θ).

It will be convenient to rewrite the above equation as in <cit.>. For that, let  be the map from ℳ_a to its complexification ℳ_a^ such that

    𝒳_γ^(ν)(a, ζ, θ) = 𝒳_γ^sf(a, ζ, ).


We'll do a modification in the construction of <cit.> as follows: We'll use the term “BPS ray” for each ray {ℓ_γ : Ω(γ,a) ≠ 0 } as in <cit.>. This terminology comes from Physics. In the language of Riemann-Hilbert problems, these are known as “anti-Stokes” rays. That is, they represent the contour Σ where a function has prescribed discontinuities.

The problem is local on ℬ, so instead of defining a Riemann-Hilbert problem using the BPS rays ℓ_γ, we will cover ℬ' with open sets {U_α : α∈Δ} such that for each α, U_α is compact, U_α⊂ V_α, with V_α open and . ℳ' |_V_α a trivial fibration. For any ray r in the ζ-plane, define ℍ_r as the half-plane of vectors making an acute angle with r. Assume that there is a pair of rays r, -r such that for all a ∈ U_α, half of the rays  lie inside ℍ_r and the other half lie in ℍ_-r. We call such rays admissible rays. If U_α is small enough, there exists admissible rays for such a neighborhood. We are allowing the case that r is a BPS ray ℓ_γ, as long as it satisfies the above condition. As a varies in U_α, some BPS rays (or anti-Stokes rays, in RH terminology) converge into a single ray (wall-crossing phenomenon) (see Figures <ref> and <ref>). 





 
 For γ∈Γ, we define γ > 0 (resp. γ < 0) as ℓ_γ∈ℍ_r (resp. ℓ_γ∈ℍ_-r). Our Riemann-Hilbert problem will have only two anti-Stokes rays, namely r and -r. The specific discontinuities at the anti-Stokes rays for the function we're trying to obtain are called Stokes factors (see <cit.>). In (<ref>), the Stokes factor was given by S^-1_ℓ.
 
 In this case, the Stokes factors are the concatenation of all the Stokes factors S^-1_ℓ in (<ref>) in the counterclockwise direction:

    S_+     = ∏^_γ > 0𝒦^Ω(γ; a)_γ
    
    S_-     = ∏^_γ < 0𝒦^Ω(γ; a)_γ


We will denote the solutions of this Riemann-Hilbert problem by 𝒴. As in (<ref>), we can write 𝒴 as

    𝒴_γ(a, ζ, θ) = 𝒳_γ^sf(a, ζ, Θ),

for Θ : ℳ_a →ℳ_a^.

A different choice of admissible pairs r', -r' gives an equivalent Riemann-Hilbert problem, where the two solutions 𝒴, 𝒴' differ only for ζ in the sector defined by the rays r,r', and one can be obtained from the other by analytic continuation. 

In the case of the Pentagon, we have two types of wall-crossing phenomenon. Namely, as a varies, ℓ_e moves in the ζ-plane until it coincides with the ℓ_m ray for some value of a in the wall of marginal stability (Fig. <ref> and <ref>). We'll call this type I of wall-crossing. In this case we have the Pentagon identity

    𝒦_e 𝒦_m = 𝒦_m 𝒦_e+m𝒦_e,

As a goes around 0, the ℓ_e ray will then intersect with the ℓ_-m ray now. Because of the monodromy γ_m ↦γ_-e+m around 0, ℓ_m becomes ℓ_-e+m. This second type (type II) of wall-crossing is illustrated in Fig. <ref> and <ref>.





This gives a second Pentagon identity

    𝒦_-e𝒦_m = 𝒦_m 𝒦_-e+m𝒦_-e



In any case, the Stokes factors above remain the same even if a is in the wall of marginal stability. The way we defined S_+, S_- makes this true for the general case also.

Specifically, in the Pentagon the two Stokes factors for the first type of wall-crossing are given by the maps:

    . [                   𝒴_m ↦𝒴_m(1-𝒴_e(1-𝒴_m))^-1;                   𝒴_e           ↦𝒴_e(1-𝒴_m) ]}    S_+ 
    and, similarly. [                      𝒴_m ↦𝒴_m(1-𝒴^-1_e(1-𝒴^-1_m));                      𝒴_e        ↦𝒴_e(1-𝒴^-1_m)^-1 ]}    S_-


For the second type:


    . [                   𝒴_m        ↦𝒴_m(1-𝒴^-1_e);                   𝒴_e ↦𝒴_e(1-𝒴_m(1-𝒴^-1_e)) ]}    S_+ 
    
    . [                      𝒴_m           ↦𝒴_m(1-𝒴_e)^-1;                      𝒴_e ↦𝒴_e(1-𝒴^-1_m(1-𝒴_e))^-1 ]}    S_-




 §.§ Solutions


In <cit.> we prove the following theorem (in fact, a more general version is proven).


There exist functions 𝒴_m(a, ζ, θ_e, θ_m),  𝒴_e(a, ζ, θ_e, θ_m) defined for a ≠ 0,  smooth on a, θ_e and θ_m. The functions are sectionally analytic on ζ and obey the jump condition

    [        𝒴^+ = S_+ 𝒴^-,    along r;        𝒴^+  = S_-𝒴^-,   along -r ]

Moreover, 𝒴_m,  𝒴_e obey the reality condition (<ref>) and the asymptotic condition <ref>.



Our construction used integrals along a fixed admissible pair r,-r and our Stokes factors are concatenation of the Stokes factors in <cit.>. Thus, the coefficients f^γ' are different here, but they are still obtained by power series expansion of the explicit Stokes factor. In particular, it may not be possible to express

    f^γ' = c_γ'γ'

for some constant c_γ'. For instance, in the pentagon, wall-crossing type I, we have, for 0≤ j≤ i and γ' = γ_ie +jm:

    f^γ' = (-1)^jij/i^2γ_ie.

Because of this, we didn't use the Cauchy-Schwarz property of the norm in Γ in the estimates above as in <cit.>. Nevertheless, the tameness condition on the Ω(γ',a) invariants still give us the desired contraction.



Observe that, since we used admissible rays, the Stokes matrices don't change at the walls of marginal stability and we were able to treat both sides of the wall indistinctly. Thus, the functions 𝒴 in Theorem <ref> are smooth across the wall.



