# Hitchin Connection on the Veech Curve

Shehryar Sikander<EMAIL_ADDRESS>
###### Abstract.

We give an expression for the pull back of the Hitchin connection from the
moduli space of genus two curves to a ten-fold covering of a Teichmüller curve
discovered by Veech. We then give an expression, in terms of iterated
integrals, for the monodromy representation of this connection. As a corollary
we obtain quantum representations of infinitely many pseudo-Anosov elements in
the genus two mapping class group.

This is author’s thesis supported in part by the center of excellence grant
’Center for Quantum Geometry of Moduli Spaces’ from the Danish National
Research Foundation (DNRF95).

###### Contents

  1. 1 Introduction
    1. 1.1 Acknowledgements
  2. 2 Moduli spaces of vector bundles and Hitchin connection in genus two
    1. 2.1 The Heisenberg group
    2. 2.2 The Hitchin connection
      1. 2.2.1 Riemann surfaces with theta structure
      2. 2.2.2 Projectively flat connections
  3. 3 Teichmüller curves and pseudo-Anosov mapping classes
    1. 3.1 Hitchin connection and Hyperlogarithms on the Veech curve
  4. 4 Generators of the (orbifold) fundamental group
    1. 4.1 Computing Monodromy

##  1\. Introduction

Let $S_{g}$ be a closed connected and oriented surface of genus $g\geqslant
2$, and consider its _mapping class group_ $\Gamma_{g}$ of orientation-
preserving diffeomorphisms up to isotopy. More precisely,

$\Gamma_{g}:=\text{Diffeo}^{+}(S_{g})/\text{Diffeo}^{+}_{0}(S_{g}),$ (1)

where $\text{Diffeo}^{+}(S_{g})$ is the group of orientation-preserving
diffeomorphisms of $S_{g}$, and $\text{Diffeo}_{0}^{+}(S_{g})$ denotes the
connected component of the identity. The mapping class group is of fundamental
importance in low-dimensional topology, and has been studied for more than a
century. Excellent surveys on recent progress and open problems related to the
group $\Gamma_{g}$ can be found in [Ham11, Far06], and a comprehensive
treatment is given in [FM11]. Recently, and perhaps surprisingly, ideas from
quantum field theory have been applied very successfully to answer questions
about the group $\Gamma_{g}$, which resisted all traditional methods.

In the seminal paper [Wit89], Witten laid the foundation of a three-
dimensional topological quantum field theory, now called the _Witten-
Reshetikhin-Turaev_ TQFT. Implicit in this TQFT is a finite-dimensional
projective linear representation of $\Gamma_{g}$, for any positive integer
$k$, which is called the _quantum representation_ at level $k$. These
representations provide a powerful tool for studying the mapping class groups.
Indeed, the collection of representations can distinguish any two mapping
classes, except the identity and the hyper-elliptic involution in genus 2.
This is the content of Turaev’s asymptotic faithfulness conjecture, which was
proved by Andersen using the theory of Toeplitz operators [And06, FWW02]. By
applying similar technology, the quantum representations were also used to
prove that $\Gamma_{g}$ does not have Kazhdan’s property (T) [And]. Finally,
the quantum representations have been used by Masbaum and Reid to show that
any finite group occurs as a quotient of a finite-index subgroup of
$\Gamma_{g}$ [MR12].

There are by now (at least) three different rigorous constructions of the
quantum representations of $\Gamma_{g}$. The first uses representation theory
of the quantum group $U_{q}(\mathfrak{sl}(2))$, with $q$ a fixed root of unity
[RT90, RT91, Tur94], the second gives a combinatorial construction using skein
theory [BHMV95], and the third provides a geometric description involving the
parallel transport of a projectively flat connection in a vector bundle over
the moduli space of smooth compact Riemann surfaces of genus $g$. The fact
that these three constructions are equivalent follows from [AU12] and [Las98].

While quantum representations have been utilized successfully to answer
questions about $\Gamma_{g}$, they are themselves far from being well
understood. Amongst all mapping classes, the quantum representations of
pseudo-Anosov mapping classes are the most mysterious. A number of problems in
this direction are discussed in [AMU06] and chapter seven of [Oht02].

Let $\mathcal{T}_{g}$ be the _Teichmüller space_ of marked Riemann surfaces of
genus $g$. It is well-known that $\mathcal{T}_{g}$ has the structure of a
complex manifold which is homeomorphic to an open ball in $\mathbb{C}^{3g-3}$.
Moreover, it follows from Royden’s work that $\mathcal{T}_{g}$ is _Kobayashi
hyperbolic_. The group $\Gamma_{g}$ has a natural and properly discontinuous
action on $\mathcal{T}_{g}$. In fact, for $g\geqslant 3$, Royden showed that
$\Gamma_{g}$ can be identified with the group of biholomorphic automorphims of
$\mathcal{T}_{g}$, but in genus two, the hyperelliptic involution $\sigma$
acts trivially, and the automorphisms of $\mathcal{T}_{2}$ amount to
$\Gamma_{2}/\langle\sigma\rangle$. The quotient
$\mathcal{M}_{g}:=\mathcal{T}_{g}/\Gamma_{g}$ is a complex orbifold called
Riemann’s moduli space, and the orbifold singularities correspond to
equivalence classes of Riemann surfaces with non-trivial automorphism group.
Since $\mathcal{T}_{g}$ is simply connected, the mapping class group
$\Gamma_{g}$ can be identified with the orbifold fundamental group
$\pi_{1}^{orb}(\mathcal{M}_{g},*)$. Morevoer, $\mathcal{M}_{g}$ is also
Kobayashi hyperbolic since the Kobayashi metric is invariant under
biholomorphic automorphisms.

Let $X$ be a compact Riemann surface of genus $g\geqslant 2$ and let
$\mathcal{M}_{X}^{B}$ be the moduli space of semi-stable vector bundles of
rank 2 with trivial determinant on $X$. Drezet and Narasimhan showed that
$\operatorname{Pic}(\mathcal{M}^{B}_{X})\cong\mathbb{Z}$. Let
$\mathcal{L}_{X}\to\mathcal{M}_{X}^{B}$ denote the ample generator of the
group $\operatorname{Pic}(\mathcal{M}^{B}_{X})$, and let $H^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{X}:=H^{0}(\mathcal{M}^{B}_{X},\mathcal{L}^{k}_{X})$
for any positive integer $k$. The dimension of this vector space is given by
the Verlinde formula (16).

From [ADPW91], [Hit90], and [Fal93], it follows that the vector spaces
$H^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{X}$ can be glued together to form a vector
bundle $\mathcal{H}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}\to\mathcal{T}_{g}$ with a projectively flat
connection in the associated projective bundle. We denote this projectively
flat vector bundle by

$(\mathbb{P}\mathcal{H}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}},{\boldsymbol{\nabla}}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}})\to\mathcal{T}_{g}.$ (2)

The action of $\Gamma_{g}$ on $\mathcal{T}_{g}$ lifts to an action on the
projective bundle $\mathbb{P}\mathcal{H}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}\to\mathcal{T}_{g}$ which preserves the
connection $\boldsymbol{\nabla}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise 1.0pt\hbox{$\scriptscriptstyle)$}}$.
Consequently, upon passing to the quotient by this action we obtain the
projectively flat vector bundle

$(\mathbb{P}\mathcal{\check{H}}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}},\check{\boldsymbol{\nabla}}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}})\to\mathcal{M}_{g}.$ (3)

Once a base point in $\mathcal{M}_{g}$ is chosen, the monodromy of (3) gives
rise to the quantum representation

$\rho^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{g}\colon\Gamma_{g}\cong\pi_{1}^{orb}(\mathcal{M}_{g},*)\to\operatorname{Aut}(\mathbb{P}\mathcal{\check{H}}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\scriptscriptstyle*}).$ (4)

A Teichmüller curve is a pair $(V,\phi)$ where $V$ is a finite area hyperbolic
surface and

$\phi\colon V\to\mathcal{M}_{g}$

is a generically injective holomorphic map which is an isometry for the
Kobayashi metrics on the domain and the codomain. Equivalently, a Teichmüller
curve is an algebraic curve in $\mathcal{M}_{g}$ which is a complex geodesic
for the Kobayashi (=Teichmüller) metric. A typical construction of Teichmüller
curves comes from _Veech surfaces_. A Veech surface generates a complex
geodesic $\mathbb{H}\to\mathcal{T}_{g}$ for the Kobayashi metric, such that
the image of $\mathbb{H}$ under the composition with the qoutient map
$\mathbb{H}\to\mathcal{T}_{g}\to\mathcal{M}_{g}$ is an algebraic curve: This
curve is naturally a complex geodesic for the Kobayashi metric and is thus a
Teichmüller curve. Suppose a Teichmüller curve $(V,\phi)$ is such that
$\phi(V)$ comes from a Veech surface, then $\pi_{1}^{orb}(V,*)$ can be
identified as the subgroup of
$\pi_{1}^{orb}(\mathcal{M}_{g},\phi(*))\cong\Gamma_{g}$ which preserves
(setwise) the image of the Kobayashi geodesic $\mathbb{H}\to\mathcal{T}_{g}$
under its action on $\mathcal{T}_{g}$. See section 3 for precise definitions
and more details.

Using the holomorphic map $\phi\colon V\to\mathcal{M}_{g}$, one can pull back
the projectively flat vector bundle (3) which we denote by

$(\mathbb{P}\mathcal{H}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\scriptscriptstyle
V},\boldsymbol{\nabla}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\scriptscriptstyle V})\to V.$ (5)

Since this is a projectively flat vector bundle on a Riemann surface,
computing its monodromy is considerably easier than computing the monodromy of
(3), and one can hope to obtain explicit matrices for the monodromy
representation of $\pi_{1}^{orb}(V,*)$ with respect to
$\boldsymbol{\nabla}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\scriptscriptstyle V}$. Assuming $\phi(V)$
comes from a Veech surface, these explicit matrices will then correspond to
the quantum representation of a subgroup of $\Gamma_{g}$. Moreover, since
$\phi(V)$ is a totally geodesic submanifold with respect to the Teichmüller
metric, it follows from Bers’ characterization of pseudo-Anosov mapping
classes that each geodesic on $V$ corresponds to a pseudo-Anosov mapping
class. Taking appropriate products of quantum representations of the
generators of $\pi_{1}^{orb}(V,*)$ will then give explicit matrices for the
quantum representations of pseudo-Anosov elements. We refer to section 3 for
further details.

In the case where $X$ has genus two, van Geemen and de Jong in [vGdJ98] gave a
rather explicit expression for the projectively flat connection (3). This
construction plays a central role in the present paper, and its details are
recalled in section 2.2.

A special feature of the genus two case is the Narasimhan-Ramanan isomorphism,
see [NR69], which implies that $S^{k}(H^{(1)}_{X})=H^{(k)}_{X}$ where $S^{k}$
denotes the $k^{\mathrm{th}}$ symmetric power of a vector space. Beauville
showed an isomorphism, for all $g\geqslant 2$, between $\mathbb{P}H^{(1)}_{X}$
and $\mathbb{P}F_{g}$, where $F_{g}$ is the vector space of complex valued
functions on the group $\mathbb{Z}_{2}^{g}$, see [Bea88]. Moreover, this
isomorphism _only_ depends on the choice of a level two theta structure on
$X$.

These two isomorphisms together imply that a choice of a level two theta
structure on $X$, where $X$ has genus two, gives a canonical isomorphism
$\mathbb{P}H^{(k)}_{X}\cong\mathbb{P}S^{k}(F_{2})$. There is a finite degree
covering $\widetilde{\mathcal{C}}_{\infty}\to\mathcal{M}_{2}$ Let us denote
the group morphism given by the monodromy of the projectively flat connection
by

$\rho^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{2}\colon\pi_{1}^{orb}(\mathcal{M}_{2},*)\to\operatorname{Aut}(\mathbb{P}S^{k}(F_{2})).$
(6)

The central object of this paper is a special Teichmüller curve

$\phi\colon\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}\to\mathcal{M}_{2}$ (7)

with orbifold fundamental group given by the Hecke traingle group of signature
$(2,5,\infty)$,

$\pi_{1}^{orb}(\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi})=\triangle(2,5,\infty)=\langle\,S,T\mid
S^{2}=(ST)^{5}=1\,\rangle.$ (8)

It is the first in family of Teichmüller curves discovered by Veech in
[Vee89]. The papers [McM06b], [BM10], and [Loc05] give an algebraic
description of $\phi$, which we recall. Let $\widetilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}:=\mathbb{C}\mathbb{P}^{1}-\mu_{5}$, where
$\mu_{5}$ is the set of fifth roots of unity. Let
$D\subset\operatorname{Aut}(\widetilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}})$ be the subgroup
generated by the maps $t\mapsto{1\over t}$ and $t\mapsto\zeta^{2}t$, where $t$
is the restriction of the global rational coordinate on
$\mathbb{C}\mathbb{P}^{1}$ to $\widetilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}$, and
$\smash{\zeta:=\exp(\frac{2\pi}{5}\sqrt{-1})}$. For any
$t\in\widetilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}$, let $F_{t}$ be the unique smooth Riemann
surface of genus two associated with the polynomial

$y^{2}=\prod_{i=1}^{5}(x-\zeta^{i}-\zeta^{-i}t)=x^{5}-5tx^{3}+5t^{2}x-t^{5}-1.$

It turns out that $F_{t}$ is isomorphic to $F_{gt}$ for all $g\in D$, and that
$\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}\cong\widetilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}/D$. Under this
identification, the map $\phi\colon\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}\to\mathcal{M}_{2}$
is simply given by $\phi([t])=[F_{t}]$, for
$[t]\in\widetilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}/D$.

In section 3, we compute the pullback of the projectively flat connection (3)
on $\mathcal{M}_{2}$ to $\chi$ and then to $\widetilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}$. This pullback
connection is defined in the trivial vector bundle
$\mathbb{P}S^{k}(F_{2})\times\widetilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}$ and is explicitly
given as

$\boldsymbol{\nabla}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\scriptscriptstyle\widetilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}}=d+\omega^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\scriptscriptstyle\widetilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}}\qquad\text{where}\qquad\omega^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\,\scriptscriptstyle\widetilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}}={\mathchar
22\relax\mkern-9.0muk}\sum_{1\leqslant i\leqslant 5}{A^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{i}\over
t-\zeta^{i}}\,dt\quad\in\Omega^{1}(\widetilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}},\operatorname{End}S^{k}(F_{2})).$ (9)

Here $\mathchar 22\relax\mkern-9.0muk:=-(16k+32)^{-1}$ and $A_{i}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}\in\operatorname{End}(S^{k}(F_{2}))$ are
explicit second-order differential operators defined in Proposition 3.4. The
connection $\boldsymbol{\nabla}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\scriptscriptstyle\widetilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}}$ extends as a
meromorphic connection on $\mathbb{C}\mathbb{P}^{1}$ with logarithmic
singularities at each $\zeta^{i}\in\mu_{5}$ and corresponding residue given by
$A_{i}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}$.

