poles along $\\{\infty\\}\times U(\xi_{0})$. Here
$\nabla:\mathcal{E}\to\mathcal{E}\otimes\Omega^{1}_{(\mathbb{C}P^{1}-\\{\infty\\})\times
U(\xi_{0})/U(\xi_{0})}$ is considered as a relative connection (it does not
differentiate in the $\xi$ direction), and
$\nabla^{\xi}=\nabla|_{\mathcal{E}^{\xi}}$.

######  Theorem 3.8.

For $U(\xi_{0})$ small enough, the holomorphic bundle with relative flat
connection $(\mathcal{E},\nabla)\to(\mathbb{C}P^{1}-\\{\infty\\})\times
U(\xi_{0})$ extends to a filtered bundle with a (relative) flat meromorphic
connection
$(\mathcal{Q}_{*}^{(\xi_{0})}\mathcal{E},\nabla)\to(\mathbb{C}P^{1}\times
U(\xi_{0}),\\{\infty\\}\times U(\xi_{0}))$. Furthermore, we have that
$(\mathcal{Q}_{a}^{(\xi_{0})}\mathcal{E},\nabla)|_{\mathbb{C}P^{1}\times\\{\xi\\}}=(\mathcal{Q}_{a}\mathcal{E}^{\xi},\nabla^{\xi})$.

###### Proof.

This follows from proposition $9.2.1$, corollary $9.2.5$, and Theorem $11.1.2$
of [Moc08]. ∎

Furthermore, we have

######  Theorem 3.9.

Let $\nu$ be a holomorphic section of
$\mathcal{Q}_{a}^{(\xi_{0})}\mathcal{E}|_{\\{\infty\\}\times U(\xi_{0})}$ such
that $\nu(\xi)$ is a compatible framing for
$(\mathcal{Q}_{a}\mathcal{E}^{\xi},\nabla^{\xi})$. Then the sectorial frames
of flat sections $\Phi_{i}$ used to build the the Stokes data associated to
$(\mathcal{Q}_{a}\mathcal{E}^{\xi},\nabla^{\xi},\nu(\xi))$ vary
holomorphically in $\xi$. Hence the Stokes data itself also varies
holomorphically in $\xi$.

###### Proof.

Given $\nu(\xi)$, we can extend it to a holomorphic local trivialization of
$\mathcal{Q}_{a}^{(\xi_{0})}$ such that in that local frame we have

$\nabla^{\xi}=d-\frac{H}{\xi}\frac{dw}{w^{3}}+\Lambda(\xi)\frac{dw}{w}+A(w,\xi)dw$
(3.78)

where $A(w,\xi)$ is holomorphic in both variables. The result now follows from
lemma $7$ and corollary $8$ of [Boa01a]. ∎

This does not prove the holomorphic dependence of the magnetic coordinate in
$\xi$, since we do not know that the compatible frames $\nu_{a}^{\xi}$ from
section 3.5.1 glue together holomorphically in $\xi$.

####  3.5.3 Proof of the holomorphic dependence

Pick $\xi_{0}\in\mathbb{C}^{*}$ and let
$(\mathcal{Q}^{(\xi_{0})}_{a}\mathcal{E},\nabla)$ be as before. We also fix
some $i=1,2,3,4$ and pick $U(\xi_{0})$ small enough so that:

  * •

There is a sector $S_{i}$ centered at $z=\infty$ such that
$S_{i}\subset\widehat{\text{Sect}}_{i}(\xi)$ for $\xi\in U(\xi_{0})$

  * •

$S_{i}$ contains the Stokes ray and two anti-Stokes rays in the interior of
$\widehat{\text{Sect}}_{i}(\xi)$.

Furthermore, let $\nu(\xi)$ be a holomorphic frame of
$\mathcal{Q}_{a}^{(\xi_{0})}\mathcal{E}|_{\\{\infty\\}\times U(\xi_{0})}$ such
that $\nu(\xi)$ is a compatible frame for
$(\mathcal{Q}_{a}\mathcal{E}^{\xi},\nabla^{\xi})$ for each $\xi\in U(\xi_{0})$
(the fact that such a frame exists follows from the third and fourth stament
of theorem $11.1.2$ of [Moc08]). Then (after a possible reordering of the
elements of the frame) $\nu(\xi)$ extends to a holomorphic local frame of
$\mathcal{Q}_{a}^{(\xi_{0})}\mathcal{E}$, where:

$\nabla^{\xi}=d-\frac{H}{\xi}\frac{dw}{w^{3}}+\Lambda(\xi)\frac{dw}{w}+A(w,\xi)dw$
(3.79)

as in theorem 3.9.

We can then construct for $\xi\in U(\xi_{0})$ the frame of flat sections
$\widetilde{\Phi}_{i}(\xi)$ on $S_{i}$ used to build the Stokes data of
$(\mathcal{Q}_{a}\mathcal{E}^{\xi},\nabla^{\xi},\nu(\xi))$. On the other hand,
we have the frame of flat sections $\Phi_{i}(\xi)$ of
$(\mathcal{Q}_{a}\mathcal{E}^{\xi},\nabla^{\xi},\nu_{a}^{\xi})$ used to build
the magnetic twistor coordinate. We clearly have that
$\widetilde{\Phi}_{i}(\xi)=\Phi_{i}(\xi)\cdot C_{i}(\xi)$ for some matrix
$C_{i}(\xi)$ that depends only on $\xi\in U(\xi_{0})$.

######  Lemma 3.10.

The matrix $C_{i}(\xi)$ is diagonal.

###### Proof.

For each $\xi\in U(\xi_{0})$, the frames $\nu_{a}^{\xi}$ and $\nu(\xi)$ are
compatible frames for the same meromorphic flat bundle
$(\mathcal{Q}_{a}\mathcal{E}^{\xi},\nabla^{\xi})$, with associated irregular
type

$-\frac{H}{\xi}\frac{dw}{w^{3}}+\Lambda(\xi)\frac{dw}{w}$ (3.80)

Hence, we must have that $\nu(\xi)|_{\infty}=\nu_{a}^{\xi}|_{\infty}\cdot
D(\xi)$ for some diagonal matrix $D(\xi)$ depending only on $\xi$.

By applying an argument like the one found in lemma 3.6, we have that

$w^{-\Lambda(\xi)}\text{exp}(-\xi^{-1}\frac{H}{2w^{2}})C_{i}(\xi)\text{exp}(\xi^{-1}\frac{H}{2w^{2}})w^{\Lambda(\xi)}\to
D(\xi)\;\;\text{as}\;\;w\to 0\;\;w\in S$ (3.81)

By the choice of sector $S$ containing a Stokes ray for each $\xi\in
U(\xi_{0})$, we have that the off-diagonal entries of $C_{i}(\xi)$ must be
$0$, and the diagonal entries must match the diagonal entries of $D(\xi)$.
Hence we conclude what we want. ∎

Next we will need the following lemma:

######  Lemma 3.11.

The asymptotics in lemma 3.5 of the frames of flat sections $\Phi_{i}(\xi)$
used to build the magnetic twistor coordinates hold uniformly in $\xi\in
U(\xi_{0})$ for $U(\xi_{0})$ small enough and bounded.

###### Proof.

The proof of this is in appendix B. ∎

######  Theorem 3.10.

The magnetic twistor coordinate $\mathcal{X}_{m}(\xi)$ depends holomorphically
on $\xi$.

###### Proof.

It is enough to prove that the frames of flat sections $\Phi_{i}(\xi)$ depend
holomorphically on $\xi$.

We know by lemma 3.10 that $\widetilde{\Phi}_{i}(\xi)=\Phi_{i}(\xi)\cdot
C_{i}(\xi)$ for $C_{i}(\xi)$ diagonal. Furthermore, we have that
$\overline{\partial}_{\xi}\widetilde{\Phi}_{i}(\xi)=0$, so if we show that
$\overline{\partial}_{\xi}C_{i}(\xi)=0$, then we would be able to conclude
what we want.

In the following, we will use the same notation of lemma 3.5. By lemma 3.11 we
then have that

$A_{i}(w,\xi)\cdot e^{-Q(\xi,w)}\to 1\;\;\text{uniformly in $\xi\in
U(\xi_{0})$ as}\;\;w\to 0,\;w\in S$ (3.82)

where $\Phi_{i}=(e_{1},e_{2})\cdot A_{i}(w,\xi)$.

Since $\widetilde{\Phi}_{i}(\xi)$ is holomorphic in $\xi$ and the frame
$(e_{1},e_{2})$ is also holomorphic in $\xi$ (since it does not depend on
$\xi$), we conclude that the matrix $A_{i}(w,\xi)\cdot C_{i}(\xi)$ is
holomorphic in $\xi$. Furthermore, $e^{Q(\xi)}$ is diagonal and also
holomorphic in $\xi$, so we have that $A_{i}(w,\xi)\cdot C_{i}(\xi)\cdot
e^{-Q(\xi,w)}=A_{i}(w,\xi)\cdot e^{-Q(\xi,w)}\cdot C_{i}(\xi)$ and
$\overline{\partial}_{\xi}(A_{i}(w,\xi)\cdot C_{i}(\xi)\cdot
e^{-Q(\xi,w)})=0$.

By shrinking $U(\xi_{0})$ if necessary, we can assume that $C_{i}(\xi)$ is a
bounded funcion of $\xi$. Hence we get that

$A_{i}(w,\xi)\cdot e^{-Q(\xi)}\cdot C_{i}(\xi)\to C_{i}(\xi)$ (3.83)

uniformly in $\xi\in U(\xi_{0})$ as $w\to 0$ along $S_{i}$. Since
$A_{i}(w,\xi)\cdot e^{-Q(\xi)}\cdot C_{i}(\xi)$ is holomorphic in $\xi$, we
conclude that $C_{i}(\xi)$ must be a holomorphic function in $\xi$, so
$\Phi_{i}(\xi)$ must depend holomorphically on $\xi$.

∎

###  3.6 Twistor coordinates Part 3: Asymptotics in the twistor parameter

In this section we compute the asymptotics of $\mathcal{X}_{m}(\xi)$ as
$\xi\to 0$ and $\xi\to\infty$. More precisely, we will verify the asymptotics
have a formula similar to equation (2.25). Along the way, we also show that
the reality condition for $\mathcal{X}_{m}(\xi)$ holds (see equation (2.26)).

The plan for computing the asymptotics will be the following:

  * •

First, we express the sectorial flat sections used in the definition of Stokes
data, in a convenient way for studying the asymptotics as $\xi\to 0$. This is
done in lemma 3.12.

  * •

Then, we apply lemma 3.12 to the actual computation of the asymptotics of
$\mathcal{X}_{m}(\xi)$ as $\xi\to 0$ in theorem 3.11.

  * •

Finally, we prove the reality condition for $\mathcal{X}_{m}(\xi)$, and use it
to compute the asymptotics as $\xi\to\infty$. This is theorem 3.12 and
corollary 3.2, respectively.

Notation: For the rest of this section, we will use the following notation:
$\text{Arg}(w)$, $\text{Arg}_{m}(w)$ and $\text{Log}_{m}(w)$ are defined as in
3.4.4; $\text{Arg}_{P}(w)$ and $\text{Log}_{P}(w)$ are the principal branches
(i.e. with $\text{Arg}_{P}(w)\in(-\pi,\pi]$). When we switch coordinates to
$z=1/w$, we denote by $\text{Log}_{p}(z)$ and $\text{Arg}_{p}(z)$ the branches
with $[-\pi,\pi)$. We then have the relations
$\text{Log}_{p}(z)=-\text{Log}_{P}(w)$ and
$\text{Arg}_{p}(z)=-\text{Arg}_{P}(w)$.

####  3.6.1 Preliminaries for the asymptotic computation

Here we develop some of the preliminary notation and computations that we will
need for the main asymptotic computation of the twistor magnetic coordinate.

We first start with some results that will help us understand the asymptotic
behaviour of the exponentially decreasing flat sections of $\nabla^{\xi}$
along certain special curves.

######  Definition 3.18.

We fix the quadratic differential $\phi:=(z^{2}+2m)dz^{2}$ over
$\mathbb{C}P^{1}$, with $m\in\mathbb{C}^{*}$. Given a phase $e^{i\theta}\in
S^{1}$, a WKB curve888This terminology comes from [GMN13]; a more common
terminology is that of horizontal trajectory for $e^{-2i\theta}\phi$. with
phase $e^{i\theta}$ is a parametrized curve $\gamma$ in $\mathbb{C}P^{1}$ such
that

$\phi(\dot{\gamma})=e^{2i\theta}$ (3.84)

For the rest of the section we fix:

  * •

$m\in\mathbb{C}^{*}$.

  * •

An element
$(E,\overline{\partial}_{E},\theta,h,(e_{1},e_{2}))\in\mathcal{H}^{\text{fr}}$
with $\text{Det}(\theta)=-(z^{2}+2m)dz^{2}$.

  * •

A WKB curve $\gamma$ with phase $e^{i\text{Arg}(m)}$.

  * •

A frame $(\eta_{1},\eta_{2})$ of eigenvectors of $\theta$ along the curve
$\gamma$, with growth near $w=1/z=0$ of the form
$|\eta_{i}|_{h}=\mathcal{O}(|w|^{a_{i}})$ for some $a_{i}\in\mathbb{R}$. For
example, we could take the frame compatible with the parabolic structure, or a
normalized frame.

With this data fixed, we consider the flatness equation of the pullback
connection $\gamma^{*}\nabla^{\xi}$:

$\frac{d}{dt}dt+\xi^{-1}\gamma^{*}\theta+\gamma^{*}A+\xi\gamma^{*}\theta^{\dagger_{h}}=0$
(3.85)

where $A$ denotes the connection form of $D(\overline{\partial}_{E},h)$ in the
frame $(\eta_{1},\eta_{2})$.

######  Proposition 3.10.

Let $a,a_{0}\in\mathbb{R}$ and let
$M(t):=\textnormal{exp}(-\int_{a}^{t}\gamma^{*}A_{\text{diag}})$ denote the
diagonal gauge transformation that gauges away the diagonal part of
$\gamma^{*}A$. Furthermore, let

$\lambda_{i}(t,\xi)=-\xi^{-1}\gamma^{*}\theta_{ii}-\xi\gamma^{*}\theta^{\dagger_{h}}_{ii}$
(3.86)

and assume that either
$\textnormal{Re}(\lambda_{1}(t,\xi))>\textnormal{Re}(\lambda_{2}(t,\xi))$ or
$\textnormal{Re}(\lambda_{1}(t,\xi))<\textnormal{Re}(\lambda_{2}(t,\xi))$
holds for all $t\in\mathbb{R}$ and for small enough $\xi\in\mathbb{H}_{m}$.
Then there is a neighborhood $U_{0}$ of $\xi=0$ such that for $\xi\in
U_{0}\cap\mathbb{H}_{m}$, we can write an exponentially decreasing flat
section of $\gamma^{*}\nabla^{\xi}$ for $t\in(a_{0},\infty)$ with the
following form:

  * •

In the case
$\textnormal{Re}(\lambda_{1}(t,\xi))>\textnormal{Re}(\lambda_{2}(t,\xi))$ we
have

$s(t,\xi)=\textnormal{exp}\Big{(}\int_{a}^{t}\lambda_{2}(\tau,\xi)d\tau\Big{)}\Big{(}E_{1}(t,\xi)\eta_{1}+(M_{22}(t)+E_{2}(t,\xi))\eta_{2}\Big{)}$
(3.87)

  * •

And in the case
$\textnormal{Re}(\lambda_{1}(t,\xi))<\textnormal{Re}(\lambda_{2}(t,\xi))$

$s(t,\xi)=\textnormal{exp}\Big{(}\int_{a}^{t}\lambda_{1}(\tau,\xi)d\tau\Big{)}\Big{(}(M_{11}(t)+E_{1}(t,\xi))\eta_{1}+E_{2}(t,\xi)\eta_{2}\Big{)}$
(3.88)

where in both cases $E_{i}(t,\xi)$ satisfies:

  * •

For fixed $t\in(a_{0},\infty)$ we have that $E_{i}(t,\xi)\to 0$ as $\xi\to 0$.

  * •

$E_{i}(t,\xi)\to 0$ as $t\to\infty$ uniformly in $\xi\in\mathbb{H}_{m}\cap
U_{0}$.

###### Proof.

In appendix C. The proof of this uses classical techniques in the theory of
ordinary differential equations depending of parameters. ∎

Let us now state the setting for our asymptotic computation:

  * •

Let $\lambda:=\sqrt{z^{2}+2m}\;dz$ be the square root of $\phi$ with branch
cut given by the line segment between $z=\pm\sqrt{-2m}$, and with power series
expansion centered at $z=\infty$ given by
$(z+m/z+...)dz=(-1/w^{3}-m/w+...)dw$.

  * •

We let $\gamma(t)$ be a WKB curve with phase $e^{i\text{Arg}(m)}$, such that
$\lambda(\dot{\gamma})=e^{i\text{Arg}(m)}$. With this choice, $\gamma(t)$
crosses the branch cut of $\lambda$. Notice that we can take (and will take)
$\gamma(t)$ of the form $\gamma(t)=g(t)e^{i\text{Arg}(m)/2}$, where $g(t)$ is
a certain real valued function satisfying $g(t)\to\pm\infty$ (or $\mp\infty)$
as $t\to\pm\infty$ (see figure 6).

Figure 6: The crosses denote $+\pm\sqrt{-2m}$, the wavy red line denotes the
branch cut of $\lambda$, and the horizontal black line the WKB path.

  * •

For $\xi\in\mathbb{H}_{m}$ and for sufficiently big $t$, $\gamma(t)$ lies
either on $\widehat{\text{Sect}}_{1}(\xi)\cap\widehat{\text{Sect}}_{4}(\xi)$
or in $\widehat{\text{Sect}}_{2}(\xi)\cap\widehat{\text{Sect}}_{3}(\xi)$. We
orient $\gamma$ such that it lies in
$\widehat{\text{Sect}}_{1}(\xi)\cap\widehat{\text{Sect}}_{4}(\xi)$ for
sufficiently big $t$.

  * •

Let $\widetilde{\lambda}$ be a branch of a square root of $\phi$ defined in a
neighborhood of $\gamma$, such that
$\widetilde{\lambda}(\dot{\gamma})=\lambda(\dot{\gamma})$ for big enough $t$.
With our choices, we then have that
$\text{Re}(\frac{1}{\xi}\widetilde{\lambda}(\dot{\gamma}))>0$ for
$\xi\in\mathbb{H}_{m}$.

  * •

We pick a frame $(\eta_{1},\eta_{2})$ of $SL(2,\mathbb{C})$ eigenvectors of
$\theta$ along $\gamma$, such that $(\eta_{1},\eta_{2})\to(e_{1},e_{2})$ as
$t\to\infty$, and $(\eta_{1},\eta_{2})\to(e_{2},-e_{1})$ as $t\to-\infty$. In
the frame $(\eta_{1},\eta_{2})$ we have

$\theta=\begin{bmatrix}\widetilde{\lambda}&0\\\
0&-\widetilde{\lambda}\end{bmatrix}$ (3.89)

  * •

We pick $t_{0}<t_{1}$, such that $t_{0}$ is a time before the crossing with
the branch cut of $\lambda$, and $t_{1}$ is a time after the crossing with the
branch cut of $\lambda$.

We now state the lemma that will allow us to compute the asymptotics of the
magnetic twistor coordinate:

######  Lemma 3.12.

In the previous setting, let $s_{i}(z,\xi)$ be the flat sections from 3.4.4.
They satisfy the following:

For small enough $\xi\in\mathbb{H}_{m}$:

$\begin{split}s_{1}(\gamma(t),\xi)&=\textnormal{exp}\Big{(}\int_{\gamma(t_{1})}^{\gamma(t)}\Big{(}-\frac{1}{\xi}\widetilde{\lambda}+\mathcal{O}(\xi)\Big{)}\Big{)}\Big{(}(M_{11}^{(1)}(\gamma(t))+E^{(1)}_{1}(\gamma(t),\xi))\eta_{1}+E^{(1)}_{2}(\gamma(t),\xi)\eta_{2}\Big{)}\beta_{1}(\gamma(t_{1}),\xi)\\\
s_{3}(\gamma(t),\xi)&=\textnormal{exp}\Big{(}\int_{\gamma(t_{0})}^{\gamma(t)}\Big{(}\frac{1}{\xi}\widetilde{\lambda}+\mathcal{O}(\xi)\Big{)}\Big{)}\Big{(}(M_{22}^{(3)}(\gamma(t))+E^{(3)}_{2}(\gamma(t),\xi))\eta_{2}+E^{(3)}_{1}(\gamma(t),\xi)\eta_{1}\Big{)}\beta_{3}(\gamma(t_{0}),\xi)\\\
\end{split}$ (3.90)

For small enough $\xi\in\mathbb{H}_{-m}$:

$\begin{split}s_{2}(\gamma(t),\xi)&=\textnormal{exp}\Big{(}\int_{\gamma(t_{1})}^{\gamma(t)}\Big{(}\frac{1}{\xi}\widetilde{\lambda}+\mathcal{O}(\xi)\Big{)}\Big{)}\Big{(}(M_{22}^{(2)}(\gamma(t))+E^{(2)}_{2}(\gamma(t),\xi))\eta_{2}+E^{(2)}_{1}(\gamma(t),\xi)\eta_{1}\Big{)}\beta_{2}(\gamma(t_{1}),\xi)\\\
s_{4}(\gamma(t),\xi)&=\textnormal{exp}\Big{(}\int_{\gamma(t_{0})}^{\gamma(t)}\Big{(}-\frac{1}{\xi}\widetilde{\lambda}+\mathcal{O}(\xi)\Big{)}\Big{)}\Big{(}(M_{11}^{(4)}(\gamma(t))+E^{(4)}_{1}(\gamma(t),\xi))\eta_{1}+E^{(4)}_{2}(\gamma(t),\xi)\eta_{2}\Big{)}\beta_{4}(\gamma(t_{0}),\xi)\end{split}$
(3.91)

where $\beta_{i}(\gamma(t_{j}),\xi)$ are normalization constants that ensure
that the flat sections have the correct asymptotics as $t\to\infty$ (resp.
$t\to-\infty$) for $i=1,2$ (resp. $i=3,4$), and where $M^{(i)}_{j}(t)$ and
$E^{(i)}_{j}(t,\xi)$ have the same role and properties as in proposition 3.10.