Let's reintroduce the solutions in <cit.>. Denote by 𝒳_e, 𝒳_m the solutions to the Riemann-Hilbert problem with jumps of the form S_ℓ^-1 at each BPS ray ℓ with the same asymptotics and reality condition as 𝒴_e, 𝒴_m. In fact, we can see that the functions 𝒴 are the analytic continuation of 𝒳 up until the admissible rays r, -r. 





In a patch U_α⊂ℬ' containing the wall of marginal stability, define the admissible ray r as the ray where ℓ_e, ℓ_m (or ℓ_e, ℓ_-m) collide. Since one is the analytic continuation of the other, 𝒳 and 𝒴 differ only in a small sector in the ζ-plane bounded by the ℓ_e, ℓ_m (ℓ_e, ℓ_-m) rays, for a not in the wall. As a approaches the wall, such a sector converges to the single admissible ray r. Thus, away from the ray where the two BPS rays collide, the solutions 𝒳 in <cit.> are continuous in a.




§ EXTENSION TO THE SINGULAR FIBERS


In this paper we will only consider the Pentagon example and in this section we will extend the Darboux coordinates 𝒳_e, 𝒳_m obtained above to the singular locus D ⊂ℬ where one of the charges Z_γ approaches zero.

Let u be a coordinate for ℬ =. We can assume that the two bad fibers of ℳ are at -2,2 in the complex u-plane. For almost all ζ∈, the BPS rays converge in a point of the wall of marginal stability away from any bad fiber:






It is assumed that lim_u → 2 Z_γ_1 exists and it is nonzero. If we denote this limit by c = |c|e^iϕ, then for ζ such that ζ→ϕ + π, the ray ℓ_γ_1 emerging from -2 approaches the other singular point u = 2 (see Figure <ref>).




When ζ = ϕ + π, the locus { u : Z_γ(u)/ζ∈_-}, for some γ such that Ω(γ;u) ≠ 0 crosses u = 2. See Figure <ref>.



As ζ keeps changing, the rays leave the singular locus, but near u = 2, the tags change due to the monodromy of γ_1 around u=2. Despite this change of labels, near u = 2 only the rays ℓ_γ_2, ℓ_-γ_2 pass through this singular point. See Figure <ref>



In the general case of Figures <ref>, <ref> or <ref>, the picture near u = 2 is like in the Ooguri-Vafa case, Figure <ref>.

In any case, because of the specific values of the invariants Ω, it is possible to analytically extend the function 𝒳_γ_1 around u = 2. The global jump coming from the rays ℓ_γ_2, ℓ_-γ_2 is the opposite of the global monodromy coming from the Picard-Lefschetz monodromy of γ_1 ↦γ_1 - γ_2 (see (<ref>)). Thus, it is possible to obtain a function 𝒳_γ_1 analytic on a punctured disk on ℬ' near u = 2 extending 𝒳_γ_1.

From this point on, we use the original formulation of the Riemann-Hilbert problem using BPS rays as in <cit.>. We also use a = Z_γ_2(u) to coordinatize a disk near u = 2, and we label {γ_1, γ_2} as {γ_m, γ_e} as in the Ooguri-Vafa case. Recall that, to shorten notation, we write ℓ_e, 𝒳_e, etc. instead of ℓ_γ_e, 𝒳_γ_e, etc.

By our work in the previous section, solutions 𝒳_γ (or, taking logs, Υ_γ) to the Riemann-Hilbert problem are continuous at the wall of marginal stability for all ζ except those in the ray ℓ_m = Z_m/ζ∈_- = ℓ_e (to be expected by the definition of the RH problem). We want to extend our solutions to the bad fiber located at a=0. We'll see that to achieve this, it is necessary to introduce new θ coordinates.




For convenience, we rewrite the integral formulas for the Pentagon in terms of Υ as in <cit.>. We will only write the part in ℬ_in, the  part is similar.

    Υ_e(a,ζ)     = θ_e -
    1/4π{∫_ℓ_mdζ'/ζ'ζ' + ζ/ζ' -ζlog[ 1 - 𝒳_m^sf(a,ζ', Υ_m) ] - ∫_ℓ_-mdζ'/ζ'ζ' + ζ/ζ' -ζlog[ 1 - 𝒳_-m^sf(a,ζ', Υ_-m)] }, 
    Υ_m(a,ζ)     = θ_m +
    1/4π{∫_ℓ_edζ'/ζ'ζ' + ζ/ζ' -ζlog[ 1 - 𝒳_e^sf(a,ζ', Υ_e) ]  - ∫_ℓ_-edζ'/ζ'ζ' + ζ/ζ' -ζlog[ 1 - 𝒳_-e^sf(a,ζ', Υ_-e) ] }











We can focus only on the integrals above, so write Υ_γ(a,ζ) = θ_γ + 14πΦ_γ(a,ζ), for γ∈{γ_m, γ_e}. To obtain the right gauge transformation of the torus coordinates θ, we'll split the integrals above into four parts and then we'll show that two of them define the right change of coordinates (in , and a similar transformation for ) that simplify the integrals and allow an extension to the singular fiber.

By Theorem <ref>, both Υ_m, Υ_e satisfy the “reality condition”, which expresses a symmetry in the behavior of the complexified coordinates Υ:

 
    Υ_γ(a, ζ) = Υ_γ(a, -1/ζ),      a ≠ 0
 
 If we write as Υ_0 (resp. Υ_∞) the asymptotic of this function as ζ→ 0 (resp. ζ→∞) so that
 
    Υ_0 = θ + 1/4πΦ_0,

 for a suitable correction Φ_0. A similar equation holds for the asymptotic as ζ→∞. By the asymptotic condition <ref>, Φ_0 is imaginary. 

Condition (<ref>) also shows that Φ_0 = - Φ_∞. This and the fact that Φ_0 is imaginary give the reality condition
 
    Υ_0 = Υ_∞

 
Split the integrals in (<ref>) into four parts as in (<ref>). For example, if we denote by ζ_e := -a/|a|, the intersection of the unit circle with the ℓ_e ray, then

    ∫_ℓ_edζ'/ζ'ζ' + ζ/ζ' -ζlog( 1 - 𝒳_e^sf(a,ζ', Υ_e) ) = 
        -∫_0^ζ_edζ'/ζ'log( 1 - 𝒳_e^sf(a,ζ', Υ_e) ) + ∫_ζ_e^ζ_e ∞dζ'/ζ'log( 1 - 𝒳_e^sf(a,ζ', Υ_e) ) 
        + ∫_0^ζ_e2 dζ'/ζ'-ζlog( 1 - 𝒳_e^sf(a,ζ', Υ_e) )  + ∫_ζ_e^ζ_e ∞ 2dζ' {1/ζ'-ζ -1/ζ'}log( 1 - 𝒳_e^sf(a,ζ', Υ_e) )


We consider the first two integrals apart from the rest. If we take the limit a → 0 the exponential decay in 𝒳_e^sf:

    exp( π R a/ζ' + π R ζ' a)

vanishes and the integrals are no longer convergent. 




