# _New formula for the prepotentials associated with Hurwitz-Frobenius
manifolds and generalized WDVV equations_

Chaabane REJEB111Université de Sherbrooke, CANADA. Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider the Hurwitz spaces of ramified coverings of $\mathbb{P}^{1}$ with
prescribed ramification profile over the point at infinity. By means of a
particular symmetric bidifferential on a compact Riemann surface, we introduce
quasi-homogeneous differentials. By following Dubrovin, we construct on
Hurwitz spaces a family of Frobenius manifold structures associated with the
quasi-homogeneous differentials. We explicitly derive new generating formulas
for the corresponding prepotentials. This produces quasi-homogeneous solutions
to the following generalized WDVV associativity equations:
$F_{i}\eta^{-1}F_{j}=F_{j}\eta^{-1}F_{i}$, where the invertible constant
matrix $\eta$ is a linear combination of the matrices $F_{j}$. In particular,
our approach provides another look at Dubrovin’s construction of semi-simple
Hurwitz-Frobenius manifolds and establishes an alternative practical method to
calculate their primary free energy functions. As applications, we use our
formalism to obtain various explicit quasi-homogeneous solutions to the WDVV
equations in genus zero and one and give a new proof of Ramanujan’s
differential equations for Eisenstein series.

Key words: WDVV equations, Frobenius manifold, Hurwitz space, Prepotential,
Riemann surface, Ramified covering, Quasi-homogeneous solution.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Frobenius manifolds
    2. 2.2 Hurwitz spaces
    3. 2.3 Symmetric bidifferentials on compact Riemann surfaces
  3. 3 Flat metrics and flat coordinates in terms of the bidifferential on simple and double Hurwitz spaces
    1. 3.1 An auxiliary result
    2. 3.2 Characterization of flat functions
    3. 3.3 Flat coordinates of Dubrovin’s flat metrics and duality relations
    4. 3.4 Flat coordinates of intersection forms
  4. 4 Hurwitz-Frobenius manifold structures
    1. 4.1 Frobenius manifold structures on simple Hurwitz spaces
      1. 4.1.1 Frobenius algebra structures
      2. 4.1.2 Primary differentials
      3. 4.1.3 Euler vector field and quasi-homogeneous differentials
      4. 4.1.4 A new formula for the prepotential
      5. 4.1.5 Prepotentials associated with holomorphic differentials: examples in high dimension
    2. 4.2 Almost Frobenius manifold structures on double Hurwitz spaces
      1. 4.2.1 Frobenius algebras associated with intersection forms
      2. 4.2.2 Prepotentials of almost Frobenius manifolds
  5. 5 Prepotentials of deformed Hurwitz-Frobenius manifold structures
    1. 5.1 Deformations of Frobenius manifold structures on simple Hurwitz spaces
    2. 5.2 Prepotentials of the $\mathbf{q}$-deformed Frobenius manifold structures
  6. 6 Examples of solutions to the WDVV equations in genus zero
  7. 7 Examples of genus one solutions to the WDVV equations
    1. 7.1 Weierstrass functions
    2. 7.2 Examples of 3-dimensional Hurwitz-Frobenius manifolds and Ramanujan identities
      1. 7.2.1 Prepotential associated with the Chazy equation
      2. 7.2.2 Ramanujan identities
      3. 7.2.3 Further examples in dimension 3
    3. 7.3 Examples of 3-dimensional deformed Hurwitz-Frobenius manifolds
  8. 8 Appendix
    1. 8.1 Appendix 1: Diagonal flat metrics
    2. 8.2 Appendix 2: Multivalued differentials and quasi-homogeneity properties
    3. 8.3 Appendix 3: Dubrovin’s bilinear pairing and prepotentials

##  1 Introduction

The Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) associativity equations are the
following system of nonlinear partial differential equations for a function
$F(t)=F(t^{1},\dots,t^{N})$:

$F_{\alpha}F_{1}^{-1}F_{\beta}=F_{\beta}F_{1}^{-1}F_{\alpha},\quad\quad\alpha,\beta=1,\dots,N,$
(1.1)

where for each $\alpha$, $F_{\alpha}$ denotes the matrix
$\big{(}F_{\alpha}\big{)}_{\gamma\delta}:=\partial_{t^{\alpha}}\partial_{t^{\gamma}}\partial_{t^{\delta}}F$
such that the matrix $F_{1}$ is constant and nondegenerate. The WDVV equations
were discovered in the context of 2D-topological field theory (2D-TFT) at the
end of 80’s [12, 44].
The concept of a Frobenius manifold was first axiomatized and thoroughly
studied by B. Dubrovin [13, 14, 15, 17, 18] as a geometric reformulation of
the WDVV associativity equations. Roughly speaking, a Frobenius manifold is a
manifold with a flat metric and a Frobenius algebra structure on the tangent
bundle with two global vector fields, the unit and the Euler vector fields,
satisfying some integrable conditions. Each Frobenius manifold gives rise to a
quasi-homogeneous solution to the WDVV equations (1.1), called the
prepotential of the Frobenius manifold. Besides 2D-TFT, Frobenius manifolds
appear in a wide variety of mathematical physics contexts, including mirror
symmetry [10, 35, 39], theory of unfolding spaces of singularities [26, 37],
quantization theory [4, 28], quantum cohomology [30], integrable systems [31,
40] and so on.
Various examples of Frobenius manifolds have been constructed. Some of them
are derived from orbit spaces of certain groups, such as finite irreducible
Coxeter groups, affine Weyl groups and Jacobi groups [9, 14, 16].
Hurwitz spaces provide another source of interesting examples of Frobenius
manifold structures [14, 41, 42, 34]. A “simple” Hurwitz space in genus $g$ is
the space of ramified coverings $(C_{g},\lambda)$ of the Riemann sphere
$\mathbb{P}^{1}$, up to an equivalence relation (see Subsection 2.2), where
$C_{g}$ is a surface of genus $g$ and
$\lambda:C_{g}\longrightarrow\mathbb{P}^{1}$ is a meromorphic function of a
fixed degree with $N$ simple branch points
$\lambda_{1},\dots,\lambda_{N}\in\mathbb{C}$ and a prescribed ramification
profile over the point at infinity. The $N$ branch points $\\{\lambda_{j}\\}$
serve as local coordinates on the “simple” Hurwitz space.
In [14] (Lecture 5), Dubrovin established that a family of $N$ specific
Darboux-Egoroff metrics give rise to $N$ semi-simple Frobenius manifold
structures on the same simple Hurwitz space of dimension $N$. Such Darboux-
Egoroff metrics were written in terms of the so-called primary differentials
defined on the underlying Riemann surfaces. Moreover, in the system of local
coordinates $\\{\lambda_{j}\\}_{j}$ consisting of simple branch points, the
Frobenius algebra multiplication of the obtained family of $N$ semi-simple
Hurwitz-Frobenius manifolds is defined by
$\partial_{\lambda_{i}}\circ\partial_{\lambda_{j}}=\delta_{ij}\partial_{\lambda_{i}}$.
Let us emphasize that the crucial tool in Duborvin’s construction was a
bilinear pairing defined on the space of differentials having some suitable
properties. Particularly, by means of such bilinear pairing, Dubrovin proposed
a formula for the corresponding prepotentials, that is, quasi-homogeneous
solutions to the WDVV equations (1.1). For more details, we refer to [14]
(Lecture 5). In Appendix 3 below, we recall the expression for the bilinear
pairing and the formula for the prepotentials obtained in [14].
By employing a generalized bilinear pairing and viewing the above described
“simple” Hurwitz spaces as real manifolds, Shramchenko [41] gave another class
of Darboux-Egoroff metrics in terms of the Schiffer and Bergman kernels and
obtained $``\text{real double}"$ semi-simple Hurwitz-Frobenius manifolds.
Furthermore, by introducing some parameters, Shramchenko extended in [42]
Dubrovin’s construction and built deformed semi-simple Hurwitz-Frobenius
manifold structures. The solutions to the WDVV equations that are associated
with the “real double” and deformed Hurwitz-Frobenius manifold are also
described by means of a Dubrovin type bilinear pairing.
Notwithstanding the fact that a general formula for the prepotential is known
for all the already mentioned Hurwitz-Frobenius manifolds, their doubles and
deformations, it seems still be tricky to apply Dubrovin’s bilinear pairing
method and to produce explicit solutions to the WDVV equations. Moreover, at
least to our knowledge, the only known examples come from three and four
dimensional Hurwitz spaces in genus zero and one [11, 14, 41, 42].

The main purpose of the current work is to develop a new formula for the
prepotentials associated with (a generalized) Hurwitz-Frobenius manifolds. The
idea of our approach is the following. First, by following [14] (see also
[41]), we construct new Frobenius manifold structures on simple Hurwitz spaces
induced by quasi-homogeneous differentials defined on the underlying Riemann
surfaces. A differential $\omega_{0}$ on a Riemann surface $C_{g}$ is called
quasi-homogeneous of degree $d_{0}\geq 0$ if $\omega_{0}$ admits the following
integral representation with respect to the canonical symmetric bidifferential
$W(P,Q)$ on $C_{g}$:

$\omega_{0}(P)=c_{0}\int_{\ell_{0}}\big{(}\lambda(Q)\big{)}^{d_{0}}W(P,Q),$
(1.2)

where $c_{0}$ is a nonzero constant and $\ell_{0}$ is a contour on the surface
$C_{g}$ satisfying some conditions. For properties of the symmetric
bidifferential $W(P,Q)$, we refer to [22, 23, 45] and Subsection 2.3 below.
Note that the language of the bidifferential $W(P,Q)$ in the context of
Frobenius manifolds was first employed by Shramchenko in [41]. Interestingly,
it was observed in [41] that Dubrovin’s primary differentials can be
conveniently rewritten in terms of the symmetric bidifferential $W(P,Q)$ as in
(1.2). This means that each primary differential $\phi_{t^{A}}$ of Dubrovin is
quasi-homogeneous and can be characterized, through formula (1.2), by the
triplet $(d_{0},c_{0},\ell_{0})=(d_{A},c_{A},\ell_{A})$ where $d_{A}$ is the
degree of $\phi_{t^{A}}$, $c_{A}$ is a constant and $\ell_{A}$ is a suitable
contour on $C_{g}$.
Let us denote by $t^{A}(\omega_{0})$ the function defined by

$t^{A}(\omega_{0}):=c_{A}\int_{\ell_{A}}\big{(}\lambda(P)\big{)}^{d_{A}}\omega_{0}(P),$
(1.3)

with $(d_{A},c_{A},\ell_{A})$ being the triplet determined by the primary
differential $\phi_{t^{A}}$ (when $\ell_{A}$ is a curve joining two poles of
$\lambda$, the integral in the right hand side of (1.3) is defined by an
appropriate principal value (see [14] and Subsection 3.3 of this paper). Then
the set consisting of the $N$ functions $\big{\\{}t^{A}(\omega_{0})\big{\\}}$
provides a system of flat coordinates of the flat metric $\eta(\omega_{0})$ of
the Frobenius manifold associated with the differential $\omega_{0}$.
Analogously to [14], the metric $\eta(\omega_{0})$ is defined by

$\eta(\omega_{0})=\sum_{j=1}^{N}\bigg{(}\underset{P_{j}}{{\rm
res}}\frac{\omega_{0}(P)^{2}}{d\lambda(P)}\bigg{)}(d\lambda_{j})^{2},$ (1.4)

where $P_{1},\dots,P_{N}$ are the simple ramification points of the covering
$(C_{g},\lambda)$. Let us denote by $\eta$ the Gram constant matrix of the
flat metric $\eta(\omega_{0})$ in flat coordinates
$\big{\\{}t^{A}(\omega_{0})\big{\\}}$. Then, by using a convenient choice of
the constants $c_{A}$ in the integral representation formula (1.2) for the
primary and quasi-homogeneous differentials $\phi_{t^{A}}$, we prove that the
constant Gram matrix $\eta$ can be obtained in such a way that all the rows of
the constant matrix $\eta$ contain exactly one nonzero entry which is 1. This
leads us to introduce a useful duality relations between flat coordinates
$\big{\\{}t^{A}(\omega_{0})\big{\\}}$ of the metric $\eta(\omega_{0})$ (this
duality was already implicitly present in [14]). More precisely, two flat
coordinates $t^{A}(\omega_{0})$ and $t^{B}(\omega_{0})$ are called
$\eta(\omega_{0})$-dual to each other if $\eta^{AB}=1$ and in this case
$t^{B}$ is denoted by $t^{A^{\prime}}$. Here $\eta^{AB}$ are the constant
entries of the contravariant metric $\eta^{*}(\omega_{0})$ induced by
$\eta(\omega_{0})$. Note that, with the help of the bidifferential $W(P,Q)$,
the contravariant metric $\eta^{*}(\omega_{0})$, as well as the introduced
duality relations, we present new arguments to prove some results obtained by
Dubrovin [14] and used as key ingredients in the construction of Hurwitz-
Frobenius manifolds. The details of these results are discussed in Sections
3.1, 3.2, 3.3 and 4.1.

Next, we establish the following new formula for the prepotentials associated
with the Hurwitz-Frobenius manifolds induced by the quasi-homogeneous
differential $\omega_{0}$ (1.2):

$\begin{split}\mathcal{F}_{\omega_{0}}&=\frac{1}{2(1+d_{0})}\sum_{A,B}\frac{\big{(}(d_{A}+d_{0})t^{A}(\omega_{0})+r_{0,A}\big{)}\big{(}(d_{B}+d_{0})t^{B}(\omega_{0})+r_{0,B}\big{)}}{1+d_{0}+d_{A^{\prime}}}t^{A^{\prime}}\big{(}\phi_{t^{B^{\prime}}}\big{)},\\\
\end{split}$ (1.5)

where $d_{0},d_{A},d_{A^{\prime}}$ are respectively the degrees of the quasi-
homogeneous differentials $\omega_{0}$, $\phi_{t^{A}}$,
$\phi_{t^{A^{\prime}}}$ and $t^{A^{\prime}}$ is the dual coordinate of
$t^{A}$, while $\\{r_{0,A}\\}_{A}$ are certain constants depending on the
differentials $\omega_{0}$ and $\phi_{t^{A}}$ and
$\big{\\{}t^{A^{\prime}}(\phi_{t^{B^{\prime}}})\big{\\}}$ denotes the set of
flat coordinates of the flat metric $\eta\big{(}\phi_{t^{B^{\prime}}}\big{)}$
(1.4) induced by the primary differential $\phi_{t^{B^{\prime}}}$ (see Theorem
4.1 and formulas (4.19), (4.27) and (4.29) below).
The prepotential $\mathcal{F}_{\omega_{0}}$ (1.5) is a quasi-homogeneous
solution to the following generalized WDVV equations:

$\left\\{\begin{array}[]{ll}F_{\alpha}\eta^{-1}F_{\beta}=F_{\beta}\eta^{-1}F_{\alpha},\quad\alpha,\beta=1,\dots,N,\\\
\\\ \eta=\sum_{j}f_{\alpha}(t)F_{\alpha},\end{array}\right.$ (1.6)

where $F_{\alpha}=\partial_{t^{\alpha}(\omega_{0})}H$ and $H$ is the Hessian
matrix of $\mathcal{F}_{\omega_{0}}$, $\eta$ is the constant $N\times
N$-matrix of the metric $\eta(\omega_{0})$ and the coefficients $f_{\alpha}$
are some functions of flat coordinates $\\{t^{\alpha}(\omega_{0})\\}$.
Note that when $\omega_{0}$ is among the $N$ primary differentials of
Dubrovin, formula (1.5) provides a novel alternative method to calculate
prepotentials associated with the aforementioned semi-simple Hurwitz-Frobenius
manifolds developed by Dubrovin and the function $\mathcal{F}_{\omega_{0}}$
becomes a solution to the WDVV equations (1.1).
As a first application, we use our approach to reconstruct the Frobenius
manifold structure associated with the Chazy equation satisfied by the
Eisenstein series $E_{2}$ and recalculate the corresponding prepotential
$\mathcal{F}_{\phi_{1}}$. This Frobenius manifold was first obtained by
Dubrovin [14]. It is constructed on the three dimensional Hurwitz space of
Weierstrass elliptic curves and induced by the normalized holomorphic
differential $\phi_{1}$. Note that, in this case, the two remaining primary
differentials $\phi_{2},\phi_{3}$ are such that $\phi_{2}$ is an Abelian
differential of the second kind and $\phi_{3}$ is a multivalued differential.
Moreover, by showing that flat coordinates of the flat metrics
$\eta(\phi_{2})$ and $\eta(\phi_{3})$ can be computed directly from the
Hessian matrix of the obtained prepotential $\mathcal{F}_{\phi_{1}}$, we
propose a new proof of the Ramanujan differential equations for the Eisenstein
series $E_{2}$, $E_{4}$ and $E_{6}$ [32].
As a second application of formula (1.5), we prove that for any positive
integer $m$, the following function
$\mathcal{F}_{\omega_{0}}=\mathcal{F}_{\omega_{0}}(t_{0},t_{1},\dots,t_{2m+1})$
involving the Jacobi odd $\theta_{1}$-function

$\begin{split}\mathcal{F}_{\omega_{0}}&=\frac{(t_{2m+1})^{2}}{2}t_{0}+t_{2m+1}\sum_{k=1}^{m}t_{k}t_{2m+1-k}+\frac{1}{2}\sum_{k=1}^{m}\big{(}t_{2m+1-k}\big{)}^{2}\log(t_{2m+1-k})\\\
&\quad+\frac{1}{2}\bigg{(}\sum_{k=1}^{m}t_{2m+1-k}\bigg{)}^{2}\log\bigg{(}\sum_{j=1}^{m}t_{2m+1-j}\bigg{)}-\sum_{k=1}^{m}\big{(}t_{2m+1-k}\big{)}^{2}\log\left(\frac{\theta_{1}\big{(}t_{k}\big{|}{2{\rm
i}\pi}t_{0}\big{)}}{\theta_{1}^{\prime}\big{(}0\big{|}2{\rm i}\pi
t_{0}\big{)}}\right)\\\ &\quad+\frac{1}{2}\sum_{j,k=1,j\neq
k}^{m}t_{2m+1-j}t_{2m+1-k}\log\bigg{(}\frac{\theta_{1}^{\prime}\big{(}0\big{|}{2{\rm
i}\pi}t_{0}\big{)}\theta_{1}\big{(}t_{j}-t_{k}\big{|}{2{\rm
i}\pi}t_{0}\big{)}}{\theta_{1}\big{(}t_{j}\big{|}{2{\rm
i}\pi}t_{0}\big{)}\theta_{1}\big{(}t_{k}\big{|}{2{\rm
i}\pi}t_{0}\big{)}}\bigg{)}\end{split}$ (1.7)

is a genus one quasi-homogeneous solution to the WDVV equations (1.6), with
$\omega_{0}$ being the normalized holomorphic differential. This explicit
solution is directly suggested by our formula (1.5) and a particular choice of
the combinatorial parameters of the Hurwitz space (here $g=1$ and the $m+1$
poles of the ramified coverings are all simple). We refer the reader to
Section 4.1.5 for details.
Note that the case $m=1$ of the prepotential (1.7) was obtained in [19, 11]
through a long computation based on Dubrovin’s bilinear pairing method.
In our case, most of the terms of expression (1.7) are obtained by simple
specialization of the general formula (1.5) to the chosen Hurwitz space.
Moreover, it is the structure of the general formula (1.5) that suggests which
Hurwitz space and primary differential to look at in order to obtain an
explicit solution while avoiding lengthy calculations. Note that all known
explicit examples obtained through Dubrovin’s bilinear pairing method require
an explicit parametrization of the Hurwitz space. Such a parametrization is
not necessary for applying formula (1.5) which allows to construct
prepotentials for a larger class of Hurwitz spaces, of which (1.7) is an
example. To our knowledge this is the first example of a prepotential of a
Dubrovin-Hurwitz-Frobenius manifold depending on a large (greater than 4)
number of variables.
In addition, the following function depending on $2m$-variables:

