# Asymptotic geometry of non-abelian Hodge theory and Riemann–Hilbert
correspondence, rank three $\widetilde{E}_{6}$ case

by Miklós Eper111Department of Algebra and Geometry, Institute of Mathematics,
Faculty of Natural Sciences, Budapest University of Technology and Economics,
Műegyetem rkp. 3., Budapest H-1111, Hungary, e-mail<EMAIL_ADDRESS>and
Szilárd Szabó222Department of Algebra and Geometry, Institute of Mathematics,
Faculty of Natural Sciences, Budapest University of Technology and Economics,
Műegyetem rkp. 3., Budapest H-1111, Hungary; Alfréd Rényi Institute of
Mathematics, Reáltanoda utca 13-15., Budapest 1053, Hungary,
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

We prove the Geometric P=W conjecture in rank 3 on the three-punctured sphere.
We describe the topology at infinity of the related character variety. We use
asymptotic abelianization of harmonic bundles away from the ramification
divisor and an equivariant approach near the branch points to find the WKB
(also known as Liouville–Green or phase-integral) expansion of the involved
maps. We analyze the Stokes phenomenon governing their behavior.

##  1 Introduction and statement of the main result

The main characters of this paper are, on the one hand, the moduli space
$\mathcal{M}_{\textrm{Dol}}(\alpha)$ of gauge-equivalence classes of rank $3$
Higgs bundles on the complex projective line $\mathbb{C}P^{1}$ with three
logarithmic points (called the Dolbeault moduli space) and, on the other hand,
the space $\mathcal{M}_{\textrm{B}}(\textbf{c})$ of representations of the
fundamental group of the thrice-punctured $\mathbb{C}P^{1}$ in
$\operatorname{SL}(3,\mathbb{C})$ up to overall conjugation (called the Betti
moduli space or character variety). These two spaces are special instances
corresponding to the extended root system $\widetilde{E}_{6}$ of a more
general setup, where the genus of the underlying curve and the structure group
of the objects may be arbitrary. It is known in general that the Dolbeault and
Betti moduli spaces are diffeomorphic to each other via a composition of the
non-abelian Hodge and Riemann–Hilbert correspondences. However, this
diffeomorphism is highly transcendental, and extensive literature is devoted
to understanding its properties, as well as to using it to translate known
facts about one of these moduli spaces to new statements about the other one.
For instance, it is known that these moduli spaces carry natural complex
algebraic variety structures (of dimension $2$ in the $\widetilde{E}_{6}$
case), but the diffeomorphism between them is not compatible with their
complex structures. Our goal is to investigate the asymptotic behavior of this
diffeomorphism $\psi$ at infinity in the special case $\widetilde{E}_{6}$
specified above (i.e., genus $0$, rank $3$, and with suitably chosen
parameters to be specified later), in the hope that this particular case will
shed light on some phenomena that continue to hold in the general case too. We
may summarize our main result as:

######  Theorem 1.1.

The Geometric P=W conjecture holds for rank 3 tame harmonic bundles over the
three-punctured sphere.

Along the way, we also give a self-contained proof for the following:

######  Theorem 1.2 (Proposition 3.1).

The $\operatorname{GL}(3,\mathbb{C})$ character variety of the three-punctured
sphere admits a smooth compactification by a curve of type $I_{1}$. In
particular, the body of its nerve complex is of homotopy type $S^{1}$.

We note that P. Etingof, A. Oblomkov, E. Rains have obtained essentially the
same result about the compactifying divisor of type $I_{1}$ using
representation theoretical techniques [9, Proposition 6.6], including in the
cases corresponding to the affine root systems $\widetilde{E}_{7}$ and
$\widetilde{E}_{8}$ (rather than just $\widetilde{E}_{6}$ studied here). We
now turn to describing the context and precise meaning of Theorem 1.1.

One motivation of this study comes from Hitchin’s WKB problem [23], which
roughly reads as follows: consider a $\mathbb{C}^{\times}$ orbit in the
Hitchin base and a smooth lift to the Dolbeault space. Consider the family of
flat connections corresponding to this lift, and determine the behaviour of
the associated transport matrices, as the point of the Hitchin base converges
to infinity along the $\mathbb{C}^{\times}$-orbit. For more about the WKB
approximation theory of the Schrödinger operator, see [51]. More recently,
[14, Section 13] and [36, Section 2.3] carried out WKB analysis of Hitchin’s
equations. It remains an actively studied area, related to different other
fields, e.g. resurgence [25, Section 6], Borel resummation [40, Section 5],
Teichmüller theory [42], cluster algebras [22], and so on. As we will point
out in the next paragraph, this article adds Hodge theory to the list.

Another source of inspiration is the so-called P=W conjecture. The Betti space
is known to be an affine algebraic variety, and as such its cohomology spaces
carry a mixed Hodge structure [7]. On the other hand, the Hitchin map endows
the cohomology spaces of the Dolbeault space with a perverse Leray filtration.
The conjecture, formulated by M. de Cataldo, T. Hausel and L. Migliorini [5],
states that the diffeomorphism $\psi$ between the spaces respects these
filtrations. Recently this has been an intensely investigated area, with
strong ties to other fields such as Cohomological Hall Algebras, the geometry
of the affine Springer fiber, and Donaldson–Thomas theory. In [5] it was
proved in the rank two case for compact curves. The complete proof for the P=W
conjecture came recently from two different sources, using different
techniques [18] and [33].

A geometric counterpart of the conjecture was proposed in [23, Conjecture 1.1]
and [47, Conjecture 11.1]. So far, this Geometric $P=W$ conjecture has
received significantly less attention than the original cohomological version.
Mauri, Mazzon and Stevenson [34] proved it in genus $1$ and type $A$. The
second author dealt with the conjecture in case of rank 2 Higgs bundles with
irregular singularities over $\mathbb{C}P^{1}$, belonging to the Painlevé
cases [50], and with rank 2 logarithmic Higgs bundles over the five-punctured
sphere, by establishing the WKB-analysis of coordinates of the character
variety [49]. A. Némethi and the second author provided a different proof for
these Painlevé cases using low-dimensional topology techniques [39]. As far as
the authors know, these are the only cases where the full assertion of the
Geometric P=W is proved. The Geometric $P=W$ conjecture provides an
understanding of the Hitchin map on the Betti side, away from a compact
subset. More precisely, it asserts the existence of the following homotopy
commutative diagram:

${\mathcal{M}_{\textrm{Dol}}\setminus
h^{-1}(B_{R}(0))}$${\mathcal{M}_{B}\setminus\psi(h^{-1}(B_{R}(0)))}$${\mathcal{H}\setminus
B_{R}(0)}$${\lvert\mathcal{D}\partial\mathcal{M}_{B}\rvert}$$\scriptstyle{\psi}$$\scriptstyle{H}$$\scriptstyle{S}$
(1)

where

$\psi=\operatorname{RH}\circ\operatorname{NAHC}$

is the composition of the non-abelian Hodge correspondence

$\operatorname{NAHC}\colon\mathcal{M}_{\textrm{Dol}}(\alpha)\to\mathcal{M}_{\textrm{dR}}(\beta,\tau)$

and the Riemann Hilbert correspondence

$\operatorname{RH}\colon\mathcal{M}_{\textrm{dR}}(\beta,\tau)\to\mathcal{M}_{\textrm{B}}(\textbf{c})$

and $H$ is the Hitchin fibration over the Hitchin base $\mathcal{H}$. The
Hitchin base will turn out to be a one-dimensional affine space in our case
(see Section 2.3 for more details), and $\mathcal{H}\setminus B_{R}(0)$ is a
neighbourhood of infinity in the base, where $R\gg 1$. Moreover
$\lvert\mathcal{D}\partial\mathcal{M}_{B}\rvert$ denotes the body of the dual
(or nerve) complex of the compactifying divisor of the Betti space, and $S$ is
Simpson’s natural map from a neighborhood of infinity in $\mathcal{M}_{B}$ to
the body of the nerve complex (see Section 2.1 and 6.2 for more details).

In [23] the conjecture about the existence of a commutative diagram up to
homotopy was stated in higher generality. In particular, the homotopy type of
the body of the nerve complex is expected to be always that of a sphere. This
homotopy sphere assertion was proved by C. Simpson in [47] over
$\mathbb{C}P^{1}$, with an arbitrary finite number of punctures in the rank
$2$ case. The investigation of the homotopy type of the topological space of
the dual boundary complex of the character variety is a basic step to deal
with the Geometric $P=W$ conjecture, and numerous results belong to this. A.
Komyo [26] proved the assertion for some $2$ and $4$ dimensional tame cases,
and it was generalized by C. Simpson [47], who showed that in the rank $2$
case for an arbitrary number $n$ of logarithmic points on $\mathbb{C}P^{1}$,
the homotopy type of the dual boundary complex is that of $S^{2n-7}$. Another
result from M. Mauri, E. Mazzon and M. Stevenson shows that that the dual
boundary complex of a log-Calabi-Yau compactification of the
$\operatorname{GL}(n,\mathbb{C})$ character variety of a 2-torus is
homeomorphic to $S^{2n-1}$, see [34].

In order to prove our result, we will employ the asymptotic abelianization
approach used in [50] to establish the Geometric P=W conjecture in rank 2
corresponding to the Painlevé cases, suitably adapted to rank 3. An important
technical difference with the rank $2$ case is that in [49], [50] the second
author made use of certain local models called fiducial solutions [11] that
are only available in rank $2$. In the present article, however, we have found
a way to get around finding specific model solutions around the branch points
by making use of recent results of T. Mochizuki and the second author [37] for
the analysis. This simplifies the presentation and clarifies the picture, thus
paving the way for potential higher-rank and higher-dimensional
generalizations of our viewpoint. As a consequence of the analysis, we find
that the asymptotic behavior of $\psi$ depends on a decomposition into sectors
of the Hitchin base, namely it is determined by different exponential terms in
each sector around infinity. To be more precise, the absolute values of the
trace coordinates are of exponential growth in the parameter, with a constant
factor that are constant on a sector. Over all sectors, these constant factors
are the integrals of the Liouville $1$-form along some loop on the spectral
curve. This kind of behavior is an instance of the Stokes phenomenon, that we
analyze in detail in this particular case. As for the phases of the trace
coordinates, we find that they are asymptotically equal on each sector to the
holonomy of the spectral sheaf along certain loops that we determine.

This paper is organized as follows. In Section 2, we recall the background
material necessary to explain our arguments. In Section 3, we first describe
the $\operatorname{GL}(3,\mathbb{C})$ (which is equivalent in this case to
$\operatorname{SL}(3,\mathbb{C})$) character variety of the three-punctured
sphere in general, and prove Theorem 1.2. We then analyze trace coordinates on
the character variety, going back to classical work of R. Fricke and F. Klein
[13], and which were generalized for $\operatorname{GL}(3,\mathbb{C})$
character varieties by S. Lawton [28], [29]. In Section 4, we use T.
Mochizuki’s asymptotic abelianization technique [36] and equivariant method of
T. Mochizuki and the second author [37] to give the large-scale analysis of
harmonic bundles. In Section 5 we use T. Mochizuki’s solution of the Hitchin
WKB problem on non-critical paths [36, Section 2.3] to describe the parallel
transport matrices, and apply Riemann–Hilbert correspondence to the previous
setup. Finally in Section 6, we investigate the asymptotic behaviour of the
trace coordinates under the Riemann–Hilbert correspondence, analyze the
emerging Stokes phenomenon, and prove Theorem 1.1.

Acknowledgement: The second author would like to thank T. Mochizuki for
collaboration and inspiring discussions about harmonic bundles.

##  2 Preparatory material

First, let us introduce the material that we will need to establish and prove
our result. The structure follows more or less [49] and [11].

###  2.1 Basic notations, definitions and results

Consider $X=\mathbb{C}P^{1}$ with the standard Riemannian-metric, and with
coordinate charts $z$ and $w=z^{-1}$. For the distinct points
$0,1,\infty\in\mathbb{C}P^{1}$, let $D$ be the simple effective divisor
$D=0+1+\infty$ (and denote by $D$ the support set of the divisor as well).
Consider furthermore a smooth vector bundle $E$ of rank $3$ and degree $0$ on
$\mathbb{C}P^{1}$. We denote by $K$ and $\mathcal{O}$ the sheaves of
holomorphic $1$-forms and functions on $\mathbb{C}P^{1}$, and by
$\Omega^{1,0}$ and $\Omega^{0,1}$ the smooth $(1,0)$\- and $(0,1)$-forms on
$\mathbb{C}P^{1}$. Then, a $1$-form valued $\mathcal{O}$-linear vector bundle
morphism $\theta$ is called a Higgs field:

$\theta:E\rightarrow E\otimes\Omega^{1,0},$

Moreover, we consider partial $(0,1)$-connections $\overline{\partial}_{E}$ on
$E$ over $\mathbb{C}P^{1}$. Together with a Hermitian metric $h$ on $E$, the
basic objects of our investigation will be Higgs bundles
$(E,\theta,\overline{\partial}_{E})$, which satisfy Hitchin’s equations:

$\begin{cases}\overline{\partial}_{E}\theta=0\\\
F_{h}+[\theta,\theta^{{\dagger}_{h}}]=0\end{cases}$

where $F_{h}$ denotes the curvature form of the Chern connection
$\nabla_{h}^{+}$, associated with $\overline{\partial}_{E}$ and $h$, and
$\theta^{{\dagger}_{h}}$ denotes the adjoint of the Higgs field with respect
to $h$ (i.e. $\theta^{{\dagger}_{h}}:E\rightarrow E\otimes\Omega^{0,1}$). If
the Hitchin’s equations are satisfied, then the bundle is called harmonic, and
$h$ is called Hermitian-Einstein metric. We also get a holomorphic structure
$\overline{\partial}_{\operatorname{det}E}$ on the complex line bundle
$\operatorname{det}E$, induced by $\overline{\partial}_{E}$, and a Hermitian
metric $h_{\operatorname{det}E}$ on it, induced by $h$. Now denote by
$\mathcal{E}$ the holomorphic vector bundle $(E,\overline{\partial}_{E})$ on
$\mathbb{C}P^{1}\setminus D$. The holomorphic vector bundle $\mathcal{E}$ is
also defined over $D$, and we assume that $\theta$ has logarithmic
singularities at the points of $D$, that is we are considering logarithmic
Higgs bundles.

Let us define the parabolic structure of Higgs bundles, based on [35], [4] and
[32]. Fix a weight vector for all $p\in D$:
$\underline{\alpha}_{P}=(\alpha_{P}^{1},\alpha_{P}^{2},\alpha_{P}^{3})$, where
$\alpha_{P}^{j}$’s lie in a unit interval for all $j=1,2,3$, and
$\alpha_{P}^{1}<\alpha_{P}^{2}<\alpha_{P}^{3}$. A quasi-parabolic structure on
$\mathcal{E}$ with divisor $D$ is a filtration on the fiber of $\mathcal{E}$
over each point of $D$:

$0=l_{P}^{3}\subset l_{P}^{2}\subset l_{P}^{1}\subset
l_{P}^{0}={\left.\kern-1.2pt\mathcal{E}\vphantom{\big{|}}\right|_{P}}.$ (2)

A parabolic structure is a quasi-parabolic structure together with a choice of
weight vectors $\underline{\alpha}_{P}$ at each $P$. It gives rise to an
$\mathbb{R}$-filtration $\mathcal{P}_{*}$ of $\mathcal{E}$. We always assume
that the Higgs field is weakly parabolic, meaning that
$\theta:l_{P}^{i}\rightarrow l_{P}^{i}\otimes K(D)$ at each $P\in D$. The
Higgs field is called strongly parabolic if $\theta:l_{P}^{i}\rightarrow
l_{P}^{i+1}\otimes K(D)$ at each $P\in D$, i.e. the residue is nilpotent with
respect to the filtration in that the action on the graded pieces
$\textrm{gr}_{l}^{\boldsymbol{\cdot}}\mathcal{E}$ of $l_{P}^{\bullet}$ is
trivial.

As usual, under stability of a Higgs bundle
$(E,\theta,\overline{\partial}_{E})$ we mean that for any proper holomorphic
subbundle $F\subset E$ which satisfies $\theta:F\rightarrow F\otimes K(D)$,
the inequality $\mu(F)<\mu(E)$ holds, where
$\mu(E)=\frac{\textrm{deg}E}{\textrm{rank}E}$ is the slope of the bundle
($\mu(F)$ defined similarly). In the parabolic setting, we speak about
$\alpha$-stability, which depends on the weight vectors
$\underline{\alpha}_{P}$, and means that for all $F$ satisfying the above
conditions

$\frac{\textrm{pdeg}_{\underline{\alpha}}E}{\textrm{rank}E}>\frac{\textrm{pdeg}_{\underline{\alpha}}F}{\textrm{rank}F},$

where the parabolic degree of the parabolic bundle (and subbundle) is:

$\displaystyle\textrm{pdeg}_{\underline{\alpha}}E=\textrm{deg}E+\sum_{P\in
D}\sum_{j=1}^{3}\alpha_{P}^{j}$ (3)
$\displaystyle\textrm{pdeg}_{\underline{\alpha}}F=\textrm{deg}F+\sum_{P\in
D}\sum_{j=1}^{3}\alpha_{P}^{j}\cdot\textrm{dim}(({\left.\kern-1.2ptF\vphantom{\big{|}}\right|_{P}}\cap
l_{P}^{j-1})/({\left.\kern-1.2ptF\vphantom{\big{|}}\right|_{P}}\cap
l_{P}^{j}))$ (4)

The Higgs bundle is called $\alpha$-polystable, if it is the direct sum of
lower rank $\alpha$-stable Higgs bundles, with the same parabolic slope as
$(E,\theta,\overline{\partial}_{E})$. By the results of Hitchin [20] and
Simpson [45], it is known that a Higgs bundle admits a unique Hermitian-
Einstein metric $h$ with $\operatorname{det}h=h_{\operatorname{det}E}$ if and
only if it is polystable.

Let $\textrm{SL}(E)$ be the principal bundle of automorphisms of $E$ which
induce the identity on $\operatorname{det}E$. Then the group of complex gauge
transformations, denoted by $\mathcal{G}$, is the group of sections of
$\textrm{SL}(E)$. Moreover its Lie algebra consists of the sections of
$\mathfrak{sl}(E)$, the vector bundle of traceless endomorphisms of $E$. The
gauge group $\mathcal{G}$ acts on the Higgs bundles via

$g.(E,\theta,\overline{\partial}_{E})=(E,g^{-1}\theta
g,g^{-1}\overline{\partial}_{E}g),\hskip 14.22636pt\forall g\in\mathcal{G}.$

######  Definition 2.1.

The moduli space of harmonic, $\alpha$-stable, strongly parabolic, meromorphic
$\textrm{SL}(3,\mathbb{C})$-Higgs bundles, with at most logarithmic
singularities, with given weight vectors $\underline{\alpha}_{P}$, up to the
complex gauge action, is called the Dolbeault moduli space, denoted by
$\mathcal{M}_{\textrm{Dol}}(\alpha)$.

See [27] for a differential geometric construction of this space, and [41] for
an algebraic geometric one.

In case of logarithmic Higgs bundles, for $\theta$ and $h$ the so called
tameness condition is satisfied, that is at each $P\in D$, $h$ admits a lift
along any ray to $P$, which grows at most polynomially in the standard metric.
(Here we consider $h$ as an equivariant harmonic map from the universal cover
of the Riemann surface to the Hermitian symmetric space
$\textrm{GL}(3,\mathbb{C})/\textrm{U}(3)$). For such a tame, harmonic bundle
$(E,\theta,\overline{\partial}_{E},h)$ the connection

$\nabla=\nabla_{h}^{+}+\theta+\theta^{{\dagger}_{h}}$ (5)

is integrable, and $\nabla^{1,0}$ has regular singularities. Fix again for all
$P\in D$ some
$\underline{\beta}_{P}=(\beta_{P}^{1},\beta_{P}^{2},\beta_{P}^{3})$ parabolic
weight vectors and the
$\underline{\tau}_{P}=(\tau_{P}^{1},\tau_{P}^{2},\tau_{P}^{3})$ eigenvalues of
the residue of the connection. The definition of $\beta$-stability and
parabolic structure of the integrable connection is just the same as for the
Higgs field (the parabolic structure of the underlying vector bundle is
already given, see also [32]). We again require that (5) is compatible with
the filtration, that is
$(\textrm{res}_{P}\nabla-\tau_{P}^{j}\operatorname{id})(l_{P}^{j})\subset
l_{P}^{j+1}$, for all $P\in D$ and $j=0,1,2$. If the eigenvalues
$\tau_{P}^{j}$ are pairwise different that this implies that
$\textrm{res}_{P}\nabla$ is diagonal with respect to some basis compatible
with the filtration. The complex gauge group action on the space of
connections is also inherited from the action on the space of Higgs bundles.

######  Definition 2.2.

The moduli space of $\beta$-stable, parabolic, integrable
$\textrm{SL}(3,\mathbb{C})$-connections, with regular singularities at the
punctures, with given weight vectors $\underline{\beta}_{P}$ and given
residues $\underline{\tau}_{P}$ at each $P\in D$, up to the complex gauge
action, is called the de Rham moduli space, denoted by
$\mathcal{M}_{\textrm{dR}}(\beta,\tau)$.