 
 By combining the two integrals with their analogues in the ℓ_-e ray we obtain:

    -∫_0^ζ_edζ'/ζ'log( 1 - 𝒳_e^sf(a,ζ', Υ_e) ) + ∫_ζ_e^ζ_e ∞dζ'/ζ'log( 1 - 𝒳_e^sf(a,ζ', Υ_e) ) 
       ∫_0^-ζ_edζ'/ζ'log( 1 - 𝒳_e^sf^-1(a,ζ', -Υ_e) ) - ∫_-ζ_e^-ζ_e ∞dζ'/ζ'log( 1 - 𝒳_e^sf^-1(a,ζ', -Υ_e) )

The parametrization in the first pair of integrals is of the form ζ' = tζ_e, and in the second pair ζ' = -tζ_e. Making the change of variables ζ' ↦ 1/ζ', we can pair up these integrals in a more explicit way as:

    -∫_0^1 dt/t{log[ 1 - exp( -π R |a| (1/t + t ) +iΥ_e(a,-te^i a)  ) ] . 
       . + log[ 1 - exp( -π R |a| (1/t + t ) - iΥ_e(a,1/t e^i a)  ) ] }
         + ∫_0^1 dt/t{log[ 1 - exp( -π R |a| (1/t + t ) +iΥ_e(a,-1/t e^i a) ) ] . 
       . + log[ 1 - exp( -π R |a| (1/t + t ) -iΥ_e(a,te^i a) ) ]  }


By (<ref>), the integrands come in conjugate pairs. Therefore, we can rewrite (<ref>) as:
 
    -2∫_0^1 dt/tRe {.    log[ 1 - exp( -π R |a| (1/t + t ) +iΥ_e(a,-te^i a) ) ] -  
       . log[ 1 - exp( -π R |a| (1/t + t ) -iΥ_e(a,te^i a)  ) ] }

 
    = -2∫_0^1 dt/tlog| 1 - exp( -π R |a| (t^-1 + t ) +iΥ_e(a,-te^i a)  )/1 - exp( -π R |a| (t^-1 + t ) -iΥ_e(a,te^i a) )|

 

Observe that (<ref>) itself suggest the correct transformation of the θ coordinates that fixes this. Indeed, for a fixed a ≠ 0 and θ_e, let Q be the map

    Q(θ_m) =  θ_m + ψ(a,θ),

where
 
    ψ_in(a,θ)    = 1/2π∫_0^1 dt/tlog| 1 - exp( -π R |a| (t^-1 + t ) +iΥ_e(a,-te^i a)  )/1 - exp( -π R |a| (t^-1 + t ) -iΥ_e(a,te^i a)  )| 
        = 1/2π∫_0^1 dt/tlog| 1 - [𝒳_e](-te^i a)/1 - [𝒳_-e](te^i a)|

 for a ∈. For a ∈ where the wall-crossing is of type I, let φ =  (Z_γ_e + γ_m(a)), with ζ'= -t e^iφ parametrizing the ℓ_e + m ray:
 
    ψ_out(a,θ)     = 1/2π∫_0^1 dt/t{log| 1 - exp( -π R |a| (t^-1 + t ) +iΥ_e(a,-te^i a)  )/1 - exp( -π R |a| (t^-1 + t ) -iΥ_e(a,te^i a)  )| . 
        + . log| 1 - exp( -π R |Z_γ_e + γ_m| (t^-1 + t ) +iΥ_e +m(a,-te^iφ)  )/1 - exp( -π R |Z_γ_e + γ_m| (t^-1 + t ) -iΥ_e+m(a,te^iφ)  )|  }
        = 1/2π∫_0^1 dt/t{log| 1 - [𝒳_e](-te^i a)/1 - [𝒳_-e](te^i a)| + log| 1 - [𝒳_e+m](-te^iφ)/1 - [𝒳_-e-m](te^iφ)| }

 
 Similarly, for wall-crossing of type II, φ =  (Z_γ_-e + γ_m(a)), with ζ'= -t e^iφ for the ℓ_-e + m ray:
  
    ψ_out(a,θ)     = 1/2π∫_0^1 dt/t{log| 1 - exp( -π R |a| (t^-1 + t ) +iΥ_e(a,-te^i a)  )/1 - exp( -π R |a| (t^-1 + t ) -iΥ_e(a,te^i a)  )| . 
        + . log| 1 - exp( -π R |Z_γ_-e + γ_m| (t^-1 + t ) +iΥ_-e +m(a,-te^iφ)  )/1 - exp( -π R |Z_γ_-e + γ_m| (t^-1 + t ) -iΥ_-e+m(a,te^iφ)  )|  }
        = 1/2π∫_0^1 dt/t{log| 1 - [𝒳_e](-te^i a)/1 - [𝒳_-e](te^i a)| + log| 1 - [𝒳_-e+m](-te^iφ)/1 - [𝒳_e-m](te^iφ)| }

 
 As a approaches the wall of marginal stability W, a →φ. We need to show the following

The two definitions ψ_in and ψ_out coincide at the wall of marginal stability.
 

  First let a approach W from the “in” region, so we're using definition (<ref>). Start with the pair of functions (𝒳_e, 𝒳_m) in the ζ-plane and let 𝒳_e denote the analytic continuation of 𝒳_e. See Figure <ref>. When they reach the ℓ_e ray, 𝒳_e jumped to 𝒳_e(1-𝒳_m) by (<ref>) and (<ref>). Thus 𝒳_e = 𝒳_e(1-𝒳_m) along the ℓ_e ray.