Let $\epsilon>0$ be a small real number and consider the map $\Phi^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\epsilon}\colon[0,1-\epsilon]\to\operatorname{Aut}(S^{k}(F_{2}))$
which satisfies the differential equation

${d\,\Phi^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\epsilon}(x)\over dx}={\mathchar
22\relax\mkern-9.0muk}\sum_{1\leqslant i\leqslant 5}{A^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{i}\over x-\zeta^{i}}\Phi^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\epsilon}(x)\qquad\qquad\Phi^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\epsilon}(0)=\operatorname{Id}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise 1.0pt\hbox{$\scriptscriptstyle)$}}.$
(10)

The limit of this map as $\epsilon\to 0$ will play a central role in proving
Theorem 1.1. Differential equations such as (10) were first considered by
Poincaré who gave explicit solutions of such equations [Poi84]. Lappo-
Danilevsky, among many other results, computed the monodromy of these
solutions, see mémoire second in [LD53]. The solutions are given in terms of
_hyperlogarithms_ which, for the specific case of (10), are defined upon the
choice of a tuple $(\zeta^{i_{1}},\dots,\zeta^{i_{r}})\in(\mu_{5})^{r}$ by the
iterated integral

$L_{0}(\zeta^{i_{1}},\dots,\zeta^{i_{r}}|1-\epsilon):=\int_{0}^{1-\epsilon}\mkern-6.0mu{1\over
s_{r}-\zeta^{i_{r}}}\int_{0}^{s_{r}}\mkern-6.0mu{1\over
s_{{r-1}}-\zeta^{{i}_{r-1}}}\cdots\int_{0}^{s_{2}}\mkern-6.0mu{1\over
s_{1}-\zeta^{{i}_{1}}}ds_{1}\cdots ds_{{r-1}}ds_{{r}}.$

The solution to (10) is then given by

$\Phi^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\epsilon}(x):=\operatorname{Id}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}+\sum_{r=1}^{\infty}{\mathchar
22\relax\mkern-9.0muk}^{r}\sum_{\mathclap{\,\,\zeta^{i_{1}},\dots,\zeta^{i_{r}}}}L_{0}(\zeta^{i_{1}},\dots,\zeta^{i_{r}}|x)A^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{i_{1}}\dots A^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{i_{r}}\quad\quad\text{for}\,\,x\in[0,1-\epsilon],\,\zeta^{i_{j}}\in\mu_{5}.$

We can now state the main theorem.

######  Theorem 1.1.

For any positive integer $k$, let $\rho_{2}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}\colon\pi_{1}^{orb}(\mathcal{M}_{2},[X])\to\operatorname{Aut}(\mathbb{P}S^{k}(F_{2}))$
be the quantum representation based at the orbifold point determined by the
compact Riemann surface $X$ associated to the polynomial $y^{2}=x^{5}-1$.
Then, under the isomorphism $\pi_{1}^{orb}(\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi})\cong\triangle(2,5,\infty)$, we have that

$\rho_{2}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}(ST)={M_{0}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}}\qquad\text{and }\qquad\rho_{2}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}(T)=(\Phi^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}})^{-1}\cdot M_{1}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}\cdot\exp\big{(}\mathchar
22\relax\mkern-9.0muk\pi iA^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{1}\big{)}\cdot\Phi^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise 1.0pt\hbox{$\scriptscriptstyle)$}},$

where,

$M_{0}=\begin{bmatrix}0&0&-1&\hphantom{-}1\\\ 0&0&\hphantom{-}0&-1\\\
1&0&-1&\phantom{-}0\\\ 1&1&-1&\hphantom{-}0\end{bmatrix}\qquad\text{and}\qquad
M_{1}=\begin{bmatrix}0&1&\hphantom{-}0&\hphantom{-}0\\\
1&0&\phantom{-}0&\hphantom{-}0\\\ 1&1&\hphantom{-}0&-1\\\
1&1&-1&\hphantom{-}0\end{bmatrix}$ (11)

and

$\Phi^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}=\operatorname{Id}+\sum_{r=1}^{\infty}\,{\mathchar
22\relax\mkern-9.0muk}^{r}\;\sum_{\mathclap{\overset{\,\,\zeta^{i_{1}},\dots,\zeta^{i_{r}}}{\\!\\!\\!\scriptscriptstyle\zeta^{i_{r}}\neq
1}}}\;L_{0}(\zeta^{i_{1}},\dots,\zeta^{i_{r}}|1)\,\,A^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{i_{1}}\cdots A^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{i_{r}}\quad\quad\in\operatorname{End}(S^{k}(F_{2}))$
(12)

where $1\leqslant i_{j}\leqslant 5$, and the Hyperlogarithm
$L_{0}(\zeta^{i_{1}},\dots,\zeta^{i_{r}}|1)$ is given by

$L_{0}(\zeta^{i_{1}},\dots,\zeta^{i_{r}}|1):=\int_{0}^{1}\mkern-6.0mu{1\over
s_{r}-\zeta^{i_{r}}}\int_{0}^{s_{r}}\mkern-6.0mu{1\over
s_{{r-1}}-\zeta^{{i}_{r-1}}}\cdots\int_{0}^{s_{2}}\mkern-6.0mu{1\over
s_{1}-\zeta^{{i}_{1}}}ds_{1}\cdots ds_{{r-1}}ds_{{r}}.$

$M_{i}^{(k)}$ denotes the k-th symmetric power of $M_{i}$.

###  1.1. Acknowledgements

I wish to thank my advisor J. E. Andersen for introducing me to the subject of
Geometric Quantization and Topological Quantum Field Theories. I wish to thank
Sasha Beilinson for sharing his preprint [BK] and his fascinating insight on
Geometric Quantization of varieties over finite fields. I wish to thank Benson
Farb for his interest and fruitful discussions on volumes of pseudo-Anosov
mapping tori. I wish to thank Don Zagier for encouragement and interest during
the early part of this work.

##  2\. Moduli spaces of vector bundles and Hitchin connection in genus two

Let $X$ be a compact Riemann surface of genus $g\geqslant 2$. Let $E\to X$ be
a holomorphic vector bundle and let

$\displaystyle\mu(E):={\text{degree}(E)\over\text{rank}(E)}.$

The rational number $\mu(E)$ is called the slope of the vector bundle $E\to
X$.

######  Definition 2.1.

A holomorphic vector bundle $E\to X$ is called stable, respectively semi-
stable, if $\mu(E)>\mu(F)$, respectively $\mu(E)\geqslant\mu(F)$, for all
proper holomorphic sub-bundles $F\subset E$.

Consider the moduli space of semi-stable bundles,

$\mathcal{M}^{B}_{X}:=\\{E\to
X\mid\wedge^{2}E\cong\mathcal{O}_{X}\,\text{and}\,E\,\text{ is semi-
stable}\\}/\\!\sim_{s},$ (13)

where $\sim_{s}$ denotes the $S$-equivalence for semi-stable vector bundles.
For an introduction to $S$-equivalence, which is weaker than holomorphic
equivalence, see section 2 of [NR69]. Seshadri proved that (13) is an
irreducible normal projective variety of dimension $3g-3$ [Ses67]. For
$g\geqslant 3$, the variety $\mathcal{M}^{B}_{X}$ has rational singularities,
see [BM08] Remark 2.7, and the singular locus, which has codimension greater
than one, corresponds to ($S$-equivalence classes of) semi-stable vector
bundles which are not stable, see theorem 1 of [NR69].

The variety $\mathcal{M}^{B}_{X}$ is locally factorial111i.e. all local rings
are unique factorization domains, see page 141 of [Har77] for more details.,
and a normal projective variety is locally factorial if and only if there is
an isomorphism between its equivalence classes of line bundles and its
equivalence classes of (Weil) divisors222It is important to distinguish
between Weil and Cartier divisors since we are in the setting of a singular
variety. We use the definition on page 130 of [Har77], which makes sense in
the setting of normal varieties.. In particular, associated with any closed
codimension one subvariety of $\mathcal{M}^{B}_{X}$ is a unique line bundle on
$\mathcal{M}^{B}_{X}$.

For any $L\in\operatorname{Pic}^{g-1}(X)$ the set of points

$\Xi_{L}:=\\{[E]\in\mathcal{M}^{B}_{X}\,|\,h^{0}(X,E^{\prime}\otimes L)\neq
0\,\text{for some}\,E^{\prime}\in[E]\\}$ (14)

is a divisor in $\mathcal{M}^{B}_{X}$, see page 90 of [DN69]. Denote the line
bundle associated with (14) by $\mathcal{O}(\Xi_{L})$333The variety
$\mathcal{M}^{B}_{X}$ is locally factorial, see théorème A in [DN69], thus to
every (Weil) divisor is associated a line bundle, see section 6 of [DN69]. The
following is théorème B from [DN69].

######  Theorem 2.2.

The line bundle $\mathcal{O}(\Xi_{L})$ has the following properties.

  1. a)

The isomorphism class of $\mathcal{O}(\Xi_{L})$ is independent of the choice
of $L\in\operatorname{Pic}^{g-1}(X)$.

  2. b)

The group $\operatorname{Pic}(\mathcal{M}^{B}_{X})$ is isomorphic to
$\mathbb{Z}$, and is generated by $\mathcal{O}(\Xi_{L})$.

From now on denote the ample generator of
$\operatorname{Pic}(\mathcal{M}^{B}_{X})$ by

$\mathcal{L}_{X}\to\mathcal{M}^{B}_{X}.$ (15)

Since $\mathcal{M}^{B}_{X}$ is projective the vector space
$H^{0}(\mathcal{M}^{B}_{X},\mathcal{L}^{k}_{X})$ is finite-dimensional. In
fact, there is the following remarkable formula, first conjectured by string
theorist E. Verlinde,

$\dim_{\mathbb{C}}H^{0}(\mathcal{M}^{B}_{X},\mathcal{L}^{k}_{X})=\bigg{(}{k+2\over
2}\bigg{)}^{g-1}\sum_{j=1}^{k+1}\big{(}\sin^{2}{j\pi\over k+2}\big{)}^{1-g}.$
(16)

We refer to [Zag96] for an excellent introduction to number-theoretical
aspects of this formula, a brief historical overview, and ten different
formulas for the right-hand side of (16). A geometric proof of (16) is given
in [Fal94] . Further references can be found in [Zag96].

Let $X^{(g-1)}$ be the $(g-1)$-fold symmetric product of $X$. The set

$\Theta_{X}:=\\{L\in\text{Pic}^{g-1}(X)|h^{0}(X,L)\neq 0\\}$ (17)

is the image of the canonical map

$C\colon X^{(g-1)}\to\operatorname{Pic}^{g-1}(X)$ (18)

sending an element $D\in X^{(g-1)}$ (considered as an effective divisor of
degree $g-1$ on $X$) to its (equivalence class of) associated line bundle
$\mathcal{O}(D)\in\operatorname{Pic}^{g-1}(X)$. The map (18) is proper and
birational onto its image, see section 11.2 of [LB92], which implies that (17)
is a closed irreducible subvariety of dimension $g-1$. In particular, (17) is
a divisor, called the canonical theta divisor associated with $X$. Let
$\mathcal{O}(2\Theta_{X})$ be the line bundle associated with the divisor
$2\Theta_{X}$, and let

$|2\Theta_{X}|:=\mathbb{P}H^{0}(\operatorname{Pic}^{1}(X),\mathcal{O}(2\Theta_{X}))$

be the projective space of divisors linearly equivalent to $2\Theta_{X}$. For
a positive integer $n$, it is a classical result that dimension of
$H^{0}(\operatorname{Pic}^{g-1}(X),\mathcal{O}(n\Theta_{X}))$ is $n^{g}$, thus
$|n\Theta_{X}|\cong\mathbb{C}\mathbb{P}^{n^{g}-1}$.

Let $X$ be of genus $g=2$. In this case the theta divisor (17) is the image of
$X$ under (18). M. S. Narasimhan and S. Ramanan, see theorem 2 of [NR69],
proved a canonical isomorphism

$\mathcal{M}^{B}_{X}\cong|2\Theta_{X}|\,\,(\cong\mathbb{C}\mathbb{P}^{3}).$
(19)

The Narasimhan-Ramanan isomorphism in particular implies that the ample
generator of $\operatorname{Pic}(\mathcal{M}^{B}_{X})$ can be identified with
the ample generator of $\operatorname{Pic}(\mathbb{C}\mathbb{P}^{3})$. This
identification gives specialization of the Verlinde formula (16) to

$\dim_{\mathbb{C}}H^{0}(\mathcal{M}_{X},\mathcal{L}^{k}_{X})={k+3\choose 3}$
(20)

and implies that

$H^{0}(\mathcal{M}_{X},\mathcal{L}^{k}_{X})=S^{k}(H^{0}(\mathcal{M}_{X},\mathcal{L}_{X}))$

where $S^{k}$ denotes the $k^{th}$ symmetric power of a vector space.

###  2.1. The Heisenberg group

Let $X$ be a compact Riemann surface of genus $g\geqslant 2$. In this section,
we study the group of automorphisms of the line bundle
$\mathcal{L}_{X}\to\mathcal{M}_{X}^{B}$ which cover a natural action of the
group of two-torsion points of $\operatorname{Pic}^{0}(X)$ on
$\mathcal{M}_{X}^{B}$. It turns out that the representation of this group of
automorphisms on $H^{0}(\mathcal{M}^{B}_{X},\mathcal{L}_{X})$ is equivalent to
the Schrödinger representation of the Heisenberg group (of level 2).

Let

$J_{X}[2]:=\\{L\in\text{Pic}^{0}(X)\mid L^{2}\cong\mathcal{O}_{X}\\}$ (21)

be the group of two-torsion points of $\operatorname{Pic}^{0}(X)$. Let $E\to
X$ be a semistable vector bundle of rank two with trivial determinant. The
tensor product $E\otimes L\to X$, for any $L\in J_{X}[2]$, is again a
semistable vector bundle of rank two with trivial determinant, and if
$E\sim_{s}E^{\prime}$, then $(E\otimes L)\sim_{s}(E^{\prime}\otimes L)$, see
[AM99] section 1 for more details. This operation of tensoring with two-
torsion points of $\operatorname{Pic}^{0}(X)$ therefore gives a well-defined
action on the moduli space of bundles,

$J_{X}[2]\times\mathcal{M}^{B}_{X}\to\mathcal{M}^{B}_{X}.$ (22)

Let $\mathcal{G}(X)$ be the group of automorphisms of the line bundle
$\mathcal{L}_{X}\to\mathcal{M}^{B}_{X}$ which cover the action (22). For all
$L\in J_{X}[2]$, we have that $L^{*}(\mathcal{L}_{X})\cong\mathcal{L}_{X}$,
see [AM99] page four or remarque 2.6 in [Bea88], which implies that the group
$\mathcal{G}(X)$ is a central extension

$1\to\mathbb{C}^{*}\to\mathcal{G}(X)\to J_{X}[2]\to 1,$ (23)

see section 2 of [AM99] for more details. Following [vGdJ98], we study the
action of $\mathcal{G}(X)$ on $H^{0}(\mathcal{M}^{B}_{X},\mathcal{L}_{X})$ by
using Beauville’s isomorphism between
$H^{0}(\mathcal{M}_{X}^{B},\mathcal{L}_{X})$ and theta functions of order two
associated with $X$. It should be mentioned that the methods developed to
study this action of $\mathcal{G}(X)$ in [AM99] are intrinsic and exploit the
geometry of the moduli space $\mathcal{M}_{X}^{B}$.

Let $A$ be a complex abelian variety and $L\to A$ an arbitrary line bundle.
Associated to $(A,L)$ is the group of automorphisms of $L$ which cover
translations in $A$. This group is called the theta group of $(A,L)$, see
section 6.1 of [LB92] for more details. Let $\mathcal{G}(2\Theta^{0}_{X})$ be
the theta group of $(J(X),\mathcal{O}(2\Theta^{0}_{X}))$ where
$J(X):=\operatorname{Pic}^{0}(X)$ is the Jacobian of $X$ and
$\Theta^{0}_{X}:=\\{L\otimes\kappa^{-1}|L\in\Theta_{X}\\}\subset J(X)$ is a
translate of the canonical theta divisor (17) by a theta characteristic
$\kappa\in\operatorname{Pic}^{g-1}(X)$, i.e. $\kappa^{2}\cong K_{X}$ where
$K_{X}$ is the canonical bundle of $X$. It is well known that the linear
equivalence class of the divisor $\Theta^{0}_{X}$ is independent of the choice
of the theta characteristic444This point needs to be discussed, it might be
that the translates by any $\beta\in\operatorname{Pic}^{g-1}(X)$ are linearly
equivalenent, see theorem 11.2.4 in [LB92], thus the isomorphism class of
$\mathcal{O}(2\Theta^{0}_{X})$ is also independent of the choice of the theta
characteristic. The theta group $\mathcal{G}(2\Theta^{0}_{X})$ is a central
extension, see [vP93] or proposition 6.1.1 of [LB92]

$1\to\mathbb{C}^{*}\to\mathcal{G}(2\Theta_{X}^{0})\to J_{X}[2]\to 1,$ (24)

and its action on $H^{0}(J(X),\mathcal{O}(2\Theta_{X}^{0}))$ is the unique (up
to isomorphism) irreducible one in which the subgroup $\mathbb{C}^{*}$ acts by
multiplication, see section 6.4 of [LB92]. In [Bea88], see also [Bea91], a
canonical isomorphism is proved

$H^{0}(\mathcal{M}_{X}^{B},\mathcal{L}_{X})\cong
H^{0}(J(X),\mathcal{O}(2\Theta_{X}^{0})),$ (25)

it turns out that $\mathcal{G}(X)\cong\mathcal{G}(2\Theta_{X}^{0})$ and
isomorphism (25) intertwines the two representations, see remarque 2.6 in
[Bea88], page 6 of [AM99] or [vP93].

