#  $P=W$ conjecture in lowest degree for rank $2$ over the $5$-punctured
sphere

Szilárd Szabó  Budapest University of Technology and Economics, 1111.
Budapest, Egry József utca 1. H épület, Hungary, and Rényi Institute of
Mathematics, 1053. Budapest, Reáltanoda utca 13-15. Hungary
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We use abelianization of Higgs bundles away from the ramification divisor and
fiducial solutions to analyze the large scale behaviour of Fenchel–Nielsen co-
ordinates on the moduli space of rank $2$ Higgs bundles on the Riemann sphere
with $5$ punctures. We solve the related Hitchin WKB problem and prove the
lowest degree weighted pieces of the $P=W$ conjecture in this case.

##  1\. Introduction and statement of the main result

In this paper we investigate the moduli space
${\mathcal{M}}_{\operatorname{Dol}}$ of Higgs bundles on $\mathbb{C}P^{1}$
with $5$ logarithmic points in rank $2$ and the corresponding character
variety ${\mathcal{M}}_{\operatorname{B}}$, subject to specific choices of
parameters. These spaces are complex varieties of dimension $4$. The first aim
of the paper is to give a complete answer (Propositions 10, 11, 13) for these
spaces to the Hitchin WKB problem raised in [19]:
Hitchin WKB problem Consider a non-trivial ${\mathbb{C}}^{\times}$-orbit in
the Hitchin base and a family of Higgs bundles lifting this orbit in the
Dolbeault moduli space; determine then the asymptotic behaviour of the
transport matrices of the associated family of representations in the
character variety.

The second, closely related goal is to use these results in order to obtain
for these spaces one extremal graded piece of the so-called $P=W$ conjecture:

######  Theorem 1.

Let ${\mathcal{M}}_{\operatorname{Dol}}$ and
${\mathcal{M}}_{\operatorname{B}}$ denote the Dolbeault moduli space and
character variety of $\mathbb{C}P^{1}$ with $5$ logarithmic points in rank
$2$. Then, for every $0\leq k\leq 4$ the regular singular Riemann–Hilbert
correspondence and the non-abelian Hodge correspondence induce an isomorphism

$\operatorname{Gr}_{P}^{-k-2}H^{k}({\mathcal{M}}_{\operatorname{Dol}},{\mathbb{Q}})\cong\operatorname{Gr}^{W}_{2k}H^{k}({\mathcal{M}}_{\operatorname{B}},{\mathbb{Q}}).$

The weights appearing in the theorem represent the lowest (respectively,
highest) possibly non-trivial weights of $P$ (respectively, $W$). For our
weight conventions, see Sections 2.3 and 2.7. Notice that since
${\mathcal{M}}_{\operatorname{B}}$ is a smooth $4$-dimensional affine variety,
by the Andreotti–Frankel theorem [1] the only degrees where it may have non-
trivial cohomology are $0\leq k\leq 4$.

Even though this paper contains a detailed study of just one special case of
the $P=W$ conjecture in lowest weight, many of our technical results are valid
for an arbitrary number $n\geq 5$ of parabolic points. We expect that it will
be possible to treat more general cases along the same lines.

Let us now give some motivational background for this study. The $P=W$
conjecture is a major open problem in non-abelian Hodge theory, formulated by
M. de Cataldo, T. Hausel and L. Migliorini [4] as a correspondence between the
(decreasing) perverse Leray filtration $P$ induced by the Hitchin map on the
cohomology of a Dolbeault moduli space and the (increasing) weight filtration
$W$ of Deligne’s mixed Hodge structure on the cohomology of the associated
character variety (Betti space). Recent years have brought intense activity
toward establishing various cases of the $P=W$ conjecture. The starting point
was the paper of de Cataldo, Hausel and Migliorini [4] where it was proven in
rank $2$ over compact curves. As an important further contribution, M. de
Cataldo, D. Maulik and J. Shen [5] established it for curves of genus $2$. C.
Felisetti and M. Mauri [10] proved it for character varieties admitting a
symplectic resolution, i.e. in genus $1$ and arbitrary rank, and in genus $2$
and rank $2$. The author has established the conjecture for complex
$2$-dimensional moduli spaces of rank $2$ Higgs bundles with irregular
singularities over $\mathbb{C}P^{1}$ corresponding to the Painlevé cases [32].
J. Shen and Z. Zhang [29] proved it for five infinite families of moduli
spaces of parabolic Higgs bundles over $\mathbb{C}P^{1}$.

The $P=W$ conjecture has been generalized in various interesting contexts. A.
Harder showed a similar statement for elliptic Lefschetz fibrations using
methods coming from toric surfaces [14, Theorem 4.5]. Z. Zhang [33] found a
related phenomenon for the weight filtration of certain $2$-dimensional
cluster varieties and the perverse filtration of elliptic fibrations with
constrained singular fibers. A motivation for the $P=W$ conjecture was the so-
called curious hard Lefschetz conjecture of T. Hausel, E. Letellier and F.
Rodriguez-Villegas [15], that has been confirmed by A. Mellit [24].

Among the various generalizations and analogues of the $P=W$ conjecture is an
intriguing geometric counterpart formulated by L. Katzarkov, A. Noll, P.
Pandit and C. Simpson [19, Conjecture 1.1] and C. Simpson [31, Conjecture
11.1] that is quite relevant to our approach; this version is now called
Geometric $P=W$ conjecture. Roughly speaking, the Geometric $P=W$ conjecture
asserts the existence of a certain homotopy commutative diagram involving the
Riemann–Hilbert map, non-abelian Hodge correspondence, the Hitchin map and the
natural map from the character variety to the topological realization of its
dual boundary complex. An immediate consequence of validity of this conjecture
is that the homotopy type of the topological space of the dual boundary
complex of the character variety is that of a sphere of given dimension,
therefore finding this homotopy type is a first consistency check of the
conjecture. The Geometric $P=W$ conjecture has also attracted considerable
attention in recent times. A. Komyo [20] used an explicit geometric
description to prove that the homotopy type of the dual boundary complex of
the character variety for $\mathbb{C}P^{1}$ with $5$ logarithmic points and
group $\operatorname{Gl}(2,{\mathbb{C}})$ (that is, the Betti space we will
deal with in this paper) is that of the $3$-sphere. C. Simpson [31]
generalized Komyo’s result to the case of arbitrarily many logarithmic points
on $\mathbb{C}P^{1}$, in rank $2$, by proving that the homotopy type of the
dual boundary complex is that of $S^{2n-7}$; for this purpose, he introduced
Fenchel–Nielsen type co-ordinates that will be widely used in this paper. T.
Mochizuki [25] solved the closely related Hitchin WKB problem for non-critical
paths. M. Mauri, E. Mazzon and M. Stevenson [22, Theorem 6.0.1] used Berkovich
space techniques to show 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 $S^{2n-1}$. L. Katzarkov, A. Harder and V.
Przyjalkowski have formulated a version of the cohomological $P=W$ conjecture
for log-Calabi–Yau manifolds and their mirror symmetric pairs, and in [18,
Section 4] discussed a geometric version thereof. The author established the
Geometric $P=W$ conjecture in the Painlevé cases in [32] via asymptotic
abelianization of solutions of Hitchin’s equations. In joint work with A.
Némethi [27], the author gave a second proof for the same cases using
different techniques, namely plumbing calculus. As far as the author is aware,
up to date these latter articles are the only ones in which the full assertion
of the Geometric $P=W$ conjecture has been confirmed, rather than just its
implication on the homotopy type of the dual boundary complex. It is
remarkable that the geometrical understanding of the moduli spaces developed
in [27, Section 6] is quite reminiscent to the description of the weight
filtration in terms of dual torus fibrations appearing in [18, Section 4] (up
to the difference that the latter paper deals with the case of a smooth
elliptic anti-canonical divisor rather than a singular one).

Previously, F. Loray and M. Saito [21] studied the algebraic geometric
structure of the moduli space that we consider, endowed with its de Rham
complex structure. R. Donagi and T. Pantev [8] investigated Hecke transforms
on this space and proved the Geometric Langlands correspondence for it. The
paths that we consider in Propositions 10, 11, 13 are homologically non-
trivial loops that do not satisfy the non-critical condition, therefore our
results do not directly follow from previous study of T. Mochizuki [25]
(though we make use of methods of that paper).

We will achieve our goals by refining the approach pioneered in our previous
paper [32]. Namely, using asymptotic abelianization we reduce the study to the
classical abelian Hodge theory and Riemann–Hilbert correspondence treated in
detail for instance in [13]. Specifically, we will make use of technical
results of T. Mochizuki [25], L. Fredrickson, R. Mazzeo, J. Swoboda and H.
Weiss [11] and R. Mazzeo, J. Swoboda, H. Weiss and F. Witt [23] describing the
large-scale behaviour of solutions of Hitchin’s equations, combined with C.
Simpson’s Fenchel–Nielsen type co-ordinates of the character variety [31]. The
studies in [11] and [23] were inspired by physical considerations pertinent to
the WKB-analysis of Hitchin’s equations given by D. Gaiotto, G. Moore and A.
Neitzke [12], where the authors stated a conjecture about the large scale
Riemannian structure of the Hodge moduli spaces. In a certain sense, our work
therefore points out a connection between two seemingly unrelated circles of
ideas: the $P=W$ conjecture on the algebraic topology of the Hodge moduli
spaces on the one hand, and the Gaiotto–Moore–Neitzke conjecture on their
Riemannian geometry on the other hand. This fits nicely into the broader
picture of topology and Riemannian geometry having influence on one another,
the bridge between them being built by geometric analysis. In particular, it
will become clear from our proofs that integrals of the canonical Liouville
$1$-form along some paths in the spectral cover govern the behaviour of the
non-abelian Hodge and the Riemann–Hilbert correspondences; the same kind of
integrals show up in [12].

Acknowledgements: The author would like to thank T. Hausel, M. Mauri, R.
Mazzeo, T. Mochizuki, A. Némethi and C. Simpson for useful discussions. During
the preparation of this manuscript, the author was supported by the _Lendület_
Low Dimensional Topology grant of the Hungarian Academy of Sciences and by the
grants K120697 and KKP126683 of NKFIH.

##  2\. Basic notions and preparatory results

###  2.1. Moduli spaces of tame harmonic bundles

Consider $X=\mathbb{C}P^{1}$ with co-ordinates $z$ and $w=z^{-1}$, endowed
with the standard Riemannian metric. We denote by ${\mathcal{O}}$ and $K$ the
sheaves of holomorphic functions and holomorphic $1$-forms respectively on
$\mathbb{C}P^{1}$. We set

$t_{0}=-\frac{1}{k},\quad t_{1}=0,\quad t_{2}=1,\quad t_{3}=-1,\quad
t_{4}=\frac{1}{k}$ (2.1)

for some $0<k<1$. These choices will not be used until Section 6.1, so the
results of all preceding sections are valid for any quintuple of distinct
points in $\mathbb{C}P^{1}$. We consider the simple effective divisor

$D=t_{0}+t_{1}+t_{2}+t_{3}+t_{4}$

and set

$L=K(D).$

By an abuse of notation, we will also denote by $D$ the support set of $D$.
Finally, we fix a point $x_{0}\in\mathbb{C}P^{1}\setminus D$. Much of the
following discussion has a straightforward generalization to simple effective
divisors of higher length too.

For $0\leq j\leq 4$ we fix

$\alpha_{j}^{-}=\frac{1}{4},\quad\alpha_{j}^{+}=\frac{3}{4}$ (2.2)

that will serve as parabolic weights in the Dolbeault complex structure.
Notice that

$\sum_{j=0}^{4}(\alpha_{j}^{-}+\alpha_{j}^{+})=5.$

We will write

$\vec{\alpha}=(\alpha_{j}^{-},\alpha_{j}^{+})_{j=0}^{4}.$

The basic object of our study will be a certain Hodge moduli space
${\mathcal{M}}_{\operatorname{Hod}}$ of _tame harmonic bundles_ [30] of rank
$2$ and parabolic degree $0$ on $\mathbb{C}P^{1}$ with parabolic structure at
$D$. We will describe this moduli space from two perspectives called the
Dolbeault and the de Rham moduli spaces. Consider a smooth vector bundle $V$
of rank $2$ and degree $-5$ over $\mathbb{C}P^{1}$. Then, the equations
defining harmonic bundles are _Hitchin’s equations_ [17]

$\displaystyle\bar{\partial}_{{\mathcal{E}}}\theta$ $\displaystyle=0$ (2.3)
$\displaystyle F_{h}+[\theta,\theta^{\dagger}]$ $\displaystyle=0$ (2.4)

for a $(0,1)$-connection $\bar{\partial}_{{\mathcal{E}}}$ on $V$, a Hermitian
metric $h$ on $V$ and a section $\theta$ of
$End(V)\otimes\Omega^{1,0}_{\mathbb{C}P^{1}}$ over $\mathbb{C}P^{1}\setminus
D$, where $F_{h}$ is the curvature of the Chern connection $\nabla_{h}$
associated to $(\bar{\partial}_{{\mathcal{E}}},h)$ and $\theta^{\dagger}$ is
the section of $End(V)\otimes\Omega^{0,1}_{\mathbb{C}P^{1}}$ obtained by
taking the adjoint of the endomorphism part of $\theta$ with respect to $h$
and the complex conjugate of its form-part. The reason of the terminology
“harmonic bundle” is the fact that the equations imply that the map $h$ is
harmonic from (the universal cover of) the Riemann surface to the symmetric
space $\mbox{Gl}_{2}({\mathbb{C}})/\mbox{U}(2)$. The behaviour of $\theta$ and
$h$ is assumed to satisfy the so-called _tameness_ condition at each $t_{j}$,
namely $h$ should admit a lift along any ray to $t_{j}$ which grows at most
polynomially in Euclidean distance.

Hitchin’s equations are presented above from the Dolbeault point of view. Let
us first describe the boundary behaviour of the data from this perspective.
Let us denote by ${\mathcal{E}}$ the holomorphic vector bundle
$(V,\bar{\partial}_{{\mathcal{E}}})$ on $\mathbb{C}P^{1}\setminus D$. It turns
out that there exists an extension of the holomorphic bundle ${\mathcal{E}}$
over $D$ such that the Higgs field has at most logarithmic poles at $D$. A
_parabolic structure_ on ${\mathcal{E}}$ at $D$ is by definition a filtration

$0\subset\ell_{j}\subset{\mathcal{E}}|_{t_{j}}$ (2.5)

of the fiber of ${\mathcal{E}}$ at every $t_{j}\in D$ that is stabilized by
$\mbox{res}_{t_{j}}\theta$. We assume that the Higgs field $\theta$ is
strongly parabolic, meaning that the action of $\mbox{res}_{t_{j}}\theta$ both
on $\ell_{j}$ and on ${\mathcal{E}}|_{t_{j}}/\ell_{j}$ is trivial. Then, in
the Dolbeault complex structure ${\mathcal{M}}_{\operatorname{Hod}}$
parameterizes $\vec{\alpha}$-stable parabolic Higgs bundles with Higgs field
having at most logarithmic poles at $D$ such that the eigenvalues of the
residue of the associated Higgs field at $t_{j}$ vanish and the parabolic
weights of the underlying holomorphic vector bundle in the Dolbeault picture
at $t_{j}$ are equal to $\alpha_{j}^{\pm}$. The latter assumption on parabolic
weights encodes a certain growth behaviour of the evaluation of the metric $h$
on elements of a local holomorphic trivialization. The moduli space of such
logarithmic parabolic Higgs bundles is known to be a ${\mathbb{C}}$-analytic
manifold

${\mathcal{M}}_{\operatorname{Dol}}(\vec{0},\vec{\alpha})$

called _Dolbeault moduli space_ , whose underlying smooth manifold is
${\mathcal{M}}_{\operatorname{Hod}}$.

Let us now turn to the de Rham point of view. It is known that if
$(\bar{\partial}_{{\mathcal{E}}},h,\theta)$ is a tame harmonic bundle then the
connection

$\nabla=\nabla_{h}+\theta+\theta^{\dagger}$

is integrable, and the underlying holomorphic vector bundle admits an
extension over $D$ with respect to which $\nabla^{1,0}$ has regular
singularities. The associated de Rham moduli space parameterizes
$\vec{\beta}$-stable parabolic integrable connections on $V$ with regular
singularities near the punctures $t_{j}$, with eigenvalues of its residue
given by

$\mu_{j}^{\pm}=\alpha_{j}^{\pm}$ (2.6)

and parabolic weights given by

$\beta_{j}^{\pm}=\alpha_{j}^{\pm}.$ (2.7)

Again, a parabolic structure on the underlying holomorphic vector bundle at
$D$ is defined as a flag of its fiber over $t_{j}\in D$ that is stabilized by
$\mbox{res}_{t_{j}}\nabla^{1,0}$ and such that its action on the first graded
piece of the filtration be $\mu_{j}^{-}$. The _de Rham moduli space_ of such
parabolic integrable connections with regular singularities will be denoted by

${\mathcal{M}}_{\operatorname{dR}}(\vec{\mu},\vec{\beta});$

it is a ${\mathbb{C}}$-analytic manifold with underlying smooth manifold
${\mathcal{M}}_{\operatorname{Hod}}$.

It follows from the above discussion that there exists a canonical
diffeomorphism

$\psi\colon{\mathcal{M}}_{\operatorname{Dol}}(\vec{0},\vec{\alpha})\to{\mathcal{M}}_{\operatorname{dR}}(\vec{\mu},\vec{\beta})$
(2.8)

called _non-abelian Hodge correspondence_.

###  2.2. Character variety, Riemann–Hilbert correspondence, dual boundary
complex

We will need a third point of view of harmonic bundles, called Betti side. The
_Betti moduli space_ (or character variety)
${\mathcal{M}}_{\operatorname{B}}(\vec{c},\vec{0})$ parameterizes filtered
local systems on $\mathbb{C}P^{1}\setminus D$ with prescribed conjugacy class
of its monodromy around every $t_{j}$ and growth order of parallel sections on
rays emanating from the punctures, up to simultaneous conjugation by elements
of $\mbox{PGl}(2,{\mathbb{C}})$. We will now describe the value and role of
parameters $\vec{c}$. Namely, the monodromy transformation of an integrable
connection $\nabla$ in
${\mathcal{M}}_{\operatorname{dR}}(\vec{\mu},\vec{\beta})$ along a positively
oriented simple loop in $\mathbb{C}P^{1}$ separating $t_{j}$ from the other
parabolic points has eigenvalues

$c_{j}^{\pm}=\exp(-2\pi\sqrt{-1}\mu_{j}^{\pm})=\exp(-2\pi\sqrt{-1}\alpha_{j}^{\pm})=\pm\sqrt{-1}$
(2.9)

and all weights of the associated filtration equal to $0$.

It is known that the map

$\operatorname{RH}\colon{\mathcal{M}}_{\operatorname{dR}}(\vec{\mu},\vec{\beta})\to{\mathcal{M}}_{\operatorname{B}}(\vec{c},\vec{0})$
(2.10)

mapping any integrable connection to its (filtered) local system of vector
spaces is a ${\mathbb{C}}$-analytic isomorphism, called _Riemann–Hilbert map_.

It is known that ${\mathcal{M}}_{\operatorname{B}}(\vec{c},\vec{0})$ is an
affine algebraic variety, which is smooth for generic choices of the
parameters. We will denote by
$\overline{{\mathcal{M}}}_{\operatorname{B}}(\vec{c},\vec{0})$ a smooth
compactification by a simple normal crossing divisor $D_{\operatorname{B}}$.
Such a compactification exists by Nagata’s compactification theorem [26]
combined with Hironaka’s theorem on the existence of resolutions of
singularities in characteristic $0$ [16].

######  Definition 1.

The _dual complex_ of $D_{\operatorname{B}}$ is the simplicial complex
$\mathbb{D}D_{\operatorname{B}}$ whose vertices are in bijection with
irreducible components of $D_{\operatorname{B}}$, and whose $k$-faces are
formed by $(k+1)$-tuples of vertices such that the intersection of the
corresponding components is non-empty. We will denote the $k$-skeleton of
$\mathbb{D}D_{\operatorname{B}}$ by $\mathbb{D}_{k}D_{\operatorname{B}}$, and
the topological realization of $\mathbb{D}D_{\operatorname{B}}$ by
$|\mathbb{D}D_{\operatorname{B}}|$.

We will require $D_{\operatorname{B}}$ to be a very simple normal crossing
divisor, meaning that any such non-empty intersection of components is
connected. The above procedure may be applied to any quasi-projective smooth
variety $X$, and an important result due to Danilov [3] states that the
homotopy type of the simplicial complex is independent of the chosen
compactification. We will apply it to
${\mathcal{M}}_{\operatorname{B}}(\vec{c},\vec{0})$, and we will call the
resulting simplicial complex its _dual boundary complex_ , denoted by
$\mathbb{D}\partial{\mathcal{M}}_{\operatorname{B}}(\vec{c},\vec{0})$. A.
Komyo [20] showed that for character varieties of rank $2$ representations
with $k=5$ parabolic points the homotopy type of the dual boundary complex is
that of the sphere $S^{3}$. C. Simpson [31] generalized this result to
character varieties of the complement of $k\geq 5$ parabolic points, by
showing that for $X={\mathcal{M}}_{\operatorname{B}}(\vec{c},\vec{0})$ the
dual boundary complex is homotopy equivalent to the sphere $S^{2k-7}$.

###  2.3. Topological description of the weights in mixed Hodge structure

Another closely related consequence of the fact that
${\mathcal{M}}_{\operatorname{B}}(\vec{c},\vec{0})$ is a smooth affine
algebraic variety is that its cohomology spaces carry a mixed Hodge structure
(MHS), defined by P. Deligne [7]. Let us recall the topological
characterization of the weights in MHS, following [9, Section 6.5]. In what
follows, we will often drop $\vec{c},\vec{0}$ from the notation and write
${\mathcal{M}}_{\operatorname{B}}$ for the character variety.