$\mathcal{F}=\sum_{k=1}^{m}t_{2m+1-k}e^{t_{k}}+\frac{1}{2}\sum_{k=1}^{m}\big{(}t_{2m+1-k}\big{)}^{2}\Big{(}t_{k}+\log\big{(}t_{2m+1-k}\big{)}\Big{)}+\frac{1}{2}\sum_{k,s=1,s\neq
k}^{m}t_{2m+1-k}t_{2m+1-s}\log\big{(}e^{t_{s}}-e^{t_{k}}\big{)}$

provides an example of explicit solution to the generalized WDVV equations
(1.6), where $\eta$ is the anti-diagonal matrix
$\eta_{\alpha\beta}=\delta_{2m+1,\alpha+\beta}$ and the coefficients
$f_{\alpha}$ are given by

$f_{\alpha}=\frac{e^{-t_{\alpha}}}{1-\sum_{r=1}^{m}t_{2m+1-r}e^{-t_{r}}},\quad\quad
f_{2m+1-\alpha}=-\frac{t_{2m+1-\alpha}e^{-t_{\alpha}}}{1-\sum_{r=1}^{m}t_{2m+1-r}e^{-t_{r}}},\quad\quad\alpha=1,\dots,m.$

This example comes from the genus zero Hurwitz-Frobenius manifold structure
determined by a quasi-homogeneous differential not belonging to the family of
Dubrovin primary differentials.
On the other hand, we extend formula (1.5) to prepotentials associated with
the already mentioned deformations of Hurwitz-Frobenius manifolds and obtained
in [42]. This brings us to give new explicit quasi-homogeneous solutions to
the WDVV equations by considering the deformations of the obtained
prepotentials.

This paper is organized as follows. In Section 2, we recall the definitions of
a Frobenius manifold structure, simple and double Hurwitz spaces and that of
the symmetric bidifferential on a Riemann surface. In Section 3, we introduce
a family of flat metrics in terms of the symmetric bidifferential and
construct their systems of flat coordinates. The study here includes the
intersection forms regarded as flat metrics on double Hurwitz spaces. The goal
of Section 4 is to describe Frobenius manifold structures induced by quasi-
homogeneous differentials and to give a formula the corresponding
prepotentials as well as examples in genus one. Sections 5 is devoted to
extending our approach to the framework of deformed Frobenius manifold
structures. In Section 6, we construct examples of Frobenius manifold
structures on $2m$-dimensional Hurwitz space in genus zero and compute the
corresponding explicit prepotentials. Ramanujan’s differential equations and
other examples of genus one quasi-homogeneous solutions to the WDVV equations
are discussed in Section 7. Finally, in the Appendix, we give details of some
results used in this work for the reader’s convenience.

##  2 Preliminaries

###  2.1 Frobenius manifolds

###### Definition 2.1

A $\mathbb{C}$-algebra $(\mathcal{A},\circ,e)$ supplied with a nondegenerate
and symmetric inner product
$\mathop{\langle\cdot,\cdot\rangle}:\mathcal{A}\times\mathcal{A}\longrightarrow\mathbb{C}$
is a Frobenius algebra if

i)

the algebra $(\mathcal{A},\circ,e)$ is associative, commutative with unity
$e$;

ii)

the multiplication $``\circ"$ is compatible with
$\mathop{\langle\cdot,\cdot\rangle}$, in the sense that

$\mathop{\langle x\circ y,z\rangle}=\mathop{\langle x,y\circ
z\rangle},\quad\quad\forall\ x,y,z\in\mathcal{A}.$

A Frobenius algebra $(\mathcal{A},\circ,e,\mathop{\langle\cdot,\cdot\rangle})$
is called semi-simple if it contains no nilpotent element.

The next definition of a Frobenius manifold differs slightly from the original
one given by Dubrovin. More precisely, in order to be consistent with the
generalized WDVV equations (1.6), the unit vector field is not required to be
flat.

###### Definition 2.2

Let $\mathcal{M}$ be a manifold of dimension N. A Frobenius manifold structure
on $\mathcal{M}$ is a data of
$\big{(}\mathcal{M},\circ,e,\mathop{\langle\cdot,\cdot\rangle},E\big{)}$ such
that each tangent space $T_{t}\mathcal{M}$ is a Frobenius algebra varying
smoothly or analytically over $\mathcal{M}$ with the additional properties:

  * _1._

the metric $\mathop{\langle\cdot,\cdot\rangle}_{t}$ on $\mathcal{M}$ is flat
(but not necessarily real and positive);

  * _2._

the tensor $\big{(}\nabla_{v}\textbf{c}\big{)}(x,y,z)$ is symmetric in four
vector fields $v,x,y,z\in T_{t}\mathcal{M}$, where $\nabla$ is the Levi-Civita
connection of the metric $\mathop{\langle\cdot,\cdot\rangle}$ and c is the
symmetric 3-tensor

$\textbf{c}(x,y,z):=\mathop{\langle x\circ y,z\rangle},\quad\forall\ x,y,z\in
T_{t}\mathcal{M};$

  * _3._

the vector field $E$ is covariantly linear, i.e. $\nabla\nabla E=0$, and

$\displaystyle\begin{split}&[E,e]=-e,\\\ &[E,x\circ y]-[E,x]\circ
y-x\circ[E,y]=x\circ y,\\\
&\big{(}Lie_{E}\mathop{\langle\cdot,\cdot\rangle}\big{)}(x,y):=E\mathop{\langle
x,y\rangle}-\mathop{\langle[E,x],y\rangle}-\mathop{\langle
x,[E,y]\rangle}=(2-D)\mathop{\langle x,y\rangle},\end{split}$ (2.1)

for some constant $D$, called the charge of the Frobenius manifold. The vector
field $E$ is called Euler vector field.

###### Definition 2.3

Let $\big{(}\mathcal{M},\circ,e,\mathop{\langle\cdot,\cdot\rangle},E\big{)}$
be a Frobenius manifold.
We shall call it normalized if the unit vector field $e$ is covariantly
constant with respect to $\nabla$, the Levi-Civita connection of
$\mathop{\langle\cdot,\cdot\rangle}$, namely $\nabla e=0$.
A Frobenius manifold is called semi-simple if the Frobenius algebra in the
tangent space at any point of $\mathcal{M}$ is semi-simple.

The flatness of the metric $\mathop{\langle\cdot,\cdot\rangle}$ implies that
locally there exist flat coordinates $\\{t^{\alpha}:\ 1\leq\alpha\leq N\\}$ so
that the matrix
$\big{(}\mathop{\langle\partial_{t^{\alpha}},\partial_{t^{\beta}}\rangle}\big{)}_{\alpha\beta}$
is constant. On the other hand, from the symmetry of the tensor
$\nabla\textbf{c}$, it follows that there is a function
$F=F(t^{1},\dots,t^{N})$ such that

$\partial_{t^{\alpha}}\partial_{t^{\beta}}\partial_{t^{\gamma}}F=\textbf{c}\big{(}\partial_{t^{\alpha}},\partial_{t^{\beta}},\partial_{t^{\gamma}}\big{)}.$
(2.2)

The function $F$ is unique up to additional quadratic and linear polynomial
functions of $t^{1},\dots,t^{N}$ and it is known as the prepotential of the
Frobenius manifold. Physicists call it the primary free energy function of the
2D-TFT [12].
Note that (LABEL:Euler_vector_def) yields that the Lie derivative, along the
Euler vector field $E$, of the symmetric 3-tensor c is such that:

$\begin{split}\big{(}Lie_{E}.\textbf{c}\big{)}(x,y,z)&:=E.\textbf{c}(x,y,z)-\textbf{c}\big{(}[E,x],y,z\big{)}-\textbf{c}\big{(}x,[E,y],z\big{)}-\textbf{c}\big{(}x,y,[E,z]\big{)}\\\
&=(3-D)\textbf{c}(x,y,z).\end{split}$ (2.3)

Let us assume that $E=\sum_{\epsilon}E^{\epsilon}(t)\partial_{t^{\epsilon}}$,
with $t=(t^{1},\dots,t^{N})$. Then the requirement $\nabla\nabla E=0$ implies
that $\partial_{t^{\alpha}}\partial_{t^{\beta}}E^{\epsilon}=0$, for all
$\alpha,\beta,\epsilon$. This fact, (2.2) and (2.3) allow us to write

$\big{(}Lie_{E}.\textbf{c}\big{)}\big{(}\partial_{t^{\alpha}},\partial_{t^{\beta}},\partial_{t^{\gamma}}\big{)}=\partial_{t^{\alpha}}\partial_{t^{\beta}}\partial_{t^{\gamma}}E.F=(3-D)\partial_{t^{\alpha}}\partial_{t^{\beta}}\partial_{t^{\gamma}}F.$

Thus the prepotential $F$ is a quasi-homogeneous function of degree $3-D$
(modulo quadratic and/or linear terms). Following Dubrovin [14] (Lecture 1),
we use a generalized quasi-homogeneity property is considered in our context.
More precisely, a function $f$ is called quasi-homogeneous of degree $\nu_{f}$
with respect to the Euler vector field
$E=\sum_{\alpha}\big{(}d_{\alpha}t^{\alpha}+r_{\alpha}\big{)}\partial_{t^{\alpha}}$
if

$\textstyle
E.f=\nu_{f}f+\sum_{\alpha,\beta}A_{\alpha\beta}t^{\alpha}t^{\beta}+\sum_{\alpha}B_{\alpha}t^{\alpha}+C.$
(2.4)

On the other hand, we emphasize that the flatness of the unit vector field $e$
(i.e. $\nabla e=0$) implies that $e$ can be taken of the form
$e=\partial_{t^{1}}$ (by making a linear change of flat coordinates). In this
case, the metric $\mathop{\langle\cdot,\cdot\rangle}$ satisfies

$\mathop{\langle\partial_{t^{\alpha}},\partial_{t^{\beta}}\rangle}=\partial_{t^{1}}\partial_{t^{\alpha}}\partial_{t^{\beta}}F.$

Otherwise, we have $e=\sum_{\epsilon}e^{\epsilon}(t)\partial_{t^{\epsilon}}$
and

$\mathop{\langle\partial_{t^{\alpha}},\partial_{t^{\beta}}\rangle}=\textstyle\sum_{\epsilon}e^{\epsilon}(t)\partial_{t^{\epsilon}}\partial_{t^{\alpha}}\partial_{t^{\beta}}F.$

Lastly, we mention some results concerning a semi-simple Frobenius manifold
$\big{(}\mathcal{M},\circ,e,\mathop{\langle\cdot,\cdot\rangle},E\big{)}$ that
were derived by Dubrovin [14] (Lecture 3): In a neighborhood of each point of
$\mathcal{M}$, there is a system of canonical coordinates
$\lambda_{1},\dots,\lambda_{N}$ such that the Frobenius algebra multiplication
is given by
$\partial_{\lambda_{i}}\circ\partial_{\lambda_{j}}=\delta_{ij}\partial_{\lambda_{i}}$,
the unit and Euler vector fields are respectively represented by
$e=\sum_{j}\partial_{\lambda_{j}}$ and
$E=\sum_{j}\lambda_{j}\partial_{\lambda_{j}}$ and, with respect to the
canonical coordinates, the flat metric $\mathop{\langle\cdot,\cdot\rangle}$
becomes of Darboux-Egoroff type (diagonal with diagonal terms generated by a
potential).

###  2.2 Hurwitz spaces

The Hurwitz space $\mathcal{H}_{g,L}$ is the moduli space of equivalence
classes of ramified (or branched) coverings $[(C_{g},\lambda)]$, where $C_{g}$
is a compact Riemann surface of genus $g$ and
$\lambda:C_{g}\longrightarrow\mathbb{P}^{1}$ is a meromorphic function on
$C_{g}$ of degree $L\geq 2$. Two ramified coverings $(C_{g},\lambda)$ and
$(\widetilde{C}_{g},\widetilde{\lambda})$ are equivalent if there is a
biholomorphic map $h:C_{g}\longrightarrow\widetilde{C}_{g}$ such that the
following diagram commutes:

---
$\textstyle{C_{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{\lambda}$$\textstyle{\widetilde{C}_{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\lambda}}$$\textstyle{\mathbb{P}^{1}}$

The ramification points of the covering $(C_{g},\lambda)$ are the zeros of the
differential $d\lambda(P)$ (here $\lambda$ is regarded as a holomorphic map)
and its branch points are the images of the ramification points by the
projection $\lambda$.
A branch point is called simple if the corresponding ramification index is
$r=1$. Here we mention that the ramification index at a point is a nonnegative
integer $r$ such that the number of sheets of the covering which are glued at
this point is $r+1$. In particular, the fiber of a simple branch point
contains exactly $L-1$ points, with $L$ being the degree of $\lambda$.
The (classical) Hurwitz space of simply branched coverings (i.e. all its
branching points are simple) of the Riemann sphere of degree $L$ and genus $g$
appeared first in the works of Clebsch [7] and Hurwitz [27], where they showed
that such space is a connected complex manifold and the simple branch points
can be taken as local coordinates on it.

Let $m,n$ be two positive integers and $\mathbf{m}=(m_{0},\dots,m_{n})$ and
$\mathbf{n}=(n_{0},\dots,n_{m})$ be two fixed vectors of nonnegative integers
such that

$\textstyle\sum_{i=0}^{n}(m_{i}+1)=\sum_{j=0}^{m}(n_{j}+1)=L.$

The simple Hurwitz space
$\mathcal{H}_{g,L}(n_{0},\dots,n_{m})=\mathcal{H}_{g,L}(\mathbf{n})$
parameterizes isomorphism classes of ramified coverings
$[(C_{g},\lambda)]\in\mathcal{H}_{g,L}$ where the branched covering
$(C_{g},\lambda)$ has $N$ simple distinct branch points
$\lambda_{1}=\lambda(P_{1}),\dots,\lambda_{N}=\lambda(P_{N})\in\mathbb{C}$ and
$m+1$ prescribed poles $\infty^{0},\dots,\infty^{m}$ of order
$n_{0}+1,\dots,n_{m}+1$, respectively. In particular, the divisor of the
differential $d\lambda$ has the following form:

$(d\lambda)=\textstyle\sum_{j=1}^{N}P_{j}-\sum_{i=0}^{m}(n_{i}+2)\infty^{i}.$

According to Fulton [25], the simple Hurwitz space
$\mathcal{H}_{g,L}(n_{0},\dots,n_{m})$ is still a connected complex manifold
of dimension $N$. Let us emphasize that, due to the Riemann-Hurwitz formula,
the number $N$ of simple and finite branch points of any covering
$[(C_{g},\lambda)]\in\mathcal{H}_{g,L}(n_{0},\dots,n_{m})$ can be expressed as
follows:

$N=2g+2L-2-\textstyle\sum_{i=0}^{m}n_{i}=2g+2m+\textstyle\sum_{i=0}^{m}n_{i}.$

As previously indicated, simple Hurwitz spaces were considered by Dubrovin
[14] and Shramchenko [41, 42] in the framework of Frobenius manifolds. Note
that the Frobenius manifold structures were constructed on open subsets of a
covering $\widehat{\mathcal{H}}_{g,L}(n_{0},\dots,n_{m})$ of the simple
Hurwitz space $\mathcal{H}_{g,L}(n_{0},\dots,n_{m})$ whose points are pairs of
$\big{\\{}\big{(}C_{g},\lambda\big{)},\\{a_{k},b_{k}\\}\big{\\}}$, where
$[(C_{g},\lambda)]$ is a point in $\mathcal{H}_{g,L}(n_{0},\dots,n_{m})$ and
$\\{a_{k},b_{k}\\}$ is a canonical homology basis of cycles on $C_{g}$. Here,
we will also work on a fixed open set
$\widehat{\mathcal{H}}_{g,L}(n_{0},\dots,n_{m})$ and we continue to call it
simple Hurwitz space for brevity.

As a generalization of simple Hurwitz spaces, we can consider equivalence
classes of ramified coverings in $\mathcal{H}_{g,L}$ with two special fibers
over $0$ and the point at infinity. These are called double Hurwitz spaces.
More precisely, the double Hurwitz space
$\mathcal{H}_{g,L}(\mathbf{n};\mathbf{m})$ consists of the classes
$[(C_{g},\lambda)]\in\mathcal{H}_{g,L}$ such that the following conditions are
satisfied:

1.

The covering $\lambda$ has $N$ simple ramification points $P_{1},\dots,P_{N}$
with distinct images
$\lambda_{1},\dots,\lambda_{N}\in\mathbb{P}^{1}\setminus\\{0,\infty\\}$;

2.