The third main object of our research is the Betti moduli space, also known as
character variety. Under the stability condition, (5) is an irreducible
integrable connection. For any choice of base point $x_{0}\notin D$, analytic
continuation of solutions provides a representation

$\rho:\pi_{1}(\mathbb{C}P^{1}\setminus
D,x_{0})\rightarrow\textrm{SL}(3,\mathbb{C})$

that is well-defined up to simultaneous conjugation by elements of
$\textrm{PGL}(3,\mathbb{C})$ (corresponding to different choices of a basis of
solutions at $x_{0}$). The eigenvalues, denoted by
$\underline{c}_{P}=(c_{P}^{1},c_{P}^{2},c_{P}^{3})$, of the local monodromy
around $P\in D$ are determined by
$(\underline{\beta}_{P},\underline{\tau}_{P})$. (As a matter of fact, the
local system admits a filtration and corresponding weights too, but we will
not need this extra structure here.)

######  Definition 2.3.

The moduli space of the above described representations is called the Betti
moduli space or character variety, denoted by
$\mathcal{M}_{\textrm{B}}(\textbf{c})$.

It is known that the Betti space is a smooth, affine algebraic variety for
generic parameters. There exists a compactification of the Betti space by a
simple normal crossing divisor $D_{B}$ (see the results of Nagata and Hironaka
[38], [19]). In our case $D_{B}$ is a complex curve. As customary, we define
its dual complex $\mathcal{D}D_{B}$ as the simplicial complex whose vertices
are the irreducible components of $D_{B}$, and whose edges corresponds to the
intersections of the components. We want to apply this to the compactification
of the Betti moduli space, therefore the resulting simplicial complex will be
called dual boundary complex, denoted by
$\mathcal{D}\partial\mathcal{M}_{B}(\textbf{c})$.

It is known from Simpson [46]that there is a connection between the above
defined parameters. With the eigenvalues of the residues of the Higgs-field
being equal to 0, it simplifies to

$\alpha_{P}^{i}=\beta_{P}^{i}=\tau_{P}^{i},\hskip 5.69046pt\textrm{and}\hskip
5.69046ptc_{P}^{i}=e^{-2\pi\sqrt{-1}\alpha_{P}^{i}},\forall P\in D,\forall
i\in\\{1,2,3\\}.$

Moreover, the following theorem holds.

######  Theorem 2.4.

Assume that the parabolic degree of $\mathcal{E}$ is 0.

  1. 1.

[3] The spaces $\mathcal{M}_{\textrm{Dol}}(\alpha)$,
$\mathcal{M}_{\textrm{dR}}(\beta,\tau)$ and
$\mathcal{M}_{\textrm{B}}(\textbf{c})$ are $\mathbb{C}$-analytic manifolds,
and there exists a diffeomorphism

$\operatorname{NAHC}:\mathcal{M}_{\textrm{Dol}}(\alpha)\rightarrow\mathcal{M}_{\textrm{dR}}(\beta,\tau)$

called the non-abelian Hodge correspondence.

  2. 2.

[21, Theorem 7.1] There exists a complex bianalytic isomorphism

$\operatorname{RH}:\mathcal{M}_{\textrm{dR}}(\beta,\tau)\rightarrow\mathcal{M}_{\textrm{B}}(\textbf{c}),$

called the Riemann–Hilbert correspondence.

###  2.2 Choice of parameters

Now let us choose the parameters introduced in the previous subsection, and
explain these choices. For all $P\in D$, set

$\displaystyle\alpha_{P}^{1}=\beta_{P}^{1}=\tau_{P}^{1}=-\frac{1}{3}$
$\displaystyle\alpha_{P}^{2}=\beta_{P}^{2}=\tau_{P}^{2}=0$
$\displaystyle\alpha_{P}^{3}=\beta_{P}^{3}=\tau_{P}^{3}=\frac{1}{3}$

Consequently

$c_{P}^{2}=1,\hskip
5.69046ptc_{P}^{3}=-\frac{1}{2}+\frac{\sqrt{3}}{2}i=\varepsilon,\hskip
5.69046ptc_{P}^{1}=-\frac{1}{2}-\frac{\sqrt{3}}{2}i=\varepsilon^{2},$

where $i=\sqrt{-1}$ and $\varepsilon$ stands for a primitive cubic root of
unity.

######  Lemma 2.5.

With these parameter values

  * i)

the parabolic degree of $\mathcal{E}$ is zero,

  * ii)

the traces of $\theta$ and $\theta^{2}$ are identically zero.

###### Proof.

  * i)

Since at all $P\in D$, the sum of the parabolic weights equal to 0, it follows
from equation (3), that the parabolic degree of $\mathcal{E}$ is 0.

  * ii)

The residues of the Higgs field are traceless at all $P\in D$.

On $\mathbb{C}P^{1}$ we have for the sheaf of holomorphic 1-forms
$K\cong\mathcal{O}(-2)$, and with the divisor $D=0+1+\infty$, we have
$K(D)\cong\mathcal{O}(1)$, via the identification
$\frac{\operatorname{d}\\!z}{z(z-1)}\leftrightarrow 1$. Then

$\displaystyle\operatorname{tr}\theta\in H^{0}(\mathbb{C}P^{1},K(D))\cong
H^{0}(\mathbb{C}P^{1},\mathcal{O}(1))\cong\mathbb{C}^{2}$
$\displaystyle\operatorname{tr}\theta^{2}\in
H^{0}(\mathbb{C}P^{1},K(D)^{\otimes 2})\cong
H^{0}(\mathbb{C}P^{1},\mathcal{O}(2))\cong\mathbb{C}^{3},$

We have three independent vanishing conditions for both
$\operatorname{tr}\theta$ and $\operatorname{tr}\theta^{2}$ at the points
$P\in D$, therefore both must be zero globally. (One condition is even
redundant in the case of $\operatorname{tr}\theta$.)

∎

###  2.3 The Hitchin fibration

The characteristic coefficients of a Higgs bundle of rank $3$ over
$\mathbb{C}P^{1}$ with logarithmic singularities at $D$ belong to the vector
space

$\displaystyle\mathcal{B}=H^{0}(\mathbb{C}P^{1},K(D))\oplus
H^{0}(\mathbb{C}P^{1},K(D)^{\otimes 2})\oplus
H^{0}(\mathbb{C}P^{1},K(D)^{\otimes 3})\cong$ $\displaystyle\cong
H^{0}(\mathbb{C}P^{1},\mathcal{O}(1))\oplus
H^{0}(\mathbb{C}P^{1},\mathcal{O}(2))\oplus
H^{0}(\mathbb{C}P^{1},\mathcal{O}(3))\cong\mathbb{C}^{2}\oplus\mathbb{C}^{3}\oplus\mathbb{C}^{4}$

According to Lemma 2.5, with the choices made in Section 2.2, the first two
components vanish. The third characteristic coefficient of $\theta$ is
$\operatorname{det}\theta\in H^{0}(\mathbb{C}P^{1},K(D)^{\otimes 3})$. Let us
use the notation $L=K(D)$, with the natural projection from the total space of
$L$, $p_{L}:\textrm{Tot}L\rightarrow\mathbb{C}P^{1}$. Define
$\zeta\frac{\operatorname{d}\\!z}{z(z-1)}$ to be the canonical section of
$p_{L}^{*}L$ over $p_{L}^{-1}(\mathbb{C})$, i.e. away from the infinity
section. With this, the characteristic polynomial of $\theta$ is

$\operatorname{det}(\zeta\operatorname{id}_{\mathcal{E}}-\theta)=\zeta^{3}\operatorname{id}^{\otimes
3}_{\mathcal{E}}+H_{\theta}$

where $H_{\theta}$ lies in the last direct summand of $\mathcal{B}$. The third
coefficient $\operatorname{det}\theta=H_{\theta}$ has 4 parameters of freedom,
but with the three independent vanishing relations at $P\in D$, this reduces
to a one-parameter family. One can see [32, Appendix A], that it has the form

$\operatorname{det}\theta=H_{\theta}=(tz(z-1)+p_{2}(z))\frac{\operatorname{d}\\!z^{\otimes
3}}{z^{3}(z-1)^{3}}$

where $p_{2}(z)=az^{2}+bz+c$ is a quadratic polynomial, and $a,b,c,t$ are the
4 parameters, from which $a,b,c$ are fixed to be $0$, and $t\in\mathbb{C}$ is
the only free parameter. The 1-dimensional subspace

$\mathcal{H}=\left\\{t\frac{\operatorname{d}\\!z^{\otimes
3}}{z^{2}(z-1)^{2}}\colon t\in\mathbb{C}\right\\}\subset\mathcal{B}$ (6)

where $\operatorname{det}\theta$ may take its values is called the Hitchin
base of $\mathcal{M}_{\textrm{Dol}}(\alpha)$, and the map

$H:\mathcal{M}_{\textrm{Dol}}(\alpha)\rightarrow\mathcal{H},\hskip
8.5359pt(\mathcal{E},\theta)\mapsto\operatorname{det}\theta$

is called the Hitchin fibration. Clearly, for any $\tau\in\mathbb{C}^{\times}$
and $(\mathcal{E},\theta)$ we have

$H((\mathcal{E},\tau\theta_{1}))=\tau^{3}H((\mathcal{E},\theta)).$

We choose the preferred point

$q_{1}=\frac{\operatorname{d}\\!z^{\otimes 3}}{z^{2}(z-1)^{2}}\in\mathcal{H}.$
(7)

For every $t\in\mathbb{C}$ the smooth curve called the spectral curve is
defined via

$\Sigma_{t}=\\{(z,\zeta)|\zeta^{3}+tz(z-1)=0\\}\subset\textrm{Tot}L.$ (8)

Notice that $\Sigma_{t}$ has maximal ramification over $0,1$ and a smooth
compactification, denoted by $\widetilde{\Sigma}_{t}$, in $\textrm{Tot}L$ at
$z=\infty$ with $(w,\zeta)=(0,0)$, where it also admits cyclic ramification.
One can easily compute from Riemann–Hurwitz formula that its genus is equal to
1.

###  2.4 Ramification of the spectral curve and the Jacobian

The equation in (8) defining $\widetilde{\Sigma}_{t}$ has three roots over
every point of $\mathbb{C}P^{1}\setminus D$:

$\displaystyle\xi_{1}=R^{1/3}e^{i\varphi/3}z^{1/3}(z-1)^{1/3},\hskip
8.5359pt\xi_{2}=\varepsilon R^{1/3}e^{i\varphi/3}z^{1/3}(z-1)^{1/3},$
$\displaystyle\xi_{3}=\varepsilon^{2}R^{1/3}e^{i\varphi/3}z^{1/3}(z-1)^{1/3},$

where we recall that $\varepsilon$ is a cubic root of unity and we switched to
polar coordinates $t=Re^{i\varphi}$. That is, $\xi_{1,2,3}$ is a three-valued
holomorphic function on $\mathbb{C}P^{1}\setminus D$, and $p_{L}$ induces a
projection map
$p_{R,\varphi}:\widetilde{\Sigma}_{R,\varphi}\rightarrow\mathbb{C}P^{1}$. This
is indeed a ramified triple cover over $\mathbb{C}P^{1}$, with ramification
points $z=0,z=1$ on chart $z$, and $w=0$ on chart $w$, independently of the
value of $t>0$. Thus we introduce the ramification divisor
$\Delta=\\{0,1,\infty\\}=D$, and denote the lift of the ramification divisor
by $\widetilde{\Delta}$ (or $\widetilde{D}$), whose points are the branch
points on $\widetilde{\Sigma}_{R,\varphi}$. We summarize properties of the
spectral curve.

######  Proposition 2.6.

$\widetilde{\Sigma}_{R,\varphi}$ is a smooth genus 1 curve, with ramification
index at all 3 points of $D$ equal to 3, i.e. all its ramifications are
cyclic.

We will use the polar coordinates

$\tau^{3}=t=Re^{i\varphi}.$ (9)

For any $\theta_{1}$ satisfying $H(\theta_{1})=q_{1}$ let us use the notation
for the cubic meromorphic differentials, with double poles at the punctures

$tq_{1}=Re^{i\varphi}q_{1}:=\operatorname{det}(\tau\theta_{1})=Re^{i\varphi}\frac{\operatorname{d}\\!z^{\otimes
3}}{z^{2}(z-1)^{2}}\in H^{0}(\mathbb{C}P^{1},K^{\otimes 3}(2D)).$ (10)

Viewed as a section of $L^{\otimes 3}$, $q_{1}$ has a simple zero. Then the
sheets of $\widetilde{\Sigma}_{R,\varphi}$ are just the cubic roots of
$tq_{1}$:

$\displaystyle Q_{1,R,\varphi}$
$\displaystyle=\tau\frac{\operatorname{d}\\!z}{z^{2/3}(z-1)^{2/3}}=R^{1/3}e^{i\varphi/3}\frac{\operatorname{d}\\!z}{z^{2/3}(z-1)^{2/3}}$
$\displaystyle Q_{2,R,\varphi}$
$\displaystyle=\varepsilon\tau\frac{\operatorname{d}\\!z}{z^{2/3}(z-1)^{2/3}}=\varepsilon
R^{1/3}e^{i\varphi/3}\frac{\operatorname{d}\\!z}{z^{2/3}(z-1)^{2/3}},$ (11)
$\displaystyle Q_{3,R,\varphi}$
$\displaystyle=\varepsilon^{2}\tau\frac{\operatorname{d}\\!z}{z^{2/3}(z-1)^{2/3}}=\varepsilon^{2}R^{1/3}e^{i\varphi/3}\frac{\operatorname{d}\\!z}{z^{2/3}(z-1)^{2/3}}.$

According to [1], the three corresponding eigenspaces determine a line bundle
$\mathcal{L}_{R,\varphi}\rightarrow\widetilde{\Sigma}_{R,\varphi}$, whose
pushforward $(p_{R,\varphi})_{*}\mathcal{L}_{R,\varphi}$ is isomorphic to
$\mathcal{E}$. The degree of $\mathcal{L}_{R,\varphi}$ can be computed via
[31, 2.3.3] or [48, Theorem 5.4]:

$\textrm{deg}\mathcal{L}=\textrm{deg}\mathcal{E}+r(1-r)\left(1-g-\frac{n}{2}\right)=3,$

where $r=3$ is the rank, $n=3$ is the number of ramification points, and $g=0$
is the genus of $\mathbb{C}P^{1}$. Thus an element of the Dolbeault moduli
space determines a spectral curve and a line bundle of degree 3 on it, and
vice versa. So, the fiber of the Hitchin fibration over the point
parameterized by a fixed $t\in\mathcal{H}$ is
$\textrm{Pic}^{3}(\widetilde{\Sigma}_{t})$, that is a 2-torus, namely a torsor
over $\operatorname{Jac}(\widetilde{\Sigma}_{t})$. In particular, the Hitchin
fibration is an elliptic fibration.

Let us discuss this correspondence between the Hitchin fibers and the Jacobian
of the spectral curve a bit more in detail. Consider the following period
lattice $\Lambda_{t}\subset H^{0,1}(\widetilde{\Sigma}_{t})\cong\mathbb{C}$,
provided by the image $\textrm{Im}(p^{0,1}\circ\iota)$, where

$\displaystyle\iota:H^{1}(\widetilde{\Sigma}_{t},2\pi i\mathbb{Z})\rightarrow
H^{1}(\widetilde{\Sigma}_{t},\mathbb{C})$ $\displaystyle
p^{0,1}:H^{1}(\widetilde{\Sigma}_{t},\mathbb{C})\rightarrow
H^{0,1}(\widetilde{\Sigma}_{t})$

are the coefficient inclusion on the first cohomology class, and the
projection of harmonic forms to their antiholomorphic part respectively. There
exists a $\mathbb{C}$-analytic isomorphism
$\operatorname{Jac}(\widetilde{\Sigma}_{t})\cong
H^{0,1}(\widetilde{\Sigma}_{t})/\Lambda_{t}$. Namely, any class in
$H^{0,1}(\widetilde{\Sigma}_{t})/\Lambda_{t}$ can be represented by a
$\mu\in\Omega^{0,1}(\widetilde{\Sigma}_{t})$, because of the abelian Hodge
correspondence. Then the connection form
$u=\mu-\overline{\mu}=2i\textrm{Im}\mu\in\Omega^{1}(\widetilde{\Sigma}_{t})$
defines a flat $U(1)$-connection. For the line bundle
$\mathcal{L}\in\operatorname{Jac}(\widetilde{\Sigma}_{t})$ given by $\mu$ and
a $1$-cycle $X\in H_{1}(\widetilde{\Sigma}_{t},\mathbb{Z})$ we call

$\operatorname{hol}_{X}(\mathcal{L})=e^{\oint_{X}u}$ (12)

the holonomy of $\mathcal{L}$ along $X$. Fix $X,Y$ 1-cycles generating
$H_{1}(\widetilde{\Sigma}_{t},\mathbb{Z})$. The abelian version of Theorem 1
can then be expressed as saying that the holonomy map

$\operatorname{hol}\colon\operatorname{Jac}(\widetilde{\Sigma}_{t})\rightarrow
T^{2}=S^{1}\times S^{1},\hskip
8.5359pt\mu\mapsto(\operatorname{hol}_{X}(\mathcal{L}),\operatorname{hol}_{Y}(\mathcal{L}))=\left(e^{\oint_{X}u},e^{\oint_{Y}u}\right),$

is an isomorphism between the Jacobian and the 2-torus. See [15, Section 4]
for more details.

###  2.5 The Hitchin section

There is a preferred line bundle $\mathcal{L}_{0}$ over
$\widetilde{\Sigma}_{t}$ giving rise to a section of $H$ analogous to the
Hitchin section. Namely, as it is well-known,

$(p_{L})_{*}\mathcal{O}_{\widetilde{\Sigma}_{t}}\cong\mathcal{O}_{\mathbb{C}P^{1}}\oplus
K_{\mathbb{C}P^{1}}(D)^{-1}\oplus
K_{\mathbb{C}P^{1}}(D)^{-2}\cong\mathcal{O}_{\mathbb{C}P^{1}}\oplus\mathcal{O}_{\mathbb{C}P^{1}}(-1)\oplus\mathcal{O}_{\mathbb{C}P^{1}}(-2),$

the direct summands being generated by $1,\zeta,\zeta^{2}$ respectively. The
preferred choice of spectral sheaf is then
$\mathcal{L}_{0}=p_{L}^{*}L\otimes\mathcal{O}_{\widetilde{\Sigma}_{t}}$.
Notice that $\mathcal{O}_{\widetilde{\Sigma}_{t}}\cong
K_{\widetilde{\Sigma}_{t}}$, because $\widetilde{\Sigma}_{t}$ is an elliptic
curve. Moreover, from the local form at $z=0$ of the equation defining
$\widetilde{\Sigma}_{t}$ we get

$\frac{\operatorname{d}\\!z}{z}=h(\zeta)\frac{\operatorname{d}\\!\zeta}{\zeta}$

for some local holomorphic function $h$ with $h(0)\neq 0$, and similarly for
the other poles $P\in D$. This means that

$p_{L}^{*}K_{\mathbb{C}P^{1}}(D)\otimes\mathcal{O}_{\widetilde{\Sigma}_{t}}=K_{\widetilde{\Sigma}_{t}}(\widetilde{D}).$

We infer

$\mathcal{L}_{0}=p_{L}^{*}L\otimes\mathcal{O}_{\widetilde{\Sigma}_{t}}\cong
K_{\widetilde{\Sigma}_{t}}(\widetilde{D})\cong\mathcal{O}_{\widetilde{\Sigma}_{t}}(\widetilde{D}).$

By the projection formula we then have

$\mathcal{E}_{0}=(p_{L})_{*}\mathcal{L}_{0}\cong K(D)\oplus\mathcal{O}\oplus
K(D)^{-1},$

in particular, the degree of $\mathcal{E}_{0}$ is equal to zero, as required.
Stable Higgs fields

$\\\ \theta_{t}:K(D)\oplus\mathcal{O}\oplus K(D)^{-1}\rightarrow
K(D)^{2}\oplus K(D)\oplus\mathcal{O}$ (13)

over $\mathcal{E}_{0}$ are of the form

$\theta_{t}=\begin{pmatrix}0&0&tq_{1}\\\ 1&0&0\\\ 0&1&0\end{pmatrix}$ (14)

where $q_{1}$ is given in (7). Applying a constant (i.e., depending only on
$t$) gauge transformation

$\begin{pmatrix}t^{-\frac{1}{3}}&0&0\\\ 0&1&0\\\
0&0&t^{\frac{1}{3}}\end{pmatrix}=\begin{pmatrix}\tau^{-1}&0&0\\\ 0&1&0\\\
0&0&\tau\end{pmatrix}$

the Higgs field gets transformed into

$\tau\theta_{1}=\tau\begin{pmatrix}0&0&q_{1}\\\ 1&0&0\\\ 0&1&0\end{pmatrix}.$

Since $q_{1}$ has a zero viewed as a section of $L^{\otimes 3}$, we get that

$\textrm{res}_{P}(\tau\theta_{1})=\begin{pmatrix}0&0&0\\\ 1&0&0\\\
0&1&0\end{pmatrix}.$

The strongly parabolic condition is that the residues of $\theta_{t}$ at the
points of $D$ are nilpotent with respect to the quasi-parabolic filtration.
Now, there exists a unique filtration of $\mathcal{E}_{0}$ over the points of
$D$ compatible with the above residue, namely

$l^{2}_{P}=\mathbb{C}\cdot\zeta^{2},\hskip
8.5359ptl^{1}_{P}=\mathbb{C}\cdot\zeta\oplus\mathbb{C}\cdot\zeta^{2}$

The parabolic weights corresponding to the generators $1,\zeta,\zeta^{2}$ are
therefore respectively equal to $-\frac{1}{3},0,\frac{1}{3}$. See also the
results from [12], [17].