###### Proof.

We will give a proof for $s_{1}(\gamma(t),\xi)$, since the others follow
similarly.

By proposition 3.10 and the setting of our computation, we can write an
exponentially decreasing flat section for $\gamma^{*}\nabla^{\xi}$ by

$s(t,\xi)=\text{exp}\Big{(}\int_{\gamma(t_{1})}^{\gamma(t)}\Big{(}-\frac{1}{\xi}\widetilde{\lambda}+\mathcal{O}(\xi)\Big{)}\Big{)}\Big{(}(M_{11}(t)+E_{1}(t,\xi))\eta_{1}+E_{2}(t,\xi)\eta_{2}\Big{)}$
(3.92)

where we are denoting $\mathcal{O}(\xi)=-\xi\theta^{\dagger_{h}}_{11}$. Note
that because of theorem 3.4, we have that
$\theta^{\dagger_{h}}_{11}=\overline{\theta_{11}}^{t}+\phi$, where $\phi$ is
exponentially decreasing as $z\to\infty$.

By our definition of $\gamma(t)$, when $t\to\infty$ we have that $\gamma(t)$
lies in $\widehat{\text{Sect}}_{1}(\xi)\cap\widehat{\text{Sect}}_{4}(\xi)$.
Since exponentially decreasing flat sections along such sectors are uniquely
determined up to scaling, we have that
$s_{1}(\gamma(t),\xi)=c(\xi,\gamma(t_{1}))s(t,\xi)$ for some number
$c(\xi,\gamma(t_{1}))$.

Now let $F(w)$ be the antiderivative of
$-\lambda=(1/w^{3}+m/w+\mathcal{O}(w))dw$ of the form
$-\frac{1}{2w^{2}}+mLog_{m}(w)+\mathcal{O}(w)$. Furthermore, let
$\widetilde{F}(w)$ be the antiderivative of the $\mathcal{O}(\xi)$ term of the
form
$-\frac{1}{2\overline{w}^{2}}+\overline{m}\overline{Log_{m}(w)}+\mathcal{O}(|w|)$.

On the other hand, it is not hard to check that

$\text{Arg}_{m}(\gamma(t))\to\text{Arg}_{m}(\sqrt{-m^{-1}})-\pi/2\;\;\text{as}\;\;t\to\infty\\\
$ (3.93)

where $\sqrt{-m^{-1}}$ uses the principal branch in the $w$-coordinate.
Furthermore, because of our conventions, the following holds (see figure 7):

$\text{Arg}_{m}(\sqrt{-m^{-1}})=\text{Arg}_{P}(\sqrt{-m^{-1}})$ (3.94)

Figure 7: The pictures show that
$\text{Arg}_{m}(\sqrt{-m^{-1}})=\text{Arg}_{P}(\sqrt{-m^{-1}})$ for
$m\in\\{1,i,-1,-i\\}$, but similar pictures hold for any $m$. The red dot in
the pictures denotes $\sqrt{-m^{-1}}$, where $\sqrt{w}$ uses the principal
branch. The wavy red line denotes the branch cut of $\text{Arg}_{m}$.

Now define the following:

$\begin{split}\beta_{1,{\xi^{-1}}}(t_{1})&:=\text{exp}\Big{(}\frac{1}{\xi}F(1/\gamma(t_{1}))\Big{)}\\\
\beta_{1,{\xi^{0}}}(t_{1})&:=\text{exp}\Big{(}-im^{(3)}(\text{Arg}_{P}(\sqrt{-m^{-1}})-\frac{\pi}{2})+\int_{t_{1}}^{\infty}\gamma^{*}A_{11}\Big{)}\\\
\beta_{1,\xi}(t_{1})&:=\text{exp}\Big{(}\xi\widetilde{F}(1/\gamma(t_{1}))\Big{)}\\\
\beta_{1}(t_{1},\xi)&:=\beta_{1,{\xi^{-1}}}(t_{1})\beta_{1,{\xi^{0}}}(t_{1})\beta_{1,\xi}(t_{1})\end{split}$
(3.95)

It is then easy to check that $\beta_{1}(t_{1},\xi)s(t,\xi)$ has the same
asymptotics as $t\to\infty$ as $s_{1}(\gamma(t),\xi)$ (see lemma 3.5). So
$s_{1}(\gamma(t),\xi)=\beta_{1}(t_{1},\xi)s(t,\xi)$.

∎

For completeness, and because we will need it in the asymptotic computation
for the magnetic twistor coordinate, we write the other normalization
constants $\beta_{i}(t_{j},\xi)$. These are:

$\begin{split}\beta_{2}(t_{1},\xi)&:=\text{exp}\Big{(}-\frac{1}{\xi}F(1/\gamma(t_{1}))+im^{(3)}(\text{Arg}_{P}(\sqrt{-m^{-1}})-\frac{\pi}{2})+\int_{t_{1}}^{\infty}\gamma^{*}A_{22}-\xi\widetilde{F}(1/\gamma(t_{1}))\Big{)}\\\
\beta_{3}(t_{0},\xi)&:=\text{exp}\Big{(}\frac{1}{\xi}F(1/\gamma(t_{0}))-im^{(3)}(\text{Arg}_{P}(\sqrt{-m^{-1}})+\frac{\pi}{2})+i\pi+\int_{t_{0}}^{-\infty}\gamma^{*}A_{22}+\xi\widetilde{F}(1/\gamma(t_{0}))\Big{)}\\\
\beta_{4}(t_{0},\xi)&:=\text{exp}\Big{(}-\frac{1}{\xi}F(1/\gamma(t_{0}))+im^{(3)}(\text{Arg}_{P}(\sqrt{-m^{-1}})+\frac{\pi}{2})+\int_{t_{0}}^{-\infty}\gamma^{*}A_{11}-\xi\widetilde{F}(1/\gamma(t_{0}))\Big{)}\\\
\end{split}$ (3.96)

where the extra $i\pi$ in $\beta_{3}$ comes because of the fact that
$\eta_{2}\to-e_{1}$ instead of $e_{1}$ as $t\to-\infty$.

We should justify why the integrals of the form

$\int_{t_{j}}^{\pm\infty}\gamma^{*}A_{ii}$ (3.97)

are finite. Let $w=re^{i\theta}$, and write
$A_{ii}=A_{ii,r}dr+A_{ii,\theta}d\theta$. By our choice of $\gamma(t)$ we have
that $d\theta(\dot{\gamma})=0$. On the other hand, $A_{ii,r}$ is regular at
$w=0$ and $dr(\dot{\gamma})\sim t^{-3/2}$ as $t\to\pm\infty$. Hence, the
integrals of the above form converge.

####  3.6.2 Computing the asymptotics when $\xi\to 0$

We are finally ready to start computing the asymptotics of the magnetic
twistor coordinate.

######  Theorem 3.11.

Consider the magnetic twistor coordinate $\mathcal{X}_{m}(\xi)$ associated to
$[(E,\overline{\partial}_{E},\theta,h,(e_{1},e_{2}))]\in\mathfrak{X}^{\text{fr}}$.
Then with the setting and notations of the previous section we have

$\mathcal{X}_{m}(\xi)\sim_{\xi\to 0}A(\xi)$ (3.98)

where

$A(\xi):=\textnormal{exp}\Big{(}-\frac{1}{\xi}m\textnormal{Log}_{p}\Big{(}-\frac{m}{2e}\Big{)}+im^{(3)}\textnormal{Arg}_{p}(-m)+i\pi+\int_{-\infty}^{\infty}\gamma^{*}A_{11}\Big{)}$
(3.99)

###### Proof.

First assume that $\xi\in\mathbb{H}_{m}$. We pick $\xi$ to be small enough so
that the intervals where the expressions for $s_{1}$ and $s_{3}$ of lemma 3.12
hold, overlap for some $t\in(t_{0},t_{1})$. By using lemma 3.12, we have that

$\begin{split}&s_{3}(\gamma(t),\xi)\wedge s_{1}(\gamma(t),\xi)\\\
&=\text{exp}\Big{(}\int_{\gamma(t_{0})}^{\gamma(t_{1})}\Big{(}\frac{1}{\xi}\widetilde{\lambda}+\mathcal{O}(\xi)\Big{)}\Big{)}\beta_{1}(\gamma(t_{1}),\xi)\beta_{3}(\gamma(t_{0}),\xi)\Big{(}(M^{(1)}_{11}+E_{1}^{(1)})(M_{22}^{(3)}+E_{2}^{(3)})-E_{1}^{(3)}E_{2}^{(1)}\Big{)}\eta_{2}\wedge\eta_{1}\\\
\end{split}$ (3.100)

It is then easy to see that the following asymptotic computation holds

$\begin{split}&\mathcal{X}_{m}(\xi)=a(\xi)=\frac{s_{3}(\gamma(t),\xi)\wedge
s_{1}(\gamma(t),\xi)}{s_{2}(\gamma(t),\xi)\wedge s_{1}(\gamma(t),\xi)}\\\
&\sim_{\xi\to
0,\xi\in\mathbb{H}_{m}}\frac{\text{exp}\Big{(}\frac{1}{\xi}\int_{\gamma(t_{0})}^{\gamma(t_{1})}\widetilde{\lambda}\Big{)}\beta_{1,\xi^{-1}}(t_{1})\beta_{1,\xi^{0}}(t_{1})\beta_{3,\xi^{-1}}(t_{0})\beta_{3,\xi^{0}}(t_{0})M_{11}^{(1)}(\gamma(t))M_{22}^{(3)}(\gamma(t))\eta_{2}\wedge\eta_{1}}{s_{2}(\gamma(t),\xi)\wedge
s_{1}(\gamma(t),\xi)}\\\ \end{split}$ (3.101)

To continue the computation, we use the following lemma:

######  Lemma 3.13.

The following holds

$\textnormal{exp}\Big{(}\frac{1}{\xi}\int_{\gamma(t_{0})}^{\gamma(t_{1})}\widetilde{\lambda}\Big{)}\beta_{1,\xi^{-1}}(t_{1})\beta_{3,\xi^{-1}}(t_{0})=\textnormal{exp}\Big{(}-\frac{1}{\xi}m\textnormal{Log}_{p}\Big{(}-\frac{m}{2e}\Big{)}\Big{)}$
(3.102)

###### Proof.

By deforming the path $\gamma$ to a path from $\gamma(t_{0})$ to
$\gamma(t_{1})$ passing through $z=\sqrt{-2m}$ (where $\sqrt{-2m}$ uses
$\text{Arg}_{p}(z)$) we get that

$\text{exp}\Big{(}\frac{1}{\xi}\int_{\gamma(t_{0})}^{\gamma(t_{1})}\widetilde{\lambda}\Big{)}=\text{exp}\Big{(}\frac{1}{\xi}\int_{\sqrt{-2m}}^{\gamma(t_{1})}\lambda-\frac{1}{\xi}\int_{\gamma(t_{0})}^{\sqrt{-2m}}\lambda\Big{)}$
(3.103)

To compute this quantity, we use the following antiderivative $\Lambda(z)$ of
$\lambda=\sqrt{z^{2}+2m}dz$:

$\Lambda(z):=\frac{z}{2}\sqrt{z^{2}+2m}+m\text{Log}_{p}(z+\sqrt{z^{2}+2m})-\frac{m}{2}-m\text{Log}_{p}(2)=\frac{z^{2}}{2}+m\text{Log}_{p}(z)+\mathcal{O}(1/z)$
(3.104)

Notice that with our conventions,
$Log_{P}(1/\gamma(t_{i}))=Log_{m}(1/\gamma(t_{i}))$, so that is easy to check
that $F(1/\gamma(t_{j}))=-\Lambda(\gamma(t_{j}))$. Hence,
$\beta_{i,\xi^{-1}}(t_{j})=\text{exp}(-\xi^{-1}\Lambda(\gamma(t_{j})))$.

Applying this to $(3.103)$, we get:

$\begin{split}\text{exp}\Big{(}\frac{1}{\xi}\int_{\gamma(t_{0})}^{\gamma(t_{1})}\widetilde{\lambda}\Big{)}\beta_{1,\xi^{-1}}(t_{1})\beta_{3,\xi^{-1}}(t_{0})&=\text{exp}\Big{(}-\frac{2}{\xi}\Lambda(\sqrt{-2m})\Big{)}\\\
&=\text{exp}\Big{(}-\frac{1}{\xi}(2m\text{Log}_{p}(\sqrt{-2m})-m-2m\text{Log}_{p}(2))\Big{)}\\\
&=\text{exp}\Big{(}-\frac{1}{\xi}m\text{Log}_{p}\Big{(}-\frac{m}{2e}\Big{)}\Big{)}\end{split}$
(3.105)

∎

Using lemma $3.13$, we get the following asymptotics as $\xi\to 0$,
$\xi\in\mathbb{H}_{m}$:

$\begin{split}&a(\xi)=\frac{s_{3}(\gamma(t),\xi)\wedge
s_{1}(\gamma(t),\xi)}{s_{2}(\gamma(t),\xi)\wedge s_{1}(\gamma(t),\xi)}\\\
&\cong_{\xi\to
0,\xi\in\mathbb{H}_{m}}\frac{\text{exp}\Big{(}-\frac{1}{\xi}m\text{Log}_{p}\Big{(}-\frac{m}{2e}\Big{)}-2im^{(3)}\text{Arg}_{P}(\sqrt{-m^{-1}})+i\pi\Big{)}\text{exp}\Big{(}\int_{t}^{\infty}\gamma^{*}A_{11}\Big{)}\text{exp}\Big{(}\int_{t}^{-\infty}\gamma^{*}A_{22}\Big{)}\eta_{2}\wedge\eta_{1}}{s_{2}(\gamma(t),\xi))\wedge
s_{1}(\gamma(t),\xi)}\\\
&=\frac{\text{exp}\Big{(}-\frac{1}{\xi}m\text{Log}_{p}\Big{(}-\frac{m}{2e}\Big{)}+im^{(3)}\text{Arg}_{p}(-m)+i\pi\Big{)}\text{exp}\Big{(}\int_{t}^{\infty}\gamma^{*}A_{11}\Big{)}\text{exp}\Big{(}\int_{t}^{-\infty}\gamma^{*}A_{22}\Big{)}\eta_{2}\wedge\eta_{1}}{s_{2}(\gamma(t),\xi))\wedge
s_{1}(\gamma(t),\xi)}\\\ \end{split}$ (3.106)

The last expression might seem to depend on $t$, but it actually does not. To
show this, notice that since $(\eta_{1},\eta_{2})$ is an $SL(2,\mathbb{C})$
frame and $D=d+A$ preserves the volume form, we have that
$\text{Tr}(\gamma^{*}A)=0$. This implies that
$\text{exp}(\int_{t}^{\infty}\gamma^{*}A_{11})\text{exp}(\int_{t}^{-\infty}\gamma^{*}A_{22})=\text{exp}(\int_{-\infty}^{\infty}\gamma^{*}A_{11})$.
On the other hand we have that $s_{2}\wedge s_{1}$ is a flat section of
$Det(\gamma^{*}\nabla^{\xi})$, which in the $SL(2,\mathbb{C})$ frame given by
$\eta_{1}\wedge\eta_{2}$ along the WKB curve has the form

$\text{Det}(\gamma^{*}\nabla^{\xi})=\frac{d}{dt}dt+\xi^{-1}\text{Tr}(\gamma^{*}\theta)+\text{Tr}(\gamma^{*}A)+\xi\text{Tr}(\gamma^{*}\theta^{\dagger_{h}})=\frac{d}{dt}dt$
(3.107)

This tells us that we can write

$s_{2}(\gamma(t),\xi)\wedge s_{1}(\gamma(t),\xi)=c\eta_{1}\wedge\eta_{2}$
(3.108)

for some constant $c$. In fact, because of the asymptotics of the flat
sections, we see that $c=-1$. Hence, putting the results together we get the
following asymptotics:

$\frac{s_{3}(\gamma(t),\xi)\wedge
s_{1}(\gamma(t),\xi)}{s_{2}(\gamma(t),\xi)\wedge
s_{1}(\gamma(t),\xi)}\cong_{\xi\to
0,\xi\in\mathbb{H}_{m}}=\text{exp}\Big{(}-\frac{1}{\xi}m\text{Log}_{p}\Big{(}-\frac{m}{2e}\Big{)}+im^{(3)}\text{Arg}_{p}(-m)+i\pi+\int_{-\infty}^{\infty}\gamma^{*}A_{11}\Big{)}$
(3.109)

So far we have computed the asymptotics as $\xi\to 0$ with
$\xi\in\mathbb{H}_{m}$. Let us compute the asymptotics when $\xi\to 0$ with
$\xi\in\mathbb{H}_{-m}$, and see that they match with the previous
asymptotics.

Now we need to compute the asymptotics of

$-\frac{1}{b(\xi)}=-\frac{s_{3}(\gamma(t),\xi)\wedge
s_{2}(\gamma(t),\xi)}{s_{4}(\gamma(t),\xi)\wedge
s_{2}(\gamma(t),\xi)}\;\;\text{where}\;\;\xi\in\mathbb{H}_{-m}.$ (3.110)

Following a similar computation from before using the relevant expressions for
the $s_{i}$ and $\beta_{i}$ for $i=2,4$, we get the following:

$\mathcal{X}_{m}(\xi)\cong_{\xi\to
0,\xi\in\mathbb{H}_{-m}}\frac{-1}{\text{exp}\Big{(}\frac{1}{\xi}m\text{Log}_{p}\Big{(}-\frac{m}{2e}\Big{)}-im^{(3)}\text{Arg}_{p}(-m)-\int_{-\infty}^{\infty}\gamma^{*}A_{11}\Big{)}}=A(\xi)$
(3.111)

Hence, the asymptotics agree on $\mathbb{H}_{m}$ and $\mathbb{H}_{-m}$, and we
proved what we want. ∎

####  3.6.3 The reality condition and the asymptotics as $\xi\to\infty$

In this section we prove the reality condition for the magnetic twistor
coordinate (see (2.26)). Once we have the reality condition, the asymptotics
when $\xi\to\infty$ automatically follow from the asymptotics when $\xi\to 0$.

The reality condition states that the following equality must hold:

$\mathcal{X}_{m}(\xi)=\overline{\mathcal{X}_{m}(-1/\overline{\xi})^{-1}}\;\;\;\text{for}\;\;\;\xi\in\mathbb{C}^{*}/l_{\pm}(-2im)$
(3.112)

Let $(E,\overline{\partial}_{E},\theta,h,g)\in\mathcal{H}^{\text{fr}}$ be a
compatibly framed wild harmonic bundle. We will denote by $(e_{1},e_{2})$ an
extension of $g$ to a $SU(2)$ frame in a neighborhood $U_{\infty}$ of
$z=\infty$, where the singularity of the associated flat connection
$\nabla^{\xi}=\xi^{-1}\theta+D(\overline{\partial}_{E},h)+\xi\theta^{\dagger}$
has the appropriate form. We write in this frame $\nabla^{\xi}=d+A(\xi)$.

We can consider the conjugate bundle
$\overline{E}\to\overline{\mathbb{C}P^{1}}$, with the induced connection
$\overline{\nabla^{\xi}}$. In the induced frame $\\{\overline{e}_{i}\\}$ the
connection has the form $\overline{\nabla^{\xi}}=d+\overline{A(\xi)}$.
Furthermore, we can consider the dual bundle
$\overline{E}^{*}\to\overline{\mathbb{C}P^{1}}$ with the induced connection
$\overline{\nabla^{\xi}}^{*}$. In the dual frame $\\{\overline{e}^{*}_{i}\\}$,
the connection $\overline{\nabla^{\xi}}^{*}$ takes the form
$d-\overline{A(\xi)}^{t}=d+A(-1/\overline{\xi})$.

Consider now the associated compatibly framed meromorphic flat bundle
$(\mathcal{PE}_{a}^{\xi},\nabla^{\xi},\tau_{a}^{\xi})$ and a fundamental
solution $(y_{1},y_{2})=\Phi_{i}(\xi)$ of $\nabla^{\xi}$ on
$\widehat{\text{Sect}}_{i}$, with the corresponding asymptotics determined by
the compatible frame. If we denote by
$e^{Q(\xi)}=\text{diag}(e^{Q_{1}(\xi)},e^{Q_{2}(\xi)})$, where:

$\begin{split}e^{Q_{1}(\xi)}&=\text{exp}\Big{(}-\xi^{-1}\Big{(}\frac{1}{2w^{2}}-m\text{Log}(w)\Big{)}-im^{(3)}\text{Arg}(w)-\xi\Big{(}\frac{1}{2\overline{w}^{2}}-\overline{m}\overline{\text{Log}(w)}\Big{)}\Big{)}\\\
e^{Q_{2}(\xi)}&=\text{exp}\Big{(}\xi^{-1}\Big{(}\frac{1}{2w^{2}}-m\text{Log}(w)\Big{)}+im^{(3)}\text{Arg}(w)+\xi\Big{(}\frac{1}{2\overline{w}^{2}}-\overline{m}\overline{\text{Log}(w)}\Big{)}\Big{)}\\\
\end{split}$ (3.113)

then we know that by theorem 3.5 that $(y_{1},y_{2})\cdot
e^{-Q(\xi)}\to(e_{1},e_{2})$ as $w\to 0$ along $\widehat{\text{Sect}}_{i}$.
Here, the $Q_{i}(\xi)$ are defined using the same branch of the logarithm and
argument as the one used to define the flat frames $\Phi_{i}(\xi)$ in section
3.4.4. We then have the following lemma:

######  Lemma 3.14.

If $(e_{1},e_{2})\cdot B=\Phi_{i}=(y_{1},y_{2})$, then
$(\overline{e}_{1}^{*},\overline{e}_{2}^{*})\cdot(B^{-1})^{\dagger}:=\overline{\Phi}_{i}^{*}=(\overline{y}_{1}^{*},\overline{y}_{2}^{*})$
is a flat frame for
$(\overline{\mathcal{PE}_{a}^{\xi}}^{*},\overline{\nabla^{\xi}}^{*})$.
Furthermore, $(\overline{y}_{1}^{*},\overline{y}_{2}^{*})\cdot
e^{-Q(-1/\overline{\xi})}\to(\overline{e}_{1}^{*},\overline{e}_{2}^{*})$ as
$w\to 0$ along $\widehat{\textnormal{Sect}}_{i}(\xi)$.