Therefore, 

    ψ_in(a,θ) = 1/2π∫_0^1 dt/tlog| 1 - [𝒳_e(1-𝒳_m)](-te^i a)/1 - [𝒳_-e(1-𝒳_m)^-1](te^i a)|


Now starting from the “out” region, and focusing on the wall-crossing of type I for the moment, we start with the pair (𝒳_e, 𝒳_m) as before. This time, 𝒳_e at the ℓ_e ray has not gone to any jump yet. See Figure <ref>. Only 𝒳_e+m undergoes a jump at the ℓ_e+m ray and it is of the form 𝒳_e+m↦𝒳_e+m(1-𝒳_e)^-1.



When a hits the wall W, φ =  a and the integrals are taken over the same ray. Thus, we can combine the logs and obtain:

 
    ψ_out(a,θ)      = 1/2π∫_0^1 dt/t{log| 1 - [𝒳_e](-te^i a)/1 - [𝒳_-e](te^i a)| + log| 1 - [𝒳_e+m(1-𝒳_e)^-1](-te^i a)/1 - [𝒳_-e-m(1-𝒳_e)](te^i a)| }
        = 1/2π∫_0^1 dt/tlog| 1 - [𝒳_e(1-𝒳_m)](-te^i a)/1 - [𝒳_-e(1-𝒳_m)^-1](te^i a)|

 
and the two definitions coincide. For the wall-crossing of type II the proof is entirely analogous.


 






 
 
 Q is a reparametrization in θ_m; that is, a diffeomorphism of ℝ/2πℤ.
 




 
 To show that Q is injective, it suffices to show that |∂ψ/∂θ_m| < 1. We will show this in the  region. The proof for the  region is similar.
 
 To simplify the calculations, write
  
    ψ(a,θ) = 2∫_0^1 dt/tlog| 1-Cf(θ_m)/1-Cg(θ_m)|

  for functions f, g of the form e^iΥ_γ for different choices of γ (they both depend on other parameters, but they're fixed here) and a factor C of the form
  
    C = exp( -π R |a| (t^-1 + t))

  Now take partials in both sides of (<ref>) and bring the derivative inside the integral. After an application of the chain rule we get the estimate
  
    | ∂ψ/∂θ_m| ≤ 2∫_0^1 dt/t |C| {|f||∂Υ_e(t)/∂θ_m|/|1-Cf| + |g||∂Θ_e(-t)/∂θ_m|/|1-Cg|}
 
 By the estimates in <cit.>, |∂Υ_e/∂θ_m| < 1. In <cit.>, we show that |f|, |g| can be bounded by 2. The part C has exponential decay so if R is big enough we can bound the above by 1 and injectivity is proved. For surjectivity, just observe that ψ(θ_m + 2π) = ψ(θ_m), so Q(θ_m + 2π) = θ_m + 2π. 
 
 
With respect to the new coordinate θ'_m, the functions Υ_e, Υ_m satisfy the equation:


    Υ_e(a,ζ)  = θ_e +
    1/4π∑_γ'Ω(γ';a) ⟨γ_e, γ' ⟩   ∫_γ'dζ'/ζ'ζ' + ζ/ζ' -ζlog[ 1 - 𝒳_γ'^sf(a,ζ', Υ_γ') ]  
    
    Υ_m(a,ζ)  = θ'_m +
    1/2π∑_γ'Ω(γ';a) ⟨γ_m, γ' ⟩{.
       ∫_0^b'dζ'/ζ' - ζlog[ 1 - 𝒳_γ'^sf(a,ζ', Υ_γ') ] + 
       .  ∫_b'^b' ∞ζ dζ'/ζ'(ζ' - ζ)log[ 1 - 𝒳_γ'^sf(a,ζ', Υ_γ') ] } ,

for b' the intersection of the unit circle with the ℓ_γ' ray. The Ω(γ';a) jump at the wall, but in the Pentagon case, the sum is finite.

In order to show that Υ converges to some function, even at a = 0, observe that the integral equations in (<ref>) and (<ref>) still make sense at the singular fiber, since in the case of (<ref>), lim_a → 0 Z_m = c ≠ 0 and the exponential decay is still present, making the integrals convergent. In the case of (<ref>), the exponential decay is gone, but the different kernel makes the integral convergent, at least for ζ∈^×. The limit function lim_a → 0Υ should be then a solution to the integral equations obtained by recursive iteration, as in <cit.>.

We have to be specially careful with the Cauchy integral in (<ref>). It will be better to obtain each iteration Υ^(ν)_m when |a| → 0 by combining the pair of rays ℓ_γ', ℓ_-γ' into a single line L_γ', where in the case of the Pentagon, γ' can be either γ_e or γ_e+m, depending on the side of the wall we're at. We formulate a boundary problem over each infinite curve L_γ' as in <ref>. As in the Ooguri-Vafa case, the jump function[Since we do iterations of boundary problems, we abuse notation and use simply G(ζ) where it should be G^(ν)(ζ). This shouldn't cause any confusion, as our main focus in this section is how to obtain any iteration of 𝒳_m] G(ζ) has discontinuities of the first kind at 0 and ∞, but we also have a new difficulty: For θ_e close to 0, the jump function G(ζ) = 1-e^iΥ^(ν - 1)_γ'(ζ) may be 0 for some values of ζ.

Since the asymptotics of Υ^(ν)_e as ζ→ 0 or ζ→∞ are θ_e ± iϕ_e ≠ 0, the jump function G(ζ) can only attain the 0 value inside a compact interval away from 0 or ∞, hence these points are isolated in L_γ'. By the symmetry relation expressed in Lemma <ref>, the zeroes of G(ζ) come in pairs in L_γ' and are of the form ζ_k, -1/ζ_k. By our choice of orientation for L_γ', one of the jumps is inverted so that G(ζ) has only zeroes along L_γ' and no poles.

Thus, as in <ref>, we have a Riemann-Hilbert problem of the form[To simplify notation, we omit the iteration index ν in the Riemann-Hilbert problem expressed. By definition, 𝒳_m = 𝒳^sf_m X_m, for any iteration ν]

    X_m^+(ζ) = G(ζ) X_m^-(ζ)


In <cit.>, we show that the solutions of (<ref>) exist and are unique, given our choice of kernel in (<ref>). We thus obtain each iteration Υ_m^(ν) of (<ref>). Moreover, since by <cit.>, 𝒳_m^+ = 0 at points ζ in the L_e ray where G(ζ) = 0, Υ_m^(ν)+ has a logarithmic singularity at such points.