##  3 Description of the Betti moduli space

In this chapter we consider the Betti space (or character variety),
parameterizing the irreducible representations of the fundamental group of
$\mathbb{C}P^{1}\setminus D$ in $\textrm{GL}(3,\mathbb{C})$, up to the
simultaneous conjugation by of $\textrm{PGL}(3,\mathbb{C})$. We pick
$z_{0}\in\mathbb{C}P^{1}\setminus\\{0,1,\infty\\}$ once and for all, and all
occurrences of fundamental group will mean with base point $z_{0}$. Since the
fundamental group of $\mathbb{C}P^{1}\setminus\\{0,1,\infty\\}$ is isomorphic
to the free group generated by two elements, this amounts to considering maps

$\rho:\pi_{1}\left(\mathbb{C}P^{1}\setminus\\{0,1,\infty\\}\right)\cong\langle
a,b\rangle\rightarrow\textrm{GL}(3,\mathbb{C})$

under the constraint that the eigenvalues of $\rho(a),\rho(b)$ and $\rho(ab)$
are fixed: $\\{\lambda_{1},\lambda_{2},\lambda_{3}\\}$,
$\\{\mu_{1},\mu_{2},\mu_{3}\\}$ and $\\{\nu_{1},\nu_{2},\nu_{3}\\}$
respectively (previously denoted by the vectors $\underline{c}_{P}$). Because
of the $\textrm{PGL}(3,\mathbb{C})$ action, we have the freedom to choose
$\rho(a)$ to be diagonal with elements
$\\{\lambda_{1},\lambda_{2},\lambda_{3}\\}$. Having achieved this, there
remains the action of the maximal torus $\left(\mathbb{C}^{\times}\right)^{2}$
of $\textrm{PGL}(3,\mathbb{C})$. The action of
$(t_{1},t_{2})\in\left(\mathbb{C}^{\times}\right)^{2}$ on a matrix
$B=[b_{ij}]$ is the standard conjugation

$(t_{1},t_{2}).B=\begin{pmatrix}b_{11}&t_{1}b_{12}&t_{1}t_{2}b_{13}\\\
t_{1}^{-1}b_{21}&b_{22}&t_{2}b_{23}\\\
t_{1}^{-1}t_{2}^{-1}b_{31}&t_{2}^{-1}b_{32}&b_{33}\end{pmatrix}.$

With the notations $\rho(a)=A,\rho(b)=B$, the constraints on the eigenvalues
are equivalent to constraints on the traces of $A^{j},B^{j}$ and $(AB)^{j}$
for $j=1,2,3$. Thus the Betti space can be written as

$\displaystyle\mathcal{M}_{B}=\\{A,B\in\textrm{GL}(3,\mathbb{C})|A=\textrm{Diag}[\lambda_{1},\lambda_{2},\lambda_{3}],B=[b_{i,j}],$
$\displaystyle\operatorname{tr}(B^{j})=\sigma_{j}(\underline{\mu}),\operatorname{tr}((AB)^{j})=\sigma_{j}(\underline{\nu}),j=1,2,3\\}/\left(\mathbb{C}^{\times}\right)^{2}$

where $\sigma_{j}$ is the degree $j$ homogeneous symmetric polynomial in 3
variables. Because of the irreducibility of the representations, we are given
that both $b_{21}$ and $b_{31}$ can not vanish simultaneously. Possibly
passing to a Zariski open subset (that does not alter validity of our
arguments), we may assume that they are both nonzero. We may then use the
$\left(\mathbb{C}^{\times}\right)^{2}$ action to remove these coefficients.
Since the assumption on $(AB)^{3}$ is redundant, this gives

$\displaystyle\mathcal{M}_{B}=\\{B=\begin{pmatrix}b_{11}&b_{12}&b_{13}\\\
1&b_{22}&b_{23}\\\
1&b_{32}&b_{33}\end{pmatrix}|\operatorname{tr}(B^{j})=\sigma_{j}(\underline{\mu}),j=1,2,3,$
$\displaystyle\operatorname{tr}((AB)^{j})=\sigma_{j}(\underline{\nu}),j=1,2\\}.$

Now, the conditions
$\operatorname{tr}(B)=b_{11}+b_{22}+b_{33}=\sigma_{1}(\underline{\mu})$, and
$\operatorname{tr}(AB)=\lambda_{1}b_{11}+\lambda_{2}b_{22}+\lambda_{3}b_{33}=\sigma_{1}(\underline{\nu})$
can be used to express $b_{22}$ and $b_{33}$ in terms of $b_{11}$:

$\displaystyle
b_{22}=\frac{\lambda_{3}-\lambda_{1}}{\lambda_{2}-\lambda_{3}}b_{11}+c_{1}(\underline{\lambda},\underline{\mu},\underline{\nu})=:Q(b_{11})$
$\displaystyle
b_{33}=\frac{\lambda_{1}-\lambda_{2}}{\lambda_{2}-\lambda_{3}}b_{11}+c_{2}(\underline{\lambda},\underline{\mu},\underline{\nu})=:P(b_{11}),$

where $c_{1},c_{2}$ are constants depending only on
$\underline{\lambda},\underline{\mu},\underline{\nu}$, while $P$ and $Q$ are
degree 1 polynomials in $b_{11}$. Switching to the notation

$\displaystyle B=\begin{pmatrix}b_{11}&b_{12}&b_{13}\\\ 1&b_{22}&b_{23}\\\
1&b_{32}&b_{33}\end{pmatrix}=\begin{pmatrix}X&Y&Z\\\ 1&Q(X)&V\\\
1&W&P(X)\end{pmatrix}$

implies

$\displaystyle\mathcal{M}_{B}=\\{(X,Y,Z,V,W)\in\mathbb{C}^{5}|\operatorname{tr}(B^{2})=\sigma_{2}(\underline{\mu}),\operatorname{tr}(B^{3})=\sigma_{3}(\underline{\mu}),\operatorname{tr}((AB)^{2})=\sigma_{2}(\underline{\nu})\\}.$

The remaining three conditions read as:

$\displaystyle\operatorname{tr}(B^{2})=X^{2}+2Y+2Z+Q^{2}(X)+P^{2}(X)+2VW=\sigma_{2}(\underline{\mu})$
$\displaystyle\operatorname{tr}((AB)^{2})=\lambda_{1}^{2}X^{2}+2\lambda_{1}\lambda_{2}Y+2\lambda_{1}\lambda_{3}Z+\lambda_{2}^{2}Q^{2}(X)+\lambda_{3}^{2}P^{2}(X)+2\lambda_{2}\lambda_{3}VW=\sigma_{2}(\underline{\nu})$
$\displaystyle\operatorname{tr}(B^{3})=X^{3}+P^{3}(X)+Q^{3}(X)+3ZW+3XZ+3YV+3XY$
$\displaystyle+3Q(X)VW+3Q(X)Y+3P(X)VW+3P(X)Z=\sigma_{3}(\underline{\mu})$

Now, the first two equations allow us to express $Y$ and $Z$, and eliminate
these variables from the third equation. We thus obtain a description of the
Betti space as a cubic surface in $\operatorname{Spec}\mathbb{C}[X,V,W]$.
Then, we can consider the homogenisation of the resulting equation. This
procedure provides us the compatifying curve of $\mathcal{M}_{B}$ as a
homogeneous cubic curve in $\mathbb{C}P^{2}$, with equation

$\begin{gathered}\frac{-3(\lambda_{1}-\lambda_{2})(\lambda_{1}-\lambda_{3})^{2}(\lambda_{2}+\lambda_{3})}{\lambda_{1}(\lambda_{2}-\lambda_{3})^{4}}X^{3}+\frac{3(\lambda_{1}-\lambda_{3})^{2}(\lambda_{1}\lambda_{3}-\lambda_{2}^{2})}{\lambda_{1}(\lambda_{2}-\lambda_{3})^{3}}X^{2}V\\\
+\frac{3(\lambda_{1}-\lambda_{2})^{2}(\lambda_{3}^{2}-\lambda_{1}\lambda_{2})}{\lambda_{1}(\lambda_{2}-\lambda_{3})^{3}}X^{2}W+\frac{-3(\lambda_{1}(\lambda_{2}-\lambda_{3})^{2}+\lambda_{2}(\lambda_{1}-\lambda_{3})^{2}+\lambda_{3}(\lambda_{1}-\lambda_{2})^{2})}{\lambda_{1}(\lambda_{2}-\lambda_{3})^{2}}XVW\\\
+\frac{3\lambda_{2}(\lambda_{1}-\lambda_{3})}{\lambda_{1}(\lambda_{3}-\lambda_{2})}VW^{2}+\frac{3\lambda_{3}(\lambda_{1}-\lambda_{2})}{\lambda_{1}(\lambda_{2}-\lambda_{3})}V^{2}W=0\end{gathered}$
(15)

###  3.1 Topology of the compactifying divisor

Although we have made special choices for the parameters
$\underline{\lambda},\underline{\mu},\underline{\nu}$, here we will see that
up to homeomorphism, the compatifying curve is the same as with general
choices. Namely, it is the genus $0$ curve with nodal singularity, also called
fishtail, and denoted by $I_{1}$ in Kodaira’s list of singular elliptic curves
[24].

######  Proposition 3.1.

The curve $C$ determined by equation (15) in $\mathbb{C}P^{2}=\\{[X:V:W]\\}$,
is of type $I_{1}$.

###### Proof.

The proof has a similar idea as Proposition 2.3.3 in [16]. Consider the pencil
of projective lines passing through the point
$P=\left[\frac{\lambda_{2}-\lambda_{3}}{\lambda_{3}-\lambda_{1}}:\frac{\lambda_{1}-\lambda_{2}}{\lambda_{3}-\lambda_{1}}:1\right]$
in $\mathbb{C}P^{2}$, and parameterize them by
$[t_{0}:t_{1}]\in\mathbb{C}P^{1}$, via

$L_{[t_{0}:t_{1}]}=\\{[X:V:W]\in\mathbb{C}P^{2}|t_{0}\left(X-\frac{\lambda_{2}-\lambda_{3}}{\lambda_{3}-\lambda_{1}}\right)=t_{1}\left(V-\frac{\lambda_{1}-\lambda_{2}}{\lambda_{3}-\lambda_{1}}\right)\\}.$

Determine the intersection points of line $L_{[t_{0}:t_{1}]}$ with $C$: if
${[t_{0}:t_{1}]}=[1:0]$, then
$X=\frac{\lambda_{2}-\lambda_{3}}{\lambda_{3}-\lambda_{1}}$, and substituting
this into (15):

$\begin{gathered}\frac{-(\lambda_{1}-\lambda_{2})^{2}(\lambda_{2}+\lambda_{3})}{(\lambda_{3}-\lambda_{1})(\lambda_{2}-\lambda_{3})}+(\lambda_{1}\lambda_{3}-\lambda_{2}^{2})V+\frac{(\lambda_{1}-\lambda_{2})^{2}(\lambda_{3}^{2}-\lambda_{1}\lambda_{2})}{(\lambda_{3}-\lambda_{1})^{2}(\lambda_{2}-\lambda_{3})}W\\\
+\frac{-(\lambda_{1}(\lambda_{2}-\lambda_{3})^{2}+\lambda_{2}(\lambda_{1}-\lambda_{3})^{2}+\lambda_{3}(\lambda_{1}-\lambda_{2})^{2})}{(\lambda_{3}-\lambda_{1})(\lambda_{2}-\lambda_{3})}VW\\\
+\frac{\lambda_{2}(\lambda_{1}-\lambda_{3})}{(\lambda_{3}-\lambda_{2})}VW^{2}+\frac{\lambda_{3}(\lambda_{1}-\lambda_{2})}{(\lambda_{2}-\lambda_{3})}V^{2}W=0\end{gathered}$

Here, if $W=0$, then
$(\lambda_{1}\lambda_{3}-\lambda_{2}^{2})V=\frac{(\lambda_{1}-\lambda_{2})^{2}(\lambda_{2}+\lambda_{3})}{(\lambda_{3}-\lambda_{1})(\lambda_{2}-\lambda_{3})}$
has a unique solution for $V$. If $W\neq 0$, then we can choose $W=1$, and the
equation simplifies:

$V^{2}-\frac{2(\lambda_{1}-\lambda_{2})}{\lambda_{3}-\lambda_{1}}V+\frac{(\lambda_{1}-\lambda_{2})^{2}}{(\lambda_{3}-\lambda_{1})^{2}}=0,$

which is a complete square, therefore has a unique solution for $V$. That is,
$L_{[1:0]}$ and $C$ have two intersection points:

$P=\left[\frac{\lambda_{2}-\lambda_{3}}{\lambda_{3}-\lambda_{1}}:\frac{\lambda_{1}-\lambda_{2}}{\lambda_{3}-\lambda_{1}}:1\right],\hskip
8.5359ptQ=\left[\frac{\lambda_{2}-\lambda_{3}}{\lambda_{3}-\lambda_{1}}:\frac{(\lambda_{1}-\lambda_{2})^{2}(\lambda_{2}+\lambda_{3})}{(\lambda_{3}-\lambda_{1})(\lambda_{2}-\lambda_{3})(\lambda_{1}\lambda_{3}-\lambda_{2}^{2})}:0\right]$

The other case, if $t_{1}\neq 0$ (assume $t_{1}=1$), then
$V=t_{0}\left(X-\frac{\lambda_{2}-\lambda_{3}}{\lambda_{3}-\lambda_{1}}\right)+\frac{\lambda_{1}-\lambda_{2}}{\lambda_{3}-\lambda_{1}}$.
If $W\neq 0$ ($W=1$), then substitute the above equation for $V$ into (15)

$\displaystyle\frac{1}{\lambda_{2}-\lambda_{3}}\left(X-\frac{\lambda_{2}-\lambda_{3}}{\lambda_{3}-\lambda_{1}}\right)^{2}\left(X\left(\frac{(\lambda_{1}-\lambda_{3})^{2}(\lambda_{1}\lambda_{3}-\lambda_{2}^{2})}{(\lambda_{2}-\lambda_{3})^{2}}t_{0}\right.\right.$
$\displaystyle\left.+\frac{-(\lambda_{1}-\lambda_{2})^{2}(\lambda_{1}-\lambda_{3})^{2}(\lambda_{2}+\lambda_{3})}{(\lambda_{2}-\lambda_{3})^{3})}\right)+\lambda_{3}(\lambda_{1}-\lambda_{2})t_{0}^{2}$
$\displaystyle+\frac{-\lambda_{1}^{2}\lambda_{2}-2\lambda_{1}^{2}\lambda_{3}+6\lambda_{1}\lambda_{2}\lambda_{3}-2\lambda_{2}^{2}\lambda_{3}-\lambda_{2}\lambda_{3}^{2}}{\lambda_{2}-\lambda_{3}}t_{0}$
$\displaystyle\left.+\frac{(\lambda_{1}-\lambda_{2})(\lambda_{1}^{2}\lambda_{2}+2\lambda_{1}^{2}\lambda_{3}-4\lambda_{1}\lambda_{2}\lambda_{3}+\lambda_{2}^{2}\lambda_{3}+\lambda_{2}\lambda_{3}^{2})}{(\lambda_{2}-\lambda_{3})^{2}}\right)=0$

This has solution $X=\frac{\lambda_{2}-\lambda_{3}}{\lambda_{3}-\lambda_{1}}$,
which provides $P$. The remaining factor is linear in $X$, so for fixed
$t_{0}$, it has a unique solution for $X$, except if the coefficient of $X$ is
$0$, i.e.:
$t_{0}^{\prime}=\frac{(\lambda_{1}-\lambda_{2})^{2}(\lambda_{2}+\lambda_{3})}{(\lambda_{2}-\lambda_{3})(\lambda_{1}\lambda_{3}-\lambda_{2}^{2})}$,
this case the only intersection point of $C$ and $L_{[t_{0}:t_{1}]}$ is $P$.
In other cases, there are one more intersection point besides $P$, except when
the root of the linear factor provides $P$ again. This happens if the
following quadratic equation satisfies for $t_{0}$:

$\displaystyle\lambda_{3}(\lambda_{1}-\lambda_{2})t_{0}^{2}+\frac{(\lambda_{1}-\lambda_{2})(-\lambda_{1}\lambda_{2}-3\lambda_{1}\lambda_{3}+3\lambda_{2}\lambda_{3}+\lambda_{3}^{2})}{\lambda_{2}-\lambda_{3}}t_{0}$
$\displaystyle+\frac{(\lambda_{1}-\lambda_{2})(2\lambda_{1}^{2}\lambda_{2}+2\lambda_{1}^{2}\lambda_{3}-\lambda_{1}\lambda_{2}^{2}-6\lambda_{1}\lambda_{2}\lambda_{3}-\lambda_{1}\lambda_{3}^{2}+2\lambda_{2}^{2}\lambda_{3}+2\lambda_{2}\lambda_{3}^{2})}{(\lambda_{2}-\lambda_{3})^{2}}=0$

This has solutions
$t_{0}^{+}=\frac{2\lambda_{1}-\lambda_{2}-\lambda_{3}}{\lambda_{2}-\lambda_{3}}$
and
$t_{0}^{-}=\frac{\lambda_{1}\lambda_{2}+\lambda_{1}\lambda_{3}-2\lambda_{2}\lambda_{3}}{\lambda_{3}(\lambda_{2}-\lambda_{3})}$.
One can check that for $t_{0}^{+}$, the line $L_{[t_{0}^{+}:1]}$ passes
through $[0:1:0]$, so it has two intersection points with $C$. But $C$ has
only two common points with the line $W=0$, namely $[0:1:0]$ and $Q$, thus
$L_{[t_{0}^{-}:1]}$ intersects $C$ only at $P$. We deduce that there are
exactly two parameters $[t_{0}^{\prime}:1]$ and $[t_{0}^{+}:1]$ for which
$L_{[t_{0}:t_{1}]}$ has only one intersection point with $C$, namely $P$. This
means that the map $\mathbb{C}P^{1}\rightarrow C$ sending $[t_{0}:t_{1}]$ to
the other intersection of $L_{[t_{0}:t_{1}]}$ and $C$ (besides $P$) is one-to-
one, except for $[t_{0}^{\prime}:1]$ and $[t_{0}^{+}:1]$. That is, $C$ is
homeomorphic with a $\mathbb{C}P^{1}$ with $[t_{0}^{\prime}:1]$ and
$[t_{0}^{+}:1]$ identified. ∎

###  3.2 The trace coordinates

We will use the so called trace coordinates on the Betti moduli space,
introduced by Lawton [28], [29]. Let $\rho$ be at the SL(3,$\mathbb{C}$)
character variety of a rank 2 free group $\langle a,b\rangle$, and consider
the character map $\mathcal{M}_{B}\rightarrow\mathbb{C}^{9}$:

$\begin{gathered}\rho\mapsto\left(\operatorname{tr}(\rho(a)),\operatorname{tr}(\rho(b)),\operatorname{tr}(\rho(a)\rho(b)),\operatorname{tr}(\rho(a)^{-1}),\operatorname{tr}(\rho(b)^{-1}),\operatorname{tr}((\rho(a)\rho(b))^{-1}),\right.\\\
\left.\operatorname{tr}(\rho(a)\rho(b)^{-1}),\operatorname{tr}(\rho(a)^{-1}\rho(b)),\operatorname{tr}(\rho(a)\rho(b)\rho(a)^{-1}\rho(b)^{-1})\right)=:(x_{1},x_{2},...,x_{9}).\end{gathered}$
(16)

This way the above map gives coordinates on $\mathcal{M}_{B}$, under the
condition $x_{9}^{2}-p(\underline{x})x_{9}+q(\underline{x})=0$, where $p$ and
$q$ are two polynomials in the variables $\\{x_{i}\\}_{i=1}^{9}$, see [28,
Section 4]. On the three-punctured sphere, where
$\gamma_{1},\gamma_{2},\gamma_{3}$ are simple loops around the punctures
($0,1,\infty$), with a common base point, $[\gamma_{1}]$ and $[\gamma_{2}]$
generate the fundamental group of the curve, while
$[\gamma_{3}]=([\gamma_{1}][\gamma_{2}])^{-1}$. With our special choice of
eigenvalues for $\rho([\gamma_{1}]),\rho([\gamma_{2}])$, described in Section
2.2, we have the following: $x_{1}=x_{2}=x_{3}=x_{4}=x_{5}=x_{6}=0$, and
$x_{7},x_{8},x_{9}$ are the nonzero coordinates. Indeed, the above character
map gives a $\mathcal{M}_{B}(\textbf{c})\rightarrow\mathbb{C}^{3}$ morphism,
and the polynomials simplify to $p(\underline{x})=x_{7}x_{8}-3$, and
$q(\underline{x})=9-6x_{7}x_{8}+x_{7}^{3}+x_{8}^{3}$. Thus the condition
$x_{9}^{2}-p(\underline{x})x_{9}+q(\underline{x})=0$ reads as

$0=x_{9}^{2}-x_{7}x_{8}x_{9}+3x_{9}+9-6x_{7}x_{8}+x_{7}^{3}+x_{8}^{3}.$

The homogenisation of this will again provide the equation of the curve at the
infinity: $0=-x_{7}x_{8}x_{9}+x_{7}^{3}+x_{8}^{3}$.