The divisor of $\lambda$ is as follows:

$(\lambda)=\textstyle\sum_{i=0}^{n}(m_{i}+1)\mathbf{z}^{i}-\sum_{i=0}^{m}(n_{i}+1)\infty^{i}.$

In this case, the Riemann-Hurwitz formula tells us that the number $N$ of
simple branch points belonging to $\mathbb{C}\setminus\\{0\\}$ is given by

$N=2g+2L-2-\textstyle\sum_{i=0}^{n}m_{i}-\sum_{i=0}^{m}n_{i}=2g+m+n.$

Contrary to simple Hurwitz spaces, double Hurwitz spaces are not in general
connected. Note that when $\mathbf{m}=(0,\dots,0)$, the double Hurwitz space
$\mathcal{H}_{g,L}(\mathbf{n};\mathbf{0})$ becomes an open subset of the
simple Hurwitz space $\mathcal{H}_{g,L}(n_{0},\dots,n_{m})$ determined by the
conditions $\lambda_{1}\neq 0,\dots,\lambda_{N}\neq 0$.
As above, we shall work with a connected (component of a) covering
$\widehat{\mathcal{H}}_{g,L}(\mathbf{n};\mathbf{m})$ of the considered double
Hurwitz space, whose elements are pairs: a point of the space
$\mathcal{H}_{g,L}(\mathbf{n};\mathbf{m})$ and a canonical homology basis of
cycles on the underlying surface.

Now, we shall introduce some notation that will be frequently used throughout
this work.
Standard local parameters near ramification points Let
$[(C_{g},\lambda)]\in\mathcal{H}_{g,L}(n_{0},\dots,n_{m})\cup\mathcal{H}_{g,L}(\mathbf{n};\mathbf{m})$.
The complex structure on the surface $C_{g}$ is defined by the covering as
follows.
The distinguished local parameter near a simple ramification point $P_{j}\in
C_{g}$ is defined by $x_{j}(P):=\sqrt{\lambda(P)-\lambda_{j}}$ provided that
$P_{j}$ is not a pole, nor a zero of the meromorphic function $\lambda$.
Near a pole $\infty^{i}$ of $\lambda$ of order $n_{i}+1$, the standard local
parameter $z_{i}(P)$ is defined by
$z_{i}(P):=\big{(}\lambda(P)\big{)}^{-\frac{1}{n_{i}+1}}$. Lastly, in a
neighborhood of a zero $\mathbf{z}^{i}$ of $\lambda$, the local parameter is
denoted by $\mathfrak{z}_{i}(P)$ and given by
$\mathfrak{z}_{i}(P):=\big{(}\lambda(P)\big{)}^{\frac{1}{m_{i}+1}}$, with
$m_{i}$ being the ramification index of $\lambda$ at the point
$\mathbf{z}^{i}$.

Evaluation of a differential at a ramification point By following Dubrovin
[14], we shall introduce an appropriate evaluation of a differential at a
ramification point. It is defined with respect to the previously mentioned
standard local parameters.
Let $\omega$ be a meromorphic differential on the surface $C_{g}$. Assume that
$\omega$ is holomorphic near a simple ramification point $P_{j}$, with

$\omega(P)\underset{P\sim
P_{j}}{=}\textstyle\Big{(}\sum_{k=0}^{\infty}a_{k}x_{j}^{k}(P)\Big{)}dx_{j}(P),\quad\quad
x_{j}(P)=\sqrt{\lambda(P)-\lambda_{j}}.$

Define the evaluation $\omega(P_{j})$ of the differential $\omega(P)$ at
$P=P_{j}$ with respect to the fixed local parameter $x_{j}(P)$ by

$\omega(P_{j})=a_{0}=\frac{\omega(P)}{dx_{j}(P)}\Big{|}_{P=P_{j}}.$ (2.5)

Since $d\lambda(P)=2x_{j}(P)dx_{j}(P)$ near the point $P_{j}$, it follows that
the quantity $\omega(P_{j})$ satisfies

$\frac{\big{(}\omega(P_{j})\big{)}^{2}}{2}=\underset{P_{j}}{{\rm res}}\
\frac{\big{(}\omega(P)\big{)}^{2}}{d\lambda(P)}.$ (2.6)

We shall define the evaluation of $\omega$ at a pole $\infty^{i}$ or at a zero
$\mathbf{z}^{i}$ of $\lambda$ as follows

$\begin{split}&\omega^{(k)}(\infty^{i})=k!\underset{\infty^{i}}{{\rm res}}\
\lambda(P)^{\frac{k+1}{n_{i}+1}}\omega(P),\quad\forall\ k\in\mathbb{N};\\\
&\omega^{(k)}(\mathbf{z}^{i})=k!\underset{\mathbf{z}^{i}}{{\rm res}}\
\lambda(P)^{-\frac{k+1}{m_{i}+1}}\omega(P),\quad\forall\
k\in\mathbb{N}.\end{split}$ (2.7)

Here, $\omega$ is not supposed to be holomorphic near $\infty^{i}$ and
$\mathbf{z}^{i}$, but if it is, then the terms $\omega^{(k)}(\infty^{i})/k!$
(resp. $\omega^{(k)}(\mathbf{z}^{i})/k!$) are nothing but the coefficients of
the Taylor expansion of $\omega$ near the pole $\infty^{i}$ (resp. the zero
$\mathbf{z}^{i}$).

###  2.3 Symmetric bidifferentials on compact Riemann surfaces

Let $C_{g}$ be a compact Riemann surfaces of genus $g\geq 1$ equipped with a
canonical homology basis $\\{a_{k},b_{k}\\}$. Corresponding to the fixed basis
of cycles $\\{a_{k},b_{k}\\}$, there is a uniquely determined basis of
holomorphic differentials $\omega_{1},\dots,\omega_{g}$, normalized by the
conditions $\oint_{a_{k}}\omega_{j}=\delta_{jk}$. Denote by $\mathbb{B}$ the
Riemann matrix of $b$-periods, i.e. $\mathbb{B}_{jk}=\oint_{b_{k}}\omega_{j}$.
Let $\alpha$ and $\beta$ be two arbitrary real $g$-dimensional vectors. The
Riemann theta function with characteristic $\Delta=[\alpha,\beta]$ and modulus
$\mathbb{B}$ is defined by

$\Theta_{\Delta}(z)=\Theta_{\Delta}(z|\mathbb{B}):=\sum_{n\in\mathbb{Z}^{g}}e^{{\rm{i}}\pi\mathop{\langle\mathbb{B}(n+\alpha),n+\alpha\rangle}+2{\rm{i}}\pi\mathop{\langle
z+\beta,n+\alpha\rangle}},\quad\quad z\in\mathbb{C}^{g}.$ (2.8)

For any integer vectors $p,q\in\mathbb{Z}^{g}$, the function $\Theta_{\Delta}$
enjoys the following quasi-periodicity properties:

$\Theta_{\Delta}(z+p+\mathbb{B}q)=e^{-{\rm{i}}\pi\mathop{\langle\mathbb{B}q,q\rangle}-2{\rm{i}}\pi\mathop{\langle
z+\beta,q\rangle}+2{\rm{i}}\pi\mathop{\langle
p,\alpha\rangle}}\Theta_{\Delta}(z).$ (2.9)

In what follows we will work with an odd and non-singular half-integer
characteristic $\Delta$ where by this we mean that
$\alpha,\beta\in\frac{1}{2}\mathbb{Z}^{g}$,
$4\mathop{\langle\alpha,\beta\rangle}\in 1+2\mathbb{Z}$ and the gradient of
the function $\Theta_{\Delta}$ at the origin is nonzero:

$\big{(}\partial_{z_{1}}\Theta_{\Delta}(0),\dots,\partial_{z_{g}}\Theta_{\Delta}(0)\big{)}\neq(0,\dots,0).$
(2.10)

Particularly, $\Theta_{\Delta}$ becomes an odd function and the holomorphic
differential

$\omega_{\Delta}(P):=\textstyle\sum_{k=1}^{g}\partial_{z_{k}}\Theta_{\Delta}(0)\omega_{k}(P)$

does not vanish identically. For more detailed treatment on the Riemann theta
function, we refer to [21, 22, 29].

The (canonical) symmetric bidifferential $W$ on $C_{g}$ is the meromorphic
differential on $C_{g}\times C_{g}$ defined by means of the
$\Delta$-characteristic Riemann theta function $\Theta_{\Delta}$ by [22, 23,
45]

$W(P,Q):=d_{P}d_{Q}\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(Q)\big{)}\Big{)}=d_{P}d_{Q}\log\big{(}E(P,Q)\big{)},$
(2.11)

where
$\mathcal{A}(P):=\big{(}\int^{P}\omega_{1},\dots,\int^{P}\omega_{g}\big{)}$
stands for the Abel map on $C_{g}$ and $E$ is the prime form.
By using the properties of Riemann theta function [21, 22, 29], we can observe
that the bidifferential $W(P,Q)$ satisfies the following properties:

i)

it is a symmetric meromorphic differential on $C_{g}\times C_{g}$:
$W(P,Q)=W(Q,P)$;

ii)

it has a pole of second order on the diagonal with biresidue 1:

$W(P,Q)\underset{P\sim
Q}{=}\left(\frac{1}{\big{(}\xi(P)-\xi(Q)\big{)}^{2}}+\text{reg}\right)d\xi(P)d\xi(Q),$
(2.12)

where $\xi$ is a local parameter near $P\sim Q$;

iii)

its $a$-periods with respect to either of the arguments vanish:

$\oint_{a_{k}}W(P,Q)=0,\quad\quad\forall\ k=1,\dots,g.$ (2.13)

We point out that properties i), ii) and iii) provide a characterization of
the canonical symmetric bidifferential $W(P,Q)$. Indeed, the difference of two
kernels that fulfill these three conditions would have no pole and would have
vanishing $a$-periods, and therefore it would vanish identically.
In genus zero, the symmetric bidifferential on the Riemann sphere
$\mathbb{P}^{1}$ is given by the following rational expression:

$W(P,Q)=\frac{dz_{P}dz_{Q}}{(z_{P}-z_{Q})^{2}}.$ (2.14)

Rauch variational formulas Let $[(C_{g},\lambda)]$ be a point in the already
described (covering of the) simple or double Hurwitz space. Then, for each
fixed $P,Q\in C_{g}$, the quantity $W(P,Q)$ becomes a function of simple
branch points $\lambda_{1},\dots,\lambda_{N}$. The dependence of the
bidifferential $W(P,Q)$ on $\\{\lambda_{j}\\}_{j}$ is specified by Rauch’s
variational formulas [23, 33, 45]:

$\partial_{\lambda_{j}}W(P,Q)=\frac{1}{2}W(P,P_{j})W(P_{j},Q)=\underset{P_{j}}{{\rm
res}}\frac{W(P,S)W(S,Q)}{d\lambda(S)}.$ (2.15)

Here $W(P,P_{j})$ denotes the evaluation of the bidifferential $W(P,Q)$ (the
point $P$ is fixed) at $Q=P_{j}$ with respect to the standard local parameter
$x_{j}(Q)=\sqrt{\lambda(Q)-\lambda_{j}}$ near the ramification point $P_{j}$:

$W(P,P_{j}):=\frac{W(P,Q)}{dx_{j}(Q)}\Big{|}_{Q=P_{j}}.$

##  3 Flat metrics and flat coordinates in terms of the bidifferential on
simple and double Hurwitz spaces

In this section, we consider a family of flat metrics associated with
differential forms $\omega$ with respect to the bidifferential $W$, i.e., let
$\omega$ be the differential form corresponding to the covering
$(C_{g},\lambda)$ defined by:

$\omega(P)=\int_{\gamma}h(\lambda(Q))W(P,Q).$ (3.1)

Here $\gamma$ and $h$ are assumed to satisfy the following conditions:

  * C1)

$\gamma$ is an arbitrary smooth contour on the compact Riemann surface $C_{g}$
not passing through any of the ramification points $P_{j}$ and such that its
projection $\lambda(\gamma)\subset\mathbb{P}^{1}$ is independent of the branch
points $\\{\lambda_{j}\\}_{j}$, where by this we mean that the contour
$\lambda(\gamma)$ does not change under small variations of
$\lambda_{1},\dots,\lambda_{N}$;

  * C2)

$h$ is a function defined in a neighborhood of $\gamma$ such that $h$ does not
depend on the branch points $\lambda_{1},\dots,\lambda_{N}$.

Motivated by the work of Dubrovin [14] (formulas (5.35a)-(5.35d)) (see also
[41, 34]), the pair $(h,\gamma)$ will be chosen in such a manner that the
differential $\omega$ on the surface $C_{g}$ is holomorphic outside the
assigned poles $\infty^{0},\dots,\infty^{m}$ and zeros
$\mathbf{z}^{0},\dots,\mathbf{z}^{n}$ of $\lambda$.
Due to the properties of the symmetric bidifferential $W(P,Q)$ and those of
the contour $\gamma$, we observe that $\omega$ is holomorphic around each
ramification point $P_{j}$. Moreover, in view of Rauch’s formulas (2.15), the
partial derivatives of $\omega(P)$ with respect to the branch points
$\\{\lambda_{j}\\}_{j}$ of the covering $(C_{g},\lambda)$ are expressed by
means of the evaluations $\big{\\{}\omega(P_{j})\big{\\}}_{j}$ as follows:

$\partial_{\lambda_{j}}\omega(P)=\frac{1}{2}W(P,P_{j})\int_{\gamma}h(\lambda(Q))W(Q,P_{j})=\frac{\omega(P_{j})}{2}W(P,P_{j}).$
(3.2)

In particular, for each $j$ we observe that if $\omega(P_{j})\neq 0$, then
$\partial_{\lambda_{j}}\omega(P)$ is an Abelian differential of the second
kind with a unique double pole at the point $P_{j}$. Moreover, their
$a$-periods are all zero.
For a fixed differential $\omega$ as in (3.1), we will work in the open domain
$\mathcal{M}_{\omega}$ in the simple Hurwitz space
$\widehat{\mathcal{H}}_{g,L}(n_{0},\dots,n_{m})$ or double Hurwitz space
$\widehat{\mathcal{H}}_{g,L}(\mathbf{n};\mathbf{m})$ determined by the
conditions

$\omega(P_{j})\neq 0,\quad\forall\ j=1,\dots,N.$ (3.3)

For $\beta_{1},\beta_{2}\in\mathbb{C}$, with $(\beta_{1},\beta_{2})\neq(0,0)$,
consider the diagonal metric $\mathbf{ds^{2}_{\beta_{1},\beta_{2}}}(\omega)$
depending on the chosen differential $\omega$ defined by:

$\mathbf{ds^{2}_{\beta_{1},\beta_{2}}}(\omega):=\sum_{j=1}^{N}\bigg{(}\frac{\omega(P_{j})^{2}}{2\big{(}\beta_{1}\lambda_{j}+\beta_{2}\big{)}}\bigg{)}(d\lambda_{j})^{2}=\sum_{j=1}^{N}\bigg{(}\underset{P_{j}}{{\rm
res}}\frac{\omega(P)^{2}}{\big{(}\beta_{1}\lambda(P)+\beta_{2}\big{)}d\lambda(P)}\bigg{)}(d\lambda_{j})^{2}.$
(3.4)

Consider the Möbius transformation:

$\textsf{M}.z:=\frac{\delta{z}-\beta_{2}}{\delta+\beta_{1}},\quad\quad\text{with}\quad\delta(\beta_{1}+\delta)=1.$

We can observe that when $\beta_{1}=0$, then $\textsf{M}.z=z\pm\beta_{2}$ and
thus $\infty$ is the unique fixed point of M. In this case, the metric
$\mathbf{ds^{2}_{0,\beta_{2}}}(\omega)$ is well defined in
$\mathcal{M}_{\omega}$ (3.3) seen as an open subset of the simple Hurwitz
space $\widehat{\mathcal{H}}_{g,L}(n_{0},\dots,n_{m})$.
When $\beta_{1}\neq 0$, then M has two fixed points which are $\infty$ and
$\xi=-\beta_{2}/\beta_{1}$. Therefore we regard
$\mathbf{ds^{2}_{\beta_{1},\beta_{2}}}(\omega)$ as a metric defined in
$\mathcal{M}_{\omega}$ viewed as an open subset of the double Hurwitz space
$\widehat{\mathcal{H}}_{g,L}(\mathbf{n};\mathbf{m}){(\xi)}$. Here the space
$\widehat{\mathcal{H}}_{g,L}(\mathbf{n};\mathbf{m}){(\xi)}$ is the image of
the already described double Hurwitz space
$\widehat{\mathcal{H}}_{g,L}(\mathbf{n};\mathbf{m})$ by the biholomorphic
mapping

$[(C_{g},\lambda)]\longmapsto[(C_{g},\lambda+\xi)],\quad\quad\xi=-\beta_{2}/\beta_{1}$

(that is the two special fibers become over the two fixed points $\infty$ and
$\xi=-\beta_{2}/\beta_{1}$ of M).
It is known that the Christoffel symbols of a diagonal metric
$\sum_{i}g_{ii}(d\lambda_{i})^{2}$ satisfy

$\Gamma_{ij}^{k}=0,\quad\Gamma_{ii}^{k}=-\frac{\partial_{k}g_{ii}}{2g_{kk}},\quad\Gamma_{ii}^{i}=\frac{\partial_{i}g_{ii}}{2g_{ii}},\quad\Gamma_{ij}^{i}=\frac{\partial_{j}g_{ii}}{2g_{ii}},\quad\quad\text{$i,j,k$
are distinct}.$ (3.5)

Therefore, by (3.2), (3.5) and a straightforward computation, we deduce that
the nonvanishing Christoffel symbols of the diagonal metric
$\mathbf{ds^{2}_{\beta_{1},\beta_{2}}}(\omega)$ are

$\begin{split}&\Gamma_{ij}^{i}=\frac{\partial_{\lambda_{j}}\omega(P_{i})}{\omega(P_{i})}=\frac{\omega(P_{j})}{2\omega(P_{i})}W(P_{i},P_{j}),\quad\quad
i\neq j;\\\
&\Gamma_{ii}^{j}=-\frac{\beta_{1}\lambda_{j}+\beta_{2}}{\beta_{1}\lambda_{i}+\beta_{2}}\Gamma_{ij}^{j},\quad\quad\quad\quad
i\neq j;\\\
&\Gamma_{ii}^{i}=-\frac{\beta_{1}}{2\beta_{1}\lambda_{i}+2\beta_{2}}+\frac{\partial_{\lambda_{i}}\omega(P_{i})}{\omega(P_{i})}.\end{split}$
(3.6)

In Appendix 1, we mainly use (3.6) as well as identity (3.7) below to prove
that the Riemann curvature tensor of $\nabla$ (the Levi-Civita connection of
$\mathbf{ds^{2}_{\beta_{1},\beta_{2}}}(\omega)$) vanishes identically. This
establishes the flatness property of the metric
$\mathbf{ds^{2}_{\beta_{1},\beta_{2}}}(\omega)$. We refer to Theorem 8.1 and
its proof for more details.
In particular, the diagonal metrics $\mathbf{ds^{2}_{0,\beta_{2}}}(\omega)$
are all Darboux-Egoroff metrics with common rotation coefficients given in
terms of the symmetric bidifferential $W(P,Q)$ by

$\kappa_{ij}:=\frac{W(P_{i},P_{j})}{2}=\frac{W(P,Q)}{2dx_{i}(P)dx_{j}(Q)}\Big{|}_{P=P_{i},Q=P_{j}},\quad
i\neq j.$

###  3.1 An auxiliary result

The following useful result describes the action of the family of the vector
fields

$\textstyle\sum_{j=1}^{N}\big{(}\beta_{1}\lambda_{j}+\beta_{2}\big{)}\partial_{\lambda_{j}},\quad\quad(\beta_{1},\beta_{2})\neq(0,0),$

on the quantities $W(P_{i},P_{j})$ and $\omega(P_{j})$ seen as functions of
the branch points $\\{\lambda_{1},\dots,\lambda_{N}\\}$ of a covering
$(C_{g},\lambda)$.