###### Proof.

Since $\Phi_{i}$ is a flat frame, and in the frame $(e_{1},e_{2})$ we have
that $\nabla^{\xi}=d+A(\xi)$, then $B$ must satisfy

$B^{-1}dB+B^{-1}AB=0$ (3.114)

If $(3.115)$ holds, then

$-B^{\dagger}A^{\dagger}(B^{-1})^{\dagger}+B^{\dagger}d(B^{-1})^{\dagger}=0$
(3.115)

And since in the frame $(\overline{e}_{1}^{*},\overline{e}_{2}^{*})$ we have
that $\overline{\nabla^{\xi}}^{*}=d-A^{\dagger}$, we conclude that
$(\overline{e}_{1}^{*},\overline{e}_{2}^{*})\cdot(B^{-1})^{\dagger}$ must be a
flat frame for $\overline{\nabla^{\xi}}^{*}$.

To check the asymptotics, just notice that $(y_{1},y_{2})\cdot
e^{-Q(\xi)}\to(e_{1},e_{2})$ implies that $Be^{-Q(\xi)}\to 1$. On the other
hand, we have that:

$\begin{split}(\overline{y}_{1}^{*},\overline{y}_{2}^{*})\cdot
e^{-Q(-1/\overline{\xi})}&=(\overline{e}_{1}^{*},\overline{e}_{2}^{*})\cdot(B^{-1})^{\dagger}e^{-Q(-1/\overline{\xi})}\\\
&=(\overline{e}_{1}^{*},\overline{e}_{2}^{*})\cdot(B^{-1})^{\dagger}e^{\overline{Q(\xi)}}\\\
&=(\overline{e}_{1}^{*},\overline{e}_{2}^{*})\cdot(e^{Q(\xi)}B^{-1})^{\dagger}\\\
&=(\overline{e}_{1}^{*},\overline{e}_{2}^{*})\cdot((Be^{-Q(\xi)})^{-1})^{\dagger}\end{split}$
(3.116)

so we conclude that $(\overline{y}_{1}^{*},\overline{y}_{2}^{*})\cdot
e^{-Q(-1/\overline{\xi})}\to(\overline{e}_{1}^{*},\overline{e}_{2}^{*})$ as
$w\to 0$ along $\widehat{\text{Sect}}_{i}(\xi)$. ∎

We will need two more easy lemmas in order to show the reality condition of
the magnetic coordinate:

######  Lemma 3.15.

If on $\widehat{\textnormal{Sect}}_{i}\cap\widehat{\textnormal{Sect}}_{i+1}$
we have $\Phi_{i+1}=\Phi_{i}\cdot S_{i}$, then we have that
$\overline{\Phi}_{i+1}^{*}=\overline{\Phi}_{i}^{*}\cdot(S_{i}^{-1})^{\dagger}$.

###### Proof.

If we write $(e_{1},e_{2})\cdot B_{i}=\Phi_{i}$ and $(e_{1},e_{2})\cdot
B_{i+1}=\Phi_{i+1}$, then we have that $B_{i}S_{i}=B_{i+1}$. Hence, we get
that $(B_{i}^{-1})^{\dagger}(S_{i}^{-1})^{\dagger}=(B_{i+1}^{-1})^{\dagger}$,
and then by the previous lemma

$\overline{\Phi}_{i+1}^{*}=(\overline{e}_{1}^{*},\overline{e}_{2}^{*})\cdot(B_{i+1}^{-1})^{\dagger}=(\overline{e}_{1}^{*},\overline{e}_{2}^{*})\cdot(B_{i}^{-1})^{\dagger}(S_{i}^{-1})^{\dagger}=\overline{\Phi}_{i}^{*}\cdot(S_{i}^{-1})^{\dagger}$
(3.117)

∎

######  Lemma 3.16.

The local bundle map defined by $e_{i}\to\overline{e_{i}}^{*}$, gives an
isomorphism
$(\overline{\mathcal{PE}_{a}^{\xi}}^{*},\overline{\nabla^{\xi}}^{*})|_{U_{\infty}}\cong(\mathcal{PE}_{a}^{-\frac{1}{\overline{\xi}}},\nabla^{-\frac{1}{\overline{\xi}}})|_{U_{\infty}}$,
where $U_{\infty}$ is a small neighborhood of $z=\infty$ where the frame
$(e_{1},e_{2})$ is defined. In particular, we have the following
correspondence between flat frames: for $i=1,2$ and $\xi\in\mathbb{H}_{m}$,
the flat frame $\overline{\Phi}_{i}^{*}(\xi)$ goes to the flat frame
$\Phi_{i+1}(-1/\overline{\xi})$ of $\nabla^{-1/\overline{\xi}}$ on
$\widehat{\textnormal{Sect}}_{i+1}(-1/\overline{\xi})$; and for $i=2,3$ and
$\xi\in\mathbb{H}_{-m}$, to the flat frame $\Phi_{i-1}(-1/\overline{\xi})$ of
$\nabla^{-1/\overline{\xi}}$ on
$\widehat{\textnormal{Sect}}_{i-1}(-1/\overline{\xi})$.

###### Proof.

For the first statement, just notice that in the frame
$(\overline{e}_{1}^{*},\overline{e}_{2}^{*})$, the induced connection looks
like $\overline{\nabla^{\xi}}^{*}=d+A(-1/\overline{\xi})$.

On the other hand, the flat frame $\Phi_{i}(-1/\overline{\xi})$ on
$\widehat{\text{Sect}}_{i}(-1/\overline{\xi})$ is uniquely characterized by
the asymptotic condition
$\Phi_{i}(-1/\overline{\xi})e^{-Q(-1/\overline{\xi})}\to(e_{1},e_{2})$ when
$w\to 0$ along $\widehat{\text{Sect}}_{i}(-1/\overline{\xi})$. If $i=1,2$ and
$\xi\in\mathbb{H}_{m}$, we have that
$\widehat{\text{Sect}_{i}}(\xi)=\widehat{\text{Sect}}_{i+1}(-1/\overline{\xi})$
and that the flat frame $\overline{\Phi}_{i}^{*}$ of
$\overline{\nabla^{\xi}}^{*}$ goes to a flat frame of
$\nabla^{-\frac{1}{\overline{\xi}}}$ satisfying the corresponding asymptotic
condition on $\widehat{\text{Sect}}_{i+1}(-1/\overline{\xi})$. We then
conclude that for $i=1,2$ and $\xi\in\mathbb{H}_{m}$ we have
$\overline{\Phi}^{*}_{i}(\xi)=\Phi_{i+1}(-1/\overline{\xi})$. The other case
similarly follows. ∎

######  Theorem 3.12.

The magnetic twistor coordinate satisfies the following reality condition:

$\mathcal{X}_{m}(\xi)=\overline{\mathcal{X}_{m}(-1/\overline{\xi})^{-1}}\;\;\;\text{for}\;\;\;\xi\in\mathbb{C}^{*}/l_{\pm}(-2im)$
(3.118)

###### Proof.

Assume first that $\xi\in\mathbb{H}_{m}$. We then want to relate
$b(-1/\overline{\xi})$ and $a(\xi)$.

By the previous two lemmas, we have that

$\Phi_{2}(-1/\overline{\xi})\cdot
S_{2}(-1/\overline{\xi})=\Phi_{3}(-1/\overline{\xi})=\overline{\Phi}^{*}_{2}(\xi)=\overline{\Phi}^{*}_{1}(\xi)\cdot(S_{1}(\xi)^{-1})^{\dagger}=\Phi_{2}(-1/\overline{\xi})\cdot(S_{1}^{-1}(\xi))^{\dagger}$
(3.119)

.

From this we conclude that
$S_{2}(-1/\overline{\xi})=(S_{1}^{-1}(\xi))^{\dagger}$ so that the Stokes
matrix elements are related by $b(-1/\overline{\xi})=-\overline{a(\xi)}$.
Hence, we have that for $\xi\in\mathbb{H}_{m}$

$\mathcal{X}_{m}(\xi)=a(\xi)=-\overline{b(-1/\overline{\xi})}=\overline{\mathcal{X}_{m}(-1/\overline{\xi})^{-1}}$
(3.120)

Similarly, if $\xi\in\mathbb{H}_{-m}$, we need to compare $b(\xi)$ and
$a(-1/\overline{\xi})$.

By the above two lemmas we have that

$\Phi_{1}(-1/\overline{\xi})\cdot
S_{1}(-1/\overline{\xi})=\Phi_{2}(-1/\overline{\xi})=\overline{\Phi}^{*}_{3}(\xi)=\overline{\Phi}^{*}_{2}(\xi)\cdot(S_{2}(\xi)^{-1})^{\dagger}=\Phi_{1}(-1/\overline{\xi})\cdot(S_{2}^{-1}(\xi))^{\dagger}$
(3.121)

so that $S_{1}(-1/\overline{\xi})=S_{2}^{-1}(\xi))^{\dagger}$ and hence the
relation among the Stokes matrix elements is
$a(-1/\overline{\xi})=-\overline{b(\xi)}$. We then conclude that for
$\xi\in\mathbb{H}_{-m}$

$\mathcal{X}_{m}(\xi)=-\frac{1}{b(\xi)}=\frac{1}{\overline{a(-1/\overline{\xi})}}=\overline{\mathcal{X}_{m}(-1/\overline{\xi})^{-1}}$
(3.122)

This proves the reality condition. ∎

######  Corollary 3.2.

Using the same notation from section 3.6.2, we have that

$\mathcal{X}_{m}(\xi)\sim_{\xi\to\infty}\overline{A(-1/\overline{\xi})^{-1}}$
(3.123)

###### Proof.

By the reality condition, we have that the asymptotics of
$\mathcal{X}_{m}(\xi)$ as $\xi\to\infty$ match the asymptotics of
$\overline{\mathcal{X}_{m}(-1/\overline{\xi})^{-1}}$ as $\xi\to\infty$. But by
proposition 3.11, we have that

$\overline{\mathcal{X}_{m}(-1/\overline{\xi})^{-1}}\sim_{\xi\to\infty}\overline{A(-1/\overline{\xi})^{-1}}$
(3.124)

so the result follows. ∎

###  3.7 Non-vanishing of Stokes data

In this section we show that given
$(E,\overline{\partial}_{E},\theta,h,g)\in\mathcal{H}^{\text{fr}}$ with
$\text{Det}(\theta)=-(z^{2}+2m)dz^{2}$ and $m\neq 0$, the magnetic twistor
coordinate $\mathcal{X}_{m}((E,\overline{\partial}_{E},\theta,h,g),\xi)$ is
actually well defined for $\xi\in\mathbb{C}^{*}-l_{\pm}(-2im)$ (i.e. that
$b(\xi)\neq 0$ for $\xi\in\mathbb{H}_{-m}$).

####  3.7.1 The case of trivial Stokes data

We begin with the following lemma dealing with the case of trivial Stokes
data:

######  Lemma 3.17.

Consider
$(E,\overline{\partial}_{E},\theta,h,g)\in\mathcal{H}^{\textnormal{fr}}$ and
its associated compatibly framed filtered flat bundle
$(\mathcal{P}^{h}_{*}\mathcal{E}^{\xi},\nabla^{\xi},\tau_{*}^{\xi})$ for
$\xi\in\mathbb{C}^{*}$. Suppose that the Stokes data associated to
$(\mathcal{P}^{h}_{*}\mathcal{E}^{\xi},\nabla^{\xi},\tau_{*}^{\xi})$ is
trivial (i.e. $S_{i}=1$ for $i=1,2,3,4$ and $M_{0}=1$). Then the parameter $m$
specifying the singularity of the Higgs field of
$(E,\overline{\partial}_{E},\theta,h,g)$ must be $0$.

###### Proof.

We divide the proof into two cases:

  * •

Suppose first that the parabolic weights of
$(\mathcal{P}^{h}_{1/2}\mathcal{E}^{\xi},\nabla^{\xi},\tau_{1/2}^{\xi})$ lie
in $(-1/2,1/2)\subset\mathbb{R}$: Consider the framed wild harmonic bundle
$(E_{0},\overline{\partial}_{E_{0}},\theta_{0},h_{0},g_{0})\in\mathcal{H}^{\text{fr}}$
given in example 3.1. Let us give a description of the associated filtered
bundle
$(\mathcal{P}^{h}_{*}\mathcal{E}^{\xi}_{0},\nabla^{\xi}_{0},\tau_{*,0}^{\xi})$.

To give a description of this filtered bundle, it is enough to specify the
parabolic flat bundle
$(\mathcal{P}^{h}_{1/2}\mathcal{E}^{\xi}_{0},\nabla^{\xi}_{0},\tau_{1/2,0}^{\xi})$.
Since $m^{(3)}=m=0$, the parabolic filtration is trivial in this case, with
the holomorphic frames describing the extension of $\mathcal{E}^{\xi}$ to
$\mathcal{PE}_{0}^{\xi}$ given by
$\tau_{0}^{\xi}=g_{0}\cdot\text{exp}((\frac{\overline{\xi}}{2w^{2}}-\frac{\xi}{2\overline{w}^{2}})H)$.
Furthermore,
$g_{0}\cdot\text{exp}((\frac{z^{2}\overline{\xi}}{2}-\frac{\overline{z}^{2}\xi}{2})H)$
gives a global holomorphic trivialization of
$\mathcal{P}^{h}_{1/2}\mathcal{E}_{0}^{\xi}$, so that
$\mathcal{P}^{h}_{1/2}\mathcal{E}_{0}^{\xi}\cong\mathcal{O}\oplus\mathcal{O}$.
In this global trivialization, the connection has the following form:

$\nabla^{\xi}_{0}=d-(\xi^{-1}+\overline{\xi})H\frac{dw}{w^{3}}$ (3.125)

This connection has a frame of flat sections defined over
$\mathbb{C}\subset\mathbb{C}P^{1}$, given by
$g_{0}\cdot\text{exp}((-\frac{z^{2}\xi^{-1}}{2}-\frac{\overline{z}^{2}\xi}{2})H)$
satisfying the appropriate asymptotics with respect to $\tau_{0}^{\xi}$ (and
$g_{0}$). From this fact, we see that the formal monodromy and the associated
Stokes matrices are trivial.

Now going back to
$(\mathcal{P}^{h}_{1/2}\mathcal{E}^{\xi},\nabla^{\xi},\tau_{1/2}^{\xi})$, the
fact that it comes from a harmonic bundle implies that
$\text{pdeg}(\mathcal{P}^{h}_{1/2}\mathcal{E}^{\xi})=0$, while the assumption
on the parabolic weights imply that
$\text{deg}(\mathcal{P}^{h}_{1/2}\mathcal{E}^{\xi})=0$. Since it also has
trivial Stokes data, the Riemann-Hilbert correspondence for framed flat
bundles presented in [BB04] implies that
$(\mathcal{P}^{h}_{1/2}\mathcal{E}^{\xi},\nabla^{\xi},\tau_{1/2}^{\xi})$ is
isomorphic to
$(\mathcal{P}^{h}_{1/2}\mathcal{E}^{\xi}_{0},\nabla^{\xi}_{0},\tau_{1/2,0}^{\xi})$
(as compatibly framed flat bundles). In particular,
$(\mathcal{P}^{h}_{1/2}\mathcal{E}^{\xi},\nabla^{\xi})$ is isomorphic with
$(\mathcal{P}^{h}_{1/2}\mathcal{E}^{\xi}_{0},\nabla^{\xi}_{0},)$, so that
$(E,\overline{\partial}_{E},\theta,h)$ is isomorphic to
$(E_{0},\overline{\partial}_{E_{0}},\theta_{0},h_{0})$. Hence,

$-(z^{2}+2m)dz^{2}=\text{Det}(\theta)=\text{Det}(\theta_{0})=-z^{2}dz^{2}$
(3.126)

which implies that $m=0$.

  * •

The parabolic weights of
$(\mathcal{P}^{h}_{1/2}\mathcal{E}^{\xi},\nabla^{\xi},\tau_{1/2}^{\xi})$ are
both equal to $1/2$: in this case, the condition on the parabolic weights and
the fact that $\text{pdeg}(\mathcal{P}^{h}_{1/2}\mathcal{E}^{\xi})=0$ implies
that $\text{deg}(\mathcal{P}^{h}_{1/2}\mathcal{E}^{\xi})=1$. Hence, the
exponent of formal monodromy $\Lambda(\xi)$ of
$(\mathcal{P}^{h}_{1/2}\mathcal{E}^{\xi},\nabla^{\xi},\tau_{1/2}^{\xi})$ must
satisfy $\text{Tr}(\Lambda(\xi))=-1$ (see [Boa01b]), which together with the
fact that the formal monodromy is trivial implies that

$\Lambda(\xi)=\begin{bmatrix}n&0\\\ 0&-n-1\\\ \end{bmatrix}$ (3.127)

for some $n\in\mathbb{Z}$.

Now consider the degree $1$ bundle
$\mathcal{O}(-n)\oplus\mathcal{O}(n+1)\to\mathbb{C}P^{1}$. In the $z$
coordinate of $\mathbb{C}\subset\mathbb{C}P^{1}$, and the usual trivialization
over that neighborhood, we consider the connection

$\nabla=d+(\xi^{-1}+\overline{\xi})Hzdz$ (3.128)

In the neighborhood $\mathbb{C}P^{1}-\\{0\\}$ with coordinate $w=1/z$ and its
usual trivialization, we have that

$\nabla=d-(\xi^{-1}+\overline{\xi})H\frac{dw}{w^{3}}+\begin{bmatrix}n&0\\\
0&-n-1\\\ \end{bmatrix}\frac{dw}{w}$ (3.129)

So $\nabla$ is a meromorphic connection with the appropriate irregular part.
The trivialization over $\mathbb{C}P^{1}-\\{0\\}$ clearly gives a compatible
framing $\tau$ over $z=\infty$, so we get a compatibly framed meromorphic flat
bundle $(\mathcal{O}(-n)\oplus\mathcal{O}(n+1),\nabla,\tau)$.

It is easy to check that the Stokes matrices and formal monodromy associated
to $(\mathcal{O}(-n)\oplus\mathcal{O}(n+1),\nabla,\tau)$ are trivial. Hence,
since $(\mathcal{O}(-n)\oplus\mathcal{O}(n+1),\nabla,\tau)$ and
$(\mathcal{P}^{h}_{1/2}\mathcal{E}^{\xi},\nabla^{\xi},\tau_{1/2}^{\xi})$ have
the same formal type and same Stokes data, we get that
$(\mathcal{P}^{h}_{1/2}\mathcal{E}^{\xi},\nabla^{\xi},\tau_{1/2}^{\xi})$ is
isomorphic to $(\mathcal{O}(-n)\oplus\mathcal{O}(n+1),\nabla,\tau)$ as framed
flat bundles (the proof of this fact follows part of the argument of the proof
of lemma D.2, for example).

On the other hand,
$(\mathcal{P}^{h}_{1/2}\mathcal{E}^{\xi},\nabla^{\xi},\tau_{1/2}^{\xi})$ has
the trivial filtration as a parabolic bundle with parabolic weights both equal
to $1/2$, so $(\mathcal{O}(-n)\oplus\mathcal{O}(n+1),\nabla,\tau)$ also gets
this parabolic structure. Now notice that $\nabla$ preserves the line bundles
$\mathcal{O}(-n)$ and $\mathcal{O}(n+1)$, and with the induced parabolic
structures on the line bundles we have that

$\begin{split}\text{pdeg}(\mathcal{O}(-n))&=-n-\frac{1}{2}\\\
\text{pdeg}(\mathcal{O}(n+1))&=n+1-\frac{1}{2}=n+\frac{1}{2}\\\ \end{split}$
(3.130)

This shows that no matter what $n\in\mathbb{Z}$ is, we have that
$(\mathcal{P}^{h}_{1/2}\mathcal{E}^{\xi},\nabla^{\xi})$ is unstable. But
$(\mathcal{P}^{h}_{1/2}\mathcal{E}^{\xi},\nabla^{\xi})$ comes from a harmonic
bundle, so it must also be polystable by theorem 3.2. This contradiction shows
that $(\mathcal{P}^{h}_{1/2}\mathcal{E}^{\xi},\nabla^{\xi},\tau_{1/2}^{\xi})$
cannot have parabolic weights equal to $1/2$ if it has trivial monodromy data.
We conclude that the only case that occurs is the previous case.

Hence, we conclude that if $(E,\overline{\partial}_{E},\theta,h,g)$ has
trivial Stokes data, we must necessarily have $m=0$. ∎

####  3.7.2 Proof of the non-vanishing of Stokes data when $m\neq 0$

Recall that our Stokes data is made out of the $2\times 2$ unipotent Stokes
matrices $S_{i}(\xi)$ for $i=1,2,3,4$ and formal monodromy $M_{0}=e^{-2\pi
i\Lambda(\xi)}$ that must satisfy the relation
$S_{1}S_{2}S_{3}S_{4}M_{0}^{-1}=1$. We will label the off-diagonal non-trivial
complex numbers of $S_{1},S_{2},S_{3}$ and $S_{4}$ by $a,b,c$ and $d$
respectively, as in section 3.4.4. Hence, $\mathcal{X}_{m}(\xi)$ is defined
using $a(\xi)$ and $-1/b(\xi)$ using the conventions from section 3.4.4.