######  Lemma 3.2.

This curve $0=-x_{7}x_{8}x_{9}+x_{7}^{3}+x_{8}^{3}$ is also of type $I_{1}$.

###### Proof.

One can easily see that $(x_{7},x_{8})=(0,0)$ is its only singular point. We
may set near that point $x_{9}=1$, and the equation becomes $x_{7}x_{8}\approx
0$ up to terms of degree $3$. This shows that the singularity is a node.
Therefore the singular cubic curve must be a singular elliptic curve, and we
can apply Kodaira’s classification, which shows that an elliptic curve with
one nodal singularity must be an $I_{1}$ curve. ∎

Figure 1: The $-x_{7}x_{8}+x_{7}^{3}+x_{8}^{3}=0$ nodal (fishtail) curve.

######  Corollary 3.3.

The fundamental group of the body of the dual boundary complex of the
compactifying divisor $D_{B}$ is cyclic, so
$\lvert\mathcal{D}\partial\mathcal{M}_{B}(\textbf{c})\rvert$ is of homotopy
type $S^{1}$.

Notice that we used two different types of coordinates, but the two pictures
coincide, as expected: for our special choice of parameters, the compactifying
divisor of the Betti space is topologically the same as in the general
description. According to Simpson [44], the Riemann Hilbert correspondence
provides equivalence between vector bundles with integrable connection and
representations of the fundamental group, via the monodromies of the
connection around the punctures. Therefore our aim is to apply the trace
coordinates to the monodromies of the connection, and investigate their
asymptotic behaviour.

##  4 Asymptotic analysis of non-abelian Hodge correspondence

Consider a Higgs bundle
$(\mathcal{E},\theta_{1})\in\mathcal{M}_{\textrm{Dol}}(\alpha)$ such that
$H(\mathcal{E},\theta_{1})=q_{1}$ where $q_{1}$ is defined in (7). For any
nonzero complex number $\tau$, the Higgs bundle
$(\mathcal{E},\overline{\partial}_{E},\tau\theta_{1})$ is also stable with the
same parabolic structure, therefore it gives another element of
$\mathcal{M}_{\textrm{Dol}}(\alpha)$. For this Higgs bundle we clearly have
$H(\mathcal{E},\tau\theta_{1})=\tau^{3}q_{1}=tq_{1}$, and the real Hitchin’s
equation reads as

$F_{h_{\tau}}+|\tau|^{2}[\theta_{1},\theta_{1}^{{\dagger}_{h_{\tau}}}]=0,$

where $h_{\tau}$ is the unique Hermitian–Einstein metric solving the equation.
We consider the 1-parameter family of Higgs bundles
$(\mathcal{E},\overline{\partial}_{E},\tau\theta_{1})$, where
$|\tau|\rightarrow\infty$, or equivalently the family
$(\mathcal{E},\overline{\partial}_{E},\theta_{t})$, where
$|tq|\rightarrow\infty$ in the Hitchin base $\mathcal{H}$ (for $\theta_{t}$,
see (14)). In this section, we analyze the limiting behavior of the
correspondence given in Theorem 2.4 1.

###  4.1 The limit of the Hermitian–Einstein metric

In [37], T. Mochizuki and the second author developed tools to understand the
limiting behaviour of the solution $h_{\tau}$ of Hitchin’s real equation, as
$|\tau|\to\infty$. The aim of this section is to describe the asymptotic
behaviour of $h_{\tau}$ under this limit in our particular case, based on
[37].

####  4.1.1 Reduction to the case $t>0$

Let us work in polar coordinates $t=Re^{i\varphi}$, see (9). Let us set333To
match the assumption of [37] completely, we should take a complex manifold for
$\mathcal{S}$; an open annulus with trivial radial dependence of the objects
is appropriate.

$\mathcal{S}=S^{1}=\\{e^{i\varphi}|\;\varphi\in[0,2\pi]\\}$

Set $X=\mathbb{C}P^{1}$ and $\mathcal{D}=\mathcal{S}\times D$. Set

$\widetilde{\mathcal{X}}=\\{(e^{i\varphi},(z,\zeta))|\;\zeta^{3}=e^{i\varphi}q(z)\\}\subset\mathcal{S}\times
T^{*}X.$

The family version [37, Theorem 1.8] applied to this family
$\widetilde{\mathcal{X}}$ and parameter $R>0$ (more precisely,
$|\tau|=R^{1/3}$) converging to $\infty$ (instead of
$\tau\in\mathbb{C}^{\times}$) shows that the results stated below hold for the
parameter $\tau$ too. In the sequel, we will replace limits $R\to\infty$ by
$|\tau|\to\infty$.

####  4.1.2 Convergence to a decoupled harmonic metric

A Hermitian metric $h$ on $E$ is said to be decoupled harmonic for
$(\mathcal{E},\theta)$ if the equations

$F_{h}=0=[\theta,\theta^{{\dagger}_{h}}]$

hold. It is clear that $h$ is decoupled harmonic for $(\mathcal{E},\theta)$ if
and only if it is decoupled harmonic for $(\mathcal{E},\tau\theta)$ for some
(equivalently, any) $\tau\neq 0$.

######  Theorem 4.1.

[37, Theorem 1.4] Fix a stable, generically semi-simple Higgs bundle
$(\mathcal{E},\overline{\partial}_{E},\theta)$ of $0$ parabolic degree and let
$h_{\tau}$ be its Hermitian–Einstein metric. If $\mathcal{E}$ is the direct
image of a line bundle on the normalization of $\Sigma_{t}$, then the sequence
$h_{\tau}$ converges to a decoupled harmonic metric $h_{\infty}$ locally on
$\mathbb{C}P^{1}\setminus D$ in the $C^{\infty}$ topology, as
$|\tau|\to\infty$.

######  Remark 4.2.

The generic semi-simplicity and direct image of a line bundle conditions
trivially hold in our particular case. A priori, the limit $h_{\infty}$ could
depend on the argument $\varphi$ of $t$. However, the interpretation in terms
of canonical decomposable filtration (see the next subsection) shows that this
is not the case.

####  4.1.3 Canonical decomposable filtration

A crucial notion for Theorem 4.1 is that of canonical decomposable filtration
$\mathcal{P}^{\star}_{*}$ for the Higgs bundle, see [37, Definition 5.10] (and
[48, Claim 2.1] for a similar condition in the unramified irregular
situation). If the Higgs bundle is generically semi-simple (which clearly
holds in the case at hand), then in general there exists a degree $\ell$
covering $\varphi_{\ell}$ of $X=\mathbb{C}P^{1}$ ramified over $D$ such that
locally near the points $D$ there exists a direct sum decomposition

$\varphi_{\ell}^{*}(\mathcal{E},\theta)\cong\bigoplus_{i=1}^{r}(\mathcal{E}_{i},\theta_{i})$

with $\mathcal{E}_{i}$ of rank $1$. (In our case, we have $\ell=r=3$.) A
filtration $\mathcal{P}_{*}$ is said to be decomposable if for any
$\alpha\in\mathbb{R}$ we have

$\varphi_{\ell}^{*}\mathcal{E}_{\alpha}=\bigoplus_{i=1}^{r}\mathcal{E}_{i,\alpha},$

where we use $\mathcal{E}_{\alpha}$ for the associated $\mathbb{R}$-parabolic
sheaves.

For our purposes, we only need the case of a maximally ramified spectral
curve, which implies that it is locally connected, i.e. $S=1$ in [37, Section
5.2.2]. Let $\mathcal{P}_{*}\operatorname{det}\,(\mathcal{E})$ be the
filtration induced by (2) on $\operatorname{det}\,(\mathcal{E})$. In this
case, [37, Proposition 5.13] states that there exists a unique decomposable
filtered bundle $\mathcal{P}^{\star}_{*}(\mathcal{E})$ such that
$\operatorname{det}\,\mathcal{P}^{\star}_{*}(\mathcal{E})=\mathcal{P}_{*}\operatorname{det}\,(\mathcal{E})$.
It is easy to see that with our choices of weights $\underline{\alpha}_{P}$,
$\mathcal{P}_{*}\operatorname{det}\,(\mathcal{E})$ is the trivial filtration
(simply because the weights at $P$ add up to $0$). We call
$\mathcal{P}^{\star}_{*}(\mathcal{E})$ the canonical decomposable filtration.

####  4.1.4 The induced Hermitian metric on the determinant line bundle

We study the Hermitian metric $h_{\tau,\operatorname{det}\,E}$ on
$\operatorname{det}\,E$ induced by $h_{\tau}$.

######  Lemma 4.3.

The induced parabolic bundle $\operatorname{det}\,\mathcal{E}$ is the trivial
line bundle with trivial parabolic filtration. With respect to a suitable
global trivialization of $\operatorname{det}\,\mathcal{E}$, we have
$h_{\operatorname{det}\,\mathcal{E}}\equiv 1$.

###### Proof.

We observe that since $\operatorname{tr}\,\theta\equiv 0$, i.e. the Higgs
field takes values in $\mathfrak{sl}(3,\mathbb{C})$, the metric
$h_{\tau,\operatorname{det}\mathcal{E}}$ turns $\operatorname{det}E$ into a
flat unitary parabolic line bundle. Moreover, the parabolic weight of
$\operatorname{det}\mathcal{E}$ at $P\in D$ is the sum of the parabolic
weights of $E$ at $P$, so it vanishes. Now, $\operatorname{det}\mathcal{E}$ is
a line bundle of degree $0$ over $\mathbb{C}P^{1}$, therefore it is isomorphic
to $\mathcal{O}_{\mathbb{C}P^{1}}$. Therefore, by uniqueness of the
Hermitian–Einstein metric, we see that with respect to a global trivialization
of $\mathcal{O}$, the metric $h_{\operatorname{det}\mathcal{E}}$ is some
constant. By rescaling the trivialization of $\mathcal{O}$, we can arrange
$h_{\operatorname{det}\mathcal{E}}\equiv 1$.∎

####  4.1.5 Limiting metric

Let us be given a Hermitian metric on a tame harmonic bundle
$(\mathcal{E},\theta)$ with parabolic divisor $D$. Let $P\in D$ and $z$ be a
holomorphic chart centered at $P$. Simpson’s theory [46] induces a metric
filtration on $\mathcal{E}$ given by

$\mathcal{P}^{h}_{a}\mathcal{E}(U)=\left\\{f\in\mathcal{E}(U\setminus
P)|\;\mbox{for
any}\;\varepsilon>0,\;|f|_{h}=O\left(|z|^{-a-\varepsilon}\right)\right\\}.$

By [37, Lemma 7.4, Theorem 7.5], the metric $h_{\infty}$ appearing in Theorem
4.1 is the unique decoupled harmonic metric satisfying the properties:

  1. 1.

$\mathcal{P}^{h_{\infty}}_{*}$ agrees with the canonical decomposable
filtration $\mathcal{P}^{\star}_{*}$ on $\mathcal{E}$,

  2. 2.

$\operatorname{det}h_{\infty}=h_{\operatorname{det}\,\mathcal{E}}$.

In Lemma 4.8 we will determine $\mathcal{P}^{\star}_{*}\mathcal{E}$.

####  4.1.6 Hermitian metric on the spectral line bundle

We use $\zeta\mapsto\tau\zeta$ with $\tau^{3}=t$ to identify $\Sigma_{t}$ with
$\Sigma_{1}$, compatibly with $p_{L}$. As previously discussed, let
$\mathcal{L}$ be the line bundle such that $\mathcal{E}=p_{*}\mathcal{L}$.

Now, let us equip $\mathcal{L}\rightarrow\Sigma_{1}$ with a parabolic
structure. For all the lifted points $\widetilde{P}\in\widetilde{D}$ on the
spectral curve let us set the parabolic weights to be
$\widetilde{\alpha}_{\widetilde{P}}=-1$.Then $\textrm{pdeg}\,\mathcal{L}=0$,
because $\textrm{deg}\,\mathcal{L}=3$. By [46], [2], there exists a unique
metric $h_{\mathcal{L}}$ that

  1. 1.

is adapted to the above parabolic structure

  2. 2.

solves the abelian Hitchin’s equation, and

  3. 3.

induces the metric $h_{\operatorname{det}\mathcal{E}}$.

######  Remark 4.4.

For
$\mathcal{L}=\mathcal{L}_{0}=\mathcal{O}_{\widetilde{\Sigma}_{t}}(\widetilde{D})$,
$h_{\mathcal{L}}$ induces on

$\mathcal{L}_{0}\otimes\mathcal{O}_{\widetilde{\Sigma}_{t}}(-\widetilde{D})\cong\mathcal{O}_{\widetilde{\Sigma}_{t}}$

a flat metric compatible with the trivial parabolic structure. It follows that
this induced metric is $h\equiv 1$ with respect to a global trivialization.

We define the orthogonal pushforward $p_{*}h_{\mathcal{L}}$ of the metric
$h_{\mathcal{L}}$ as the unique metric on $p_{*}\mathcal{L}$ for which

  1. 1.

the eigenspaces of $\theta$ are orthogonal to each other, i.e. for any local
sections $l_{1},l_{2}$ of $\mathcal{L}$ supported on different sheets of
$\widetilde{\Sigma}_{t}$, we have

$p_{*}h_{\mathcal{L}}((p_{L})_{*}l_{1},(p_{L})_{*}l_{2})=0,$

  2. 2.

for any local section $l$ of $\mathcal{L}$ supported on a single sheet of
$\widetilde{\Sigma}_{t}$, we have

$p_{*}h_{\mathcal{L}}((p_{L})_{*}l,(p_{L})_{*}l)=h_{\mathcal{L}}(l,l)$

By [37, Proposition 2.3], there exists some flat Hermitian metric on
$\mathcal{L}$ whose orthogonal pushforward is $h_{\infty}$. We next show that
these definitions of $h_{\infty}$ and $h_{\mathcal{L}}$ are compatible.

######  Lemma 4.5.

The metric $h_{\infty}$ agrees with the orthogonal pushforward
$p_{*}h_{\mathcal{L}}$ of $h_{\mathcal{L}}$.

###### Proof.

The filtration induced on $\mathcal{E}$ by the parabolic filtration of
$\mathcal{L}$ defined above is the canonical decomposable one (see Lemma 4.8).
The statement therefore essentially reduces to [37, Proposition 5.34]. ∎

####  4.1.7 Cover of the underlying curve

We now set $N=\overline{B}_{\delta}(P)$ for $P\in D$ and some $0<\delta\ll 1$
(see Figure 2). In the sequel, we will make use of a cover of the surface
$\mathbb{C}P^{1}\setminus D$ by open regions $U_{1},U_{2}$ where

$\displaystyle U_{1}$ $\displaystyle=\mathbb{C}P^{1}\setminus\bigcup_{P\in
D}\overline{B}_{\delta}(P)$ (17) $\displaystyle U_{2}$
$\displaystyle=\bigcup_{P\in D}B_{2\delta}(P).$ (18)

We pick a loop $\gamma_{P}$ in the annulus $U_{1}\cap
U_{2}=B_{2\delta}(P)\setminus\overline{B}_{\delta}(P)$ winding around $P$ once
in positive direction. On the other hand, we pick a path $\eta_{P}$ in $U_{1}$
from the base point $z_{0}$ to the starting point of $\gamma_{P}$, and also a
path $\sigma_{P}$ from $P$ to the starting point of $\gamma_{P}$, see Figure
2.

Figure 2: The base point $z_{0}=\frac{1}{2}$ and path $\eta_{P}$ approaching
the point $P\in D$, and the setup near $P$: the loop $\gamma_{P}$ lying in the
annulus $B_{2\delta}(P)\setminus\overline{B}_{\delta}(P)$, and $\sigma_{P}$
connecting $P$ and the starting point of $\gamma_{P}$, where $P\in\\{0,1\\}$.

###  4.2 Analysis away from the ramification divisor

In this section we will work over the set (17). Let $U\subset U_{1}$ be any
simply connected open subset such that

$p_{L}^{-1}(U)\cap\widetilde{\Sigma}_{t}=V_{1}\sqcup V_{2}\sqcup V_{3},$

the restriction of $p_{L}$ to each $V_{i}$ being a homeomorphism onto $U$.

####  4.2.1 Induced unitary frame of $\mathcal{E}$

Over $U$, the relation $(p_{L})_{*}\mathcal{L}=\mathcal{E}$ reads as

${\left.\kern-1.2pt\mathcal{E}\vphantom{\big{|}}\right|_{U}}={\left.\kern-1.2pt\mathcal{L}\vphantom{\big{|}}\right|_{V_{1}}}\oplus{\left.\kern-1.2pt\mathcal{L}\vphantom{\big{|}}\right|_{V_{2}}}\oplus{\left.\kern-1.2pt\mathcal{L}\vphantom{\big{|}}\right|_{V_{3}}}.$

Theorem 4.1 states in particular that
$[\theta,\theta^{{\dagger}_{h_{\infty}}}]=0$. Let $\mathbf{l}_{i}$ be a
$h_{\mathcal{L}}$-unitary trivialization of
${\left.\kern-1.2pt\mathcal{L}\vphantom{\big{|}}\right|_{V_{i}}}$ and define

$\mathbf{f}_{i}=(p_{L})_{*}\mathbf{l}_{i}.$

By Lemma 4.5,

$(\mathbf{f}_{1},\mathbf{f}_{2},\mathbf{f}_{3})$ (19)

is a smooth $h_{\infty}$-unitary trivialization of
${\left.\kern-1.2ptE\vphantom{\big{|}}\right|_{U}}$. We will use this frame to
express $\tau\theta_{1},\nabla_{\tau}$ throughout this section. In particular,
$\theta,\theta^{{\dagger}_{h_{\infty}}}$ both have diagonal matrices, and the
matrix of $h_{\infty}$ is the $3\times 3$ identity.

####  4.2.2 Flat unitary connection induced by $h_{\infty},\mathcal{E}$

The holomorphic structure on the line bundle $\mathcal{L}$ is provided by the
operator $\overline{\partial}_{\mathcal{L}}=\overline{\partial}+\mu$, for some
$\mu\in\Omega^{0,1}(\widetilde{\Sigma}_{t})$. With respect to (19), the
holomorphic structure of
${\left.\kern-1.2pt\mathcal{E}\vphantom{\big{|}}\right|_{U}}$ reads as

$\overline{\partial}_{E}=\overline{\partial}+\begin{pmatrix}\mu_{1}&0&0\\\
0&\mu_{2}&0\\\ 0&0&\mu_{3}\end{pmatrix},$

with $\mu_{i}$ the restriction of $\mu$ to $V_{i}$. The metric
$h_{\mathcal{L}}$ is Hermitian–Einstein, thus the associated unitary
connection $\nabla_{h_{\mathcal{L}}}^{+}$ is flat on each $V_{i}$. Let us
denote its connection form by

$u_{i}=\mu_{i}-\overline{\mu}_{i}\in\Omega^{1}(V_{i},\sqrt{-1}\mathbb{R}).$

Then the flat $\textrm{U(1)}^{\times 3}$-connection form on
${\left.\kern-1.2pt\mathcal{E}\vphantom{\big{|}}\right|_{U}}$, associated with
$h_{\infty}$ is

$\nabla_{h_{\infty}}^{+}=\begin{pmatrix}u_{1}&0&0\\\ 0&u_{2}&0\\\
0&0&u_{3}\end{pmatrix}.$

####  4.2.3 Family of flat connections induced by $h_{\infty},\theta$

As we have already mentioned, the Higgs field is diagonal in the frame (19),
with its eigenvalues (11) as diagonal elements. We are ready to express the
integrable connection associated to the decoupled metric $h_{\infty}$ over $U$
with respect to the unitary frame (19):

$\begin{gathered}\nabla_{\tau}^{\operatorname{appr}}=\nabla_{h_{\infty}}^{+}+\tau\theta_{1}+(\tau\theta_{1})^{{\dagger}_{h_{\infty}}}=\nabla_{h_{\infty}}^{+}+\tau\theta_{1}+h_{\infty}^{-1}\overline{\tau\theta_{1}}^{T}h_{\infty}=\\\
=\operatorname{d}+\left(\begin{smallmatrix}u_{1}&0&0\\\ 0&u_{2}&0\\\
0&0&u_{3}\end{smallmatrix}\right)+\left(\begin{smallmatrix}\tau
Q_{1}+\overline{\tau Q_{1}}&0&0\\\ 0&\tau Q_{2}+\overline{\tau Q_{2}}&0\\\
0&0&\tau Q_{3}+\overline{\tau Q_{3}}\end{smallmatrix}\right)=\\\
=\operatorname{d}+\left(\begin{smallmatrix}u_{1}+2\operatorname{Re}(\tau
Q_{1})&0&0\\\ 0&u_{2}+2\operatorname{Re}(\varepsilon\tau Q_{1})&0\\\
0&0&u_{3}+2\operatorname{Re}(\varepsilon^{2}\tau
Q_{1})\end{smallmatrix}\right)\end{gathered}$ (20)

Notice that the formula

$u_{i}+2\operatorname{Re}(\varepsilon^{i-1}Q_{t})$

is a decomposition into imaginary and real part.