The representation of $\mathcal{G}(2\Theta_{X}^{0})$ on
$H^{0}(J(X),\mathcal{O}(2\Theta_{X}^{0}))$ has another description which we
recall. Let

$H_{g}:=\mathbb{C}^{*}\times\mathbb{Z}_{2}^{g}\times\mathbb{Z}_{2}^{g}$ (26)

and denote by

$\text{E}\colon(\mathbb{Z}_{2}^{g}\times\mathbb{Z}_{2}^{g})^{2}\to\mathbb{C}^{*},$
(27)

the standard alternating form, see e.g. page 160 of [LB92]. The set $H_{g}$
admits a group structure with multiplication given by

$(\alpha,x_{1},x_{2})\cdot(\beta,y_{1},y_{2})=(\alpha\beta\,E(x_{1},y_{2}),x_{1}+y_{1},x_{2}+y_{2}).$
(28)

This is a non-abelian group with centre $\\{(a,0,0)\in H_{g}\\}$. This group
is called the Heisenberg group (of level 2). It is isomorphic to
$\mathcal{G}(2\Theta_{X}^{0})$ , and an isomorphism which is the identity on
the subgroups $\mathbb{C}^{*}$, i.e.

$\alpha\colon\mathcal{G}(2\Theta_{X}^{0})\mathrel{\mathop{\rightarrow}^{\mathrm{\cong}}_{\mathrm{}}}H_{g},\quad\quad\alpha|_{\mathbb{C}^{*}}=\operatorname{Id}|_{\mathbb{C}^{*}},$
(29)

is called a _theta structure_. There are only finitely many choices of theta
structures, see theorem 6.6.7 of [LB92] or [vP93].

Let

$F_{g}:=\\{f\colon\mathbb{Z}_{2}^{g}\to\mathbb{C}\\}$ (30)

be the $2^{g}$ dimensional complex vector space of complex valued functions on
the group $\mathbb{Z}_{2}^{g}$. The group $H_{g}$ has a unique irreducible
representation on $F_{g}$, called the Schrödinger representation, in which the
centre $\\{(a,0,0)\\}$ of $H_{g}$ acts by multiplication, see [LB92] page 164
or [vP93] for an explicit construction of this representation. We denote this
representation by

$S\colon H_{g}\to\mathrm{GL}(F_{g}).$ (31)

The main result concerning this representation is that given a choice of a
theta structure $\alpha\colon\mathcal{G}(2\Theta_{X}^{0})\to H_{g}$, there
exists a unique (up to scaler multiplication) isomorphism

$H^{0}(J(X),\mathcal{O}(2\Theta^{0}_{X}))\cong F_{g},$ (32)

which intertwines the two representations, see section 6.7 of [LB92] or
[vP93]. The isomorphisms (25) and (32) together give the isomorphism

$\mathbb{P}H^{0}(\mathcal{M}^{B}_{X},\mathcal{L}_{X})\cong\mathbb{P}F_{g}$
(33)

which only depends on the choice of a theta structure. Since $F_{g}$ has a
canonical basis given by the delta functions, a choice of a theta structure
thus gives a canonical basis (up to scaling) of
$H^{0}(\mathcal{M}^{B}_{X},\mathcal{L}_{X})$.

Consider the following subgroup of automorphims of $H_{g}$,

$A(H_{g}):=\\{\beta\in\operatorname{Aut}(H_{g})\mid\beta(a,0,0)=a\\}.$ (34)

The Schrödinger representation (31) is the unique irreducible representation
in which the centre $\\{(a,0,0)\\}$ of $H_{g}$ acts by multiplication. Given
any $h\in A(H_{g})$, the representation $S\circ h$ enjoys the same property.
Schur’s lemma, see [vGdJ98] section 4.2.4 for more details, then provides a
projective representation

$T_{g}\colon A(H_{g})\to\mathbb{P}\mathrm{GL}(F_{g}).$ (35)

We gather some results on the structure of the group $A(H_{g})$ which will be
used later. The following is well known, see [LB92] lemma 6.6.6 or [vGdJ98]
page 217,

$0\to\mathbb{Z}_{2}^{2g}\to A(H_{g})\to\mathrm{Sp}(2g,\mathbb{Z}_{2})\to 0.$
(36)

Another explicit description of $A(H_{g})$ is given in [Igu66], see also
[Sas99]. Recall Igusa’s subgroup (of level (2, 4)) of
$\mathrm{Sp}(2g,\mathbb{Z})$

$\Gamma_{g}(2,4)=\left\\{\begin{pmatrix}I\\!+\\!2A&\\!2B\\\
2C&\\!I\\!+\\!2D\\\
\end{pmatrix}\in\mathrm{Sp}(2g,\mathbb{Z})\mid\text{diag}(B)\equiv\text{diag}(C)\equiv
0\pmod{2}\right\\}.$ (37)

The group (37) is a normal subgroup of $\mathrm{Sp}(2g,\mathbb{Z})$, and the
quotient is isomorphic to

$\mathrm{Sp}(2g,\mathbb{Z})/\Gamma_{g}(2,4)\cong A(H_{g}),$ (38)

For an explicit construction of this isomorphism, see [Sas99] page 336.

Assume that $X$ has genus $g=2$. In this case, recall from the previous
section that the Narasimhan-Ramanan isomorphism implies that

$S^{k}(H^{0}(\mathcal{M}^{B}_{X},\mathcal{L}_{X}))=H^{0}(\mathcal{M}^{B}_{X},\mathcal{L}^{k}_{X}).$
(39)

It follows from (33) and (39) that a choice of a theta structure gives a
canonical isomorphism

$\mathbb{P}H^{0}(\mathcal{M}^{B}_{X},\mathcal{L}^{k}_{X})\cong\mathbb{P}S^{k}(F_{g}).$
(40)

Another special feature of genus two is the isomorphism

$\mathrm{Sp}(4,\mathbb{Z}_{2})\cong S_{6}$ (41)

where $S_{6}$ is the symmetric group, see [CF96]. This isomorphism and (36)
imply that $A(H_{2})$ can be described as an extension of $S_{6}$ by
$\mathbb{Z}^{4}_{2}$.

###  2.2. The Hitchin connection

In this section we study the construction of the moduli space
$\mathcal{M}^{B}_{X}$ over a family of genus two compact Riemann surfaces. Let

$\mathcal{C}_{\infty}:=\\{(z_{1},...,z_{6})\in(\mathbb{C}\mathbb{P}^{1})^{6}\mid
z_{i}\neq z_{j}\,\,\text{for any}\,\,i\neq j\\}.$ (42)

Let $\mathcal{C}$ be the open subset
$\\{(z_{1},...,z_{6})\in\mathcal{C}_{\infty}\mid z_{i}\neq\infty\,\,\text{for
any}\,\,1\leqslant i\leqslant 6\\}$ of $\mathcal{C}_{\infty}$. Any $\bf{z}$
$=(z_{1},\dots,z_{6})\in\mathcal{C}_{\infty}$ defines the following affine
curve

$X^{A}_{\bf{z}}:=\bigg{\\{}(x,y)\in\mathbb{C}^{2}\mid\,\,y^{2}=\bigg{\\{}\begin{array}[]{l
l}\prod_{i}^{6}(x-z_{i})&\quad\text{if $\bf{z}\in\mathcal{C}$}\\\
\prod_{i}^{5}(x-z_{i})&\quad\text{if ${\bf{z}}\notin\mathcal{C}$ with
$z_{i}=\infty$ excluded}\end{array}\bigg{\\}}.$ (43)

Associated to this curve is a unique genus two compact Riemann surface, see
section VII of [FK92] for more details, which we denote by $X_{\bf{z}}$. For
any Riemann surface $X_{\bf{z}}$, we have the associated moduli space
$\mathcal{M}^{B}_{X_{\bf{z}}}$ of semistable vector bundles of rank two with
trivial determinant on $X_{\bf{z}}$, defined in (13). From [vGdJ98] section
2.5, it follows that there exists a proper holomorphic submersion,

$p\colon\mathcal{M}\to\mathcal{C}_{\infty},$ (44)

with fiber $p^{-1}({\bf{z}})=\mathcal{M}^{B}_{X_{\bf{z}}}$. Moreover, there
exists a line bundle $\mathcal{L}\to\mathcal{M}$ such that
$\mathcal{L}|_{p^{-1}({\bf{z}})}=\mathcal{L}_{X_{{\bf{z}}}}$; the ample
generator of $\operatorname{Pic}(\mathcal{M}^{B}_{X_{\bf{z}}})$. For every
positive integer $k$, denote the vector bundle
$p_{*}(\mathcal{L}^{k}_{X_{\bf{z}}})\to\mathcal{C}_{\infty}$ by

$\mathcal{V}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}\to\mathcal{C}_{\infty}.$ (45)

By definition, the fiber of this bundle is given by $\mathcal{V}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{{\bf{z}}}=H^{0}(\mathcal{M}^{B}_{X_{\bf{z}}},\mathcal{L}^{k}_{X_{{\bf{z}}}})$,
and by Narasimhan-Ramanan isomorphism, (39), ensure that $\mathcal{V}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{{\bf{z}}}=S^{k}(\mathcal{V}^{(1)}_{{\bf{z}}})$.

####  2.2.1. Riemann surfaces with theta structure

Recall from section 2.2 that any compact Riemann surface $X$ is has an
associated theta group $\mathcal{G}(2\Theta^{0}_{X})$, and that a theta
structure is a choice of an isomorphism between $\mathcal{G}(2\Theta^{0}_{X})$
and the (level 2) Heisenberg group. A Riemann surface with a (level 2) theta
structure is a pair $(X,\alpha)$, where $X$ is a compact Riemann surface and
$\alpha$ is a theta structure as defined in (29). The moduli spaces of such
pairs are known to exist and turn out to be finite-degree coverings of the
moduli spaces of compact Riemann surfaces, see [Igu66].

In [vGdJ98], an unramified sixteen-fold covering
$P\colon\widetilde{\mathcal{C}}_{\infty}\to\mathcal{C}_{\infty}$ is shown to
exist such that each fiber $P^{-1}(\bf{z})$ is in bijection with the set of
all (level 2) theta structures on the Riemann surface $X_{\bf{z}}$. In
particular, every ${\widetilde{\bf{z}}}\in\widetilde{\mathcal{C}}_{\infty}$
corresponds to a pair $(X_{P(\widetilde{\bf{z}})},\alpha)$, i.e. the Riemann
surface $X_{P(\widetilde{\bf{z}})}$ with a choice of a theta structure
$\alpha$. On this covering the pull back of the projectivization of (45)
trivializes, see lemma 2.4. We recall this covering.

The symmetric group $S_{6}$ acts on $\mathcal{C}_{\infty}$ by permuting
coordinates. The quotient
$\overline{\mathcal{C}}_{\infty}:=\mathcal{C}_{\infty}/S_{6}$ is the space of
unordered configurations of six points on $\mathbb{C}\mathbb{P}^{1}$. It is
well known that $\pi_{1}(\overline{\mathcal{C}}_{\infty})$ is $SB_{6}$, where
$SB_{6}$ denotes the spherical braid group of degree six, see section 1.5 of
[Bir75] for an introduction to spherical braid groups. We have the following
diagram of group morphisms,

$\mathrm{Sp}(4,\mathbb{Z})/\Gamma_{2}(2,4)\cong
A(H(2))$$SB_{6}$$\mathrm{Sp}(4,\mathbb{Z})$$\mathrm{Sp}(4,\mathbb{Z}_{2})\cong
S_{6}$$\tau$$a$$b$$c$ (46)

where the construction of $\tau$, which is surjective, is given on page 79 of
[CB88] 555There is a typo in this reference, the inverse sign on two of the
generators ($T_{2}$ and $T_{4}$) is missing., and $a$, $b$, and $c$ are
naturally induced. The two isomorphisms follow from (38) and (41). Since all
the morphisms are surjective and $b=c\circ a$, see page 13 of [Sas99], it
follows that $\text{Ker}(b\circ\tau)\supset\text{Ker}(a\circ\tau)$ and
$\text{Ker}(b\circ\tau)/\text{Ker}(a\circ\tau)\cong\text{Ker}(c)$. This gives
the short exact sequence

$1\to\text{Ker}(a\circ\tau)\to\text{Ker}(b\circ\tau)\to\text{Ker}(c)\to 1.$
(47)

By definition of $\overline{\mathcal{C}}_{\infty}$, we have that
$\text{Ker}(b\circ\tau)\cong\pi_{1}(\mathcal{C}_{\infty})$. This isomorphism
together with (47) implies that there exists a covering
$P\colon\widetilde{\mathcal{C}}_{\infty}\to\mathcal{C}_{\infty}$ with
$\pi_{1}(\widetilde{\mathcal{C}}_{\infty})\cong\text{Ker}(a\circ\tau)$ and
deck group $\text{Ker}(c)$. From (36) we know that
$\text{Ker}(c)\cong\mathbb{Z}^{4}_{2}$, which implies that
$\widetilde{\mathcal{C}}_{\infty}$ is a sixteen-fold cover.

######  Definition 2.3.

Let $V\to M$ be a holomorphic vector bundle over a complex manifold $M$.
Denote by $\mathbb{P}(V)\to M$ the bundle of projective spaces with fibers
$\mathbb{P}(V)_{x}:=\mathbb{P}(V_{x})$. We call $\mathbb{P}(V)\to M$ the
projective bundle associated with $V\to M$.

The following is part of theorem 4.4.2 in [vGdJ98].

######  Lemma 2.4.

The pull back of the projective bundle associated with (45) to the covering
${P}\colon\widetilde{\mathcal{C}}_{\infty}\to\mathcal{C}_{\infty}$
trivializes, in particular,

${P}^{*}(\mathbb{P}(\mathcal{V}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}))\cong\mathbb{P}(S^{k}(F_{2}))\times\widetilde{\mathcal{C}}_{\infty},$
(48)

where the vector space $F_{2}$ is defined in (30).

###### Proof.

Each $\tilde{\bf{z}}\in\widetilde{\mathcal{C}}_{\infty}$ corresponds to a pair
$(X_{P(\tilde{\bf{z}})},\alpha)$ where $\alpha$ is a (level 2) theta structure
on the Riemann surface $X_{P(\tilde{\bf{z}})}$, see section 4.5 of [vGdJ98]
for details. A theta structure on $X_{P(\tilde{\bf{z}})}$ gives a canonical
isomorphism $\mathbb{P}(\mathcal{V}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{X_{P(\tilde{\bf{z}})}})\cong\mathbb{P}(S^{k}(F_{2}))$,
see (40). This implies that each fiber of
${P}^{*}(\mathbb{P}(\mathcal{V}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise 1.0pt\hbox{$\scriptscriptstyle)$}}))$
is canonically isomorphic to $\mathbb{P}(S^{k}(F_{2}))$ and the proof follows.
∎

####  2.2.2. Projectively flat connections

We recall the (Čech) description of a projectively flat connection, see
section 2.3.3 and 2.3.6 of [vGdJ98], page 369 of [Hit90], and section 3.3 of
[Fel96].