 §.§ Estimates and a new gauge transformation






















	


















As we've seen in the Ooguri-Vafa case, we expect our solutions lim_a → 0Υ to be unbounded in the ζ variable. 
Define a Banach space X as the completion under the sup norm of the space of functions Φ: ^××𝕋× U →^2n  that are piecewise holomorphic on ^×, smooth on 𝕋× U, for U an open subset of ℬ containing 0 and such that (<ref>), (<ref>) hold. 

Like in the Ooguri-Vafa case, let a → 0 fixing a. We will later get rid of this dependence on a with another gauge transformation of θ_m. The following estimates on Υ^(ν) will clearly give us that the sequence converges to some limit Υ^(ν).


In the Pentagon case, at the bad fiber a = 0:

    Υ_e^(ν + 1)    = Υ_e^(ν) + O( e^-2πν R |Z_m|), ν≥ 2 
    Υ_m^(ν + 1)    = Υ_m^(ν) + O( e^-2πν R |Z_m|), ν≥ 1




As before, we prove this by induction. Note that Υ^(1)_m = Υ^OV, the extension of the Ooguri-Vafa case obtained in (<ref>), and Υ^(1)_m differs considerably from θ_m because of the logζ term. Hence the estimates cannot start at ν = 0. Because of this reason, Υ^(2)_e differs considerably from Υ^(1)_e since this is the first iteration where Υ^(1)_m is considered.

Let ν = 1. The integral equations for Υ_e didn't change in this special case. By Lemma 3.3 in <cit.>, we have for the general case:


    Υ^(1)_e = θ_e + ∑_γ'Ω(γ',a) ⟨γ_e, γ' ⟩e^-2π R |Z_γ'|/4π i √(R |Z_γ'|)ζ_γ' + ζ/ζ_γ' - ζ e^iθ_γ' + O( e^-2π R |Z_γ'|/R)

where ζ_γ' = -Z_γ'/|Z_γ'| is the saddle point for the integrals in (<ref>), and ζ is not ζ_γ'. Note that there is no divergence if ζ→ 0 or ζ→∞. If ζ = ζ_γ', again by Lemma 3.3 in <cit.>, we obtain estimates as in (<ref>) except for the √(R) terms in the denominator.

In any case, for the Pentagon, the γ' in (<ref>) are only γ_± m,  γ_± (e+m), depending on the side of the wall of marginal stability. At a = 0, Z_e+m = Z_m, so (<ref>)  gives that log[1 - e^i Υ^(1)_e] = log[1 - e^iθ_e] + O(e^-2π R |Z_m|) along the ℓ_e ray, and a similar estimate holds for log[1 - e^-i Υ^(1)_e] along the ℓ_-e ray. Plugging in this in (<ref>), we get (<ref>) for ν = 1.

For general ν, a saddle point analysis on Υ^(ν)_e can still be performed and obtain as in (<ref>):

    Υ^(ν+1)_e = θ_e + e^-2π R |Z_m|/4π i √(R |Z_m|){ζ_m + ζ/ζ_m - ζ e^iΥ^(ν)_m(ζ_m) -  ζ_m - ζ/ζ_m + ζ e^-iΥ^(ν)_m(-ζ_m)} + O( e^-2π R |Z_γ'|/R),

from one side of the wall. On the other side (for type I) it will contain the extra terms

    e^-2π R |Z_m|/4π i √(R |Z_m|){ζ_m + ζ/ζ_m - ζ e^i(Υ^(ν)_m(ζ_m) + Υ^(ν)_e(ζ_m)) -  ζ_m - ζ/ζ_m + ζ e^-i(Υ^(ν)_m(-ζ_m) - Υ^(ν)_e(-ζ_m))}.

Observe that for this approximation we only need Υ^(ν) at the point ζ_m. By the previous part, for ν = 2,

    e^iΥ^(2)_m(ζ_m) = e^iΥ^(1)_m(ζ_m)(1 + O( e^-2π R |Z_m|) )

Thus, for ν = 2,

    Υ^(3)_e     = θ_e + e^-2π R |Z_m|/4π i √(R |Z_m|){ζ_m + ζ/ζ_m - ζ e^iΥ^(1)_m(ζ_m)(1 + O( e^-2π R |Z_m|) ) . 
        -  . ζ_m - ζ/ζ_m + ζ e^-iΥ^(1)_m(-ζ_m)(1 + O( e^-2π R |Z_m|) ) } + O( R^1/2)
        = Υ^(2)_e + O( e^-4π R|Z_m|)

and similarly in the other side of the wall. For general ν, the same arguments show that (<ref>), (<ref>) hold after the appropriate ν.






There is still one problem: the limit of 𝒳_m we obtained as a → 0 for the analytic continuation of 𝒳_m was only along a fixed ray a = constant. To get rid of this dependence, it is necessary to perform another gauge transformation on the torus coordinates θ. Recall that we are restricted to the Pentagon case. Let a → 0 fixing a. Let ζ_γ denote Z_γ/|Z_γ|. In particular, ζ_e = a/|a| and this remains constant since we're fixing a. Also, ζ_m = Z_m/|Z_m| and this is independent of a since Z_m has a limit as a → 0. The following lemma will allow us to obtain the correct gauge transformation.
 

For the limit . 𝒳_m|_a=0 obtained above, its imaginary part is independent of the chosen ray a = c along which a → 0.


Let Υ_m denote the analytic continuation of Υ_m yielding 𝒳_m. Start with a fixed value a ≡ρ_0, for ρ_0 different from Z_m(0),  (-Z_m(0)). For another ray a ≡ρ, we compute . Υ_m|_a=0
 a = ρ - . Υ_m|_a=0
 a = ρ_0 (without analytic continuation for the moment).