Because of the reality condition satisfied by the Stokes data, we have that
for $\xi\in\mathbb{H}_{-m}$ the equality
$-\overline{a(-1/\overline{\xi})}=b(\xi)$ holds. Hence, if we want to show
that $\mathcal{X}_{m}(\xi)$ is well defined, it is enough to show that
$a(\xi)$ does not vanish for $\xi\in\mathbb{H}_{m}$. To show this, we will
need the following lemma, whose proof is easy:

######  Lemma 3.18.

Let $U_{\pm}$ be the set of upper (lower, respectively) unipotent $2\times 2$
matrices, and $T\subset SL(2,\mathbb{C})$ the subset of diagonal matrices.
Then the set

$\mathcal{M}=\\{(S_{1},S_{2},S_{3},S_{4},M_{0})\in(U_{-}\times
U_{+})^{2}\times T\;\;|\;\;S_{1}S_{2}S_{3}S_{4}M_{0}^{-1}=1\\}$ (3.131)

is a complex $2$ dimensional manifold. Furthermore, if we denote by
$\mathcal{S}$ the subset of $\mathcal{M}$ defined by

$\mathcal{S}=\\{(S_{1},S_{2},S_{3},S_{4},M_{0})\in\mathcal{M}\;\;|\;\;S_{1}\neq
S_{2}\;\;\text{and either}\;\;S_{1}=1\;\;\text{or}\;\;S_{2}=1\\}$ (3.132)

then $\mathcal{S}$ is a complex $1$ dimensional submanifold with $2$
components. The components are determined by whether $S_{1}=1$ (i.e. $a=0$) or
$S_{2}=1$ (i.e. $b=0$).

######  Proposition 3.11.

Let $(E,\overline{\partial}_{E},\theta,h,g)\in\mathcal{H}^{\textnormal{fr}}$
with associated parameters $m^{(3)}$ and $m\neq 0$. Then for
$\xi\in\mathbb{H}_{m}$ we have that
$a([(E,\overline{\partial}_{E},\theta,h,g)],\xi)\neq 0$.

###### Proof.

Let us first see what condition we get if $a(\xi)=0$ for
$\xi\in\mathbb{H}_{m}$. If $a(\xi)=0$, then by the relation
$1+a(\xi)b(\xi)=\mu^{-1}(\xi)$ from (3.53) we conclude that $\mu^{-1}=1$. In
particular, by equation (3.57) we must have

$\xi^{-1}m-m^{(3)}-\overline{m}\xi=k$ (3.133)

for some $k\in\mathbb{Z}$.

Solving for $\xi$, we get two solutions $\xi^{\pm}_{k}$ for each
$k\in\mathbb{Z}$ given by

$\xi_{k}^{\pm}=\Bigg{(}\frac{-(k+m^{(3)})\pm\sqrt{(k+m^{(3)})^{2}+4|m|^{2}}}{2|m|^{2}}\Bigg{)}m$
(3.134)

The values $\xi_{k}^{\pm}$ are the only possible values of the twistor
parameter for which we could have $a(\xi)=0$ (for our fixed
$(E,\overline{\partial}_{E},\theta,h,g)\in\mathcal{H}^{\text{fr}}$).

In particular, the solutions contained in $\mathbb{H}_{m}$ are given by

$\xi_{k}^{+}=\Bigg{(}\frac{-(k+m^{(3)})+\sqrt{(m^{(3)}+k)^{2}+4|m|^{2}}}{2|m|^{2}}\Bigg{)}m$
(3.135)

and from the formula we see that $\xi_{k}^{+}\to 0$ as $k\to\infty$, and that
$\xi_{k}^{+}\to\infty$ as $k\to-\infty$. Furthermore, all $\xi_{k}^{+}$ are
contained in the ray determined by $0$ and $m$.

Because of the asymptotic behavior of $a(\xi)$ as $\xi\to 0$ from section
3.6.2, we see that $a(\xi_{k}^{+})$ cannot be $0$ for all $k$, otherwise the
asymptotics would not hold. Hence, $a(\xi_{k}^{+})\neq 0$ for some $k$
sufficiently big. Similarly, we get from the asymptotics as $\xi\to\infty$
that $a(\xi_{k}^{+})\neq 0$ for some $k$ sufficiently negative.

We will now show that the fact that $a(\xi_{k}^{+})\neq 0$ for at least one
$k\in\mathbb{Z}$ implies that $a(\xi_{n}^{+})\neq 0$ for all $n\in\mathbb{Z}$.

To show this, we will need the following lemma:

######  Lemma 3.19.

Let $(E,\overline{\partial}_{E},\theta,h,g)\in\mathcal{H}^{\textnormal{fr}}$
with associated parameters $m^{(3)}$ and $m\neq 0$, and consider the curve
$\xi(\widetilde{m^{(3)}}):\mathbb{R}\to\mathbb{H}_{m}$ given by

$\xi(\widetilde{m^{(3)}})=\Bigg{(}\frac{-(k+\widetilde{m^{(3)}})+\sqrt{(\widetilde{m^{(3)}}+k)^{2}+4|m|^{2}}}{2|m|^{2}}\Bigg{)}m$
(3.136)

Furthermore, let
$(\mathcal{P}_{*}^{h}\mathcal{E}^{\xi},\nabla^{\xi},\tau_{*}^{\xi})$ denote
the associated compatibly framed flat bundle of
$(E,\overline{\partial}_{E},\theta,h,g)$ for $\xi\in\mathbb{C}^{*}$, and
$\mathcal{S}$, $\mathcal{M}$ be as in lemma 3.18. Then we can find a
continuous map $\tau:\mathbb{R}\to\mathcal{S}\subset\mathcal{M}$ such that
$\tau(m^{(3)}+n)$ is the Stokes data associated
$(\mathcal{P}_{*}^{h}\mathcal{E}^{\xi(m^{(3)}+n)},\nabla^{\xi(m^{(3)}+n)},\tau_{*}^{\xi(m^{(3)}+n)})$
for all $n\in\mathbb{Z}$.

###### Proof.

(of lemma:) Let $(\mathcal{P}_{*}^{h}\mathcal{E}^{0},\theta)$ be the
associated filtered Higgs bundle to $(E,\overline{\partial}_{E},\theta,h)$,
and let
$(E,\theta)=(\mathcal{P}_{*}^{h}\mathcal{E}^{0},\theta)|_{\mathbb{C}P^{1}-\\{\infty\\}}$.
Near $z=\infty$, we choose a splitting of $(E,\theta)$ in eigenlines, such
that $\theta$ has the form of equation (3.13). We then have that $m^{(3)}$
specifies the filtered/parabolic structure of
$(\mathcal{P}_{*}^{h}\mathcal{E}^{0},\theta)$ (see for example the
construction of lemma 3.1). By varying $m^{(3)}$ in $\mathbb{R}$, we thus vary
the parabolic structure and hence the filtered Higgs bundle of parabolic
degree $0$ we obtain from $(E,\theta)$ by following lemma 3.1. We will denote
by $(\mathcal{P}_{*}^{h}\mathcal{E}^{0}(\widetilde{m^{(3)}}),\theta)$ the
filtered Higgs bundles of parabolic degree $0$ that we obtain from
$(E,\theta)$ with parabolic structure specified by
$\widetilde{m^{(3)}}\in\mathbb{R}$.

Notice that the filtered/parabolic structure only depends on
$\widetilde{m^{(3)}}$ mod $1$, so if $h(\widetilde{m^{3}})$ denotes a curve of
harmonic metrics adapted to
$(\mathcal{P}_{*}^{h}\mathcal{E}^{0}(\widetilde{m^{(3)}}),\theta)$, we get by
the uniqueness of harmonic metrics that for all $n\in\mathbb{Z}$, we have
$h(\widetilde{m^{(3)}}+n)=c(n)h(\widetilde{m^{(3)}})$ for some constant
$c(n)>0$.

We denote by $(E,\overline{\partial}_{E},\theta,h(\widetilde{m^{(3)}}))$ the
corresponding curve of wild harmonic bundles, and by

$\sigma(\widetilde{m^{(3)}})=[(E(\widetilde{m^{(3)}}),\overline{\partial}_{E(\widetilde{m^{(3)}})},\theta,h(\widetilde{m^{(3)}}),g(\widetilde{m^{(3)}}))]$
(3.137)

the corresponding curve of equivalence classes of framed wild harmonic bundles
obtained by applying the construction of proposition 3.2. Notice that
$g(\widetilde{m^{(3)}})$ satisfies that
$g(\widetilde{m^{(3)}}+n)=g(\widetilde{m^{(3)}})\cdot\sqrt{c(n)^{-1}}$ and
that
$(E(\widetilde{m^{(3)}}),\overline{\partial}_{E(\widetilde{m^{(3)}})})=(E(\widetilde{m^{(3)}}+n),\overline{\partial}_{E(\widetilde{m^{(3)}}+n)})$
for all $n\in\mathbb{Z}$. Hence, by example 3.2 we have that
$\sigma(\widetilde{m^{(3)}})=\sigma(\widetilde{m^{(3)}}+n)$ for all
$n\in\mathbb{Z}$.

Now consider the continuous map
$\tau(\widetilde{m^{3}}):\mathbb{R}\to\mathcal{M}$, where
$\tau(\widetilde{m^{(3)}})$ is the Stokes data of the associated compatibly
framed flat bundle
$[(\mathcal{P}^{h(\widetilde{m^{(3)}})}_{*}\mathcal{E}^{\xi(\widetilde{m^{(3)}})},\nabla^{\xi(\widetilde{m^{(3)}})},\tau_{*}^{\xi(\widetilde{m^{(3)}})})]$
of $\sigma(\widetilde{m^{(3)}})$.

By lemma 3.17, we have that $\tau$ does not go through
$(1,1,1,1,1)\in\mathcal{M}$. Hence, by the choice of
$\xi(\widetilde{m^{(3)}})$ we have that $\tau$ lands in
$\mathcal{S}\subset\mathcal{M}$. Furthermore, since
$\sigma(\widetilde{m^{(3)}}+n)=\sigma(\widetilde{m^{(3)}})$ for all
$n\in\mathbb{Z}$, we see that the same holds for $\tau$. We conclude that
$\tau$ satisfies the required properties.

∎

Going back to the proof of the proposition, let $\tau$ be a curve like in the
previous lemma, associated to our chosen
$(E,\overline{\partial}_{E},\theta,h,g)\in\mathcal{H}^{\text{fr}}$. This curve
must then be contained in one of the two components of $\mathcal{S}$, and it
goes through all the points where $a(\xi)$ could be $0$ (i.e. through the
points $\tau(m^{(3)}+n)$ with $n\in\mathbb{Z}$). Since the components of
$\mathcal{S}$ are determined by whether $a\neq 0$ or $b\neq 0$, and we know
that
$a([(E,\overline{\partial}_{E},\theta,h,g)],\xi_{n}^{+})=a([(E,\overline{\partial}_{E},\theta,h,g)],\xi(m^{(3)}+n))\neq
0$ for at least one $n$, we conclude that
$a([(E,\overline{\partial}_{E},\theta,h,g)],\xi_{n}^{+})\neq 0$ for all
$n\in\mathbb{Z}$. Since these are all the points in $\mathbb{H}_{m}$ where
$a([(E,\overline{\partial}_{E},\theta,h,g)],\xi)$ could be $0$, we then have
that $a([(E,\overline{\partial}_{E},\theta,h,g)],\xi)\neq 0$ for all
$\xi\in\mathbb{H}_{m}$. ∎

By the remark at the begining of the section we then conclude:

######  Corollary 3.3.

Given $[(E,\overline{\partial}_{E},\theta,h,g)]\in\mathfrak{X}^{\text{fr}}$
with parameter $m\neq 0$, the magnetic twistor coordinate
$\mathcal{X}_{m}([(E,\overline{\partial}_{E},\theta,h,g)],\xi)$ is well
defined for $\xi\in\mathbb{C}^{*}-l_{\pm}(-2im)$ and it takes values in
$\mathbb{C}^{*}$.

##  4 $\mathfrak{X}^{\text{fr}}$ and the Ooguri-Vafa space

In this section we identify a subset of $\mathfrak{X}^{\text{fr}}$ with the
Ooguri-Vafa space. In particular, we show that under the identification and a
proper choice of cut-off, the twistor coordinates we built for
$\mathfrak{X}^{\text{fr}}$ match the twistor coordinates of the Ooguri-Vafa
space.

We will use an “ov” superscript to distinguish between the Ooguri-Vafa twistor
coordinates and the twistor coordinates for harmonic bundles. For example, the
electric twistor coordinate of the Ooguri-Vafa space will be denoted by
$\mathcal{X}_{e}^{\text{ov}}(\xi)$.

###  4.1 Matching parameters with the Ooguri-Vafa space

We start by comparing the electric twistor coordinates:

$\begin{split}\mathcal{X}_{e}^{\text{ov}}(\xi)&=\exp[\pi\xi^{-1}z+i\theta_{e}+\pi\xi\overline{z}]\\\
\mathcal{X}_{e}(\xi)&=\text{exp}[-2\pi
i(\xi^{-1}m-m^{(3)}-\overline{m}\xi)]\\\ &=\text{exp}[\pi\xi^{-1}(-2im)+i(2\pi
m^{(3)})+\pi\xi\overline{(-2im)}]\end{split}$ (4.1)

so that we get the correspondence:

$\begin{split}z&\iff-2im\\\ \theta_{e}&\iff 2\pi m^{(3)}\\\ \end{split}$ (4.2)

To find the analog of the magnetic angle, we will use the asymptotics of
$\mathcal{X}_{m}(\xi)$ from proposition 3.11:

$\mathcal{X}_{m}(\xi)\sim_{\xi\to
0}\text{exp}\Big{(}-\frac{1}{\xi}m\text{Log}_{p}\Big{(}-\frac{m}{2e}\Big{)}+im^{(3)}\text{Arg}_{p}(-m)+i\pi+\int_{-\infty}^{\infty}\gamma^{*}A_{11}\Big{)}$
(4.3)

which we can rewrite as:

$\mathcal{X}_{m}(\xi)\sim_{\xi\to
0}\text{exp}\Big{(}-\frac{i}{2\xi}\Big{(}(-2im)\text{Log}_{p}\Big{(}-\frac{2im}{4i}\Big{)}-(-2im)\Big{)}+im^{(3)}\text{Arg}_{p}(-m)+i\pi+\int_{-\infty}^{\infty}\gamma^{*}A_{11}\Big{)}$
(4.4)

Comparing the last expression with the asymptotics of
$\mathcal{X}_{m}^{\text{ov}}(\xi)$:

$\mathcal{X}_{m}^{\text{ov}}(\xi)\sim_{\xi\to
0}\text{exp}\Big{(}-\frac{i}{2\xi}(z\text{Log}_{p}(z/\Lambda)-z)+i\theta_{m}+\frac{1}{2\pi
i}\sum_{s\neq
0,s\in\mathbb{Z}}\frac{1}{s}e^{is\theta_{e}}K_{0}(2\pi|sz|)\Big{)}$ (4.5)

we see that under the correspondence $z\iff-2im$, the $\xi^{-1}$ term matches
if we pick $\Lambda=4i$. On the other hand, when comparing the $\xi^{0}$ term,
we see that by matching the imaginary part we get:

$\theta_{m}\iff
m^{(3)}\text{Arg}_{p}(-m)+\pi+\text{Im}\Big{(}\int_{-\infty}^{\infty}\gamma^{*}A_{11}\Big{)}\;\;\;(\text{mod}\;2\pi)$
(4.6)

####  4.1.1 The magnetic angle on $\mathfrak{X}^{\text{fr}}$

The matching of parameters from above will let us define an analog of the
Ooguri-Vafa magnetic angle on $\mathfrak{X}^{\text{fr}}$. Before defining it,
we will need the following lemmas:

######  Lemma 4.1.

Suppose that
$(E_{i},\overline{\partial}_{E_{i}},\theta_{i},h_{i},g_{i})\in\mathcal{H}^{\textnormal{fr}}$
for $i=1,2$ have parameters $m_{i}$ and $m^{(3)}_{i}$ describing the
singularity of $\theta_{i}$ and $\overline{\partial}_{E_{i}}$ at $z=\infty$
(as in (3.13) and (3.14)). If there is an isomorphism between these two
elements, then $m_{1}=m_{2}$ and $m^{(3)}_{1}=m^{(3)}_{2}$. In particular, it
makes sense to associate $m$ and $m^{(3)}$ to elements of
$\mathfrak{X}^{\textnormal{fr}}$.

###### Proof.

If the two elements of $\mathcal{H}^{\text{fr}}$ are isomorphic, then

$-(z^{2}+2m_{1})dz^{2}=\text{Det}(\theta_{1})=\text{Det}(\theta_{2})=-(z^{2}+2m_{2})dz^{2}$
(4.7)

so that $m_{1}=m_{2}$. On the other hand, if we donote the compatible frames
(and their extensions to local frames around $z=\infty$) by
$g_{1}=(e_{1},e_{2})$, $g_{2}=(f_{1},f_{2})$, and the isomorphism by $T$, we
then have:

$m^{(3)}_{2}f_{1}=\overline{\partial}_{E_{2}}(-2\overline{w}\partial_{\overline{w}})(f_{1})|_{w=0}=T(\overline{\partial}_{E_{1}}(-2\overline{w}\partial_{\overline{w}})(e_{1})|_{w=0})=T(m^{(3)}_{1}e_{1})=m^{(3)}_{1}f_{1}$
(4.8)

so that $m^{(3)}_{1}=m^{(3)}_{2}$. ∎

######  Definition 4.1.

Let $m\in\mathbb{C}$ and $m^{(3)}\in(-1/2,1/2]$. We will denote by
$\mathfrak{X}^{\textnormal{fr}}(m,m^{(3)})\subset\mathfrak{X}^{\textnormal{fr}}$
the set of equivalence classes whose singularity with respect to the
compatible framing is described by $m$ and $m^{(3)}$.

For the following proposition, we will need the next lemma:

######  Lemma 4.2.

Let $m\in\mathbb{C}$ and $m^{(3)}\in(-1/2,1/2]\subset\mathbb{R}$. Then up to
equivalence, there is a unique polystable filtered Higgs bundle
$(\mathcal{E}_{*},\theta)$ with $\text{Tr}(\theta)=0$,
$\text{Det}(\theta)=-(z^{2}+2m)dz^{2}$, and parabolic weights determined by
$m^{(3)}$ as follows:

  * •

if $m^{(3)}\in(-1/2,1/2)$, then for the eigenline decomposition near $\infty$
of the induced $1/2$-parabolic Higgs bundle $(\mathcal{E}_{1/2},\theta)$, we
have that $\pm m^{(3)}$ is the weight associated to the line corresponding to
the eigenvalue $\pm(z+m/z+...)dz$.

  * •

if $m^{(3)}=1/2$ then the parabolic structure of the induced $1/2$-parabolic
structure $(\mathcal{E}_{1/2},\theta)$ is the trivial filtration with weight
$1/2$.

###### Proof.

In appendix D. ∎

######  Proposition 4.1.

We have a $U(1)$-action on $\mathfrak{X}^{\textnormal{fr}}$ given by

$e^{i\theta}\cdot[(E,\overline{\partial}_{E},\theta,h,g)]=[(E,\overline{\partial}_{E},\theta,h,e^{\frac{i\theta}{2}}\cdot
g)]$ (4.9)

where if $g=(e_{1},e_{2})$, then
$e^{\frac{i\theta}{2}}\cdot(e_{1},e_{2})=(e^{\frac{i\theta}{2}}e_{1},e^{-\frac{i\theta}{2}}e_{2})$.

Furthermore, for $m\neq 0$ we have that
$\mathfrak{X}^{\textnormal{fr}}(m,m^{(3)})$ is a $U(1)$-torsor under this
action.

###### Proof.

First notice that
$(E,\overline{\partial}_{E},\theta,h,g)\in\mathcal{H}^{\text{fr}}$ is
isomorphic to $(E,\overline{\partial}_{E},\theta,h,e^{i\pi}\cdot g)$ by the
morphism $-\text{I}_{E}$ (where $\text{I}_{E}:E\to E$ denotes the identity
map), so that the map
$U(1)\times\mathcal{H}^{\text{fr}}\to\mathfrak{X}^{\text{fr}}$ given by

$e^{i\theta}\cdot(E,\overline{\partial}_{E},\theta,h,g)=[(E,\overline{\partial}_{E},\theta,h,e^{\frac{i\theta}{2}}\cdot
g)]$ (4.10)

is well defined.

On the other hand if $T$ is an isomorphism between
$(E_{1},\overline{\partial}_{E_{1}},\theta_{1},h_{1},g_{1})$ and
$(E_{2},\overline{\partial}_{E_{2}},\theta_{2},h_{2},g_{2})$, then clearly $T$
also gives an isomorphism between
$(E_{1},\overline{\partial}_{E_{1}},\theta_{1},h_{1},e^{\frac{i\theta}{2}}\cdot
g_{1})$ and
$(E_{2},\overline{\partial}_{E_{2}},\theta_{2},h_{2},e^{\frac{i\theta}{2}}\cdot
g_{2})$. Hence we get an $U(1)$-action
$U(1)\times\mathfrak{X}^{\text{fr}}\to\mathfrak{X}^{\text{fr}}$, defined by
equation (4.9).