######  Remark 4.6.

Over the Hitchin section, i.e. for
$\mathcal{L}=\mathcal{L}_{0}=\mathcal{O}_{\widetilde{\Sigma}_{t}}(\widetilde{D})$,
we have $\mu\equiv 0$ for the $(0,1)$-partial connection defining
$\mathcal{L}_{0}(-\widetilde{D})$. This then implies $u_{i}\equiv 0$ for all
$1\leq i\leq 3$, and the limiting connection simplifies to

$\operatorname{d}+2\operatorname{Re}\left(\begin{smallmatrix}\tau Q_{1}&0&0\\\
0&\varepsilon\tau Q_{1}&0\\\ 0&0&\varepsilon^{2}\tau
Q_{1}\end{smallmatrix}\right).$

####  4.2.4 Estimate for the associated integrable connection

Let $h_{\tau}$ denote the metric solving the Hitchin’s equations for
$(\mathcal{E},\tau\theta_{1})$, and $\nabla_{\tau}$ the associated integrable
connection (5). On the other hand, we have the approximate flat connection
(20). We consider the Sobolev norm $\|\cdot\|_{L^{2}_{l}(U,h_{\infty})}$ on
sections of $End(E)\otimes\Omega^{1}$ defined using the standard Riemannian
metric $|\operatorname{d}\\!z|^{2}$ on $\mathbb{C}P^{1}$, the fiber metric
$h_{\infty}$ and the Killing form.

######  Proposition 4.7.

Fix $l\in\mathbb{N}$. Over any simply connected open subset $U$ of $U_{1}$,
there exist $C_{1}>0,\epsilon_{1},R_{1}>0$ depending only on $l$ and
$\delta>0$ (used to define $U_{1}$), such that for all $|\tau|>R_{1}$

$\|\nabla_{\tau}-\nabla_{\tau}^{\operatorname{appr}}\|_{L^{2}_{l}(U,h_{\infty})}\leq
C_{1}e^{-\epsilon_{1}|\tau|}.$

In particular, there exists $C_{2}>0,\epsilon_{2},R_{1}>0$ depending only on
$\delta>0$ such that for all $|\tau|>R_{1}$

$\|\nabla_{\tau}-\nabla_{\tau}^{\operatorname{appr}}\|_{C^{0}(U,h_{\infty})}\leq
C_{2}e^{-\epsilon_{2}|\tau|}.$

###### Proof.

Let $s(h_{\infty},h_{\tau})$ be defined by $h_{\tau}=h_{\infty}\cdot
s(h_{\infty},h_{\tau})$. We then have

$\displaystyle\|\nabla_{h_{\tau}}^{+}-\nabla_{h_{\infty}}^{+}\|_{L^{2}_{l}(U)}$
$\displaystyle=\|\partial\left(\log h_{\tau}-\log
h_{\infty}\right)\|_{L^{2}_{l}(U)}$ $\displaystyle=\|\partial\log
s(h_{\infty},h_{\tau})\|_{L^{2}_{l}(U)}$ $\displaystyle\leq
2\|\partial(s(h_{\infty},h_{\tau})-\operatorname{id})\|_{L^{2}_{l}(U)}$
$\displaystyle\leq
2\|s(h_{\infty},h_{\tau})-\operatorname{id}\|_{L^{2}_{l+1}(U)}$
$\displaystyle\leq C_{1}e^{-\epsilon_{1}|\tau|},$

where $\log$ stands for matrix logarithm and the last estimate follows from
[37, Theorem 7.14].

For the other term, up to increasing $C_{1}$ and decreasing $\epsilon_{1}$ we
have

$\displaystyle\|h_{\infty}^{-1}\overline{\tau\theta_{1}}^{T}h_{\infty}-h_{\tau}^{-1}\overline{\tau\theta_{1}}^{T}h_{\tau}\|_{L^{2}_{l}(U)}^{2}$
$\displaystyle=\|-(\operatorname{Ad}(s(h_{\infty},h_{\tau})^{-1})-\operatorname{id})\operatorname{Ad}(h_{\infty}^{-1})(\overline{\tau\theta_{1}}^{T})\|_{L^{2}_{l}(U)}^{2}$
$\displaystyle\leq
4\|s(h_{\infty},h_{\tau})-\operatorname{id}\|_{L^{2}_{l}(U)}\cdot\|\operatorname{Ad}(h_{\infty}^{-1})(\overline{\tau\theta_{1}}^{T})\|_{L^{2}_{l}(U)}$
$\displaystyle\leq
C_{1}e^{-\epsilon_{1}|\tau|}\|\operatorname{Ad}(h_{\infty}^{-1})(\overline{\tau\theta_{1}}^{T})\|_{L^{2}_{l}(U)}$

by Cauchy–Schwartz. Now, the second factor of the last line can be bounded
from above by $C_{1}^{\prime}|\tau|$ for some $C_{1}^{\prime}>0$, so it can be
absorbed by the first term (possibly up to increasing $C_{1}$ again).

The last statement immediately follows from Sobolev’s embedding theorem. ∎

###  4.3 Near the ramification divisor

In this section we describe asymptotic behaviour of solutions of Hitchin’s
equations on (18). Obviously, it is sufficient to deal with $B_{2\delta}(P)$
for a given $P\in D$, see Figure 2, where $0<\delta\ll 1$. Let us consider the
cyclic triple cover

$\varphi_{3}\colon\widetilde{B}_{2\delta}(P)\to B_{2\delta}(P)$

defined by $\widetilde{\Sigma}_{1}$, i.e.

$\zeta^{3}=z(z-1),$

where we use the trivialization

$\frac{\operatorname{d}\\!z}{z(z-1)}$ (21)

of $L=K(D)$.

####  4.3.1 Frames and parabolic structure

Let us fix a trivialization
$\mathcal{L}\cong\mathcal{O}_{\widetilde{B}_{2\delta}(P)}$, i.e. a nowhere
vanishing holomorphic section $\mathbf{l}$ of $\mathcal{L}$ over
$\widetilde{B}_{2\delta}(P)$. Then, over
$B_{2\delta}(P)\subset\mathbb{C}P^{1}$, $\mathcal{E}$ can be decomposed as

${\left.\kern-1.2pt\mathcal{E}\vphantom{\big{|}}\right|_{B_{2\delta}(P)}}\cong\mathcal{O}_{B_{2\delta}(P)}\oplus\mathcal{O}_{B_{2\delta}(P)}\cdot\zeta\oplus\mathcal{O}_{B_{2\delta}(P)}\cdot\zeta^{2}$
(22)

(see (13)), and $\theta$ is the operation induced by multiplication by
$\zeta$. Its residue is nilpotent with respect to the filtration

$0\subset\mathcal{O}_{B_{2\delta}(P)}\cdot\zeta^{2}\subset\mathcal{O}_{B_{2\delta}(P)}\cdot\zeta\oplus\mathcal{O}_{B_{2\delta}(P)}\cdot\zeta^{2}\subset{\left.\kern-1.2pt\mathcal{E}\vphantom{\big{|}}\right|_{B_{2\delta}(P)}}.$

A metric $h$ is compatible with the parabolic structure if and only if

$|f_{1}(z)+f_{2}(z)\cdot\zeta+f_{3}(z)\cdot\zeta^{2}|_{h}^{2}\approx|f_{1}(z)|^{2}|z|^{-2/3}+|f_{2}(z)|^{2}+|f_{3}(z)|^{2}|z|^{2/3}.$

Recall the definition of the metric $h_{\mathcal{L}}$ from Section 4.1.6. Let
us set

$h\colon\widetilde{B}_{2\delta}(P)\to\mathbb{R},\quad
h(\zeta)=h_{\mathcal{L}}(\mathbf{l}(\zeta),\mathbf{l}(\zeta)).$ (23)

Recall that

$h\approx|\zeta|^{-2}$

because the parabolic weight of $\mathcal{L}$ is $-1$. With respect to the
decomposition (22), the metric $h_{\infty}$ is then given by the matrix

$h\begin{pmatrix}1&\zeta&\zeta^{2}\\\
\bar{\zeta}&\bar{\zeta}\zeta&\bar{\zeta}\zeta^{2}\\\
\bar{\zeta}^{2}&\bar{\zeta}^{2}\zeta&\bar{\zeta}^{2}\zeta^{2}\end{pmatrix}.$

####  4.3.2 Diagonalizing trivialization

For $0<\delta\ll 1$, with respect to the trivialization (21) of $L$ let us use
the approximations up to higher order terms on $\gamma_{0},\gamma_{1}$, near
the points $0,1\in\mathbb{C}P^{1}$:

${\left.\kern-1.2pt\theta_{t}\vphantom{\big{|}}\right|_{\gamma_{0}}}\approx\begin{pmatrix}0&0&tz\\\
1&0&0\\\ 0&1&0\end{pmatrix},\hskip
8.5359pt{\left.\kern-1.2pt\theta_{t}\vphantom{\big{|}}\right|_{\gamma_{1}}}\approx\begin{pmatrix}0&0&t(z-1)\\\
1&0&0\\\ 0&1&0\end{pmatrix}.$

We focus on the first one, i.e. the neighbourhood of 0, but a similar analysis
holds near $1$ too.

The aim of this section is to introduce a smooth approximately unitary frame

$(\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3})$ (24)

defined over $\widetilde{B}_{2\delta}(P)$ in which the Higgs field is
diagonal. We will expand the vectors (24) in coordinates with respect to (22).
They can be determined uniquely (up to phase) as the unit length eigenvectors
of ${\left.\kern-1.2pt\theta_{t}\vphantom{\big{|}}\right|_{\gamma_{0}}}$. A
diagonalizing frame is given by

$\displaystyle\mathbf{e}_{1}^{\prime}$
$\displaystyle=\begin{pmatrix}(tz)^{2/3}\\\ (tz)^{1/3}\\\ 1\end{pmatrix}$
$\displaystyle\mathbf{e}_{2}^{\prime}$
$\displaystyle=\begin{pmatrix}\varepsilon^{2}(tz)^{2/3}\\\
\varepsilon(tz)^{1/3}\\\ 1\end{pmatrix}$
$\displaystyle\mathbf{e}_{3}^{\prime}$
$\displaystyle=\begin{pmatrix}\varepsilon(tz)^{2/3}\\\
\varepsilon^{2}(tz)^{1/3}\\\ 1\end{pmatrix}$

where we have used the cubic root of unity $\varepsilon=e^{2\pi i/3}$. Then,
the expression of the frame (24) with respect to (22) is:

$\displaystyle\mathbf{e}_{1}$
$\displaystyle=\frac{1}{|\mathbf{e}_{1}^{\prime}|_{h_{\infty}}}\begin{pmatrix}(tz)^{2/3}\\\
(tz)^{1/3}\\\ 1\end{pmatrix}$ $\displaystyle\mathbf{e}_{2}$
$\displaystyle=\frac{1}{|\mathbf{e}_{2}^{\prime}|_{h_{\infty}}}\begin{pmatrix}\varepsilon^{2}(tz)^{2/3}\\\
\varepsilon(tz)^{1/3}\\\ 1\end{pmatrix}$ $\displaystyle\mathbf{e}_{3}$
$\displaystyle=\frac{1}{|\mathbf{e}_{3}^{\prime}|_{h_{\infty}}}\begin{pmatrix}\varepsilon(tz)^{2/3}\\\
\varepsilon^{2}(tz)^{1/3}\\\ 1\end{pmatrix}.$

####  4.3.3 Identifying the canonical decomposable filtration

We are ready to proving:

######  Lemma 4.8.

With our choices, the canonical decomposable filtration
$\mathcal{P}^{\star}_{*}\mathcal{E}$ is the trivial filtration

$\mathcal{P}^{\star}_{a}\mathcal{E}=z^{-[a]}\cdot\mathcal{E},$

where

$[a]=\max\\{n\in\mathbb{Z}|\quad n\leq a\\}$

is the usual floor function.

###### Proof.

We focus on $P=0$. Set $U=B_{2\delta}(0),U^{(3)}=\widetilde{B}_{2\delta}(0)$
and let

$\varphi_{3}\colon U^{(3)}\to U$

be the covering $\zeta^{3}=z(z-1)$, as before. Let

$U^{(3),*}=\varphi_{3}^{-1}(U^{*})=\varphi_{3}^{-1}(U-\\{0\\}).$

By the formulas of the previous section, we have

$\mathbf{e}_{1}\wedge\mathbf{e}_{2}\wedge\mathbf{e}_{3}=zg(z)$

in $U$ for some holomorphic function $g$ with $g(0)\neq 0$.

The sum of the parabolic weights being $0$, the induced filtration
$\mathcal{P}_{*}\operatorname{det}\,\mathcal{E}$ is trivial. So, we have

$\mathbf{e}_{1}\wedge\mathbf{e}_{2}\wedge\mathbf{e}_{3}\in\varphi_{3}^{*}(\mathcal{P}_{1}\operatorname{det}\,\mathcal{E})$

and

$\mathbf{e}_{1}\wedge\mathbf{e}_{2}\wedge\mathbf{e}_{3}\notin\varphi_{3}^{*}(\mathcal{P}_{b}\operatorname{det}\,\mathcal{E})$

for any $b<1$. The formula after [37, Lemma 5.2] specializes to

$\mathcal{P}_{a}(\varphi_{3}^{*}\operatorname{det}\,\mathcal{E})=\sum_{b\in\mathbb{R},k\in\mathbb{Z},rb+k\leq
a}\zeta^{-k}\varphi_{3}^{*}(\mathcal{P}_{b}\operatorname{det}\mathcal{E}).$

Let us apply it with $k=-3,b=1$ to obtain that

$\mathbf{e}_{1}\wedge\mathbf{e}_{2}\wedge\mathbf{e}_{3}\in\mathcal{P}_{0}(\varphi_{3}^{*}\operatorname{det}\mathcal{E}),$

and

$\mathbf{e}_{1}\wedge\mathbf{e}_{2}\wedge\mathbf{e}_{3}\notin\mathcal{P}_{a}(\varphi_{3}^{*}\operatorname{det}\mathcal{E})$

for any $a<0$. This gives the value $0$ for the constant $b$ in the proof of
[37, Proposition 5.13]. The canonical decomposable filtration on the $i$-th
component is then defined by

$\mathcal{P}^{\star}_{a}(\mathcal{O}_{U^{(3),*}}\mathbf{e}_{i})=\zeta^{-[a]}\mathcal{O}_{U^{(3)}}\mathbf{e}_{i}.$

Said differently, the canonical decomposable filtration on the $i$-th
component is the trivial filtration. Next, this is extended to
$\varphi_{3}^{*}\mathcal{E}$ by direct sum:

$\mathcal{P}^{\star}_{a}(\varphi_{3}^{*}\mathcal{E})=\bigoplus_{i=1}^{3}\mathcal{P}^{\star}_{a}(\mathcal{O}_{U^{(3),*}}\mathbf{e}_{i}).$

It is immediate that this is the trivial filtration, because we have the
trivial filtration on all summands. At last, we define the canonical
decomposable filtration on $E$ as the Galois descent of this filtration. The
statement follows. ∎

####  4.3.4 Equivariance and scaling properties

We discuss some further consequences of the formula for (24).

First, for given $|t|=R$ the choice of radius $r=R^{-1}$ guarantees that the
coordinates of (24) remain bounded, namely if we switch to local polar
coordinates on the circle $\gamma_{0}(t)$ of radius $r=\frac{3\delta}{2}$
centered at $0$, with angle coordinate $\vartheta\in[0,2\pi]$, they are given
by the following (recalling again that $\varepsilon=e^{2\pi\sqrt{-1}/3}$)

$\displaystyle\mathbf{e}_{1}$
$\displaystyle\approx\frac{1}{\sqrt{3}}\begin{pmatrix}e^{2\sqrt{-1}(\vartheta+\varphi)/3}\\\
e^{\sqrt{-1}(\vartheta+\varphi)/3}\\\ 1\end{pmatrix}$
$\displaystyle\mathbf{e}_{2}$
$\displaystyle\approx\frac{1}{\sqrt{3}}\begin{pmatrix}e^{2\sqrt{-1}(\vartheta+\varphi+2\pi)/3}\\\
e^{\sqrt{-1}(\vartheta+\varphi+2\pi)/3}\\\ 1\end{pmatrix}$
$\displaystyle\mathbf{e}_{3}$
$\displaystyle\approx\frac{1}{\sqrt{3}}\begin{pmatrix}e^{2\sqrt{-1}(\vartheta+\varphi+4\pi)/3}\\\
e^{\sqrt{-1}(\vartheta+\varphi+4\pi)/3}\\\ 1\end{pmatrix}$

Our second observation relates to the deck transformation group
$\operatorname{Gal}(\varphi_{3})\cong\mu_{3}$ of $\varphi_{3}$. Let
$\gamma\in\operatorname{Gal}(\varphi_{3})$ be a generator.

######  Proposition 4.9.

Let $\mathcal{L}=\mathcal{L}_{0}$, i.e. we assume that $(\mathcal{E},\theta)$
lies on the Hitchin section. Then, the metric $h_{\infty}$ is
$\operatorname{Gal}(\varphi_{3})$-invariant, namely
$h(\gamma\cdot\zeta)=h(\zeta)$ for every $\zeta\in U^{(3)}$.

###### Proof.

There exists a canonical isomorphism
$\gamma^{*}\mathcal{O}_{\widetilde{\Sigma}_{t}}\cong\mathcal{O}_{\widetilde{\Sigma}_{t}}$.
It induces a canonical isomorphism
$\gamma^{*}\mathcal{O}_{\widetilde{\Sigma}_{t}}(\widetilde{D})\cong\mathcal{O}_{\widetilde{\Sigma}_{t}}(\widetilde{D})$
because $\widetilde{D}$ is fixed by $\operatorname{Gal}(\varphi_{3})$. The
metric $\gamma^{*}h_{\mathcal{L}_{0}}$ is Hermitian–Einstein on
$\mathcal{L}_{0}$, and compatible with the same parabolic structure. By
uniqueness, we get $\gamma^{*}h_{\mathcal{L}_{0}}=c\cdot h_{\mathcal{L}_{0}}$
for some $c\in\mathbb{R}_{+}$. Because $\gamma^{3}=1$, we have $c^{3}=1$. The
conclusion follows. ∎

##  5 Asymptotic analysis of the Riemann Hilbert correspondence

In this section, our aim is to apply the Riemann–Hilbert map to the integrable
connection associated by the non-abelian Hodge correspondence to a Higgs
bundle in a Hitchin fiber close to infinity. The asymptotic connection form of
such integrable connections was obtained in Section 4. The procedure consists
in determining the monodromy on a positively oriented loop around the
punctures. More precisely, let us fix some base point
$z_{0}\in\mathbb{C}P^{1}\setminus D$, and consider the concatenations of paths
$\eta_{P}*\gamma_{P}*\eta_{P}^{-1}$ where, as usual, $\eta_{P}^{-1}$ is the
path $\eta_{P}$ with opposite orientation (see Fig.2). This is clearly a
simple loop around the puncture $P\in D$ separating it from the other two
punctures. We need to determine the parallel transport maps along these paths
of the flat connections whose approximations were given in Section 4, and
determine the monodromies by integrating the connection forms on the paths and
exponentiating. We will do so separately on the paths $\gamma_{P}$ and
$\eta_{P}$, and the mondoromy on the concatenation will be the triple product
of the parallel transport maps corresponding to the above decomposition into
paths. From now on, the notation ”$\approx$” will mean ”agrees up to
exponentially small terms”, or more precisely:

$A\approx B\iff AB^{-1}=\operatorname{id}+O(e^{-\epsilon R^{1/3}})$

for some $\epsilon>0$. This implies in particular that if $B$ has an
asymptotic expansion as $R\to\infty$, then the same holds for $A$ too with the
same expansion. Notice that by Baker–Campbell–Dynkin–Hausdorff, $A\approx B$
is equivalent to requiring

$B^{-1}A=\operatorname{id}+O(e^{-\epsilon R^{1/3}}).$

###  5.1 Hitchin WKB problem

In this section we recall results of [36, Section 2.4]. We use the simplifying
convention that the values of $C,\epsilon>0$ may change from one occurrence to
the other. Moreover, in this section for simplicity we replace $R^{1/3}$ by
$R$.

####  5.1.1 The dilation spectrum

Let $(V_{0},h_{0}),(V_{1},h_{1})$ be finite dimensional Hermitian vector
spaces, of dimension $r$ over $\mathbb{C}$. Let $\Pi\colon V_{0}\to V_{1}$ be
a linear isomorphism. According to the spectral theorem applied to the
positive symmetric matrix $h_{1}(e_{i}(0),e_{j}(0))$ where $e_{i}(0)$ is any
orthonormal basis of $(V_{0},h_{0})$, there exist orthonormal bases
$(q_{1}(s),\ldots q_{r}(s))$ of $(V_{s},h_{s})$ ($s\in\\{0,1\\}$) such that

$\Pi(q_{j}(0))=e^{\beta_{j}}q_{j}(1)\quad 1\leq j\leq r$

for some real numbers

$\beta_{1}\geq\beta_{2}\geq\cdots\beta_{r}.$

The vector

$(\beta_{1},\ldots,\beta_{r})\in\mathbb{R}^{r}$

is called the _dilation spectrum_ of $\Pi$, see [23, Definition 4.2, Lemma
4.5].