The integrals in (<ref>) are of two types. One type is of the form

    ∫_0^ζ_± edζ'/ζ' - ζlog[ 1 - e^iΥ_± e(ζ')] +  ∫_ζ_± e^ζ_± e∞ζ dζ'/ζ'(ζ' - ζ)log[ 1 - e^iΥ_± e(ζ')]

The other type appears only in the outside part of the wall of marginal stability. Since Z : Γ→ is a homomorphism, Z_γ_e + γ_m = Z_γ_e + Z_γ_m. At a = 0, Z_e = a = 0, so Z_e+m = Z_m. Hence, ℓ_m = ℓ_e+m at the singular fiber. This second type of integral is thus of the form

    ∫_0^ζ_± mdζ'/ζ' - ζlog[ 1 - e^iΥ_± (e+m)(ζ')] +  ∫_ζ_± m^ζ_± m∞ζ dζ'/ζ'(ζ' - ζ)log[ 1 - e^iΥ_± (e+m)(ζ')]

Since the ℓ_m stays fixed at a = 0 independently of a, (<ref>) does not depend on a, so this has a well-defined limit as a → 0. We should focus then only on integrals of the type (<ref>). For a different a, ζ_e changes to another point ζ_e in the unit circle. See Figure <ref>. The paths of integration change accordingly. We have two possible outcomes: either ζ lies outside the sector determined by the two paths, or ζ lies inside the region.



In the first case (ζ_1 on Figure <ref>), the integrands

    log[1-e^iΥ_± e(ζ')]/ζ'-ζ, ζlog[1-e^iΥ_± e(ζ')]/ζ'(ζ'-ζ)

are holomorphic on ζ' in the sector between the two paths. By Cauchy's formula, the difference between the two integrals is just the integration along a path C_± e between the two endpoints ζ_± e, ζ_± e. If f(s) parametrizes the path C_e, let C_-e = -1/f(s). The orientation of C_e in the contour containing ∞ is opposite to the contour containing 0. Similarly for C_-e. Thus, the difference of Υ_m for these two values of a is the integral along C_e, C_-e of the difference of kernels (<ref>), namely:

    ∫_C_edζ'/ζ'log[1-e^iΥ_e(ζ')] - ∫_C_-edζ'/ζ'log[1-e^-iΥ_e(ζ')]
 

Even if e^iΥ_e(ζ') = 1 for ζ' in the contour, the integrals in (<ref>) are convergent, so this is well-defined for any values of θ_e ≠ 0. By symmetry of C_e, C_-e and the reality condition (<ref>), the second integral is the conjugate of the first one. Thus (<ref>) is only real.

When ζ hits one of the contours, ζ coincides with one of the ℓ_e or ℓ_-e rays, for some value of a. The contour integrals jump since ζ lies now inside the contour (ζ_2 in Figure <ref>). The jump is by the residue of the integrands (<ref>). This gives the jump of 𝒳_m that the analytic continuation around a = 0 cancels. Therefore, only the real part of Υ_m depends on a.


By the previous lemma, . Υ_m|_a=0
 a = ρ - . Υ_m|_a=0
 a = ρ_0 is real and is given by (<ref>). Define then a new gauge transformation:

    θ_m = θ'_m - 1/2π{∫_C_edζ'/ζ'log[1-e^iΥ_e(ζ')] + ∫_C_-edζ'/ζ'log[1-e^-iΥ_e(ζ')] }


This eliminates the dependence on a for the limit . 𝒳_m |_a=0. As we did in <ref> in Theorem <ref>, we can extend the torus fibration ℳ' by gluing a S^1-fiber bundle of the form D × (0, 2π) × S^1 for D a disk around a = 0, θ_e ∈ (0,2π) and θ_m the new coordinate of the S^1 fibers. Using Taub-NUT space as a local model for this patch, the trivial S^1 bundle can be extended to θ_e = 0 where the fiber degenerates into a point (nevertheless, in Taub-NUT coordinates the space is still locally isomorphic to ^2). Since 𝒳_m ≡ 0 if θ_e = 0 as in <ref>, in this new manifold ℳ we  thus obtain a well defined function 𝒳_m.





 §.§ Extension of the derivatives

Extension of the derivatives@Extension of the derivatives

So far we were able to extend the functions 𝒳_e, 𝒳_m to ℳ. Unfortunately, we can no longer bound uniformly on ν the derivatives of 𝒳_m near a = 0, so the Arzela-Ascoli arguments no longer work here. Since there's no difference on the definition of 𝒳_e at a = 0 from that of the regular fibers, this function extends smoothly to a = 0.

We have to obtain the extension of all derivatives of 𝒳_m directly from its definition. It suffices to extend the derivatives of 𝒳_m only, as the analytic continuation doesn't affect the symplectic form ϖ(ζ) (see below).


log𝒳_m extends smoothly to ℳ, for θ_e ≠ 0.


For convenience, we rewrite Υ_m with the final magnetic coordinate θ_m:

    Υ_m     = θ_m + 1/2π{∫_C_edζ'/ζ'log[1-e^i Υ_e(ζ')] -
    ∫_C_-edζ'/ζ'log[1-e^-i Υ_e(ζ')] }
        + 1/2π∑_γ'Ω(γ';a) ⟨γ_m, γ' ⟩{∫_0^ζ_γ'dζ'/ζ' - ζlog[ 1 - 𝒳_γ'^sf(a,ζ', Υ_γ') ] . +  
       .  ∫_ζ_γ'^ζ_γ'∞ζ dζ'/ζ'(ζ' - ζ)log[ 1 - 𝒳_γ'^sf(a,ζ', Υ_γ') ] }

where e^i Υ_e(ζ') is evaluated only at a = 0. For γ' of the type ±γ_e ±γ_m, 𝒳_γ' and its derivatives still have exponential decay along the ℓ_γ' ray, so these parts in Υ_m extend to a =0 smoothly. It thus suffices to extend only


    Υ_m     = θ_m + 1/2π{∫_C_edζ'/ζ'log[1-e^i Υ_e(ζ')] -
    ∫_C_-edζ'/ζ'log[1-e^-i Υ_e(ζ')] . 
        +  ∫_0^ζ_edζ'/ζ' - ζlog[ 1 - 𝒳_e^sf(a,ζ', Υ_e) ] + ∫_ζ_e^ζ_e∞ζ dζ'/ζ'(ζ' - ζ)log[ 1 - 𝒳_e^sf(a,ζ', Υ_e) ] 
        -. ∫_0^-ζ_edζ'/ζ' - ζlog[ 1 - 𝒳_e^sf^-1(a,ζ', -Υ_e) ] -  ∫_-ζ_e^-ζ_e∞ζ dζ'/ζ'(ζ' - ζ)log[ 1 - 𝒳_e^sf^-1(a,ζ', -Υ_e) ]}

together with the semiflat part π R Z_m/ζ + π R ζZ_m, which we assume is as in the Generalized Ooguri-Vafa case, namely:

    𝒳_m =  exp( -i R /2ζ(alog a - a + f(a)) + i Υ_m + iζ R/2 (aloga - a + f(a) ))

for a holomorphic function f near a = 0 and such that f(0) ≠ 0. The derivatives of the terms involving f(a) clearly extend to a = 0, so we focus on the rest, as in <ref>.