Now let $\mathfrak{X}^{\text{fr}}(m,m^{(3)})\subset\mathfrak{X}^{\text{fr}}$
be as in definition $4.1$. Clearly the $U(1)$-action on
$\mathfrak{X}^{\text{fr}}$ restricts to an action on
$\mathfrak{X}^{\text{fr}}(m,m^{(3)})$. Let us now check that this action is
freely transitive:

  * •

The action is free: Let $e^{i\theta}\neq 1$. From the way Stokes data
transforms under changes of compatible framing, it is easy to check that

$e^{i\theta}\mathcal{X}_{m}(\xi)([(E,\overline{\partial}_{E},\theta,h,g)])=\mathcal{X}_{m}(\xi)(e^{i\theta}\cdot[(E,\overline{\partial}_{E},\theta,h,g)])$
(4.11)

i.e. $\mathcal{X}_{m}(\xi)$ is equivariant with respect to the $U(1)$-action
on $\mathfrak{X}^{\text{fr}}$ and the natural $U(1)$-action on $\mathbb{C}$.
On the other hand, since $\mathcal{X}_{m}(\xi)$ is valued in $\mathbb{C}^{*}$,
equation 4.11 and the fact that $e^{i\theta}\neq 1$ implies that
$\mathcal{X}_{m}(\xi)([(E,\overline{\partial}_{E},\theta,h,g)])\neq\mathcal{X}_{m}(\xi)(e^{i\theta}\cdot[(E,\overline{\partial}_{E},\theta,h,g)])$.
Since Stokes data is an isomorphism invariant, we must have
$[(E,\overline{\partial}_{E},\theta,h,g)]\neq
e^{i\theta}\cdot[(E,\overline{\partial}_{E},\theta,h,g)]$.

  * •

The action is transitive: Let
$[(E_{i},\overline{\partial}_{E_{i}},\theta_{i},h_{i},g_{i})]\in\mathfrak{X}^{\text{fr}}(m,m^{(3)})$
for $i=1,2$. By lemma 4.2, we have that the underlying filtered Higgs bundles
$(\mathcal{P}^{h_{i}}_{*}E_{i},\theta_{i})$ are isomorphic. But then we must
have that $(E_{1},\overline{\partial}_{E_{1}},\theta_{1},h_{1})$ is isomorphic
to $(E_{2},\overline{\partial}_{E_{2}},\theta_{2},h_{2})$. Following the
construction of proposition 3.2, we then find an isomorphism between
$(E_{1},\overline{\partial_{1}},\theta_{1},h_{1},g_{1})$ and
$(E_{2},\overline{\partial_{2}},\theta_{2},h_{2},\widetilde{g}_{2})$ for some
compatible frame $\widetilde{g}_{2}$. Now $g_{2}$ and $\widetilde{g_{2}}$ must
be $SU(2)$ eigenframes of $\theta_{2}(w^{3}\partial_{w})|_{w=0}$ with the
corresponding fixed order of the eigenvalues given by the form of the
singularity, so we must have $g_{2}=e^{i\theta}\cdot\widetilde{g}_{2}$ for
some $e^{i\theta}\in U(1)$. Hence,

$e^{2i\theta}\cdot[(E_{1},\overline{\partial_{1}},\theta_{1},h_{1},g_{1})]=e^{2i\theta}\cdot[(E_{2},\overline{\partial_{2}},\theta_{2},h_{2},\widetilde{g}_{2})]=[(E_{2},\overline{\partial_{2}},\theta_{2},h_{2},g_{2})]$
(4.12)

∎

Recall that for $m\neq 0$ we have the correspondence

$\theta_{m}\iff
m^{(3)}\text{Arg}_{p}(-m)+\pi+\text{Im}\Big{(}\int_{-\infty}^{\infty}\gamma^{*}A_{11}\Big{)}\;\;\;(\text{mod}\;2\pi)\\\
$ (4.13)

Because of the equivariance of $\mathcal{X}_{m}(\xi)$ under the $U(1)$-actions
on $\mathfrak{X}^{\text{fr}}$ and $\mathbb{C}^{*}$, we see that acting on
$[(E,\overline{\partial}_{E},\theta,h,g)]\in\mathfrak{X}^{\text{fr}}(m,m^{(3)})$
by $e^{i\theta}$ shifts the quantity to the right of equation (4.13) by
$\theta$. This motivates the following definitions:

######  Definition 4.2.

Let $m\neq 0$. The marked point of the $U(1)$-torsor
$\mathfrak{X}^{\textnormal{fr}}(m,m^{(3)})$ is the unique element
$[(E,\overline{\partial}_{E},\theta,h,g_{0})]\in\mathfrak{X}^{\textnormal{fr}}(m,m^{(3)})$
such that

$\textnormal{exp}\Big{(}im^{(3)}\textnormal{Arg}_{p}(-m)+i\pi+i\textnormal{Im}\Big{(}\int_{-\infty}^{\infty}\gamma^{*}A_{11}\Big{)}\Big{)}=1$
(4.14)

######  Definition 4.3.

For $m\neq 0$, the magnetic angle $\theta_{m}$ of
$[(E,\overline{\partial}_{E},\theta,h,g)]\in\mathfrak{X}^{\textnormal{fr}}(m,m^{(3)})$
is defined to be the unique real number mod $2\pi$ such that:

$[(E,\overline{\partial}_{E},\theta,h,g)]=e^{i\theta_{m}}\cdot[(E,\overline{\partial}_{E},\theta,h,g_{0})]$
(4.15)

where $[(E,\overline{\partial}_{E},\theta,h,g_{0})]$ is the marked point of
$\mathfrak{X}^{\textnormal{fr}}(m,m^{(3)})$.

Notice that our definition of marked point uses a branch of the Arg function
(with $\text{Arg}(z)\in[-\pi,\pi)$), so the magnetic angle is not a priori a
global continuous function of $m$. In the following, we will compute how the
magnetic angle jumps when we go around a loop in the $m$ parameter. We will
see that it will match the jump of the magnetic angle of the Ooguri-Vafa
space.

######  Definition 4.4.

We will say that a map $\gamma:[0,1]\to\mathcal{H}^{\textnormal{fr}}$ is a
continuous path, if the elements
$\gamma(t):=(E_{t},\overline{\partial}_{E_{t}},\theta_{t},h_{t},g_{t})$
satisfy:

  * •

The vector bundles $E_{t}\to\mathbb{C}P^{1}$ fit into a continuous vector
bundle $E\to[0,1]\times\mathbb{C}P^{1}$, with
$E|_{\\{t\\}\times\mathbb{C}P^{1}}=E_{t}$, and with trivializations having
transition functions depending smoothly on the points of $\mathbb{C}P^{1}$ and
continuously on $t\in[0,1]$.

  * •

In the above trivializations, the structures $\overline{\partial}_{E_{t}}$,
$\theta_{t}$ and $h_{t}$ vary continuously in $t$.

  * •

The elements of the frames $g_{t}$ give a continuous section of
$E|_{[0,1]\times\\{\infty\\}}$.

Furthermore, we will say that a map
$\gamma:[0,1]\to\mathfrak{X}^{\textnormal{fr}}$ is a continuous path if there
is a lift to a continuous path
$\widetilde{\gamma}:[0,1]\to\mathcal{H}^{\textnormal{fr}}$.

Remark: a way to construct such paths is, for example, by employing
proposition $4.9$ of [Moc19] and the constructions of lemma 3.1 and
proposition 3.2.

######  Proposition 4.2.

Let $l:=\\{m\in\mathbb{C}\;|\;m\in\mathbb{R}_{>0}\\}$, and let
$\gamma:[0,1]\to\mathfrak{X}^{\textnormal{fr}}$ be a loop in
$\mathfrak{X}^{\textnormal{fr}}$ such that the associated curve $m(\gamma(t))$
of $m$ parameters of $\gamma(t)$ gives a counterclockwise loop around $m=0$
with $m(\gamma(0))\in l$. Then

$\lim_{t\to 1}\theta_{m}(\gamma(t))=\theta_{m}(\gamma(0))+2\pi m^{(3)}-\pi$
(4.16)

###### Proof.

We will use our usual notations for Stokes data as in section 3.4.4.

The magnetic angle is defined in terms of the $\xi^{0}$ term of the
asymptotics of the Stokes data $a(\gamma(t),\xi)$ as $\xi\to 0$ along
$\xi\in\mathbb{H}_{m(\gamma(t))}$, or by the $\xi^{0}$ term of the asymptotics
of the Stokes data $-1/b(\gamma(t),\xi)$ as $\xi\to 0$ along
$\xi\in\mathbb{H}_{-m(\gamma(t))}$. We will look at the monodromy of
$a(\gamma(t),\xi)$, but a similar argument holds if we use
$-1/b(\gamma(t),\xi)$.

As in section 3.4.4, we denote by $\Phi_{i}$ the frame of flat sections of
$\nabla^{\xi}$ defined on the extended sector
$\widehat{\text{Sect}}_{i}(\xi)$. As we move around the loop $\gamma(t)$, the
labelings of the sectors move in a clockwise manner (recall the conventions of
the $m$ dependence of the labelings of section 3.4.4). After going around the
loop, we get that the labelings moved in such a way that the following sectors
are interchanged:
$\widehat{\text{Sect}}_{1}(\xi)\iff\widehat{\text{Sect}}_{3}(\xi)$ and
$\widehat{\text{Sect}}_{2}(\xi)\iff\widehat{\text{Sect}}_{4}(\xi)$. If we
denote by $\widetilde{\Phi}_{i}$ the frame of flat sections obtained after
going around the loop, it is easy to see that we get the following relations:

$\begin{split}\widetilde{\Phi}_{1}&=\Phi_{3}\cdot M_{0}^{-1}\\\
\widetilde{\Phi}_{2}&=\Phi_{4}\cdot M_{0}^{-1}\\\
\widetilde{\Phi}_{3}&=\Phi_{1}\\\ \widetilde{\Phi}_{4}&=\Phi_{2}\\\
\end{split}$ (4.17)

Now recall the relations that we have among the entries of the Stokes matrices
(3.53) and among the flat sections (3.54). Using these relations and (4.17),
we find that:

$\widetilde{a}=\frac{\widetilde{s}_{3}\wedge\widetilde{s}_{1}}{\widetilde{s}_{2}\wedge\widetilde{s}_{1}}=\mu^{-1}\frac{s_{1}\wedge
s_{3}}{s_{4}\wedge s_{3}}=\mu^{-2}c=-\mu^{-1}a$ (4.18)

Since $-\mu^{-1}(\gamma(0),\xi)=\text{exp}(\pi\xi^{-1}(-2im)+i(2\pi
m^{(3)}-\pi)+\pi\xi\overline{(-2im)})$, from the asymptotics of
$-\mu^{-1}(\gamma(0),\xi)a(\gamma(0),\xi)$ as $\xi\to 0$ along
$\mathbb{H}_{m}$ we conclude what we want.

∎

Hence, we see that the magnetic angle has the same monodromy as the usual
Ooguri-Vafa magnetic angle.

###  4.2 Matching the twistor coordinates

By taking $z=-2im$ and $\theta_{e}=2\pi m^{(3)}$ we clearly have that for
$[(E,\overline{\partial}_{E},\theta,h,g)]\in\mathfrak{X}^{\text{fr}}(m,m^{(3)})$,
the following holds:

$\mathcal{X}_{e}(\xi)([(E,\overline{\partial}_{E},\theta,h,g)])=\mathcal{X}_{e}^{\text{ov}}(\xi)(z,\theta_{e})$
(4.19)

(recall equation 4.1).

So the remaining question is whether the magnetic twistor coordinate
$\mathcal{X}_{m}(\xi)$ on $\mathfrak{X}^{\text{fr}}$ matches
$\mathcal{X}_{m}^{\text{ov}}(\xi)$, under the appropriate matching of
parameters.

######  Theorem 4.1.

Fix
$[(E,\overline{\partial}_{E},\theta,h,g)]\in\mathfrak{X}^{\textnormal{fr}}(m,m^{(3)})$
and let $\mathcal{X}_{m}^{\textnormal{ov}}(\xi)$ be the magnetic twistor
coordinate of the Ooguri-Vafa space with cut-off $\Lambda=4i$. Then by taking
$z=-2im$, $\theta_{e}=2\pi m^{(3)}$ and
$\theta_{m}=\theta_{m}([(E,\overline{\partial}_{E},\theta,h,g)])$ we have that
$\mathcal{X}_{m}(\xi)([(E,\overline{\partial}_{E},\theta,h,g)])=\mathcal{X}_{m}^{\textnormal{ov}}(\xi)(z,\theta_{e},\theta_{m})$
for all $\xi\in\mathbb{C}^{*}-l_{\pm}(-2im)$.

###### Proof.

We will abbreviate the notation and just write $\mathcal{X}_{m}(\xi)$ and
$\mathcal{X}_{m}^{\text{ov}}(\xi)$.

We consider the quotient
$\mathcal{X}_{m}(\xi)/\mathcal{X}_{m}^{\text{ov}}(\xi)$ as a function of
$\xi\in\mathbb{C}^{*}-l_{\pm}(-2im)$. Because both functions have the same
jumping behavior along $l_{\pm}(-2im)$, we have that
$\mathcal{X}_{m}(\xi)/\mathcal{X}_{m}^{\text{ov}}(\xi)$ extends to a
continuous function on $\mathbb{C}^{*}$ that is holomorphic in
$\xi\in\mathbb{C}^{*}-l_{\pm}(-2im)$.

Furthermore, because of our matching of parameters and the asymptotics of both
functions, we have that

$\lim_{\xi\to
0}\frac{\mathcal{X}_{m}(\xi)}{\mathcal{X}_{m}^{\text{ov}}(\xi)}=r$ (4.20)

where $r$ is a real number given by the quotient of the real part of the
$\xi^{0}$ term of the asymptotics of both functions as $\xi\to 0$. More
specifically,

$r=\text{exp}\Big{(}\text{Re}\Big{(}\int_{-\infty}^{\infty}\gamma^{*}A_{11}\Big{)}-\frac{1}{2\pi
i}\sum_{s\neq
0,s\in\mathbb{Z}}\frac{1}{s}e^{is\theta_{e}}K_{0}(2\pi|sz|)\Big{)}$ (4.21)

Using the reality condition of both coordinates, we also get that

$\lim_{\xi\to\infty}\frac{\mathcal{X}_{m}(\xi)}{\mathcal{X}_{m}^{\text{ov}}(\xi)}=\lim_{\xi\to\infty}\frac{\overline{\mathcal{X}_{m}^{-1}(-1/\overline{\xi})}}{\overline{\mathcal{X}_{m}^{\text{ov}-1}(-1/\overline{\xi})}}=r^{-1}$
(4.22)

Hence, we can extend $\mathcal{X}_{m}(\xi)/\mathcal{X}_{m}^{\text{ov}}(\xi)$
to a continuous function on $\mathbb{C}P^{1}$ which is holomorphic on
$\mathbb{C}^{*}-l_{\pm}(-2im)$. An application of Morera’s theorem then shows
that $\mathcal{X}_{m}(\xi)/\mathcal{X}_{m}^{\text{ov}}(\xi)$ is a holomorphic
function on $\mathbb{C}P^{1}$, and hence constant.

In particular, we conclude that

$\frac{\mathcal{X}_{m}(\xi)}{\mathcal{X}_{m}^{\text{ov}}(\xi)}=r=r^{-1}$
(4.23)

so that $r=\pm 1$ and
$\mathcal{X}_{m}(\xi)=\pm\mathcal{X}_{m}^{\text{ov}}(\xi)$.

To fix the sign, notice that $r$ is an exponential of a real number (recall
equation (4.21)), so that we conclude that $r=1$ and then
$\mathcal{X}_{m}(\xi)=\mathcal{X}_{m}^{\text{ov}}(\xi)$ for all
$\xi\in\mathbb{C}^{*}-l_{\pm}(-2im)$.

∎

###  4.3 The Hyperkähler structure on $\mathfrak{X}^{\text{fr}}$

######  Definition 4.5.

We will denote by $\mathfrak{X}^{\textnormal{fr}}_{*}(4i)$ the subset of
$[(E,\overline{\partial}_{E},\theta,h,g)]\in\mathfrak{X}^{\textnormal{fr}}$
such that if $\text{Det}(\theta)=-(z^{2}+2m)dz^{2}$, then $m\neq 0$ and
$z=-2im$ lies in the base $\mathcal{B}$ of
$\mathcal{M}^{\textnormal{ov}}(4i)$. Furthermore, we denote by
$\mathcal{M}_{*}^{\textnormal{ov}}(4i)\subset\mathcal{M}^{\textnormal{ov}}(4i)$
the points of the Ooguri-Vafa space with $z\neq 0$, and by $\mathcal{B}_{*}$
the punctured basis $\mathcal{B}-\\{0\\}$.

The results of sections 4.1 and 4.2 then show:

######  Theorem 4.2.

We can identify $\mathfrak{X}^{\textnormal{fr}}_{*}(4i)$ with
$\mathcal{M}^{\textnormal{ov}}_{*}(4i)$. Under this identification,
$\mathfrak{X}^{\textnormal{fr}}_{*}(4i)$ acquires the structure of a
hyperkähler manifold. For $\xi\in\mathbb{C}^{*}$, the twistor family
$\Omega(\xi)$ of holomorphic symplectic forms on
$\mathfrak{X}^{\textnormal{fr}}_{*}(4i)$ is given by

$\Omega(\xi)=\frac{d\mathcal{X}_{e}(\xi)}{\mathcal{X}_{e}(\xi)}\wedge\frac{d\mathcal{X}_{m}(\xi)}{\mathcal{X}_{m}(\xi)}$
(4.24)

###### Proof.

Let $U_{1}:=\\{z\in\mathcal{B}_{*}\;|\;\text{Arg}(z)\neq-\pi/2\;\\}$ and
$U_{2}:=\\{z\in\mathcal{B}_{*}\;|\;\text{Arg}(z)\neq\pi/2\;\\}$. We then can
write $U_{1}\cap U_{2}=V_{1}\cup V_{2}$, where
$V_{1}=\\{z\in\mathcal{B}_{*}\;|\;\text{Arg}(z)\in(\pi/2,3\pi/2)\;\text{(mod
$2\pi$)}\\}$ and
$V_{2}=\\{z\in\mathcal{B}_{*}\;|\;\text{Arg}(z)\in(-\pi/2,\pi/2)\;\text{(mod
$2\pi$)}\\}$. Furthermore, let
$\pi:\mathfrak{X}_{*}^{\text{fr}}(4i)\to\mathcal{B}_{*}\times S^{1}$ be
defined by $\pi([(E,\overline{\partial}_{E},\theta,h,g)])=(-2im,e^{2\pi
im^{(3)}})$, where $m$ and $m^{(3)}$ are the associated parameters of
$[(E,\overline{\partial}_{E},\theta,h,g)]$ specifying the singularity.

We then have that $(-2im,e^{2\pi im^{(3)}},e^{i\theta_{m}})$ gives coordinates
for $\pi^{-1}(U_{1}\times S^{1})\subset\mathfrak{X}_{*}^{\text{fr}}(4i)$.
Similarly, proposition 4.2 implies that we can find another magnetic angle
coordinate $\widetilde{\theta}_{m}$ over $U_{2}\times S^{1}$ such that
$(-2im,e^{2\pi im^{(3)}},e^{i\widetilde{\theta}_{m}})$ gives coordinates for
$\pi^{-1}(U_{2}\times S^{1})$, and such that the transition function
$\phi:(U_{1}\cap U_{2})\times S^{1}\to U(1)$ from $\theta_{m}$ to
$\widetilde{\theta}_{m}$ is given by:

$\phi(-2im,e^{2\pi im^{(3)}})=\begin{cases}1\;\;\text{if}\;\;-2im\in V_{1}\\\
e^{2\pi im^{(3)}-i\pi}\;\;\text{if}\;\;-2im\in V_{2}\end{cases}$ (4.25)

This allow us to put the structure of an $U(1)$-principal bundle on
$\pi:\mathfrak{X}_{*}^{\text{fr}}(4i)\to\mathcal{B}_{*}\times S^{1}$,
isomorphic to $\mathcal{M}^{\text{ov}}_{*}(4i)\to\mathcal{B}_{*}\times S^{1}$.

The identification above induces a hyperkähler structure on
$\mathfrak{X}_{*}^{\text{fr}}(4i)$, and by theorem 4.1 and equation 2.22, the
twistor family of holomorphic symplectic forms is given by the formula of
equation 4.24 (to get the holomorphic symplectic forms corresponding to
$\xi=0$ and $\xi=\infty$ we put $\xi\Omega(\xi)|_{\xi=0}$ and
$\xi^{-1}\Omega(\xi)|_{\xi=\infty}$, respectively). ∎

However, this is not the end of the story, since the hyperkähler structure of
$\mathcal{M}^{\text{ov}}_{*}(4i)$ actually extends to the points where $z=0$.
From section 2.2 we know that it extends to the points where $z=0$ and
$\theta_{e}\neq 0$. The set where $z=\theta_{e}=0$ degenerates into a point,
and the hyperkähler structure also extends to this point (see for example
section $4.1$ of [GMN10]).

The picture of the fiber $z=0$ of
$\mathcal{M}^{\text{ov}}(\Lambda)\to\mathcal{B}$ is that of a torus with a
node, with the point responsible of the node corresponding to
$z=\theta_{e}^{\textnormal{ov}}=0$, with the circle corresponding to the
magnetic angle degenerating into a point.

In the following, we show that under our identification of parameters, we get
the same picture for the elements of $\mathfrak{X}^{\text{fr}}$ with $m=0$
(and hence, $z=0$).

####  4.3.1 The central fiber of $\mathfrak{X}^{\text{fr}}$

Recall that that given $m\in\mathbb{C}$ and $m^{(3)}\in(-1/2,1/2]$, we denote
by $\mathfrak{X}^{\text{fr}}(m,m^{(3)})$ the subset of equivalence classes of
$\mathfrak{X}^{\text{fr}}$ with corresponding parameters $m$ and $m^{(3)}$
describing the singularities.

Strictly speaking, we have only shown that
$\mathfrak{X}^{\text{fr}}(m,m^{(3)})$ is not empty when $m\neq 0$ (see lemma
3.1 and proposition 3.2). In appendix E we show that
$\mathfrak{X}^{\text{fr}}(0,m^{(3)})$ is also not empty for every
$m^{(3)}\in(-1/2,1/2]$.