###### Theorem 3.1

Let $(C_{g},\lambda)$ be a branched covering of $\mathbb{P}^{1}$ with $N$
simple critical points $P_{1},\dots,P_{N}$ and $\omega$ be a differential on
$C_{g}$ defined by (3.1). Then for all $i,j=1,\dots,N$, $i\neq j$, we have

$\displaystyle\sum_{k=1}^{N}(\beta_{1}\lambda_{k}+\beta_{2})\partial_{\lambda_{k}}W(P_{i},P_{j})=-\beta_{1}W(P_{i},P_{j});$
(3.7)
$\displaystyle\sum_{k=1}^{N}(\beta_{1}\lambda_{k}+\beta_{2})\partial_{\lambda_{k}}\omega(P_{j})=-\frac{\beta_{1}}{2}\omega(P_{j})+\int_{\gamma}\Big{(}\big{(}\beta_{1}\lambda(Q)+\beta_{2}\big{)}h^{\prime}(\lambda(Q))W(Q,P_{j})\Big{)}.$
(3.8)

_Proof:_ We distinguish two cases depending on wether $\beta_{1}$ is zero or
not.
_First case: $\beta_{1}=0$_. Consider the family of branched coverings
$\big{\\{}(C_{g},\lambda_{\varepsilon})\big{\\}}_{\varepsilon}$ with

$\lambda_{\varepsilon}(P):=\lambda(P)+\varepsilon.$ (3.9)

We observe that all the branched coverings $(C_{g},\lambda_{\varepsilon})$
share the set of ramification points and then, by (3.9) we deduce that the
standard local parameter $x_{j}^{\varepsilon}(P)$ near the point $P_{j}$,
induced by the covering $\lambda_{\varepsilon}$, is given by

$x_{j}^{\varepsilon}(P):=\sqrt{\lambda_{\varepsilon}(P)-\lambda_{\varepsilon}(P_{j})}=x_{j}(P).$

Moreover, we have a one-parameter group
$(\chi_{\varepsilon})_{\varepsilon\in\mathbb{C}}$, a subgroup of the
automorphism group of the simple Hurwitz space
$\mathcal{H}_{g,L}(n_{0},\dots,n_{m})$, defined by

$\chi_{\varepsilon}\big{(}[(C_{g},\lambda)]\big{)}=[(C_{g},\lambda_{\varepsilon})].$

Denote by $\big{(}(\chi_{-\varepsilon})^{*}W\big{)}(P,Q)$ the pullback of the
function $W(P,Q)$ by the map $\chi_{-\varepsilon}=\chi_{\varepsilon}^{-1}$
(when $P,Q$ are fixed, we view $W(P,Q)$ as a function on the Hurwitz space).
Then the quantity $\big{(}(\chi_{-\varepsilon})^{*}W\big{)}(P,Q)$ defines a
bidifferential on the same surface $C_{g}$ equipped with complex structure
induced by the covering $(C_{g},\lambda_{-\varepsilon})$. Since for any
$\varepsilon$, the complex structure on surface $C_{g}$ defined by the
coverings $(C_{g},\lambda_{\varepsilon})$ and $(C_{g},\lambda)$ is the same,
by uniqueness of the symmetric bidifferential it follows that
$\big{(}(\chi_{-\varepsilon})^{*}W\big{)}(P,Q)=W(P,Q)$, $\forall\
\varepsilon$.
On the other hand, given that the differential $\omega$ in (3.1) is defined by
the pair $(h(\lambda),\gamma)$ and the contour $\gamma$ is on the surface
$C_{g}$ (hence $\gamma$ is independent of $\lambda$ and
$\lambda_{\varepsilon}$), we deduce that the $\chi_{-\varepsilon}$-pullback of
$\omega$ is defined by $(h(\lambda_{-\varepsilon}),\gamma)$ and thus the
differential $(\chi_{-\varepsilon})^{*}\omega$ is given by

$\big{(}(\chi_{-\varepsilon})^{*}\omega\big{)}(P)=\int_{\gamma}h\big{(}\lambda_{-\varepsilon}(Q)\big{)}W(P,Q)=\int_{\gamma}h\big{(}\lambda(Q)-\varepsilon\big{)}W(P,Q).$

This brings us to write:

$\displaystyle\frac{\big{(}(\chi_{-\varepsilon})^{*}W\big{)}(P,Q)}{dx_{i}^{\varepsilon}(P)dx_{j}^{\varepsilon}(Q)}=\frac{W(P,Q)}{dx_{i}(P)dx_{j}(Q)};$
$\displaystyle\frac{\big{(}(\chi_{-\varepsilon})^{*}\omega\big{)}(P)}{dx_{j}^{\varepsilon}(P)}=\int_{\gamma}h\big{(}\lambda(Q)-\varepsilon\big{)}\frac{W(P,Q)}{dx_{j}(P)}.$

Recalling that the ramification points of $\lambda$ and
$\lambda_{-\varepsilon}$ coincide, we perform the evaluation at
$(P,Q)=(P_{i},P_{j})$ in the first equality and $P=P_{j}$ in the second one
and recalling notation (2.5) we get

$\displaystyle\big{(}(\chi_{-\varepsilon})^{*}W\big{)}(P_{i},P_{j})=W(P_{i},P_{j}),\quad
i\neq j;$
$\displaystyle\big{(}(\chi_{-\varepsilon})^{*}\omega\big{)}(P_{j})=\int_{\gamma}h\big{(}\lambda(Q)-\varepsilon\big{)}W(P_{j},Q).$

Now differentiating these two relations with respect to $\varepsilon$ at
$\varepsilon=0$ and using the fact that the flow associated with the one-
parameter group $(\chi_{\varepsilon})_{\varepsilon}$ is generated by the
vector field $\sum_{k=1}^{N}\partial_{\lambda_{k}}$, we obtain

$\displaystyle\textstyle\sum_{k=1}^{N}\partial_{\lambda_{k}}W(P_{i},P_{j})=-\frac{d}{d\varepsilon}\big{(}(\chi_{-\varepsilon})^{*}W\big{)}(P_{i},P_{j})\Big{|}_{\varepsilon=0}=0;$
$\displaystyle\textstyle\sum_{k=1}^{N}\partial_{\lambda_{k}}\omega(P_{j})=-\frac{d}{d\varepsilon}\big{(}(\chi_{-\varepsilon})^{*}\omega\big{)}(P_{j})\Big{|}_{\varepsilon=0}=\int_{\gamma}h^{\prime}(\lambda(Q))W(P_{j},Q).$

_Second case: $\beta_{1}\neq 0$._ Let us consider the family of branched
coverings $(C_{g},\lambda_{\varepsilon})$ of $\mathbb{P}^{1}$ defined by

$\lambda_{\varepsilon}(P)=e^{\varepsilon\beta_{1}}\big{(}\lambda(P)-\xi\big{)}+\xi,\quad\quad\text{with}\quad\xi:=-\beta_{2}/\beta_{1}.$
(3.10)

As in the first case, we can see that the family of transformations
$\\{\varphi_{\varepsilon}\\}_{\varepsilon}$, with

$\varphi_{\varepsilon}\big{(}[(C_{g},\lambda)]\big{)}:=[(C_{g},\lambda_{\varepsilon})],$

gives rise to a local one-parameter subgroup of automorphisms of the Hurwitz
space.
Similarly to the first case, we have
$\big{(}(\varphi_{-\varepsilon})^{*}W\big{)}(P,Q)=W(P,Q)$ and

$\big{(}(\varphi_{-\varepsilon})^{*}\omega\big{)}(P)=\int_{\gamma}h\big{(}\lambda_{-\varepsilon}(Q)\big{)}W(P,Q)=\int_{\gamma}h\big{(}e^{-\varepsilon\beta_{1}}\big{(}\lambda(Q)-\xi\big{)}+\xi\big{)}W(P,Q).$

In addition, (3.10) implies that $d\lambda_{\varepsilon}(P)=0$ if and only if
$d\lambda(P)=0$ and thus the ramification points of the coverings
$(C_{g},\lambda_{\varepsilon})$ coincide with those of $(C_{g},\lambda)$.
Consequently, the standard local parameter $x_{j}^{\varepsilon}(P)$ near the
critical point $P_{j}$ and induced by $\lambda_{-\varepsilon}$ becomes

$x_{j}^{\varepsilon}(P):=\sqrt{\lambda_{-\varepsilon}(P)-\lambda_{-\varepsilon}(P_{j})}=e^{-\varepsilon\frac{\beta_{1}}{2}}x_{j}(P).$

Therefore
$dx_{j}^{\varepsilon}(P)=e^{-\varepsilon\frac{\beta_{1}}{2}}dx_{j}(P)$ and

$\displaystyle\frac{\big{(}(\varphi_{-\varepsilon})^{*}W\big{)}(P,Q)}{dx_{i}^{\varepsilon}(P)dx_{j}^{\varepsilon}(Q)}=e^{\varepsilon\beta_{1}}\frac{W(P,Q)}{dx_{i}(P)dx_{j}(Q)};$
$\displaystyle\frac{\big{(}(\varphi_{-\varepsilon})^{*}\omega\big{)}(P)}{dx_{j}^{\varepsilon}(P)}=e^{\varepsilon\frac{\beta_{1}}{2}}\int_{\gamma}h\big{(}e^{-\varepsilon\beta_{1}}\big{(}\lambda(Q)-\xi\big{)}+\xi\big{)}\frac{W(P,Q)}{dx_{j}(P)}.$

Plugging $(P,Q)=(P_{i},P_{j})$ and $P=P_{j}$ in these equalities,
respectively, and using introduced notation (2.5), we deduce that they can be
rewritten as follows

$\displaystyle\big{(}(\varphi_{-\varepsilon})^{*}W\big{)}(P_{i},P_{j})=e^{\varepsilon\beta_{1}}W(P_{i},P_{j});$
$\displaystyle\big{(}(\varphi_{-\varepsilon})^{*}\omega\big{)}(P_{j})=e^{\varepsilon\frac{\beta_{1}}{2}}\int_{\gamma}h\big{(}e^{-\varepsilon\beta_{1}}\big{(}\lambda(Q)-\xi\big{)}+\xi\big{)}W(P_{j},Q).$

Let us now differentiate both sides with respect to $\varepsilon$ at
$\varepsilon=0$. Since the vector field
$\sum_{j}\big{(}\beta_{1}\lambda_{j}-\beta_{1}\xi\big{)}\partial_{\lambda_{j}}$
generates the local flow induced by the local one-parameter group
$(\varphi_{\varepsilon})_{\varepsilon}$, it follows that

$\displaystyle\sum_{k}\big{(}\beta_{1}\lambda_{k}-\beta_{1}\xi\big{)}\partial_{\lambda_{k}}W(P_{i},P_{j})=-\frac{d}{d\varepsilon}\big{(}(\varphi_{-\varepsilon})^{*}W\big{)}(P_{i},P_{j})\Big{|}_{\varepsilon=0}=-\beta_{1}W(P_{i},P_{j});$
$\displaystyle\sum_{k}\big{(}\beta_{1}\lambda_{k}-\beta_{1}\xi\big{)}\partial_{\lambda_{k}}\omega(P_{j})=-\frac{\beta_{1}}{2}\omega(P_{j})+\int_{\gamma}\bigg{(}\big{(}\beta_{1}\lambda(Q)-\beta_{1}\xi)\big{)}h^{\prime}(\lambda(Q))W(P_{j},Q)\bigg{)}.$

Finally replacing $\xi$ by $-\beta_{2}/\beta_{1}$, we conclude that formulas
(3.7) and (3.8) hold true.

$\square$

###  3.2 Characterization of flat functions

Flat coordinates of a flat metric $\Lambda$ are a set of coordinates
$\\{t^{A}\\}$ in which coefficients of the metric are all constant. In these
coordinates the Christoffel symbols vanish and the covariant derivative
$\nabla_{\partial_{t^{A}}}$ is nothing but the usual partial derivative
$\partial_{t^{A}}$ in the direction of the flat coordinate $t^{A}$. We refer
to the monograph [38] for details.
It is known that the flat coordinates are solutions to the partial
differential equations

$\partial_{\lambda_{i}}\partial_{\lambda_{j}}t=\textstyle\sum_{k}\Gamma_{ij}^{k}\partial_{\lambda_{k}}t,$

where $\Gamma_{ij}^{k}$ are the Christoffel symbols of the flat metric
$\Lambda$.
In the case where the considered flat metric $\Lambda$ is diagonal, by (3.5)
we see that these equations can be rewritten in the form:

$\displaystyle\partial_{\lambda_{i}}\partial_{\lambda_{j}}t=\Gamma_{ij}^{i}\partial_{\lambda_{i}}t+\Gamma_{ij}^{j}\partial_{\lambda_{j}}t,\quad\quad
i\neq j,$ (3.11)
$\displaystyle\partial_{\lambda_{j}}^{2}t=\displaystyle\sum_{s}\Gamma_{jj}^{s}\partial_{\lambda_{s}}t.$
(3.12)

We shall call a solution to equations (3.11)-(3.12) a flat function with
respect to the metric $\Lambda$ or more shortly a $\Lambda$-flat function.

Consider the diagonal flat metric
$\mathbf{ds^{2}_{\beta_{1},\beta_{2}}}(\omega)$ defined by (3.4), where
$\omega$ is a fixed differential with respect to the kernel $W$ given by (3.1)
and (3.3). We are going to construct flat coordinates $t(\omega)$ for this
metric of the following specific type:

$t(\omega):=\int_{\ell}f(\lambda(P))\omega(P),$ (3.13)

where the contour $\ell$ and the function $f$ enjoy the conditions C1) and C2)
in (3.1), i.e., $\lambda(\ell)$ and $f$ are independent of the branch points
$\lambda_{1},\dots,\lambda_{N}$.
Note the pair $(h,\gamma)$ defining the differential $\omega$ in (3.1) gives
rise to a function of type (3.13). Conversely, given that a function
$t(\omega)$ of type (3.13), then the properties of $(f,\ell)$ allow us to
attach it to the following differential $\phi_{t}(P)$ of the form(3.1):

$\phi_{t}(P):=\int_{\ell}f(\lambda(Q))W(P,Q).$ (3.14)

Below a characterization of the
$\mathbf{ds^{2}_{\beta_{1},\beta_{2}}(\omega)}$-flatness of the function
$t(\omega)$ will be given with the help of its associated differential form
$\phi_{t}$.

Our choice of functions of (3.13) is firstly motivated by the following
result.

###### Proposition 3.1

The function $t(\omega)$ in (3.13) is a solution to equations (3.11) for the
flat metrics $\mathbf{ds^{2}_{\beta_{1},\beta_{2}}}(\omega)$ for all
$(\beta_{1},\beta_{2})\neq(0,0)$.

_Proof:_ Since by (3.6) the Christoffel coefficients $\Gamma_{ij}^{i}$ are
independent of $\beta_{1},\beta_{2}$ when $i\neq j$, the system (3.11) is the
same for all the considered flat metrics
$\mathbf{ds^{2}_{\beta_{1},\beta_{2}}}(\omega)$.
Now, the properties of $\ell$ permit us to differentiate under the integral
sign and thanks to Rauch formulas (3.2) we have

$\forall\
j,\quad\partial_{\lambda_{j}}t(\omega)=\frac{\omega(P_{j})}{2}\phi_{t}(P_{j})=\underset{P_{j}}{{\rm
res}}\frac{\omega(P)\phi_{t}(P)}{d\lambda(P)}.$ (3.15)

Here, the second equality is an immediate consequence of the holomorphic
property of the differentials $\omega$ and $\phi_{t}$ near the point $P_{j}$
(a simple zero of $d\lambda(P)$).
Differentiating in (3.15) once again with respect to $\lambda_{i}$, $i\neq j$,
we obtain

$\displaystyle\partial_{\lambda_{i}}\partial_{\lambda_{j}}t(\omega)$
$\displaystyle=\frac{1}{4}W(P_{i},P_{j})\phi_{t}(P_{i})\omega(P_{j})+\frac{1}{4}W(P_{i},P_{j})\phi_{t}(P_{j})\omega(P_{i}).$

On the other hand, using (3.6) and (3.15), we deduce that

$\displaystyle\Gamma_{ij}^{i}\partial_{\lambda_{i}}t(\omega)+\Gamma_{ij}^{j}\partial_{\lambda_{j}}t(\omega)=\frac{1}{4}\phi_{t}(P_{i})W(P_{i},P_{j})\omega(P_{j})+\frac{1}{4}\phi_{t}(P_{j})W(P_{i},P_{j})\omega(P_{i}),\quad
i\neq j.$

This shows that the function $t(\omega)$ obeys equations (3.11) as desired.