In this section we adopt the point of view of [9] and consider homology groups
rather than cohomology; application of the standard duality operation is
implicitly meant whenever we compare a homology group with a cohomology group.
This involves switching the signs of the degrees of the weight filtration. Let
$\overline{{\mathcal{M}}}_{\operatorname{B}}$ be a smooth compactification of
${\mathcal{M}}_{\operatorname{B}}$ by a simple normal crossing divisor
$D_{\operatorname{B}}$. We spell out the general construction of the mixed
Hodge structure of $X\setminus Y$ given in [9] for
$X=\overline{{\mathcal{M}}}_{\operatorname{B}}$ and $Y=D_{\operatorname{B}}$.

The filtration is the abutment of the spectral sequence associated to a double
complex $A_{**}$ endowed with a filtration $W$. For any $p\geq 1$ we denote by
$\tilde{D}^{p}$ the disjoint union of the $p$-fold intersections of the
irreducible components of $D_{\operatorname{B}}$, and set
$\tilde{D}^{0}=\overline{{\mathcal{M}}}_{\operatorname{B}}$. We denote by
$C_{t}^{\pitchfork}(\tilde{D}^{s})$ the free abelian group generated by
dimensionally transverse $t$-cycles in $\tilde{D}^{s}$, i.e. cycles for the
$0$-perversity function. We let

$A_{s,t}=C_{t}^{\pitchfork}(\tilde{D}^{-s}),$

where $s\leq 0,t\geq 0$. The filtration $W$ is defined by

$W_{s}=\bigoplus_{p\leq s}A_{p,t}.$

There exists a well-defined intersection morphism

$\cap\colon C_{t}^{\pitchfork}(\tilde{D}^{s})\to
C_{t-2}^{\pitchfork}(\tilde{D}^{s+1})$

compatible with $W$, turning $A_{**}$ into a filtered double complex. It is
shown in [9, Theorem 1.5] that the associated spectral sequence $E_{st}^{r}$
degenerates at page $r=2$ and abuts to the filtration

$E_{st}^{\infty}\otimes{\mathbb{Q}}=\operatorname{Gr}^{W}_{-t}H_{s+t}({\mathcal{M}}_{\operatorname{B}},{\mathbb{Q}}).$

The filtration $W$ on the right-hand side is then equal to Deligne’s weight
filtration.

The topological representatives of $\operatorname{Gr}^{W}_{-2k}H_{k}$
corresponding to the choices $t=2k$ and $s=-k$ are generated by classes of the
following form (for the similar cases $k=1$ and $k=2$ over surfaces see [9,
Example 6.9]). Take a generic point $Q$ in the $k$-fold intersections of the
divisors

$Q\in\tilde{D}^{k}\setminus\tilde{D}^{k+1}.$

Let the corresponding divisor components be denoted without loss of generality
$Y_{1},\ldots,Y_{k}$. The preimage $\Pi^{-1}(Q)$ of $Q$ in the normal bundle
of $Y_{1}\cap\cdots\cap Y_{k}$ in $X\setminus
Y={\mathcal{M}}_{\operatorname{B}}$ deformation retracts onto a
$k$-dimensional real torus (the boundary of a tubular neighbourhood of
$Y_{1}\cap\cdots\cap Y_{k}$). If one considers all $k$-tuple intersections of
divisor components, then the classes of these tori generate
$\operatorname{Gr}^{W}_{-2k}H_{k}$, and the dual cohomology classes generate
$\operatorname{Gr}^{W}_{2k}H^{k}$.

###  2.4. Hitchin maps and bases

######  Proposition 2.

  1. (1)

For every strongly parabolic $\vec{\alpha}$-stable Higgs bundle
$({\mathcal{E}},\theta)$ with logarithmic singularities at $D$ we have

$\mbox{tr}(\theta)\equiv 0.$

  2. (2)

There exists a linear subspace

$\mathcal{B}=\\{q:\quad q(t_{j})=0\;\;\mbox{for all}\;\;0\leq j\leq
4\\}\subset H^{0}(\mathbb{C}P^{1},L^{\otimes 2})\cong{\mathbb{C}}^{7}$

of dimension $2$ over ${\mathbb{C}}$ such that an $\vec{\alpha}$-stable Higgs
bundle $({\mathcal{E}},\theta)$ with logarithmic singularities at $D$ is
strongly parabolic if and only if

$\det(\theta)\in\mathcal{B}.$

###### Proof.

We have

$\displaystyle\mbox{tr}(\theta)$ $\displaystyle\in
H^{0}(\mathbb{C}P^{1},L)\cong{\mathbb{C}}^{4}$ $\displaystyle\det(\theta)$
$\displaystyle\in H^{0}(\mathbb{C}P^{1},L^{\otimes 2})\cong{\mathbb{C}}^{7}.$

The requirement on the eigenvalues of the residues of $\theta$ together
imposes $5$ linear relations on $\mbox{tr}(\theta)$; however, one of these
conditions expresses that the sum of the eigenvalues is 0, and is therefore
redundant. So, $\mbox{tr}(\theta)$ is uniquely determined as $0\in
H^{0}(\mathbb{C}P^{1},L)$, proving the first assertion. On the other hand, the
same requirements impose $5$ independent linear relations on $\det(\theta)$.
The second assertion follows. ∎

From now on, we will often let ${\mathcal{M}}_{\operatorname{Dol}}$ stand for
${\mathcal{M}}_{\operatorname{Dol}}(\vec{0},\vec{\alpha})$. It follows from
the Proposition that we have a well-defined map

$\displaystyle H\colon{\mathcal{M}}_{\operatorname{Dol}}$
$\displaystyle\to\mathcal{B}$ (2.11) $\displaystyle({\mathcal{E}},\theta)$
$\displaystyle\mapsto-\det(\theta)$

called the _Hitchin map_. (The negative sign will simplify our later
formulas.) The target space $\mathcal{B}$ of $h$ is called the _Hitchin base_.
The generic element of $\mathcal{B}$ will be denoted as

$q\in H^{0}(\mathbb{C}P^{1},L^{\otimes 2}).$

We fix the isomorphism ${\mathcal{O}}(3)\cong L$ given on the affine open
subset $w\neq 0$ by

$s(z,w)=\sum_{i=0}^{3}s_{i}z^{3-i}w^{i}\mapsto
S(z)=s(z,1)\frac{\mbox{d}z}{\prod_{j=0}^{4}(z-t_{j})}.$ (2.12)

Under this isomorphism, the value $s(t_{j},1)$ for $0\leq j\leq 4$ is equal to
the some non-zero multiple (only depending on the divisor $D$ and $j$) of the
residue $\mbox{res}_{t_{j}}(S)$. The isomorphism (2.12) induces the
isomorphism ${\mathcal{O}}(6)\cong L^{\otimes 2}$ given by

$q(z,w)=\sum_{i=0}^{6}q_{i}z^{6-i}w^{i}\mapsto
Q(z)=q(z,1)\frac{\mbox{d}z^{\otimes 2}}{\prod_{j=0}^{4}(z-t_{j})^{2}}.$ (2.13)

###  2.5. Spectral curve, Jacobian variety

We consider the total space $\mbox{Tot}(L)$ of $L$ with the natural projection

$p_{L}\colon\mbox{Tot}(L)\to\mathbb{C}P^{1}.$

We denote by $\zeta$ the canonical section of $p_{L}^{*}L$. For any

$q\in H^{0}(\mathbb{C}P^{1},L^{\otimes 2})$

we denote for simplicity $p_{L}^{*}q$ by $q$.

We endow $\mathcal{B}$ with a scalar product (we will be more precise in
(2.23)), pick $R>0$ and let $S^{3}_{R}$ denote the sphere of radius $R$ in
$\mathcal{B}\cong{\mathbb{C}}^{2}$. For $q\in S^{3}_{1}$ we write
$\zeta_{\pm}(Rq,z)$ for the roots of

$\zeta^{2}-Rq=0,$

specifically

$\zeta_{\pm}(Rq,z)=\pm\sqrt{Rq(z,1)}.$ (2.14)

We denote by

$X_{Rq}=\\{([z:w],\pm\sqrt{Rq(z,w)})\\}\subset\mbox{Tot}(L)$ (2.15)

the Riemann surface of the bivalued function $\zeta_{\pm}(Rq,z)$. For a
generic choice of $q$ this curve is smooth and of genus

$g(X_{Rq})=2.$

For generic $q\in S^{3}_{1}$, the fiber $H^{-1}(q)$ is smooth, and known to be
isomorphic to an abelian variety of dimension $2$ over ${\mathbb{C}}$, namely
(a torsor over) the Jacobian $\operatorname{Jac}(X_{q})$ of $X_{q}$:

$H^{-1}(q)\cong\operatorname{Jac}(X_{q})=H^{0,1}(X_{q})/\Lambda_{q}$ (2.16)

for the period lattice $\Lambda_{q}\subset
H^{0,1}(X_{q})\cong{\mathbb{C}}^{2}$ of $X_{Rq}$. Recall that

$\Lambda_{q}=\operatorname{Im}\left(p^{0,1}\circ\iota\right)$

where the map

$\iota\colon H^{1}(X_{q},2\pi\sqrt{-1}{\mathbb{Z}})\to
H^{1}(X_{q},{\mathbb{C}})$

is induced by the coefficient inclusion
$2\pi\sqrt{-1}{\mathbb{Z}}\to{\mathbb{C}}$ and the map

$p^{0,1}\colon H^{1}(X_{q},{\mathbb{C}})\to H^{0,1}(X_{q})$ (2.17)

is projection of harmonic forms to their antiholomorphic part. Then, for given
$\mu_{1},\mu_{2}\in H^{0,1}(X_{q})$ the relation

$\mu_{1}-\mu_{2}\in\Lambda_{q}$

is equivalent to the following condition: for every $1$-cycle $A$ on $X_{q}$
with coefficients in ${\mathbb{Z}}$ we have

$\int_{A}(\mu_{1}-\mu_{2})\in 2\pi\sqrt{-1}{\mathbb{Z}}.$

The abelian version of the Hodge correspondence $\psi$ of (2.8) on $X_{q}$
states that any class in $H^{0,1}(X_{q})$ may be represented by an anti-
holomorphic form, i.e. $\mu\in\Omega^{0,1}(X_{q})$ satisfying $\partial\mu=0$,
and that then the $\operatorname{U}(1)$-connection on the trivial line bundle
defined by the connection form

$B=\mu-\bar{\mu}\in\Omega^{1}(X_{q})$

is flat, see [13, Proposition 4.1.5]. With this notation, fixing any basis
$A_{1},A_{2},B_{1},B_{2}\in H_{1}(X_{q},{\mathbb{Z}})$, the abelian version of
$\operatorname{RH}\circ\psi$ (where $\operatorname{RH}$ is the Riemann–Hilbert
correspondence (2.10)) is then the diffeomorphism between the Jacobian and the
$4$-torus given by

$\displaystyle\operatorname{Jac}(X_{q})$ $\displaystyle\to T^{4}=(S^{1})^{4}$
$\displaystyle\mu$
$\displaystyle\mapsto\left(e^{\int_{A_{1}}B},e^{\int_{A_{2}}B},e^{\int_{B_{1}}B},e^{\int_{B_{2}}B}\right).$

###  2.6. Ramification of spectral curve

Clearly, $X_{Rq}$ is ramified over $D$. Let $\tilde{D}$ denote the
corresponding branch divisor, so $\tilde{D}$ consists of the preimages of the
points of $D$ on $X_{Rq}$, all counted with multiplicity $1$. We again let

$Z_{\pm}(Rq,z)$ (2.18)

stand for the bivalued meromorphic differentials corresponding to (2.14) over
the chart $z$ under the isomorphism (2.12). In concrete terms, we have

$Z_{\pm}(Rq,z)=\pm\sqrt{Rq(z,1)}\frac{\mbox{d}z}{\prod_{j=0}^{4}(z-t_{j})}$
(2.19)

We set

$\Delta_{q}=\\{z\in{\mathbb{C}}\colon\quad q(z)=0\\}.$

Regardless of the value of $R>0$, $\Delta_{q}$ is the ramification divisor of
the projection map

$p_{Rq}:X_{Rq}\to\mathbb{C}P^{1}$ (2.20)

induced by $p_{L}$. $\Delta_{q}$ contains the points of $D$ by Proposition 2,
and is of cardinality $6$ because $\mbox{deg}(L^{\otimes 2})=6$. It follows
that it is of the form

$\Delta_{q}=\\{t_{0},t_{1},t_{2},t_{3},t_{4},t(q)\\}$ (2.21)

for some $t(q)\in\mathbb{C}P^{1}$. In case $t(q)=t_{j_{0}}$ for some $0\leq
j_{0}\leq 4$, we assign multiplicity $2$ to $t_{j_{0}}$ in $\Delta_{q}$. On
the other hand, for any fixed $t\in\mathbb{C}P^{1}$ we denote by $\Delta_{t}$
the set of $q\in S^{3}_{1}$ such that $t\in\Delta_{q}$. We denote by
$\tilde{\Delta}_{q}$ the corresponding ramification points on $X_{q}$, and
similarly by $\tilde{t}(q),\tilde{D}$ the lifts of $t(q)$ and of the divisor
$D$, respectively.

######  Proposition 3.

For any fixed $t\in\mathbb{C}P^{1}$, the set $\Delta_{t}$ is diffeomorphic to
$S^{1}$, and the map

$\displaystyle t\colon S^{3}_{1}$ $\displaystyle\to\mathbb{C}P^{1}$
$\displaystyle q$ $\displaystyle\mapsto t(q)$

defined by (2.21) is the Hopf fibration.

###### Proof.

The section $q$ is a homogeneous polynomial of degree $6$, vanishing at the
points of $D$ by Proposition 2, hence is of the form

$q(z,w)=(az-bw)\prod_{j=0}^{4}(z-t_{j}w)$

for some $(a,b)\in{\mathbb{C}}^{2}\setminus\\{(0,0)\\}$. Using the isomorphism
(2.13), the corresponding meromorphic quadratic differential reads as

$Q(z)=\frac{(az-b)\mbox{d}z^{\otimes 2}}{\prod_{j=0}^{4}(z-t_{j})}.$ (2.22)

The coefficients $(a,b)$ describe natural co-ordinates of the space
$\mathcal{B}\cong{\mathbb{C}}^{2}$. For any fixed
$[z_{0}:w_{0}]\in\mathbb{C}P^{1}$ the condition $az_{0}-bw_{0}=0$ is linear in
$(a,b)$, hence $\Delta_{t}$ is the link of a line passing through $0$ in
$\mathcal{B}$. This shows the first assertion.

The map appearing in the second assertion is

$(a,b)\mapsto[b:a].$

This is just the canonical map from ${\mathbb{C}}^{2}\setminus\\{(0,0)\\}$ to
$\mathbb{C}P^{1}$. The result follows. ∎

Using the factorization (2.22) we will assume that the norm of $\mathcal{B}$
is

$|q|=\sqrt{|a|^{2}+|b|^{2}}.$ (2.23)

We use standard Hopf co-ordinates

$a=\cos(\theta)e^{\sqrt{-1}(\varphi-\phi)},\quad
b=\sin(\theta)e^{\sqrt{-1}(\varphi+\phi)}$ (2.24)

with $\theta\in[0,\frac{\pi}{2}]$ and $\varphi\in[0,2\pi]$ and
$\phi\in[0,\pi]$. Then, on the chart $\mbox{Spec}({\mathbb{C}}[z])$ the map
$t$ reads as

$t\colon(a,b)\mapsto\frac{b}{a}=\tan(\theta)e^{2\sqrt{-1}\phi}.$

The parameter of the Hopf circles is $\varphi$.

###  2.7. Perverse Leray filtration

Consider a general quasi-projective variety $Y$ and denote by
$D^{b}(Y,{\mathbb{Q}})$ the derived category of bounded complexes of
${\mathbb{Q}}$-vector spaces $K$ on $Y$ with constructible cohomology sheaves
of finite rank. Beilinson, Bernstein and Deligne [2] defined truncation
functors

${}^{\mathfrak{p}}\\!\tau_{\leq
i}:D^{b}(Y,{\mathbb{Q}})\to^{\mathfrak{p}}\\!\\!D^{\leq i}(Y,{\mathbb{Q}})$

encoding the support condition for the middle perversity function, giving rise
to a system of truncations

$0\to\cdots\to\,^{\mathfrak{p}}\\!\tau_{\leq-p}K\to\,^{\mathfrak{p}}\\!\tau_{\leq-p+1}K\to\cdots\to
K.$

This gives rise to the _perverse filtration_

$P^{p}H(Y,K)=\operatorname{Im}(H(Y,^{\mathfrak{p}}\\!\tau_{\leq-p}K)\to
H(Y,K)).$

We will apply the above results to the following setup. Consider the right
derived direct image functor

$\mathbf{R}H_{*}\colon
D^{b}({\mathcal{M}}_{\operatorname{Dol}},{\mathbb{Q}})\to
D^{b}(\mathcal{B},{\mathbb{Q}})$

and denote by $R^{l}H_{*}$ the $l$’th right derived direct image sheaf. Let
${\mathbf{H}}$ denote hypercohomology of a complex of sheaves and $H$ stand
for cohomology of a single sheaf. (We hope that the two different usages of
the symbol $H$ for the Hitchin map and for cohomology groups will not lead to
confusion.) Let
$\underline{{\mathbb{Q}}}_{{\mathcal{M}}_{\operatorname{Dol}}}$ denote the
constant sheaf with fibers ${\mathbb{Q}}$ on
${\mathcal{M}}_{\operatorname{Dol}}$. With these notations, we will be
interested in the perverse filtration on
$K=\mathbf{R}H_{*}\underline{{\mathbb{Q}}}_{{\mathcal{M}}_{\operatorname{Dol}}}$
over $Y=\mathcal{B}$. We then have

${\mathbf{H}}^{n}(\mathcal{B},\mathbf{R}H_{*}\underline{{\mathbb{Q}}}_{{\mathcal{M}}_{\operatorname{Dol}}})\cong
H^{n}({\mathcal{M}}_{\operatorname{Dol}},{\mathbb{Q}}).$

We will make use of a geometric characterization of the perverse filtration
provided by M. de Cataldo and L. Migliorini in [6, Theorem 4.1.1] in terms of
the flag filtration $F$. Namely, let

$Y_{-2}\subset Y_{-1}\subset Y=\mathcal{B}$ (2.25)

be a generic full affine flag in $\mathcal{B}$, namely $Y_{-1}$ a generic line
and $Y_{-2}$ a generic point within $Y_{-1}$. We then have the equality

$\displaystyle
P^{p}{\mathbf{H}}^{n}(Y,\mathbf{R}H_{*}\underline{{\mathbb{Q}}}_{{\mathcal{M}}_{\operatorname{Dol}}})$
$\displaystyle=F^{p+n}{\mathbf{H}}^{n}(Y,\mathbf{R}H_{*}\underline{{\mathbb{Q}}}_{{\mathcal{M}}_{\operatorname{Dol}}})$
$\displaystyle=\operatorname{Ker}({\mathbf{H}}^{n}(Y,\mathbf{R}H_{*}\underline{{\mathbb{Q}}}_{{\mathcal{M}}})\to{\mathbf{H}}^{n}(Y_{p+n-1},\mathbf{R}H_{*}\underline{{\mathbb{Q}}}_{{\mathcal{M}}}|_{Y_{p+n-1}})),$

where $F^{\bullet}$ stands to denote the flag filtration. It follows
immediately from $Y_{-3}=\varnothing$ that $P^{1-n}{\mathbf{H}}^{n}=0$ and
$P^{-n-2}{\mathbf{H}}^{n}={\mathbf{H}}^{n}$, so the only possibly non-trivial
graded pieces live in degrees $-n-2,-n-1,-n$. Notice that for $p=-1-n$ we get

${\mathbf{H}}^{*}(Y_{-2},\mathbf{R}H_{*}\underline{{\mathbb{Q}}}_{{\mathcal{M}}}|_{Y_{-2}})\cong
H^{*}(H^{-1}(Y_{-2}),{\mathbb{Q}})\cong\Lambda^{*}H^{1}(H^{-1}(Y_{-2}),{\mathbb{Q}}),$

the exterior algebra over $H^{1}(T^{4},{\mathbb{Q}})\cong{\mathbb{Q}}^{4}$.
Moreover, by the isomorphism theorem

$\operatorname{Gr}_{P}^{-n-2}H^{n}({\mathcal{M}}_{\operatorname{Dol}},{\mathbb{Q}})\cong\operatorname{Im}(H^{n}({\mathcal{M}}_{\operatorname{Dol}},{\mathbb{Q}})\to
H^{n}(H^{-1}(Y_{-2}),{\mathbb{Q}})).$

##  3\. Large scale behaviour of solutions of Hitchin’s equations

###  3.1. Asymptotic abelianization, limiting configuration

We fix a generic element $q\in S^{3}_{1}$ and consider
$({\mathcal{E}},\theta)\in{\mathcal{M}}_{\operatorname{Dol}}(\vec{0},\vec{\alpha})$
such that

$H({\mathcal{E}},\theta)=q.$ (3.1)

As we have explained in Section 2.5, choosing such a Higgs bundle
$({\mathcal{E}},\theta)$ amounts to fixing a point in an abelian variety of
complex dimension $2$. Then, for any $t\in{\mathbb{C}}^{\times}$ we have
$({\mathcal{E}},t\theta)\in{\mathcal{M}}_{\operatorname{Dol}}(\vec{0},\vec{\alpha})$,
i.e. ${\mathbb{C}}^{\times}$ acts on
${\mathcal{M}}_{\operatorname{Dol}}(\vec{0},\vec{\alpha})$. Obviously,

$H({\mathcal{E}},t\theta)=t^{2}q\in S^{3}_{|t|^{2}}.$

For any fixed value of $t$, there exists a unique solution $h_{t}$ of the real
Hitchin’s equation (2.4) associated to the pair $({\mathcal{E}},t\theta)$. We
will summarize some results of [11] (partly based on [23] and [25]) regarding
the asymptotic behaviour of the tame harmonic bundle associated to
$({\mathcal{E}},\sqrt{R}\theta)$ (the parameter $t>0$ of [11] thus being
replaced by $\sqrt{R}$ with $R>0$). The analysis in [11] relies on the
assumption that $\theta$ is generically regular semisimple. This holds for
generic $q\in S^{3}_{1}$. Indeed, if $\theta$ is not generically regular
semisimple then the curve (2.15) is a section $s$ of $p_{L}$ with multiplicity
$2$, which is clearly not the case generically.