We now prove the following lemma, which is analogous to proposition 4.1:

######  Lemma 4.3.

For $m^{(3)}\neq 0$ we have that $\mathfrak{X}^{\textnormal{fr}}(0,m^{(3)})$
is a $U(1)$-torsor under the $U(1)$-action defined on proposition 4.1.

###### Proof.

Let
$[(E,\overline{\partial}_{E},\theta,h,g)]\in\mathfrak{X}^{\text{fr}}(0,m^{(3)})$
for $m^{(3)}\neq 0$.

Let us now show that the $U(1)$-action is freely transitive:

  * •

The action is free: by the taking the associated Stokes data, we see that the
formal monodromy of the flat connection $\nabla^{\xi}$ turns out to be
$e^{2\pi im^{(3)}}\neq 1$ (notice that it does not depend on
$\xi\in\mathbb{C}^{*}$). Using the notation from 3.4.1, we then see that the
relation $1+a(\xi)b(\xi)=\mu^{-1}(\xi)=e^{2\pi im^{(3)}}$ implies that
$a(\xi)\neq 0$ and $b(\xi)\neq 0$ for all $\xi\in\mathbb{C}^{*}$. On the other
hand let $e^{i\theta}\neq 1$, it is then easy to check that

$a(\xi,e^{i\theta}\cdot[(E,\overline{\partial}_{E},\theta,h,g)])=e^{i\theta}a(\xi,[(E,\overline{\partial}_{E},\theta,h,g)])$
(4.26)

and since $a(\xi)\neq 0$, we must have that

$a(\xi,e^{i\theta}\cdot[(E,\overline{\partial}_{E},\theta,h,g)])\neq
a(\xi,[(E,\overline{\partial}_{E},\theta,h,g)])$ (4.27)

Since Stokes data is an isomorphism invariant, we conclude that

$[(E,\overline{\partial}_{E},\theta,h,g)]\neq
e^{i\theta}\cdot[(E,\overline{\partial}_{E},\theta,h,g)]$ (4.28)

so the action is free.

  * •

The action is transitive: the same proof as proposition 4.1.

∎

On the other hand, we have:

######  Lemma 4.4.

$\mathfrak{X}^{\textnormal{fr}}(0,0)$ is just a point.

###### Proof.

Let
$[(E_{0},\overline{\partial}_{E_{0}},\theta_{0},h_{0},g_{0})]\in\mathfrak{X}^{\text{fr}}(0,0)$
be the framed wild harmonic bundle from example 3.1, and let
$[(E,\overline{\partial}_{E},\theta,h,g)]\in\mathfrak{X}^{\text{fr}}(0,0)$ be
any other framed wild harmonic bundle.

By the same proof of the trasitivity of the $U(1)$-action of proposition 4.1,
we have that

$[(E,\overline{\partial}_{E},\theta,h,g)]=e^{i\theta}\cdot[(E_{0},\overline{\partial}_{E_{0}},\theta_{0},h_{0},g_{0})]$
(4.29)

for some $e^{i\theta}\in U(1)$.

But from the description of
$[(E_{0},\overline{\partial}_{E_{0}},\theta_{0},h_{0},g_{0})]$ in example 3.1
it is easy to see that
$[(E_{0},\overline{\partial}_{E_{0}},\theta_{0},h_{0},g_{0})]=e^{i\theta}\cdot[(E_{0},\overline{\partial}_{E_{0}},\theta_{0},h_{0},g_{0})]$
for every $e^{i\theta}\in U(1)$, the isomorphism being the map $T:E_{0}\to
E_{0}$ described in the global canonical frame $(e_{1},e_{2})$ by

$T=\begin{bmatrix}e^{i\frac{\theta}{2}}&0\\\
0&e^{-i\frac{\theta}{2}}\end{bmatrix}$ (4.30)

Hence,
$[(E_{0},\overline{\partial}_{E_{0}},\theta_{0},h_{0},g_{0})]=[(E,\overline{\partial}_{E},\theta,h,g)]$
and $\mathfrak{X}^{\text{fr}}(0,0)$ is just a point. ∎

From these results we see that the set of elements of
$\mathfrak{X}^{\text{fr}}(0,m^{(3)})$ have the same picture as the central
fiber of $\mathcal{M}^{\text{ov}}(\Lambda)$: it looks like a torus with a
node, with the angle $\theta_{e}=2\pi im^{(3)}$ parametrizing the “big”
circle, and the $U(1)$-torsors $\mathfrak{X}^{\text{fr}}(0,m^{(3)})$
degenerating at $m^{(3)}=0$. Furthermore, under our correspondence of
parameters with the Ooguri-Vafa space, we see that the node in
$\mathcal{M}^{\text{ov}}(4i)$ (with $z=\theta_{e}=0$) goes to the point
$\mathfrak{X}^{\text{fr}}(0,0)$ (with $m=m^{(3)}=0$).

######  Definition 4.6.

We will denote by $\mathfrak{X}^{\textnormal{fr}}(4i)$ the subset of
$[(E,\overline{\partial}_{E},\theta,h,g)]\in\mathfrak{X}^{\textnormal{fr}}$
such that if $z=-2im$, then $z$ lies in the base $\mathcal{B}$ of
$\mathcal{M}^{\textnormal{ov}}(4i)$.

We then deduce the following extension of theorem 4.2

######  Theorem 4.3.

Under our correspondence of parameters, we can identify
$\mathfrak{X}^{\textnormal{fr}}(4i)$ with $\mathcal{M}^{\textnormal{ov}}(4i)$.

##  Appendix A Estimates for the connection form

We use the setting and notation of the beginning of the proof of proposition
3.2.

We prove the following:

######  Lemma A.1.

After orthonormalizing the frame $(v_{1},v_{2})\to(e_{1},e_{2})$, the Chern
connection is expressed in this orthonormal frame as:

$D=d-\frac{1}{2}\begin{bmatrix}a(v_{1})&0\\\ 0&a(v_{2})\\\
\end{bmatrix}\Big{(}\frac{dw}{w}-\frac{d\overline{w}}{\overline{w}}\Big{)}+\text{regular
terms at $w=0$}$ (A.1)

###### Proof.

The gauge transformation $g$ that satisfies $(v_{1},v_{2})\cdot
g=(e_{1},e_{2})$ is given by

$g=\begin{bmatrix}\frac{1}{|v_{1}|_{h}}&-\frac{h(v_{1},v_{2})}{|v_{1}|^{2}_{h}|v_{2}-h(v_{1},v_{2})|v_{1}|_{h}^{-2}v_{1}|_{h}}\\\
0&\frac{1}{|v_{2}-h(v_{1},v_{2})|v_{1}|_{h}^{-2}v_{1}|_{h}}\\\ \end{bmatrix}$
(A.2)

Furthermore, we know that $|v_{i}|_{h}^{2}=|w|^{-2a(v_{i})}f_{i}(w)$, where
$f_{i}(w)$ is a positive real function that is bounded near $w=0$ (this is a
consequence of theorem 3.5).

If $A$ denotes the connection matrix of $D$ in the frame $(e_{1},e_{2})$, then
we have that

$A=g^{-1}dg+g^{-1}\Big{(}\begin{bmatrix}-a(v_{1})&0\\\ 0&-a(v_{2})\\\
\end{bmatrix}\frac{dw}{w}+\text{regular}\Big{)}g$ (A.3)

By using the fact that the off-diagonal terms of $g$ and the off-diagonal
terms of the regular terms of the connection matrix of $D$ in the frame
$(v_{1},v_{2})$ go to 0 exponentially as $w\to 0$, it is easy to check that

$g^{-1}\Big{(}\begin{bmatrix}-a(v_{1})&0\\\ 0&-a(v_{2})\\\
\end{bmatrix}\frac{dw}{w}+\text{regular}\Big{)}g=\begin{bmatrix}-a(v_{1})&0\\\
0&-a(v_{2})\\\ \end{bmatrix}\frac{dw}{w}+\text{regular}$ (A.4)

On the other hand, the fact that in the frame $(v_{1},v_{2})$ we have that

$D=D_{0}+\text{regular}=d+H^{-1}\partial H$ (A.5)

where $H$ is the matrix of the hermitian metric in the frame $(v_{1},v_{2})$,
implies that the functions $f_{i}^{-1}\partial_{z}f_{i}$ are regular at $w=0$.
Since the $f_{i}$ are real, we get that
$f_{i}^{-1}\partial_{\overline{z}}f_{i}$ is also regular, so that
$f_{i}^{-1}df_{i}=-f_{i}df_{i}^{-1}$ is regular at $w=0$. From this fact we
conclude that:

$g^{-1}dg=\frac{1}{2}\begin{bmatrix}a(v_{1})&0\\\ 0&a(v_{2})\\\
\end{bmatrix}\Big{(}\frac{dw}{w}+\frac{d\overline{w}}{\overline{w}}\Big{)}+\text{regular}$
(A.6)

Hence, in the frame $(e_{1},e_{2})$ we get that:

$D=d+A=d-\frac{1}{2}\begin{bmatrix}a(v_{1})&0\\\ 0&a(v_{2})\\\
\end{bmatrix}\Big{(}\frac{dw}{w}-\frac{d\overline{w}}{\overline{w}}\Big{)}+\text{regular}$
(A.7)

∎

##  Appendix B Proof of lemma 3.11

The goal of this appendix is to show that the asymptotics of lemma 3.5 hold
uniformly in $\xi\in U(\xi_{0})$, where $\xi_{0}\in\mathbb{C}^{*}$ and
$U(\xi_{0})$ is some small bounded neighborhood of $\xi_{0}$.

Because of the expression of the flat frames $\Phi_{i}(\xi)$ in terms of
extensions of the compatible frames $g=(e_{1},e_{2})$ of the harmonic bundles,
it is easy to check that it is enough to show that:

  * •

$\Sigma_{i}(\widehat{F}(\xi))\to 1$ uniformly in $\xi\in U(\xi_{0})$ as $w\to
0$ with $w\in S_{i}\subset\widehat{\text{Sect}}_{i}(\xi)$. Here $S_{i}$ is the
sector defined at the beginning of section 3.5.3.

  * •

$g_{\xi}(w)\to 1$ uniformly in $\xi\in U(\xi_{0})$ as $w\to 0$.

We will only prove the second statement, since the first one follows from the
proof of theorem $6$ of [Boa01a]. For the proof of the second statement we
will follow similar arguments and notations as those found in [BB04], where
they construct $g_{\xi}(w)$ for $\xi=1$.

Let us begin with some notation: we will denote by $D\subset\mathbb{C}P^{1}$
the unit disc centered at $w=1/z=0$; we also denote by $E$ the vector bundle
corresponding to an element
$(E,\overline{\partial}_{E},\theta,h,g)\in\mathcal{H}^{\text{fr}}$,
trivialized over $D$ by an extension of the compatible framing at $w=0$ to a
local $SU(2)$ framing. For $\delta>0$ we then define the weighted Sobolev
spaces, as in [BB04]:

$\begin{split}L_{\delta}^{p}&=\Big{\\{}f\in\Gamma(D,\text{End(E)})\;\;\Big{|}\;\;\frac{f}{r^{\delta+2/p}}\in
L^{p}(D,\text{End}(E))\Big{\\}}\\\
L_{\delta}^{p,k}&=\Big{\\{}f\in\Gamma(D,\text{End(E)})\;\;\Big{|}\;\;\frac{D(\overline{\partial}_{E},h)^{j}f_{i}}{r^{i(k-j)}}\in
L^{p}_{\delta}\;\;\text{for}\;\;i=0,3\;\;0\leq j\leq k\Big{\\}}\end{split}$
(B.1)

where $r$ is the radial coordinate on the disk, and where $f_{0}$ and $f_{3}$
denote the diagonal and off-diagonal components of $f\in\text{End}(E)$. The
reason for the strange indexing notation for the diagonal and off-diagonal
part is so that our notation agrees with [BB04]. The highest order pole of the
singularity (cubic order in our case) acts non-trivially via the adjoint
action on the off-diagonal part of a section $f\in\Gamma(D,\text{End}(E))$
(hence the “$3$” subscript), while the whole singular part acts trivially on
the diagonal part of $f$ (hence the “$0$” subscript).

Similarly, we have the Banach spaces $C^{k}_{\delta}$ defined by:

$C^{k}_{\delta}=\Big{\\{}f\in\Gamma(\overline{D},\text{End(E)})\;\;\Big{|}\;\;\frac{f}{r^{\delta}}\in
C^{k}(\overline{D},\text{End}(E))\Big{\\}}$ (B.2)

The gauge transformation $g_{\xi}$ is built as a solution to the following
problem: with respect to an extension of the compatible framing, we have the
expression

$(\nabla^{\xi})^{0,1}=\overline{\partial}-\xi
H\frac{d\overline{w}}{\overline{w}^{3}}-(\xi\overline{m}+\frac{m^{(3)}}{2})H\frac{d\overline{w}}{\overline{w}}+a_{\text{reg}}^{0,1}+\xi\theta_{\text{reg}}^{\dagger_{h}}$
(B.3)

where $a_{\text{reg}}^{0,1}$ and $\theta_{\text{reg}}^{\dagger_{h}}$ denote
the regular parts of $D(\overline{\partial}_{E},h)^{(0,1)}$ and
$\theta^{\dagger_{h}}$ respectively. The gauge transformation $g_{\xi}$ then
satisfies:

$\begin{split}g_{\xi}\cdot(\nabla^{\xi})^{0,1}&=\overline{\partial}-\xi
H\frac{d\overline{w}}{\overline{w}^{3}}-(\xi\overline{m}+\frac{m^{(3)}}{2})H\frac{d\overline{w}}{\overline{w}}\\\
g_{\xi}(w=0)&=1\\\ g_{\xi,0}-1\in C^{0}_{\delta}&\;\;g_{\xi,3}\in
C^{0}_{2+\delta}\;\;\text{for some}\;\;\delta>0\\\ \end{split}$ (B.4)

In order to show the existence of such a $g_{\xi}$ we will extend some of the
results in chapter $7$ of [BB04] in order to get statements for families in
$\xi$:

######  Lemma B.1.

Take $\delta\in\mathbb{R}-\mathbb{Z}$ and $p>2$. On the unit disk, the problem

$\frac{\partial f}{\partial\overline{w}}=g$ (B.5)

has a solution $f=T(g)$ such that $|f|_{C^{0}_{-1+\delta}}\leq
c|g|_{L^{p}_{-2+\delta}}$.

Furthermore, if $\lambda(\xi)$ is a continuous function of $\xi\in
U(\xi_{0})$, then the same is true if
$\delta-\text{Re}(\lambda(\xi))\in\mathbb{R}-\mathbb{Z}$ for the problem

$\frac{\partial
f}{\partial\overline{w}}-\frac{\lambda(\xi)}{2\overline{w}}f=g$ (B.6)

By picking $U(\xi_{0})$ small enough, we have
$|T_{\xi}(g)|_{C^{0}_{-1+\delta}}\leq c|g|_{L^{p}_{-2+\delta}}$ for all
$\xi\in U(\xi_{0})$, for a uniform constant $c$.

###### Proof.

We will only show the last two statements of the lemma, since the first is the
same as lemma $7.2$ of [BB04].

By the same argument given in lemma $7.2$ of [BB04], we can assume for our
problem that $\delta-\text{Re}(\lambda(\xi))\in(0,1)$ for $\xi\in U(\xi_{0})$.
By shrinking $U(\xi_{0})$ further if necessary, we can assume that
$\delta-\text{Re}(\lambda(\xi))\in(\delta_{0},\delta_{1})$ for
$0<\delta_{0}<\delta_{1}<1$ and $\xi\in U(\xi_{0})$.

If $T$ denotes the solution operator for the first in-homogeneous Cauchy-
Riemann problem (B.5), then
$T_{\xi}(g):=r^{\lambda(\xi)}T(r^{-\lambda(\xi)}g)$ is the solution operator
for (B.6).

Now notice that by applying Hölder’s inequality, we get

$|T_{\xi}(g)(w)|=\Big{|}r^{\lambda(\xi)}\int_{D}\frac{|u|^{-\lambda(\xi)}g(u)}{w-u}|d^{2}u|\Big{|}\leq
r^{\text{Re}(\lambda(\xi))}|g|_{L^{p}_{-2+\delta}}\Big{(}\int_{D}\frac{|d^{2}u|}{|u|^{2-(\delta-\text{Re}(\lambda(\xi)))\frac{p}{p-1}}|w-u|^{\frac{p}{p-1}}}\Big{)}^{\frac{p-1}{p}}$
(B.7)

We will denote $\delta(\xi)=\delta-\text{Re}(\lambda(\xi))$. By our conditions
on $\delta(\xi)$ and $p$, we have

$\int_{\mathbb{C}}\frac{|d^{2}u|}{|u|^{2-\delta(\xi)\frac{p}{p-1}}|w-u|^{\frac{p}{p-1}}}<\infty$
(B.8)

so if $D_{1/|w|}$ denotes the disk centered at the origin of radius $1/|w|$,
we can write

$\begin{split}\int_{D}\frac{|d^{2}u|}{|u|^{2-\delta(\xi)\frac{p}{p-1}}|w-u|^{\frac{p}{p-1}}}&=\frac{1}{|w|^{(1-\delta(\xi))\frac{p}{p-1}}}\int_{D_{1/|w|}}\frac{|d^{2}u|}{|u|^{2-\delta(\xi)\frac{p}{p-1}}|1-u|^{\frac{p}{p-1}}}\\\
&\leq\frac{1}{|w|^{(1-\delta(\xi))\frac{p}{p-1}}}\int_{\mathbb{C}}\frac{|d^{2}u|}{|u|^{2-\delta(\xi)\frac{p}{p-1}}|1-u|^{\frac{p}{p-1}}}\\\
&=\frac{1}{|w|^{(1-\delta(\xi))\frac{p}{p-1}}}c(\xi)\end{split}$ (B.9)

where $c(\xi):U(\xi_{0})\to\mathbb{R}$ is defined by:

$c(\xi):=\int_{\mathbb{C}}\frac{|d^{2}u|}{|u|^{2-\delta(\xi)\frac{p}{p-1}}|1-u|^{\frac{p}{p-1}}}$
(B.10)

Now it is easy to check that $c(\xi)$ depends continuously on $\xi$. To show
this, consider the function $h(u)$ defined in the following way:

$\begin{split}h(u)&=\frac{1}{|u|^{2-\delta_{0}\frac{p}{p-1}}|1-u|^{\frac{p}{p-1}}}\;\;\text{if}\;\;|u|<1\\\
h(u)&=\frac{1}{|u|^{2-\delta_{1}\frac{p}{p-1}}|1-u|^{\frac{p}{p-1}}}\;\;\text{if}\;\;|u|>1\\\
\end{split}$ (B.11)

then we get that $h\in L^{1}(\mathbb{C})$ and furthermore

$\frac{1}{|u|^{2-\delta(\xi)\frac{p}{p-1}}|1-u|^{\frac{p}{p-1}}}\leq
h(u)\;\;\text{for every}\;\;\xi\in U(\xi_{0})\;\;\text{and for almost
every}\;\;u\in\mathbb{C}$ (B.12)

From this fact, it follows from general theorems about integrals depending on
parameters, that $c(\xi)$ must be a continuous function of $\xi\in
U(\xi_{0})$. In particular, by further restricting $U(\xi_{0})$ if necessary,
we can find a constant $C$ such that $c(\xi)\leq C$ for $\xi\in U(\xi_{0})$.

Hence, by going back (B.7), we conclude that

$|T_{\xi}(g)(w)|\leq|g|_{L^{p}_{-2+\delta}}\frac{|w|^{\text{Re}(\lambda(\xi))}C^{\frac{p-1}{p}}}{|w|^{1-\delta(\xi)}}=|g|_{L^{p}_{-2+\delta}}\frac{C^{\frac{p-1}{p}}}{|w|^{1-\delta}}$
(B.13)

so we finally get that

$|T_{\xi}(g)|_{C^{0}_{-1+\delta}}\leq|g|_{L^{p}_{-2+\delta}}C^{\frac{p-1}{p}}\;\;\;\text{for
all}\;\;\;\xi\in U(\xi_{0})$ (B.14)

∎

######  Lemma B.2.

Let $U_{\delta}:=\\{u\in\Gamma(D,End(E))\;\;|\;\;u_{3}\in
C^{0}_{2+\delta},u_{0}\in C^{0}_{\delta}\\}$, and
$A_{\delta}:=\\{a\in\Gamma(D,End(E))\;\;|\;\;a_{3}\in
L^{p}_{\delta+1},a_{0}\in L^{p}_{\delta-1}\\}$. Furthermore, let
$\overline{\partial}_{0}=\overline{\partial}-(\xi\overline{m}+\frac{m^{(3)}}{2})H\frac{d\overline{w}}{\overline{w}}$.
Then for some $\delta^{\prime}<\delta$ there is a continuous map
$T_{\xi}:A_{\delta^{\prime}}\to U_{\delta^{\prime}}$ such that
$\overline{\partial}_{0}(T_{\xi}(c))=c$. If we pick $U(\xi_{0})$ sufficiently
small, the family of solution maps $T_{\xi}$ has a uniform bound in $\xi$.

###### Proof.

This follows from the beginning of the proof of lemma $7.1$ in [BB04] and our
previous lemma B.1. ∎

######  Theorem B.1.

For a sufficiently small disk $D_{\lambda}$ centered at $w=0$ and $U(\xi_{0})$
bounded and sufficiently small, we have a solution $g_{\xi}(w)$ of the problem
described in (B.4) that is defined on $D_{\lambda}\times U(\xi_{0})$ and
depends continuously on $\xi$. Furthermore, $g_{\xi}(w)\to 1$ uniformly in
$\xi$ as $w\to 0$.

###### Proof.

We will follow mostly the same argument as in the proof of lemma $7.1$ of
[BB04]. We put it here just to emphasize the behavior in families parametrized
by $\xi$, which is not done in the aforementioned paper.