######  Definition 2.5.

Let $\mathbb{P}(V)\to M$ be the projective bundle associated with a
holomorphic vector bundle $V\to M$ over a complex manifold $M$. A projective
connection in $\mathbb{P}(V)\to M$ is the data $\\{U_{i},\nabla_{i}\\}$, where
$\\{U_{i}\\}$ is an open covering of $M$ and $\nabla_{i}$ is a connection in
the restriction $V|_{U_{i}}$ such that on any intersection $U_{i}\cap U_{j}$
the difference $\nabla_{i}-\nabla_{j}$ is scalar multiple of the identity,
i.e.

$\nabla_{i}-\nabla_{j}=\alpha\cdot\operatorname{Id}$

where $\alpha$ is a holomorphic 1-form on $U_{i}\cap U_{j}$ and
$\operatorname{Id}$ is the multiplicative identity of the ring
$\Gamma(U_{i}\cap U_{j},\operatorname{End}(V)_{|U_{i}\cap U_{j}})$. Moreover,
a projective connection $\\{U_{i},\nabla_{i}\\}$ is a projectively flat
connection if each $\nabla_{i}$ is flat666Usually, a weaker condition that
each $\nabla_{i}$ is _projectively_ flat is imposed..

Two projective connections $\\{U_{i},\nabla_{i}\\}$ and
$\\{U^{\prime}_{i},\nabla^{\prime}_{i}\\}$ are called equivalent if on any
intersection $U_{i}\cap U^{\prime}_{j}$ the difference
$\nabla_{i}-\nabla^{\prime}_{j}$ is a scaler multiple of the identity. If $G$
is a group acting by automorphisms on $M$, then a projective connection
$\\{U_{i},\nabla_{i}\\}$ is invariant under the action of $G$ if
$\\{U_{i},\nabla_{i}\\}$ and $\\{g^{*}(U_{i}),g^{*}(\nabla_{i})\\}$ are
equivalent for all $g\in G$.

A flat connection in a holomorphic vector bundle $V\to M$ induces a unique
projectively flat connection in $\mathbb{P}(V)\to M$. Two flat connections
induce equivalent projectively flat connections if their difference is a
scaler multiple of the identity. Upon the choice of a base point $x\in M$, the
monodromy of a projectively flat connection provides a group homomorphism

$\mathfrak{M}\colon\pi_{1}(M,x)\to\operatorname{Aut}(\mathbb{P}V_{x}),$ (49)

see the before mentioned references for details.

Recall the open subset
$\mathcal{C}:=\\{(z_{1},\dots,z_{6})\in\mathcal{C}_{\infty}\mid
z_{i}\neq\infty\\}$. We outline the construction of the flat connection, see
(56), in the bundle

$S^{k}(F_{2})\times\mathcal{C}$ (50)

carried out in [vGdJ98]. This connection is then used to prove proposition 2.7
and corollary 2.8.

Denote the delta function basis of $F_{2}$ by $\langle
x_{1},x_{2},x_{3},x_{4}\rangle$. Let $\partial_{i}$ be the differential
operator defined by

$\partial_{i}(x_{j})=\delta_{ij}:=\left\\{\begin{array}[]{l l}1\,\,\,\text{if
$i=j$}\\\ 0\,\,\,\text{if $i\neq j$ }\end{array}\right.$ (51)

For all $1\leqslant i<j\leqslant 6$ consider the differential operators

$\displaystyle\Omega^{1,2}:=-(x_{1}\partial_{1}+x_{2}\partial_{2}-x_{3}\partial_{3}-x_{4}\partial_{4})^{2}\quad\quad\Omega^{1,3}:=-(x_{1}\partial_{4}-x_{2}\partial_{3}-x_{3}\partial_{2}+x_{4}\partial_{1})^{2}$
$\displaystyle\Omega^{1,4}:=+(x_{1}\partial_{4}+x_{2}\partial_{3}-x_{3}\partial_{2}-x_{4}\partial_{1})^{2}\quad\quad\Omega^{1,5}:=+(x_{1}\partial_{3}-x_{2}\partial_{4}-x_{3}\partial_{1}+x_{4}\partial_{2})^{2}$
$\displaystyle\Omega^{1,6}:=-(x_{1}\partial_{3}+x_{2}\partial_{4}+x_{3}\partial_{1}+x_{4}\partial_{2})^{2}\quad\quad\Omega^{2,3}:=+(x_{1}\partial_{4}-x_{2}\partial_{3}+x_{3}\partial_{2}-x_{4}\partial_{1})^{2}$
$\displaystyle\Omega^{2,4}:=-(x_{1}\partial_{4}+x_{2}\partial_{3}+x_{3}\partial_{2}+x_{4}\partial_{1})^{2}\quad\quad\Omega^{2,5}:=-(x_{1}\partial_{3}-x_{2}\partial_{4}+x_{3}\partial_{1}-x_{4}\partial_{2})^{2}$
$\displaystyle\Omega^{2,6}:=+(x_{1}\partial_{3}+x_{2}\partial_{4}-x_{3}\partial_{1}-x_{4}\partial_{2})^{2}\quad\quad\Omega^{3,4}:=-(x_{1}\partial_{1}-x_{2}\partial_{2}+x_{3}\partial_{3}-x_{4}\partial_{4})^{2}$
$\displaystyle\Omega^{3,5}:=-(x_{1}\partial_{2}+x_{2}\partial_{1}+x_{3}\partial_{4}+x_{4}\partial_{3})^{2}\quad\quad\Omega^{3,6}:=+(x_{1}\partial_{2}-x_{2}\partial_{1}-x_{3}\partial_{4}+x_{4}\partial_{3})^{2}$
$\displaystyle\Omega^{4,5}:=+(x_{1}\partial_{2}-x_{2}\partial_{1}+x_{3}\partial_{4}-x_{4}\partial_{3})^{2}\quad\quad\Omega^{4,6}:=-(x_{1}\partial_{2}+x_{2}\partial_{1}-x_{3}\partial_{4}-x_{4}\partial_{3})^{2}$
$\displaystyle\Omega^{5,6}:=-(x_{1}\partial_{1}-x_{2}\partial_{2}-x_{3}\partial_{3}+x_{4}\partial_{4})^{2},$

where the composition law is

$(x_{i}\partial_{j})\circ(x_{k}\partial_{l})=x_{i}x_{k}\partial_{j}\partial_{l}+\delta_{jk}x_{i}\partial_{l}.$
(52)

Let $\widehat{\Omega}^{i,j}$ denote the degree two part of $\Omega^{i,j}$. The
operator (51) has an induced action on any monomial
$x_{1}^{a}x_{2}^{b}x_{3}^{c}x_{4}^{d}$, see section 4.5 of [vGdJ98]. For a
positive integer $k$, choose the basis of $S^{k}(F_{2})$ given by monomials
$x_{1}^{a}x_{2}^{b}x_{3}^{c}x_{4}^{d}$ such that $a+b+c+d=k$. For all
$1\leqslant i<j\leqslant 6$, the induced action of the operator
$\widehat{\Omega}^{i,j}$ on $S^{k}(F_{2})$ preserves $S^{k}(F_{2})$, see
section 3.2.5 of [vGdJ98], and we denote this induced action by
$\widehat{\Omega}^{i,j}_{k}\in\operatorname{End}(S^{k}(F_{2}))$. These
endomorphisms satisfy the infinitesimal braid relations, i.e.

$[\widehat{\Omega}^{s,t}_{k},\widehat{\Omega}^{u,v}_{k}]=0,\quad\quad[\widehat{\Omega}^{s,u}_{k},\widehat{\Omega}^{s,t}_{k}+\widehat{\Omega}^{t,u}_{k}]=0$
(53)

where $s,t,u,v$ are all distinct, see proposition 3.2.4 of [vGdJ98].

For a positive integer $k$, let

$\mathchar 22\relax\mkern-9.0muk:={-1\over 16(k+2)},$ (54)

and consider the following $\operatorname{End}(S^{k}(F_{2}))$ valued
holomorphic 1-form on $\mathcal{C}$

$\omega^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}:=\mathchar
22\relax\mkern-9.0muk\sum_{1\leqslant i<j\leqslant
6}\widehat{\Omega}^{i,j}_{k}{dz_{i}-dz_{j}\over z_{i}-z_{j}}.$ (55)

Since (50) is a trivial vector bundle, the expression

$\boldsymbol{\nabla}_{0}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}:=d+{\omega}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise 1.0pt\hbox{$\scriptscriptstyle)$}},$
(56)

where $d$ denotes the trivial connection, defines a connection in (50).
Moreover, using the relations (53) and Arnold’s trick, see proposition XIX.2.1
of [Kas95], it can be shown that the curvature 2-form of the connection (56)
vanishes, i.e.

$-d\omega^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}+\omega^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}\wedge\omega^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}=0.$ (57)

We note the following relations, called the Ward identities in two-dimensional
conformal field theory, see [Koh02], which will be used later.

######  Proposition 2.6.

For a positive integer $k$, we have that

$\displaystyle\qquad\quad 1)\,\sum_{i<j}\widehat{\Omega}^{i,j}_{k}=3\mathchar
22\relax\mkern-9.0mukk^{2}\operatorname{Id}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise 1.0pt\hbox{$\scriptscriptstyle)$}}$
$\displaystyle\qquad\quad 2)\,\sum_{i\neq
j}\widehat{\Omega}^{i,j}_{k}=\mathchar
22\relax\mkern-9.0mukk^{2}\operatorname{Id}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}\qquad\text{for any fixed $j$}$
$\displaystyle\qquad\quad
3)\,\sum_{i<j}\widehat{\Omega}^{i,j}_{k}(z_{i}+z_{j})=6\mathchar
22\relax\mkern-9.0mukk^{2}\operatorname{Id}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}\sum_{i<j}z_{i}+z_{j}$

where $\operatorname{Id}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}$ is the multiplicative identity of the ring
$\operatorname{End}(S^{k}(F_{2}))$.

###### Proof.

Direct computations, notice that $(2)$ implies $(3)$. ∎

The automorphism group of $\mathbb{C}\mathbb{P}^{1}$ acts on
$\mathcal{C}_{\infty}$ by automorphisms. It is well-known that the group of
automorphisms of $\mathbb{C}\mathbb{P}^{1}$ is isomorphic to
$\mathbb{P}\mathrm{SL}(2,\mathbb{C})$ and generated by the following three
transformations, see page 10 of [Mas88],

$T(z)=z+a,\quad D(z)=bz,\quad and\quad I(z)={1\over z}$ (58)

where $z\in\mathbb{C}\mathbb{P}^{1}$, $a\in\mathbb{C}$, and
$b\in\mathbb{C}^{*}$. These three transformations are called translations,
dilations, and inversion respectively.

######  Proposition 2.7.

The connection (56) in the bundle (50) determines a unique (up to equivalence)
projectively flat connection $\boldsymbol{\nabla}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}:=\\{U_{i},\nabla^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{i}\\}$ in the projective bundle

$\mathbb{P}(S^{k}(F_{2}))\times\mathcal{C}_{\infty}.$ (59)

Moreover, this projectively flat connection $\boldsymbol{\nabla}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise 1.0pt\hbox{$\scriptscriptstyle)$}}$
is invariant under the $\mathbb{P}\mathrm{SL}(2,\mathbb{C})$ action on
$\mathcal{C}_{\infty}$.

###### Proof.

We must produce the data $\\{U_{i},\nabla^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{i}\\}$ satisfying the conditions of
definition 2.5. It is easy to construct an open covering $\\{U_{i}\\}$ of
$\mathcal{C}_{\infty}$ such that each $U_{i}=\tau_{i}^{*}(\mathcal{C})$ (WE
NEED TO GIVE $\tau_{i}^{\prime}s$) for some
$\tau_{i}\in\mathbb{P}\mathrm{SL}(2,\mathbb{C})$. Let $\tau_{0}$ be the
identity. Then $\boldsymbol{\nabla}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{0}$, defined in (56), is a flat connection
in $S^{k}(F_{2})\times U_{0}$. For each $U_{i}$, we define

$\nabla^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{i}:=\tau_{i}^{*}(\boldsymbol{\nabla}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{0})=d+\tau_{i}^{*}(\omega^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise 1.0pt\hbox{$\scriptscriptstyle)$}})$
(60)

which gives a connection in the restriction $S^{k}(F_{2})\times U_{i}$.
Moreover, since $\tau_{i}$ is a biholomorphism, the curvature of
$\nabla^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{i}$ is the pull back of the curvature of
$\boldsymbol{\nabla}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{0}$, since the curvature of
$\boldsymbol{\nabla}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{0}$ vanishes, it follows that
$\nabla^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{i}$ is flat. To show that $\nabla^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{i}-\nabla^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{j}$ is a scaler multiple of the identity
on the intersections $U_{i}\cap U_{j}$, we compute that on the intersection
$\mathcal{C}\cap\tau^{*}(\mathcal{C})$,

$\boldsymbol{\nabla}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{0}-\tau^{*}(\boldsymbol{\nabla}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{0})=0$

when $\tau$ is a translation or a dilation, and

$\displaystyle\boldsymbol{\nabla}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{0}-\tau^{*}(\boldsymbol{\nabla}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{0})$ $\displaystyle=\mathchar
22\relax\mkern-9.0muk\sum_{1\leqslant i<j\leqslant
6}\widehat{\Omega}^{i,j}_{k}\big{(}{dz_{i}\over z_{i}}+{dz_{j}\over
z_{j}}\big{)}$ $\displaystyle=6\mathchar
22\relax\mkern-9.0mukk\operatorname{Id}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}\sum_{1\leqslant i<j\leqslant
6}\big{(}{dz_{i}\over z_{i}}+{dz_{j}\over z_{j}}\big{)}$

when $\tau$ is the inversion. The second equality follows from identity (3) of
proposition 2.6, notice also that the 1-form $6\mathchar
22\relax\mkern-9.0mukk\sum_{1\leqslant i<j\leqslant 6}\big{(}{dz_{i}\over
z_{i}}+{dz_{j}\over z_{j}}\big{)}$ is holomorphic on the intersection. These
three cases suffice since $\mathbb{P}\mathrm{SL}(2,\mathbb{C})$ is generated
by translations, dilations, and the inversion.

Moreover, these calculations also show that two different choices of the set
of automorphisms $\\{\tau_{i}\\}$ give equivalent projectively flat
connections, thus the connection (56) uniquely determines (up to equivalence)
the projectively flat connection. Lastly, by its construction, it is clear
that $\\{U_{i},\nabla^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{i}\\}$ is invariant under the
$\mathbb{P}\mathrm{SL}(2,\mathbb{C})$ action. ∎

Recall the covering
${P}\colon\widetilde{\mathcal{C}}_{\infty}\to\mathcal{C}_{\infty}$.

######  Corollary 2.8.

The pull back

${P}^{*}(\boldsymbol{\nabla}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}})=\\{{P}^{*}(U_{i}),{P}^{*}(\nabla^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{i})\\}$

defines a unique projectively flat connection in
${P}^{*}(\mathbb{P}(\mathcal{V}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise 1.0pt\hbox{$\scriptscriptstyle)$}})$.

###### Proof.

Follows from lemma 2.4. ∎

##  3\. Teichmüller curves and pseudo-Anosov mapping classes

In this section we recall some basic results in Teichmüller theory and
Teichmüller curves. We end this section with a discussion of our main object
of interest: the Veech curve. Let $S_{g}$ be a closed, connected and oriented
surface of genus $g\geqslant 2$. A _marked_ Riemann surface is a pair $(X,f)$
where $X$ is a smooth compact connected Riemann surface and $f\colon S_{g}\to
X$ is an orientation preserving diffeomorphism. Let

$\mathcal{T}_{g}:=(X,f)/\sim,$ (61)

where $(X,f)\sim(X^{\prime},f^{\prime})$ if there exists a biholomorphism
$g\colon X\to X^{\prime}$ such that $g\circ f$ is isotopic to $f^{\prime}$.
The group $\Gamma_{g}$ acts on $\mathcal{T}_{g}$ as follows. For any
$[\Psi]\in\Gamma_{g}$ let

$[\Psi]\cdot[(X,f)]:=[(X,f\circ\Psi^{-1})].$ (62)

The class $[(X,f\circ\Psi^{-1})]$ is independent of the choice of the
representative in $[\Psi]$ as any two isotopic markings are equivalent, this
implies that (62) gives a well defined group action of $\Gamma_{g}$ on
$\mathcal{T}_{g}$.