Let ${\mathcal{L}}_{{\mathcal{E}}}\in\operatorname{Jac}(X_{q})$ be the line
bundle such that

${\mathcal{E}}=p_{q*}{\mathcal{L}}_{{\mathcal{E}}}$ (3.2)

(see (2.20)) and for any $R>0$ let us denote by

$\rho\colon X_{Rq}\to X_{Rq}$

the involution exchanging $Z_{+}(Rq,z)$ and $Z_{-}(Rq,z)$ (see (2.18)). As
$\rho$ is the restriction to $X_{Rq}$ of an algebraic involution defined over
all $\mbox{Tot}(L)$, we will omit $Rq$ from its notation. Then, there exists a
short exact sequence of sheaves on $X_{Rq}$

$0\to
p_{Rq}^{*}{\mathcal{E}}\to{\mathcal{L}}_{{\mathcal{E}}}\oplus\rho^{*}{\mathcal{L}}_{{\mathcal{E}}}\to{\mathcal{O}}_{\Delta_{q}}\to
0.$ (3.3)

Notice that for any $R>0$ there is an isomorphism

$X_{Rq}\cong X_{q}$

commuting with $p_{L}$; we deduce that the restriction of the Hitchin map $H$
to the ${\mathbb{R}}^{+}$-orbit of $q$ is canonically isomorphic to a product

${\mathbb{R}}^{+}\times H^{-1}(q).$

Therefore, in the sequel we will often identify $H^{-1}(Rq)$ and $H^{-1}(q)$.

Let $\tilde{t}(q)\in X_{Rq}$ be the preimage of $t(q)$ under $p_{Rq}$.

######  Proposition 4.

Formulas (2.19) define univalued holomorphic differentials on $X_{Rq}$,
vanishing to order $2$ at $\tilde{t}(q)$.

###### Proof.

The fact that the functions are univalued is clear, as $X_{Rq}$ is by
definition their Riemann surface.

For simplicity, let us work on the chart $z$ of $\mathbb{C}P^{1}$ and set
$[z:1]=z$, a similar analysis works over the chart $w$. Furthermore, in this
proof we use the notation

$\zeta_{i}=\zeta_{i}(Rq,z)$

with $i\in\\{\pm\\}$. A holomorphic chart of $X_{Rq}$ near $t_{j}$ is given by
$\zeta_{i}$, with local equation

$\zeta_{i}^{2}=R(z-t_{j})h_{j}(z)$

for some holomorphic function $h_{j}$ (depending on $q$) such that
$h_{j}(t_{j})\neq 0$. This shows that

$2\zeta_{i}\mbox{d}\zeta_{i}=R\mbox{d}zh_{j}(z)+R(z-t_{j})\mbox{d}h_{j}.$

We derive that the $1$-form $\omega$ defined by

$\displaystyle\omega=\frac{\mbox{d}z}{\zeta_{i}}$
$\displaystyle=\frac{1}{h_{j}}\left(\frac{2\mbox{d}\zeta_{i}}{R}-\frac{z-t_{j}}{\zeta_{i}}\mbox{d}h_{j}\right)$
$\displaystyle=\frac{1}{h_{j}}\left(\frac{\mbox{d}\zeta_{i}}{R}-\zeta_{i}\frac{\mbox{d}h_{j}}{h_{j}}\right)$

is holomorphic in $\zeta_{i}$. The formula shows that $\omega$ is holomorphic
near $z=t(q)$ too. At any point away from the ramification divisor
$\Delta_{q}$ the form $\omega$ is obviously regular.

Now, by (2.19) we have

$\displaystyle Z_{i}$
$\displaystyle=\pm\sqrt{R}\sqrt{az-b}\frac{\mbox{d}z}{\prod_{j=0}^{4}\sqrt{z-t_{j}}}$
$\displaystyle=\sqrt{R}(az-b)\frac{\mbox{d}z}{\zeta_{i}}=\pm\sqrt{R}(az-b)\omega,$

where the root of the polynomial $az-b$ is $t(q)$. The first assertion
immediately follows. For the second assertion, it is sufficient to notice that
near $z=t(q)$ we have

$az-b=\zeta_{i}^{2}h(\zeta_{i})$

for some non-vanishing holomorphic function $h$. This finishes the proof. ∎

Fix some $q\in S^{3}_{1}$ and consider a Higgs bundle $({\mathcal{E}},\theta)$
satisfying (3.1), and recall the notation (2.19). Let
${\mathcal{L}}_{{\mathcal{E}}}$ be the line bundle satisfying (3.2). By
abelian Hodge theory, there exists (up to multiplication by a constant) a
unique Hermitian metric $h_{\det({\mathcal{E}})}$ on $\det({\mathcal{E}})$
over $\mathbb{C}P^{1}$ satisfying:

  * •

the associated unitary connection $\nabla_{h_{\det({\mathcal{E}})}}$ on
$\det({\mathcal{E}})$ is flat (i.e.,$h_{\det({\mathcal{E}})}$ is
Hermitian–Einstein),

  * •

for some local holomorphic trivialization $\vec{e}_{1}\wedge\vec{e}_{2}$ of
$\det({\mathcal{E}})$ at $t_{j}$ we have

$\lim_{z\to
t_{j}}|z-t_{j}|^{-1}|\vec{e}_{1}\wedge\vec{e}_{2}|_{h_{\det({\mathcal{E}})}}=1$

for every $0\leq j\leq 4$.

Notice that the last condition is imposed by the choice of parabolic weights
(2.2).

Moreover, there exists (up to a scalar) a unique abelian Hermitian metric
$h_{{\mathcal{L}}_{{\mathcal{E}}}}$ on ${\mathcal{L}}_{{\mathcal{E}}}$ over
$X_{q}$ with parabolic points at $\tilde{\Delta}_{q}$ such that

  * •

the associated unitary connection $\nabla_{h_{{\mathcal{L}}_{{\mathcal{E}}}}}$
on ${\mathcal{L}}_{{\mathcal{E}}}$ is flat (i.e.,
$h_{{\mathcal{L}}_{{\mathcal{E}}}}$ is Hermitian–Einstein),

  * •

we have
$h_{{\mathcal{L}}_{{\mathcal{E}}}}\otimes\rho^{*}h_{{\mathcal{L}}_{{\mathcal{E}}}}=p_{q}^{*}h_{\det{\mathcal{E}}}$
over $\mathbb{C}P^{1}\setminus\Delta_{q}$ (see (3.3)),

  * •

for some trivialization $\vec{l}$ of ${\mathcal{L}}_{{\mathcal{E}}}$ at each
point and some local chart $\zeta$ of $X_{Rq}$ centered at
$\tilde{t}_{j}\in\tilde{D}$ we have

$\lim_{\zeta\to
0}|\zeta|^{-1}|\vec{l}|_{h_{{\mathcal{L}}_{{\mathcal{E}}}}}^{2}=1,$

  * •

for some trivialization $\vec{l}$ of ${\mathcal{L}}_{{\mathcal{E}}}$ at each
point and some local chart $\zeta$ of $X_{Rq}$ centered at $\tilde{t}(q)$ we
have

$\lim_{\zeta\to 0}|\zeta||\vec{l}|_{h_{{\mathcal{L}}_{{\mathcal{E}}}}}^{2}=1,$

Let $h_{{\mathcal{E}},\infty}$ be the orthogonal push-forward of
$h_{{\mathcal{L}}_{{\mathcal{E}}}}$ by $p_{q}$ over
$\mathbb{C}P^{1}\setminus\Delta_{q}$, so that in the direct sum decomposition
(3.3) the summands
${\mathcal{L}}_{{\mathcal{E}}},\rho^{*}{\mathcal{L}}_{{\mathcal{E}}}$ are
orthogonal to each other and the restrictions of $h_{{\mathcal{E}},\infty}$ to
these summands are respectively
$h_{{\mathcal{L}}_{{\mathcal{E}}}},\rho^{*}h_{{\mathcal{L}}_{{\mathcal{E}}}}$.
Let $\nabla_{h_{{\mathcal{E}},\infty}}$ be the flat
$\operatorname{U}(1)\times\operatorname{U}(1)$-connection on ${\mathcal{E}}$
associated to $h_{{\mathcal{E}},\infty}$ over
$\mathbb{C}P^{1}\setminus\Delta_{q}$. Over any simply connected subset of
$\mathbb{C}P^{1}\setminus\Delta_{q}$, let $p_{q,\ast}$ stand for the inverse
of $p_{q}^{*}$ on either branch of $X_{q}$. Let

$B_{\det({\mathcal{E}})}\in\Omega^{1}(\mathbb{C}P^{1}\setminus
D,{\mathbb{C}}),\qquad\frac{1}{2}p_{q}^{*}B_{\det({\mathcal{E}})}+B_{{\mathcal{L}}_{{\mathcal{E}}}}\in\Omega^{1}(X_{q}\setminus\tilde{\Delta}_{q},{\mathbb{C}})$

stand for the connection forms of the flat abelian
$\operatorname{U}(1)$-connections
$\nabla_{h_{\det({\mathcal{E}})}},\nabla_{h_{{\mathcal{L}}_{{\mathcal{E}}}}}$
with respect to some smooth unitary frames. The action of $\rho$ on the second
connection form is given by

$\frac{1}{2}p_{q}^{*}B_{\det({\mathcal{E}})}+B_{{\mathcal{L}}_{{\mathcal{E}}}}\mapsto\frac{1}{2}p_{q}^{*}B_{\det({\mathcal{E}})}-B_{{\mathcal{L}}_{{\mathcal{E}}}}.$

By the above properties, the connection form of
$\nabla_{h_{{\mathcal{E}},\infty}}$ with respect to a smooth unitary
trivialization of $V$ compatibe with the decomposition (3.3) reads as

$\begin{pmatrix}\frac{1}{2}B_{\det({\mathcal{E}})}+p_{q,\ast}B_{{\mathcal{L}}_{{\mathcal{E}}}}&0\\\
0&\frac{1}{2}B_{\det({\mathcal{E}})}-p_{q,\ast}B_{{\mathcal{L}}_{{\mathcal{E}}}}\end{pmatrix}.$
(3.4)

Moreover, if one denotes by
$\mu_{\det({\mathcal{E}})},\frac{1}{2}\mu_{\det({\mathcal{E}})}+\mu_{{\mathcal{L}}_{{\mathcal{E}}}}$
the $(0,1)$-forms of the $\bar{\partial}$-operators of the corresponding line
bundles with respect to smooth unitary frames, then we have

$\displaystyle B_{\det({\mathcal{E}})}$
$\displaystyle=\mu_{\det({\mathcal{E}})}-\bar{\mu}_{\det({\mathcal{E}})},$
(3.5) $\displaystyle B_{{\mathcal{L}}_{{\mathcal{E}}}}$
$\displaystyle=\mu_{{\mathcal{L}}_{{\mathcal{E}}}}-\bar{\mu}_{{\mathcal{L}}_{{\mathcal{E}}}}.$
(3.6)

We then obviously have
$p^{0,1}B_{\det({\mathcal{E}})}=\mu_{\det({\mathcal{E}})}$ and
$p^{0,1}B_{{\mathcal{L}}_{{\mathcal{E}}}}=\mu_{{\mathcal{L}}_{{\mathcal{E}}}}$,
see (2.17). We call $({\mathcal{E}},\theta,h_{{\mathcal{E}},\infty})$ the
limiting configuration associated to $({\mathcal{E}},\theta)$. We introduce
the model integrable connection

$\nabla_{\sqrt{R}}^{\operatorname{limiting}}=\nabla_{h_{{\mathcal{E}},\infty}}+\begin{pmatrix}2\Re
Z_{+}(Rq,z)&0\\\ 0&2\Re Z_{-}(Rq,z)\end{pmatrix}$ (3.7)

with respect to the same trivializations as above. On the other hand, we
denote by $h_{\sqrt{R}}$ the solution of (2.4) and $\nabla_{\sqrt{R}}$ the
Hermitian–Einstein metric and integrable connection associated to
$({\mathcal{E}},\sqrt{R}\theta)$.

######  Theorem 2 (T. Mochizuki).

[25, Corollary 2.13] Over any simply connected compact set
$K\subset{\mathbb{C}}\setminus\Delta_{q}$ there exists a gauge transformation
$g_{\sqrt{R}}$ such that

$g_{\sqrt{R}}\cdot\nabla_{\sqrt{R}}-\nabla_{\sqrt{R}}^{\operatorname{limiting}}\to
0$

(measured with respect to $h_{\sqrt{R}}$) as $R\to\infty$, uniformly over $K$.
More precisely, there exist $c_{2},C_{2}>0$ (depending on $K,q$) such that for
any $z\in K$ we have

$|g_{\sqrt{R}}\cdot\nabla_{\sqrt{R}}(z)-\nabla_{\sqrt{R}}^{\operatorname{limiting}}(z)|_{h}<C_{2}e^{-c_{2}\sqrt{R}}.$

###  3.2. Fiducial solution, approximate solutions

We will equally need the asymptotic form of the solution of Hitchin’s
equations near the points of $\Delta_{q}$, where Theorem 2 does not apply.
Such a description is provided by R. Mazzeo, J. Swoboda, H. Weiss, F. Witt in
[23] over a smooth projective curve $X$ of arbitrary genus. This decription is
extended by L. Fredrickson, R. Mazzeo, J. Swoboda, H. Weiss in [11] to the
case of a smooth projective curve $X$ of arbitrary genus for solutions of
Hitchin’s equations with a finite number of logarithmic singularities and
adapted parabolic structure. In accordance with our notations, we let
$\sqrt{R}$ be the rescaling parameter of the Higgs field, equal to the
parameter $t$ of [23] and [11]. We denote the standard holomorphic co-ordinate
of ${\mathbb{C}}$ by $\tilde{z}$ and work in a fixed disc
$B_{r_{0}}(0)=\\{|\tilde{z}|\leq r_{0}\\}$ for some $r_{0}>0$. We write
$\tilde{z}=\tilde{r}e^{\sqrt{-1}\tilde{\varphi}}$ for polar co-ordinates of a
point.

We first describe the solution in the case $0$ is a logarithmic singularity of
a harmonic bundle, with Dolbeault parabolic weights denoted by
$\alpha^{\pm}\in[0,1)$ as given in (2.2). Let

$m_{\sqrt{R}}\colon{\mathbb{R}}_{+}\to{\mathbb{R}}$

be the unique solution of the Painlevé III type equation

$\left(\frac{\mbox{d}^{2}}{\mbox{d}\tilde{r}^{2}}+\frac{1}{\tilde{r}}\frac{\mbox{d}}{\mbox{d}\tilde{r}}\right)m_{\sqrt{R}}=8R\tilde{r}^{-1}\sinh(2m_{\sqrt{R}})$

satisfying the boundary behaviours

$\displaystyle m_{\sqrt{R}}(\tilde{r})$
$\displaystyle\approx\left(\frac{1}{2}+\alpha_{j}^{+}-\alpha_{j}^{-}\right)\log(\tilde{r})=\log(\tilde{r}),\quad\tilde{r}\to
0+$ $\displaystyle m_{\sqrt{R}}(\tilde{r})$
$\displaystyle\approx\frac{1}{\pi}K_{0}(8\sqrt{R\tilde{r}})\approx\frac{1}{2\pi\sqrt{2}\sqrt[4]{R\tilde{r}}}e^{-8\sqrt{R\tilde{r}}},\quad\tilde{r}\to\infty$
(3.8)

where the sign $\approx$ stands for complete asymptotic expansion and $K_{0}$
is the modified Bessel function (or Bessel function of imaginary argument) of
order $0$. Furthermore, let us set

$F_{\sqrt{R}}(\tilde{r})=-\frac{1}{8}+\frac{1}{4}\tilde{r}\partial_{\tilde{r}}m_{\sqrt{R}}.$
(3.9)

Making use of the values fixed in (2.2), the _fiducial solution_ of Hitchin’s
equations on $B_{r_{0}}(0)$, introduced in [11, Proposition 3.9], is defined
in a fixed unitary frame as

$\displaystyle A_{\sqrt{R}}^{\operatorname{fid}}$
$\displaystyle=\left(\frac{\alpha^{+}+\alpha^{-}}{4}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}+F_{\sqrt{R}}(\tilde{r})\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\right)2\sqrt{-1}\mbox{d}\tilde{\varphi}$ (3.10)
$\displaystyle=\left(\frac{1}{4}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}+F_{\sqrt{R}}(\tilde{r})\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\right)2\sqrt{-1}\mbox{d}\tilde{\varphi},$ (3.11)
$\displaystyle\theta_{\sqrt{R}}^{\operatorname{fid}}$
$\displaystyle=\begin{pmatrix}0&\tilde{r}^{-1/2}e^{m_{\sqrt{R}}(\tilde{r})}\\\
\tilde{z}^{-1}\tilde{r}^{1/2}e^{-m_{\sqrt{R}}(\tilde{r})}&0\end{pmatrix}\mbox{d}\tilde{z}.$
(3.12)

The unitary frame with respect to which the above formulas hold is called
_fiducial frame_ , and denoted by

$(e_{1}^{\operatorname{fid}},e_{2}^{\operatorname{fid}}).$ (3.13)

There is a similar family of solutions for the ramification point $t(q)$ of
the spectral curve $X_{q}$ of the Higgs field. We fix a holomorphic chart
$\tilde{z}$ centered at $t(q)$ with associated polar coordinates denoted by
$\tilde{r},\tilde{\varphi}$. Then, with respect to a unitary frame again
denoted by

$(e_{1}^{\operatorname{fid}},e_{2}^{\operatorname{fid}})$ (3.14)

one can introduce

$\displaystyle A_{\sqrt{R}}^{\operatorname{fid}}$
$\displaystyle=\left(\frac{1}{8}+\frac{1}{4}\tilde{r}\partial_{\tilde{r}}\ell_{\sqrt{R}}\right)\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}2\sqrt{-1}\mbox{d}\tilde{\varphi}$ (3.15)
$\displaystyle\theta_{\sqrt{R}}^{\operatorname{fid}}$
$\displaystyle=\begin{pmatrix}0&\tilde{r}^{1/2}e^{\ell_{\sqrt{R}}(\tilde{r})}\\\
\tilde{z}\tilde{r}^{-1/2}e^{-\ell_{\sqrt{R}}(\tilde{r})}&0\end{pmatrix}\mbox{d}\tilde{z},$
(3.16)

where $\ell_{\sqrt{R}}$ is the solution of the equation

$\left(\frac{\mbox{d}^{2}}{\mbox{d}\tilde{r}^{2}}+\frac{1}{\tilde{r}}\frac{\mbox{d}}{\mbox{d}\tilde{r}}\right)\ell_{\sqrt{R}}=8R\tilde{r}\sinh(2\ell_{\sqrt{R}})$

satisfying the boundary behaviours

$\displaystyle\ell_{\sqrt{R}}(\tilde{r})$
$\displaystyle\approx-\frac{1}{2}\log(\tilde{r}),\quad\tilde{r}\to 0+$
$\displaystyle\ell_{\sqrt{R}}(\tilde{r})$
$\displaystyle\approx\frac{1}{\pi}K_{0}\left(\frac{8}{3}\sqrt{R\tilde{r}^{3}}\right)\approx\frac{\sqrt{3}}{2\pi\sqrt{2}\sqrt[4]{R\tilde{r}^{3}}}e^{-\frac{8}{3}\sqrt{R\tilde{r}^{3}}},\quad\tilde{r}\to\infty.$

The _limiting fiducial solution_ is obtained by letting $R\to\infty$ in the
above formulas, specifically

$\displaystyle A_{\infty}^{\operatorname{fid}}$
$\displaystyle=\frac{1}{8}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}2\sqrt{-1}\mbox{d}\tilde{\varphi}$
$\displaystyle\theta_{\infty}^{\operatorname{fid}}$
$\displaystyle=\begin{pmatrix}0&\tilde{r}^{1/2}\\\
\tilde{r}^{1/2}e^{\sqrt{-1}\tilde{\varphi}}&0\end{pmatrix}\mbox{d}\tilde{z}.$

In order to assemble the limiting configuration and the fiducial solutions
into a family of approximate solutions $h_{Rq}^{\operatorname{app}}$, [11]
perform a gluing construction. We describe this construction.