For $\lambda>0$, let $h_{\lambda}$ be the homothety $h_{\lambda}(w)=\lambda
w$, and let
$\varphi(w,\xi)=\text{exp}((\frac{\overline{\xi}}{2w^{2}}-\frac{\xi}{2\overline{w}^{2}})H)$.
Furthermore, we denote
$c(w,\xi)=-a_{\text{reg}}^{0,1}(w)-\xi\theta_{\text{reg}}^{\dagger_{h}}(w)$.

If we write $g_{\xi}(w)=1+u(w,\xi)$, the problem (B.4) that we are trying to
solve can be rephrased as the problem of finding $u(w,\xi)$ such that

$\begin{split}\overline{\partial}_{0}(u(w,\xi))&=[\xi
H\frac{d\overline{w}}{\overline{w}^{3}},u(w,\xi)]+c(w,\xi)(1+u(w,\xi))\\\
u(0,\xi)&=0\\\ u_{0}\in C^{0}_{\delta^{\prime}},&\;\;\;u_{3}\in
C^{0}_{2+\delta^{\prime}}\;\;\text{for}\;\;\delta^{\prime}>0\;\;\text{as
before}.\\\ \end{split}$ (B.15)

The last two equations can be satisfied if $u\in U_{\delta^{\prime}}$. On the
other hand, as in the proof of lemma $7.1$ of [BB04], to solve the first
equation is enough to find a fixed point of the map
$\widetilde{T}_{\xi}:U_{\delta^{\prime}}\to U_{\delta^{\prime}}$ given by:

$\widetilde{T}_{\xi}(u)=h^{*}_{\lambda}(\varphi)\cdot
T_{\xi}((h^{*}_{\lambda}(\varphi))^{-1}\cdot[h^{*}_{\lambda}(c(\xi))(1+u)])$
(B.16)

where $``\cdot"$ denotes the action by conjugation.

Indeed, we have that if $v$ is a fixed point, then

$\begin{split}\overline{\partial}_{0}(v)=\overline{\partial}_{0}(\widetilde{T}_{\xi}(v))=\Big{[}\xi
H\frac{d\overline{w}}{\lambda^{2}\overline{w}^{3}},v\Big{]}+h^{*}_{\lambda}(c)(1+v)\end{split}$
(B.17)

Now since $\overline{\partial}_{0}$ is invariant under rescaling, if we put
$\widetilde{w}=\lambda w$, then
$u(\widetilde{w},\xi)=v(\widetilde{w}/\lambda,\xi)$ satisfies:

$\overline{\partial}_{0}(u(\widetilde{w},\xi))=\Big{[}\xi
H\frac{d\overline{\widetilde{w}}}{\overline{\widetilde{w}}^{3}},u(\widetilde{w},\xi)\Big{]}+c(\widetilde{w},\xi)(1+u(\widetilde{w},\xi))$
(B.18)

and hence $1+u$ is the solution we seek.

Now to find the fixed point, we need to show that $\widetilde{T}_{\xi}$ is a
contraction. We have:

$|\widetilde{T}_{\xi}(u)-\widetilde{T}_{\xi}(v)|_{U_{\delta}}\leq
C|u-v|_{U_{\delta}}|h^{*}_{\lambda}(c(\xi))|_{A_{\delta}}\leq
C|u-v|_{U_{\delta}}(|h^{*}_{\lambda}(a_{\text{reg}})^{0,1}|_{A_{\delta}}+D|h^{*}_{\lambda}(\theta_{\text{reg}}^{\dagger_{h}})|_{A_{\delta}})$
(B.19)

where $C$ does not depend on $\xi$; and since $U(\xi_{0})$ is bounded, we have
the last bound with $D$ independent of $\xi\in U(\xi_{0})$.

Furthermore by the proof of lemma $7.1$ of [BB04], we have that
$|h^{*}_{\lambda}(a_{\text{reg}}^{0,1})|_{A_{\delta}}=\lambda^{\delta}|a_{\text{reg}}^{0,1}|_{A_{\delta}}$
and
$|h^{*}_{\lambda}(\theta^{\dagger_{h}}_{\text{reg}})|_{A_{\delta}}=\lambda^{\delta}|\theta^{\dagger_{h}}_{\text{reg}}|_{A_{\delta}}$,
so that

$|\widetilde{T}_{\xi}(u)-\widetilde{T}_{\xi}(v)|_{U_{\delta}}\leq\lambda^{\delta}C|u-v|_{U_{\delta}}(|a_{\text{reg}}^{0,1}|_{A_{\delta}}+D|\theta_{\text{reg}}^{\dagger_{h}}|_{A_{\delta}})$
(B.20)

hence, for $\lambda$ small enough $\widetilde{T}_{\xi}$ becomes a contraction,
so we can find a fixed point.

Since $\widetilde{T}_{\xi}(u)$ is a continuous function of $w$ and $\xi$, and
the rightmost term in the last inequality does not depend on $\xi$, we
actually get that the fixed point must be a continuous function of both
variables defined on $D_{\lambda}\times U(\xi_{0})$.

Finally, by shrinking $U(\xi_{0})$ and $D_{\lambda}$ if necessary, we have
that our solution $g_{\xi}(w)=1+u(w,\xi)$ to problem (B.4) is uniformly
continuous on $D_{\lambda}\times U(\xi_{0})$. Hence, given $\epsilon>0$, we
can find $\delta_{1},\delta_{2}>0$ such that
$|g_{\xi}(w)-g_{\xi^{\prime}}(w^{\prime})|<\epsilon$ as long as
$|w-w^{\prime}|<\delta_{1}$ and $|\xi-\xi^{\prime}|<\delta_{2}$. In
particular, if $|w|<\delta_{1}$, we have that

$|g_{\xi}(w)-1|=|g_{\xi}(w)-g_{\xi}(0)|<\epsilon\;\;\text{for all}\;\;\xi\in
U(\xi_{0})$ (B.21)

so $g_{\xi}(w)\to 1$ as $w\to 0$ uniformly in $\xi\in U(\xi_{0})$.

∎

##  Appendix C Proof of proposition 3.10

Here we prove the first case of proposition 3.10, where
$\text{Re}(\lambda_{1}(t,\xi))>\text{Re}(\lambda_{2}(t,\xi))$, since the other
case is similar. We will use the following notation:

$\begin{split}\lambda_{i}(t,\xi)&=-\xi^{-1}\gamma^{*}\theta_{ii}-\xi\gamma^{*}\theta^{\dagger_{h}}_{ii}\\\
\lambda_{ij}(t,\xi)&=\lambda_{i}(t,\xi)-\lambda_{j}(t,\xi)\\\
R&=-\gamma^{*}A-\xi\gamma^{*}\theta^{\dagger_{h}}_{od}\\\ \end{split}$ (C.1)

where $\theta^{\dagger_{h}}_{\text{od}}$ denotes the off-diagonal part of
$\theta^{\dagger_{h}}$. If $\gamma^{*}A_{d}$ denotes the diagonal part of
$\gamma^{*}A$, then after doing a diagonal gauge transformation of the form
$(\eta_{1},\eta_{2})\to(\eta_{1},\eta_{2})\cdot\text{exp}(-\int_{a}^{t}\gamma^{*}A_{d})$,
we can gauge away the diagonal part of $\gamma^{*}A$, so we will assume from
the beginning that we are in this gauge. We then have that $R$ only has off-
diagonal elements, while $\lambda_{i}(t,\xi)$ is still the same as before.

The first thing we want to show is that there is a continuous solution to the
following integral equation:

$y_{i}(t,\xi)=\delta_{i2}\text{exp}\Big{(}\int_{a}^{t}\lambda_{2}(\tau,\xi)d\tau\Big{)}-\int_{t}^{\infty}\text{exp}\Big{(}\int_{\tau}^{t}\lambda_{i}(s,\xi)ds\Big{)}R_{ij}(\tau,\xi)y_{j}(\tau,\xi)d\tau\;\;\text{for}\;\;i,j=1,2\;\;i\neq
j$ (C.2)

on any interval $(a_{0},\infty)$, as long as $\xi$ is restricted to lie in a
small enough neighborhood of $0$. It is easy to check that a solution of C.2,
gives a solution to the original flatness equation in our chosen gauge. More
explicitly, if we denote $M(t)=\text{exp}(-\int_{a}^{t}\gamma^{*}A_{d})$, then
in the frame
$(\widetilde{\eta}_{1},\widetilde{\eta}_{2})=(\eta_{1},\eta_{2})\cdot M$ we
have a solution of the form
$s(t,\xi)=y_{1}(t,\xi)\widetilde{\eta}_{1}+y_{2}(t,\xi)\widetilde{\eta}_{2}$.

If we perform the change:

$y_{i}(t,\xi)=z_{i}(t,\xi)\text{exp}\Big{(}\int_{a}^{t}\lambda_{2}(\tau,\xi)d\tau\Big{)}$
(C.3)

we obtain the following integral equation for the $z_{i}(t,\xi)$:

$z_{i}(t,\xi)=\delta_{i2}-\int_{t}^{\infty}\text{exp}\Big{(}\int_{\tau}^{t}\lambda_{i2}(s,\xi)ds\Big{)}R_{ij}(\tau,\xi)z_{j}(\tau,\xi)d\tau\;\;\text{for}\;\;i,j=1,2\;\;i\neq
j$ (C.4)

If we use the integral relation twice and change the order of integration we
get the following integral equations for the $z_{i}(t,\xi)$:

$\begin{split}z_{1}(t,\xi)&=-\int_{t}^{\infty}\text{exp}\Big{(}\int_{t}^{\tau}\lambda_{21}\Big{)}R_{12}(\tau)d\tau+\int_{t}^{\infty}\Big{(}\int_{t}^{s}\text{exp}\Big{(}\int_{t}^{\tau}\lambda_{21}\Big{)}R_{12}(\tau)d\tau\Big{)}R_{21}(s)z_{1}(s)ds\\\
&:=\delta(t,\xi)+\int_{t}^{\infty}\epsilon_{1}(t,s,\xi)z_{1}(s,\xi)ds\\\
z_{2}(t,\xi)&=1+\int_{t}^{\infty}\Big{(}\int_{t}^{s}R_{21}(\tau)\text{exp}\Big{(}\int_{\tau}^{s}\lambda_{21}\Big{)}d\tau\Big{)}R_{12}(s)z_{2}(s)ds\\\
&:=1+\int_{t}^{\infty}\epsilon_{2}(t,s,\xi)z_{2}(s,\xi)ds\\\ \end{split}$
(C.5)

where we have defined

$\begin{split}\delta(t,\xi)&:=-\int_{t}^{\infty}\text{exp}\Big{(}\int_{t}^{\tau}\lambda_{21}\Big{)}R_{12}(\tau)d\tau\\\
\epsilon_{1}(t,s,\xi)&:=\Big{(}\int_{t}^{s}\text{exp}\Big{(}\int_{t}^{\tau}\lambda_{21}\Big{)}R_{12}(\tau)d\tau\Big{)}R_{21}(s)\\\
\epsilon_{2}(t,s,\xi)&:=\Big{(}\int_{t}^{s}\text{exp}\Big{(}\int_{\tau}^{s}\lambda_{21}\Big{)}R_{21}(\tau)d\tau\Big{)}R_{12}(s)\\\
\end{split}$ (C.6)

Before showing that there is a solution $z_{i}(t,\xi)$ for the integral
equations obtained above, we will say a few things about the functions
$\delta(t,\xi)$ and $\epsilon_{i}(t,s,\xi)$. We will use the following
notation:
$\lambda_{21}(t,\xi)=\xi^{-1}\lambda_{21,\xi^{-1}}+\xi\lambda_{21,\xi}$, where
$\lambda_{21,\xi^{-1}}=\gamma^{*}(-\theta_{22}+\theta_{11})$ and
$\lambda_{21,\xi}=\gamma^{*}(-\theta^{\dagger_{h}}_{22}+\theta^{\dagger_{h}}_{11})$.

Notice that by our choice of gauge and WKB path, the term
$\lambda_{21,\xi^{-1}}$ is constant. Hence, after integration by parts, we
find the following expression for $\delta(t,\xi)$:

$\delta(t,\xi)=\frac{\xi}{{\lambda}_{21,\xi^{-1}}}R_{12}(t,\xi)+\frac{\xi}{\lambda_{21,\xi^{-1}}}\int_{t}^{\infty}\text{exp}\Big{(}\int_{t}^{\tau}\xi^{-1}\lambda_{21,\xi^{-1}}ds\Big{)}\frac{d}{d\tau}\Big{(}\text{exp}\Big{(}\int_{t}^{\tau}\xi\lambda_{21,\xi}(s)ds\Big{)}R_{12}(\tau,\xi)\Big{)}d\tau$
(C.7)

where we used the fact that the elements of $R$ go to $0$ (exponentially fast)
as $t\to\infty$. Hence, using this exponential decrease of the terms of $R$,
we conclude that $|\delta(t,\xi)|\leq|\xi|f(t,\xi)$, where $f(t,\xi)\to 0$ as
$t\to\infty$ uniformly in $\xi$ for small enough $\xi$ (restricted to the
corresponding half-plane $\mathbb{H}_{m}$). Furthermore, for fixed $t$, we
have $\delta(t,\xi)\to 0$ as $\xi\to 0$ with $\xi\in\mathbb{H}_{m}$.

Similarly, we have the following expressions for $\epsilon_{i}$ after
integration by parts:

$\begin{split}\epsilon_{1}(t,s,\xi)&=\xi\text{exp}\Big{(}\int_{t}^{\tau}\lambda_{21}\Big{)}\frac{R_{12}(\tau,\xi)}{\lambda_{21,\xi^{-1}}}\Bigg{|}_{\tau=t}^{\tau=s}R_{21}(s,\xi)\\\
&-\xi\frac{R_{21}(s,\xi)}{\lambda_{21,\xi^{-1}}}\int_{t}^{s}\text{exp}\Big{(}\int_{t}^{\tau}\xi^{-1}\lambda_{21,\xi^{-1}}ds\Big{)}\frac{d}{d\tau}\Big{(}\text{exp}\Big{(}\int_{t}^{\tau}\xi\lambda_{21,\xi}(s)ds\Big{)}R_{12}(\tau,\xi)\Big{)}d\tau\\\
\epsilon_{2}(t,s,\xi)&=-\xi\text{exp}\Big{(}\int_{\tau}^{s}\lambda_{21}\Big{)}\frac{R_{21}(\tau,\xi)}{\lambda_{21,\xi^{-1}}}\Bigg{|}_{\tau=t}^{\tau=s}R_{12}(s,\xi)\\\
&+\xi\frac{R_{12}(s,\xi)}{\lambda_{21,\xi^{-1}}}\int_{t}^{s}\text{exp}\Big{(}\int_{\tau}^{s}\xi^{-1}\lambda_{21,\xi^{-1}}ds\Big{)}\frac{d}{d\tau}\Big{(}\text{exp}\Big{(}\int_{\tau}^{s}\xi\lambda_{21,\xi}(s)ds\Big{)}R_{21}(\tau,\xi)\Big{)}d\tau\\\
\end{split}$ (C.8)

In the expressions above we always assume $t\leq s$, since this is the range
of interest for the problem. We can conclude that
$|\epsilon_{i}(t,s,\xi)|\leq|\xi|g_{i}(t,\xi)$, where $g_{i}(t,\xi)\to 0$ as
$t\to\infty$ uniformly in $\xi$ for small enough $\xi$. For this statement we
use again the fact that the components of $R$ go to $0$ exponentially fast as
$t\to\infty$. Furthermore, for fixed $t$, we have $\epsilon_{i}(t,\xi)\to 0$
as $\xi\to 0$ with $\xi\in\mathbb{H}_{m}$.

Now we are ready to prove that the integral equation $(\ref{IE3})$ has a
continuous solution on the interval $[a_{0},\infty)$.

Because of the expressions given above for $\epsilon_{i}$, by restricting the
$\xi\in\mathbb{H}_{m}$ to lie in a small neighborhood $U_{0}$ of $\xi=0$, we
can ensure that for some $c\in(0,1)$:

$\int_{t}^{\infty}|\epsilon_{i}(t,s,\xi)|ds<c<1\;\;\text{for
all}\;\;t\in[a_{0},\infty)\;\;\text{and}\;\;\xi\in U_{0}$ (C.9)

Now let $z^{(0)}_{2}(t,\xi):=1$,
$z^{(0)}_{1}(t,\xi):=\delta(t,\xi)=-\int_{t}^{\infty}\text{exp}(\int_{t}^{\tau}\lambda_{21})R_{12}(\tau,\xi)d\tau$.
We define $z^{(m)}_{i}$ recursively by plugging $z^{(m-1)}_{i}$ into the right
side of the integral equation (C.5); at each step we get a continuous and
bounded function for $(t,\xi)\in[a_{0},\infty)\times U_{0}$.

For some $M>0$, we clearly have $|z^{(0)}_{i}|_{C^{0}}<M$, where
$|\;\;|_{C^{0}}$ denotes the uniform norm on $C^{0}([t_{0},\infty)\times
U_{0})$. Furthermore, assume inductively that we have shown that
$|z^{(m)}_{i}-z^{(m-1)}_{i}|_{C^{0}}<c^{m}M$. We then have that:

$|z^{(m+1)}_{i}-z^{(m)}_{i}|_{C^{0}}<\Big{|}\int_{t}^{\infty}|\epsilon_{i}(t,s,\xi)|ds\Big{|}_{C^{0}}|z^{(m)}_{i}-z^{(m-1)}_{i}|_{C^{0}}<c^{m+1}M$
(C.10)

From this we see that $z_{i}^{(m)}$ converges uniformly in
$(t,\xi)\in[a_{0},\infty)\times U_{0}$ to a function $z_{i}\in
C^{0}([a_{0},\infty)\times U_{0})\cap L^{\infty}([a_{0},\infty)\times U_{0})$.
The $z_{i}$ clearly satisfy the integral equation $(\ref{IE3})$. We claim that
it is also a solution of the integral equation (C.4).

We are trying to find a solution of an integral equation of the form:

$z=e+T(z)$ (C.11)

where $e_{i}=\delta_{i2}$ and $T$ is the linear integral operator part of the
integral equation (C.4). We have found a solution to the integral equation
$z=e+T(e)+T^{2}(z)$. This last solution must be unique, since if $z$ and
$z^{\prime}$ are solutions, then
$|z-z^{\prime}|_{C^{0}}=|T^{2}(z-z^{\prime})|_{C^{0}}<\delta|z-z^{\prime}|_{C^{0}}$,
which implies that $z=z^{\prime}$. Now notice that:

$e+T(e)+T^{2}(e+T(z))=e+T(e+T(e)+T^{2}(z))=e+T(z)$ (C.12)

so $e+T(z)$ is also a solution of (C.5). By the uniqueness, we conclude that
$z=e+T(z)$, so that $z$ also solves (C.4).

With the same notation from above, notice that (C.5), (C.7) and (C.8) allow us
to conclude that for fixed $t\in[a_{0},\infty)$ we have $T(z)\to 0$ as $\xi\to
0$; and furthermore, we have that $T(z)\to 0$ as $t\to\infty$ uniformly in
$\xi\in U_{0}$.

Finally, recalling the statement and notations of proposition 3.10; setting
$E_{1}(t,\xi):=M_{11}(t)z_{1}(t,\xi)$ and
$E_{2}(t,\xi):=M_{22}(t)(z_{2}(t,\xi)-1)$; and using that $M(t)$ is bounded in
$t\in[a_{0},\infty)$ (see the remark related to equation (3.97)), we conclude
the result of proposition 3.10.

##  Appendix D Proof of lemma 4.2

In this appendix we prove the following lemma:

######  Lemma D.1.

Let $m\in\mathbb{C}$ and $m^{(3)}\in(-1/2,1/2]\subset\mathbb{R}$. Then up to
equivalence, there is a unique polystable filtered Higgs bundle
$(\mathcal{E}_{*},\theta)\to(\mathbb{C}P^{1},\infty)$ with
$\textnormal{Tr}(\theta)=0$, $\textnormal{Det}(\theta)=-(z^{2}+2m)dz^{2}$,
$\textnormal{pdeg}(\mathcal{E}_{*})=0$, and parabolic weights determined by
$m^{(3)}$ as follows:

  * •

if $m^{(3)}\in(-1/2,1/2)$, then for the eigenline decomposition near $\infty$
of the induced $1/2$-parabolic Higgs bundle $(\mathcal{E}_{1/2},\theta)$, we
have that $\pm m^{(3)}$ is the weight associated to the line corresponding to
the eigenvalue $\pm(z+m/z+...)$.

  * •

if $m^{(3)}=1/2$ then the parabolic structure of the induced $1/2$-parabolic
structure $(\mathcal{E}_{1/2},\theta)$ is the trivial filtration with weight
$1/2$.

Before we give the proof we will need some notation and another lemma.

Recall that a compatibly framed connection
$(E,\nabla,\tau)\to(\mathbb{C}P^{1},\infty)$ determines uniquely a formal type
$(Q,\Lambda)$ (recall definition 3.13 and lemma 3.4). If we denote by
$U_{\pm}\subset GL(2,\mathbb{C})$ the upper (resp. lower) unipotent matrices,
and we fix a formal type $(Q,\Lambda)$, we will denote

$\mathcal{S}(Q,\Lambda):=\\{(S_{1},S_{2},S_{3},S_{4})\in(U_{-}\times
U_{+})^{2}\;\;|\;\;\text{Stokes matrices of}\;(E,\nabla,\tau)\;\text{with
formal type}\;(Q,\Lambda)\\}$ (D.1)

Now if $(E,\nabla,\tau)\to(\mathbb{C}P^{1},\infty)$ has formal type
$(Q,\Lambda)$, notice that if $T\subset GL(2,\mathbb{C})$ denotes the set of
diagonal matrices, then the set of possible compatible frames $\tau^{\prime}$
for $(E,\nabla,\tau^{\prime})\to(\mathbb{C}P^{1},\infty)$ with formal type
$(Q,\Lambda)$ is a $T$-torsor, where $t\in T$ acts on the framing in the
obvious way. Furthermore, if $(E,\nabla,\tau)\to(\mathbb{C}P^{1},\infty)$ has
formal type $(Q,\Lambda)$ and we act on $\tau$ by $t\in T$, then the
corresponding Stokes matrices in $\mathcal{S}(Q,\Lambda)$ get conjugated by
$t$. Hence, we get a $T$-action on $\mathcal{S}(Q,\Lambda)$, and we will
denote the orbits by $\mathcal{S}(Q,\Lambda)/T$.