$\square$

###### Proposition 3.2

Let $(\beta_{1},\beta_{2})\neq(0,0)$ and $t(\omega)$ be the function defined
by (3.13) and $\phi_{t}$ be the corresponding differential (3.14), where the
pair $(f,\ell)$ enjoys the two conditions C1) and C2) as in (3.1).
Then the following are equivalent:

i)

The function $t(\omega)$ satisfies equations (3.12) for the metric
$\mathbf{ds^{2}_{\beta_{1},\beta_{2}}}(\omega)$;

ii)

for all $j=1,\dots,N$, we have

$\textstyle\sum_{k=1}^{N}\big{(}\beta_{1}\lambda_{k}+\beta_{2}\big{)}\partial_{\lambda_{k}}\phi_{t}(P_{j})=-\frac{\beta_{1}}{2}\phi_{t}(P_{j});$

iii)

for all $j=1,\dots,N$, the following holds:

$\int_{\ell}\big{(}\beta_{1}\lambda(P)+\beta_{2}\big{)}f^{\prime}(\lambda(P))W(P,P_{j})=0.$
(3.16)

_Proof:_ The equivalence between items ii) and iii) follows immediately from
(3.8) applied to the differential form $\phi_{t}$ (3.14).
On the other hand, we have

$\displaystyle\sum_{s}\Gamma_{jj}^{s}\partial_{\lambda_{s}}t(\omega)=\sum_{s\neq
j}\Gamma_{jj}^{s}\partial_{\lambda_{s}}t(\omega)+\Gamma_{jj}^{j}\partial_{\lambda_{j}}t(\omega)$
$\displaystyle=-\frac{\omega(P_{j})}{4\beta_{1}\lambda_{j}+4\beta_{2}}\sum_{s\neq
j}\big{(}\beta_{1}\lambda_{s}+\beta_{2})W(P_{s},P_{j})\phi_{t}(P_{s})+\left(-\frac{\beta_{1}\omega(P_{j})}{4\beta_{1}\lambda_{j}+4\beta_{2}}+\frac{1}{2}\partial_{\lambda_{j}}\omega(P_{j})\right)\phi_{t}(P_{j})$
$\displaystyle=-\frac{\omega(P_{j})}{2\beta_{1}\lambda_{j}+2\beta_{2}}\sum_{s\neq
j}\big{(}\beta_{1}\lambda_{s}+\beta_{2})\partial_{\lambda_{s}}\phi_{t}(P_{j})+\left(-\frac{\beta_{1}\omega(P_{j})}{4\beta_{1}\lambda_{j}+4\beta_{2}}+\frac{1}{2}\partial_{\lambda_{j}}\omega(P_{j})\right)\phi_{t}(P_{j})$
$\displaystyle=-\frac{\omega(P_{j})}{2\beta_{1}\lambda_{j}+2\beta_{2}}\sum_{s=1}^{N}\big{(}\beta_{1}\lambda_{s}+\beta_{2})\partial_{\lambda_{s}}\phi_{t}(P_{j})+\frac{\omega(P_{j})}{2}\partial_{\lambda_{j}}\phi_{t}(P_{j})+\left(-\frac{\beta_{1}\omega(P_{j})}{4\beta_{1}\lambda_{j}+4\beta_{2}}+\frac{1}{2}\partial_{\lambda_{j}}\omega(P_{j})\right)\phi_{t}(P_{j})$
$\displaystyle=\frac{\beta_{1}\omega(P_{j})\phi_{t}(P_{j})}{4\beta_{1}\lambda_{j}+4\beta_{2}}-\frac{\omega(P_{j})}{2\beta_{1}\lambda_{j}+2\beta_{2}}\int_{\ell}\bigg{(}\big{(}\beta_{1}\lambda(P)+\beta_{2}\big{)}f^{\prime}(\lambda(P))W(P,P_{j})\bigg{)}+\frac{\omega(P_{j})}{2}\partial_{\lambda_{j}}\phi_{t}(P_{j})$
$\displaystyle\quad+\left(-\frac{\beta_{1}\omega(P_{j})}{4\beta_{1}\lambda_{j}+4\beta_{2}}+\frac{1}{2}\partial_{\lambda_{j}}\omega(P_{j})\right)\phi_{t}(P_{j})$
$\displaystyle=\partial_{\lambda_{j}}^{2}t(\omega)-\frac{\omega(P_{j})}{2\beta_{1}\lambda_{j}+2\beta_{2}}\int_{\ell}\bigg{(}(\beta_{1}\lambda(P)+\beta_{2}\big{)}f^{\prime}(\lambda(P))W(P,P_{j})\bigg{)},$

where we have used relations (3.6) and (3.15) in the second equality, Rauch’s
formulas (3.2) in the third equality and relation (3.8) in the fifth one.
From the last equality we conclude that $t(\omega)$ is a solution to the
system of partial differential equations
$\partial_{\lambda_{j}}^{2}t(\omega)=\sum_{s}\Gamma_{jj}^{s}\partial_{\lambda_{s}}t(\omega)$
if and only if conditions (3.16) are satisfied. This finishes the proof of the
proposition.

$\square$

Note that the $N$ conditions (3.16) do not depend on the choice of the
differential $\omega$ that we work with. In addition, by making use of the
properties of the symmetric bidifferential $W(P,Q)$ (2.11), Proposition 3.2
offers a method to construct flat coordinates of the flat metric
$\mathbf{ds^{2}_{\beta_{1},\beta_{2}}}(\omega)$. More precisely, this method
consists in finding functions of type (3.13) and adequately choosing the
corresponding pairs $(f,\ell)$ in such a way that the $N$ conditions in (3.16)
are fulfilled. This method will be applied in the next two subsections.

The next useful result gives practical techniques to calculate the entries of
the constant Gram matrix of the contravariant metric
$\left(\mathbf{ds^{2}_{\beta_{1},\beta_{2}}}(\omega)\right)^{*}$.
Here we recall that if $\Lambda=\sum_{i,j}\Lambda_{ij}dx^{i}dx^{j}$ is a
metric on a manifold $\mathcal{M}$, then $\Lambda$ induces a dual metric
$\Lambda^{*}$ in the cotangent spaces. The components of $\Lambda^{*}$ are
given by

$\Lambda^{*}(dx^{i},dx^{j})=\Lambda^{ij},$

where $\Lambda^{ij}$ are the entries of the inverse matrix of
$(\Lambda_{ij})$.

###### Theorem 3.2

Let $t^{j}(\omega)$ and $\phi_{t^{j}}(P)$ be respectively the functions and
the corresponding differentials defined by

$t^{j}(\omega):=\int_{\ell_{j}}f_{j}(\lambda(P))\omega(P),\quad\quad\phi_{t^{j}}(P):=\int_{\ell_{j}}f_{j}(\lambda(Q))W(P,Q),\quad
j=1,2.$

Here, we assume that the pairs $(f_{j},\ell_{j})$ obey the same conditions C1)
and C2) satisfied by the pair $(h,\gamma)$ defining the differential $\omega$
(3.1), but the functions $t^{1}(\omega)$ and $t^{2}(\omega)$ are not
necessarily supposed to be flat with respect to the flat metric
$\mathbf{ds^{2}_{\beta_{1},\beta_{2}}}(\omega)$.
Then we have the following results:

$\displaystyle\left(\mathbf{ds^{2}_{\beta_{1},\beta_{2}}}(\omega)\right)^{*}\left(dt^{1}(\omega),dt^{2}(\omega)\right)$
$\displaystyle=\frac{1}{2}\sum_{j=1}^{N}\big{(}\beta_{1}\lambda_{j}+\beta_{2}\big{)}\phi_{t^{1}}(P_{j})\phi_{t^{2}}(P_{j})$
(3.17)
$\displaystyle=\sum_{j=1}^{N}\big{(}\beta_{1}\lambda_{j}+\beta_{2}\big{)}\partial_{\lambda_{j}}t^{1}\left(\phi_{t^{2}}\right)$
(3.18) $\displaystyle=\sum_{j=1}^{N}\underset{P_{j}}{{\rm
res}}\bigg{(}\big{(}\beta_{1}\lambda(P)+\beta_{2}\big{)}\frac{\phi_{t^{1}}(P)\phi_{t^{2}}(P)}{d\lambda(P)}\bigg{)}.$
(3.19)

where $t^{1}\left(\phi_{t^{2}}\right)$ is the function defined as in (3.13).

_Proof:_ We have

$\displaystyle\left(\mathbf{ds^{2}_{\beta_{1},\beta_{2}}}(\omega)\right)^{*}(dt^{1}(\omega),dt^{2}(\omega))$
$\displaystyle=\sum_{i,j}\partial_{\lambda_{i}}t^{1}(\omega)\partial_{\lambda_{j}}t^{2}(\omega)\left(\mathbf{ds^{2}_{\beta_{1},\beta_{2}}}(\omega)\right)^{*}(d\lambda_{i},d\lambda_{j})$
$\displaystyle=\sum_{j}\frac{2\beta_{1}\lambda_{j}+2\beta_{2}}{\omega(P_{j})^{2}}\partial_{\lambda_{j}}t^{1}(\omega)\partial_{\lambda_{j}}t^{2}(\omega)$
$\displaystyle=\frac{1}{2}\sum_{j}\big{(}\beta_{1}\lambda_{j}+\beta_{2}\big{)}\phi_{t^{1}}(P_{j})\phi_{t^{2}}(P_{j}),$

where we have used the chain rule formula in the first equality, (3.4) in the
second one and (3.15) in the third equality.
Formula (3.18) is direct consequence of (3.17) combined with the Rauch partial
derivatives (3.15) for $\omega=\phi_{t_{2}}$:

$\partial_{\lambda_{j}}t^{1}\left(\phi_{t^{2}}\right)=\frac{1}{2}\phi_{t^{1}}(P_{j})\phi_{t^{2}}(P_{j}).$

Moreover, by using (3.17), we see that relation (3.19) holds.

$\square$

###### Remark 3.1

Formula (3.17) has been derived by Romano [34] in the particular case where
$\beta_{1}=1$ and $\beta_{2}=0$.

###  3.3 Flat coordinates of Dubrovin’s flat metrics and duality relations

In this subsection, we assume that $\beta_{1}=0$ and $\beta_{2}=1$ and then we
shall work in open subsets of the simple Hurwitz space
$\widehat{\mathcal{H}}_{g,L}(n_{0},\dots,n_{m})$.
Inspired by Dubrovin’s work [14] (see also [41]), we shall give a system of
flat coordinates of the Darboux-Egoroff metric

$\eta(\omega):=\mathbf{ds^{2}_{0,1}}(\omega)=\frac{1}{2}\sum_{j}\big{(}\omega(P_{j})\big{)}^{2}(d\lambda_{j})^{2}=\sum_{j}\bigg{(}\underset{P_{j}}{{\rm
res}}\frac{\omega(P)^{2}}{d\lambda(P)}\bigg{)}(d\lambda_{j})^{2},$ (3.20)

where $\omega$ is a differential satisfying (3.1) and (3.3). As already
mentioned in the introduction, we allow $\omega$ to belong to a family of
differentials larger than the finite list of primary differentials given by
Dubrovin in [14] (Lecture 5).
As in the Dubrovin case, among the list of flat coordinates of the metric
$\eta(\omega)$, there is a particular subfamily of flat functions involving an
appropriate principal value. More precisely, the principal value in the
function

$s^{i}(\omega):=p.v.\int_{\infty^{0}}^{\infty^{i}}\omega(P)$

is defined by omitting the divergent part of the integral as a function of the
local parameters $z_{0}(P):=\lambda(P)^{-\frac{1}{n_{0}+1}}$,
$z_{i}(P):=\lambda(P)^{-\frac{1}{n_{i}+1}}$ near the points $\infty^{0}$ and
$\infty^{i}$, respectively.
In Theorem 3.3 below, we pay particular attention to the contribution of this
specific principal value when $\omega$ is the Abelian differential of the
third kind $\Omega_{\infty^{0}\infty^{j}}$.

###### Proposition 3.3

Let $\omega$ be a differential satisfying (3.1) and (3.3) and $\eta(\omega)$
be the corresponding Dubrovin metric (3.20).
Then the following $N$ functions are all $\eta(\omega)$-flat :

$\begin{array}[]{lllll}t^{i,\alpha}(\omega):=\displaystyle\frac{\sqrt{n_{i}+1}}{\alpha}\underset{\infty^{i}}{{\rm
res}\ }\lambda(P)^{\frac{\alpha}{n_{i}+1}}\omega(P),&\quad
i=0,\dots,m,\quad\alpha=1,\dots,n_{i};\\\ \\\
v^{i}(\omega):=\displaystyle\underset{\infty^{i}}{{\rm res}\
}\lambda(P)\omega(P),&\quad i=1,\dots,m;\\\ \\\ s^{i}(\omega):=\displaystyle
p.v.\int_{\infty^{0}}^{\infty^{i}}\omega(P),&\quad i=1,\dots,m;\\\ \\\
\rho^{k}(\omega):=\displaystyle\frac{1}{2{\rm{i}}\pi}\oint_{b_{k}}\omega(P),&\quad
k=1,\dots,g;\\\ \\\
u^{k}(\omega):=\displaystyle\oint_{a_{k}}\lambda(P)\omega(P),&\quad
k=1,\dots,g.\end{array}$ (3.21)

_Proof:_ The functions $t^{i,\alpha}(\omega)$, $v^{i}(\omega)$,
$\rho^{k}(\omega)$, $u^{k}(\omega)$ in (3.21) are of the form (3.13).
Furthermore, by employing the properties of the symmetric bidifferential, we
can directly check that necessary and sufficient conditions (3.16) are
satisfied. This establishes their flatness property.
Now let us deal with the functions $s^{i}(\omega)$, $i=1,\dots,m$. If the
differential $\omega$ has no poles (within a fundamental polygon of the
surface $C_{g}$), then we have
$s^{i}(\omega)=\int_{\infty^{0}}^{\infty^{i}}\omega(P)$ (here the integral
converges). Hence $s^{i}(\omega)$ is of the form (3.13) and obeys the
conditions in (3.16).
Assume now that the poles of $\omega$ are among the assigned ramification
points $\infty^{0},\dots,\infty^{m}$. Note that (3.16) is valid for $\omega$
such that Rauch’s formula (3.15) holds. In order to ensure the validity of
formula (3.15) for the functions
$s^{i}(\omega)=p.v.\int_{\infty^{0}}^{\infty^{i}}\omega(P)$, we always assume
that the singular parts of $\omega$ near its poles do not depend on the branch
points $\lambda_{1},\dots,\lambda_{N}$ (which serve as local coordinates in
the considered simple Hurwitz space). Therefore the differential (3.14)
corresponding to the function $s^{i}(\omega)$ is
$\phi_{s^{i}}(P)=\int_{\infty^{0}}^{\infty^{i}}W(P,Q)$ and thus (3.16) is
valid for the function $s^{i}(\omega)$.

$\square$

We now move on to study some characteristic properties of the differentials
$\phi_{t^{A}}$ associated with the flat functions $t^{A}$ (3.21) in the sense
of (3.13) and (3.14). These properties will be used as a crucial ingredient to
calculate the entries of the Gram matrix of the contravariant metric
$\eta^{*}(\omega)$.
According to the correspondence between (3.13) and (3.14), the family of the
$\eta(\omega)$-flat functions

$\mathcal{S}(\omega):=\Big{\\{}t^{i,\alpha}(\omega),\ v^{i}(\omega),\
s^{i}(\omega),\ \rho^{k}(\omega),\ u^{k}(\omega)\Big{\\}}$ (3.22)

defined by (3.21) brings us to the following differentials:

$\begin{array}[]{lllll}\textbf{\emph{1.}}\quad\phi_{t^{i,\alpha}}(P):=\displaystyle\frac{\sqrt{n_{i}+1}}{\alpha}\underset{\infty^{i}}{{\rm
res}\
}\lambda(Q)^{\frac{\alpha}{n_{i}+1}}W(P,Q),&i=0,\dots,m;\quad\alpha=1,\dots,n_{i};\\\
\\\
\textbf{\emph{2.}}\quad\phi_{v^{i}}(P):=\displaystyle\underset{\infty^{i}}{{\rm
res}\ }\lambda(Q)W(P,Q),&i=1,\dots,m;\\\ \\\
\textbf{\emph{3.}}\quad\phi_{s^{i}}(P):=\displaystyle\int_{\infty^{0}}^{\infty^{i}}W(P,Q),&i=1,\dots,m;\\\
\\\
\textbf{\emph{4.}}\quad\phi_{\rho^{k}}(P):=\displaystyle\frac{1}{2{\rm{i}}\pi}\oint_{b_{k}}W(P,Q),&k=1,\dots,g;\\\
\\\
\textbf{\emph{5.}}\quad\phi_{u^{k}}(P):=\displaystyle\oint_{a_{k}}\lambda(Q)W(P,Q),&k=1,\dots,g.\end{array}$
(3.23)

###### Remark 3.2

i) For later use, we emphasize that all the differentials listed in (3.23)
share the following particular integral representation with respect to the
kernel $W(P,Q)$:

$\phi_{t^{A}}(P)=c_{A}\int_{\ell_{A}}\big{(}\lambda(Q)\big{)}^{d_{A}}W(P,Q),$
(3.24)

where $c_{A}$ is a nonzero complex number and $d_{A}\geq 0$. As we will see
later, the nonnegative constants $d_{A}$ are of particular importance and play
the role of the quasi-homogeneous degrees of the differentials $\phi_{t^{A}}$
(see Subsections 4.1.3 and 4.1.4).
As an example, if $t^{A}=t^{i,\alpha}$ we have
$c_{i,\alpha}=\frac{\sqrt{n_{i}+1}}{\alpha}$,
$d_{i,\alpha}=\frac{\alpha}{n_{i}+1}$ and $\ell_{i,\alpha}$ is a small contour
around the pole $\infty^{i}$.
ii) Consider a flat function $t^{A}(\omega)\in\mathcal{S}(\omega)$ from the
list (3.21)-(3.22). If $t^{A}(\omega)\neq s^{i}(\omega)$, then $t^{A}(\omega)$
also admits the following integral representation:

$t^{A}(\omega)=c_{A}\int_{\ell_{A}}\big{(}\lambda(P)\big{)}^{d_{A}}\omega(P),$

where $c_{A}$ and $d_{A}$ are as above.
iii) The list (3.23) coincides (up to multiplicative constants) with
Dubrovin’s primary differentials given by Shramchenko [41] (Theorem 2) in
terms of the bidifferential $W(P,Q)$. We have chosen different multiplicative
constants in order to introduce a convenient duality relation between flat
coordinates (see Definition 3.1 below).

###### Proposition 3.4

The following statements hold:

  * _1)_

For $i=0,\dots,m$ and $\alpha=1,\dots,n_{i}$, the differential
$\phi_{t^{i,\alpha}}$ is an Abelian differential of the second kind. It has
the only pole at $\infty^{i}$ with the principal part:

$\phi_{t^{i,\alpha}}(P)\underset{P\sim\infty^{i}}{=}\left(\frac{\sqrt{n_{i}+1}}{z_{i}(P)^{\alpha+1}}+O(1)\right)dz_{i}(P),\quad\quad\lambda(P)=z_{i}(P)^{-n_{i}-1}.$
(3.25)

  * _2)_

For $i=1,\dots,m$, the differential $\phi_{v^{i}}$ is an Abelian differential
of the second kind having the only pole at $\infty^{i}$ with principal part:

$\phi_{v^{i}}(P)\underset{P\sim\infty^{i}}{=}\left(\frac{n_{i}+1}{z_{i}(P)^{n_{i}+2}}\
+\ O(1)\right)dz_{i}(P)=-d\lambda(P)+\
\text{holomorphic},\quad\quad\lambda(P)=z_{i}(P)^{-n_{i}-1}.$ (3.26)

  * _3)_

For $i=1,\dots,m$, the differential $\phi_{s^{i}}$ is an Abelian differential
of the third kind having simple poles at the points $\infty^{i}$ and
$\infty^{0}$ with residues $1$, $-1$ respectively. Furthermore, the following
items hold true:

    * i)

In genus zero, we have

$\phi_{s^{i}}(P)=\Omega_{\infty^{0}\infty^{i}}(P)=\left(\frac{1-\delta_{\infty^{i},\infty}}{z_{P}-\infty^{i}}-\frac{1-\delta_{\infty^{0},\infty}}{z_{P}-\infty^{0}}\right)dz_{P},$
(3.27)

with $\infty\in\mathbb{P}^{1}$ being the point at infinity and
$\delta_{\infty^{i},\infty}$ the Kronecker symbol.