$\mathcal{T}_{g}$ has the structure of a complex manifold which is
homeomorphic to an open ball in $\mathbb{C}^{3g-3}$. Traditional approach to
constructing the complex structure is by Bers embedding of $\mathcal{T}_{g}$
into the complex vector space of holomorphic quadratic differentials on a
(reference) compact Riemann surface of genus $g$, see chapter 6 of [IT92] for
details. A differential geometric approach to the complex structure is given
by the _Weil-Petersson_ metric, see section 3 and theorem 3.3 in [Ham13]. In
the algebro-geometric approach one shows that for all
$[(X,f)]\in\mathcal{T}_{g}$ there exists a neighbourhood which can be
identified with the base space of a Kuranishi family of (equivalence classes
of) marked compact Riemann surfaces of genus $g$ with central fiber isomorphic
to $[(X,f)]$. By the universal property of these Kuranishi families, and the
fact that a marked Riemann surface has trivial automorphisms, these local
neighborhoods can be glued together into a single Kuranishi family, thus
giving $\mathcal{T}_{g}$ the structure of a complex manifold, see [ACG11]
chapter XV for details. In all the three topologies induced by these three
complex structures, it can be shown that $\mathcal{T}_{g}$ is an open ball in
$\mathbb{C}^{3g-3}$. From now on we consider $\mathcal{T}_{g}$ as a complex
manifold with complex structure induced from the Bers embedding.

H. L. Royden, see [Roy71], showed that the action (62) of $\Gamma_{g}$ on
$\mathcal{T}_{g}$ is by biholomorphic automorphisms and the group of
biholomorphic automorphims of $\mathcal{T}_{g}$, for genus $g\geqslant 3$, can
be identified with $\Gamma_{g}.$ The group of biholomorphic automorphims of
$\mathcal{T}_{2}$ is equivalent to $\Gamma_{2}$ modulo the hyperelliptic
involution. Moreover, the action of $\Gamma_{g}$ on $\mathcal{T}_{g}$ is
properly discontinuous but not free, see theorem 6.18 in [IT92], or
proposition 1.13 in [Ham13] for the proof. The quotient

$\mathcal{M}_{g}:=\mathcal{T}_{g}/\Gamma_{g}$ (63)

is the moduli space of compact Riemann surfaces of genus $g$. Since the action
(62) is not free, the space $\mathcal{M}_{g}$ is a complex orbifold of
dimension $3g-3$, the orbifold singularities correspond to (equivalence
classes of) Riemann surfaces which admit non-trivial automorphisms. Since
$\mathcal{T}_{g}$ is simply connected, we have the identification

$\Gamma_{g}\cong\pi_{1}^{orb}(\mathcal{M}_{g},*).$ (64)

Like any complex manifold, $\mathcal{T}_{g}$ can be equipped with the
_Kobayashi pseudo-metric_ , see [Kob98] for a comprehensive introduction, see
also page 167 of [IT92]. If the Kobayashi pseudo-metric is complete (or non-
degenerate), see page 168 of [Kob98], then the complex manifold is called
_Kobayashi hyperbolic_ , or a _hyperbolic complex space_. More classically, a
metric on $\mathcal{T}_{g}$ is introduced by using (the existence and
uniqueness of) minimal dilation quasi-conformal maps between compact Riemann
surfaces equipped with holomorphic quadratic differentials of the same
type777i.e, the number and the multiplicity of the zeroes of the quadratic
differentials match, this metric is called the _Teichmüller metric_ , see
section 5.1.3 of [IT92] for more details. H. L Royden showed that the
Kobayashi pseudo-metric is equivalent to the Teichmüller metric, see [Roy71]
or theorem 6.21 in [IT92], and since the Teichmüller metric is complete, see
theorem 5.4 in [IT92], it follows that $\mathcal{T}_{g}$ is Kobayashi
hyperbolic.

Since the Kobayashi metric is always invariant under the group of
biholomorphic automorphisms, see [Kob98] or [IT92], the complex orbifold
$\mathcal{M}_{g}$ is also Kobayashi hyperbolic. Let $V$ be a hyperbolic
Riemann surface, i.e. $V\cong\mathbb{H}/G$ where $G$ is a discrete subgroup of
$\mathbb{P}\mathrm{SL}(2,\mathbb{R})$, and let

$\phi\colon V\to\mathcal{M}_{g}$ (65)

be a generically injective holomorphic map. The pair $(V,\phi)$ is called a
_Teichmüller curve_ if $V$ has finite hyperbolic area and $\phi$ is an
isometry with respect to the Kobayashi metric on the domain888 On
$\mathbb{H}$, and thus on $V$, the Kobayashi metric is equivalent to the
Poincaré metric normalized such that its curvature is -4, see page 20 of
[Kob98] and section 1 of [McM06b]. and the codomain. In other words, a
Teichmüller curve is an irreducible algebraic curve in $\mathcal{M}_{g}$ which
is a totally geodesic sub manifold for the Kobayashi (=Teichmüller) metric.

The simplest example of a Teichmüller curve is
$(\mathcal{M}_{1},\operatorname{Id})$, where $\mathcal{M}_{1}$ is the moduli
space of compact Riemann surfaces of genus one. For $g\geqslant 2$, the
existence of Teichmüller curves already follows from W. Thurston’s
construction of pseudo-Anosov mapping classes, see section 8 of [Thu88]. From
[HS06] it follows that for any $g\geqslant 2$ there exist countably many
Teichmüller curves in $\mathcal{M}_{g}$. C. T. McMullen, see [McM03],
constructed an infinite and explicit family of (primitive)999i.e. These
Teichmüller curves don’t cover Teichmüller curves in lower genera. Teichmüller
curves in $\mathcal{M}_{2}$, he also discovered that the image of any
Teichmüller curve in $\mathcal{M}_{2}$ always lies on a Hilbert modular
surface101010Hilbert modular surfaces, see Hirzebruch’s seminal paper [Hir73]
and the book [van88] for a general introduction, live in $\mathcal{A}_{2}$,
the moduli space of abelian surfaces. Here we mean the image of the
Teichmüller curve under the Torelli map $\mathcal{M}_{2}\to\mathcal{A}_{2}$
always lies in a Hilbert modular surface.. He then extended his construction
to genus $g=3\,\text{and}\,4$ in [McM06a]. A complete classification of
(primitive) Teichmüller curves in $\mathcal{M}_{2}$ follows from the work of
[McM03], [McM05], [McM06b] and [Möl06], we refer to section 5.5 in [Möl13] for
more details. W. Veech, in [Vee89], constructed two Teichmüller curves for
each $g\geqslant 2$, the image of both of these lie in the hyperelliptic locus
of $\mathcal{M}_{g}$. In [BM10], using algebro-geometric methods a large
number of Teichmüller curves are produced, the strong point of this approach
being that the family of Riemann surfaces that these Teichmüller curves
parametrize can be described explicitly as a (complex) one parameter family of
polynomials. For further details on this approach we refer to the excellent
article [Möl13]. More recently, the authors in [MZ] have innovated a new
description of (certain) Teichmüller curves in terms of theta functions, from
their viewpoint these Teichmüller curves lead to a genuinely new type of
modular forms.

Typical construction of Teichmüller curves comes from _Veech surfaces._ If
$(X,f)$ is a marked Riemann surface, then a holomorphic quadratic differential
$q\in H^{0}(X,(\Omega^{1}_{X})^{\otimes 2})$ generates an embedding

$\widetilde{\phi}_{(X,q)}\colon\mathbb{H}\to\mathcal{T}_{g},$ (66)

which is an isometry for the Kobayashi metrics on the domain and the codomain.
The image $\widetilde{\phi}_{(X,q)}(\mathbb{H})$ is a complex geodesic for the
Kobayashi metric on $\mathcal{T}_{g}$. Such an embedding is called the
_Teichmüller disc_ generated by the pair $(X,q)$ 111111Since the marking does
not play any role in what follows, we drop it from the notation. A pair
$(X,q)$ is also called a half translation surface., see [Thu88] and [Vee89]
for the construction of such discs. Let
$\operatorname{Stab}(\widetilde{\phi}_{(X,q)})$ be the subgroup of
$\Gamma_{g}$ which preserves the image $\widetilde{\phi}_{(X,q)}(\mathbb{H})$
set wise. The embedding (66), upon passing to the quotient by the action of
$\Gamma_{g}$, induces the complex geodesic

$f\colon\mathbb{H}\to\mathcal{T}_{g}\to\mathcal{M}_{g}$ (67)

which factors through
$\mathbb{H}/\operatorname{Stab}(\widetilde{\phi}_{(X,q)})$. If
$\operatorname{Stab}(\widetilde{\phi}_{(X,q)})$ is a lattice in
$\operatorname{Aut}(\widetilde{\phi}_{(X,q)}(\mathbb{H}))(=\mathbb{P}\mathrm{SL}(2,\mathbb{R}))$,
i.e. $\mathbb{H}/\operatorname{Stab}(\widetilde{\phi}_{(X,q)})$ has finite
hyperbolic area, then the induced map, which we also denote by $f$;

$f\colon\mathbb{H}/\operatorname{Stab}(\widetilde{\phi}_{(X,q)})\to\mathcal{M}_{g}$
(68)

is proper and generically injective. It follows that the pair
$(\mathbb{H}/\operatorname{Stab}(\widetilde{\phi}_{(X,q)}),f)$ is a
Teichmüller curve. For further details on the structure of the group
$\operatorname{Stab}(\widetilde{\phi}_{(X,q)})$, several examples of groups
which arise as $\operatorname{Stab}(\widetilde{\phi}_{(X,q)})$, and open
problems related to the group $\operatorname{Stab}(\widetilde{\phi}_{(X,q)})$
we refer to the excellent survey article [Möl]. A pair $(X,q)$ such that
$\operatorname{Stab}(\widetilde{\phi}_{(X,q)})$ is a lattice is called a Veech
surface. Veech surfaces are intimately related with polygonal billiard tables
which have optimal dynamics, for dynamical aspects of Veech surfaces we refer
to [Vee89].

Assume that $(\mathbb{H}/\operatorname{Stab}(\widetilde{\phi}_{(X,q)}),f)$ is
a Teichmüller curve. Our interest in
$\operatorname{Stab}(\widetilde{\phi}_{(X,q)})$ lies in the fact that this
subgroup of $\Gamma_{g}$ contains infinitely many pseudo-Anosov mapping
classes which can be described explicitly. Denote by
$V:=\mathbb{H}/\operatorname{Stab}(\widetilde{\phi}_{(X,q)})$, and let
$\gamma$ be a closed geodesic (with respect to the hyperbolic metric) in $V$.
Since $\pi_{1}^{orb}(V)\cong\operatorname{Stab}(\widetilde{\phi}_{(X,q)})$ it
follows that the homotopy class $[\gamma]\in\pi_{1}^{orb}(V)$ corresponds to a
unique (up to conjugation) element in
$\operatorname{Stab}(\widetilde{\phi}_{(X,q)})$, denote this element by
$\varphi_{\gamma}$. Let $\widetilde{\gamma}$ be a lift of $\gamma$ such that
$\widetilde{\gamma}\subset\widetilde{\phi}_{(X,q)}(\mathbb{H})$, c.f. (66).
From the theory of hyperbolic Riemann surfaces, we know that the action of
$\varphi_{\gamma}$ on $\widetilde{\phi}_{(X,q)}(\mathbb{H})$ preserves the
geodesic $\widetilde{\gamma}$.

L. Bers, see section 6.5 of [IT92], showed that an element of $\Gamma_{g}$ is
pseudo-Anosov if and only if its action on $\mathcal{T}_{g}$ preserves a
(real) geodesic with respect to the Teichmüller, or equivalently Kobayashi,
metric. By construction, $\widetilde{\gamma}$ is a (real) geodesic with
respect to the Kobayashi metric which is preserved by $\varphi_{\gamma}$. It
follows that $\varphi_{\gamma}$ is a pseudo-Anosov element in $\Gamma_{g}$. In
this way, to any closed geodesic in $V$ we can associate a pseudo-Anosov
element in $\Gamma_{g}$. Moreover, if the generators of
$\operatorname{Stab}(\widetilde{\phi}_{(X,q)})$ are known, then any pseudo-
Anosov associated to a closed geodesic in $V$ can be written as a unique word
(up to conjugation) in these generators.

Let $n=2g+1$ and $n^{\prime}=2g+2$ with $g\geqslant 2$. For all $n$, let
$\operatorname{Stab}({\widetilde{\phi}}_{n}):=\operatorname{Stab}({\widetilde{\phi}}_{(X_{n},q_{n})})$
where $X_{n}$ is the unique compact hyperelliptic Riemann surface defined by
the polynomial $y^{2}=x^{n}+1$ and $q_{n}=c_{n}{dx^{2}\over y^{2}}$, here
$c_{n}$ is a positive constant such that $\lVert q_{n}\rVert=1$121212see, for
example, [Vee89] for the definition of this norm. Define
$\operatorname{Stab}({\widetilde{\phi}}_{n^{\prime}})$ by replacing $n$ with
$n^{\prime}$ in the definition of
$\operatorname{Stab}({\widetilde{\phi}}_{n})$. In [Vee89] it is shown that the
group $\operatorname{Stab}({\widetilde{\phi}_{n}})$ is isomorphic to the Hecke
triangle group of signature $(2,n,\infty)$, and that the group
$\operatorname{Stab}({\widetilde{\phi}}_{n^{\prime}})$ is isomorphic to the
Hecke triangle group of signature $({n^{\prime}\over 2},\infty,\infty)$.

Denote the Teichmüller curve corresponding to the lattice
$\operatorname{Stab}({\widetilde{\phi}_{n}})$ by $(\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}_{n},\phi_{n})$. An
algebraic description of the map

$\phi_{n}\colon\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}_{n}\to\mathcal{M}_{g},$ (69)

given in [BM10], [McM06b], and [Loc05], is as follows. Let
$\widetilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}_{n}:=(\mathbb{C}\mathbb{P}^{1}-\\{\mu_{n}\\})$,
where $\mu_{n}$ denotes the set of $n$’th roots of unity. Let $t$ be the
restriction of the global rational coordinate on $\mathbb{C}\mathbb{P}^{1}$ to
$\widetilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}_{n}$ and let
$D_{n}\subset\operatorname{Aut}(\widetilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}_{n})$ denote the
dihedral group generated by $t\mapsto\zeta_{n}^{2}t$ and $t\mapsto 1/t$ where
$\zeta_{n}:=\exp({2\pi\sqrt{-1}\over n})$. For all
$t\in\tilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}_{n}$ let $F_{t}$ be the unique compact
hyperelliptic Riemann surface associated with the polynomial

$\displaystyle y^{2}=\prod_{i=0}^{n}(x-\zeta_{n}^{i}-t\zeta_{n}^{-i}).$

It turns out that $\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}_{n}\cong\widetilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}_{n}/D_{n}$ and the
isomorphism class of $Y_{t}$ depends only on $[t]\in\tilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}/D$. The map (69)
is then simply given, after realizing $\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}_{n}$ as the
quotient $\widetilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}_{n}/D_{n}$, by
$\phi_{n}([t])=[F_{t}]\in\mathcal{M}_{g}$.