We start by describing normal forms of the solution of Hitchin’s equations
near the points of $\Delta_{q}$. By [11, Proposition 3.4] there exists a
unique holomorphic co-ordinate $\tilde{z}_{t}$ defined in a neighbourhood of
the ramification point $t=t(q)$ such that

$q(\tilde{z}_{t})=-\tilde{z}_{t}(\mbox{d}\tilde{z}_{t})^{2}.$ (3.17)

Furthermore, there exists a holomorphic gauge of ${\mathcal{E}}$ near $t(q)$
with respect to which one has

$\theta=\begin{pmatrix}0&1\\\
\tilde{z}_{t}&0\end{pmatrix}\mbox{d}\tilde{z}_{t},\quad
h_{{\mathcal{E}},\infty}=Q_{t}(\tilde{z}_{t})\begin{pmatrix}|\tilde{z}_{t}|^{\frac{1}{2}}&0\\\
0&|\tilde{z}_{t}|^{-\frac{1}{2}}\end{pmatrix}$

where $Q_{t}$ is a locally-defined smooth function completely determined by
$h_{\det{\mathcal{E}}}$ and $q$. Similarly, for any $0\leq j\leq 4$ [11,
Proposition 3.5] shows that there exists some holomorphic co-ordinate
$\tilde{z}_{j}$ of ${\mathcal{E}}$ near $t_{j}$ such that we have

$q(\tilde{z}_{j})=-\tilde{z}_{j}^{-1}(\mbox{d}\tilde{z}_{j})^{2}.$ (3.18)

Furthermore, there exists a holomorphic gauge of ${\mathcal{E}}$ near $t_{j}$
with respect to which one has

$\theta=\begin{pmatrix}0&1\\\
\tilde{z}_{j}^{-1}&0\end{pmatrix}\mbox{d}\tilde{z}_{j},\quad
h_{{\mathcal{E}},\infty}=Q_{j}|\tilde{z}_{j}|^{\alpha_{j}^{+}+\alpha_{j}^{-}}\begin{pmatrix}|\tilde{z}_{j}|^{-\frac{1}{2}}&0\\\
0&|\tilde{z}_{j}|^{\frac{1}{2}}\end{pmatrix}=Q_{j}\begin{pmatrix}|\tilde{z}_{j}|^{\frac{1}{2}}&0\\\
0&|\tilde{z}_{j}|^{\frac{3}{2}}\end{pmatrix}$

where $Q_{j}$ is a locally-defined function completely determined by
$h_{\det{\mathcal{E}}}$ and $q$. Let us fix a cutoff function
$\chi\colon[0,\infty)\to[0,1]$ such that $\chi(\tilde{r})=1$ for all
$\tilde{r}\leq\frac{1}{2}$ and $\chi(\tilde{r})=0$ for all $\tilde{r}\geq 1$.
Define the smooth Hermitian metric $h_{\sqrt{R}}^{\operatorname{app}}$ to be
equal

  * •

to

$Q_{t}(\tilde{z}_{t})\begin{pmatrix}|\tilde{z}_{t}|^{\frac{1}{2}}e^{\ell_{\sqrt{R}}(|\tilde{z}_{t}|)\chi(|\tilde{z}_{t}|)}&0\\\
0&|\tilde{z}_{t}|^{-\frac{1}{2}}e^{\ell_{\sqrt{R}}(|\tilde{z}_{t}|)\chi(|\tilde{z}_{t}|)}\end{pmatrix}$

on $|\tilde{z}_{t}|\leq 1$ in a holomorphic co-ordinate and gauge with respect
to which the above normal form holds ;

  * •

to

$Q_{j}(\tilde{z}_{j})\begin{pmatrix}|\tilde{z}_{j}|^{\frac{1}{2}}e^{m_{\sqrt{R}}(|\tilde{z}_{j}|)\chi(|\tilde{z}_{j}|)}&0\\\
0&|\tilde{z}_{j}|^{\frac{3}{2}}e^{m_{\sqrt{R}}(|\tilde{z}_{j}|)\chi(|\tilde{z}_{j}|)}\end{pmatrix}$

on $|\tilde{z}_{j}|\leq 1$ in a holomorphic co-ordinate and gauge with respect
to which the above normal form holds ;

  * •

to $h_{{\mathcal{E}},\infty}$ on the complement of the above discs.

Fix a background Hermitian metric $h_{0}$ on $V$ and let us denote by
$H_{\sqrt{R}},H_{\sqrt{R}}^{\operatorname{app}}$ the $h_{0}$-Hermitian
sections of $End(V)$ satisfying

$h_{\sqrt{R}}(v,w)=h_{0}\left((H_{\sqrt{R}})^{\frac{1}{2}}v,(H_{\sqrt{R}})^{\frac{1}{2}}w\right)$
(3.19)

and similarly

$h_{\sqrt{R}}^{\operatorname{app}}(v,w)=h_{0}\left((H_{\sqrt{R}}^{\operatorname{app}})^{\frac{1}{2}}v,(H_{\sqrt{R}}^{\operatorname{app}})^{\frac{1}{2}}w\right).$

Then, for a fixed Higgs bundle $({\mathcal{E}},\theta)$ one may look for
solutions $({\mathcal{E}},{\sqrt{R}}\theta,h_{\sqrt{R}})$ of Hitchin’s
equations (i.e., the Hermite–Einstein equation for the pair
$({\mathcal{E}},{\sqrt{R}}\theta)$) in the form

$(H_{\sqrt{R}})^{\frac{1}{2}}=e^{\gamma_{\sqrt{R}}}(H_{\sqrt{R}}^{\operatorname{app}})^{\frac{1}{2}}$
(3.20)

for some
$\sqrt{-1}{\mathfrak{su}}(V,h_{\sqrt{R}}^{\operatorname{app}})$-valued section
$\gamma_{\sqrt{R}}$.

######  Theorem 3.

[23, Theorem 6.7],[11, Theorem 6.2] Assume that all the zeroes of $q$ are
simple. Then, there exists $C,\mu>0$ and a unique section $\gamma_{\sqrt{R}}$
such that the Hermitian metric (3.19) with (3.20) satisfies the
Hermite–Einstein equation, and

$\|\gamma_{\sqrt{R}}\|_{\mathcal{C}^{2,\alpha}_{b}}\leq Ce^{-(\mu/2)\sqrt{R}}$

for an appropriate Hölder norm $\mathcal{C}^{2,\alpha}_{b}$.

The practical implication of this result for our purpose is that one may
perturb the approximate solution by a term exponentially small in $\sqrt{R}$
so as to obtain the solution of Hitchin’s equations. We will denote by
$\nabla_{\sqrt{R}}$ the flat connection associated to the solution
$({\mathcal{E}},{\sqrt{R}}\theta,h_{\sqrt{R}})$, i.e.

$\nabla_{\sqrt{R}}=\bar{\partial}_{{\mathcal{E}}}+\partial^{h_{\sqrt{R}}}+\sqrt{R}\theta+\sqrt{R}\theta^{\dagger,h_{\sqrt{R}}}$
(3.21)

where $\dagger,h_{\sqrt{R}}$ stands for adjoint with respect to
$h_{\sqrt{R}}$. Then, $\nabla_{\sqrt{R}}$ is approximated up to exponentially
decreasing error terms in $R$ by

$\nabla_{\sqrt{R}}^{\operatorname{app}}=\bar{\partial}_{{\mathcal{E}}}+\partial^{h_{\sqrt{R}}^{\operatorname{app}}}+\sqrt{R}\theta+\sqrt{R}\theta^{\dagger,h_{\sqrt{R}}^{\operatorname{app}}}.$

##  4\. Simpson’s Fenchel–Nielsen co-ordinates

Simpson has defined in [31, Section 10] co-ordinates of
${\mathcal{M}}_{\operatorname{B}}(\vec{c},\vec{\gamma})$. In this section, we
will recall the definition of these co-ordinates. The general element of the
Betti moduli space is a local system $V$ on $\mathbb{C}P^{1}\setminus D$,
given by a representation $\chi$ of its fundamental group, with eigenvalues
around the punctures $t_{j}$ equal to $c_{j}^{\pm}=\pm\sqrt{-1}$. For each
$2\leq i\leq 3$ there are two different co-ordinates: $l_{i}\in{\mathbb{C}}$
and $[p_{i}:q_{i}]\in\mathbb{C}P^{1}$. By analogy with classical Teichmüller
theory, we will call co-ordinates of the first type $l_{i}$ the _complex
length co-ordinates_ and those of the second type $[p_{i}:q_{i}]$ the _complex
twist co-ordinates_. Indeed, the traditional length co-ordinates in
Teichmüller space belong to ${\mathbb{R}}$ and ${\mathbb{C}}$ is its
complexification; similarly, the twist co-ordinates take values in $S^{1}$,
which is the real part ${\mathbb{R}}P^{1}$ of $\mathbb{C}P^{1}$ for the
canonical real structure.

######  Remark 1.

The construction of the co-ordinates depend on some choices, most importanty
the radii $r_{0}$ of discs around the punctures. However, as our main concern
is the homotopy type of the diffeomorphism $\operatorname{RH}\circ\psi$ and
different choices obviously result in homotopic maps, the actual choices do
not fundamentally affect our arguments. As a matter of fact, throughout we
will let $r_{0}\to 0$.

###  4.1. Complex length co-ordinates

We fix disjoint open discs $D_{j}$ around the points $t_{j}$ for $0\leq j\leq
4$; to fix our ideas we pick
$D_{j}=B_{r_{0}}(t_{j})^{o}=\\{|z-t_{j}|<r_{0}\\}$ for some $0<r_{0}\ll 1$ so
that the different discs $D_{j}$ are disjoint. We then set

$S=\mathbb{C}P^{1}\setminus(D_{0}\cup\cdots\cup D_{4}).$ (4.1)

Then $S$ is a smooth surface with boundary, inheriting an orientation from
$\mathbb{C}P^{1}$. Let us denote by $\xi_{j}$ the boundary component $\partial
D_{j}$, taken with the orientation induced from $S$. Specifically, we let

$\xi_{j}(\varphi)=t_{j}+r_{0}e^{\sqrt{-1}\varphi}\quad\mbox{for}\;\varphi\in[0,2\pi].$
(4.2)

Fix a simple loop $\rho_{2}$ in $S$ separating the boundary components
$\xi_{1},\xi_{2}$ from the remaining boundary components
$\xi_{3},\xi_{4},\xi_{0}$, and a simple loop $\rho_{3}$ in $S$ separating the
boundary components $\xi_{4},\xi_{0}$ from the remaining boundary components
$\xi_{1},\xi_{2},\xi_{3}$, so that $\rho_{2}$ and $\rho_{3}$ be disjoint from
each other. These curves then decompose $S$ into the union

$S=S_{2}\cup S_{3}\cup S_{3}$ (4.3)

of three pairs of pants:

  * •

$S_{2}$ with boundary components $\xi_{1},\xi_{2},\rho_{2}$;

  * •

$S_{3}$ with boundary components $\xi_{3},\rho_{2},\rho_{3}$;

  * •

and $S_{3}$ with boundary components $\xi_{4},\xi_{0},\rho_{3}$.

This decomposition gives rise to a decomposition of $\mathbb{C}P^{1}$ into the
three closed connected analytic subsets

$\displaystyle X_{2}$ $\displaystyle=S_{2}\cup D_{1}\cup D_{2}$ (4.4)
$\displaystyle X_{3}$ $\displaystyle=S_{3}\cup D_{3}$ (4.5) $\displaystyle
X_{4}$ $\displaystyle=S_{3}\cup D_{4}\cup D_{0}.$ (4.6)

Furthermore, we fix

  * •

base points $x_{i}\in\mbox{int}(S_{i})$ and $s_{i}\in\rho_{i}$;

  * •

paths $\psi_{i}$ connecting $x_{i}$ to $x_{i+1}$ passing through $s_{i}$;

  * •

paths $\eta_{1},\eta_{2}$ connecting $x_{2}$ respectively to the base points
of $\xi_{1},\xi_{2}$;

  * •

a path $\eta_{3}$ connecting $x_{3}$ to the base point of $\xi_{3}$;

  * •

paths $\eta_{4},\eta_{0}$ connecting $x_{4}$ respectively to the base points
of $\xi_{4},\xi_{0}$.

As the $D_{j}$ will actually depend on its radius $r_{0}$, we need to make a
coherent choice for the paths $\eta_{0},\ldots,\eta_{4}$. We achieve this for
instance for $\eta_{1}$ by first fixing a path starting at $x_{2}$ and ending
at $t_{1}$, and then restricting this fixed path to the (uniquely determined)
sub-interval of its domain such that the restriction connects $x_{2}$ to the
base point of $\xi_{1}$. We apply a similar procedure to $\eta_{j}$ for all
$0\leq j\leq 4$. We set $\rho_{1}=\xi_{1}$ and $\rho_{4}=\xi_{0}$.

$x_{2}$$x_{3}$$x_{4}$$s_{2}$$s_{3}$$\rho_{1}=\xi_{1}$$D_{1}$$\xi_{2}$$D_{2}$$\xi_{3}$$D_{3}$$\xi_{0}=\rho_{4}$$D_{0}$$\xi_{4}$$D_{4}$$\rho_{3}$$\rho_{2}$$\psi_{2}$$\psi_{3}$$\eta_{1}$$\eta_{2}$$\eta_{3}$$\eta_{0}$$\eta_{4}$$S_{2}$$S_{3}$$S_{3}$
Figure 1. Decomposition of $S$ into three pairs of pants, indicating base
points and paths. The shaded regions do not belong to $S$.

Following [31], for $2\leq i\leq 3$ we set $l_{i}(V)=l_{i}(\chi)$ for the
trace of $\chi$ evaluated on the class of the loop $\rho_{i}$:

$l_{i}(V)=\operatorname{tr}\chi[\rho_{i}]$

By definition, $l_{i}$ is the $i$’th _complex length co-ordinate_.

###  4.2. Complex twist co-ordinates

Twist co-ordinates are only defined over the part
${\mathcal{M}}_{\operatorname{B}}(\vec{c},\vec{\gamma})^{\prime}$ of the
moduli space where we have $|l_{i}|\neq 2$ (equivalently, the eigenvalues of
$\chi[\rho_{i}]$ are distinct) for both $2\leq i\leq 3$, for the complex
length co-ordinates $l_{i}$ introduced in Section 4.1, and a further stability
condition holds (see [31, Definition 5.1]). It is proven in [31, Corollary
9.2] that the homotopy type of the dual boundary complex of
${\mathcal{M}}_{\operatorname{B}}(\vec{c},\vec{\gamma})$ agrees with the one
of ${\mathcal{M}}_{\operatorname{B}}(\vec{c},\vec{\gamma})^{\prime}$.

Let us introduce the scalar quantities

$\displaystyle l_{1}$ $\displaystyle=c_{1}^{+}+c_{1}^{-}$ $\displaystyle
l_{4}$ $\displaystyle=c_{4}^{+}+c_{4}^{-}$ $\displaystyle u_{i}$
$\displaystyle=\frac{l_{i-1}-c_{i}^{-}l_{i}}{c_{i}^{+}-c_{i}^{-}}$
$\displaystyle w_{i}$ $\displaystyle=u_{i}(l_{i}-u_{i})-1$

for $2\leq i\leq 4$, where $l_{i}$ are the complex length co-ordinates
associated to $V$ as in Section 4.1. Furthermore, introduce the matrices

$\displaystyle A_{i}$ $\displaystyle=\begin{pmatrix}c_{i}^{+}&0\\\
0&c_{i}^{-}\end{pmatrix}$ $\displaystyle R_{i}$
$\displaystyle=\begin{pmatrix}u_{i}&1\\\ w_{i}&(l_{i}-u_{i})\end{pmatrix}$
$\displaystyle R^{\prime}_{i-1}$
$\displaystyle=A_{i}R_{i}=\begin{pmatrix}c_{i}^{+}u_{i}&c_{i}^{+}\\\
c_{i}^{-}w_{i}&c_{i}^{-}(l_{i}-u_{i})\end{pmatrix}$ $\displaystyle T_{i}$
$\displaystyle=\begin{pmatrix}0&1\\\ -1&l_{i}\end{pmatrix}$ $\displaystyle
U_{i}$ $\displaystyle=\begin{pmatrix}1&0\\\ u_{i}&1\end{pmatrix}.$

These quantities are all determined by the fixed constants $c_{i}^{\pm}$ and
the length co-ordinates $l_{2},l_{3}$.

Let $V_{i}(l_{i-1},l_{i})$ denote the local system on $S_{i}$ whose monodromy
matrices around $\rho_{i-1},\rho_{i}$ and $\xi_{i}$, acting on its fiber over
$x_{i}$, are respectively $R^{\prime}_{i-1},R_{i},A_{i}$. [31, Corollary 10.3]
implies that if $V|_{S_{i}}$ is stable then there exists a unique (up to a
scalar) isomorphism

$h_{i}\colon V|_{S_{i}}\to V_{i}(l_{i-1},l_{i}).$ (4.7)

By an abuse of notation, we denote by

$\psi_{i}\colon V_{x_{i}}\to V_{x_{i+1}}$ (4.8)

the parallel transport map of $V$ along the path $\psi_{i}$. Introduce

$P_{i}=h_{i+1}\circ\psi_{i}\circ h_{i}^{-1}\colon
V_{i}(l_{i-1},l_{i})_{x_{i}}\to V_{i+1}(l_{i},l_{i+1})_{x_{i+1}}$ (4.9)

and

$Q_{i-1}=A_{i}^{-\frac{1}{2}}U_{i}P_{i-1}U_{i-1}^{-1},$ (4.10)

for any choice of the square root of $A_{i}$. It turns out that one has

$Q_{i}=\begin{pmatrix}p_{i}&q_{i}\\\ -q_{i}&p_{i}+l_{i}q_{i}\end{pmatrix}$
(4.11)

for some $[p_{i}:q_{i}]\in\mathbb{C}P^{1}$ satisfying

$p_{i}^{2}+l_{i}p_{i}q_{i}+q_{i}^{2}\neq 0.$

By definition, $[p_{i}:q_{i}]\in\mathbb{C}P^{1}$ for $i\in\\{2,3\\}$ is the
$i$’th _complex twist co-ordinate_. Notice that a scalar factor on $Q_{i}$ has
no impact on $[p_{i}:q_{i}]$. Let us introduce

$\mathbf{Q}=\\{(l,[p:q])\in({\mathbb{C}}\setminus\\{\pm
2\\})\times\mathbb{C}P^{1}\;\mbox{satisfying}\;p^{2}+lpq+q^{2}\neq 0\\}.$

According to [31, Theorem 10.6], the map

$\displaystyle{\mathcal{M}}_{\operatorname{B}}(\vec{c},\vec{\gamma})^{\prime}$
$\displaystyle\to\mathbf{Q}^{2}$ $\displaystyle V$
$\displaystyle\mapsto((l_{2},[p_{2}:q_{2}]),(l_{3},[p_{3}:q_{3}]))$

is a diffeomorphism.

###  4.3. Homotopy type of compactifying divisor

According to [31, Lemma 10.7] and [28, Lemma 6.2] we have homotopy
equivalences

$\displaystyle\mathbb{D}\partial\mathbf{Q}$ $\displaystyle\sim S^{1}$ (4.12)
$\displaystyle\mathbb{D}\partial\mathbf{Q}^{2}$ $\displaystyle\sim S^{1}\ast
S^{1}\sim S^{3},$ (4.13)

where $X\ast Y$ stands for the join of the topological spaces $X,Y$. Combining
these arguments, [31, Corollary 10.8] shows that

$\mathbb{D}\partial{\mathcal{M}}_{\operatorname{B}}(\vec{c},\vec{\gamma})\sim
S^{3}.$

Let us spell out explicitly the homotopy equivalence (4.12). We now consider
two copies of

$\mathbf{Q}\subset\mathbb{C}P^{1}\times\mathbb{C}P^{1},$

that we will denote by $\mathbf{Q}_{i}$ for $i\in\\{2,3\\}$. A
compactification of $\mathbf{Q}_{i}$ is
$\mathbb{C}P^{1}\times\mathbb{C}P^{1}$, an open affine of the first component
being parametrised by $l_{i}$, and the second component being parametrised by
$[p_{i}:q_{i}]$. Let us denote by $F_{i,+},F_{i,-},F_{i,\infty}$ the fibers of
the first projection over $2,-2$ and $\infty$ respectively. The irreducible
decomposition of the compactifying divisor of $\mathbf{Q}_{i}$ in
$\mathbb{C}P^{1}\times\mathbb{C}P^{1}$ reads as

$\partial\mathbf{Q}_{i}=\mathbb{C}P^{1}\times\mathbb{C}P^{1}\setminus\mathbf{Q}_{i}=C_{i}\cup
F_{i,+}\cup F_{i,-}\cup F_{i,\infty}$

where $C_{i}$ is the quadric defined by
$p_{i}^{2}+l_{i}p_{i}q_{i}+q_{i}^{2}=0$, see Figure 2.

$F_{i,-}$$F_{i,+}$$F_{i,\infty}$$C_{i}$$[0:1]$$[1:0]$ Figure 2. Compactifying
divisor in $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$.

Clearly, $C_{i}$ is generically $2:1$ over $\mathbb{C}P^{1}_{t}$, with
ramification points in the fibers $F_{i,+},F_{i,-}$. Therefore, the
compactifying divisor in $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$ is not normal
crossing. To remedy this failure, we consider the blow up of
$\mathbb{C}P^{1}\times\mathbb{C}P^{1}$ in the intersection points $(2,[1:-1])$
and $(-2,[1:1])$, see Figure 3.

$F_{i,-}$$F_{i,+}$$F_{i,\infty}$$E_{i,-}^{1}$$E_{i,+}^{1}$$C_{i}$$[0:1]$$[1:0]$
Figure 3. Compactifying divisor in first blow-up.

We continue to denote by $C_{i},F_{i,+},F_{i,-},F_{i,\infty}$ the proper
transforms in $X$ of the named divisors, and we denote by
$E_{i,+}^{1},E_{i,-}^{1}$ the exceptional divisors. The compactifying divisor
in the blow-up is

$C_{i}\cup F_{i,+}\cup F_{i,-}\cup F_{i,\infty}\cup E_{i,+}^{1}\cup
E_{i,-}^{1}.$

However, this is still not simple normal crossing, because of the triple
intersection points of $C_{i},E_{i,+}^{1}$ and $F_{i,+}$ on the one hand, and
$C_{i},E_{i,-}^{1}$ and $F_{i,-}$ on the other hand. Therefore, we need to
blow up again in these intersection points, see Figure 4.

$F_{i,-}$$F_{i,+}$$F_{i,\infty}$$E_{i,-}^{1}$$E_{i,+}^{1}$$E_{i,-}^{2}$$E_{i,+}^{2}$$C_{i}$$[0:1]$$[1:0]$
Figure 4. Compactifying divisor in second blow-up.

The compactifying divisor in this surface is of normal crossing. Dropping the
subscripts $i$ for simplicity, its dual complex is

$F_{-}$$E_{-}^{2}$$E_{+}^{2}$$F_{+}$$F_{\infty}$$C$$E_{-}^{1}$$E_{+}^{1}$$e_{\infty}$$e_{0}$

Obviously, this graph deformation retracts to the cycle defined by the
vertices $F_{i,\infty},C_{i}$ together with the edges $e_{0},e_{\infty}$
connecting them. Notice that in $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$, this
cycle reduces to the normal crossing components $F_{i,\infty},C_{i}$.