    * ii)

If $g\geq 1$, then the differential $\phi_{s^{i}}$ is given by

$\begin{split}\phi_{s^{i}}(P)&=\Omega_{\infty^{0}\infty^{i}}(P)=d_{P}\log\left(\frac{\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(\infty^{i})\big{)}}{\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(\infty^{0})\big{)}}\right)\\\
&=\sum_{k=1}^{g}\omega_{k}(P)\bigg{(}\frac{\partial_{z_{k}}\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(\infty^{i})\big{)}}{\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(\infty^{i})\big{)}}-\frac{\partial_{z_{k}}\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(\infty^{0})\big{)}}{\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(\infty^{0})\big{)}}\bigg{)}.\end{split}$
(3.28)

Here $\mathcal{A}$ denotes the Abel map and $\Theta_{\Delta}$ is the Riemann
theta function (2.8) corresponding to an odd and non-singular half integer
characteristic $\Delta$.

  * _4)_

For $k=1,\dots,g$, the differentials $\phi_{\rho^{k}}=\omega_{k}$ are
holomorphic.

  * _5)_

For $k=1,\dots,g$, $\phi_{u^{k}}(P)$ is a multivalued differential with a jump
along the cycle $b_{k}$ as follows:

$\phi_{u^{k}}(P+b_{k})-\phi_{u^{k}}(P)=-2{\rm{i}}\pi d\lambda(P).$ (3.29)

Furthermore, the $a$-periods of $\phi_{\rho^{k}}$ are as follows
$\displaystyle\oint_{a_{l}}\phi_{\rho^{k}}(P)=\delta_{kl}$, while the other
differentials have zero $a$-periods.

These analytic properties of the differentials (3.23) were essentially stated
in [14] (Lecture 5) and [41] (Theorem 2). Since they will be important in
several subsequent results, we include detailed proofs for the reader’s
convenience.

_Proof:_ 1) and 2) Let $P\in C_{g}\setminus\\{\infty^{i}\\}$ be fixed, where
$\infty^{i}$ is among the assigned poles $\infty^{0},\dots,\infty^{m}$ of the
covering $(C_{g},\lambda)$. Since the differential $W(P,\cdot)$ is holomorphic
on $C_{g}\setminus\\{P\\}$, by notation (2.7), we can write

$\displaystyle\frac{1}{\alpha}\underset{\infty^{i}}{{\rm
res}}\lambda(Q)^{\frac{\alpha}{n_{i}+1}}W(P,Q)$
$\displaystyle=\frac{1}{\alpha!}W^{(\alpha-1)}(P,\infty^{i}).$

In particular, $\phi_{t^{i,\alpha}}$ is a holomorphic differential on
$C_{g}\setminus\\{\infty^{i}\\}$.
On the other hand, from (2.12) we know that

$W(P,Q)\underset{Q\sim\infty^{i}}{\underset{P\sim\infty^{i}}{=}}\bigg{(}\frac{1}{(z_{i}(P)-z_{i}(Q))^{2}}+O(1)\bigg{)}dz_{i}(P)dz_{i}(Q).$

Differentiating $(\alpha-1)$-times this relation w.r.t. $z_{i}(Q)$ and
evaluating it at $Q=\infty^{i}$ we obtain

$W^{(\alpha-1)}(P,\infty^{i})=\Big{(}\frac{\alpha!}{z_{i}(P)^{\alpha+1}}+O(1)\Big{)}dz_{i}(P).$

This gives us the stated result of the first item. The second one can be
proven in the same way taking $n_{i}+1$ instead of $\alpha$.
3) When $g=0$, we obtain formula (3.27) for the differential $\phi_{s^{i}}$ by
integrating the genus zero bidifferential (2.14) along a path from
$\infty^{0}$ to $\infty^{i}$. If $g\geq 1$, then formula (2.11) defining the
symmetric bidifferential $W(P,Q)$ in terms of the Abel map and Riemann theta
function brings us to write:

$\displaystyle\phi_{s^{i}}(P)$
$\displaystyle=d_{P}\log\left(\frac{\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(\infty^{i})\big{)}}{\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(\infty^{0})\big{)}}\right)$
$\displaystyle=\sum_{k=1}^{g}\omega_{k}(P)\bigg{(}\frac{\partial_{z_{k}}\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(\infty^{i})\big{)}}{\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(\infty^{i})\big{)}}-\frac{\partial_{z_{k}}\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(\infty^{0})\big{)}}{\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(\infty^{0})\big{)}}\bigg{)}.$

We thus arrive at (3.28). Moreover the function
$\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(\infty^{j})\big{)}$ is
holomorphic near the point $\infty^{j}$ and has the following behavior:

$\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(\infty^{j})\big{)}\underset{P\sim\infty^{j}}{=}\bigg{(}\sum_{l=1}^{g}\omega_{l}(\infty^{j})\partial_{z_{l}}\Theta_{\Delta}(0)\bigg{)}z_{j}(P)+o\big{(}z_{j}(P)\big{)},\quad
j=0,i.$

This shows the desired analytic properties of the differential $\phi_{s^{i}}$.
4) The fact that $\phi_{\rho^{k}}$ coincides with the normalized holomorphic
differential $\omega_{k}$ follows directly from the quasi-periodicity property
(2.9) of the function $\Theta_{\Delta}$:

$\phi_{\rho^{k}}(P)=\frac{1}{2i\pi}d_{P}\Big{[}\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(Q)-\mathbb{B}e_{k}\big{)}\Big{)}-\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(Q)\big{)}\Big{)}\Big{]}=\omega_{k}(P).$

5) Now we are going to prove the last assertion. Our arguments here are
different from those used in [41]. By (2.11), we can write

$\displaystyle\lambda(Q)W(P,Q)$
$\displaystyle=\lambda(Q)d_{P}d_{Q}\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(Q)\big{)}\Big{)}$
$\displaystyle=d_{Q}\Big{[}\lambda(Q)d_{P}\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(Q)\big{)}\Big{)}\Big{]}-d\lambda(Q)d_{P}\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(Q)\big{)}\Big{)}.$

Therefore

$\phi_{u^{k}}(P)=-\oint_{a_{k}}d_{P}\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(Q)\big{)}\Big{)}d\lambda(Q).$

Consider the function

$F_{u^{k}}(P):=\int^{P}\phi_{u^{k}}=-\oint_{a_{k}}\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(Q)\big{)}\Big{)}d\lambda(Q).$

Using the quasi-periodicity properties (2.9) of the
$\Theta_{\Delta}$-function, we conclude that $F_{u^{k}}$ is a multivalued
function with

$\displaystyle F_{u^{k}}(P+a_{j})-F_{u^{k}}(P)$
$\displaystyle=\oint_{a_{j}}\phi_{u^{k}}=\oint_{a_{k}}\lambda(Q)\left(\oint_{P\in
a_{j}}W(P,Q)\right)=0;$ $\displaystyle F_{u^{k}}(P+b_{j})-F_{u^{k}}(P)$
$\displaystyle=\oint_{b_{j}}\phi_{u^{k}}=-\oint_{a_{k}}\log\bigg{(}\frac{\Theta_{\Delta}\big{(}\mathcal{A}(P+b_{j})-\mathcal{A}(Q)\big{)}}{\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(Q)\big{)}}\bigg{)}d\lambda(Q)$
$\displaystyle=-\oint_{a_{k}}\bigg{(}-{\rm{i}}\pi\mathop{\langle\mathbb{B}e_{j},e_{j}\rangle}-2{\rm{i}}\pi\mathop{\langle\beta,e_{j}\rangle}-2{\rm{i}}\pi\int_{Q}^{P}\omega_{j}\bigg{)}d\lambda(Q)$
$\displaystyle=2{\rm{i}}\pi\oint_{a_{k}}\Big{(}\int_{Q}^{P}\omega_{j}\Big{)}d\lambda(Q)$
$\displaystyle=2{\rm{i}}\pi\oint_{a_{k}}\bigg{(}d_{Q}\Big{(}\lambda(Q)\int_{Q}^{P}\omega_{j}\Big{)}+\lambda(Q)\omega_{j}(Q)\bigg{)}$
$\displaystyle=2{\rm{i}}\pi\bigg{(}\lambda(Q)\int_{Q}^{P}\omega_{j}\bigg{)}\bigg{|}_{Q=P_{0}}^{Q=P_{0}+a_{k}}+2{\rm{i}}\pi\oint_{a_{k}}\lambda(Q)\omega_{j}(Q),$

where $P_{0}$ is a starting point of the contour $a_{k}$ which we keep
arbitrary for now.
Thus we arrive at

$F_{u^{k}}(P+b_{j})-F_{u^{k}}(P)=\oint_{b_{j}}\phi_{u^{k}}=-2{\rm{i}}\pi\lambda(P_{0})\delta_{jk}+2{\rm{i}}\pi\oint_{a_{k}}\lambda(Q)\omega_{j}(Q).$
(3.30)

On the other hand, note that when considering the difference
$F_{u^{k}}(P+b_{j})-F_{u^{k}}(P)$ we implicitly consider the point $P$ as a
starting point of the cycle $b_{j}$. In the case $j=k$, let us take the
starting points of $a_{k}$ and $b_{k}$ to coincide with their intersection
point. That is let us choose $P_{0}=P$. Now, given that
$\phi_{u^{k}}(P)=dF_{u^{k}}(P)$, by (3.30) we deduce that $\phi_{u^{k}}$ has
jumps only along the cycle $b_{k}$ and we have

$\phi_{u^{k}}(P+b_{k})-\phi_{u^{k}}(P)=dF_{u^{k}}(P+b_{k})-dF_{u^{k}}(P)=-2{\rm{i}}\pi{d\lambda(P)}.$

$\square$

The goal of the following theorem is to elucidate the principal value of the
integral $\int_{\infty^{0}}^{\infty^{j}}$ of the Abelian differential of the
third kind $\phi_{s^{i}}(P)=\Omega_{\infty^{0}\infty^{i}}(P)$ given by
(3.27)-(3.28). In other words, according to the third type of flat functions
(3.21), we are going to calculate the functions $s_{ij}:=s^{j}(\phi_{s^{i}})$.

###### Theorem 3.3

Let $(C_{g},\lambda)$ be a covering of $\mathbb{P}^{1}$ with prescribed poles
$\infty^{0},\dots,\infty^{m}$ of order $n_{0}+1,\dots,n_{m}+1$, respectively.

_i)_ Let $g=0$ and choose the point $\infty^{0}$ in such a way that
$\infty^{0}\in\mathbb{C}$ (without loss of generality, this may always be
assumed).
Then

$\begin{array}[]{lllll}s_{ij}=\displaystyle\log\left(\frac{\infty^{i}-\infty^{j}}{(\infty^{j}-\infty^{0})(\infty^{i}-\infty^{0})}\right)+\frac{\log\big{(}\mu_{0}(\infty^{0})\big{)}}{n_{0}+1},&\hbox{if}\quad
i\neq j\ \text{and}\ \infty^{i},\infty^{j}\in\mathbb{C};\\\ \\\
s_{ij}=\displaystyle-\log(\infty^{j}-\infty^{0})+\frac{\log\big{(}\mu_{0}(\infty^{0})\big{)}}{n_{0}+1},&\hbox{if}\quad
i\neq j\ \text{and}\ \infty^{i}=\infty;\\\ \\\
s_{ij}=\displaystyle-\log(\infty^{0}-\infty^{i})+\frac{\log\big{(}\mu_{0}(\infty^{0})\big{)}}{n_{0}+1},&\hbox{if}\quad
i\neq j\ \text{and}\ \infty^{j}=\infty;\\\ \\\
s_{ij}=\displaystyle\frac{\log\big{(}\mu_{i}(\infty^{i})\big{)}}{n_{i}+1}+\frac{\log\big{(}\mu_{0}(\infty^{0})\big{)}}{n_{0}+1}-2\log(\infty^{i}-\infty^{0})-\log(-1),&\hbox{if}\quad
i=j\ \text{and}\ \infty^{i}\in\mathbb{C};\\\ \\\
s_{ij}=\displaystyle\frac{\log\big{(}\mu_{i}(\infty^{i})\big{)}}{n_{i}+1}+\frac{\log\big{(}\mu_{0}(\infty^{0})\big{)}}{n_{0}+1},&\hbox{if}\quad
i=j\ \text{and}\ \infty^{i}=\infty,\end{array}$

where, for $i=0,\dots,m$, $\mu_{i}$ denotes the meromorphic function on
$\mathbb{P}^{1}$ defined by

$\mu_{i}(P)=\left\\{\begin{array}[]{ll}(z_{P}-\infty^{i})^{n_{i}+1}\lambda(P),&\quad\hbox{if}\quad\infty^{i}\in\mathbb{C};\\\
\\\
z_{P}^{-n_{i}-1}\lambda(P),&\quad\hbox{if}\quad\infty^{i}=\infty.\end{array}\right.$
(3.31)

_ii)_ Assume that $g\geq 1$. Then

$\begin{split}&s_{ij}=\log\left(\frac{\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{j})-\mathcal{A}(\infty^{i})\big{)}}{\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{j})-\mathcal{A}(\infty^{0})\big{)}\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{i})-\mathcal{A}(\infty^{0})\big{)}}\right)+\log\left(\sum_{k=1}^{g}\omega_{k}(\infty^{0})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{)}\right)-\log(-1),\quad
i\neq j;\\\
&s_{ii}=\log\left(\sum_{k=1}^{g}\omega_{k}(\infty^{i})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{)}\right)+\log\left(\sum_{k=1}^{g}\omega_{k}(\infty^{0})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{)}\right)-2\log\left(\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{i})-\mathcal{A}(\infty^{0})\big{)}\right)-\log(-1).\end{split}$
(3.32)

Here, as above, $\Theta_{\Delta}$ denotes the Riemann theta function (2.8)
corresponding to an odd and non-singular half integer characteristic $\Delta$.

###### Remark 3.3

By definition, the function $\mu_{i}$ in (3.31) is holomorphic near
$\infty^{i}$ and $\mu_{i}(\infty^{i})\in\mathbb{C}\setminus\\{0\\}$.
When $g\geq 1$, the k-th component of the vector
$\mathcal{A}(\infty^{i})-\mathcal{A}(\infty^{0})\in\mathbb{C}^{g}$ is
$\int_{\infty^{0}}^{\infty^{i}}\omega_{k}=s^{i}(\omega_{k})$.

_Proof:_ i) Below, for any $i_{0}=0,\dots,m$, the point
$\infty^{i_{0},\varepsilon}$ is taken near the pole $\infty^{i_{0}}$ of
$\lambda$ such that

$\infty^{i_{0},\varepsilon}=\left\\{\begin{array}[]{ll}\infty^{i_{0}}+\varepsilon,&\quad\hbox{if}\quad\infty^{i_{0}}\in\mathbb{C};\\\
\\\
1/{\varepsilon},&\quad\hbox{if}\quad\infty^{i_{0}}=\infty,\end{array}\right.$

where $\varepsilon=\varepsilon(i_{0})$ is a complex number such that
$|\varepsilon|$ is small enough.
This yields that the image of $\infty^{i_{0},\varepsilon}$ by $\mu_{i_{0}}$ is
given by

$\mu_{i_{0}}(\infty^{i_{0},\varepsilon})=\varepsilon^{n_{i}+1}\lambda(\infty^{i_{0},\varepsilon})=\Big{(}\varepsilon/{z_{i_{0}}(\infty^{i_{0},\varepsilon})}\Big{)}^{n_{i}+1},$
(3.33)

with $z_{i_{0}}(P)$ being the standard local parameter near the pole
$\infty^{i_{0}}$.
To demonstrate our result, we consider two cases, depending on wether
$\infty^{i}$ is finite or infinite.
First case: $\infty^{i}\neq\infty$. This implies that
$\Omega_{\infty^{0}\infty^{i}}(P)$ is as follows:

$\Omega_{\infty^{0}\infty^{i}}(P)=\frac{dz_{P}}{z_{P}-\infty^{i}}-\frac{dz_{P}}{z_{P}-\infty^{0}}.$

If $\infty^{j}\neq\infty$ and $i\neq j$, then by (3.33) we have

$\displaystyle\int_{\infty^{0,\varepsilon}}^{\infty^{j}}\Omega_{\infty^{0}\infty^{i}}(P)$
$\displaystyle=\int_{\varepsilon+\infty^{0}}^{\infty^{j}}\frac{dz_{P}}{z_{P}-\infty^{i}}-\int_{\varepsilon+\infty^{0}}^{\infty^{j}}\frac{dz_{P}}{z_{P}-\infty^{0}}$
$\displaystyle=\log(\infty^{j}-\infty^{i})-\log(\infty^{0}+\varepsilon-\infty^{i})+\log(\varepsilon)-\log(\infty^{j}-\infty^{0})$
$\displaystyle=\log\Big{(}\frac{\infty^{j}-\infty^{i}}{\infty^{j}-\infty^{0}}\Big{)}-\log(\infty^{0}+\varepsilon-\infty^{i})+\frac{\log\big{(}\mu_{0}(\infty^{0,\varepsilon})\big{)}}{n_{0}+1}+\log\big{(}z_{0}(\infty^{0,\varepsilon})\big{)}.$

Omitting the term $\log\big{(}z_{0}(\infty^{0,\varepsilon})\big{)}$ which
diverges and letting $\varepsilon\longrightarrow 0$, we arrive at

$p.v.\int_{\infty^{0}}^{\infty^{j}}\Omega_{\infty^{0}\infty^{i}}(P)=\log\bigg{(}\frac{\infty^{j}-\infty^{i}}{(\infty^{j}-\infty^{0})(\infty^{0}-\infty^{i})}\bigg{)}+\frac{\log\big{(}\mu_{0}(\infty^{0})\big{)}}{n_{0}+1}.$