We restrict to the case $g=2$ and $n=5$ and drop $n$ as a subscript from the
notation. A map $\widetilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}\to{\mathcal{C}}_{\infty}$ is given by

$\tilde{\phi}(t)=(1+t,\zeta+\zeta^{-1}t,\zeta^{2}+\zeta^{-2}t,\zeta^{3}+\zeta^{-3}t,\zeta^{4}+\zeta^{-4}t,\infty).$
(70)

######  Definition 3.1.

Let $S_{n}$ be the symmetric group of degree $n$. Define $\nu,\nu^{\prime}\in
S_{5}\triangleleft S_{6}$ by
$\nu(z_{1},\dots,z_{6})=(z_{5},z_{1},z_{2},z_{3},z_{4},z_{6})$ and
$\nu^{\prime}(z_{1},\dots,z_{6})=(z_{1},z_{5},z_{4},z_{3},z_{2},z_{6}).$

It is clear from definition that $\nu^{5}=\nu^{\prime 2}=\operatorname{Id}$.
We have the following relationship between $\nu$ and $R$ and between
$\nu^{\prime}$ and $I$ see [Loc05].

######  Lemma 3.2.

$\tilde{\phi}(R(t))=\zeta\nu(\tilde{\phi}(t))$ and $\tilde{\phi}(I(t))={1\over
t}\nu^{\prime}(\tilde{\phi}(t))$. Notice that the second equality is only
defined for $t$ not equal to $0$ or $\infty$.

###### Proof.

We compute

$\displaystyle\zeta^{-1}\tilde{\phi}(R(t))$
$\displaystyle=\zeta^{-1}(1+\zeta^{2}t,\zeta+\zeta
t,\zeta^{2}+t,\zeta^{3}+\zeta^{-1}t,\zeta^{4}+\zeta^{-2}t,\infty)$
$\displaystyle=(\zeta^{-1}+\zeta
t,1+t,\zeta+\zeta^{-1}t,\zeta^{2}+\zeta^{-2}t,\zeta^{3}+\zeta^{-3}t,\infty).$

Now, the last tuple is the same as

$(\zeta^{4}+\zeta^{-4}t,1+t,\zeta+\zeta^{-1}t,\zeta^{2}+\zeta^{-2}t,\zeta^{3}+\zeta^{-3}t,\infty)=\nu(\tilde{\phi}(t)).$
(71)

By a similar calculation, we obtain that

$t\tilde{\phi}(I(t))=(1+t,\zeta^{4}+\zeta^{-4}t,\zeta^{3}+\zeta^{-3}t,\zeta^{2}+\zeta^{-2}t,\zeta+\zeta^{-1}t,\infty)=\nu^{\prime}(\tilde{\phi}(t)).$
(72)

∎

Recall that $\mathcal{M}_{2}$ is the quotient of $\mathcal{C}_{\infty}$ by the
product group $\mathbb{P}\mathrm{SL}(2,\mathbb{C})\times S_{6}$. Since
multiplication by $\zeta$ and ${1\over t}$ are both elements of the dilation
subgroup of $\mathbb{P}\mathrm{SL}(2,\mathbb{C})$ this lemma immediately shows
that the map (70) descends to give a well defined map
$\phi\colon\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}\to\mathcal{M}_{2}$ and that we get the
following commutative diagram

$\tilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}$$\mathcal{C}_{\infty}$$\chi$$\mathcal{M}_{2}$$\pi_{\tilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}}$${\phi}$$\tilde{\phi}$$\pi_{\mathcal{M}_{2}}$
(73)

###  3.1. Hitchin connection and Hyperlogarithms on the Veech curve

Consider the space

$\tilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}_{0}:=\mathbb{C}-\mu_{10}$

and the embedding

$\tilde{\psi}\colon\tilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}_{0}\to\mathcal{C}_{\infty}$ (74)

defined by

$\tilde{\psi}(t)=\bigg{(}{1\over\zeta+\zeta^{-1}t},{1\over\zeta^{2}+\zeta^{-2}t},{1\over\zeta^{3}+\zeta^{-3}t},{1\over\zeta^{4}+\zeta^{-4}t},{1\over
1+t},0\bigg{)},\quad\text{for}\,\text{all}\,t\in\tilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}_{0}.$ (75)

######  Lemma 3.3.

We have that $\pi_{\mathcal{M}_{2}}(\tilde{\psi}(\tilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}_{0}))\subset\phi(\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi})$.

###### Proof.

Notice that $\tilde{\psi}$ is given by applying the involution ${1\over
z}\in\mathbb{P}\mathrm{SL}(2,\mathbb{C})$ and a permutation (which permutes
the first and the second last coordinate) to $\tilde{\phi}$. Since
$\mathcal{M}_{2}$ is defined as the quotient of $\mathcal{C}_{\infty}$ by the
product group $\mathbb{P}\mathrm{SL}(2,\mathbb{C})\times S_{6}$, it follows
that $\pi_{\mathcal{M}_{2}}(\tilde{\phi}(\tilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}))\supset\pi_{\mathcal{M}_{2}}(\tilde{\psi}(\tilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}_{0}))$ and by the
commutativity of (73) we know that
$\pi_{\mathcal{M}_{2}}(\tilde{\phi}(\tilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}))=\phi(\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi})$. ∎

######  Proposition 3.4.

We compute that the pull back $\tilde{\psi}^{*}(\omega^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise 1.0pt\hbox{$\scriptscriptstyle)$}})$
is projectively equivalent to the $\operatorname{End}(S^{k}(F_{2}))$ valued
1-form

$\omega^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\tilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}}:={\mathchar
22\relax\mkern-9.0muk}\sum_{1\leqslant i\leqslant 5}{A_{i}\,dt\over
t-\zeta^{i}}$ (76)

where $A_{i}=\widehat{\Omega}^{a,b}+\widehat{\Omega}^{c,d}$ for $1\leqslant
a<b,c<d\leqslant 5$ such that $[a+b]=[c+d]=[i]$ and $[x]:=x\pmod{5}$.

######  Remark 3.5.

For any $1\leqslant i\leqslant 5$, there exists only one solution for $A_{i}$
with the given constraints. From the expression it is obvious that the 1-form
(76) has no poles at $-\mu_{5}$ and extends as a holomorphic 1-form to
$\tilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}$, which justifies the subscript in (76).

###### Proof.

Let $D^{0}_{6}\subset\mathcal{C}_{\infty}$ be

$D^{0}_{6}:=\\{(z_{1},\dots,z_{6})\in\mathcal{C}|z_{6}=0\\}.$ (77)

Then $\tilde{\psi}(\tilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}_{0})\subset D^{0}_{6}$ and

$\displaystyle\omega^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{|D_{6}^{0}}$ $\displaystyle={\mathchar
22\relax\mkern-9.0muk}\bigg{(}\sum_{1\leqslant i<j\leqslant
5}\widehat{\Omega}^{i,j}{dz_{i}-dz_{j}\over
z_{i}-z_{j}}+\sum_{i=1}^{5}\widehat{\Omega}^{i,6}{dz_{i}\over z_{i}}\bigg{)}$
$\displaystyle={\mathchar 22\relax\mkern-9.0muk}\bigg{(}\sum_{1\leqslant
i<j\leqslant
5}\widehat{\Omega}^{i,j}d\log(z_{i}-z_{j})+\sum_{i=1}^{5}\widehat{\Omega}^{i,6}d\log(z_{i})\bigg{)}.$

Thus

$\displaystyle\tilde{\psi}^{*}(\omega^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}})={\mathchar
22\relax\mkern-9.0muk}\bigg{(}\sum_{1\leqslant i<j\leqslant
5}\widehat{\Omega}^{i,j}d\log$
$\displaystyle\bigg{(}{1\over\zeta^{i}+\zeta^{-i}t}-{1\over\zeta^{j}+\zeta^{-j}t}\bigg{)}$
$\displaystyle-\sum_{i=1}^{5}\widehat{\Omega}^{i,6}d\log(\zeta^{i}+\zeta^{-i}t)\bigg{)}.$

Now

$\displaystyle
d\log\bigg{(}{1\over\zeta^{i}+\zeta^{-i}t}-{1\over\zeta^{j}+\zeta^{-j}t}\bigg{)}=$
$\displaystyle d\log(\zeta^{i}+\zeta^{-i}t-\zeta^{j}-\zeta^{-j}t)$
$\displaystyle-d\log(\zeta^{i}+\zeta^{-i}t)-d\log(\zeta^{j}+\zeta^{-j})$

which implies that

$\displaystyle\tilde{\psi}^{*}(\omega^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}})={\mathchar
22\relax\mkern-9.0muk}\bigg{(}\sum_{1\leqslant i<j\leqslant
5}\widehat{\Omega}^{i,j}d\log(\zeta^{i}+\zeta^{-i}t-\zeta^{j}-\zeta^{-j}t)$
$\displaystyle-\sum_{1\leqslant i<j\leqslant
5}\widehat{\Omega}^{i,j}d\log(\zeta^{i}+\zeta^{-i}t)-\sum_{1\leqslant
i<j\leqslant
5}\widehat{\Omega}^{i,j}d\log(\zeta^{j}+\zeta^{-j}t))-\sum_{i=1}^{5}\widehat{\Omega}^{i,6}d\log(\zeta^{i}+\zeta^{-i}t)\bigg{)}.$

The last three terms can be simplified by

$\sum_{1\leqslant i<j\leqslant
5}\widehat{\Omega}^{i,j}d\log(\zeta^{i}+\zeta^{-i}t)+\sum_{1\leqslant
i<j\leqslant
5}\widehat{\Omega}^{i,j}d\log(\zeta^{j}+\zeta^{-j}t))+\sum_{i=1}^{5}\widehat{\Omega}^{i,6}d\log(\zeta^{i}+\zeta^{-i}t)=$
(78) $\sum_{1\leqslant i<j\leqslant
6}\widehat{\Omega}^{i,j}d\log(\zeta^{i}+\zeta^{-i}t)+\sum_{1\leqslant
i<j\leqslant
6}\widehat{\Omega}^{i,j}d\log(\zeta^{j}-\zeta^{-j}t))=-k^{2}\operatorname{Id}\sum_{i=5}^{6}d\log(\zeta^{i}+\zeta^{-i}t)$
(79)

where the last equality follows from (3) of propsition 2.6. This gives us that

$\tilde{\psi}^{*}(\omega^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}})={\mathchar
22\relax\mkern-9.0muk}\bigg{(}\sum_{1\leqslant i<j\leqslant
5}\widehat{\Omega}^{i,j}d\log(\zeta^{i}+\zeta^{-i}t-\zeta^{j}-\zeta^{-j}t)+k^{2}\operatorname{Id}\sum_{i=5}^{6}d\log(\zeta^{i}+\zeta^{-i}t)\bigg{)}$
(80)

Now,

$\displaystyle d\log(\zeta^{i}+\zeta^{-i}t-\zeta^{j}-\zeta^{-j}t)$
$\displaystyle={(\zeta^{-i}-\zeta^{-j})dt\over(\zeta^{i}-\zeta^{j})+(\zeta^{-i}-\zeta^{-j})t}$
$\displaystyle={dt\over{\zeta^{i}-\zeta^{j}\over\zeta^{-i}-\zeta^{-j}}+t}.$

This can be simplified by

$-{\zeta^{i}-\zeta^{j}\over\zeta^{-i}-\zeta^{-j}}={\zeta^{i}-\zeta^{j}\over\zeta^{-j}-\zeta^{-i}}{.}{\zeta^{i+j}\over\zeta^{i+j}}=\zeta^{i+j},$
(81)

the result being

$d\log(\zeta^{i}+\zeta^{-i}t-\zeta^{j}-\zeta^{-j}t)={dt\over t-\zeta^{i+j}}.$
(82)

Notice that

$\zeta^{i+j}=\zeta^{k+l}$ (83)

if $i+j=k+l(mod5)$. Similarly,

$\displaystyle d\log(\zeta^{i}+\zeta^{-i}t)$
$\displaystyle={\zeta^{-i}dt\over\zeta^{i}+\zeta^{-i}t}$
$\displaystyle={dt\over t+\zeta^{2i}}.$

This implies that

$\tilde{\psi}^{*}(\omega^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}})={\mathchar
22\relax\mkern-9.0muk}\bigg{(}\sum_{1\leqslant i\leqslant 5}{A_{i}\,dt\over
t-\zeta^{i}}+k^{2}\operatorname{Id}\sum_{1\leqslant i\leqslant 5}{dt\over
t+\zeta^{2i}}\bigg{)}$ (84)

which is projectively equivalent to (76). ∎

We define a connection in the trivial bundle
$H^{0}(\mathbb{C}\mathbb{P}^{3},\mathcal{O}_{\mathbb{C}\mathbb{P}^{3}}(k))\times\tilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}$ as follows

$\boldsymbol{\nabla}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\tilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}}:=d+\omega^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\tilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}}.$ (85)

The connection (85) can be considered as a meromorphic connection on
$\mathbb{C}$ with logarithmic singularities at $\mu_{5}$. The residue at each
singularity $\zeta^{i}\in\mu_{5}$ is $A_{i}$. See [Del70] for an excellent
introduction to meromorphic connections with logarithmic singularities.

We give an explicit expression of the parallel transport of (85) along paths
in $\tilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}$ in terms of iterated integrals of 1-forms on
$\tilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}$. The 1-forms ${dt\over t-\zeta^{i}}$, where
$\zeta^{i}\in\mu_{5}$, give a basis for $H^{1}_{dR}(\tilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}},\mathbb{C})$. Let
$\gamma\colon[a,b]\to\tilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}$ be a smooth map and for each $\zeta^{i}$ let

$f_{i}(s_{i})ds_{i}:=\gamma^{*}\big{(}{dt\over t-\zeta^{i}}\big{)}.$ (86)

The ordinary line integral of this 1-form is given by

$\int_{\gamma}{dt\over t-\zeta^{i}}=\int_{a}^{b}f_{i}(s_{i})ds_{i}$ (87)

which is independent of the parametrization of $\gamma$. Now choose some
r-tuple $(\zeta^{i_{1}},\dots,\zeta^{i_{r}})\in(\mu_{5})^{r}$ and consider the
iterated integral

$L_{a}(\zeta^{i_{1}},\dots,\zeta^{i_{r}}|b):=\int_{a}^{b}f_{i_{r}}(s_{i_{r}})\bigg{(}\int_{a}^{s_{i_{r}}}f_{i_{r-1}}(s_{i_{r-1}})\dots\bigg{(}\int_{a}^{s_{i_{2}}}f_{i_{1}}(s_{i_{1}})ds_{i_{1}}\bigg{)}\dots
ds_{i_{r-1}}\bigg{)}ds_{i_{r}}$ (88)

where

$f_{i_{r}}(s_{i_{r}})ds_{i_{r}}=\gamma^{*}\big{(}{dt\over
t-\zeta^{i_{r}}}\big{)}.$

This integral is independent of the choice of the parametrization of $\gamma$
and depends on the homotopy class of $\gamma$. Such integrals, under the name
of Hyperlogarithms, were introduced in [Poi84] and extensively studied in
[LD53].

The iterated integrals $L_{0}(\zeta^{i_{1}},\dots,\zeta^{i_{r}}|1)$ appear in
the study of the motivic fundamental group of $\mathbb{C}^{*}-\mu_{n}$ (when
specialized to n=5) carried out in [DG]. These iterated integrals also appear
in the construction of the cyclotomic Drinfel’d associator carried out in
[Enr].