Next, let us be more precise about the homotopy equivalence (4.13) following
[28, Lemma 6.2]. The compactifying divisor of
$\mathbf{Q}^{2}=\mathbf{Q}_{2}\times\mathbf{Q}_{3}$ in

$(\mathbb{C}P^{1})^{4}=(\mathbb{C}P^{1})^{2}\times(\mathbb{C}P^{1})^{2}$

can be given as

$(\partial\mathbf{Q}_{2}\times(\mathbb{C}P^{1})^{2})\cup((\mathbb{C}P^{1})^{2}\times\partial\mathbf{Q}_{3}).$

As we have explained above, up to homotopy of
$\mathbb{D}\partial\mathbf{Q}_{i}$ we only need to consider the divisor
components $C_{i}$ and $F_{i,\infty}$ of $\partial\mathbf{Q}_{i}$ and the
edges connecting them; in the rest of this section, we will thus replace
$\mathbb{D}\partial\mathbf{Q}_{i}$ by this subcomplex without changing the
notation. Our notation in the dual graph of $\partial\mathbf{Q}_{i}$ is that
$e_{i,\infty}$ stands for the edge corresponding to the point

$(l_{i},[p_{i}:q_{i}])=(\infty,[1:0])\in(\mathbb{C}P^{1})^{2},$

and $e_{i,0}$ for the edge corresponding to

$(l_{i},[p_{i}:q_{i}])=(\infty,[0:1])\in(\mathbb{C}P^{1})^{2}.$

Now, to each of the four points

$\displaystyle(\infty,[1:0]),(\infty,[1:0])$
$\displaystyle(\infty,[1:0]),(\infty,[0:1])$
$\displaystyle(\infty,[0:1]),(\infty,[1:0])$
$\displaystyle(\infty,[0:1]),(\infty,[0:1])$ (4.14)

of $(\mathbb{C}P^{1})^{4}$, there corresponds in
$\mathbb{D}\partial\mathbf{Q}^{2}$ a $3$-dimensional simplex, namely the join
of the edges in $\mathbb{D}\partial\mathbf{Q}_{i}$ corresponding to each
component. Thus, the natural $\Delta$-complex structure of
$\mathbb{D}\partial\mathbf{Q}^{2}\sim S^{3}$ contains these four
$3$-simplices, which in order are

$\displaystyle e_{2,\infty}$ $\displaystyle\ast e_{3,\infty}$ $\displaystyle
e_{2,\infty}$ $\displaystyle\ast e_{3,0}$ $\displaystyle e_{2,0}$
$\displaystyle\ast e_{3,\infty}$ $\displaystyle e_{2,0}$ $\displaystyle\ast
e_{3,0}.$

##  5\. Asymptotic behaviour of Fenchel–Nielsen co-ordinates

In this section we will determine the asymptotic behaviour of the co-ordinates
reviewed in Section 4 as $R\to\infty$, for fixed $q\in S^{3}_{1}$. The
constants we will find in this section may all depend on the divisor $D$. On
the other hand, their dependence on $q$ is crucial, hence we will indicate
when a constant depends on $q$.

###  5.1. Monodromy of diagonalizing frames

For our purpose, we first need to determine the monodromy transformation of a
diagonalizing frame of the solution to Hitchin’s equations along a loop around
a logarithmic point or a ramification point.

Clearly, the gauge transformation $g_{\sqrt{R}}$ provided by Theorem 2 is
unique up to a reducible transformation, i.e. one preserving the decomposition
of (3.7) into abelian summands. Consider now any simple loop

$\gamma\colon[0,1]\to{\mathbb{C}}\setminus\Delta_{q}.$

######  Definition 5.

Let $k(\gamma,q)\in{\mathbb{Z}}/(2)$ be the number of points of $\Delta_{q}$
contained in one of the connected components of
$\mathbb{C}P^{1}\setminus\gamma$, counted with multiplicity and modulo $2$.

Notice that the number of points of $\Delta_{q}$ in the two connected
components of $\mathbb{C}P^{1}\setminus\gamma$ add up to $6$, so $k(\gamma,q)$
is independent of the chosen component.

The loop $\gamma$ may be covered by a finite union of compact discs
$K_{1},\ldots,K_{N}$ as in Theorem 2, so we get for each $K_{l}$ a local
holomorphic trivialization $(\vec{e}_{1}^{l},\vec{e}_{2}^{l})$ of
${\mathcal{E}}$ specified by the local gauges $g_{R}$. We assume that for each
$1\leq l\leq N$ we have $K_{l}\cap K_{l+1}\cap\gamma([0,1])\neq\varnothing$
(where $l=N+1$ is identified with $l=1$), and pick any point
$\gamma(\tau_{l})\in K_{l}\cap K_{l+1}$. For $1\leq l\leq N-1$, up to applying
a constant gauge transformation over $K_{l+1}$ we may assume that

$(\vec{e}_{1}^{l}(\gamma(\tau_{l})),\vec{e}_{2}^{l}(\gamma(\tau_{l})))=(\vec{e}_{1}^{l+1}(\gamma(\tau_{l})),\vec{e}_{2}^{l+1}(\gamma(\tau_{l}))).$

Let $M(\gamma,Rq)$ be the monodromy transformation of the local
trivializations, defined by

$(\vec{e}_{1}^{N}(\gamma(\tau_{N})),\vec{e}_{2}^{N}(\gamma(\tau_{N})))=(\vec{e}_{1}^{1}(\gamma(\tau_{N})),\vec{e}_{2}^{1}(\gamma(\tau_{N})))M(\gamma,Rq).$

Let $T$ stand for the transposition matrix

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

######  Proposition 6.

For any simple loop $\gamma$ we have

$M(\gamma,Rq)=\begin{pmatrix}\alpha(\gamma,Rq)&0\\\
0&\delta(\gamma,Rq)\end{pmatrix}T^{k(\gamma,q)}$

for some $\alpha(\gamma,Rq),\delta(\gamma,Rq)\in{\mathbb{C}}^{\times}$.

###### Proof.

Recall from (2.21) that $\Delta_{q}=D\cup\\{t(q)\\}$. Assume first that
$t(q)\notin D$. Then $X_{Rq}$ is smooth. Now, since all ramification points of
$p|_{L}\colon X_{Rq}\to\mathbb{C}P^{1}$ are of index $2$, the lift
$\tilde{\gamma}$ of $\gamma$ to $X_{Rq}$ is a loop if and only if
$k(\gamma,q)=0$. Let $\tilde{\zeta}_{\pm}$ be a continuous lift of
$\zeta_{\pm}$ to $\tilde{\gamma}$. Then, we have

$\tilde{\zeta}_{\pm}(Rq,\tilde{\gamma}(1))=(-1)^{k(\gamma,q)}\tilde{\zeta}_{\pm}(Rq,\tilde{\gamma}(0)).$

As $\vec{e}_{1}^{l}$ and $\vec{e}_{2}^{l}$ belong to the $\tilde{\zeta}_{+}$\-
and $\tilde{\zeta}_{-}$-eigenspaces of $\theta$ respectively over $K_{l}$, and
eigenvectors $\vec{e}_{1}^{0}$ and $\vec{e}_{1}^{N}$ in case $k(\gamma,q)=0$
(respectively, $\vec{e}_{1}^{0}$ and $\vec{e}_{2}^{N}$ in case
$k(\gamma,q)=1$) for the eigenvalue $\tilde{\zeta}_{+}$ only differ by some
nonzero scalar $\alpha$, we get the result.

In case $t(q)\in D$, i.e. $t(q)=t_{j}$ for some $0\leq j\leq 4$, the curve
$X_{Rq}$ has an ordinary double point at $(t_{j},0)$, hence the form $\omega$
is unramified over $t_{j}$. If $\gamma$ is a loop enclosing $t_{j}$ and no
other point of $D$ then $M(\gamma)=\mbox{I}$ and $k(\gamma,q)\equiv 0\pmod{2}$
(because the points of $D$ are counted with multiplicity), so we conclude by
the equality

$T^{2}=\mbox{I}.$

In the case of an arbitrary loop $\gamma$, one concludes by a combination of
the above arguments. ∎

Now, assume that $t(q)\notin D$. Using the notations of (2.22), let us set

$\tau_{j}=\tau_{j}(q)=\operatorname{res}_{z=t_{j}}(\partial_{z}^{\otimes
2}\angle Q(z))=\frac{at_{j}-b}{\prod_{0\leq k\leq 4,k\neq
j}(t_{j}-t_{k})}\in{\mathbb{C}}$ (5.1)

(where $\angle$ stands for contraction of tensor fields) and introduce the
local holomorphic co-ordinate

$\tilde{z}_{j}=\tau_{j}(z-t_{j}).$ (5.2)

This is indeed a local co-ordinate by the assumption that the root $t(q)$ of
the linear functional $az-b$ does not belong to $D$, which means $\tau_{j}\neq
0$. Notice that as $\tau_{j}$ depends continuously on $q$, there exists some
$M>0$ only depending on $t_{0},\ldots,t_{4}$ such that for all $q\in
S^{3}_{1}(\vec{0})$ and all $0\leq j\leq 4$ we have

$|\tau_{j}(q)|\leq M.$ (5.3)

Then, a simple computation shows that up to holomorphic terms in $z-t_{j}$ we
have

$\frac{\mbox{d}\tilde{z}_{j}^{\otimes 2}}{\tilde{z}_{j}}\approx Q(z).$

We write $\tilde{z}_{j}=\tilde{r}_{j}e^{\sqrt{-1}\tilde{\varphi}_{j}}$ for the
polar co-ordinates of the local parameter. With respect to these polar co-
ordinates, the circle $\xi_{j}$ of radius $r_{0}$ centered at $t_{j}$ then has
the equation

$\tilde{r}_{j}=|\tau_{j}|r_{0}.$ (5.4)

More precisely, it follows from (5.1) that we have

$\arg(\tilde{z}_{j})=\arg(\tau_{j})+\arg(z-t_{j}),$

so the parameterization (4.2) of $\xi_{j}$ becomes

$\xi_{j}(\tilde{\varphi}_{j})=t_{j}+r_{0}e^{\sqrt{-1}\tilde{\varphi}_{j}}\quad\mbox{for}\;\tilde{\varphi}_{j}\in[\arg(\tau_{j}),2\pi+\arg(\tau_{j})].$
(5.5)

Let $\gamma$ denote the positively oriented simple loop around $t_{j}$ defined
by $\tilde{r}_{j}=r_{j}$ for some fixed $0<r_{j}\ll 1$ chosen so that one of
the connected components of $\mathbb{C}P^{1}\setminus\gamma$ contains no other
point of $\Delta_{q}$ than $t_{j}$. (In the case $r_{j}=|\tau_{j}|r_{0}$ we
get $\gamma=\xi_{j}$, however we are not guaranteed that for a given $q$ this
choice of $r_{j}$ satisfies the above requirement.) Let us define the unit
norm trivialization

$\displaystyle\vec{f}_{1}(\tilde{z}_{j})$
$\displaystyle=\frac{1}{\sqrt{e^{2m_{\sqrt{R}}(\tilde{r}_{j})}+1}}\begin{pmatrix}e^{m_{\sqrt{R}}(\tilde{r}_{j})}\\\
e^{-\sqrt{-1}\tilde{\varphi}_{j}/2}\end{pmatrix}$ (5.6)
$\displaystyle\vec{f}_{2}(\tilde{z}_{j})$
$\displaystyle=\frac{1}{\sqrt{e^{2m_{\sqrt{R}}(\tilde{r}_{j})}+1}}\begin{pmatrix}e^{m_{\sqrt{R}}(\tilde{r}_{j})}\\\
-e^{-\sqrt{-1}\tilde{\varphi}_{j}/2}\end{pmatrix}$ (5.7)

in the disc $\tilde{r}_{j}\leq r_{j}$ with respect to the fiducial frame
(3.13).

######  Proposition 7.

Let $\gamma$ be the positive simple loop $\tilde{r}_{j}=r_{j}$.

  1. (1)

We have

$h^{\operatorname{fid}}(\vec{f}_{1},\vec{f}_{2})\to 0\quad(R\to\infty).$

  2. (2)

The frame $\vec{f}_{1},\vec{f}_{2}$ diagonalizes the fiducial Higgs field
(3.12) with eigenvalues

$\pm\sqrt{R}\tilde{r}_{j}^{-\frac{1}{2}}e^{-\sqrt{-1}\tilde{\varphi}_{j}/2}\mbox{d}\tilde{z}_{j},$
(5.8)

where we take the determination of the angle $\tilde{\varphi}_{j}\in[0,2\pi)$.

  3. (3)

The corresponding factors found in Proposition 6 fulfill

$\alpha(\gamma,Rq)=1=\delta(\gamma,Rq).$

###### Proof.

For part (1), as the vectors $\vec{f}_{1},\vec{f}_{2}$ are written in a
unitary frame, we simply compute

$h^{\operatorname{fid}}(\vec{f}_{1},\vec{f}_{2})=\frac{e^{2m_{\sqrt{R}}(\tilde{r}_{j})}-1}{e^{2m_{\sqrt{R}}(\tilde{r}_{j})}+1}.$

Now, observe that by (3.8) we have

$\displaystyle e^{m_{\sqrt{R}}(r_{j})}$
$\displaystyle\approx\exp\left(\frac{1}{\pi}K_{0}(8\sqrt{Rr_{j}})\right)$
$\displaystyle\approx\exp\left(\frac{1}{2\pi\sqrt{2}\sqrt[4]{Rr_{j}}}e^{-8\sqrt{Rr_{j}}}\right)$
$\displaystyle\to 1$ (5.9)

as $R\to\infty$, since

$(Rr_{j})^{-\frac{1}{4}}e^{-8\sqrt{Rr_{j}}}\to 0.$

For part (2), we first need to determine the eigendirections of the fiducial
Higgs field $\sqrt{R}\theta_{\sqrt{R}}^{\operatorname{fid}}$ with respect to
the fiducial frame. We need to find the eigenvalues $\lambda_{\pm}$ of

$\displaystyle\sqrt{R}\begin{pmatrix}0&\tilde{r}_{j}^{-1/2}e^{m_{\sqrt{R}}(\tilde{r}_{j})}\\\
\tilde{z}_{j}^{-1}\tilde{r}_{j}^{1/2}e^{-m_{\sqrt{R}}(\tilde{r}_{j})}&0\end{pmatrix}\mbox{d}\tilde{z}_{j}$
$\displaystyle=\sqrt{R}\begin{pmatrix}0&\tilde{r}_{j}^{-1/2}e^{m_{\sqrt{R}}(\tilde{r}_{j})}\\\
\tilde{r}_{j}^{-1/2}e^{-\sqrt{-1}\tilde{\varphi}_{j}-m_{\sqrt{R}}(\tilde{r}_{j})}&0\end{pmatrix}\mbox{d}\tilde{z}_{j}.$

A direct computation gives that $\lambda_{\pm}$ are given by (5.8), with
corresponding eigenspaces spanned by unit vectors $\vec{f}_{1},\vec{f}_{2}$
introduced in (5.6), (5.7).

For part (3), we fix $\tilde{r}_{j}=r_{j}$ and let $\tilde{\varphi}_{j}$ range
over $[0,2\pi]$, with a branch cut at $\tilde{\varphi}_{j}=0$, and we write

$\vec{f}_{i}(\tilde{\varphi}_{j})=\vec{f}_{i}(\tilde{z}_{j}).$

We find

$\displaystyle\vec{f}_{1}(2\pi)$
$\displaystyle=\frac{1}{\sqrt{e^{2m_{\sqrt{R}}(\tilde{r}_{j})}+1}}\begin{pmatrix}e^{m_{\sqrt{R}}(\tilde{r}_{j})}\\\
-1\end{pmatrix}=\vec{f}_{2}(0)$ $\displaystyle\vec{f}_{2}(2\pi)$
$\displaystyle=\frac{1}{\sqrt{e^{2m_{\sqrt{R}}(\tilde{r}_{j})}+1}}\begin{pmatrix}e^{m_{\sqrt{R}}(\tilde{r}_{j})}\\\
1\end{pmatrix}=\vec{f}_{1}(0).$

∎

Finally, let us study the neighbourhood of the ramification point
$t=t(q)=\frac{b}{a}$. Here, using the notation of (2.22), we introduce the
local holomorphic co-ordinate

$\tilde{z}_{t}=\left(\frac{a}{\prod_{j=0}^{4}(t-t_{j})}\right)^{\frac{1}{3}}\left(z-\frac{b}{a}\right).$

Then, up to at least quadratic terms in $\tilde{z}_{t}$ we have

$\tilde{z}_{t}\mbox{d}\tilde{z}_{t}^{\otimes 2}\approx Q(z).$

We then write $\tilde{z}_{t}=\tilde{r}_{t}e^{\sqrt{-1}\tilde{\varphi}_{t}}$
for polar co-ordinates. Let $\gamma$ be the simple positive loop defined by
$\tilde{r}_{t}=r_{5}$ for some $0<r_{5}\ll 1$ so that $\gamma$ separates
$t(q)$ from the logarithmic points. Finally, introduce the unit length
trivialization

$\displaystyle\vec{g}_{1}(\tilde{z}_{t})$
$\displaystyle=\frac{1}{\sqrt{e^{2\ell_{\sqrt{R}}(\tilde{r}_{t})}+1}}\begin{pmatrix}e^{\ell_{\sqrt{R}}(\tilde{r}_{t})}\\\
e^{\sqrt{-1}\tilde{\varphi}_{t}/2}\end{pmatrix},$
$\displaystyle\vec{g}_{2}(\tilde{z}_{t})$
$\displaystyle=\frac{1}{\sqrt{e^{2\ell_{\sqrt{R}}(\tilde{r}_{t})}+1}}\begin{pmatrix}e^{\ell_{\sqrt{R}}(\tilde{r}_{t})}\\\
-e^{\sqrt{-1}\tilde{\varphi}_{t}/2}\end{pmatrix}$

over the disc $\tilde{r}_{t}\leq r_{5}$ with respect to the fiducial frame
(3.14).

######  Proposition 8.

Let $\gamma$ be a simple loop enclosing the ramification point $t(q)$ in
counterclockwise direction such that the component of
$\mathbb{C}P^{1}\setminus\gamma$ containing $t(q)$ contains no logarithmic
point $t_{j}$. Then the frame $\vec{g}_{1},\vec{g}_{2}$ diagonalizes the
fiducial Higgs field (3.16), and the corresponding factors found in
Proposition 6 fulfill

$\alpha(\gamma,Rq)=1=\delta(\gamma,Rq).$

###### Proof.

Similar to Proposition (7), up to the following modifications: the eigenvalues
of the fiducial Higgs field are
$\pm\sqrt{R\tilde{r}_{t}}e^{\sqrt{-1}\tilde{\varphi}_{t}/2}\mbox{d}z$, with
corresponding eigendirections $\vec{g}_{1},\vec{g}_{2}$. ∎

###  5.2. Asymptotics of complex length co-ordinates

Here, we will study the behaviour of the complex length co-ordinates
$(l_{2},l_{3})$ of ${\mathcal{M}}_{\operatorname{B}}(\vec{c},\vec{0})$
introduced in Section 4.1 for the local systems obtained by applying the non-
abelian Hodge and Riemann–Hilbert correspondences to a Higgs bundle in a
Hitchin fiber close to infinity. More precisely, we set

$l_{i}({\mathcal{E}},\sqrt{R}\theta)=\operatorname{tr}\operatorname{RH}(\nabla_{\sqrt{R}})[\rho_{i}].$
(5.10)

Notice that the connection $\nabla_{\sqrt{R}}$ depends on
$({\mathcal{E}},\sqrt{R}\theta)$, hence it is justified to include the
dependence of $l_{i}$ on $({\mathcal{E}},\sqrt{R}\theta)$ in the notation.
However, to lighten notation we will sometimes simply write $l_{i}$. With
these notations, we will determine the asymptotic behaviour as $R\to\infty$ of
$l_{2}({\mathcal{E}},\sqrt{R}\theta),l_{3}({\mathcal{E}},\sqrt{R}\theta)$ for
any $({\mathcal{E}},\theta)\in H^{-1}(q)$, where $q\in S^{3}_{1}$ is fixed.

According to Theorem 2 for $R\gg 0$ and at any point in the complement of
$\Delta_{q}$ there exists a $1$-parameter family of frames that asymptotically
diagonalize $\theta$. The family is obtained by rescaling a given frame by
diagonal elements of $\operatorname{Sl}(2,{\mathbb{C}})$. (One may
additionally apply the only non-trivial element $T$ of the Weyl group, with
the effect of exchanging the two trivializations of the frame.) It follows
that there exists (up to permutation and the action of the Cartan subgroup
$S^{1}\subset\operatorname{Sl}(2,{\mathbb{C}})$) a unique such orthonormal
frame. For any loop $\gamma$ in $\mathbb{C}P^{1}\setminus\Delta_{q}$ let us
write

$\operatorname{RH}(\nabla_{\sqrt{R}})[\gamma]=\begin{pmatrix}a(\gamma,({\mathcal{E}},\sqrt{R}\theta))&b(\gamma,({\mathcal{E}},\sqrt{R}\theta))\\\
c(\gamma,({\mathcal{E}},\sqrt{R}\theta))&d(\gamma,({\mathcal{E}},\sqrt{R}\theta))\end{pmatrix}$
(5.11)

with respect to this (essentially) unique orthonormal base of the fiber
$V|_{\gamma(0)}$ of the underlying smooth vector bundle $V$ at $\gamma(0)$.
Notice that the effect of the action by the Cartan subgroup means that the
off-diagonal entries are only defined up to a common phase factor. Our aim in
this section is to study the asymptotic behaviour of the entries of
$\operatorname{RH}(\nabla_{\sqrt{R}})[\rho_{i}]$ for $2\leq i\leq 3$, and in
particular their trace. Clealy, the set of eigenvalues (hence the trace) is
invariant with respect to the action of the Weyl group. In order to achieve
this, we will decompose the class of $\rho_{i}$ in
$\pi_{1}(\mathbb{C}P^{1}\setminus\Delta_{q},\rho_{i}(0))$ into a concatenation
of several loops (see Figure 1). The number of loops appearing in this
decomposition will be $2$ or $3$, depending on the position of the
ramification point $t(q)$ with respect to the decomposition of $S$ into pairs
of pants (4.3). Around each of the loops appearing in the decomposition we
will explicitly determine the monodromy, and the monodromy around $\rho_{i}$
is essentially the product of the monodromies of the consituent loops.