If $\infty^{j}=\infty$, then by applying the above case to a point $R$
belonging to a small neighborhood of $\infty$, we find

$p.v.\int_{\infty^{0}}^{R}\Omega_{\infty^{0}\infty^{i}}(P)=\log\bigg{(}\frac{z_{R}-\infty^{i}}{(z_{R}-\infty^{0})(\infty^{0}-\infty^{i})}\bigg{)}+\frac{\log\big{(}\mu_{0}(\infty^{0})\big{)}}{n_{0}+1}.$

Moreover

$\displaystyle\int_{R}^{\infty^{j}}\Omega_{\infty^{0}\infty^{i}}(P)$
$\displaystyle=\int_{0}^{1/{z_{R}}}\bigg{(}\frac{1}{\tau(1-\tau\infty^{i})}-\frac{1}{\tau(1-\tau\infty^{0})}\bigg{)}d\tau=\log\Big{(}\frac{z_{R}-\infty^{0}}{z_{R}-\infty^{i}}\Big{)}.$

Therefore

$p.v.\int_{\infty^{0}}^{\infty^{j}}\Omega_{\infty^{0}\infty^{i}}(P)=-\log\big{(}\infty^{0}-\infty^{i})+\frac{\log\big{(}\mu_{0}(\infty^{0})\big{)}}{n_{0}+1}.$

We now treat the case $i=j$. Using twice (3.33), we get

$\displaystyle\int_{\infty^{0,\varepsilon}}^{\infty^{i,\epsilon}}\Omega_{\infty^{0}\infty^{i}}(P)$
$\displaystyle=\int_{\varepsilon+\infty^{0}}^{\epsilon+\infty^{i}}\frac{dz_{P}}{z_{P}-\infty^{i}}-\int_{\varepsilon+\infty^{0}}^{\epsilon+\infty^{i}}\frac{dz_{P}}{z_{P}-\infty^{0}}$
$\displaystyle=\log(\epsilon)-\log(\infty^{0}+\varepsilon-\infty^{i})+\log(\varepsilon)-\log(\epsilon+\infty^{i}-\infty^{0})$
$\displaystyle=\frac{\log\big{(}\mu_{i}(\infty^{i,\epsilon})\big{)}}{n_{i}+1}+\log\big{(}z_{i}(\infty^{i,\varepsilon})\big{)}-\log(\infty^{0}+\varepsilon-\infty^{i})$
$\displaystyle\quad+\frac{\log\big{(}\mu_{0}(\infty^{0,\varepsilon})\big{)}}{n_{0}+1}+\log\big{(}z_{0}(\infty^{0,\varepsilon}))-\log(\epsilon+\infty^{i}-\infty^{0}).$

Accordingly, we arrive at

$p.v.\int_{\infty^{0}}^{\infty^{i}}\Omega_{\infty^{0}\infty^{i}}(P)=\frac{\log\big{(}\mu_{i}(\infty^{i})\big{)}}{n_{i}+1}+\frac{\log\big{(}\mu_{0}(\infty^{0})\big{)}}{n_{0}+1}-2\log(\infty^{i}-\infty^{0})-\log(-1).$

Second case: $\infty^{i}=\infty$. In this case, we have
$\displaystyle\Omega_{\infty^{0}\infty^{i}}(P)=-\frac{dz_{P}}{z_{P}-\infty^{0}}$.
If $\infty^{j}\in\mathbb{C}$, then

$\displaystyle\int_{\infty^{0,\varepsilon}}^{\infty^{j}}\Omega_{\infty^{0}\infty^{i}}(P)$
$\displaystyle=-\int_{\varepsilon+\infty^{0}}^{\infty^{j}}\frac{dz_{P}}{z_{P}-\infty^{0}}=\log(\varepsilon)-\log(\infty^{j}-\infty^{0})$
$\displaystyle=\frac{\log\big{(}\mu_{0}(\infty^{0,\varepsilon})\big{)}}{n_{0}+1}+\log\big{(}z_{0}(\infty^{0,\varepsilon})\big{)}-\log(\infty^{j}-\infty^{0})$

and thus

$p.v.\int_{\infty^{0}}^{\infty^{j}}\Omega_{\infty^{0}\infty^{i}}(P)=\frac{\log(\mu_{0}(\infty^{0})\big{)}}{n_{0}+1}-\log(\infty^{j}-\infty^{0}).$

If $\infty^{j}=\infty=\infty^{i}$, then we have

$\displaystyle\int_{\infty^{0,\varepsilon}}^{\infty^{i,\epsilon}}\Omega_{\infty^{0}\infty^{i}}(P)$
$\displaystyle=-\int_{\varepsilon+\infty^{0}}^{1/\epsilon}\frac{dz_{P}}{z_{P}-\infty^{0}}=\log(\epsilon)-\log(1-\epsilon\infty^{0})+\log(\varepsilon)$
$\displaystyle=\frac{\log\big{(}\mu_{i}(\infty^{i,\epsilon})\big{)}}{n_{i}+1}+\log\big{(}z_{i}(\infty^{i,\varepsilon})\big{)}-\log(1-\epsilon\infty^{0})+\frac{\log\big{(}\mu_{0}(\infty^{0,\varepsilon})\big{)}}{n_{0}+1}+\log\big{(}z_{0}(\infty^{0,\varepsilon})\big{)}.$

Therefore

$p.v.\int_{\infty^{0}}^{\infty^{i}}\Omega_{\infty^{0}\infty^{i}}(P)=\frac{\log\big{(}\mu_{i}(\infty^{i})\big{)}}{n_{i}+1}+\frac{\log\big{(}\mu_{0}(\infty^{0})\big{)}}{n_{0}+1}.$

ii) Let $i=0,\dots,m$ and $\infty^{i,\varepsilon}$ be a point belonging to a
small neighborhood of $\infty^{i}$ such that
$z_{i}(\infty^{i,\varepsilon}):=\lambda(\infty^{i,\varepsilon})^{-\frac{1}{n_{i}+1}}=\varepsilon$.
Here, as usual, $z_{i}(P)$ denotes the standard local parameter near the point
$\infty^{i}$. We first claim the following asymptotic behavior:

$\begin{split}\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{i,\varepsilon})-\mathcal{A}(\infty^{i})\big{)}&=\varepsilon\sum_{k=1}^{g}\omega_{k}(\infty^{i})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{)}+\frac{\varepsilon^{2}}{2}\sum_{k=1}^{g}\omega_{k}^{\prime}(\infty^{i})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{)}\\\
&\quad+\frac{\varepsilon^{2}}{2}\sum_{k,l=1}^{g}\omega_{k}(\infty^{i})\omega_{l}(\infty^{i})\partial_{z_{k}}\partial_{z_{l}}\Theta_{\Delta}\big{(}0\big{)}+o(\varepsilon^{2}),\end{split}$
(3.34)

where $\omega_{k}(\infty^{i})$ and $\omega_{k}^{\prime}(\infty^{i})$ are the
evaluations defined by (2.7).
In particular, the following holds:

$\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{i,\varepsilon})-\mathcal{A}(\infty^{i})\big{)}\Big{)}=\log(\varepsilon)+\log\left(\sum_{k=1}^{g}\omega_{k}(\infty^{i})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{)}+O(\varepsilon)\right).$
(3.35)

Indeed, for all $k=1,\dots,g$, we observe that

$\displaystyle\chi_{k}(\varepsilon)$
$\displaystyle:=\int_{\infty^{i}}^{\infty^{i,\varepsilon}}\omega_{k}(P)=\int_{0}^{\varepsilon}\Big{(}\omega_{k}(\infty^{i})+\omega_{k}^{\prime}(\infty^{i})z_{i}(P)+\dots\Big{)}dz_{i}(P)$
$\displaystyle=\varepsilon\omega_{k}(\infty^{i})+\frac{\varepsilon^{2}}{2}\omega_{k}^{\prime}(\infty^{i})+o(\varepsilon^{2}),$

and we can write

$\displaystyle f(\varepsilon)$
$\displaystyle:=\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{i,\varepsilon})-\mathcal{A}(\infty^{i})\big{)}=\Theta_{\Delta}\Big{(}\chi_{1}(\varepsilon),\
\dots\ ,\chi_{g}(\varepsilon)\Big{)}.$

Differentiating $f$ with respect to $\varepsilon$ at $\varepsilon=0$, we
obtain

$\displaystyle
f^{\prime}(0)=\sum_{k=1}^{g}\omega_{k}(\infty^{i})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{)};$
$\displaystyle
f^{\prime\prime}(0)=\sum_{k=1}^{g}\omega_{k}^{\prime}(\infty^{i})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{)}+\sum_{k,l=1}^{g}\omega_{k}(\infty^{i})\omega_{l}(\infty^{i})\partial_{z_{k}}\partial_{z_{l}}\Theta_{\Delta}\big{(}0\big{)}.$

On the other hand, since $\Theta_{\Delta}$ is odd, it follows that
$f(0)=\Theta_{\Delta}(0)=0$. We thus get (3.34) expanding $f$ in the Taylor
series near $\varepsilon=0$.
We now move on to prove the stated formulas for the functions $s_{ij}$.
First case: $j\neq i$. Take a point $\infty^{0,\varepsilon}$ belonging to a
small neighborhood of $\infty^{0}$ with
$z_{0}(\infty^{0,\varepsilon}):=\lambda(\infty^{0,\varepsilon})^{-\frac{1}{n_{0}+1}}=\varepsilon$.
In view of (3.28) and (3.35), we obtain

$\displaystyle\int_{\infty^{0,\varepsilon}}^{\infty^{j}}\Omega_{\infty^{0}\infty^{i}}(P)$
$\displaystyle=\log\left(\frac{\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{j})-\mathcal{A}(\infty^{i})\big{)}}{\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{j})-\mathcal{A}(\infty^{0})\big{)}}\right)-\log\left(\frac{\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{0,\varepsilon})-\mathcal{A}(\infty^{i})\big{)}}{\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{0,\varepsilon})-\mathcal{A}(\infty^{0})\big{)}}\right)$
$\displaystyle=\log\left(\frac{\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{j})-\mathcal{A}(\infty^{i})\big{)}}{\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{j})-\mathcal{A}(\infty^{0})\big{)}}\right)-\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{0,\varepsilon})-\mathcal{A}(\infty^{i})\big{)}\Big{)}$
$\displaystyle\quad+\log(\varepsilon)+\log\bigg{(}\sum_{k=1}^{g}\omega_{k}(\infty^{0})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{)}+O(\varepsilon)\bigg{)}.$

Therefore by subtracting the divergent term
$\log(\varepsilon)=\log\big{(}z_{0}(\infty^{0,\varepsilon})\big{)}$ and
letting $\varepsilon\longrightarrow 0$, we deduce that

$\displaystyle
p.v.\int_{\infty^{0}}^{\infty^{j}}\Omega_{\infty^{0}\infty^{i}}(P)$
$\displaystyle=\log\bigg{(}\frac{\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{j})-\mathcal{A}(\infty^{i})\big{)}}{\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{j})-\mathcal{A}(\infty^{0})\big{)}\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{i})-\mathcal{A}(\infty^{0})\big{)}}\bigg{)}$
$\displaystyle\quad+\log\bigg{(}\sum_{k=1}^{g}\omega_{k}(\infty^{0})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{)}\bigg{)}-\log(-1).$

Second case: $j=i$. Take $\infty^{0,\varepsilon}$ as in the first case and
choose $\infty^{i,\epsilon}$ in neighborhood of $\infty^{i}$ such that
$z_{i}(\infty^{i,\epsilon}):=\lambda(\infty^{i,\epsilon})^{-\frac{1}{n_{i}+1}}=\epsilon$.
Again by virtue of (3.28) and (3.35), we have

$\displaystyle\int_{\infty^{0,\varepsilon}}^{\infty^{i,\epsilon}}\Omega_{\infty^{0}\infty^{i}}(P)$
$\displaystyle=\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{i,\epsilon})-\mathcal{A}(\infty^{i})\big{)}\Big{)}-\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{0,\varepsilon})-\mathcal{A}(\infty^{i})\big{)}\Big{)}$
$\displaystyle\quad-\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{i,\epsilon})-\mathcal{A}(\infty^{0})\big{)}\Big{)}+\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{0,\varepsilon})-\mathcal{A}(\infty^{0})\big{)}\Big{)}$
$\displaystyle=\log(\epsilon)+\log\bigg{(}\sum_{k=1}^{g}\omega_{k}(\infty^{i})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{)}+O(\epsilon)\bigg{)}-\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{0,\varepsilon})-\mathcal{A}(\infty^{i})\big{)}\Big{)}$
$\displaystyle\quad-\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{i,\epsilon})-\mathcal{A}(\infty^{0})\big{)}\Big{)}+\log(\varepsilon)+\log\bigg{(}\sum_{k=1}^{g}\omega_{k}(\infty^{0})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{)}+O(\varepsilon)\bigg{)}.$

Thus, by arguing as above we can see that

$\displaystyle
s_{ii}:=p.v.\int_{\infty^{0}}^{\infty^{i}}\Omega_{\infty^{0}\infty^{i}}(P)$
$\displaystyle=\log\bigg{(}\sum_{k=1}^{g}\omega_{k}(\infty^{i})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{)}\bigg{)}+\log\bigg{(}\sum_{k=1}^{g}\omega_{k}(\infty^{0})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{)}\bigg{)}$
$\displaystyle\quad-2\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{i})-\mathcal{A}(\infty^{0})\big{)}\Big{)}-\log(-1).$

The proof of the theorem is now complete.

$\square$

###### Remark 3.4

Assume that the surface $C_{g}$ is of genus $g\geq 1$ and consider the
holomorphic differentials $\omega_{k}$ and the multivalued differentials
$\phi_{u^{k}}$ (3.29). Similarly to (3.32), we can also express the flat
functions $s^{i}(\omega_{k})$ and $s^{i}\big{(}\phi_{u^{k}})$ in terms of the
Riemann Theta function and the Abel map. More precisely, for any
$k=1,\dots,g$, we have

$\displaystyle
s^{i}(\omega_{k})=\int_{\infty^{0}}^{\infty^{i}}\omega_{k}=\mathcal{A}_{k}(\infty^{i})-\mathcal{A}_{k}(\infty^{0});$
$\displaystyle
s^{i}(\phi_{u^{k}})=-\oint_{a_{k}}\log\bigg{(}\frac{\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{i})-\mathcal{A}(Q)\big{)}}{\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{i})-\mathcal{A}(Q)\big{)}}\bigg{)}d\lambda(Q).$

In addition, if we make use of (3.28) as well as the quasi-periodicity
properties of the Riemann theta function, then we observe that:

$s^{i}(\omega_{k})=s^{i}(\phi_{\rho^{k}})=\rho^{k}(\phi_{s^{i}})\quad\quad\text{and}\quad\quad
s^{i}(\phi_{u^{k}})=u^{k}(\phi_{s^{i}}),\quad\quad\forall\ i,k.$

Let us define the $N\times N$-matrix $H_{AB}:=t^{A}(\phi_{t^{B}})$ for the $N$
operations $\\{t^{A}\\}$ in (3.21) and (3.22) and the corresponding
differentials $\\{\phi_{t^{A}}\\}$ given by (3.23). The next result studies
the symmetry property of this matrix.

###### Proposition 3.5

Let $P_{0}$ be a marked point on $C_{g}$ such that $\lambda(P_{0})=0$. Assume
that all the cycles $\\{a_{k},b_{k}\\}$ start at the point $P_{0}$. Then the
following symmetry property holds:

$t^{A}(\phi_{t^{B}})=t^{B}(\phi_{t^{A}})+\log(-1)\sum_{i,j=1}^{m}(1-\delta_{ij})\delta_{s^{i},t^{A}}\delta_{s^{j},t^{B}},$
(3.36)

where $t^{A}$ and $t^{B}$ are from the list of flat functions (3.21)-(3.22).