Let us specialize to the path $[0,1-\epsilon]\subset\tilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}$, where $\epsilon$
is an arbitrarily small positive real number. In this case, for any
$(\zeta^{i_{1}},\dots,\zeta^{i_{r}})\in(\mu_{5})^{r}$, (88) is

$L_{0}(\zeta^{i_{1}},\dots,\zeta^{i_{r}}|1-\epsilon)=\int_{0}^{1-\epsilon}{1\over
s_{i_{r}}-\zeta^{i_{r}}}\bigg{(}\int_{0}^{s_{i_{r}}}{1\over
s_{i_{r-1}}-\zeta^{{i}_{r-1}}}\dots\bigg{(}\int_{0}^{s_{i_{2}}}{1\over
s_{i_{1}}-\zeta^{{i}_{1}}}ds_{i_{1}}\bigg{)}\dots
ds_{i_{r-1}}\bigg{)}ds_{i_{r}}.$ (89)

Define $\Phi^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\epsilon}\colon[0,1-\epsilon]\to\operatorname{End}(H^{0}(\mathbb{C}\mathbb{P}^{3},\mathcal{O}_{\mathbb{C}\mathbb{P}^{3}}\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise 1.0pt\hbox{$\scriptscriptstyle)$}))$
by

$\Phi^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\epsilon}(x)=\operatorname{Id}+\sum_{r=1}^{\infty}{\mathchar
22\relax\mkern-9.0muk}^{r}\,\,\,\sum_{\overset{\zeta^{i_{1}},\dots,\zeta^{i_{r}}\in(\mu_{5})^{r}}{}}L_{0}(\zeta^{i_{1}},\dots,\zeta^{i_{r}}|x)A_{i_{1}}\dots
A_{i_{r}}\quad\quad\quad x\in[0,1-\epsilon],$

where we consider $L_{0}(\zeta^{i_{1}},\dots,\zeta^{i_{r}}|x)$ as a function
in one variable: the upper limit of integration. One easily proves that, see
[Kas95] page 481 or [LD53] page 164, that $\Phi^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}(x)$ is convergent for all
$x\in[0,1-\epsilon]$ and

${d\,\Phi^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\epsilon}\over dx}={\mathchar
22\relax\mkern-9.0muk}\sum_{1\leqslant i\leqslant 5}{\Phi^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\epsilon}\,\,A_{i}\over x-\zeta^{i}}.$
(90)

Since the connection (85) is given by the 1-form (76), (90) implies that
$\Phi^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\epsilon}$ gives the parallel transport of
the connection (85) along the path $[0,1-\epsilon]$.

The integral (89) converges as $\epsilon\to 0$ if and only if
$\zeta^{i_{r}}\neq 1$, thus the integral
$L_{0}(\zeta^{i_{1}},\dots,\zeta^{i_{r}}|1)$ is well defined if and only if
$\zeta^{i_{r}}\neq 1$. Let

$\Phi^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}:=\operatorname{Id}+\sum_{r=1}^{\infty}{\mathchar
22\relax\mkern-9.0muk}^{r}\,\,\,\sum_{\overset{\zeta^{i_{1}},\dots,\zeta^{i_{r}}\in(\mu_{5})^{r}}{\
\zeta^{i_{r}}\neq 1}}L_{0}(\zeta^{i_{1}},\dots,\zeta^{i_{r}}|1)A_{i_{1}}\dots
A_{i_{r}}\,\,\,\,\in\operatorname{End}(H^{0}(\mathbb{C}\mathbb{P}^{3},\mathcal{O}_{\mathbb{C}\mathbb{P}^{3}}\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise 1.0pt\hbox{$\scriptscriptstyle)$})).$
(91)

The endomorphism (91) will play a crucial role in the monodromy computation
carried out in the next section.

We now give a result on the asymptotic behavior of the parallel transport of
(85) in arbitrarily small neighborhoods of the singularities $\mu_{5}$. Let
$D^{*}_{i}:=\\{b_{i}\in\mathbb{C}|0<|b_{i}|<\epsilon\\}$, and consider the
embedding

$B_{i}\colon D^{*}_{i}\to\tilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}},\quad\quad\quad b_{i}\mapsto\zeta^{i}+b_{i}.$
(92)

An easy calculation shows

$B^{*}_{i}(\omega^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\tilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}})=\mathchar
22\relax\mkern-9.0muk\bigg{(}{A_{i}\over b_{i}}+\sum_{\overset{1\leqslant
j\leqslant 5}{\ i\neq j}}{A_{j}\,\,\,\over
b_{i}-(\zeta^{j}-\zeta^{i})}\bigg{)}db_{i}.$ (93)

Let $Y_{i}(b_{i})$ be an
$\operatorname{End}(H^{0}(\mathbb{C}\mathbb{P}^{3},\mathcal{O}_{\mathbb{C}\mathbb{P}^{3}}(k)))$
valued function over $D^{*}_{i}$. Then $Y_{i}$ is a solution of (93) if

$Y_{i}^{\prime}=\mathchar 22\relax\mkern-9.0muk\bigg{(}{A_{i}\over
b_{i}}+\sum_{\overset{1\leqslant j\leqslant 5}{\ i\neq j}}{A_{j}\,\,\,\over
b_{i}-(\zeta^{j}-\zeta^{i})}\bigg{)}Y_{i}.$ (94)

######  Proposition 3.6.

For all $1\leqslant i\leqslant 5$, there exists a unique $Y_{i}$ satisfying
(94) such that

$Y_{i}(b_{i})=Q_{i}(b_{i})\,b_{i}^{\mathchar 22\relax\mkern-9.0mukA_{i}}$ (95)

where $Q_{i}(b_{i})=\sum_{r\geqslant 0}q^{\raise
1.0pt\hbox{$\scriptscriptstyle($}i\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{r}b_{i}^{r}$, with $q_{r}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}i\raise
1.0pt\hbox{$\scriptscriptstyle)$}}\in\operatorname{End}(H^{0}(\mathbb{C}\mathbb{P}^{3},\mathcal{O}_{\mathbb{C}\mathbb{P}^{3}}(k)))$,
and $Q_{i}(0)=q_{0}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}i\raise
1.0pt\hbox{$\scriptscriptstyle)$}}=\operatorname{Id}$.

######  Remark 3.7.

For the expression $b_{i}^{\mathchar 22\relax\mkern-9.0mukA_{i}}$ to make
sense on $D^{*}_{i}$, we must choose a branch cut for the logarithm. We make
the choice of the positive real axis in $D^{*}_{i}$ as the branch cut for the
logarithm.

###### Proof.

By the theory of ordinary differential equations, see [Was87], there exists a
solution $Y_{i}$ satisfying (94). We show that there exits a family
$q_{r}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}i\raise
1.0pt\hbox{$\scriptscriptstyle)$}}$ satisfying the above stated requirements
such that $Y_{i}$ is of the form (95). We write out

$Y_{i}^{\prime}(b_{i})=\bigg{(}Q_{i}^{\prime}(b_{i})+\mathchar
22\relax\mkern-9.0muk{Q_{i}(b_{i})A_{i}\over b_{i}}\bigg{)}b_{i}^{\mathchar
22\relax\mkern-9.0mukA_{i}}=\mathchar 22\relax\mkern-9.0muk\bigg{(}{A_{i}\over
b_{i}}+\sum_{\overset{1\leqslant j\leqslant 5}{\ i\neq j}}{A_{j}\,\,\,\over
b_{i}-(\zeta^{j}-\zeta^{i})}\bigg{)}Q_{i}(b_{i})\,b_{i}^{\mathchar
22\relax\mkern-9.0mukA_{i}}.$ (96)

This can be rewritten as

$b_{i}Q_{i}^{\prime}(b_{i})-\mathchar
22\relax\mkern-9.0muk[A_{i},Q_{i}(b_{i})]=-\mathchar
22\relax\mkern-9.0muk\,\,\sum_{\overset{1\leqslant j\leqslant 5}{\ i\neq
j}}A_{j}{b_{i}\,Q_{i}(b_{i})\over(\zeta^{j}-\zeta^{i})-b_{i}}.$ (97)

Expanding the above in powers of $b_{i}$, we get $[A_{i},q_{0}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}i\raise
1.0pt\hbox{$\scriptscriptstyle)$}}]=0$, and for $r>0$

$rq^{\raise 1.0pt\hbox{$\scriptscriptstyle($}i\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{r}-\mathchar
22\relax\mkern-9.0muk[A_{i},q^{\raise 1.0pt\hbox{$\scriptscriptstyle($}i\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{r}]=-\mathchar
22\relax\mkern-9.0muk\sum_{l=1}^{r}\bigg{(}\sum_{\overset{1\leqslant
j\leqslant 5}{\ i\neq
j}}{A_{j}\over(\zeta^{j}-\zeta^{i})^{l}}\bigg{)}q^{\raise
1.0pt\hbox{$\scriptscriptstyle($}i\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{l-1}$ (98)

The above equations, for every $r$, have a solution. Indeed, if we take
$q^{\raise 1.0pt\hbox{$\scriptscriptstyle($}i\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{0}=\operatorname{Id}$, then $q^{\raise
1.0pt\hbox{$\scriptscriptstyle($}i\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{r}$ is uniquely determined by $q^{\raise
1.0pt\hbox{$\scriptscriptstyle($}i\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{0},\dots,q^{\raise
1.0pt\hbox{$\scriptscriptstyle($}i\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{r-1}$ due to the fact that the operator
$r\operatorname{Id}-\mathchar 22\relax\mkern-9.0muk\,\operatorname{ad}(A_{i})$
is invertible with inverse equal to

${1\over r}\sum_{n\geqslant 0}{\mathchar 22\relax\mkern-9.0muk^{n}\over
r^{n}}\operatorname{ad}(A_{i})^{n}.$ (99)

The convergence of $Q_{i}(b_{i})$ results from the general fact that a formal
solution of a regular singular equation is necessarily convergent, see [Kas95]
and [Was87]. ∎

##  4\. Generators of the (orbifold) fundamental group

The curve $\chi$ is defined as the quotient $\tilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}_{\infty}/D$, where
$D$ was generated by the automorphisms $R,I\colon\tilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}_{\infty}\to\tilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}_{\infty}$. The
rotation $R$ is of order five and has two fixed points $0$ and $\infty$, the
involution $I$ has the fixed point $-1$, it follows that $\chi$ is
topologically a sphere with two orbifold points
$a:=\pi_{\tilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}}(0)$, $b:=\pi_{\tilde{\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}}}(-1)$ and one
puncture. The stabilizer group of the orbifold point $a$ is $\langle
R\rangle$, and the stabilizer group of the orbifold point $b$ is $I$.

We choose explicit paths based at the order five orbifold point of $\chi$ such
that they generate the group $\pi_{1}^{orb}(\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi},a)$. Let
$\tilde{\gamma}_{0}\subset\tilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}$ be the path starting from $0$ and running
along the real axis to $1-\epsilon$, where $\epsilon$ is an arbitrarily small
positive real number. Let $\tilde{\gamma}_{1}$ be the semi-circle starting
from $1-\epsilon$ moving around $1$ in an anti clockwise direction until it
reaches $(1-\epsilon)^{-1}$. We will also need $\tilde{\gamma}_{0}^{-1}$ which
is the path running from $1-\epsilon$ to $0$ along the real axis.

The end points of the semi-circle $\tilde{\gamma}_{1}$ are identified with
each other under the involution $I$, which implies that the projection

$\pi_{\tilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}}(\tilde{\gamma}_{1})=:\gamma_{1}\subset\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}$

is a homotopically non-trivial closed loop. Consider the projection of the
path $\tilde{\gamma}_{0}$,

$\pi_{\tilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}}(\tilde{\gamma}_{0})=:\gamma_{0}\subset\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}$

then the composition

$\gamma_{0}^{-1}\cdot\gamma_{1}\cdot\gamma_{0}=:\gamma\subset\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}$ (100)

is a closed, connected, homotopically non-trivial loop with the property that
it starts and ends at the order five orbifold point $a\in\mathchoice{\raise
1.5pt\hbox{$\chi$}}{\raise 1.5pt\hbox{$\chi$}}{\chi}{\chi}$.

The following is part of proposition 4.11 from [Loc05].

######  Proposition 4.1.

The loop $\gamma$ and the orbifold stabilizer group $\langle R\rangle$
together generate $\pi_{1}^{orb}(\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi},a)$, and under the isomorphism
$\pi_{1}^{orb}(\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi},a)\cong\triangle(2,5,\infty)$ R is identified
with $U:=S\circ T$ and $\gamma$ is identified with $T$.

Let

$H:\pi_{1}^{orb}(\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi},a)\to
H_{1}(\phi(a),\mathbb{Z})\cong\mathrm{Sp}(4,\mathbb{Z})$

be the monodromy representation provided by the Gauss-Manin connection in the
bundle of the first de Rahm cohomology groups of the genus two surfaces
parametrized by $\chi$.

######  Lemma 4.2.

The elements $\langle R\rangle$ and $\gamma$ map to $M_{0}$ and $M_{1}$
respectively where

$M_{0}:=\begin{bmatrix}0&0&-1&1\\\ 0&0&0&-1\\\ 1&0&-1&0\\\
1&1&-1&0\end{bmatrix}\quad M_{1}:=\begin{bmatrix}0&1&0&0\\\ 1&0&0&0\\\
1&1&0&-1\\\ 1&1&-1&0\end{bmatrix}.$ (101)

under the homomorphism $H$.

###### Proof.

Let $X_{\sqrt{5}}$ be the Hilbert modular surface for totally real quadratic
field $\mathbb{Q}(\sqrt{5})$ considered in [MZ]. The surface $X_{\sqrt{5}}$ is
the moduli space of abelian surfaces which admit real multiplication by the
field $\mathbb{Q}(\sqrt{5})$. In [MZ] the image of Gauss-Mannin connection
$H^{\prime}:\pi^{orb}_{1}(X_{\sqrt{5}},*)\to
H_{1}(A_{*},\mathbb{Z})\cong\mathrm{Sp}(4,\mathbb{Z})$ is described
explicitly. In [McM06b] it is shown that the Teichmüller curve
$\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}\hookrightarrow X_{\sqrt{5}}$ embeds in the
hilbert modular surface; In particular an embedding
$\pi_{1}^{orb}(\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi},a)\hookrightarrow\pi_{1}^{orb}(X_{\sqrt{5}},*)$
is provided. We simply look at the image of $<R>$ and $\gamma$ under the
composition of this embedding with $H^{\prime}$. ∎

Let us choose $\phi(a)\in\mathcal{M}_{2}$ as the base point for the orbifold
fundamental group of $\mathcal{M}_{2}$. Then
$\phi(\gamma)\in\pi_{1}^{orb}(\mathcal{M}_{2},\phi(a))$. In fact, see [Vee89]
or [Loc05], the loop $\phi(\gamma)$ corresponds to a Dehn twist along two non-
separating simple closed loops on a genus two surface, and $R$ corresponds to
an order five diffeomorphism. For each integer $k>0$ we have the projective
vector bundle with the flat connection. The monodromy of this flat connection
provides us with a group homomorphism

$\rho^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}\colon\pi_{1}^{orb}(\mathcal{M}_{2},\phi(a))\to\operatorname{End}(\mathbb{P}\overline{\mathbb{V}}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\phi(a)})$ (102)

where the fiber $\operatorname{End}(\mathbb{P}\overline{\mathbb{V}}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\phi(a)})$ can be identified with
$\operatorname{End}(\mathbb{P}H^{0}(\mathbb{C}\mathbb{P}^{3},\mathcal{O}_{\mathbb{C}\mathbb{P}^{3}}(k)))$.
In the next section, we compute the image of $\phi(\gamma)$ and $R$ under the
morphism (102), and prove theorem 1.1.

###  4.1. Computing Monodromy

In order to compute the image of $\phi(\gamma)$ and $R$ under (102) we need
two connected paths, $\tilde{\gamma}$ and $p_{R}$, based at some point
$\bf{z}\in\mathcal{C}$ such that $\pi_{\mathcal{M}_{2}}($$\bf{z}$$)=\phi(a)$
and $\pi_{\mathcal{M}_{2}}(\tilde{\gamma})=\phi(\gamma)$,
$\pi_{\mathcal{M}_{2}}(p_{R})=\phi(a)$.

We have the following well known isomorphism, see [Bir75] page 188,

$\pi_{1}^{orb}(\mathcal{M}_{2},*)/(\mathbb{Z}/2)\cong SB_{6}$ (103)

where $\mathbb{Z}/2=\langle\sigma\rangle$ and $\sigma$ is the hyperelliptic
involution.

Consider the following map

$p^{\prime}_{I}\colon[1,(1-\epsilon)^{-1}]\to\mathbb{P}\mathrm{SL}(2,\mathbb{C})\quad
s\mapsto\begin{bmatrix}s^{1\over 2}&0\\\ 0&s^{-{1\over 2}}\end{bmatrix}.$
(104)

Let $p_{I}\subset\mathcal{C}$ be the path obtained by applying the family of
Möbius transformations $p^{\prime}_{I}$ to the point
$\tilde{\psi}(1-\epsilon)\in\mathcal{C}$.