We then have the following lemma:

######  Lemma D.2.

For any $m\in\mathbb{C}$ and $m^{(3)}\in(-1/2,1/2]$, consider
$[(E_{i},\overline{\partial}_{E_{i}},\theta_{i},h_{i},g_{i})]\in\mathfrak{X}^{\textnormal{fr}}(m,m^{(3)})$
for $i=1,2$. Furthermore, let
$(\mathcal{P}^{h}_{c}\mathcal{E}^{\xi}_{i},\nabla^{\xi}_{i},\tau_{c,i}^{\xi})\to(\mathbb{C}P^{1},\infty)$
be the associated compatibly framed $c$-parabolic bundles for some fixed
$\xi\in\mathbb{C}^{*}$ and some fixed $c\in\mathbb{R}$. If we denote
$Q(\xi):=\frac{1+|\xi|^{2}}{\xi}\frac{H}{2w^{2}}$ and let $\Lambda(\xi,c)$ be
as in equation (3.35), then:

  * •

The
$(\mathcal{P}^{h}_{c}\mathcal{E}^{\xi}_{i},\nabla^{\xi}_{i},\tau_{c,i}^{\xi})\to(\mathbb{C}P^{1},\infty)$
for $i=1,2$ have Stokes data in $\mathcal{S}(Q(\xi),\Lambda(\xi,a))$.

  * •

The
$(\mathcal{P}^{h}_{c}\mathcal{E}^{\xi}_{i},\nabla^{\xi}_{i},\tau_{c,i}^{\xi})\to(\mathbb{C}P^{1},\infty)$
for $i=1,2$ are isomorphic as compatibly framed $c$-parabolic flat bundles if
and only they have the same Stokes matrices.

  * •

The
$(\mathcal{P}^{h}_{c}\mathcal{E}^{\xi}_{i},\nabla^{\xi}_{i})\to(\mathbb{C}P^{1},\infty)$
for $i=1,2$ are isomorphic as $c$-parabolic flat bundles if and only if, after
taking some compatible framings specifying the formal type
$(Q(\xi),\Lambda(\xi,a))$, their Stokes data lies in the same $T$-orbit of
$\mathcal{S}(Q(\xi),\Lambda(\xi,c))$.

###### Proof.

The fact that the Stokes matrices of
$(\mathcal{P}^{h}_{c}\mathcal{E}^{\xi}_{i},\nabla^{\xi}_{i},\tau_{c,i}^{\xi})$
lie in $\mathcal{S}(Q(\xi),\Lambda(\xi,c))$ follows from equation (3.34).

Now suppose that
$(\mathcal{P}^{h}_{c}\mathcal{E}^{\xi}_{i},\nabla^{\xi}_{i},\tau_{c,i}^{\xi})$
have the same Stokes matrices, which we denote by $S_{j}$ for $j=1,2,3,4$,
following the conventions of section 3.4.4. Let $\Phi_{j,i}$ denote the
corresponding sectorial frames of flat sections for
$(\mathcal{P}^{h}_{c}\mathcal{E}^{\xi}_{i},\nabla^{\xi}_{i},\tau_{c,i}^{\xi})\to(\mathbb{C}P^{1},\infty)$
defined on $\widehat{\text{Sect}}_{j}(\xi)$, and define
$T_{j}:\mathcal{P}^{h}_{c}\mathcal{E}^{\xi}_{1}|_{\widehat{\text{Sect}}_{j}(\xi)}\to\mathcal{P}^{h}_{c}\mathcal{E}^{\xi}_{2}|_{\widehat{\text{Sect}}_{j}(\xi)}$
by $T_{j}(\Phi_{j,1})=\Phi_{j,2}$. The fact that the Stokes matrices and the
formal monodromy are the same implies that the $T_{j}$ glue into a covariantly
constant morphism $T$ over a punctured neighborhood $U_{\infty}^{*}$ of
$\infty$, and by parallel transport, we get a covariantly constant morphism
$T:\mathcal{P}^{h}_{c}\mathcal{E}^{\xi}_{1}|_{\mathbb{C}P^{1}-\\{\infty\\}}\to\mathcal{P}^{h}_{c}\mathcal{E}^{\xi}_{2}|_{\mathbb{C}P^{1}-\\{\infty\\}}$.
Furthermore, by the expressions
$\Phi_{j,i}=\tau_{a,i}^{\xi}\cdot\Sigma_{j}(\widehat{F}_{i})w^{-\Lambda(\xi)}e^{-Q(\xi)}$
(where we abuse notation and denote by $\tau_{a,i}^{\xi}$ any local extension
of the framing at $\infty$ given by $\tau_{a,i}^{\xi}$) we see that for any
$j=1,2,3,4$

$T(\tau_{c,1}^{\xi})=\tau_{c,2}^{\xi}\cdot\Sigma_{j}(\widehat{F}_{2})\Sigma_{j}(\widehat{F}_{1})^{-1}\;\;\text{on}\;\;\widehat{\text{Sect}}_{j}(\xi)$
(D.2)

so the $\Sigma_{j}(\widehat{F}_{2})\Sigma_{j}(\widehat{F}_{1})^{-1}$ glue
together in $U_{\infty}^{*}$. Since $\Sigma_{j}(\widehat{F}_{i})\to 1$ as
$w\to 0$ for all $j=1,2,3,4$ and $i=1,2$, we see that $T$ extends over the
puncture to a morphism satisfying
$T(\tau_{c,1}^{\xi}|_{\infty})=\tau_{c,2}^{\xi}|_{\infty}$. Finally, since the
parabolic structures of
$(\mathcal{P}^{h}_{c}\mathcal{E}^{\xi}_{i},\nabla^{\xi}_{i},\tau_{c,i}^{\xi})$
are compatible with their irregular decompositions (see equation (3.6)), it is
easy to check that $T$ preserves the parabolic structures. Hence, $T$ gives an
isomorphism between
$(\mathcal{P}^{h}_{c}\mathcal{E}^{\xi}_{i},\nabla^{\xi}_{i},\tau_{c,i}^{\xi})$
for $i=1,2$ as compatibly framed $c$-parabolic flat bundles. The other
implication is trivial.

Going now to the last statement, assume that after picking compatible frames,
$(\mathcal{P}^{h}_{c}\mathcal{E}^{\xi}_{i},\nabla^{\xi}_{i},\tau_{c,i}^{\xi})$
have Stokes matrices in the same $T$-orbit of
$\mathcal{S}(Q(\xi),\Lambda(\xi,c))$. Then by the previous result, for some
$t\in T$, we have that
$(\mathcal{P}^{h}_{c}\mathcal{E}^{\xi}_{1},\nabla^{\xi}_{1},\tau_{c,1}^{\xi})$
is isomorphic to
$(\mathcal{P}^{h}_{c}\mathcal{E}^{\xi}_{2},\nabla^{\xi}_{2},\tau_{c,2}^{\xi}\cdot
t)$. Hence, by the previous argument, we get that they are isomorphic as
$c$-parabolic flat bundles (after forgetting about the framing). The other
implication also follows trivially.

∎

Now we use the previous lemma to prove lemma D.1:

###### Proof.

(of lemma D.1) We divide the proof in three cases:

  * •

$m\neq 0$: We start by picking $\xi\in\mathbb{H}_{m}$. Notice that in the case
$m\neq 0$, we have that $(\mathcal{E}_{*},\theta)$ must be stable (by the same
argument given in lemma 3.1), so by the wild non-abelian Hodge correspondence
from [BB04], we get a bijective correspondence between equivalence classes of
the filtered Higgs Bundles $(\mathcal{E}_{*},\theta)$ we wish to count, and
equivalence classes of the associated filtered flat bundles
$(\mathcal{P}^{h}_{*}\mathcal{E}^{\xi},\nabla^{\xi})$ for some fixed
$\xi\in\mathbb{C}^{*}$.

By taking the associated $c$-parabolic flat bundles and applying lemma D.2, we
can then obtain an injection of the set we wish to count into the orbits
$\mathcal{S}(Q(\xi),\Lambda(\xi,c))/T$.

Now let $a(\xi)$ and $b(\xi)$ be the non-trivial off-diagonal elements of the
unipotent matrices $S_{1},S_{2}$, respectively. For $m\neq 0$ and
$\xi\in\mathbb{H}_{m}$, we know that after taking the compatible framings
$\tau_{c}^{\xi}$, $a(\xi)\neq 0$ for any of our
$(\mathcal{P}^{h}_{c}\mathcal{E}^{\xi},\nabla^{\xi})$ (see proposition 3.11).
Since all the points of $\mathcal{S}(Q(\xi),\Lambda(\xi,c))$ with $a(\xi)\neq
0$ lie in the same $T$-orbit (recall the Stokes relations in (3.53)), we
conclude that there is only one
$(\mathcal{P}^{h}_{c}\mathcal{E}^{\xi},\nabla^{\xi})$ up to equivalence, and
hence only one of the original $(\mathcal{E}_{*},\theta)$ we started with, up
to equivalence.

  * •

$m=0$, $m^{(3)}\neq 0$: Let first check that we cannot have a strictly
polystable filtered Higgs bundle in this case. If $(\mathcal{E}_{*},\theta)$
is strictly polystable, then
$(\mathcal{E}_{*},\theta)=(\mathcal{E}_{*,1},\theta_{1})\oplus(\mathcal{E}_{*,2},\theta_{2})$,
with $\text{pdeg}(\mathcal{E}_{*,1})=\text{pdeg}(\mathcal{E}_{*,2})=0$, and it
is easy to check that this cannot occur unless $m^{(3)}=0$. Hence, if
$m^{(3)}\neq 0$, all our corresponding $(\mathcal{E}_{*},\theta)$ are stable,
and by [BB04], they are in bijective correspondence with equivalence classes
of the associated flat filtered bundles
$(\mathcal{P}^{h}_{*}\mathcal{E}^{\xi},\nabla^{\xi})$ (for some fixed
$\xi\in\mathbb{C}^{*}$). Notice that in this case, the formal monodromy of all
of our elements turns out to be given by $M_{0}=\text{exp}(-2\pi
im^{(3)}H)\neq 1$, so by the Stokes relations (3.53) we conclude that
$\mathcal{S}(Q(\xi),\Lambda(\xi,c))/T$ is just a point, so we are done by
lemma D.2.

  * •

$m=m^{(3)}=0$: in this case we have $M_{0}=1$, so by the relations (3.53) we
have that $\mathcal{S}(Q(\xi),\Lambda(\xi,c))/T$ consists of $3$ points,
depending on whether $a\neq 0$ and $b=0$, $b\neq 0$ and $a=0$, or $a=b=0$.

The case $a=b=0$ corresponds to trivial Stokes data, and it is easy to check
that the filtered Higgs bundle induced from example 3.1 gives rise to this
case. Furthermore, it is also easy to check that this is the only possible
strictly polystable filtered Higgs bundle with $m=m^{(3)}=0$. Hence, the
problem reduces to showing that there are no stable filtered Higgs bundles
with $m=m^{(3)}=0$, giving rise to either the case with $a\neq 0$ and $b=0$,
or the case $a=0$ and $b\neq 0$.

Again by the Stokes relations (3.53), the case $a=0$ and $b\neq 0$ (resp.
$a\neq 0$ and $b=0$) gives rise to purely upper-triangular (resp. lower
triangular) Stokes matrices, and hence to non-stable “Stokes representations”
(see [Boa11]). By the remarks in page $50$ of [Boa11], we conclude that these
upper-triangular (resp. lower-triangular) cases cannot correspond to stable
filtered Higgs bundles with $m=m^{(3)}=0$. Hence, there is only one polystable
filtered Higgs bundle with $m=m^{(3)}=0$.

∎

##  Appendix E Constructing polystable parabolic Higgs bundles in the case
$m=0$

Here we explain how to construct polystable $0$-parabolic Higgs bundles whose
Higgs field $\theta$ satisfies $\text{Tr}(\theta)=0$ and
$\text{Det}(\theta)=-z^{2}dz^{2}$.

The case where the parabolic structure consists of the trivial filtration with
parabolic weights equal to $0$ is already explained in example 3.1. In this
case, the parabolic Higgs bundle that we find is polystable.

The next proposition deals with the rest of the possible parabolic structures:

######  Proposition E.1.

Let $m^{(3)}\in(-1,0)$. There is a stable 0-parabolic Higgs bundle
$(E^{m^{(3)}},\theta)\to(\mathbb{C}P^{1},\infty)$ with
$\textnormal{pdeg}(E^{m^{(3)}})=0$, parabolic weights specified by $m^{(3)}$
and $-1-m^{(3)}$, $\textnormal{Tr}(\theta)=0$, and
$\textnormal{Det}(\theta)=-z^{2}dz^{2}$.

###### Proof.

We start by considering
$E=\mathcal{O}\oplus\mathcal{O}(-1)\to\mathbb{C}P^{1}$. We denote by $e_{1}$
and $e_{2}$ the usual frames over $\mathbb{C}\subset\mathbb{C}P^{1}$ of
$\mathcal{O}$ and $\mathcal{O}(-1)$ respectively. We have that $e_{1}$ gives a
global trivilization of $\mathcal{O}$, while the usual frame of
$\mathcal{O}(-1)$ over $\mathbb{C}P^{1}-\\{0\\}$ will be denoted by $f_{2}$.
Hence, if $z$ denotes the coordinate of $\mathbb{C}\subset\mathbb{C}P^{1}$,
then $z^{-1}e_{2}=f_{2}$.

In the frame $(e_{1},e_{2})$ over $\mathbb{C}\subset\mathbb{C}P^{1}$ we
define:

$\theta=\begin{bmatrix}z&0\\\ 2&-z\\\ \end{bmatrix}dz$ (E.1)

Over $\mathbb{C}$, we can write eigenvectors corresponding to the eigenvalues
$z$ and $-z$ respectively by $v_{z}=ze_{1}+e_{2}$ and $v_{-z}=e_{2}$. On the
other hand, over $\mathbb{C}P^{1}-\\{z=0\\}$ we can write
$\widetilde{v}_{z}=e_{1}+f_{2}$ and $\widetilde{v}_{-z}=f_{2}$. These are
eigenvectors of $\theta$ for $z\in\mathbb{C}^{*}\subset\mathbb{C}P^{1}$, with
eigenvalues $z$ and $-z$ respectively.

We put a parabolic structure at $\infty$ by putting the weight
$m^{(3)}\in(-1,0)$ to the line generated by $e_{1}+f_{2}|_{\infty}$ and the
weight $-1-m^{(3)}\in(-1,0)$ to the line generated by $f_{2}|_{\infty}$.
Denote $E$ with this parabolic structure by $E^{m^{(3)}}$.

Notice that with this parabolic structure, we have that

$\text{pdeg}(E^{m^{(3)}})=\text{deg}(E)-m^{(3)}-(-1-m^{(3)})=0$ (E.2)

We claim that $(E^{m^{(3)}},\theta)\to(\mathbb{C}P^{1},\infty)$ is a stable
$0$-parabolic Higgs bundle.

To check this, notice that on $\mathbb{C}^{*}\subset\mathbb{C}P^{1}$ we have
the following relations:

$\begin{split}z^{-1}v_{z}&=\widetilde{v}_{z}\\\
z^{-1}v_{-z}&=\widetilde{v}_{-z}\\\ \end{split}$ (E.3)

Hence, $v_{z}$ and $\widetilde{v}_{z}$ define a holomorphic line
$L_{z}\cong\mathcal{O}(-1)$ while $v_{-z}$ and $\widetilde{v}_{-z}$ define
$L_{-z}\cong\mathcal{O}(-1)$ ($L_{-z}$ is the same $\mathcal{O}(-1)$ summand
from the splitting in the definition of $E$). These line bundles are of course
preserved by $\theta$, and they are the only line bundles that can be
preserved by $\theta$.

We denote by $L^{m^{(3)}}_{\pm z}$ the line bundles $L_{\pm z}$ with the
induced parabolic structure from $E^{m^{(3)}}$. We then have that

$\begin{split}\text{pdeg}(L^{m^{(3)}}_{z})&=\text{deg}(L_{z})-m^{(3)}=-1-m^{(3)}<0\\\
\text{pdeg}(L^{m^{(3)}}_{-z})&=\text{deg}(L_{-z})-(-1-m^{(3)})=m^{(3)}<0\\\
\end{split}$ (E.4)

so we conclude that $(E^{m^{(3)}},\theta)$ is a stable 0-parabolic Higgs
bundle with parabolic degree 0.
∎

Finally, recalling that the role of $m^{(3)}$ in determining the filtered
structure of the Higgs bundles is periodic mod $1$, we conclude:

######  Corollary E.1.

For every $m^{(3)}\in(-1/2,1/2]$, the set
$\mathfrak{X}^{\text{fr}}(0,m^{(3)})$ is not empty.

###### Proof.

The case with $m^{(3)}=0$ is true by example 3.1. For the rest of the cases,
consider the filtered Higgs bundles associated to the $0$-parabolic Higgs
bundles constructed in proposition E.1. By applying theorem 3.3 we obtain an
adapted harmonic metric for them. Then after applying the construction of
proposition 3.2 we obtain elements of $\mathcal{H}^{\text{fr}}$ that define
equivalence classes on $\mathfrak{X}^{\text{fr}}(0,m^{(3)})$ for
$m^{(3)}\in(-1/2,1/2]-\\{0\\}$. ∎

## References

  * [BB04] O. Biquard and P. Boalch,  Wild non-abelian Hodge theory on curves,  Compositio Mathematica  140(1), 179–204 (2004).
  * [Boa01a] P. Boalch,  G-bundles, isomonodromy and quantum Weyl groups,  (2001), arXiv:math/0108152.
  * [Boa01b] P. Boalch,  Symplectic Manifolds and Isomonodromic Deformations,  Advances in Mathematics  163, 137–205 (11 2001).
  * [Boa11] P. P. Boalch,  Geometry and braiding of Stokes data; Fission and wild character varieties,  (2011), arXiv:1111.6228.
  * [DN18] D. Dumas and A. Neitzke,  Asymptotics of Hitchin’s metric on the Hitchin section,  To appear in Comm. Math. Phys (2018), arXiv:1802.07200.
  * [Fre18] L. Fredrickson,  Exponential Decay for the Asymptotic Geometry of the Hitchin Metric,  (2018), arXiv:1810.01554.
  * [GMN10] D. Gaiotto, G. W. Moore and A. Neitzke,  Four-dimensional wall-crossing via three-dimensional field theory,  Commun. Math. Phys. 299:163-224 (2010), arXiv:0807.4723.
  * [GMN13] D. Gaiotto, G. W. Moore and A. Neitzke,  Wall-crossing, Hitchin Systems, and the WKB Approximation,  Adv. Math. (2013), arXiv:0907.3987.
  * [GW00] M. Gross and P. M. H. Wilson,  Large Complex Structure Limits of K3 Surfaces,  J. Differential Geom.  55(3), 475–546 (07 2000).
  * [HKLR87] N. J. Hitchin, A. Karlhede, U. Lindström and M. Rocek,  Hyperkähler metrics and supersymmetry,  Communications in Mathematical Physics  108, 535–589 (1987).
  * [IV10] A. I. Vagabov,  On the asymptotics with respect to a parameter of solutions of differential systems with coefficients in the class Lq,  Differential Equations  46, 17–23 (03 2010).
  * [Kos70] B. Kostant,  Quantization and unitary representations,  Lectures in Modern Analysis and Applications III. Lecture Notes in Mathematics, vol 170. Springer, Berlin, Heidelberg, 1970.
  * [Lev48] N. Levinson,  The asymptotic nature of solutions of linear systems of differential equations,  Duke Math. J.  15(1), 111–126 (03 1948).
  * [Moc08] T. Mochizuki,  Wild Harmonic Bundles and Wild Pure Twistor D-modules,  Asterisque- Societe Mathematique de France (03 2008).
  * [Moc11] T. Mochizuki,  The Stokes Structure of a good meromorphic flat bundle,  Journal of the Institute of Mathematics of Jussieu  10(3), 675–712 (2011).
  * [Moc15] T. Mochizuki,  Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces,  Journal of Topology  9 (08 2015).
  * [Moc19] T. Mochizuki,  Good wild harmonic bundles and good filtered Higgs bundles,  (2019), arXiv:1902.08298.
  * [MSWW17] R. Mazzeo, J. Swoboda, H. Weiss and F. Witt,  Asymptotic Geometry of the Hitchin Metric,  Comm. Math. Phys. 367 (2019), no. 1, 151-191 (2017), arXiv:1709.03433.
  * [Nei] A. Neitzke,  Moduli spaces of Higgs bundles class notes,  https://web.ma.utexas.edu/users/neitzke/teaching/392C-higgs-bundles/higgs-bundles.pdf.
  * [Nei13] A. Neitzke,  Notes on a new construction of hyperkahler metrics,  In proceedings of Mirror Symmetry and Tropical Geometry, Cetraro 2011 (2013), arXiv:1308.2198.
  * [OV96] H. Ooguri and C. Vafa,  Summing up D instantons,  Phys. Rev. Lett.  77, 3296–3298 (1996), hep-th/9608079.
  * [Sim90] C. T. Simpson,  Harmonic Bundles on Noncompact Curves,  Journal of the American Mathematical Society  3(3), 713–770 (1990).
  * [Was87] W. Wasow,  Asymptotic expansions of ODE’s,  Dover, New York City, 1987.
  * [Wit07] E. Witten,  Gauge Theory And Wild Ramification,  (2007), arXiv:0710.0631.