######  Proposition 9.

For any fixed $q\in S^{3}_{1}$ such that $t(q)\notin D_{j}$, the connection
form of the flat connection associated to the fiducial solution (3.10), (3.12)
restricted to the curve $\xi_{j}$ (given by $\tilde{r}_{j}=r_{j}$) with
respect to the unit diagonalizing frame (5.6), (5.7) of the Higgs field reads
as

$\begin{pmatrix}\frac{3}{4}+2\sqrt{-1}\sqrt{Rr_{j}}\sin\left(\frac{\tilde{\varphi}_{j}}{2}\right)&-\frac{1}{2}r_{j}\partial_{\tilde{r}}m_{\sqrt{R}}(r_{j})\\\
-\frac{1}{2}r_{j}\partial_{\tilde{r}}m_{\sqrt{R}}(r_{j})&\frac{3}{4}-2\sqrt{-1}\sqrt{Rr_{j}}\sin\left(\frac{\tilde{\varphi}_{j}}{2}\right)\end{pmatrix}\sqrt{-1}\mbox{d}\tilde{\varphi}_{j}.$

###### Proof.

This follows from a straightforward computation. Set

$F=\frac{1}{\sqrt{e^{2m_{\sqrt{R}}(r_{j})}+1}}\begin{pmatrix}e^{m_{\sqrt{R}}(r_{j})}&e^{m_{\sqrt{R}}(r_{j})}\\\
e^{-\sqrt{-1}\tilde{\varphi}_{j}/2}&-e^{-\sqrt{-1}\tilde{\varphi}_{j}/2}\end{pmatrix}$

for the matrix formed by the restrictions of the column vectors (5.6), (5.7)
to $\xi_{j}$, with determinant

$\det(F)=-\frac{2e^{m_{\sqrt{R}}(r_{j})-\sqrt{-1}\tilde{\varphi}_{j}/2}}{e^{2m_{\sqrt{R}}(\tilde{r}_{j})}+1}$

and inverse matrix given by

$F^{-1}=\frac{\sqrt{e^{2m_{\sqrt{R}}(r_{j})}+1}}{2e^{m_{\sqrt{R}}(r_{j})-\sqrt{-1}\tilde{\varphi}_{j}/2}}\begin{pmatrix}e^{-\sqrt{-1}\tilde{\varphi}_{j}/2}&e^{m_{\sqrt{R}}(r_{j})}\\\
e^{-\sqrt{-1}\tilde{\varphi}_{j}/2}&-e^{m_{\sqrt{R}}(r_{j})}\end{pmatrix}.$

Since on $\xi_{j}$ we have $\mbox{d}\tilde{r}_{j}=0$, we need to compute the
$\mbox{d}\tilde{\varphi}_{j}$-part of

$F^{-1}\cdot(\mbox{d}+A_{\sqrt{R}}^{\operatorname{fid}}+\theta_{\sqrt{R}}^{\operatorname{fid}}+(\theta_{\sqrt{R}}^{\operatorname{fid}})^{\dagger})=-F^{-1}\mbox{d}F+\mbox{Ad}_{F^{-1}}(A_{\sqrt{R}}^{\operatorname{fid}}+\theta_{\sqrt{R}}^{\operatorname{fid}}+(\theta_{\sqrt{R}}^{\operatorname{fid}})^{\dagger}).$

Note first that $\mbox{Ad}_{F^{-1}}$ acts trivially on the central part of
(3.10). On the other hand, a computation shows that

$\displaystyle F^{-1}\begin{pmatrix}F_{\sqrt{R}}(r_{j})&0\\\
0&-F_{\sqrt{R}}(r_{j})\end{pmatrix}F$
$\displaystyle=\frac{1}{2e^{m_{\sqrt{R}}(r_{j})-\sqrt{-1}\tilde{\varphi}_{j}/2}}\begin{pmatrix}0&-2e^{m_{\sqrt{R}}(r_{j})-\sqrt{-1}\tilde{\varphi}_{j}/2}F_{\sqrt{R}}(r_{j})\\\
-2e^{m_{\sqrt{R}}(r_{j})-\sqrt{-1}\tilde{\varphi}_{j}/2}F_{\sqrt{R}}(r_{j})&0\end{pmatrix}$
$\displaystyle=\begin{pmatrix}0&-F_{\sqrt{R}}(r_{j})\\\
-F_{\sqrt{R}}(r_{j})&0\end{pmatrix}.$

By Proposition 7,
$\mbox{Ad}_{F^{-1}}(\theta_{\sqrt{R}}^{\operatorname{fid}}+(\theta_{\sqrt{R}}^{\operatorname{fid}})^{\dagger})$
is diagonal with eigenvalues given by

$\pm\sqrt{Rr_{j}}\sqrt{-1}(e^{\sqrt{-1}\tilde{\varphi}_{j}/2}-e^{-\sqrt{-1}\tilde{\varphi}_{j}/2})\mbox{d}\tilde{\varphi}_{j}=\mp
2\sqrt{Rr_{j}}\sin\left(\frac{\tilde{\varphi}_{j}}{2}\right)\mbox{d}\tilde{\varphi}_{j}.$

Lastly, restricted to $\xi_{j}$ we find

$\displaystyle F^{-1}\mbox{d}F=$
$\displaystyle\frac{1}{2e^{m_{\sqrt{R}}(r_{j})-\sqrt{-1}\tilde{\varphi}_{j}/2}}\begin{pmatrix}e^{-\sqrt{-1}\tilde{\varphi}_{j}/2}&e^{m_{\sqrt{R}}(r_{j})}\\\
e^{-\sqrt{-1}\tilde{\varphi}_{j}/2}&-e^{m_{\sqrt{R}}(r_{j})}\end{pmatrix}$
$\displaystyle\cdot\begin{pmatrix}0&0\\\
-\frac{\sqrt{-1}}{2}e^{-\sqrt{-1}\tilde{\varphi}_{j}/2}&\frac{\sqrt{-1}}{2}e^{-\sqrt{-1}\tilde{\varphi}_{j}/2}\end{pmatrix}\mbox{d}\tilde{\varphi}_{j}$
$\displaystyle=$ $\displaystyle\frac{\sqrt{-1}}{4}\begin{pmatrix}-1&1\\\
1&-1\end{pmatrix}\mbox{d}\tilde{\varphi}_{j}.$

We conclude combining the above computations and using the identity (see
(3.9))

$2F_{\sqrt{R}}(r_{j})+\frac{1}{4}=\frac{1}{2}r_{j}\partial_{\tilde{r}}m_{\sqrt{R}}(r_{j}).$

∎

The behaviour of the entries of the matrix (5.11) for $R\gg 0$ and the choice
$\gamma=\xi_{j}$ is given by the following.

######  Proposition 10.

Fix any $q\in S^{3}_{1}$ and consider the loop $\gamma=\xi_{j}$.

  1. (1)

The behaviour of the diagonal entries of (5.11) as $R\to\infty$ is given by
the limits

$\displaystyle a(\xi_{j},({\mathcal{E}},\sqrt{R}\theta))$ $\displaystyle\to 0$
$\displaystyle d(\xi_{j},({\mathcal{E}},\sqrt{R}\theta))$ $\displaystyle\to 0$

as $R\to\infty$, at exponential rate in $\sqrt{R}$.

  2. (2)

The behaviour of the off-diagonal entries of (5.11) as $R\to\infty$ is given
by the limits

$\displaystyle
b(\xi_{j},({\mathcal{E}},\sqrt{R}\theta))e^{8\Re\sqrt{\tau_{j}}\sqrt{Rr_{0}}}$
$\displaystyle\to\sqrt{-1}$ $\displaystyle
c(\xi_{j},({\mathcal{E}},\sqrt{R}\theta))e^{-8\Re\sqrt{\tau_{j}}\sqrt{Rr_{0}}}$
$\displaystyle\to\sqrt{-1}$

where $\tau_{j}$ is defined in (5.1) and $r_{0}>0$ is the radius of $\xi_{j}$
in the Euclidean metric.

###### Proof.

Let us set $r_{j}=|\tau_{j}|r_{0}$ (see (5.4)). Recall the reparameterization
(5.5) of $\xi_{j}$ with respect to the polar co-ordinates of the local
holomorphic chart (5.2). Integrating the connection form of the flat
connection found in Proposition 9 from $\arg(\tau_{j})$ to
$2\pi+\arg(\tau_{j})$ with respect to $\tilde{\varphi}_{j}$ we find the matrix

$\begin{pmatrix}\sqrt{-1}\frac{3\pi}{2}-8\cos\left(\frac{\arg(\tau_{j})}{2}\right)\sqrt{Rr_{j}}&-\sqrt{-1}\pi
r_{j}\partial_{\tilde{r}}m_{\sqrt{R}}(r_{j})\\\ -\sqrt{-1}\pi
r_{j}\partial_{\tilde{r}}m_{\sqrt{R}}(r_{j})&\sqrt{-1}\frac{3\pi}{2}+8\cos\left(\frac{\arg(\tau_{j})}{2}\right)\sqrt{Rr_{j}}\end{pmatrix}.$
(5.12)

There are two cases to consider depending on whether $\Re\sqrt{\tau_{j}}=0$ or
$\Re\sqrt{\tau_{j}}\neq 0$.

We first treat the case $\Re\sqrt{\tau_{j}}\neq 0$. This condition is
equivalent to the pair of conditions $|\tau_{j}|\neq 0$ (equivalently,
$t(q)\notin D$) and

$\cos\left(\frac{\arg(\tau_{j})}{2}\right)\neq 0,$

equivalently $\arg(\tau_{j})\neq\pi+2k\pi$ for $k\in{\mathbb{Z}}$. The matrix
(5.12) is then of the form

$\begin{pmatrix}A-C&B\\\ B&A+C\end{pmatrix}$

with

$\displaystyle A$ $\displaystyle=A_{j}=\sqrt{-1}\frac{3\pi}{2}$ $\displaystyle
B$ $\displaystyle=B_{j}=-\sqrt{-1}\pi
r_{j}\partial_{\tilde{r}}m_{\sqrt{R}}(r_{j}),$ $\displaystyle C$
$\displaystyle=C_{j}=8\cos\left(\frac{\arg(\tau_{j})}{2}\right)\sqrt{Rr_{j}}=8\Re\sqrt{\tau_{j}}\sqrt{Rr_{0}}\neq
0,$

where we have used (5.4) in the third line. A straightforward computation
shows that setting $D=\sqrt{B^{2}+C^{2}}$ the exponential of the negative of
the above matrix is

$\frac{-\sqrt{-1}}{2D}\begin{pmatrix}(-C+D)e^{-D}+(C+D)e^{D}&-2B\sinh D\\\
-2B\sinh D&(C+D)e^{-D}+(-C+D)e^{D}\end{pmatrix}.$ (5.13)

According to (3.8) as $R\to\infty$ with $r_{j}$ freezed, we have

$\displaystyle B\frac{e^{8\sqrt{Rr_{j}}}}{\sqrt{\pi}\sqrt[4]{Rr_{j}}}$
$\displaystyle\to\sqrt{-1}$ (5.14) $\displaystyle\frac{C}{\sqrt{Rr_{j}}}$
$\displaystyle\to\upsilon_{j}$ (5.15)

where we have set

$\upsilon_{j}=8\cos\left(\frac{\arg(\tau_{j})}{2}\right)\in[-8,8]\setminus\\{0\\}.$
(5.16)

As a consequence we find

$D\approx|C|.$

Moreover, notice that up to higher order terms we have

$\sqrt{B^{2}+C^{2}}=C\sqrt{1+\frac{B^{2}}{C^{2}}}\approx
C\left(1+\frac{B^{2}}{2C^{2}}\right),$

implying

$\frac{-C+\sqrt{B^{2}+C^{2}}}{2\sqrt{B^{2}+C^{2}}}e^{\pm\sqrt{B^{2}+C^{2}}}\approx\frac{B^{2}}{4C^{2}}e^{\pm
C}\approx-\frac{\pi}{4\upsilon_{j}^{2}}\frac{e^{(-16\pm\upsilon_{j})\sqrt{Rr_{j}}}}{\sqrt{Rr_{j}}}.$

We infer that the leading order terms of the matrix (5.13) are equal to

$\displaystyle\sqrt{-1}\begin{pmatrix}\frac{B^{2}}{4C^{2}}e^{-C}+e^{C}&-\frac{B}{C}e^{|C|}\\\
-\frac{B}{C}e^{|C|}&e^{-C}+\frac{B^{2}}{4C^{2}}e^{C}\end{pmatrix}$
$\displaystyle\approx-\sqrt{-1}\cdot$
$\displaystyle\cdot\begin{pmatrix}\frac{\pi}{4\upsilon_{j}^{2}\sqrt{Rr_{j}}}e^{(-16-\upsilon_{j})\sqrt{Rr_{j}}}-e^{\upsilon_{j}\sqrt{Rr_{j}}}&\frac{\sqrt{-1}\sqrt{\pi}e^{-(8-|\upsilon_{j}|)\sqrt{Rr_{j}}}}{\upsilon_{j}\sqrt[4]{Rr_{j}}}\\\
\frac{\sqrt{-1}\sqrt{\pi}e^{-(8-|\upsilon_{j}|)\sqrt{Rr_{j}}}}{\upsilon_{j}\sqrt[4]{Rr_{j}}}&-e^{-\upsilon_{j}\sqrt{Rr_{j}}}+\frac{\pi}{4\upsilon_{j}^{2}\sqrt{Rr_{j}}}e^{(-16+\upsilon_{j})\sqrt{Rr_{j}}}\end{pmatrix}$

This matrix describes the action of
$\operatorname{RH}(\nabla_{\sqrt{R}})[\xi_{j}]$ with respect to the bases

$\vec{f}_{1}(0),\vec{f}_{2}(0)\quad\mbox{and}\quad\vec{f}_{1}(2\pi),\vec{f}_{2}(2\pi)$

of the fiber $V|_{\xi_{j}(0)}=V|_{\xi_{j}(1)}$. In order to find the matrix of
$\operatorname{RH}(\nabla_{\sqrt{R}})[\xi_{j}]$ with respect to the single
basis $\vec{f}_{1}(0),\vec{f}_{2}(0)$, we need to multiply the above matrix
from the left by the inverse of $M(\xi_{j},Rq)$. By Propositions 6 and 7,
$M(\xi_{j},Rq)=T$ and the product is

$\displaystyle\operatorname{RH}(\nabla_{\sqrt{R}})[\xi_{j}]=-\sqrt{-1}\cdot$
(5.17)
$\displaystyle\cdot\begin{pmatrix}\frac{\sqrt{-1}\sqrt{\pi}e^{-(8-|\upsilon_{j}|)\sqrt{Rr_{j}}}}{\upsilon_{j}\sqrt[4]{Rr_{j}}}&-e^{-\upsilon_{j}\sqrt{Rr_{j}}}+\frac{\pi}{4\upsilon_{j}^{2}\sqrt{Rr_{j}}}e^{(-16+\upsilon_{j})\sqrt{Rr_{j}}}\\\
\frac{\pi}{4\upsilon_{j}^{2}\sqrt{Rr_{j}}}e^{(-16-\upsilon_{j})\sqrt{Rr_{j}}}-e^{\upsilon_{j}\sqrt{Rr_{j}}}&\frac{\sqrt{-1}\sqrt{\pi}e^{-(8-|\upsilon_{j}|)\sqrt{Rr_{j}}}}{\upsilon_{j}\sqrt[4]{Rr_{j}}}\end{pmatrix}.$

By (5.16) we have $8-|\upsilon_{j}|\geq 0$, whence we immediately get part (1)
(in the case $8-|\upsilon_{j}|=0$, the assertion follows from the factor
$\sqrt[4]{R}$ in the denominator). On the other hand, (5.16) also shows that
$-16-\upsilon_{j}\leq\upsilon_{j}$, with equality if and only if
$\cos\left(\frac{\arg(\tau_{j})}{2}\right)=-1$. In the case where
$-16-\upsilon_{j}<\upsilon_{j}$, the first term of
$c(\xi_{j},({\mathcal{E}},\sqrt{R}\theta))$ is negligible compared to the
second one, and we get (2) for $c(\xi_{j},({\mathcal{E}},\sqrt{R}\theta))$. In
case $\cos\left(\frac{\arg(\tau_{j})}{2}\right)=-1$, the exponential factors
in the two terms of $c(\xi_{j},({\mathcal{E}},\sqrt{R}\theta))$ agree, however
the polynomial term in $R$ converges to $0$ for the first term, while is
constant for the second term, again implying (2) for
$c(\xi_{j},({\mathcal{E}},\sqrt{R}\theta))$. A similar argument may be applied
to get (2) for $b(\xi_{j},({\mathcal{E}},\sqrt{R}\theta))$.

We now turn to the case $\Re\sqrt{\tau_{j}}=0$. In this case, (5.12)
simplifies to

$\sqrt{-1}\begin{pmatrix}\frac{3\pi}{2}&-\pi
r_{j}\partial_{\tilde{r}}m_{\sqrt{R}}(r_{j})\\\ -\pi
r_{j}\partial_{\tilde{r}}m_{\sqrt{R}}(r_{j})&\frac{3\pi}{2}\end{pmatrix}.$

The diagonal entries of this matrix are constant, and its off-diagonal ones
converge to $0$ as $R\to\infty$. By continuity, the matrix exponential of the
negative of this matrix converges to

$\sqrt{-1}\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}.$

In order to obtain the matrix of
$\operatorname{RH}(\nabla_{\sqrt{R}})[\xi_{j}]$ with respect to the single
basis $\vec{f}_{1}(0),\vec{f}_{2}(0)$, we again need to multiply by the
transposition matrix $T$, and this gives the desired formulas. ∎

Next, we will consider the loop based at $x_{2}$

$\zeta_{2}=\eta_{1}\ast\xi_{1}\ast\eta_{1}^{-1}\ast\eta_{2}\ast\xi_{2}\ast\eta_{2}^{-1}$

enclosing the punctures $t_{1},t_{2}$ once in counterclockwise direction.
Clearly, the classes

$[\rho_{2}]\in\pi_{1}(\mathbb{C}P^{1}\setminus
D,s_{2})\quad\mbox{and}\quad[\zeta_{2}]\in\pi_{1}(\mathbb{C}P^{1}\setminus
D,x_{2})$

are conjugate to each other by $\psi_{2}$, so that
$\operatorname{RH}(\nabla_{\sqrt{R}})[\zeta_{2}]$ is conjugate in
$\operatorname{Sl}(2,{\mathbb{C}})$ to
$\operatorname{RH}(\nabla_{\sqrt{R}})[\rho_{2}]$ by the parallel transport map
of $\nabla_{\sqrt{R}}$ along $\psi_{2}$. In particular, this implies that the
trace of $\operatorname{RH}(\nabla_{\sqrt{R}})[\zeta_{2}]$ agrees with the
trace $l_{2}({\mathcal{E}},\sqrt{R}\theta)$ of
$\operatorname{RH}(\nabla_{\sqrt{R}})[\rho_{2}]$. Our next aim is to compute
the asymptotic behaviour of the coefficients of
$\operatorname{RH}(\nabla_{\sqrt{R}})[\zeta_{2}]$ as $R\to\infty$. In order to
state the result, we need some preparation. We will consider the part lying
between $t_{j}$ and $t_{j}+r_{j}$ of the ray emanating out from $t_{j}$ with
direction parallel to the positive real line, and denote this path by
$\sigma_{j}$ (see Figure 5). We then set for $j\in\\{1,2\\}$

$\pi_{j}=\pi_{j}(q)=\int_{x_{2}}^{t_{j}}Z_{+}(q,z),$ (5.18)

the contour of the line integral being $\eta_{j}*\sigma_{j}^{-1}$. Notice that
this is a convergent improper integral; indeed, by (2.19) and (2.22) the
integrand grows as $|z-t_{j}|^{-\frac{1}{2}}$ near $t_{j}$. We will indicate
the dependence of $\pi_{j}$ on $q$ whenever we vary it. Notice that by its
definition (2.19), $Z_{+}$ (hence $\pi_{j}$) is only defined up to a sign. We
take $Z_{+}$ to be the square root that is the continuous extension to
$\eta_{j}$ of the square root corresponding to the negative sign in (5.8). We
will return to our choice of sign in (5.23).

######  Proposition 11.

Fix $q\in S^{3}_{1}$ and consider the loop $\gamma=\rho_{2}$. In case
$\Re(\pi_{1}-\pi_{2})\neq 0$ we have the limit

$l_{2}({\mathcal{E}},\sqrt{R}\theta)^{-1}2\cosh\left(2\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}+4\sqrt{R}\Re(\pi_{2}-\pi_{1})\right)\to-1$

as $R\to\infty$. In case $\Re(\pi_{1}-\pi_{2})=0$ the limit of
$l_{2}({\mathcal{E}},\sqrt{R}\theta)$ as $R\to\infty$ exists and is finite.

$x_{2}$$t_{1}$$t_{2}$$\rho_{1}=\xi_{1}$$\xi_{2}$$\eta_{1}$$\eta_{2}$$\sigma_{2}$$\sigma_{1}$
Figure 5. Paths $\sigma_{1},\sigma_{2}$.