######  Proposition 4.3.

The following

$\tilde{\gamma}:=\nu^{\prime}(\tilde{\psi}((\tilde{\gamma}_{0})^{-1})))\cdot
p_{I}\cdot\tilde{\psi}(\tilde{\gamma}_{1})\cdot\tilde{\psi}(\tilde{\gamma}_{0})\subset\mathcal{C}$
(105)

is a connected path with initial point $\tilde{\psi}(0)$, moreover
$\pi_{\mathcal{M}_{2}}(\tilde{\gamma})=\phi(\gamma)$.

###### Proof.

Let us first analyze the image $\tilde{\psi}(\tilde{\gamma}_{0})$. The initial
point of the path $\tilde{\psi}(\tilde{\gamma}_{0})$ is $\tilde{\psi}(0)$
which is shown in Figure 1. Here the six points form a symmetric configuration
by taking the positions at the fifth roots of unity and zero in the specified
order.

Figure 1. The configuration at $\tilde{\psi}(0)$. The crosses represent the
positions of the points and $t_{i}$ are the generators of the braid group

In Figure 2 the dashed lines show the trajectory of the six points as one
travels along $\tilde{\psi}(\tilde{\gamma}_{0})$. At the end point of
$\tilde{\psi}(\tilde{\gamma}_{0})$, the points $z_{1}$ and $z_{4}$ are a small
distance (depends on $\epsilon$) away from the positive real number ${1\over
2\mathrm{Re}(\zeta)}$ and a quick calculation shows that at the end points
$z_{1}$ and $z_{4}$ are conjugate to each other, and both have their real
parts bigger than ${1\over 2\mathrm{Re}(\zeta)}$. Moreover, $z_{1}$ lies in
lower half plane of $\mathbb{C}$ and $z_{4}$ lies in the upper half plane.
This local picture at ${1\over 2\mathrm{Re}(\zeta)}$ is shown in Figure 3. A
similar story holds for the pair $z_{2}$ and $z_{3}$ around the negative real
number ${1\over 2\mathrm{Re}(\zeta^{2})}$. The point $z_{5}$ moves to ${1\over
2}$ and $z_{6}$ remains fixed at zero. Notice also that the end point of
$\tilde{\psi}(\tilde{\gamma}_{0})$ is the initial point of
$\tilde{\psi}(\tilde{\gamma}_{1})$.

Figure 2.

We now look at the configuration at the end point of
$\tilde{\psi}(\tilde{\gamma}_{1})$. Recall that $\tilde{\gamma}_{1}$ was a
semi-circle traversed anti-clockwise around one from $(1-\epsilon)$ to
$(1-\epsilon)^{-1}$. At the end point of $\tilde{\psi}(\tilde{\gamma}_{1})$,
the points $z_{1}$ and $z_{4}$ are still conjugate to each other but both have
real parts smaller than ${1\over 2\mathrm{Re}(\zeta)}$ and $z_{1}$ lies in the
upper half plane and $z_{4}$ lies in the lower half plane. In effect, both
$z_{1}$ and $z_{4}$ move in an anti-clockwise semi-circle around ${1\over
2\mathrm{Re}(\zeta)}$. This is also shown in figure 3. A similar story holds
for the pair $z_{2}$ and $z_{3}$ around the negative real number ${1\over
2\mathrm{Re}(\zeta^{2})}$. The point $z_{5}$ moves to ${1\over 2}$ and $z_{6}$
remains fixed at zero. Notice also that the end point of
$\tilde{\psi}(\tilde{\gamma}_{1})$ is not the initial point of
$\tilde{\psi}(\tilde{\gamma}_{0})^{-1}$. However, initial point of $p_{I}$ is
the end point of $\tilde{\psi}(\tilde{\gamma}_{1})$ and it follows from

$\tilde{\psi}(1-\epsilon)=\nu^{\prime}(1-\epsilon)^{-1}(\tilde{\psi}(1-\epsilon)^{-1})$

that the configuration at the end point of $p_{I}$ differs from the
configuration at the initial point of $\tilde{\psi}(\tilde{\gamma}_{0})^{-1}$
only by the permutation $\nu^{\prime}$.

The trajectory of $z_{4}$ and $z_{1}$ along the path $p_{I}$ is shown in red
in figure 3. A similar story holds for $z_{2}$ and $z_{3}$. The point $z_{5}$
moves a little to the right and $z_{6}$ is fixed at zero.

Figure 3.

Since the end point of $p_{I}$ differs from the initial point of
$(\tilde{\psi}(\tilde{\gamma}_{0})^{-1})$ by the permutation $\nu^{\prime}$,
it follows that $\nu^{\prime}(\tilde{\psi}(\tilde{\gamma}_{0})^{-1}))\cdot
p_{I}$ is a connected path with initial point
$\tilde{\psi}((1-\epsilon)^{-1})$ and end point
$\nu^{\prime}(\tilde{\psi}(0)$.

Since the initial point of
$\nu^{\prime}(\tilde{\psi}(\tilde{\gamma}_{0})^{-1}))\cdot p_{I}$ is the end
point of
$\tilde{\psi}(\tilde{\gamma}_{1})\cdot\tilde{\psi}(\tilde{\gamma}_{0})$ it
follows that $\tilde{\gamma}$ is a connected path in $\mathcal{C}$ with
initial point $\tilde{\psi}(0)$ and end point $\nu^{\prime}(\tilde{\psi}(0))$.

Since $p_{I}$ is entirely contained in the
$\mathbb{P}\mathrm{SL}(2,\mathbb{C})$ orbit, it follows that
$\pi_{\mathcal{M}_{2}}(p_{I})$ is a constant loop in $\mathcal{M}_{2}$. This
fact combined with lemma 3.3 implies

$\pi_{\mathcal{M}_{2}}(\tilde{\gamma})=\phi(\gamma).$ (106)

∎

Consider the following path in $\mathbb{P}\mathrm{SL}(2,\mathbb{C})$

$p_{R}\colon[0,{1\over 5}]\to\mathbb{P}\mathrm{SL}(2,\mathbb{C}),\quad\quad
s\mapsto\begin{bmatrix}e^{-\pi is}&0\\\ 0&e^{\pi
is}\end{bmatrix}\,\,\,s\in[0,{1\over 5}].$ (107)

Denote also by $p_{R}$ the path in $\mathcal{C}$ given by the action of
$p_{R}$ on the point $\psi(0)$.

######  Lemma 4.4.

The path $p_{R}\subset\mathcal{C}_{\infty}$ is connected with initial point
$\widetilde{\psi}(0)$, moreover $\pi_{\mathcal{M}_{2}}(p_{R})=\phi(a)$.

###### Proof.

At both the initial and the end point of this path the first five points sit
at the fifth roots of unity (sixth at zero) but with different ordering. The
ordering differs by $\nu^{-1}$. Since the end points of $p_{R}$ are related by
an element in $S_{6}$, $P_{\infty}(p_{R})\subset\overline{\mathcal{C}}$ is a
closed loop. Moreover, since $p_{R}$ is entirely contained in the
$\mathbb{P}\mathrm{SL}(2,\mathbb{C})$ orbit, $(\overline{P}_{\infty}\cdot
P_{\infty})(p_{R})$ is a constant path in $\mathcal{M}_{2}$, namely $\phi(a)$.
∎

Figure 4. The loop $\overline{\gamma}$ illustrated as the braid
$t_{4}t_{2}t_{3}t_{4}t_{2}t_{5}t_{5}t_{1}t_{1}t_{3}$ from top to bottom

We now prove theorem 1.1 from the introduction.

###### Proof.

(Theorem 1.1)

We compute the parallel transport of $\boldsymbol{\nabla}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\mathcal{C}}$ along $\tilde{\gamma}$ and
$p_{R}$. Recall from (105)

$\tilde{\gamma}:=\nu^{\prime}(\psi((\tilde{\gamma}_{0})^{-1})))\cdot
p_{I}\cdot\psi(\tilde{\gamma}_{1})\cdot\psi(\tilde{\gamma}_{0})\subset\mathcal{C}$
(108)

and $p_{R}$ is given in lemma 4.4.

We first observe the following

######  Lemma 4.5.

The parallel transport of $\boldsymbol{\nabla}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\mathcal{C}}$ along the paths $p_{I}$ and
$p_{R}$ is projectively trivial.

###### Proof.

Recall that $\boldsymbol{\nabla}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\mathcal{C}}:=d+\omega^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise 1.0pt\hbox{$\scriptscriptstyle)$}}$
and $\omega^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}$ is projectively invariant under the
$\mathbb{P}\mathrm{SL}(2,\mathbb{C})$ on $\mathcal{C}_{\infty}$. Lemma follows
since both $p_{R}$ and $p_{I}$ are contained in the orbit of the dilation
subgroup of $\mathbb{P}\mathrm{SL}(2,\mathbb{C})$. ∎

In order to compute the parallel transport along $\tilde{\gamma}$ we must
compute the parallel transport along all the paths separately appearing in the
expression above. We compute the parallel transport along
$\widetilde{\psi}(\tilde{\gamma}_{1})$. The semi circle
$\tilde{\gamma_{1}}\subset\tilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}$ is contained in the image of $B_{1}$,
defined in (92), which is a punctured disk of radius $\epsilon$ based at
$1\in\tilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}$.

######  Proposition 4.6.

The parallel transport along $(\widetilde{\psi}\circ
B_{1})^{*}(\psi(\tilde{\gamma}_{1}))$ with respect to $(\widetilde{\psi}\circ
B_{1})^{*}\boldsymbol{\nabla}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}$ is

$Q_{1}(-i\epsilon)\cdot\exp\big{(}-\pi i\mathchar
22\relax\mkern-9.0mukA_{1}\big{)}\cdot Q_{1}(i\epsilon),$ (109)

where $Q_{1}(b_{1})$ is the power series from proposition 3.6.

######  Remark 4.7.

We assume the $\epsilon$ in the definition of $B_{1}$ to be bigger than the
$\epsilon$ in the definition of $\tilde{\gamma}_{1}$. The $\epsilon$ appearing
in the proposition refers to the $\epsilon$ for $\tilde{\gamma}_{1}$.

###### Proof.

From proposition 3.6 we know that the parallel transport of
$(\widetilde{\psi}\circ B_{1})^{*}\boldsymbol{\nabla}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise 1.0pt\hbox{$\scriptscriptstyle)$}}$
is given by a solution of the form $Y_{1}(b_{1})$ shown in (95). The pull back
$(\widetilde{\psi}\circ B_{1})^{*}(\psi(\tilde{\gamma}_{1}))$ is a semi circle
in $D_{1}$ traveling in an anti-clockwise direction from $i\epsilon$ to
$-i\epsilon$, avoiding the positive real axis in $D_{1}$ (which was our choice
for the branch cut of the logarithm on $D_{1}$). Thus the parallel transport
is given by

$\displaystyle Y_{1}(-i\epsilon)(Y_{1}(i\epsilon))^{-1}$
$\displaystyle=Q_{1}(-i\epsilon)(-i\epsilon)^{\mathchar
22\relax\mkern-9.0mukA_{1}}(i\epsilon^{-\mathchar
22\relax\mkern-9.0mukA_{1}})Q_{1}(i\epsilon)^{-1}$
$\displaystyle=Q_{1}(-i\epsilon)\exp(log(-i\epsilon)\mathchar
22\relax\mkern-9.0mukA_{1})\exp(-log(i\epsilon)\mathchar
22\relax\mkern-9.0mukA_{1})Q_{1}(i\epsilon)^{-1}$
$\displaystyle=Q_{1}(-i\epsilon)\exp((log(i\epsilon)+\pi i)\mathchar
22\relax\mkern-9.0mukA_{1})\exp(-log(i\epsilon)\mathchar
22\relax\mkern-9.0mukA_{1})Q_{1}(i\epsilon)^{-1}$
$\displaystyle=Q_{1}(-i\epsilon)\exp(\pi i\mathchar
22\relax\mkern-9.0mukA_{1})Q_{1}(i\epsilon)^{-1},$

where the first equality is from (95) and third equality follows from our
choice of the branch cut for the logarithm. ∎

For the parallel transport of $\boldsymbol{\nabla}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\mathcal{C}}$ along
$\psi(\tilde{\gamma}_{0})$, where recall that
$\tilde{\gamma}_{0}\subset\tilde{\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi}}$ is the interval $[0,1-\epsilon]$, we take
the pull back $\psi^{*}(\boldsymbol{\nabla}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise 1.0pt\hbox{$\scriptscriptstyle)$}})$,
which is given in (76), restrict it to the interval $[0,1-\epsilon]$, and
compute the monodromy there. This monodromy was computed in section 3.1 and is
given by $\Phi^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{\epsilon}(1-\epsilon)$ c.f. (89).

Now, it follows that under the map (102) we have that

$\rho^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}(R)=M^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{0}$ (110)

since parallel transport along $p_{R}$ is projectively trivial. Here
$M_{0}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}$ denotes the $k$-th symmetric power of the
endomorphism $M_{0}$ from lemma 4.2. Likewise we have that

$\displaystyle\rho^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}(\phi(\gamma))$ $\displaystyle=$
$\displaystyle M^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{1}\cdot\nu^{\prime}(\Phi_{\epsilon}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}(1-\epsilon))^{-1}\cdot
Q_{1}(-i\epsilon)\cdot exp\big{(}-\pi i\mathchar
22\relax\mkern-9.0mukA_{1}\big{)}\cdot(Q_{1}(i\epsilon))^{-1}\cdot\Phi_{\epsilon}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}(1-\epsilon)$ $\displaystyle=$
$\displaystyle(\Phi_{\epsilon}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}(1-\epsilon))^{-1}\cdot M_{1}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}\bigg{(}Q_{1}(-i\epsilon)\cdot
exp\big{(}-\pi i\mathchar 22\relax\mkern-9.0mukA_{1}\big{)}\cdot
Q_{1}(i\epsilon)^{-1}\bigg{)}\cdot\Phi_{\epsilon}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}(1-\epsilon)$

where we used the fact that $M_{1}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise 1.0pt\hbox{$\scriptscriptstyle)$}}$
commutes with the parallel transport, see [vGdJ98]. This commutation also had
the effect of removing $\nu^{\prime}$ from
$\nu^{\prime}(\Phi_{\epsilon}^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}(1-\epsilon))^{-1})$ because $M_{1}$ maps to
$\nu^{\prime}$ under the map $A(G)\to\mathcal{S}_{6}$.

In [Enr], see also [Kas95] and [LD53], it is shown that

$\displaystyle\lim_{\epsilon\to 0}$
$\displaystyle\Big{(}(\Phi_{\epsilon}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}(1-\epsilon))^{-1}\cdot M_{1}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}\bigg{(}Q_{1}(-i\epsilon)\cdot
exp\big{(}-\pi i\mathchar 22\relax\mkern-9.0mukA_{1}\big{)}\cdot
Q_{1}(i\epsilon)^{-1}\bigg{)}\cdot\Phi_{\epsilon}^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}(1-\epsilon)\Big{)}$ $\displaystyle=$
$\displaystyle(\Phi^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}})^{-1}\cdot\bigg{(}(M^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}_{1})^{-1}\cdot exp\big{(}-\pi i\mathchar
22\relax\mkern-9.0mukA_{i}\big{)}\bigg{)}\cdot\Phi^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise 1.0pt\hbox{$\scriptscriptstyle)$}}$
$\displaystyle=$ $\displaystyle\rho^{\raise
1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}(\phi(\gamma))$

where $\Phi^{\raise 1.0pt\hbox{$\scriptscriptstyle($}k\raise
1.0pt\hbox{$\scriptscriptstyle)$}}$ is defined in (91).

Replacing $R$ with $U$ and $\phi(\gamma)$ with $T$ under the isomorphism
$\pi_{1}^{orb}(\mathchoice{\raise 1.5pt\hbox{$\chi$}}{\raise
1.5pt\hbox{$\chi$}}{\chi}{\chi},a)\cong\triangle(2,5,\infty)$ we get the
statement of theorem 1.1. ∎

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