We show first that ∂log𝒳_m∂_θ_e, ∂log𝒳_m∂_θ_m extend to a = 0. Since there is no difference in the proof between electric or magnetic coordinates, we'll denote by ∂_θ a derivative with respect to any of these two variables.

We have:

    ∂/∂θlog𝒳_m     = -i/2π{∫_C_edζ'/ζ'e^i Υ_e(ζ')/1-e^i Υ_e(ζ')∂Υ_e(ζ')/∂θ - ∫_C_-edζ'/ζ'e^-i Υ_e(ζ')/1-e^-i Υ_e(ζ')∂Υ_e(ζ')/∂θ.
        +  ∫_0^ζ_edζ'/ζ' - ζ𝒳_e(ζ')/1-𝒳_e(ζ')∂Υ_e(ζ')/∂θ + ∫_ζ_e^ζ_e∞ζ dζ'/ζ'(ζ' - ζ)𝒳_e(ζ')/1-𝒳_e(ζ')∂Υ_e(ζ')/∂θ
       . +  ∫_0^-ζ_edζ'/ζ' - ζ𝒳^-1_e(ζ')/1-𝒳^-1_e(ζ')∂Υ_e(ζ')/∂θ + ∫_-ζ_e^ζ_e∞ζ dζ'/ζ'(ζ' - ζ)𝒳^-1_e(ζ')/1-𝒳^-1_e(ζ')∂Υ_e(ζ')/∂θ}



when a → 0, 𝒳_e(ζ')1-𝒳_e(ζ')→e^i Υ_e(ζ')1-e^i Υ_e(ζ'). The integrals along C_e and C_-e represent a difference of integrals along the contour in the last integrals and a fixed contour, as in Figure <ref>. Thus, when a = 0,


    . 2π i ∂/∂θlogΥ_m|_a =0    =  ∫_0^bdζ'/ζ' - ζ𝒳_e(ζ')/1-𝒳_e(ζ')∂Υ_e(ζ')/∂θ + ∫_b^b ∞ζ dζ'/ζ'(ζ' - ζ)𝒳_e(ζ')/1-𝒳_e(ζ')∂Υ_e(ζ')/∂θ
       . +  ∫_0^-bdζ'/ζ' - ζ𝒳^-1_e(ζ')/1-𝒳^-1_e(ζ')∂Υ_e(ζ')/∂θ + ∫_-b^-b ∞ζ dζ'/ζ'(ζ' - ζ)𝒳^-1_e(ζ')/1-𝒳^-1_e(ζ')∂Υ_e(ζ')/∂θ}

for a fixed point b in the unit circle, independent of a. If Υ_e(ζ') = 1 for a point c in the line L passing through the origin and b, then as seen in <cit.>, the function 𝒳_m develops a zero on the right side of such line. Nevertheless, the analytic continuation 𝒳_m around a = 0 introduces a factor of the form (1 - 𝒳_e)^-1 when a changes from region III to region I in Figure <ref>, so the pole at c on the right side of L for the derivative ∂∂θlogΥ_m coming from the integrand in (<ref>) is canceled by analytic continuation. Hence, the integrals are well defined and thus the left side has an extension to a = 0.

Now, for the partials with respect to a, a, there are two different types of dependence: one is the dependence of the contours, the other is the dependence of the integrands. The former dependence is only present in (<ref>), as the contours in Figure <ref> change with a. A simple application of the Fundamental Theorem of Calculus in each integral in (<ref>) gives that this change is:


    . -2π i  ∂/∂ alogΥ_m|_a =0    = log[1-e^-i Υ_e(ζ_e)]  - log[1-e^-i Υ_e(ζ_e)]
        - log[1-e^-i Υ_e(ζ_e)] + log[1-e^-i Υ_e(ζ_e)] = 0,

where we again used the fact that the integrals along C_e and C_-e represent the difference between the integrals in the other pairs with respect to two different rays, one fixed. By continuity on parameters, the terms are still 0 if Υ_e(ζ_e) = 0. Compare this with (<ref>), where we obtained this explicitly.

Then there is the dependence on a, a on the integrands and the semiflat part. Focusing on a only, we take partials on log𝒳_m in (<ref>) (ignoring constants and parts that clearly extend to a = 0). This is:


    log a/ζ + ∫_0^ζ_edζ'/ζ'(ζ' - ζ)𝒳_e/1-𝒳_e + ∫_0^-ζ_edζ'/ζ'(ζ' - ζ)𝒳^-1_e/1-𝒳^-1_e


This is the equivalent of (<ref>) in the general case. In the limit a → 0, we can do an asymptotic expansion of e^i Υ_e(ζ')1-e^i Υ_e(ζ') = e^i Υ_e(0)1-e^i Υ_e(0) + O(ζ'). Clearly when we write this expansion in (<ref>), the only divergent term at a = 0 is the first degree approximation in the integral. Thus, we can focus on that and assume that the 𝒳_e1-𝒳_e (resp. 𝒳^-1_e1-𝒳^-1_e) factor is constant. If we do the partial fraction decomposition, we can run the same argument as in Eqs. (<ref>) up to (<ref>) and obtain that (<ref>) is actually 0 at a = 0. The only identity needed is

    1/1-e^iΥ_e(0) +  1/1-e^-iΥ_e(0)  =  1


The argument also works for the derivative with respect to a, now with an asymptotic expansion around ∞ of Υ_e.

This shows that 𝒳_m extends in a C^1 way to a = 0. For the C^∞ extension, derivatives with respect to any θ coordinate work in the same way, all that was used was the specific form of the contours C_e, C_-e. The same thing applies to the dependence on the contours C_e, C_-e. For derivatives with respect to a, a in the integrands, we can again do an asymptotic expansion of Υ_e at 0 or ∞ and compare it to the asymptotic of the corresponding derivative of a log a - a as a → 0.