####  5.1.2 Non-critical paths

Let $X$ be any Riemann surface (in our case, an open subset of
$\mathbb{C}P^{1}\setminus D$). Let us fix any smooth path

$\gamma\colon[0,1]\to X.$

Consider maps

$\displaystyle A$ $\displaystyle\colon X\to\mathfrak{gl}(n,\mathbb{C})$
$\displaystyle U$ $\displaystyle\colon X\to\mathfrak{gl}(n,\mathbb{R})$

satisfying $[A(z),U(z)]=0$ for every $z\in X$. Then, locally on $[0,1]$ there
exists a common diagonalizing frame of $\gamma^{*}A$ and $\gamma^{*}U$. We may
assume that this is the standard basis, so that
$\gamma^{*}A,\sqrt{-1}\gamma^{*}U$ are diagonal matrices. Let

$a_{1}(t)\operatorname{d}\\!t,\ldots,a_{n}(t)\operatorname{d}\\!t\in\Omega^{1}([0,1],\mathbb{C})$

stand for the eigenvalues of $A(t)$,

$u_{1}(t)\operatorname{d}\\!t,\ldots,u_{n}(t)\operatorname{d}\\!t\in\Omega^{1}([0,1],\sqrt{-1}\mathbb{R})$

stand for those of $\sqrt{-1}U(t)$ with respect to this basis. We will write
$\mathfrak{t}$ for the Cartan subalgebra of diagonal matrices and $T$ for the
corresponding maximal torus.

We make the assumption that $\gamma$ is non-critical in the following sense.

######  Assumption 1.

For any $t\in[0,1]$ the matrix $\operatorname{Re}A(\gamma(t))$ is regular
semi-simple, i.e. for any $1\leq j<j^{\prime}\leq n$ we have

$\operatorname{Re}a_{j}(t)\neq\operatorname{Re}a_{j^{\prime}}(t).$

Up to a permutation, we may and will assume

$\operatorname{Re}a_{j}(t)<\operatorname{Re}a_{j^{\prime}}(t)\quad\mbox{for
all}\;1\leq j<j^{\prime}\leq n.$

Let us set

$\alpha_{j}=-\int_{0}^{1}\operatorname{Re}a_{j}(t)\operatorname{d}\\!t\in\mathbb{R}.$

####  5.1.3 Dilation exponents of parallel transport of harmonic bundles

We consider the Higgs bundle over $X$ of the form

$(\mathcal{E},R\theta)\cong\bigoplus_{j=1}^{r}(\mathcal{E}_{j},R\phi_{j})$
(25)

where $\mathcal{E}_{j}$ is a holomorphic line bundle. We assume

$\gamma^{*}\phi_{j}=a_{j}(t)\operatorname{d}\\!t$

and that Assumption 1 holds. Let $h_{R}$ be a harmonic metric for (25) and
$\nabla_{R}$ be the associated integrable connection. Let

$B\colon[1,\infty]\times X\to\mathfrak{gl}(n,\mathbb{C})$

be the connection matrix of $\nabla_{R}$ with respect to trivializations of
$\mathcal{E}_{j}$. We define the fundamental solution

$Y\colon\mathbb{C}^{\times}\times[0,1]\to\operatorname{GL}(r,\mathbb{C})$

as the solution of the initial value problem

$\frac{\operatorname{d}\\!Y(R,t)}{\operatorname{d}\\!t}=B(R,\gamma(t))Y(R,t),\quad
Y(R,0)=\operatorname{id}.$

We define

$\Pi_{\gamma}(R)=Y(R,1)\in\operatorname{GL}(r,\mathbb{C}).$

We define $(V_{s},h_{s})=(E_{\gamma(s)},(h_{R})_{\gamma(s)})$ for
$s\in\\{0,1\\}$. Then, $\Pi_{\gamma}(R)$ can be thought of as a linear map in
$\operatorname{Hom}(V_{0},V_{1})$. Let

$(\beta_{1}(R),\ldots,\beta_{r}(R))$

be the dilation spectrum of $\Pi_{\gamma}(R)$.

######  Theorem 5.1.

[36, Theorem 2.17] There exist $R_{0}\in\mathbb{R}_{+}$ such that for all
$R>R_{0}$ we have

$\left|\frac{1}{R}(\beta_{1}(R),\ldots,\beta_{r}(R))-(\alpha_{1},\ldots,\alpha_{r})\right|\leq
Ce^{-\epsilon R}.$

####  5.1.4 Explicit description of the parallel transform

The proof of Theorem 5.1 is obtained via an explicit description of the
parallel transform $\Pi_{\gamma}(R)$. Recall the diagonal connection form
(20). It is proven in [36, Theorem 2.17] that there exist

  1. 1.

an orthonormal frame $\mathbf{p}_{1}(0),\ldots,\mathbf{p}_{r}(0)$ of
${\left.\kern-1.2pt\mathcal{E}\vphantom{\big{|}}\right|_{\gamma(0)}}$,
connected to

$\mathbf{f}_{1}(\gamma(0)),\cdots,\mathbf{f}_{r}(\gamma(0))$

(see (19) for the particular case $r=3$) by some matrix $L(0)$,

  2. 2.

an orthonormal frame $\mathbf{p}_{1}(1),\ldots,\mathbf{p}_{r}(1)$ of
${\left.\kern-1.2pt\mathcal{E}\vphantom{\big{|}}\right|_{\gamma(1)}}$,
connected to

$\mathbf{f}_{1}(\gamma(1)),\cdots,\mathbf{f}_{r}(\gamma(1))$

by some matrix $L(1)$,

  3. 3.

$H\in\Omega^{1}([0,1],\mathfrak{t})$, with $j$-th diagonal entry denoted by
$h_{j}(t)\operatorname{d}\\!t$,

all these quantities depending on $R$, and fulfilling the following
properties:

  1. 1.

$\|L(s)-\operatorname{id}\|\leq Ce^{-\epsilon R}$ for $s\in\\{0,1\\}$,

  2. 2.

$\|H(t)\|\leq Ce^{-\epsilon R}$ for all $t\in[0,1]$,

  3. 3.

the matrix of $\Pi_{\gamma}(R)$ with respect to the bases
$\mathbf{p}_{1}(s),\ldots,\mathbf{p}_{r}(s)$ is given by the diagonal matrix

$\operatorname{diag}\left(\exp\left(R\alpha_{j}-\int_{0}^{1}u_{j}(t)+h_{j}(t)\operatorname{d}\\!t\right)\right)_{j=1}^{r},$

(see Section 5.1.2 for $\alpha_{j}$).

###  5.2 Monodromy of the diagonalizing frame

Using the results of Section 5.1, we now compute the monodromy along tiny
loops winding around the punctures once in positive direction. We will again
only treat the case of the loop $\gamma_{0}$ (see Figure 2), but the same
argument works for $\gamma_{1}$ too.

######  Proposition 5.2.

The monodromy matrix $M_{0,R,\varphi}$ of $\nabla_{t}$ with respect to the
frame (19) is given by $M_{0,R,\varphi}=TN_{0,R,\varphi}$, where

$N_{0,R,\varphi}=\left(\begin{smallmatrix}\exp(2(Rr)^{1/3}\operatorname{Re}(e^{i\varphi/3}3(\varepsilon-1)))&0&0\\\
0&\exp(2(Rr)^{1/3}\operatorname{Re}(\varepsilon
e^{i\varphi/3}3(\varepsilon-1)))&0\\\
0&0&\exp(2(Rr)^{1/3}\operatorname{Re}(\varepsilon^{2}e^{i\varphi/3}3(\varepsilon-1)))\end{smallmatrix}\right)$

and

$T=\begin{pmatrix}0&0&1\\\ 1&0&0\\\ 0&1&0\end{pmatrix}.$

###### Proof.

Obviously, the pull-back of the Higgs bundle to the spectral curve decomposes
as a direct sum of Higgs line bundles

$\varphi_{3}^{*}(\mathcal{E},\tau\theta_{1})\cong\bigoplus_{i=1}^{3}(\mathcal{O}_{U^{(3),*}}\mathbf{e}_{i},Q_{i,R,\varphi}),$

the summands being the restrictions of $\mathcal{L}$ to the sheets of
$\Sigma_{t}$. Recall the filtration $\mathcal{P}^{h_{\infty}}_{*}$ induced by
$h_{\infty}$. According to Lemma 4.8, $\mathcal{P}^{h_{\infty}}_{*}$ is the
trivial filtration. According to Lemma 4.5, $h_{\infty}$ is the direct image
of $h_{\mathcal{L}}$. It follows that $h_{\mathcal{L}}$ induces the trivial
filtration on $\mathcal{L}$. We infer that the pull-back by $\varphi_{3}$ of
the parallel transport of $\nabla_{t}$ is diagonal. In order to find its
eigenvalues, we need to find a fundamental solution of the local system.
Equivalently, we need to integrate the connection form (20) on $\gamma_{0}$,
that is:

$\left(\begin{smallmatrix}2R^{1/3}\operatorname{Re}(e^{i\varphi/3}\int_{\gamma_{0}}z^{-2/3}\operatorname{d}\\!z)&0&0\\\
0&2R^{1/3}\operatorname{Re}(\varepsilon
e^{i\varphi/3}\int_{\gamma_{0}}z^{-2/3}\operatorname{d}\\!z)&0\\\
0&0&2R^{1/3}\operatorname{Re}(\varepsilon^{2}e^{i\varphi/3}\int_{\gamma_{0}}z^{-2/3}\operatorname{d}\\!z)\end{smallmatrix}\right)+U_{0}.$

where $U_{0}$ is the diagonal matrix with diagonal entries
$\int_{\gamma_{0}}u_{i}$. The integrals appearing here can be determined
explicitly, because $\gamma_{0}$ is just the boundary of a disc of radius $r$
around $0\in\mathbb{C}P^{1}$:

$\int_{\gamma_{0}}z^{-2/3}\operatorname{d}\\!z=\int_{0}^{2\pi}\frac{rie^{i\vartheta}\operatorname{d}\\!\vartheta}{r^{2/3}e^{2i\vartheta/3}}=3r^{1/3}(\varepsilon-1)$

Now, to get the parallel transport map of $\varphi_{3}^{*}\nabla_{t}$, we take
the exponential of this matrix, and we obtain $N_{0}$. However this is not yet
the monodromy of the flat connection $\nabla_{t}$ on $\gamma_{0}$ with respect
to any basis, because when we transport the frame (24) around the loop in
positive direction, the effect on the frame is a cyclic permutation. Indeed,
one can easily check that

$\begin{gathered}\mathbf{e}_{1}(\vartheta=0)=\mathbf{e}_{3}(\vartheta=2\pi)\\\
\mathbf{e}_{3}(\vartheta=0)=\mathbf{e}_{2}(\vartheta=2\pi)\\\
\mathbf{e}_{2}(\vartheta=0)=\mathbf{e}_{1}(\vartheta=2\pi)\end{gathered}$

Thus the monodromy of the diagonalizing frame
$(\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3})$ is the permutation matrix
$T$, and we get the desired result

$M_{0,R,\varphi}=TN_{0,R,\varphi}e^{U_{0}}.$

∎

Notice that the monodromy $M_{1,R,\varphi}$ on $\gamma_{1}$ near the puncture
$1\in\mathbb{C}P^{1}$ is just the same, because the integrals coincide:

$\int_{\gamma_{0}}z^{-2/3}\operatorname{d}\\!z=\int_{\gamma_{1}}(z-1)^{-2/3}\operatorname{d}\\!z$

###  5.3 Parallel transport to the punctures and the whole monodromy

####  5.3.1 Choices of non-critical paths

We now fix our choice of the base point: we set
$z_{0}=\frac{1}{2}\in\mathbb{C}$, as well as the paths $\eta_{0},\eta_{1}$: we
let $\eta_{0}(t)=(1-t)\frac{1}{2}+tr$ and
$\eta_{1}(t)=(1-t)\frac{1}{2}+t(1-r)$. In words, we let $\eta_{0},\eta_{1}$ be
the straight line segments connecting $z_{0}$ to $r$ and $1-r$, respectively.
Recall the notion of a non-critical path from Assumption 1. Recall from the
polar coordinates (9) that $\operatorname{arg}(\tau)=\varphi/3$.

######  Lemma 5.3.

These paths $\eta_{0},\eta_{1}$ are non-critical for $\tau\theta_{1}$ if and
only if $\varphi=k\pi$ for some $k\in\mathbb{Z}$.

###### Proof.

Recall from (11) that the eigenvalues of $\tau\theta_{1}$ are equal to

$\varepsilon^{j-1}R^{1/3}e^{i\varphi/3}\frac{\operatorname{d}\\!z}{z^{2/3}(z-1)^{2/3}}.$

Clearly, $\eta_{0}^{*}\operatorname{d}\\!z,\eta_{1}^{*}\operatorname{d}\\!z$
are some nonzero real multiple of $\operatorname{d}\\!t$. Moreover, $t(t-1)<0$
for $0<t<1$ implies $t^{-2/3}(t-1)^{-2/3}>0$. The constant $R^{1/3}$ is real
too. So, we need to solve the equation

$\operatorname{Re}\left(\varepsilon^{j-1}e^{i\varphi/3}\right)=\operatorname{Re}\left(\varepsilon^{j^{\prime}-1}e^{i\varphi/3}\right)$

for $0\leq j\neq j^{\prime}\leq 2$. This happens if and only if $\varphi/3$ is
an integer multiple of $\pi/3$. ∎

Therefore, Assumption 1 for $\gamma=\eta_{0},\eta_{1}$ translates into:

$\varphi\notin\mathbb{Z}\cdot\pi.$ (26)

We say that the angles $\varphi\in\mathbb{Z}\cdot\pi$ are _critical angles of
the first kind_.

####  5.3.2 Product of parallel transport maps

Under the assumption (26), it follows from Theorem 5.1 that the matrices of
the parallel transport maps of (20) along $\eta_{0}$ and $\eta_{1}$ with
respect to the frame (19) are respectively asymptotic to

$\begin{gathered}L_{0,t}=\left(\begin{smallmatrix}\exp(\int_{\eta_{0}}u_{1}+2\operatorname{Re}\int_{\eta_{0}}Q_{t})&0&0\\\
0&\exp(\int_{\eta_{0}}u_{2}+2\operatorname{Re}(\varepsilon\int_{\eta_{0}}Q_{t}))&0\\\
0&0&\exp(\int_{\eta_{0}}u_{3}+2\operatorname{Re}(\varepsilon^{2}\int_{\eta_{0}}Q_{t}))\end{smallmatrix}\right)\\\
L_{1,t}=\left(\begin{smallmatrix}\exp(\int_{\eta_{1}}u_{1}+2\operatorname{Re}\int_{\eta_{1}}Q_{t})&0&0\\\
0&\exp(\int_{\eta_{1}}u_{2}+2\operatorname{Re}(\varepsilon\int_{\eta_{1}}Q_{t}))&0\\\
0&0&\exp(\int_{\eta_{1}}u_{3}+2\operatorname{Re}(\varepsilon^{2}\int_{\eta_{1}}Q_{t}))\end{smallmatrix}\right)\end{gathered}$

Over $\eta_{0}$ we had smooth unitary frame (19), and over $\gamma_{0}$ we had
another smooth unitary frame (24). These two frames both diagonalize $\theta$,
and are connected by a unitary transformation. Given that
$\theta(\eta_{0}(1))$ is regular semi-simple, we may therefore assume that
these two frames agree at $\eta_{0}(1)=r>0$. Denoting the monodromy of
$\nabla_{\tau}$ along $\eta_{0}*\gamma_{0}*\eta_{0}^{-1}$ and
$\eta_{1}*\gamma_{1}*\eta_{1}^{-1}$ by $A_{t}$ and $B_{t}$ respectively, we
have

$\begin{gathered}A_{t}=A_{R,\varphi}=L_{0,t}^{-1}TN_{0,R,\varphi}e^{U_{0}}L_{0,t}\\\
B_{t}=B_{R,\varphi}=L_{1,t}^{-1}TN_{1,R,\varphi}e^{U_{1}}L_{1,t}.\end{gathered}$
(27)

After we execute the matrix multiplications, we get that both matrices have
the following form similar to a companion matrix:

$A_{t}=A_{R,\varphi}=\begin{pmatrix}0&0&A_{3}(t)\\\ A_{1}(t)&0&0\\\
0&A_{2}(t)&0\end{pmatrix},\hskip
5.69046ptB_{t}=B_{R,\varphi}=\begin{pmatrix}0&0&B_{3}(t)\\\ B_{1}(t)&0&0\\\
0&B_{2}(t)&0\end{pmatrix}.$

####  5.3.3 Asymptotic behaviour of monodromy maps and period integrals

######  Proposition 5.4.

For the nonzero elements of $A_{t}$, as $t\rightarrow\infty$ we have

$\begin{gathered}A_{1}(t)\approx\operatorname{exp}\left(R^{1/3}\operatorname{Re}(e^{i\varphi/3}\pi_{0})+\int_{\eta_{0}}(u_{1}-u_{2})+\int_{\gamma_{0}}u_{1}\right)\\\
A_{2}(t)\approx\operatorname{exp}\left(R^{1/3}\operatorname{Re}(e^{i\varphi/3}\varepsilon\pi_{0})+\int_{\eta_{0}}(u_{2}-u_{3})+\int_{\gamma_{0}}u_{2}\right)\\\
A_{3}(t)\approx\operatorname{exp}\left(R^{1/3}\operatorname{Re}(e^{i\varphi/3}\varepsilon^{2}\pi_{0})+\int_{\eta_{0}}(u_{3}-u_{1})+\int_{\gamma_{0}}u_{3}\right)\end{gathered}$

where
$\pi_{0}=2(\varepsilon-1)\int^{z_{0}}_{0}\frac{\operatorname{d}\\!z}{z^{2/3}(z-1)^{2/3}}$
along $\sigma_{0}*\eta_{0}^{-1}$. Similarly for $B_{t}$ we have

$\begin{gathered}B_{1}(t)\approx\operatorname{exp}\left(R^{1/3}\operatorname{Re}(e^{i\varphi/3}\pi_{1})+\int_{\eta_{1}}(u_{1}-u_{2})+\int_{\gamma_{1}}u_{1}\right)\\\
B_{2}(t)\approx\operatorname{exp}\left(R^{1/3}\operatorname{Re}(e^{i\varphi/3}\varepsilon\pi_{1})+\int_{\eta_{1}}(u_{2}-u_{3})+\int_{\gamma_{1}}u_{2}\right)\\\
B_{3}(t)\approx\operatorname{exp}\left(R^{1/3}\operatorname{Re}(e^{i\varphi/3}\varepsilon^{2}\pi_{1})+\int_{\eta_{1}}(u_{3}-u_{1})+\int_{\gamma_{1}}u_{3}\right)\end{gathered}$

where
$\pi_{1}=2(\varepsilon-1)\int^{z_{0}}_{1}\frac{\operatorname{d}\\!z}{z^{2/3}(z-1)^{2/3}}$
along $\sigma_{1}*\eta_{1}^{-1}$.