###### Proof.

As mentioned above, it is sufficient to find the asymptotic behaviour of the
diagonal entries of $\operatorname{RH}(\nabla_{\sqrt{R}})[\zeta_{2}]$. By
Theorem 2, parallel transport map of $\nabla_{\sqrt{R}}$ along $\eta_{j}$ with
respect to a diagonalizing frame of the Higgs field is approximated by the
matrix

$P_{j}({\mathcal{E}},\sqrt{R}\theta)=\begin{pmatrix}e^{\int_{\eta_{j}}\frac{1}{2}B_{\det({\mathcal{E}})}+B_{{\mathcal{L}}_{{\mathcal{E}}}}+\sqrt{R}\Re
Z_{+}(q,z)}&0\\\
0&e^{\int_{\eta_{j}}\frac{1}{2}B_{\det({\mathcal{E}})}-B_{{\mathcal{L}}_{{\mathcal{E}}}}+\sqrt{R}\Re
Z_{-}(q,z)}\end{pmatrix}.$ (5.19)

Parallel transport $P_{j}({\mathcal{E}},\sqrt{R}\theta)$ identifies the unit
length diagonalizing frame

$(\vec{e}_{+}(\eta_{j}(0)),\vec{e}_{-}(\eta_{j}(0)))$

of $\theta_{\sqrt{R}}$ at $\eta_{j}(0)$ with a diagonalizing frame

$(\vec{e}_{+}(\eta_{j}(1)),\vec{e}_{-}(\eta_{j}(1)))$ (5.20)

at $\eta_{j}(1)$. This latter, however, is not of unit length; instead, the
lengths of its vectors are given for $i\in\\{\pm\\}$ by

$|\vec{e}_{i}(\eta_{j}(1))|=e^{\sqrt{R}\int_{\eta_{j}}\Re Z_{i}(q,z)}.$

So, by suitably picking the phases of the frame vectors $e_{+},e_{-}$, we may
assume that the matrix expressing the basis elements of (5.20) with respect to
(5.6) is given by

$Q_{j}({\mathcal{E}},\sqrt{R}\theta)=\begin{pmatrix}e^{\int_{\eta_{j}}\sqrt{R}\Re
Z_{+}(q,z)}&0\\\ 0&e^{\int_{\eta_{j}}\sqrt{R}\Re Z_{-}(q,z)}\end{pmatrix}.$

It follows that the monodromy matrix of $\nabla_{\sqrt{R}}$ along the loop
$\eta_{j}\ast\xi_{j}\ast\eta_{j}^{-1}$ with respect to the frame (5.20) is
equal to

$\operatorname{Ad}_{P_{j}({\mathcal{E}},\sqrt{R}\theta)^{-1}}\circ\operatorname{Ad}_{Q_{j}({\mathcal{E}},\sqrt{R}\theta)^{-1}}(\operatorname{RH}(\nabla_{\sqrt{R}})[\xi_{j}]).$

In view of Proposition 10 we find

$\displaystyle
a(\eta_{j}\ast\xi_{j}\ast\eta_{j}^{-1},({\mathcal{E}},\sqrt{R}\theta))e^{(8-\upsilon_{j})\sqrt{Rr_{j}}}\upsilon_{j}\sqrt[4]{Rr_{j}}$
$\displaystyle\to\sqrt{\pi}$ $\displaystyle
b(\eta_{j}\ast\xi_{j}\ast\eta_{j}^{-1},({\mathcal{E}},\sqrt{R}\theta))e^{\sqrt{Rr_{j}}\upsilon_{j}+4\int_{\eta_{j}}\sqrt{R}\Re
Z_{+}(q,z)}$
$\displaystyle\to\sqrt{-1}e^{-2\int_{\eta_{j}}B_{{\mathcal{L}}_{{\mathcal{E}}}}}$
(5.21) $\displaystyle
c(\eta_{j}\ast\xi_{j}\ast\eta_{j}^{-1},({\mathcal{E}},\sqrt{R}\theta))e^{-\sqrt{Rr_{j}}\upsilon_{j}-4\int_{\eta_{j}}\sqrt{R}\Re
Z_{+}(q,z)}$
$\displaystyle\to\sqrt{-1}e^{2\int_{\eta_{j}}B_{{\mathcal{L}}_{{\mathcal{E}}}}}$
(5.22) $\displaystyle
d(\eta_{j}\ast\xi_{j}\ast\eta_{j}^{-1},({\mathcal{E}},\sqrt{R}\theta))e^{(8-\upsilon_{j})\sqrt{Rr_{j}}}\upsilon_{j}\sqrt[4]{Rr_{j}}$
$\displaystyle\to\sqrt{\pi}$

with

$\displaystyle\upsilon_{j}$
$\displaystyle=8\cos\left(\frac{\arg(\tau_{j})}{2}\right),$ $\displaystyle
r_{j}$ $\displaystyle=|\tau_{j}|r_{0}.$

Given that $B_{{\mathcal{L}}_{{\mathcal{E}}}}$ is a
$\sqrt{-1}{\mathbb{R}}$-valued $1$-form, the above limiting values are of
length $1$ and $\sqrt{\pi}$ respectively; the phases appearing in (5.21) and
(5.22) will play a fundamental role in Section 6.

We now make the observation that by (2.19) and (2.22) for $0<r_{0}\ll 1$ we
have

$\displaystyle\int_{\sigma_{j}}Z_{+}(q,z)$
$\displaystyle=\pm\int_{\sigma_{j}}\sqrt{\frac{(az-b)}{\prod_{k=0}^{4}(z-t_{k})}}\mbox{d}z$
$\displaystyle\approx\pm\sqrt{\frac{at_{j}-b}{\prod_{0\leq k\leq 4,k\neq
j}(t_{j}-t_{k})}}\int_{t_{j}}^{t_{j}+r_{0}}\frac{\mbox{d}z}{\sqrt{z-t_{j}}}$
$\displaystyle=\pm 2\sqrt{\tau_{j}r_{0}}$

where we have used (5.1) in the last line. Notice that we have a freedom of
sign in choosing both $Z_{+}(q,z)$ and $\sqrt{\tau_{j}}$. We require that the
sign of $Z_{+}$ is chosen so that the precise form of the above equality be

$\int_{\sigma_{j}}Z_{+}(q,z)=-2\sqrt{\tau_{j}r_{0}}.$ (5.23)

We infer that the integral appearing in the exponent of the off-diagonal terms
(5.21)–(5.22) is of the form

$\displaystyle\sqrt{Rr_{j}}\upsilon_{j}+4\int_{\eta_{j}}\sqrt{R}\Re
Z_{+}(q,z)$
$\displaystyle=8\sqrt{Rr_{0}}\Re\sqrt{\tau_{j}}+4\sqrt{R}\int_{\eta_{j}}\Re
Z_{+}(q,z)$ $\displaystyle=4\sqrt{R}\left(-\int_{\sigma_{j}}\Re
Z_{+}(q,z)+\int_{\eta_{j}}\Re Z_{+}(q,z)\right)$
$\displaystyle=4\sqrt{R}\int_{x_{2}}^{t_{j}}\Re Z_{+}(q,z)$
$\displaystyle=4\sqrt{R}\Re\pi_{j},$

which allows us to recast the limits (5.21)–(5.22) as

$\displaystyle
b(\eta_{j}\ast\xi_{j}\ast\eta_{j}^{-1},({\mathcal{E}},\sqrt{R}\theta))e^{4\sqrt{R}\Re\pi_{j}}$
$\displaystyle\to\sqrt{-1}e^{-2\int_{\eta_{j}}B_{{\mathcal{L}}_{{\mathcal{E}}}}}$
$\displaystyle
c(\eta_{j}\ast\xi_{j}\ast\eta_{j}^{-1},({\mathcal{E}},\sqrt{R}\theta))e^{-4\sqrt{R}\Re\pi_{j}}$
$\displaystyle\to\sqrt{-1}e^{2\int_{\eta_{j}}B_{{\mathcal{L}}_{{\mathcal{E}}}}}.$

Now, we have

$\displaystyle\operatorname{RH}(\nabla_{\sqrt{R}})[\zeta_{2}]$ (5.24)
$\displaystyle=\operatorname{RH}(\nabla_{\sqrt{R}})[\eta_{1}\ast\xi_{1}\ast\eta_{1}^{-1}]\operatorname{RH}(\nabla_{\sqrt{R}})[\eta_{2}\ast\xi_{2}\ast\eta_{2}^{-1}].$

By definition, $l_{2}({\mathcal{E}},\sqrt{R}\theta)$ is the trace of this
matrix, hence we need to compute the diagonal entries of the product (5.24).

Its entry $a(\zeta_{2},({\mathcal{E}},\sqrt{R}\theta))$ of index $(1,1)$ is
the sum

$a(\eta_{1}\ast\xi_{1}\ast\eta_{1}^{-1},({\mathcal{E}},\sqrt{R}\theta))a(\eta_{2}\ast\xi_{2}\ast\eta_{2}^{-1},({\mathcal{E}},\sqrt{R}\theta))+b(\eta_{1}\ast\xi_{1}\ast\eta_{1}^{-1},({\mathcal{E}},\sqrt{R}\theta))c(\eta_{2}\ast\xi_{2}\ast\eta_{2}^{-1},({\mathcal{E}},\sqrt{R}\theta)).$

According to (5.17) the leading order term of the asymptotic expansion of its
first term as $R\to\infty$ is given by

$\pi\frac{e^{-(8-\upsilon_{1})\sqrt{Rr_{1}}-(8-\upsilon_{2})\sqrt{Rr_{2}}}}{\upsilon_{1}\upsilon_{2}\sqrt[4]{r_{1}r_{2}R^{2}}}.$
(5.25)

The leading order term of the asymptotic expansion of the second term of
$a(\zeta_{2},({\mathcal{E}},\sqrt{R}\theta))$ is

$-\exp\left(2\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}+4\sqrt{R}\Re(\pi_{2}-\pi_{1})\right).$
(5.26)

We again emphasize that this formula gives the polar decomposition of the
corresponding term, as
$\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}$ is purely
imaginary and $\pi_{j}\in{\mathbb{R}}$.

The terms of the $(2,2)$-entry $d(\zeta_{2},({\mathcal{E}},\sqrt{R}\theta))$
of (5.24) are similar to those of
$a(\zeta_{2},({\mathcal{E}},\sqrt{R}\theta))$, up to exchanging the subscripts
$j=1$ and $j=2$ of $\upsilon_{j},r_{j},\eta_{j}$. Namely, the term coming from
the product of diagonal entries of the factors has leading order term in its
asymptotic expansion given by (5.25), and the leading order term of its other
term is

$-\exp\left(2\int_{\eta_{1}-\eta_{2}}B_{{\mathcal{L}}_{{\mathcal{E}}}}+4\sqrt{R}\Re(\pi_{1}-\pi_{2})\right).$
(5.27)

Notice that the product of (5.26) and (5.27) equals $1$.

Now, we observe that by (5.16) the coefficient of $\sqrt{R}$ in the exponent
of (5.25) is never positive (as it has already been pointed out in the proof
of Proposition 10). On the other hand, at least one of the coefficients of
$\sqrt{R}$ in the exponent of (5.26) and in the exponent of (5.27) is non-
negative. In the extreme case where the coefficients of $\sqrt{R}$ in the
exponent of all terms (5.25), (5.26) and (5.27) vanish, then the $\sqrt{R}$ in
the denominator of (5.25) guarantees that it is negligible compared to the sum
of the other two terms. To sum up, this shows that in the trace, the leading-
order term may not be (5.25), rather it is equal to (5.26) or (5.27) according
as $\Re(\pi_{1}-\pi_{2})<0$ or $\Re(\pi_{1}-\pi_{2})>0$, and to the sum of
these terms if $\Re(\pi_{1}-\pi_{2})=0$. In any case, the term (5.25)
converges to $0$ as $R\to\infty$, and if $\Re(\pi_{1}-\pi_{2})>0$
(respectively, $\Re(\pi_{1}-\pi_{2})<0$) then the same limit holds for the
term (5.26) (respectively, (5.27)). Finally, we conclude using that for a ray
${\mathbb{C}}$ of the form $te^{\sqrt{-1}\phi_{0}}$ with fixed
$-\frac{\pi}{2}<\phi_{0}<\frac{\pi}{2}$ and variable $t>0$ we have

$\lim_{t\to\infty}2\cosh(te^{\sqrt{-1}\phi_{0}})e^{-te^{\sqrt{-1}\phi_{0}}}=1.$

∎

Recall Hopf co-ordinates (2.24) on $S^{3}_{1}$, $\varphi$ being the co-
ordinate along the Hopf fibers.

######  Proposition 12.

Fix $q\in S^{3}_{1}$ and assume $\pi_{1}(q)\neq\pi_{2}(q)$. Then there exists
a unique $\varphi_{2}\in[0,2\pi)$ such that for every
${\mathcal{L}}_{{\mathcal{E}}}\in\operatorname{Jac}(X_{q})$ the co-ordinate
$l_{2}({\mathcal{E}},e^{\sqrt{-1}\varphi_{2}}\sqrt{R}\theta)$ remains bounded
as $R\to\infty$.

###### Proof.

According to Proposition (11),
$l_{2}({\mathcal{E}},e^{\sqrt{-1}\varphi_{2}}\sqrt{R}\theta)$ is bounded as
$R\to\infty$ if and only if the equation

$\Re(\pi_{1}(e^{\sqrt{-1}\varphi}q)-\pi_{2}(e^{\sqrt{-1}\varphi}q))=0$

holds for the variable $\varphi\in[0,2\pi)$. This quantity is the horizontal
projection of

$\int_{t_{2}}^{t_{1}}Z_{+}(e^{\sqrt{-1}\varphi}q,z),$

where the contour of integration is
$\sigma_{1}*\eta_{1}^{-1}*\eta_{2}*\sigma_{2}^{-1}$. Now, taking into account
the definition (2.19), we have

$Z_{+}(e^{\sqrt{-1}\varphi}q,z)=e^{\sqrt{-1}\varphi/2}Z_{+}(q,z).$

Clearly, there exists a unique value $\varphi_{2}\in[0,2\pi)$ satisfying the
property that the non-zero complex number $\pi_{1}(q)-\pi_{2}(q)$ multiplied
by the unit length complex number $e^{\sqrt{-1}\varphi_{2}/2}$ has horizontal
projection equal to $0$. ∎

The results of this subsection have been stated for the case of
$l_{2}({\mathcal{E}},\sqrt{R}\theta)$. We now proceed to stating the results
analogous to Propositions 11 and 12 for the case of
$l_{3}({\mathcal{E}},\sqrt{R}\theta)$, whose proofs are straightforward
modifications of the ones proven so far. For $j\in\\{0,4\\}$ introduce

$\pi_{j}=\pi_{j}(q)=\int_{x_{4}}^{t_{j}}Z_{+}(q,z)$

along some path in $S_{3}$.

######  Proposition 13.

Fix $q\in S^{3}_{1}$.

  1. (1)

In case $\Re(\pi_{4}(q)-\pi_{0}(q))\neq 0$ we have the limit

$l_{3}({\mathcal{E}},\sqrt{R}\theta)2\cosh\left(2\int_{\eta_{4}-\eta_{0}}B_{{\mathcal{L}}_{{\mathcal{E}}}}+4\sqrt{R}\Re(\pi_{4}-\pi_{0})\right)\to-1$

as $R\to\infty$.

  2. (2)

In case $\Re(\pi_{4}(q)-\pi_{0}(q))=0$ the limit of
$l_{3}({\mathcal{E}},\sqrt{R}\theta)$ as $R\to\infty$ exists and is finite.

  3. (3)

If $\pi_{4}(q)\neq\pi_{0}(q)$, then there exists a unique
$\varphi_{3}\in[0,2\pi)$ such that for every
${\mathcal{L}}_{{\mathcal{E}}}\in\operatorname{Jac}(X_{q})$ the co-ordinate
$l_{3}({\mathcal{E}},e^{\sqrt{-1}\varphi_{3}}\sqrt{R}\theta)$ remains bounded
as $R\to\infty$.

###  5.3. Asymptotic behaviour of complex twist co-ordinates

For $i\in\\{2,3\\}$ we let

$[p_{i}({\mathcal{E}},\sqrt{R}\theta):q_{i}({\mathcal{E}},\sqrt{R}\theta)]$

stand for the complex twist co-ordinates $[p_{i}:q_{i}]$ introduced in Section
4.2 of the local system
$\operatorname{RH}\circ\psi({\mathcal{E}},\sqrt{R}\theta)$. In this section we
will determine the asymptotic behaviour of these co-ordinates as $R\to\infty$.
For this purpose, we first determine the asymptotic behaviour of the
quantities

$u_{i},w_{i},A_{i},R_{i},R^{\prime}_{i-1},T_{i},U_{i},h_{i},\psi_{i},P_{i},Q_{i}$

introduced in Section 4.2. For ease of notation, we will often omit to
indicate the dependence of the co-ordinates on
$({\mathcal{E}},\sqrt{R}\theta)$.

Before continuing, we add an important tweak to our construction. Namely, so
far we have assumed that the radii (with respect to Euclidean distance) of the
circles $\xi_{j}$ were all equal to each other. We now lift this constraint
and from now on we merely assume the radii to be _pairwise_ equal to each
other: the radii of $\xi_{1},\xi_{2}$ are equal to $s_{2}>0$, however the
radius of $\xi_{3}$ is some possibly different real number $s_{3}>0$, while
the radii of $\xi_{0},\xi_{4}$ are equal to yet another real number $s_{4}>0$.
From this point on, these numbers take over the role that used to be that of
$r_{0}$ up to this point. We will asjust their values according to our needs
in Section 6.1. The results proven up to now all remain valid, because their
assertions and proofs did not depend on the actual value of $r_{0}$.

In what follows, for $2\times 2$ matrices $A,B$ with non-vanishing entries
depending on a parameter $R\in{\mathbb{R}}$ we write $A\approx B$ whenever the
limit of each entry of $A$ divided by the corresponding entry of $B$ converges
to $1$ as $R\to\infty$. Similarly, for two scalar quantities $a,b$ depending
on $R\in{\mathbb{R}}$ we write $a\approx b$ to express that $\frac{a}{b}\to 1$
as $R\to\infty$. We say $a$ is negligible compared to $b$ if $\frac{a}{b}\to
0$ as $R\to\infty$.

Recall from Section 2.2 that $c_{j}^{\pm}=\pm\sqrt{-1}$. We start by recording
some asymptotic behaviours as $R\to\infty$ ensuing from Sections 4.2 and 5.2
in the case $\Re(\pi_{1}-\pi_{2})\neq 0$

$\displaystyle l_{1}$ $\displaystyle=c_{1}^{+}+c_{1}^{-}=0$ $\displaystyle
l_{2}$
$\displaystyle\approx-2\cosh\left(4\sqrt{R}\Re(\pi_{2}-\pi_{1})+2\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}\right)$
(5.28) $\displaystyle u_{2}$
$\displaystyle=\frac{l_{1}-c_{2}^{-}l_{2}}{c_{2}^{+}-c_{2}^{-}}\approx-\cosh\left(4\sqrt{R}\Re(\pi_{2}-\pi_{1})+2\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}\right)$
(5.29) $\displaystyle U_{2}$ $\displaystyle\approx\begin{pmatrix}1&0\\\
-\cosh\left(4\sqrt{R}\Re(\pi_{2}-\pi_{1})+2\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}\right)&1\end{pmatrix}$
(5.30) $\displaystyle A_{2}$ $\displaystyle=\begin{pmatrix}c_{2}^{+}&0\\\
0&c_{2}^{-}\end{pmatrix}=\sqrt{-1}\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$
(5.31) $\displaystyle R_{2}$
$\displaystyle\approx\begin{pmatrix}-\cosh\left(4\sqrt{R}\Re(\pi_{2}-\pi_{1})+2\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}\right)&1\\\
*&-\cosh\left(4\sqrt{R}\Re(\pi_{2}-\pi_{1})+2\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}\right)\end{pmatrix}$
(5.32) $\displaystyle R^{\prime}_{1}$
$\displaystyle\approx\begin{pmatrix}-\sqrt{-1}\cosh\left(4\sqrt{R}\Re(\pi_{2}-\pi_{1})+2\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}\right)&\sqrt{-1}\\\
*&\sqrt{-1}\cosh\left(4\sqrt{R}\Re(\pi_{2}-\pi_{1})+2\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}\right)\end{pmatrix}$
(5.33)

where the entries marked by $*$ may be determined by the (known) determinant
of the matrices, but we refrain from spelling them out as they will be
irrelevant for our purposes.

######  Proposition 14.

For $2\leq i\leq 4$ we have the asymptotic behaviours

$h_{i}^{-1}\approx\begin{pmatrix}v_{i}&e^{-8\sqrt{Rs_{i}}\Re\sqrt{\tau_{i}}}w_{i}\\\
-e^{8\sqrt{Rs_{i}}\Re\sqrt{\tau_{i}}}v_{i}&w_{i}\end{pmatrix}$

and

$h_{i}\approx\frac{1}{2v_{i}w_{i}}\begin{pmatrix}w_{i}&-e^{-8\sqrt{Rs_{i}}\Re\sqrt{\tau_{i}}}w_{i}\\\
e^{8\sqrt{Rs_{i}}\Re\sqrt{\tau_{i}}}v_{i}&v_{i}\end{pmatrix}$

for some $v_{i},w_{i}\in{\mathbb{C}}$ and the same choice of $\sqrt{\tau_{i}}$
as in (5.17).