Nothing we have done in this section is particular of the Pentagon example. We only needed the specific values of Ω(γ;u) given in (<ref>) to obtain the Pentagon identities at the wall and to perform the analytic continuation of 𝒳_m around u = 2.  For any integrable systems data as in section <ref>  with suitable invariants Ω(γ;u) allowing the wall-crossing formulas and analytic continuation, we can do the same isomonodromic deformation of putting all the jumps at a single admissible ray, perform saddle-point analysis and obtain the same extensions of the Darboux coordinates 𝒳_γ. This finishes the proof of Theorem <ref>.

What is exclusive of the Pentagon case is that we have a well-defined hyperkähler metric g_OV that we can use as a local model of the metric to be constructed here.

The extension of the holomorphic symplectic form ϖ(ζ) is now straightforward. We proceed as in <cit.> by first writing:

    ϖ(ζ) = -1/4π^2 Rd𝒳_e/𝒳_e∧d𝒳_m/𝒳_m
 

Where we used the fact that the jumps of the functions 𝒳_γ are via the symplectomorphisms 𝒦_γ' of the complex torus T_a (see (<ref>)) so ϖ(ζ) remains the same if we take 𝒳_m or its analytic continuation 𝒳_m.

We need to show that ϖ(ζ) is of the form

    -i/2ζω_+ + ω_3 -i ζ/2ϖ_-

that is, ϖ(ζ) must have simple poles at ζ = 0 and ζ = ∞, even at the singular fiber where a = 0.

By definition, 𝒳_e = exp(π R a/ζ + iΥ_e + π R ζa). Thus


    d𝒳_e(ζ)/𝒳_e(ζ) = π R da/ζ + i dΥ_e(ζ) +π R ζ da


By (<ref>), and since lim_a → 0 Z_m ≠ 0, 𝒳_m (resp. 𝒳_-m) of the form exp(π R Z_m(a)/ζ + iΥ_m + π R ζZ_m(a)) still has exponential decay when ζ lies in the ℓ_m ray (resp. ℓ_-m), even if a = 0. The differential d Υ_e(ζ) thus exists for any ζ∈ since the integrals defining it converge for any ζ.

As in <cit.>, we can write

    d𝒳_e/𝒳_e∧d𝒳_m/𝒳_m = d𝒳_e/𝒳_e∧( d𝒳^sf_m/𝒳^sf_m + ℐ_±),

for ℐ_± denoting the corrections to the semiflat function. By the form of 𝒳^sf =  exp(π R Z_m(a)/ζ + iθ_m + π R ζZ_m(a)), the wedge involving only the semiflat part has only simple poles at ζ = 0 and ζ = ∞, so we can focus on the corrections. These are of the form

    d𝒳_e(ζ)/𝒳_e(ζ)∧ℐ_±    = -i/2π{∫_0^ζ_edζ'/ζ'-ζ𝒳_e(ζ')/1-𝒳_e(ζ')d𝒳_e(ζ)/𝒳_e(ζ)∧d𝒳_e(ζ')/𝒳_e(ζ').
        + ∫_ζ_e^ζ_e ∞ζ dζ'/ζ'(ζ'-ζ)𝒳_e(ζ')/1-𝒳_e(ζ')d𝒳_e(ζ)/𝒳_e(ζ)∧d𝒳_e(ζ')/𝒳_e(ζ')
        + ∫_0^-ζ_edζ'/ζ'-ζ𝒳^-1_e(ζ')/1-𝒳^-1_e(ζ')d𝒳_e(ζ)/𝒳_e(ζ)∧d𝒳_e(ζ')/𝒳_e(ζ')
        + . ∫_-ζ_e^-ζ_e ∞ζ dζ'/ζ'(ζ'-ζ)𝒳^-1_e(ζ')/1-𝒳^-1_e(ζ')d𝒳_e(ζ)/𝒳_e(ζ)∧d𝒳_e(ζ')/𝒳_e(ζ')}


In the “inside” part of the wall of marginal stability. A similar equation holds in the other side. We can simplify the wedge products above by taking instead

    d𝒳_e(ζ)/𝒳_e(ζ)∧(d𝒳_e(ζ)/𝒳_e(ζ) - d𝒳_e(ζ')/𝒳_e(ζ')) = π R [ ( 1/ζ - 1/ζ')da + (ζ - ζ')da] +i ( dΦ_e(ζ) - dΦ_e(ζ') )


Recall that Φ_e represents the corrections to θ_e, so Υ_e = θ_e + Φ_e. By <ref>, Φ_e and dΦ_e are defined for ζ = 0 ζ = ∞ even if a = 0, since lim_a → 0 Z_m(a) ≠ 0 and the exponential decay in 𝒳_m^sf still present guarantees convergence of the integrals in <ref>. Hence, the terms involving dΦ_e(ζ) - dΦ_e(ζ') are holomorphic for any ζ∈. It thus suffices to consider the other terms. After simplifying the integration kernels, we obtain


    π R da/ζ∫_0^ζ_edζ'/ζ'𝒳_e(ζ')/1-𝒳_e(ζ')    +π R da ∫_ζ_e^ζ_e ∞dζ'/(ζ')^2𝒳_e(ζ')/1-𝒳_e(ζ')
    π R da/ζ∫_0^-ζ_edζ'/ζ'𝒳^-1_e(ζ')/1-𝒳^-1_e(ζ')    +π R da ∫_-ζ_e^-ζ_e ∞dζ'/(ζ')^2𝒳^-1_e(ζ')/1-𝒳^-1_e(ζ')
    
    -π R da∫_0^ζ_e dζ' 𝒳_e(ζ')/1-𝒳_e(ζ')    -π R ζ da∫_ζ_e^ζ_e ∞dζ'/ζ'𝒳_e(ζ')/1-𝒳_e(ζ')
    
    -π R da∫_0^ζ_e dζ' 𝒳^-1_e(ζ')/1-𝒳^-1_e(ζ')    -π R ζ da∫_ζ_e^ζ_e ∞dζ'/ζ'𝒳^-1_e(ζ')/1-𝒳^-1_e(ζ')


The only dependence on ζ is in the factors ζ, 1/ζ. Thus ϖ(ζ) has only simple poles at ζ = 0 and ζ = ∞.

Finally, the estimates in Lemma <ref> show that if we recover the hyperkähler metric g from the holomorphic symplectic form ϖ(ζ) as in <ref> and <ref>, we obtain that the hyperkähler metric for the Pentagon case is the metric obtained in <ref> for the Ooguri-Vafa case plus smooth corrections near a = 0, θ_e = 0, so it extends to this locus.

 This gives Theorem <ref>.








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