###### Proof.

From the matrix multiplication (27) we get

$\displaystyle
A_{1}(t)=\exp\left(-\int_{\eta_{0}}u_{2}-2\operatorname{Re}\left(\varepsilon\int_{\eta_{0}}Q_{t}\right)+\int_{\eta_{0}}u_{1}+2\operatorname{Re}\left(\int_{\eta_{0}}Q_{t}\right)\right)\cdot$
$\displaystyle\cdot\,\exp\left(2R^{1/3}r^{1/3}\operatorname{Re}(e^{i\varphi/3}3(\varepsilon-1))+\int_{\gamma_{0}}u_{1}\right)$

Now, using the notation
$Q_{t}=R^{1/3}e^{i\varphi/3}\frac{\operatorname{d}\\!z}{z^{2/3}(z-1)^{2/3}}$
from (11), we can compute the improper integral:

$\int_{\sigma_{0}}Q_{t}\approx
R^{1/3}e^{i\varphi/3}\int_{0}^{r}\frac{\operatorname{d}\\!z}{z^{2/3}}=3r^{1/3}$

So in the above formula, the real part of the exponent of $A_{1}(t)$ reads as:

$\displaystyle-2\operatorname{Re}\left(\varepsilon\int_{\eta_{0}}Q_{t}\right)+2\operatorname{Re}\left(\int_{\eta_{0}}Q_{t}\right)+2R^{1/3}\operatorname{Re}(e^{i\varphi/3}3r^{1/3}(\varepsilon-1))\approx$
$\displaystyle\approx
R^{1/3}2\operatorname{Re}\left(e^{i\varphi/3}(\varepsilon-1)\int_{\eta_{0}^{-1}}\frac{\operatorname{d}\\!z}{z^{2/3}(z-1)^{2/3}}+e^{i\varphi/3}(\varepsilon-1)\int_{\sigma_{0}}\frac{\operatorname{d}\\!z}{z^{2/3}(z-1)^{2/3}}\right),$

where we used that
$(1-\varepsilon)\int_{\eta_{0}}Q_{t}=(\varepsilon-1)\int_{\eta_{0}^{-1}}Q_{t}$.
Now, using the notation $\pi_{0}$ given in the assertion and the fact that the
concatenation $\sigma_{0}*\eta_{0}^{-1}$ is a path form $0$ to $z_{0}$, the
formula for $A_{1}(t)$ becomes

$\displaystyle
A_{1}(t)\approx\operatorname{exp}\left(R^{1/3}\operatorname{Re}(e^{i\varphi/3}\pi_{0})+\int_{\eta_{0}}(u_{1}-u_{2})+\int_{\gamma_{0}}u_{1}\right)$

Similarly

$\displaystyle
A_{2}(t)=\exp\left(-\int_{\eta_{0}}u_{3}-2\operatorname{Re}\left(\varepsilon^{2}\int_{\eta_{0}}Q_{t}\right)+\int_{\eta_{0}}u_{2}+2\operatorname{Re}\left(\varepsilon\int_{\eta_{0}}Q_{t}\right)\right)\cdot$
$\displaystyle\cdot\,\exp\left(2R^{1/3}r^{1/3}\operatorname{Re}(e^{i\varphi/3}3\varepsilon(\varepsilon-1))+\int_{\gamma_{0}}u_{2}\right)\approx$
$\displaystyle\approx\operatorname{exp}\left(R^{1/3}\operatorname{Re}(e^{i\varphi/3}\varepsilon\pi_{0})+\int_{\eta_{0}}(u_{2}-u_{3})+\int_{\gamma_{0}}u_{2}\right)$
$\displaystyle
A_{3}(t)=\exp\left(-\int_{\eta_{0}}u_{1}-2\operatorname{Re}\left(\int_{\eta_{0}}Q_{t}\right)+\int_{\eta_{0}}u_{3}+2\operatorname{Re}\left(\varepsilon^{2}\int_{\eta_{0}}Q_{t}\right)\right)\cdot$
$\displaystyle\cdot\,\exp\left(2R^{1/3}r^{1/3}\operatorname{Re}(e^{i\varphi/3}3\varepsilon^{2}(\varepsilon-1))+\int_{\gamma_{0}}u_{3}\right)\approx$
$\displaystyle\approx\operatorname{exp}\left(R^{1/3}\operatorname{Re}(e^{i\varphi/3}\varepsilon^{2}\pi_{0})+\int_{\eta_{0}}(u_{3}-u_{1})+\int_{\gamma_{0}}u_{3}\right)$

The same computations go along for the elements of the $B_{t}$ matrix, with
integrals on $\sigma_{1}*\eta_{1}^{-1}$. Notice that the results do not depend
on the parameter $r$, as we obtained the complete improper integrals from $0$
and $1$ to $z_{0}$. ∎

##  6 WKB asymptotics of the trace coordinates and proof of the Geometric P=W
conjecture

Finally, in this section we will compute the asymptotic behaviour of the trace
coordinates introduced in Section 3.2 based on the approximations for the
monodromy matrices determined in Section 5.3. Curiously, we will find that a
structure reminiscent to the Stokes phenomenon governs their behaviour.
Namely, depending on sectors in the Hitchin base, different exponential terms
of the coordinates dominate.

###  6.1 A topological reformulation of the Geometric P=W conjecture

Remember from (9) that the polar coordinates
$(R,\varphi)\in\mathbb{R_{+}}\times S^{1}$ parameterize the Hitchin base
(except for its origin), and the Dolbeault moduli space admits the Hitchin
fibration over the Hitchin base. Now consider $1\ll R$ fixed, and $\varphi$
ranging over $[0,2\pi]$, parameterizing the circle $S^{1}_{R}$ of radius $R$
in the base. The Hitchin section provides an algebraic lift

$\sigma\colon S^{1}_{R}\to\mathcal{M}_{\textrm{Dol}}(\alpha)$

of the loop $Re^{i\varphi}$ to the Dolbeault space for the Hitchin fibration
$H$.

######  Proposition 6.1.

Assume that
$\operatorname{dim}_{\mathbb{R}}\mathcal{M}_{\textrm{Dol}}(\alpha)=4$. Assume
that $S\circ\psi\circ\sigma$ induces an isomorphism

$\mathbb{Z}\cong\pi_{1}(S^{1}_{R})\to\pi_{1}(\lvert\mathcal{D}\partial\mathcal{M}_{B}\rvert)\cong\mathbb{Z}$

for $R\gg 1$. Then the diagram (1) is commutative up to homotopy.

###### Proof.

The argument of [39, Section 5] applies verbatim. It only depends on the
condition that the moduli space is a $4$-manifold, not on the rank nor the
type of singularities of the underlying Higgs bundles. ∎

Notice that in this proof we do not even need algebraicity (or even
analyticity) of $\sigma$. Therefore, the proof of Theorem 1.1 reduces to the
following.

######  Proposition 6.2.

The map $S\circ\psi\circ\sigma$ induces an isomorphism

$\mathbb{Z}\cong\pi_{1}(S^{1}_{R})\to\pi_{1}(\lvert\mathcal{D}\partial\mathcal{M}_{B}\rvert)\cong\mathbb{Z}$

for $R\gg 1$.

We devote the rest of this section to proving this assertion. Notice that the
$S^{1}$ on the left-hand side does have a natural orientation coming from the
complex structure of the Hitchin base, but the one on the right hand side does
not have a preferred orientation.

###  6.2 Asymptotics of the trace coordinates

####  6.2.1 Trace coordinates

From the shape of the above determined matrices $A_{t}$ and $B_{t}$, it
trivially follows that six out of the nine coordinates from (16) are zero.

######  Proposition 6.3.

The only nonzero trace coordinates are $\operatorname{tr}(A_{t}B_{t}^{-1})$,
$\operatorname{tr}(A_{t}^{-1}B_{t})$ and
$\operatorname{tr}(A_{t}B_{t}A_{t}^{-1}B_{t}^{-1})$.

For convenience, instead of $x_{7},x_{8},x_{9}$ appearing in Section 3 we
introduce the notations

$\begin{gathered}X_{R,\varphi}=\operatorname{tr}(A_{R,\varphi}B_{R,\varphi}^{-1})=\operatorname{tr}(A_{t}B_{t}^{-1})\\\
Y_{R,\varphi}=\operatorname{tr}(A_{R,\varphi}^{-1}B_{R,\varphi})=\operatorname{tr}(A_{t}^{-1}B_{t})\\\
Z_{R,\varphi}=\operatorname{tr}(A_{R,\varphi}B_{R,\varphi}A_{R,\varphi}^{-1}B_{R,\varphi}^{-1})=\operatorname{tr}(A_{t}B_{t}A_{t}^{-1}B_{t}^{-1})\end{gathered}$

(These coordinates are not to be confused with $X,Y,Z$ appearing in Section 3,
as those will no longer be used in the sequel.) We will refer to $X,Y,Z$ as
trace coordinates.

######  Proposition 6.4.

$X_{R,\varphi},Y_{R,\varphi},Z_{R,\varphi}$ depend $\mathbb{R}$-analytically
with $R,\varphi$.

###### Proof.

It is known that $\psi$ is a $\mathbb{R}$-analytic map because it is a
composition of the $\mathbb{R}$-analytic map $\operatorname{NAHC}$ and the
$\mathbb{C}$-analytic map $\operatorname{RH}$. Moreover, the Hitchin section
$\sigma$ is $\mathbb{C}$-algebraic, in particular $\mathbb{C}$-analytic too.
Finally, the trace coordinates are clearly $\mathbb{C}$-algebraic too on the
character variety, and we conclude. ∎

######  Proposition 6.5.

We have the approximations for the three trace coordinates:

$\begin{gathered}X_{R,\varphi}\approx\exp\left(R^{1/3}\operatorname{Re}(e^{i\varphi/3}(\pi_{0}-\pi_{1}))+\int_{\eta_{0}-\eta_{1}}u_{3}-u_{1}+\int_{\gamma_{0}-\gamma_{1}}u_{3}\right)+\\\
+\exp\left(R^{1/3}\operatorname{Re}(e^{i\varphi/3}\varepsilon(\pi_{0}-\pi_{1}))+\int_{\eta_{0}-\eta_{1}}u_{1}-u_{2}+\int_{\gamma_{0}-\gamma_{1}}u_{1}\right)+\\\
+\exp\left(R^{1/3}\operatorname{Re}(e^{i\varphi/3}\varepsilon^{2}(\pi_{0}-\pi_{1}))+\int_{\eta_{0}-\eta_{1}}u_{2}-u_{3}+\int_{\gamma_{0}-\gamma_{1}}u_{2}\right),\\\
Y_{R,\varphi}\approx\exp\left(R^{1/3}\operatorname{Re}(e^{i\varphi/3}(\pi_{1}-\pi_{0}))-\int_{\eta_{0}-\eta_{1}}u_{3}-u_{1}-\int_{\gamma_{0}-\gamma_{1}}u_{3}\right)+\\\
+\exp\left(R^{1/3}\operatorname{Re}(e^{i\varphi/3}\varepsilon(\pi_{1}-\pi_{0}))-\int_{\eta_{0}-\eta_{1}}u_{1}-u_{2}-\int_{\gamma_{0}-\gamma_{1}}u_{1}\right)+\\\
+\exp\left(R^{1/3}\operatorname{Re}(e^{i\varphi/3}\varepsilon^{2}(\pi_{1}-\pi_{0}))-\int_{\eta_{0}-\eta_{1}}u_{2}-u_{3}-\int_{\gamma_{0}-\gamma_{1}}u_{2}\right),\\\
Z_{R,\varphi}\approx\exp\left(R^{1/3}\operatorname{Re}(e^{i\varphi/3}(1-\varepsilon)(\pi_{0}-\pi_{1}))+\int_{\eta_{0}-\eta_{1}}2u_{3}-u_{1}-u_{2}+\int_{\gamma_{0}-\gamma_{1}}u_{3}-u_{2}\right)+\\\
+\exp\left(R^{1/3}\operatorname{Re}(e^{i\varphi/3}(1-\varepsilon)\varepsilon(\pi_{0}-\pi_{1}))+\int_{\eta_{0}-\eta_{1}}2u_{1}-u_{2}-u_{3}+\int_{\gamma_{0}-\gamma_{1}}u_{1}-u_{3}\right)+\\\
+\exp\left(R^{1/3}\operatorname{Re}(e^{i\varphi/3}(1-\varepsilon)\varepsilon^{2}(\pi_{0}-\pi_{1}))+\int_{\eta_{0}-\eta_{1}}2u_{2}-u_{3}-u_{1}+\int_{\gamma_{0}-\gamma_{1}}u_{2}-u_{1}\right)\end{gathered}$
(28)

###### Proof.

Straightforward matrix multiplications using Proposition 5.4. ∎

######  Remark 6.6.

  1. 1.

We have

$\pi_{0}-\pi_{1}=2(\varepsilon-1)\int_{0}^{1}\frac{\operatorname{d}\\!t}{t^{2/3}(t-1)^{2/3}}\neq
0$

because the integrand is positive. In addition, we have

$\operatorname{arg}(\pi_{0}-\pi_{1})=\operatorname{arg}(\varepsilon-1)=\frac{5\pi}{6}.$

  2. 2.

The terms $\int_{\eta_{0}}u_{i}-u_{i+1}$, and $\int_{\gamma_{0}}u_{i}$
$i=1,2,3$ are purely imaginary, thus they have no contribution to the absolute
values of the coordinates. On the other hand, they affect the behaviour of the
phases of the coordinates of Betti space.

####  6.2.2 Maximal dilation exponents of the trace coordinates

According to Proposition 6.1 there remains to answer the question: where does
the Riemann–Hilbert correspondence map the loop $S^{1}_{R}$, as
$R\rightarrow\infty$?

For fixed $1\ll R$, and $\varphi$ ranging over $[0,2\pi]$ the coordinates
$X_{R,\varphi}$, $Y_{R,\varphi}$, $Z_{R,\varphi}$ describe the image of the
above mentioned loop $t=Re^{i\varphi}$ in the Hitchin base (6) under the
Riemann–Hilbert map. To prove the statement of the Geometric P=W conjecture,
we next need to investigate the growth order of the coordinates
$X_{R,\varphi}$, $Y_{R,\varphi}$, $Z_{R,\varphi}$ with respect to each other.

Introducing the notation

$x=e^{i\varphi/3}(\pi_{0}-\pi_{1})=a+b\sqrt{-1},$ (29)

we have

$\begin{gathered}\operatorname{Re}(x)=a,\hskip
11.38092pt\operatorname{Re}(\varepsilon
x)=-\frac{1}{2}a-\frac{\sqrt{3}}{2}b,\hskip
11.38092pt\operatorname{Re}(\varepsilon^{2}x)=-\frac{1}{2}a+\frac{\sqrt{3}}{2}b\\\
\operatorname{Re}(-x)=-a,\hskip 11.38092pt\operatorname{Re}(-\varepsilon
x)=\frac{1}{2}a+\frac{\sqrt{3}}{2}b,\hskip
11.38092pt\operatorname{Re}(-\varepsilon^{2}x)=-\frac{1}{2}a-\frac{\sqrt{3}}{2}b\\\
\operatorname{Re}((1-\varepsilon)x)=\frac{3}{2}a+\frac{\sqrt{3}}{2}b,\hskip
5.69046pt\operatorname{Re}((1-\varepsilon)\varepsilon x)=-\sqrt{3}b,\hskip
5.69046pt\operatorname{Re}((1-\varepsilon)\varepsilon^{2}x)=-\frac{3}{2}a+\frac{\sqrt{3}}{2}b\end{gathered}$
(30)

Consider $\lvert X_{R,\varphi}\rvert$ from (28). It consists of three
exponential terms. We are interested in the dominance properties among these
terms as $R\rightarrow\infty$, and similarly for $\lvert
Y_{R,\varphi}\rvert,\lvert Z_{R,\varphi}\rvert$. Recall from Lemma 5.3 that
the critical angles of the first kind are integer multiples of $\pi$.

######  Proposition 6.7.

Fix any $\varphi\in(0,\pi)\cup(\pi,2\pi)$. As $R\to\infty$, we have

$\displaystyle\frac{1}{R}\ln\lvert X_{R,\varphi}\rvert$
$\displaystyle\to\max(\operatorname{Re}(x),\operatorname{Re}(\varepsilon
x),\operatorname{Re}(\varepsilon^{2}x)),$ $\displaystyle\frac{1}{R}\ln\lvert
Y_{R,\varphi}\rvert$
$\displaystyle\to\max(\operatorname{Re}(-x),\operatorname{Re}(-\varepsilon
x),\operatorname{Re}(-\varepsilon^{2}x)),$ $\displaystyle\frac{1}{R}\ln\lvert
Z_{R,\varphi}\rvert$
$\displaystyle\to\max(\operatorname{Re}((1-\varepsilon)x),\operatorname{Re}((1-\varepsilon)\varepsilon
x),\operatorname{Re}((1-\varepsilon)\varepsilon^{2}x)).$

###### Proof.

This is a straightforward consequence of the formulas (28) and (30). ∎

######  Remark 6.8.

The proposition indicates that the affine coordinates of (32) near its node,
namely the quotients $X/Z$ and $Y/Z$, obey the rules of the tropical (i.e.,
max-plus) semi-ring. See [49, Conjecture 1].

####  6.2.3 Sectorial decomposition

Proposition 6.7 leads us to introduce 12 open sectors in the complex plane
parameterized by $x=a+b\sqrt{-1}$, determined by the terms that achieve the
respective maxima, and defined as follows (see also Figure 3):

$S_{j}=\left\\{x\in\mathbb{C}\colon\quad\frac{\pi(j-1)}{6}<\operatorname{arg}(x)<\frac{\pi
j}{6}\right\\},\hskip 8.5359ptj=1,2,\ldots,12$

Notice that by (29) and Remark 6.6, one has

$\operatorname{arg}(x)=\frac{\varphi}{3}+\frac{5\pi}{6}.$ (31)

As we see on Figure 3, the dividing lines between two neighboring sectors are
the lines $b=\pm\sqrt{3}a$, $b=\pm\frac{1}{\sqrt{3}}a$ and the two axes.
Define $R_{j}$ to be the ray between sectors $S_{j}$ and $S_{j+1}$ (where
$j+1$ is understood modulo 12). We will refer to these as the _Stokes rays_ ,
and the angles $\varphi$ defining them as _critical angles of the second
kind_. Namely, for the $j$’th critical angle of the second kind $\varphi_{j}$
we have

$\frac{\varphi_{j}}{3}=\frac{\pi j}{6},$

equivalently

$\varphi_{j}=\frac{\pi j}{2}.$

Crucially, any critical of the first kind (see (26)) is also critical of the
second kind. From now on, _critical_ will stand for critical of the second
kind.

####  6.2.4 Dominance of terms within the trace coordinates

Since our aim is to analyze the asymptotic behaviour of the coordinates, we
will make use of the usual concept of dominance: we say that function $F(R)$
dominates the function $G(R)$ if and only if $\frac{\lvert G(R)\rvert}{\lvert
F(R)\rvert}\rightarrow 0$ as $R\rightarrow\infty$. We denote this relation by
$F(R)\gg G(R)$. Crucially, by Theorem 5.1, all such convergences to follow are
exponentially suppressed with $R^{1/3}$.

######  Proposition 6.9.

Fix a non-critical $\varphi\in[0,2\pi]$, so that $x$ belongs to one of the
open sectors $S_{j}$. The dominant term of the coordinates $X,Y,Z$ in each of
the sectors $S_{j}$ is given by the following table, where the notation
$\approx$ implies convergent asymptotic expansions.

Sector | Dominant term of the trace coordinates
---|---
$S_{1}$ | $\lvert X_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(x))=\exp(R^{1/3}a)$
| $\lvert Y_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(-\varepsilon
x))=\exp(R^{1/3}(\frac{1}{2}a+\frac{\sqrt{3}}{2}b))$
| $\lvert
Z_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}((1-\varepsilon)x))=\exp(R^{1/3}(\frac{3}{2}a+\frac{\sqrt{3}}{2}b))$
$S_{2}$ | $\lvert X_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(x))=\exp(R^{1/3}a)$
| $\lvert Y_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(-\varepsilon
x))=\exp(R^{1/3}(\frac{1}{2}a+\frac{\sqrt{3}}{2}b))$
| $\lvert
Z_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}((1-\varepsilon)x))=\exp(R^{1/3}(\frac{3}{2}a+\frac{\sqrt{3}}{2}b))$
$S_{3}$ | $\lvert X_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(\varepsilon^{2}x))=\exp(R^{1/3}(-\frac{1}{2}a+\frac{\sqrt{3}}{2}b))$
| $\lvert Y_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(-\varepsilon
x))=\exp(R^{1/3}(\frac{1}{2}a+\frac{\sqrt{3}}{2}b))$
| $\lvert
Z_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}((1-\varepsilon)x))=\exp(R^{1/3}(\frac{3}{2}a+\frac{\sqrt{3}}{2}b))$
$S_{4}$ | $\lvert X_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(\varepsilon^{2}x))=\exp(R^{1/3}(-\frac{1}{2}a+\frac{\sqrt{3}}{2}b))$
| $\lvert Y_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(-\varepsilon
x))=\exp(R^{1/3}(\frac{1}{2}a+\frac{\sqrt{3}}{2}b))$
| $\lvert
Z_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}((1-\varepsilon)\varepsilon^{2}x))=\exp(R^{1/3}(-\frac{3}{2}a+\frac{\sqrt{3}}{2}b))$
$S_{5}$ | $\lvert X_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(\varepsilon^{2}x))=\exp(R^{1/3}(-\frac{1}{2}a+\frac{\sqrt{3}}{2}b))$
| $\lvert
Y_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(-x))=\exp(R^{1/3}(-a))$
| $\lvert
Z_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}((1-\varepsilon)\varepsilon^{2}x))=\exp(R^{1/3}(-\frac{3}{2}a+\frac{\sqrt{3}}{2}b))$
$S_{6}$ | $\lvert X_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(\varepsilon^{2}x))=\exp(R^{1/3}(-\frac{1}{2}a+\frac{\sqrt{3}}{2}b))$
| $\lvert
Y_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(-x))=\exp(R^{1/3}(-a))$
| $\lvert
Z_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}((1-\varepsilon)\varepsilon^{2}x))=\exp(R^{1/3}(-\frac{3}{2}a+\frac{\sqrt{3}}{2}b))$
$S_{7}$ | $\lvert X_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(\varepsilon x))=\exp(R^{1/3}(-\frac{1}{2}a-\frac{\sqrt{3}}{2}b))$
| $\lvert
Y_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(-x))=\exp(R^{1/3}(-a))$
| $\lvert
Z_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}((1-\varepsilon)\varepsilon^{2}x))=\exp(R^{1/3}(-\frac{3}{2}a+\frac{\sqrt{3}}{2}b))$
$S_{8}$ | $\lvert X_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(\varepsilon x))=\exp(R^{1/3}(-\frac{1}{2}a-\frac{\sqrt{3}}{2}b))$
| $\lvert
Y_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(-x))=\exp(R^{1/3}(-a))$
| $\lvert
Z_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}((1-\varepsilon)\varepsilon
x))=\exp(R^{1/3}(-\sqrt{3}b))$
$S_{9}$ | $\lvert X_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(\varepsilon x))=\exp(R^{1/3}(-\frac{1}{2}a-\frac{\sqrt{3}}{2}b))$
| $\lvert
Y_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(-\varepsilon^{2}x))=\exp(R^{1/3}(\frac{1}{2}a-\frac{\sqrt{3}}{2}b))$
| $\lvert
Z_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}((1-\varepsilon)\varepsilon
x))=\exp(R^{1/3}(-\sqrt{3}b))$
$S_{10}$ | $\lvert X_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(\varepsilon x))=\exp(R^{1/3}(-\frac{1}{2}a-\frac{\sqrt{3}}{2}b))$
| $\lvert
Y_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(-\varepsilon^{2}x))=\exp(R^{1/3}(\frac{1}{2}a-\frac{\sqrt{3}}{2}b))$
| $\lvert
Z_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}((1-\varepsilon)\varepsilon
x))=\exp(R^{1/3}(-\sqrt{3}b))$
$S_{11}$ | $\lvert X_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(x))=\exp(R^{1/3}(a))$
| $\lvert
Y_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(-\varepsilon^{2}x))=\exp(R^{1/3}(\frac{1}{2}a-\frac{\sqrt{3}}{2}b))$
| $\lvert
Z_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}((1-\varepsilon)\varepsilon
x))=\exp(R^{1/3}(-\sqrt{3}b))$
$S_{12}$ | $\lvert X_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(x))=\exp(R^{1/3}(a))$
| $\lvert
Y_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}(-\varepsilon^{2}x))=\exp(R^{1/3}(\frac{1}{2}a-\frac{\sqrt{3}}{2}b))$
| $\lvert
Z_{R,\varphi}\rvert\approx\exp(R^{1/3}\operatorname{Re}((1-\varepsilon)x))=\exp(R^{1/3}(\frac{3}{2}a+\frac{\sqrt{3}}{2}b))$

####  6.2.5 Dominance between different trace coordinates within sectors

Next, we need to address the dominance properties among the three trace
coordinates.