###### Proof.

As the monodromy matrix of $V_{i}(l_{i-1},l_{i})$ around $\xi_{i}$ is the
diagonal matrix $A_{i}$, in order to determine $h_{i}^{-1}$ we need to find
the eigenvectors of (5.17) for $j=i$, taking into account the relation

$\sqrt{r_{i}}\upsilon_{i}=8\Re\sqrt{\tau_{i}s_{i}},$

see (5.16),(5.4). As we have shown in Proposition 10, the diagonal entries of
(5.17) converge to $0$ and its determinant converges to $1$. A direct
computation then shows that $h_{i}^{-1}$ is of the desired form. We conclude
by taking matrix inverse. ∎

######  Proposition 15.

Assume $\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})\neq 0$. Then the large scale
behaviour as $R\to\infty$ of the isomorphism $h_{2}^{-1}$ identifying
$V_{i}(l_{i-1},l_{i})$ to the restriction of $V$ to $S_{i}$ written with
respect to the unit diagonalizing frame of the local system $V$ is

$h_{2}^{-1}\approx\begin{pmatrix}\sinh\left(8\sqrt{Rs_{2}}\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})\right)&-1\\\
-e^{8\sqrt{Rs_{2}}\Re\sqrt{\tau_{2}}}\sinh\left(8\sqrt{Rs_{2}}\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})\right)&-e^{8\sqrt{Rs_{2}}\Re\sqrt{\tau_{2}}}\end{pmatrix}$

###### Proof.

By Proposition 14, we just need to find the values of $v_{2},w_{2}$; for this
purpose, we will use the monodromy around the loop $\xi_{1}$. Indeed, it is
required that the $(1,2)$-entry of
$h_{2}\operatorname{RH}(\nabla_{\sqrt{R}})[\xi_{1}]h_{2}^{-1}$ be equal to
$\sqrt{-1}$. After elementary algebra, this is equivalent to the relation

$\frac{w_{2}}{v_{2}}=-\frac{\exp(8\sqrt{Rs_{2}}\Re\sqrt{\tau_{2}})}{\sinh\left(8\sqrt{Rs_{2}}\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})\right)},$

up to super-exponential error terms in the numerator and denominator. The
pairs of solutions $[v_{2}:w_{2}]\in\mathbb{C}P^{1}$ of this equation are
defined up to simultaneous rescaling by some element of
${\mathbb{C}}^{\times}$, and such a rescaling does not affect the complex
twist co-ordinates (4.11). Hence, we may set

$\displaystyle v_{2}$
$\displaystyle=\sinh\left(8\sqrt{Rs_{2}}\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})\right)$
$\displaystyle w_{2}$ $\displaystyle=-\exp(8\sqrt{Rs_{2}}\Re\sqrt{\tau_{2}})$

and with these values we find the desired result. ∎

######  Proposition 16.

Fix $q\in S^{3}_{1}$ such that $\Re(\pi_{2}-\pi_{1})\neq 0$. Then, the complex
twist co-ordinate $[p_{2}:q_{2}]$ associated to
$({\mathcal{E}},\sqrt{R}\theta)$ converges to $[0:1]$ as $R\to\infty$ if the
conditions

$\displaystyle\int_{\psi_{2}}\Re Z_{+}$
$\displaystyle<2\Re(2\sqrt{s_{2}\tau_{2}}-\sqrt{s_{2}\tau_{1}}-\sqrt{s_{3}\tau_{3}})$
$\displaystyle|\Re(\pi_{1}-\pi_{2})|$
$\displaystyle=2\sqrt{s_{2}}\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})$

hold for one choice of a square root $Z_{+}$ of $Q$ and

$\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}\equiv
0\pmod{\pi\sqrt{-1}}.$

Specifically, we then have

$\frac{p_{2}}{q_{2}}\approx\exp\left(-8\sqrt{R}\Re(2\sqrt{s_{2}\tau_{2}}-\sqrt{s_{2}\tau_{1}}-\sqrt{s_{3}\tau_{3}})+2\int_{\psi_{2}}\left(B_{{\mathcal{L}}_{{\mathcal{E}}}}+2\sqrt{R}\Re
Z_{+}\right)\right).$

Similarly, $[p_{2}:q_{2}]\to[0:1]$ as $R\to\infty$ if

$\displaystyle\int_{\psi_{2}}\Re Z_{+}$
$\displaystyle<2\Re(2\sqrt{s_{2}\tau_{2}}-\sqrt{s_{2}\tau_{1}}-\sqrt{s_{3}\tau_{3}})$
$\displaystyle|\Re(\pi_{1}-\pi_{2})|$
$\displaystyle=-2\sqrt{s_{2}}\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})$

hold for one choice of a square root $Z_{+}$ of $Q$ and

$\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}\equiv\frac{\pi\sqrt{-1}}{2}\pmod{\pi\sqrt{-1}}.$

On the other hand, under the condition

$|\Re(\pi_{1}-\pi_{2})|\neq 2\sqrt{s_{2}}\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})$

$[p_{2}:q_{2}]$ converges to $[1:0]$.

###### Proof.

Our task is to compute

$P_{2}=h_{3}\circ\psi_{2}\circ h_{2}^{-1},$ (5.34)

where $\psi_{2}$ stands for parallel transport map along the path $\psi_{2}$.
Since constant factors multiplying $Q_{i}$ (or $P_{i}$) have no influence on
the definition (4.11) of the complex twist co-ordinates, from now on we will
ignore constant factors; said differently, the formulas of the rest of this
section hold in $\mbox{PGl}(2,{\mathbb{C}})$. In particular, the exact value
of the constanst $v_{3},w_{3}$ appearing in Proposition 14 will not be
relevant, the essential information is that their ratio is well-defined. Just
as in the proof of Proposition 11, with respect to suitable diagonalizing
bases we have

$\psi_{2}\approx\begin{pmatrix}e^{\int_{\psi_{2}}\left(\frac{1}{2}B_{\det({\mathcal{E}})}+B_{{\mathcal{L}}_{{\mathcal{E}}}}+\sqrt{R}\Re
Z_{+}(q,z)\right)}&0\\\
0&e^{\int_{\psi_{2}}\left(\frac{1}{2}B_{\det({\mathcal{E}})}-B_{{\mathcal{L}}_{{\mathcal{E}}}}-\sqrt{R}\Re
Z_{+}(q,z)\right)}\end{pmatrix}.$

We take the frame at $\psi_{2}(0)=x_{2}$ to consist of unit vectors, and then
the above matrix expresses the action of parallel transport with respect to a
basis at $\psi_{2}(1)=x_{3}$ whose first and second vectors are respectively
of length

$\exp\left(\pm\sqrt{R}\int_{x_{2}}^{x_{3}}\Re Z_{+}(q,z)\right).$

It follows that the action of parallel transport along $\psi_{2}$, written in
unit-length diagonalizing bases both at $x_{2},x_{3}$, is described by the
matrix

$\begin{pmatrix}e^{\int_{\psi_{2}}\left(\frac{1}{2}B_{\det({\mathcal{E}})}+B_{{\mathcal{L}}_{{\mathcal{E}}}}+2\sqrt{R}\Re
Z_{+}(q,z)\right)}&0\\\
0&e^{\int_{\psi_{2}}\left(\frac{1}{2}B_{\det({\mathcal{E}})}-B_{{\mathcal{L}}_{{\mathcal{E}}}}-2\sqrt{R}\Re
Z_{+}(q,z)\right)}\end{pmatrix}.$

For ease of notation, from now on we will drop the argument of $Z_{+}$ and the
term $\frac{1}{2}B_{\det({\mathcal{E}})}$ in the argument of the exponential
(remember that we work in $\mbox{PGl}(2,{\mathbb{C}})$).

Now, by Proposition 15 the growth orders of the entries of
$\psi_{2}h_{2}^{-1}$ are given by

$\begin{pmatrix}\sinh\left(8\sqrt{Rs_{2}}\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})\right)e^{\int_{\psi_{2}}\left(B_{{\mathcal{L}}_{{\mathcal{E}}}}+2\sqrt{R}\Re
Z_{+}\right)}&-e^{\int_{\psi_{2}}\left(B_{{\mathcal{L}}_{{\mathcal{E}}}}+2\sqrt{R}\Re
Z_{+}\right)}\\\
-e^{8\sqrt{Rs_{2}}\Re\sqrt{\tau_{2}}-\int_{\psi_{2}}\left(B_{{\mathcal{L}}_{{\mathcal{E}}}}+2\sqrt{R}\Re
Z_{+}\right)}\sinh\left(8\sqrt{Rs_{2}}\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})\right)\\\
&-e^{8\sqrt{Rs_{2}}\Re\sqrt{\tau_{2}}-\int_{\psi_{2}}\left(B_{{\mathcal{L}}_{{\mathcal{E}}}}+2\sqrt{R}\Re
Z_{+}\right)}\end{pmatrix}.$

Using Proposition 14, the $(1,1)$-entry of (5.34) is then of the form

$\displaystyle\frac{1}{2v_{3}}e^{\int_{\psi_{2}}\left(B_{{\mathcal{L}}_{{\mathcal{E}}}}+2\sqrt{R}\Re
Z_{+}\right)}\sinh\left(8\sqrt{Rs_{2}}\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})\right)$
(5.35)
$\displaystyle+\frac{1}{2v_{3}}e^{8\sqrt{R}\Re(\sqrt{s_{2}\tau_{2}}-\sqrt{s_{3}\tau_{3}})-\int_{\psi_{2}}\left(B_{{\mathcal{L}}_{{\mathcal{E}}}}+2\sqrt{R}\Re
Z_{+}\right)}\sinh\left(8\sqrt{Rs_{2}}\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})\right),$
(5.36)

and its $(1,2)$-entry is

$-\frac{1}{2v_{3}}e^{\int_{\psi_{2}}\left(B_{{\mathcal{L}}_{{\mathcal{E}}}}+2\sqrt{R}\Re
Z_{+}\right)}+\frac{1}{2v_{3}}e^{8\sqrt{R}\Re(\sqrt{s_{2}\tau_{2}}-\sqrt{s_{3}\tau_{3}})-\int_{\psi_{2}}\left(B_{{\mathcal{L}}_{{\mathcal{E}}}}+2\sqrt{R}\Re
Z_{+}\right)}.$ (5.37)

We now turn to computing the ratio of the entries of the first row of the
matrix

$Q_{2}=A_{3}^{-\frac{1}{2}}U_{3}P_{2}U_{2}^{-1}.$ (5.38)

Since $A_{3}^{-\frac{1}{2}}U_{3}$ is lower triangular, left multiplication by
this matrix does not affect the quotient of the entries in the first row, so
we may ignore this factor. On the other hand, if $\Re(\pi_{1}-\pi_{2})\neq 0$
then we have by (5.30)

$U_{2}^{-1}=\begin{pmatrix}1&0\\\
-u_{2}&1\end{pmatrix}\approx\begin{pmatrix}1&0\\\
\cosh\left(4\sqrt{R}\Re(\pi_{2}-\pi_{1})+2\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}\right)&1\end{pmatrix}.$

The $(1,1)$-entry of (5.38) is of the form

$\displaystyle p_{2}=$
$\displaystyle\frac{1}{2v_{3}}e^{\int_{\psi_{2}}\left(B_{{\mathcal{L}}_{{\mathcal{E}}}}+2\sqrt{R}\Re
Z_{+}\right)}\sinh\left(8\sqrt{Rs_{2}}\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})\right)$
(5.39)
$\displaystyle+\frac{1}{2v_{3}}e^{8\sqrt{R}\Re(\sqrt{s_{2}\tau_{2}}-\sqrt{s_{3}\tau_{3}})-\int_{\psi_{2}}\left(B_{{\mathcal{L}}_{{\mathcal{E}}}}+2\sqrt{R}\Re
Z_{+}\right)}\sinh\left(8\sqrt{Rs_{2}}\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})\right)$
(5.40)
$\displaystyle+\frac{1}{2v_{3}}\cosh\left(4\sqrt{R}\Re(\pi_{2}-\pi_{1})+2\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}\right)e^{\int_{\psi_{2}}\left(B_{{\mathcal{L}}_{{\mathcal{E}}}}+2\sqrt{R}\Re
Z_{+}\right)}$ (5.41)
$\displaystyle-\frac{1}{2v_{3}}\cosh\left(4\sqrt{R}\Re(\pi_{2}-\pi_{1})+2\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}\right)e^{8\sqrt{R}\Re(\sqrt{s_{2}\tau_{2}}-\sqrt{s_{3}\tau_{3}})-\int_{\psi_{2}}\left(B_{{\mathcal{L}}_{{\mathcal{E}}}}+2\sqrt{R}\Re
Z_{+}\right)},$ (5.42)

and its $(1,2)$-entry $q_{2}$ agrees with (5.37). We now analyze cases
depending on the possible relationships of the coefficients of $\sqrt{R}$ in
the arguments. Fix some $0<\varepsilon<\frac{\pi}{2}$ and let
$c\in{\mathbb{C}}$ satisfy

$\operatorname{Arg}(\pm
c)\in\left[-\frac{\pi}{2}+\varepsilon,\frac{\pi}{2}-\varepsilon\right].$

We will then use that as $|\Re c|\to\infty$ we have

$\displaystyle 2\cosh(c)$ $\displaystyle\approx e^{|\Re c|}(\cos(\Im
c)+\sqrt{-1}\operatorname{sgn}(\Re c)\sin(\Im c))$ $\displaystyle 2\sinh(c)$
$\displaystyle\approx e^{|\Re c|}(\operatorname{sgn}(\Re c)\cos(\Im
c)+\sqrt{-1}\sin(\Im c)),$

where $\approx$ is used to denote expressions having the same asymptotic
expansion as $R\to\infty$.

  1. (1)

If the inequalities

$\displaystyle\int_{\psi_{2}}\Re Z_{+}$
$\displaystyle<2\Re(\sqrt{s_{2}\tau_{2}}-\sqrt{s_{3}\tau_{3}})$ $\displaystyle
2\sqrt{s_{2}}|\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})|$
$\displaystyle<|\Re(\pi_{1}-\pi_{2})|$

hold then the second term of (5.37) dominates its first term in absolute
value, and (5.42) dominates (5.39), (5.40) and (5.41). In this case, we have

$\displaystyle[p_{2}:q_{2}]$
$\displaystyle\approx\left[\cosh\left(4\sqrt{R}\Re(\pi_{2}-\pi_{1})+2\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}\right):-1\right]$
$\displaystyle\approx\left[e^{4\sqrt{R}|\Re(\pi_{2}-\pi_{1})|}\left(\cos\left(\frac{2}{\sqrt{-1}}\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}\right)+\sqrt{-1}\operatorname{sgn}(\Re(\pi_{2}-\pi_{1}))\sin\left(\frac{2}{\sqrt{-1}}\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}\right)\right):-2\right]$
$\displaystyle\to[1:0].$

  2. (2)

In case the inequalities

$\displaystyle\int_{\psi_{2}}\Re Z_{+}$
$\displaystyle<2\Re(\sqrt{s_{2}\tau_{2}}-\sqrt{s_{3}\tau_{3}})$
$\displaystyle|\Re(\pi_{1}-\pi_{2})|$
$\displaystyle<2\sqrt{s_{2}}|\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})|,$

hold then the second term of (5.37) dominates its first term in absolute
value, and the dominant term of $p_{2}$ is (5.40). In this case, we get

$[p_{2}:q_{2}]\approx\left[\sinh\left(8\sqrt{Rs_{2}}\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})\right):1\right]\to[1:0].$

  3. (3)

In the case

$\displaystyle 2\Re(\sqrt{s_{2}\tau_{2}}-\sqrt{s_{3}\tau_{3}})$
$\displaystyle<\int_{\psi_{2}}\Re Z_{+}$ $\displaystyle
2\sqrt{s_{2}}|\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})|$
$\displaystyle<|\Re(\pi_{1}-\pi_{2})|$

the first term of (5.37) dominates its second term in absolute value, and the
dominant term of $p_{2}$ is (5.41). In this case, we get

$\displaystyle[p_{2}:q_{2}]$ $\displaystyle\approx-u_{2}$
$\displaystyle\approx\left[\cosh\left(4\sqrt{R}\Re(\pi_{2}-\pi_{1})+2\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}\right):-1\right]$
$\displaystyle\approx\left[e^{4\sqrt{R}|\Re(\pi_{2}-\pi_{1})|}\left(\cos\left(\frac{2}{\sqrt{-1}}\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}\right)+\sqrt{-1}\operatorname{sgn}(\Re(\pi_{2}-\pi_{1}))\sin\left(\frac{2}{\sqrt{-1}}\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}\right)\right):-2\right]$
$\displaystyle\to[1:0].$

  4. (4)

In the case

$\displaystyle 2\Re(\sqrt{s_{2}\tau_{2}}-\sqrt{s_{3}\tau_{3}})$
$\displaystyle<\int_{\psi_{2}}\Re Z_{+}$ $\displaystyle|\Re(\pi_{1}-\pi_{2})|$
$\displaystyle<2\sqrt{s_{2}}|\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})|$

the first term of (5.37) dominates its second term in absolute value, and the
dominant term of $p_{2}$ is (5.39), so we have

$[p_{2}:q_{2}]\approx\left[\sinh\left(8\sqrt{Rs_{2}}\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})\right):1\right]\to[1:0].$

  5. (5)

In the case

$\displaystyle\int_{\psi_{2}}\Re Z_{+}$
$\displaystyle<2\Re(\sqrt{s_{2}\tau_{2}}-\sqrt{s_{3}\tau_{3}})$ (5.43)
$\displaystyle|\Re(\pi_{1}-\pi_{2})|$
$\displaystyle=2\sqrt{s_{2}}\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})$ (5.44)
$\displaystyle\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}$
$\displaystyle\equiv 0\pmod{\pi\sqrt{-1}}$ (5.45)

the dominant term of $q_{2}$ is the second term in (5.37), and the terms
(5.40) and (5.42) cancel each other up to terms of order

$o(e^{8\sqrt{R}\Re(\sqrt{s_{2}\tau_{2}}-\sqrt{s_{3}\tau_{3}})-2\int_{\psi_{2}}\sqrt{R}\Re
Z_{+}})$

(see Proposition 11), while the terms (5.39) and (5.41) are of the same order
of growth, and add up to the double of each of them. We then have

$\frac{p_{2}}{q_{2}}\approx\frac{\exp\left(4\sqrt{R}|\Re(\pi_{1}-\pi_{2})|+\int_{\psi_{2}}\left(B_{{\mathcal{L}}_{{\mathcal{E}}}}+2\sqrt{R}\Re
Z_{+}\right)\right)}{\exp\left(8\sqrt{R}\Re(\sqrt{s_{2}\tau_{2}}-\sqrt{s_{3}\tau_{3}})-\int_{\psi_{2}}\left(B_{{\mathcal{L}}_{{\mathcal{E}}}}+2\sqrt{R}\Re
Z_{+}\right)\right)}+o(1).$

Now, assuming the inequality

$2\Re(\sqrt{s_{2}\tau_{2}}-\sqrt{s_{3}\tau_{3}})-\int_{\psi_{2}}\Re
Z_{+}>|\Re(\pi_{1}-\pi_{2})|=2\sqrt{s_{2}}\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}}),$

or equivalently,

$\int_{\psi_{2}}\Re
Z_{+}<2\Re(2\sqrt{s_{2}\tau_{2}}-\sqrt{s_{2}\tau_{1}}-\sqrt{s_{3}\tau_{3}}),$
(5.46)

then we get

$\displaystyle[p_{2}:q_{2}]$
$\displaystyle\approx\left[\exp\left(-8\sqrt{R}\Re(2\sqrt{s_{2}\tau_{2}}-\sqrt{s_{2}\tau_{1}}-\sqrt{s_{3}\tau_{3}})+2\int_{\psi_{2}}\left(B_{{\mathcal{L}}_{{\mathcal{E}}}}+2\sqrt{R}\Re
Z_{+}\right)\right):1\right]$ $\displaystyle\to[0:1]$

Notice that in view of the assumption $\Re(\pi_{1}-\pi_{2})\neq 0$ and (5.44),
the inequality (5.46) is stronger than (5.43).

  6. (6)

In the case

$\displaystyle\int_{\psi_{2}}\Re Z_{+}$
$\displaystyle<2\Re(\sqrt{s_{2}\tau_{2}}-\sqrt{s_{3}\tau_{3}})$
$\displaystyle|\Re(\pi_{1}-\pi_{2})|$
$\displaystyle=2\sqrt{s_{2}}\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})$
$\displaystyle\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}$
$\displaystyle\not{\\!\\!\equiv}0\pmod{\pi\sqrt{-1}}$

the cancellation of case (5) does not occur and we have

$[p_{2}:q_{2}]\approx-u_{2}\to[1:0].$

  7. (7)

In the case

$\displaystyle\int_{\psi_{2}}\Re Z_{+}$
$\displaystyle<2\Re(\sqrt{s_{2}\tau_{2}}-\sqrt{s_{3}\tau_{3}})$
$\displaystyle|\Re(\pi_{1}-\pi_{2})|$
$\displaystyle=-2\sqrt{s_{2}}\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})$
$\displaystyle\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}$
$\displaystyle\equiv\frac{\pi\sqrt{-1}}{2}\pmod{\pi\sqrt{-1}}$

the analysis is similar to case (5) up to sign differences. Namely, the
dominant term of $q_{2}$ is the second term in (5.37), but now we have

$\exp\left(2\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}\right)=-1.$

The terms (5.40) and (5.42) again cancel each other and (5.39), (5.41) are
asymptotically equal to each other, hence the behaviour of $[p_{2}:q_{2}]$
agrees (up to a global sign) with the one given in case (5). Namely, under
condition (5.46) we get $[p_{2}:q_{2}]\to[0:1]$ with the same exponential
decay rate as in case (5).

  8. (8)

In the case

$\displaystyle\int_{\psi_{2}}\Re Z_{+}$
$\displaystyle<2\Re(\sqrt{s_{2}\tau_{2}}-\sqrt{s_{3}\tau_{3}})$
$\displaystyle|\Re(\pi_{1}-\pi_{2})|$
$\displaystyle=-2\sqrt{s_{2}}\Re(\sqrt{\tau_{1}}-\sqrt{\tau_{2}})$
$\displaystyle\int_{\eta_{2}-\eta_{1}}B_{{\mathcal{L}}_{{\mathcal{E}}}}$
$\displaystyle\not{\\!\\!\equiv}\frac{\pi\sqrt{-1}}{2}\pmod{\pi\sqrt{-1}}$

the dominant term of $q_{2}$ is the second term of (5.37), while the terms
(5.40) and (5.42) are of the same order of growth and dominate (5.39) and
(5.41). Fixing an arbitrary $\varepsilon>0$, for all $q$ such that