######  Lemma 6.10.

We fix some angle coordinate that is not critical of the first kind,
$\varphi\notin\mathbb{Z}\cdot\pi$.

  * i)

$\lvert Z_{R,\varphi}\rvert\gg\lvert X_{R,\varphi}\rvert$ and $\lvert
Z_{R,\varphi}\rvert\gg\lvert Y_{R,\varphi}\rvert$.

  * ii)

On sectors $S_{j}$, $j=1,4,5,8,9,12$ $\lvert X_{R,\varphi}\rvert\gg\lvert
Y_{R,\varphi}\rvert$, while on sectors $S_{j}$, $j=2,3,6,7,10,11$ $\lvert
Y_{R,\varphi}\rvert\gg\lvert X_{R,\varphi}\rvert$.

Figure 3: The complex plane $(a,b)$, sectors $S_{j}$, dominant coordinate
$\lvert X_{R,\varphi}\rvert$ or $\lvert Y_{R,\varphi}\rvert$ depending on the
sector.

###### Proof.

A straightforward comparison of the exponents from the table in each sector
shows this result. Namely:

  1. 1.

On $S_{1}$:
$\frac{3}{2}a+\frac{\sqrt{3}}{2}b>a>\frac{1}{2}a+\frac{\sqrt{3}}{2}b$, that is
$\lvert Z_{R,\varphi}\rvert\gg\lvert X_{R,\varphi}\rvert\gg\lvert
Y_{R,\varphi}\rvert$.

  2. 2.

On $S_{2}$:
$\frac{3}{2}a+\frac{\sqrt{3}}{2}b>\frac{1}{2}a+\frac{\sqrt{3}}{2}b>a$, that is
$\lvert Z_{R,\varphi}\rvert\gg\lvert Y_{R,\varphi}\rvert\gg\lvert
X_{R,\varphi}\rvert$.

  3. 3.

On $S_{3}$:
$\frac{3}{2}a+\frac{\sqrt{3}}{2}b>\frac{1}{2}a+\frac{\sqrt{3}}{2}b>-\frac{1}{2}a+\frac{\sqrt{3}}{2}b$,
that is $\lvert Z_{R,\varphi}\rvert\gg\lvert Y_{R,\varphi}\rvert\gg\lvert
X_{R,\varphi}\rvert$.

  4. 4.

On $S_{4}$:
$-\frac{3}{2}a+\frac{\sqrt{3}}{2}b>-\frac{1}{2}a+\frac{\sqrt{3}}{2}b>\frac{1}{2}a+\frac{\sqrt{3}}{2}b$,
that is $\lvert Z_{R,\varphi}\rvert\gg\lvert X_{R,\varphi}\rvert\gg\lvert
Y_{R,\varphi}\rvert$.

  5. 5.

On $S_{5}$:
$-\frac{3}{2}a+\frac{\sqrt{3}}{2}b>-\frac{1}{2}a+\frac{\sqrt{3}}{2}b>-a$, that
is $\lvert Z_{R,\varphi}\rvert\gg\lvert X_{R,\varphi}\rvert\gg\lvert
Y_{R,\varphi}\rvert$.

  6. 6.

On $S_{6}$:
$-\frac{3}{2}a+\frac{\sqrt{3}}{2}b>-a>-\frac{1}{2}a+\frac{\sqrt{3}}{2}b$, that
is $\lvert Z_{R,\varphi}\rvert\gg\lvert Y_{R,\varphi}\rvert\gg\lvert
X_{R,\varphi}\rvert$.

  7. 7.

On $S_{7}$:
$-\frac{3}{2}a+\frac{\sqrt{3}}{2}b>-a>-\frac{1}{2}a-\frac{\sqrt{3}}{2}b$, that
is $\lvert Z_{R,\varphi}\rvert\gg\lvert Y_{R,\varphi}\rvert\gg\lvert
X_{R,\varphi}\rvert$.

  8. 8.

On $S_{8}$: $-\sqrt{3}b>-\frac{1}{2}a-\frac{\sqrt{3}}{2}b>-a$, that is $\lvert
Z_{R,\varphi}\rvert\gg\lvert X_{R,\varphi}\rvert\gg\lvert
Y_{R,\varphi}\rvert$.

  9. 9.

On $S_{9}$:
$-\sqrt{3}b>-\frac{1}{2}a-\frac{\sqrt{3}}{2}b>\frac{1}{2}a-\frac{\sqrt{3}}{2}b$,
that is $\lvert Z_{R,\varphi}\rvert\gg\lvert X_{R,\varphi}\rvert\gg\lvert
Y_{R,\varphi}\rvert$.

  10. 10.

On $S_{10}$:
$-\sqrt{3}b>\frac{1}{2}a-\frac{\sqrt{3}}{2}b>-\frac{1}{2}a-\frac{\sqrt{3}}{2}b$,
that is $\lvert Z_{R,\varphi}\rvert\gg\lvert Y_{R,\varphi}\rvert\gg\lvert
X_{R,\varphi}\rvert$.

  11. 11.

On $S_{11}$: $-\sqrt{3}b>\frac{1}{2}a-\frac{\sqrt{3}}{2}b>a$, that is $\lvert
Z_{R,\varphi}\rvert\gg\lvert Y_{R,\varphi}\rvert\gg\lvert
X_{R,\varphi}\rvert$.

  12. 12.

On $S_{12}$:
$\frac{3}{2}a+\frac{\sqrt{3}}{2}b>a>\frac{1}{2}a-\frac{\sqrt{3}}{2}b$, that is
$\lvert Z_{R,\varphi}\rvert\gg\lvert X_{R,\varphi}\rvert\gg\lvert
Y_{R,\varphi}\rvert$.

∎

We need to understand the behaviour of
$X_{R,\varphi},Y_{R,\varphi},Z_{R,\varphi}$ as $1\ll R$ is fixed, and
$\varphi$ ranges over $[0,2\pi]$. Lemma 6.10 shows that $\lvert
Z_{R,\varphi}\rvert$ dominates the other two coordinates on each open sector.

######  Proposition 6.11.

As $t\rightarrow\infty$ in any of the sectors $S_{j}$, the image of $\psi$
converges to the point $[0:0:1]$.

Notice that this limit point is actually the nodal point of the compactifying
curve $C$ determined in Lemma 3.2, namely

$X^{3}+Y^{3}-XYZ=0.$ (32)

####  6.2.6 Phases of the trace coordinates in terms of holonomy of the
spectral line bundle

In Lemma 6.10, we determined the dominance properties of the absolute values
of the trace coordinates along rays in any open sector $S_{j}$. We have found
that $X/Z\to 0$ and $Y/Z\to 0$ as $R\to\infty$. These quantities constitute
local affine coordinates near the nodal point of $C$. Here, we focus on the
angles of these affine coordinates. Notice that a punctured neighborhood of
the nodal point in the character variety admits a two-dimensional torus
$T^{2}=S^{1}\times S^{1}$ as a deformation retract, whose $S^{1}$-factors
correspond precisely to the angles of $X/Z$ and $Y/Z$. We find that the angles
can be interpreted as the holonomy of $\mathcal{L}$ along some loops in
$\Sigma$, depending on the given sector. Due to the $2\pi/3$ periodicity of
the diagram of Figure 3, it is sufficient to deal with four consecutive
sectors.

First, let us discuss the homology of $\widetilde{\Sigma}$ in some detail.
Consider the straight line segment $[r,1-r]\subset\mathbb{R}$. Let us
temporarily denote its lifts to $\Sigma$ by $f_{1},f_{2},f_{3}$, where $f_{j}$
lies on the sheet $\Sigma_{j}$ determined by the $Q_{j}$-eigenspace of
$\theta$ (see (11)). On the other hand, the three lifts of the loop
$\gamma_{0}$ cyclically connect the three sheets. For instance, one of these
lifts connects $\Sigma_{1}$ to $\Sigma_{2}$. Denote this lift by
$\tilde{\gamma}_{0}$. Moreover, let $\tilde{\tilde{\gamma}}_{0}$ denote the
lift of $\gamma_{0}$ connecting the sheet $\Sigma_{2}$ to $\Sigma_{3}$.
Similarly, one of the lifts $\tilde{\gamma}_{1}$ of $\gamma_{1}$ connects the
sheet $\Sigma_{1}$ to $\Sigma_{2}$, and another one
$\tilde{\tilde{\gamma}}_{1}$ connects $\Sigma_{2}$ to $\Sigma_{3}$. It follows
that the paths $f_{1}*\tilde{\gamma}_{0}*(-f_{2})*(-\tilde{\gamma}_{1})$ and
$f_{2}*\tilde{\tilde{\gamma}}_{0}*(-f_{3})*(-\tilde{\tilde{\gamma}}_{1})$
represent singular $1$-cycles on $\widetilde{\Sigma}$. Let us denote these
cycles in order by $A,B$. See Figure 4.

$0$$r$$1$$1-r$$\eta_{0}-\eta_{1}$$\gamma_{0}$$\gamma_{1}$$f_{1}$$f_{2}$$f_{3}$$\tilde{\gamma}_{0}$$\tilde{\gamma}_{1}$$\tilde{\tilde{\gamma}}_{0}$$\tilde{\tilde{\gamma}}_{1}$$A$$B$
Figure 4: The cycles $A,B$ on the spectral curve.

######  Lemma 6.12.

The cycles $A,B$ form a $\mathbb{Z}$-basis of
$H_{1}(\widetilde{\Sigma},\mathbb{Z})$.

###### Proof.

This is likely standard, yet we include it for sake of completeness. Let us
consider straight lines between the marked points
$0,1,\infty\in\mathbb{C}P^{1}=S^{2}$. They form a triangulation of $S^{2}$
with $3$ vertices, $3$ edges and $2$ faces. Up to homeomorphism,
$\widetilde{\Sigma}_{t}$ can be obtained by gluing three copies of such
triangulated $S^{2}$, by identifying their points $0$ with each other, and
similarly for their points $1$ and $\infty$. The segments $[0,1]$ in the three
copies of $S^{2}$ are (extensions of) the lifts $f_{1},f_{2},f_{3}$. The
resulting triangulation of $T^{2}$

  * •

admits $3$ vertices $v_{1},v_{2},v_{3}$, $9$ edges $e_{1},\ldots,e_{9}$ and
$6$ faces

  * •

has $6$ edges incident to every vertex,

  * •

has $3$ edges between each pair of distinct vertices,

  * •

has three differently labelled vertices on each face.

An example of such a triangulation is shown on Figure 5. Now, if we label
vertices so that $v_{1}$ corresponds to the identification of the points $0\in
S^{2}$ and $v_{2}$ to $1\in S^{2}$, then the three lifts $f_{1},f_{2},f_{3}$
correspond to the three edges of the triangulation connecting $v_{1},v_{2}$.
(On Figure 5, these edges are labelled by $e_{2},e_{4},e_{9}$. The concrete
bijection between $f_{1},f_{2},f_{3}$ and $e_{2},e_{4},e_{9}$ is irrelevant
for the argument.) The loop $A$ is the concatenation of $f_{1},f_{2}$, and $B$
is the concatenation of $f_{2},f_{3}$ (given suitable orientations so that the
concatenations make sense). Now, since $f_{1}$ and $f_{3}$ are not parallel
(i.e., they do not form a bigon), the cycles $A,B$ are linearly independent
over $\mathbb{Q}$, and generate $H_{1}(\widetilde{\Sigma},\mathbb{Z})$. ∎

$e_{1}$$e_{6}$$e_{1}$$e_{6}$$e_{2}$$e_{4}$$e_{2}$$e_{4}$$v_{1}$$v_{1}$$v_{1}$$v_{1}$$v_{2}$$v_{2}$$v_{3}$$v_{3}$$e_{8}$$e_{3}$$e_{5}$$e_{7}$$e_{9}$
Figure 5: A possible triangulation of the two-torus.

In the sequel we will use the notation of the holonomy isomorphism (12).

######  Proposition 6.13.

  1. 1.

On $S_{2}$, we have

$\frac{X/Z}{|X/Z|}\approx\operatorname{hol}_{B}(\mathcal{L}),\quad\frac{Y/Z}{|Y/Z|}\approx\operatorname{hol}_{2B}(\mathcal{L})$

  2. 2.

On $S_{3}$, we have

$\frac{X/Z}{|X/Z|}\approx\operatorname{hol}_{A+3B}(\mathcal{L}),\quad\frac{Y/Z}{|Y/Z|}\approx\operatorname{hol}_{2B}(\mathcal{L})$

  3. 3.

On $S_{4}$, we have

$\frac{X/Z}{|X/Z|}\approx\operatorname{hol}_{A}(\mathcal{L}),\quad\frac{Y/Z}{|Y/Z|}\approx\operatorname{hol}_{-B}(\mathcal{L})$

  4. 4.

On $S_{5}$, we have

$\frac{X/Z}{|X/Z|}\approx\operatorname{hol}_{A}(\mathcal{L}),\quad\frac{Y/Z}{|Y/Z|}\approx\operatorname{hol}_{2A}(\mathcal{L})$

###### Proof.

We only prove the first statement of the first item and leave the rest to the
readers, because the proofs follow the same line. We apply formulas (28)
combined with Proposition 6.9. On $S_{2}$, the dominant term of $X,Y,Z$ are
respectively their first, second and first terms. The effect of the recessive
terms on the angles is negligible, and they only amount to a homotopy. We find

$\sqrt{-1}\operatorname{Im}(\ln(X/Z))\approx\int_{\eta_{0}-\eta_{1}}u_{2}-u_{3}+\int_{\gamma_{0}-\gamma_{1}}u_{2}$

Now, the path $\eta_{0}-\eta_{1}$ is precisely the straight line segment
$[r,1-r]\subset\mathbb{R}$, with opposite orientation (i.e., from $1-r$ to
$r$). Then,

$\exp\left(\int_{\eta_{0}-\eta_{1}}u_{2}\right)=\exp\left(\int_{f_{2}}u_{2}\right)$

is the holonomy of $\mathcal{L}$ along the path $f_{2}$, because $u_{2}$ is
its connection $1$-form on the corresponding sheet (the sign comes from the
opposite orientation of $\eta_{0}-\eta_{1}$ and $[r,1-r]$). Similarly,

$\exp\left(-\int_{\eta_{0}-\eta_{1}}u_{3}\right)=\exp\left(-\int_{f_{3}}u_{3}\right)$

is the inverse of the holonomy of $\mathcal{L}$ along the path $f_{3}$. We
infer that the integral term appearing in the argument of $X/Z$ is equal to
the holonomy of $\mathcal{L}$ along the closed loop
$B=f_{2}*\tilde{\tilde{\gamma}}_{0}*(-f_{3})*(-\tilde{\tilde{\gamma}}_{1})$ on
$\widetilde{\Sigma}$. ∎

####  6.2.7 Asymptotics over the Stokes rays

We see from Lemma 6.10 that when $x$ crosses a critical angle of the second
kind that is not of the first kind at the same time, then the dominance
properties of the terms _within_ a trace coordinate change, but the dominance
_between_ the coordinates $X,Y$ remain unchanged. Therefore, for the purpose
of computing the limit of the ratios of the coordinates as $R\to\infty$, it is
sufficient to focus on the critical angles of the first kind, which are
integer multiples of $\pi$.

######  Proposition 6.14.

Let $R\to\infty$ along a ray in the Hitchin base.

  1. 1.

If $\varphi=(2k-1)\pi$ for some $k\in\mathbb{Z}$ then we have

$\displaystyle\lim\frac{\lvert X_{R,\varphi}\rvert}{\lvert
Y_{R,\varphi}\rvert}$ $\displaystyle=1,$ $\displaystyle\lim\frac{\lvert
X_{R,\varphi}\rvert}{\lvert Z_{R,\varphi}\rvert}$
$\displaystyle\geq\frac{1}{2},$ $\displaystyle\lim\frac{\lvert
Y_{R,\varphi}\rvert}{\lvert Z_{R,\varphi}\rvert}$
$\displaystyle\geq\frac{1}{2},$

including the possibility that the latter two limits are both $\infty$ (but
only simultaneously).

  2. 2.

If $\varphi=2k\pi$ then we have

$\displaystyle\frac{\lvert X_{R,\varphi}\rvert}{\lvert Y_{R,\varphi}\rvert}$
$\displaystyle\to 1,$ $\displaystyle\frac{\lvert X_{R,\varphi}\rvert}{\lvert
Z_{R,\varphi}\rvert}$ $\displaystyle\to 0,$ $\displaystyle\frac{\lvert
Y_{R,\varphi}\rvert}{\lvert Z_{R,\varphi}\rvert}$ $\displaystyle\to 0.$

###### Proof.

According to (31), the condition of part 1 is equivalent to that $x$ lies on
one of the Stokes rays $R_{3},R_{7},R_{11}$. Similarly, the condition of part
2 is equivalent to that $x$ lies on one of the Stokes rays
$R_{1},R_{5},R_{9}$.

Let us start by treating $R_{3}$, defined by $a=0,b>0$. We use the table
obtained in Proposition 6.9. The analytic functions
$X_{R,\varphi},Y_{R,\varphi},Z_{R,\varphi}$ have asymptotic expansions on both
sides of $R_{3}$, i.e. for $\varphi=(2k-1)\pi\pm\varepsilon$ as $R\to\infty$,
for any $0<\varepsilon<\frac{\pi}{12}$. The term $R^{1/3}\frac{\sqrt{3}b}{2}$
in the maximal dilation exponents of the coordinates is common (and positive),
therefore it does not change the limit of their quotients. On the other hand,
the dependence of the maximal dilation exponents on $a$ of
$X_{R,\varphi},Y_{R,\varphi},Z_{R,\varphi}$ is in order

$-\frac{a}{2},\quad\frac{a}{2},\quad\left|\frac{3a}{2}\right|.$

In particular, this dependence is continuous. In order to find desired limits,
we need the following statement.

######  Lemma 6.15.

The asymptotic expansions of (28) hold over the Stokes rays too.

In view of Proposition 6.4, this would follow from the following more
technical statement, applied individually to all diagonal entries appearing in
$X_{R,\varphi},Y_{R,\varphi},Z_{R,\varphi}$ (up to replacing $\infty$ by $0$
and setting $r=R^{-1}$).

######  Lemma 6.16.

Let

$f\colon\\{z\in\mathbb{C}|\;0<|z|<1\\}\to\mathbb{C}$

be an $\mathbb{R}$-analytic function. Assume that for each
$\varphi\in(0,2\pi)\setminus\\{\pi\\}$ we have that

$f\left(re^{\sqrt{-1}\varphi}\right)\to 1\quad as\;r\to 0.$

Then we also have

$f(\pm r)\to 1\quad as\;r\to 0.$

This property should be a consequence of the basic rigidity of analytic
functions, however as yet we lack an appropriate reference or a
counterexample.

Using the above result gives that the maximal dilation exponents of the trace
coordinates agree:

$\displaystyle\lvert
X_{R,\varphi}\rvert\approx\exp\left(R^{1/3}\frac{\sqrt{3}}{2}b\right),$
$\displaystyle\lvert
Y_{R,\varphi}\rvert\approx\exp\left(R^{1/3}\frac{\sqrt{3}}{2}b\right),$
$\displaystyle\lvert
Z_{R,\varphi}\rvert\approx\exp\left(R^{1/3}\frac{\sqrt{3}}{2}b\right)\left|e^{\sqrt{-1}\varpi}+e^{\sqrt{-1}\varpi^{\prime}}\right|$

for some $\varpi,\varpi^{\prime}\in\mathbb{R}$, coming from the purely
imaginary terms in the argument of the first and third terms of the expansion
of $Z$ in (28). The statement for $R_{3}$ follows. The argument is similar for