_Proof:_ According to Theorem 3.3, we know that

$s^{j}(\phi_{s^{i}})=s_{ij}=s_{ji}+\log(-1)=s^{i}(\phi_{s^{j}})+\log(-1),\quad\forall\
i\neq j.$

When $t^{A}$ or $t^{B}$ is not of type $s^{i}$, then it suffices to justify
(3.36) when the contours defining the functions $t^{A}$ and $t^{B}$ have an
intersection point. Thus we need to discuss the following three cases

$\rho^{k}\big{(}\phi_{u^{k}}\big{)}=u^{k}\big{(}\phi_{\rho^{k}}\big{)},\quad\quad
v^{i}\big{(}\phi_{s^{i}}\big{)}=s^{i}\big{(}\phi_{v^{i}}\big{)}\quad\quad\text{and}\quad\quad
t^{i,\alpha}\big{(}\phi_{s^{i}}\big{)}=s^{i}\big{(}\phi_{t^{i,\alpha}}\big{)}.$

Equality (3.30) tells us that

$\rho^{j}\big{(}\phi_{u^{k}}\big{)}=u^{k}\big{(}\phi_{\rho^{j}}\big{)}-\lambda(P_{0})\delta_{jk},\quad
j,k=1,\dots,g,$

with $P_{0}$ being the intersection point of the cycles $a_{k}$ and $b_{k}$.
Thus the first equality holds true under the added assumption on the cycles
$a_{k}$ and $b_{k}$.
Let us prove the second equality. Denote by $\gamma_{i}$ a curve on the
surface $C_{g}$ from $\infty^{0}$ to $\infty^{i}$ not passing through any of
the ramification points $P_{j}$ or any of the $\infty^{j}$ except of
extremities.
Let
i) $U$ be a small connected open subset of $C_{g}$ containing the point
$\infty^{i}$ such that

$\forall\ P,Q\in U,\quad
W(P,Q)=\Big{(}\frac{1}{(z_{i}(P)-z_{i}(Q))^{2}}+\sum_{j,l\geq
0}c_{j,l}z_{i}^{j}(P)z_{i}^{l}(Q)\Big{)}dz_{i}(P)dz_{i}(Q),$

where $z_{i}(P)$ is the standard local parameter near $\infty^{i}$ defined by
$\lambda(P)=z_{i}^{-n_{i}-1}(P)$;
ii) $\mathfrak{c}_{i}$ be a cycle inside $U$ such that

$\underset{Q=\infty^{i}}{{\rm
res}}\lambda(Q)W(P,Q)=\frac{1}{2{\rm{i}}\pi}\oint_{Q\in\mathfrak{c}_{i}}\lambda(Q)W(P,Q);$

iii) $P_{0,i}$ be a point in $U\cap\gamma_{i}$ chosen in such a way that there
is no intersection point between the cycle $\mathfrak{c}_{i}$ and the part of
the curve $\gamma_{i}$ from $\infty^{0}$ to $P_{0,i}$.
With these assumptions, it suffices to prove that

$p.v.\int_{P_{0,i}}^{\infty^{i}}\bigg{(}\underset{\infty^{i}}{{\rm
res}}\lambda(Q)W(P,Q)\bigg{)}=\underset{\infty^{i}}{{\rm
res}}\bigg{(}\lambda(Q)\int_{P_{0,i}}^{\infty^{i}}W(P,Q)\bigg{)}.$

We have, using the notation introduced in (2.7),

$\displaystyle\forall\ P\in U,\quad\underset{\infty^{i}}{{\rm
res}}\lambda(Q)W(P,Q)$
$\displaystyle=\frac{1}{n_{i}!}W^{(n_{i})}(P,\infty^{i})$
$\displaystyle=\Big{(}\frac{n_{i}+1}{z_{i}(P)^{n_{i}+2}}+\sum_{j\geq
0}c_{j,n_{i}}z_{i}^{j}(P)\Big{)}dz_{i}(P)$
$\displaystyle=-d\lambda(P)+\Big{(}\sum_{j\geq
0}c_{j,n_{i}}z_{i}^{j}(P)\Big{)}dz_{i}(P)$

and then

$\displaystyle\int_{P_{0,i}}^{\infty^{i,\varepsilon}}\underset{\infty^{i}}{{\rm
res}}\lambda(Q)W(P,Q)$
$\displaystyle=\lambda(P_{0,i})-\lambda(\infty^{i,\varepsilon})+\int_{z_{i}(P_{0,i})}^{\varepsilon}\Big{(}\sum_{j\geq
0}c_{j,n_{i}}z_{i}^{j}(P)\Big{)}dz_{i}(P)$
$\displaystyle=\lambda(P_{0,i})-\lambda(\infty^{i,\varepsilon})+\sum_{j\geq
0}\frac{c_{j,n_{i}}}{j+1}\big{(}\varepsilon^{j+1}-z_{i}^{j+1}(P_{0,i})\big{)},$

where
$\varepsilon=z_{i}(\infty^{i,\varepsilon})=\Big{(}\lambda(\infty^{i,\varepsilon})\Big{)}^{-\frac{1}{n_{i}+1}}$.
Letting $\varepsilon\to 0$ and omitting the divergent term, we arrive at

$p.v.\int_{P_{0,i}}^{\infty^{i}}\bigg{(}\underset{\infty^{i}}{{\rm
res}}\lambda(Q)W(P,Q)\bigg{)}=\lambda(P_{0,i})-\sum_{j\geq
0}\frac{c_{j,n_{i}}}{j+1}z_{i}^{j+1}(P_{0,i}).$

On the other hand

$\displaystyle\int_{P_{0,i}}^{\infty^{i}}W(P,Q)$
$\displaystyle=\left(\int_{z_{i}(P_{0,i})}^{0}\bigg{(}\frac{1}{(z_{i}(P)-z_{i}(Q))^{2}}+\sum_{j,l\geq
0}c_{j,l}z_{i}^{j}(P)z_{i}^{l}(Q)\bigg{)}dz_{i}(P)\right)dz_{i}(Q)$
$\displaystyle=\bigg{(}\frac{1}{z_{i}(Q)}+\frac{1}{z_{i}(P_{0,i})-z_{i}(Q)}-\sum_{j,l\geq
0}\frac{c_{j,l}}{j+1}z_{i}^{j+1}(P_{0,i})z_{i}^{l}(Q)\bigg{)}dz_{i}(Q).$

Therefore

$\displaystyle\underset{\infty^{i}}{{\rm
res}}\bigg{(}\lambda(Q)\int_{P_{0,i}}^{\infty^{i}}W(P,Q)\bigg{)}$
$\displaystyle=\underset{0}{\rm
res}\left(z_{i}^{-n_{i}-1}(Q)\bigg{(}\frac{1}{z_{i}(Q)}+\frac{1}{z_{i}(P_{0,i})-z_{i}(Q)}-\sum_{j,l\geq
0}\frac{c_{j,l}}{j+1}z_{i}^{j+1}(P_{0,i})z_{i}^{l}(Q)\bigg{)}dz_{i}(Q)\right)$
$\displaystyle=\frac{1}{z_{i}^{n_{i}+1}(P_{0,i})}-\sum_{j\geq
0}\frac{c_{j,n_{i}}}{j+1}z_{i}^{j+1}(P_{0,i})$
$\displaystyle=\lambda(P_{0,i})-\sum_{j\geq
0}\frac{c_{j,n_{i}}}{j+1}z_{i}^{j+1}(P_{0,i}).$

Thus we have established that
$v^{i}\big{(}\phi_{s^{i}}\big{)}=s^{i}\big{(}\phi_{v^{i}}\big{)}$. The
symmetry
$t^{{i,\alpha}}\big{(}\phi_{s^{i}}\big{)}=s^{i}\big{(}\phi_{t^{i,\alpha}}\big{)}$
can be shown similarly.

$\square$

###### Theorem 3.4

Let $\big{\\{}\big{(}C_{g},\lambda\big{)},\\{a_{k},b_{k}\\}\big{\\}}$ be a
point of (a fixed covering of) the simple Hurwitz space
$\widehat{\mathcal{H}}_{g,L}(n_{0},\dots,n_{m})$ and $P_{0}$ be a marked point
on $C_{g}$ such that $\lambda(P_{0})=0$. Assume that all the cycles
$\\{a_{k},b_{k}\\}$ start at the point $P_{0}$.
Then the $N$ flat functions of the set $\mathcal{S}(\omega)$ defined by
(3.21)-(3.22) give a system of flat coordinates of the Dubrovin metric
$\eta(\omega)$.
Moreover, in these coordinates, the nonzero entries of the constant matrix of
the contravariant metric $\eta^{*}(\omega)$ are as follows:

$\begin{split}&\eta^{*}(\omega)(dt^{i,\alpha}(\omega),dt^{j,\beta}(\omega))=\delta_{ij}\delta_{\alpha+\beta,n_{j}+1};\\\
&\eta^{*}(\omega)(dv^{i}(\omega),ds^{j}(\omega))=\delta_{ij};\\\
&\eta^{*}(\omega)(d\rho^{i}(\omega),du^{j}(\omega))=\delta_{ij}.\end{split}$
(3.37)

_Proof:_ We already know that the functions of the set $\mathcal{S}(\omega)$
are all flat with respect to the metric $\eta(\omega)$. Let us calculate the
matrix of $\eta^{*}(\omega)$ in these coordinates. The proof relies on the
following tools:
\- the characteristic properties of the differentials (3.23) that were
described in Proposition 3.4;
\- the following particular versions of formulas (3.18)-(3.19) for
$\eta^{*}(\omega)=\left(\mathbf{ds^{2}_{0,1}}(\omega)\right)^{*}$:

$\eta^{*}(\omega)\left(dt^{A}(\omega),dt^{B}(\omega)\right)=e.t^{A}(\phi_{t^{B}})=\sum_{k=1}^{N}\underset{P_{k}}{{\rm
res}}\frac{\phi_{t^{A}}(P)\phi_{t^{B}}(P)}{d\lambda(P)},$ (3.38)

where $e$ is the vector field $e:=\sum_{j=1}^{N}\partial_{\lambda_{j}}$.
Note that the symmetry property (3.36) implies that
$e.t^{A}(\phi_{t^{B}})=e.t^{B}(\phi_{t^{A}})$.
From their analytic properties, the differentials $\phi_{t^{i,\alpha}}$,
$\phi_{v^{i}}$, $\phi_{s^{i}}$ and $\phi_{\rho^{k}}=\omega_{k}$ are all
Abelian differentials on the surface $C_{g}$ and they are all holomorphic near
any simple ramification point $P_{k}$. Therefore, by analysing the poles of
the differential $\frac{\phi_{t^{A}}(P)\phi_{t^{B}}(P)}{d\lambda(P)}$ and
applying the residue theorem, it follows that the right hand side of (3.38) is
straightforwardly computed provided that $t^{A}$ and $t^{B}$ are among the
four types of operations
$\big{\\{}t^{i,\alpha},v^{i},s^{i},\rho^{k}\big{\\}}$.
For example if we consider the case $t^{A}=t^{i,\alpha}$ and
$t^{B}=t^{i,\beta}$, with $i=0,\dots,m$ and $\alpha,\beta=1,\dots,n_{i}$, then
the residue theorem and the behavior

$\phi_{t^{i,\alpha}}(P)\underset{P\sim\infty^{i}}{=}\Big{(}\sqrt{n_{i}+1}z_{i}(P)^{-\alpha-1}+\text{reg}\Big{)}dz_{i}(P),\quad\quad\lambda(P)=z_{i}(P)^{-n_{i}-1},$

imply that

$\displaystyle\eta^{*}(\omega)\left(dt^{A}(\omega),dt^{B}(\omega)\right)=e.t^{i,\alpha}(\phi_{t^{i,\beta}})=-\underset{\infty^{i}}{{\rm
res}}\frac{\phi_{t^{i,\alpha}}(P)\phi_{t^{i,\beta}}(P)}{d\lambda(P)}$
$\displaystyle=\frac{1}{n_{i}+1}\underset{0}{{\rm
res}}\bigg{(}z_{i}(P)^{n_{i}+2}\Big{(}\sqrt{n_{i}+1}z_{i}(P)^{-\alpha-1}+\text{reg}\Big{)}\Big{(}\sqrt{n_{i}+1}z_{i}(P)^{-\beta-1}+\text{reg}\Big{)}\bigg{)}dz_{i}(P)$
$\displaystyle=\delta_{\alpha+\beta,n_{i}+1}.$

However, the use of (3.38) in the case where $\phi_{t^{A}}=\phi_{u^{k}}$ is
the multivalued differential having jumps along the cycle $b_{k}$ needs some
details.
For $k=1,\dots,g$, let us consider the differential

$\Phi_{k}(P):=\frac{\phi(P)}{d\lambda(P)}\phi_{u^{k}}(P),$ (3.39)

where $\phi$ is one of the differentials listed in (3.23). Then $\Phi_{k}$ is
a multivalued differential and by (3.29), its jumps are as follows:

$\displaystyle\begin{array}[]{llll}\Phi_{k}(P+a_{j})-\Phi_{k}(P)=0,&\hbox{if}\quad
j=1,\dots,g;\\\ \\\
\Phi_{k}(P+b_{k})-\Phi_{k}(P)=-2{\rm{i}}\pi\phi(P),&\hbox{if}\quad\phi\neq\phi_{u^{k}};\\\
\\\
\Phi_{k}(P+b_{k})-\Phi_{k}(P)=-4{\rm{i}}\pi\phi_{u^{k}}(P)-4\pi^{2}d\lambda(P),&\hbox{if}\quad\phi=\phi_{u^{k}};\\\
\\\
\Phi_{k}(P+b_{j})-\Phi_{k}(P)=-2{\rm{i}}\pi\phi_{u^{k}}(P),&\hbox{if}\quad\phi=\phi_{u^{j}}\quad\text{and}\quad
j\neq k;\\\ \\\ \Phi_{k}(P+b_{j})-\Phi_{k}(P)=0,&\hbox{if}\quad\text{$\phi$ is
Abelian and $j\neq k$}.\end{array}$

In addition, since the multivalued differentials
$\phi_{u^{1}},\dots,\phi_{u^{g}}$ are all single valued and holomorphic inside
the fundamental polygon $F_{g}$ associated with a compact surface $C_{g}$, it
follows that the differential $\Phi_{k}$ is single valued and meromorphic
inside $F_{g}$ and has simple poles at the ramifications points
$P_{1},\dots,P_{N}$.
Furthermore, by using the distinctive analytic properties of the differentials
(3.23) (see Proposition 3.4) and keeping in mind the presence of the factor
$d\lambda(P)$, we deduce that $\Phi_{k}$ has no other poles inside the
fundamental polygon $F_{g}$.
Thus, by applying the residue theorem and recalling that $\partial
F_{g}=\sum_{j=1}^{g}\big{(}a_{j}+b_{j}+a_{j}^{-1}+b_{j}^{-1}\big{)}$ and
taking into account the mentioned jumps of $\Phi_{k}$, we obtain

$\displaystyle e.u^{k}(\phi)$
$\displaystyle=\sum_{s=1}^{N}\underset{P_{s}}{{\rm
res}}\Phi_{k}(P)=\frac{1}{2{\rm{i}}\pi}\oint_{\partial{F_{g}}}\Phi_{k}(P)$
$\displaystyle=\frac{1}{2{\rm{i}}\pi}\sum_{j=1}^{g}\bigg{(}\oint_{a_{j}}\Phi_{k}(P)+\oint_{a_{j}^{-1}}\Phi_{k}(Q)\bigg{)}+\frac{1}{2{\rm{i}}\pi}\sum_{j=1}^{g}\bigg{(}\oint_{b_{j}}\Phi_{k}(P)+\oint_{b_{j}^{-1}}\Phi_{k}(Q)\bigg{)}$
$\displaystyle=\frac{1}{2{\rm{i}}\pi}\sum_{j=1}^{g}\bigg{(}\oint_{a_{j}}\Phi_{k}(P)-\oint_{a_{j}}\Phi_{k}(P+b_{j})\bigg{)}+\frac{1}{2{\rm{i}}\pi}\sum_{j=1}^{g}\bigg{(}\oint_{b_{j}}\Phi_{k}(P)-\oint_{b_{j}}\underbrace{\Phi_{k}(P+a_{j}^{-1})}_{=\Phi_{k}(P)}\bigg{)}$
$\displaystyle=\bigg{(}\sum_{j=1,j\neq
k}^{g}\delta_{\phi,\phi_{u^{j}}}\oint_{a_{j}}\phi_{u^{k}}\bigg{)}+\big{(}1-\delta_{\phi,\phi_{u^{k}}}\big{)}\oint_{a_{k}}\phi+\delta_{\phi,\phi_{u^{k}}}\oint_{a_{k}}\Big{(}2\phi_{u^{k}}(P)-2{\rm{i}}\pi
d\lambda(P)\Big{)}$
$\displaystyle=\big{(}1-\delta_{\phi,\phi_{u^{k}}}\big{)}\delta_{\phi,\phi_{\rho^{k}}}=\delta_{\phi,\phi_{\rho^{k}}},$

where $\delta_{\phi,\phi_{t^{B}}}$ denotes the Kronecker delta function.

$\square$

###### Remark 3.5

Observe that if $g\geq 1$, then by using explicit formulas (3.32) for the
functions $s_{ij}$, we see that the action of the vector field
$e:=\sum_{j=1}^{N}\partial_{\lambda_{j}}$ on $s_{ij}$ is completely determined
by knowing its action on the following functions:

$\mathbb{B}_{kl}=2{\rm{i}}\pi\rho^{l}(\omega_{k}),\quad\omega_{k}(\infty^{i})=\frac{t^{i,1}(\omega_{k})}{\sqrt{n_{i}+1}},\quad\mathcal{A}_{k}(\infty^{i})-\mathcal{A}_{k}(\infty^{0})=\int_{\infty^{0}}^{\infty^{i}}\omega_{k}=s^{i}(\omega_{k}).$

This provides an alternative way to show that $e.s_{ij}=0$.

Using (3.37), we remark that all the rows of the constant matrix of the
Darboux-Egoroff metric $\eta^{*}(\omega)$ contain exactly one nonzero entry
which is 1.This permits us to introduce the following natural duality relation
between flat coordinates of $\eta(\omega)$:

###### Definition 3.1

For each
$t^{A}\in\mathcal{S}(\omega):=\Big{\\{}t^{i,\alpha}(\omega),v^{i}(\omega),s^{i}(\omega),\rho^{k}(\omega),u^{k}(\omega)\Big{\\}}$,
let us denote by $t^{A^{\prime}}$ the unique flat function belonging to the
set $\mathcal{S}(\omega)$ such that

$\eta^{*}(\omega)\big{(}dt^{A}(\omega),dt^{A^{\prime}}(\omega)\big{)}=1.$
(3.40)

The flat coordinates $t^{A}$ and $t^{A^{\prime}}$ are called dual to each
other with respect to the metric $\eta(\omega)$.

Let us mention the following remarks:

  * •

From (3.37) and (3.38), it follows that the introduced $\eta(\omega)$-duality
does not depend on the choice of the differential $\omega$ that we work with.

  * •

For any $A$, we have $t^{(A^{\prime})^{\prime}}(\omega)=t^{A}(\omega)$.

  * •

Due to (3.37), the five families of operations
$t^{i,\alpha},v^{i},s^{i},\rho^{k}$ and $u^{k}$ are connected by:

$\begin{split}&t^{({i,\alpha})^{\prime}}(\omega)=t^{i,n_{i}+1-\alpha}(\omega),\quad\quad\quad\quad
v^{{i}^{\prime}}(\omega)=s^{i}(\omega),\quad\quad\
\quad\rho^{k^{\prime}}(\omega)=u^{k}(\omega);\\\
&\phi_{t^{({i,\alpha})^{\prime}}}(P)=\phi_{t^{i,n_{i}+1-\alpha}}(P),\quad\quad\quad\phi_{v^{i^{\prime}}}(P)=\phi_{s^{i}}(P),\quad\quad\phi_{\rho^{k^{\prime}}}(P)=\phi_{u^{k}}(P).\end{split}$
(3.41)

###### Proposition 3.6

The derivatives of the canonical coordinates $\\{\lambda_{j}\\}_{1\leq j\leq
N}$ with respect to $\eta(\omega)$-flat coordinates $t^{A}(\omega)$ belonging
to the system (3.21) are as follows

$\displaystyle\frac{\partial\lambda_{j}}{\partial
t^{A}(\omega)}=\frac{\phi_{t^{A^{\prime}}}(P_{j})}{\omega(P_{j})},$ (3.42)

where $\phi_{t^{A^{\prime}}}$ is the differential (3.14) associated with the
$\eta(\omega)$-flat function $t^{A^{\prime}}(\omega)$.

_Proof:_ In [14], Dubrovin has proven (3.42) by using a thermodynamical type
identity. We suggest a different proof based on Rauch’s formulas, (3.37) as
well as some basic facts from linear algebra.
Let us denote by $(\Lambda_{AB})$ the constant matrix of $\eta(\omega)$ in
flat coordinates $\\{t^{A}(\omega)\\}$ and $(\eta_{jj})$ the diagonal matrix
of $\eta(\omega)$ in the coordinates $\\{\lambda_{j}\\}$.
On one hand, Rauch’s formula (3.15) yields that

$dt^{A}(\omega)=\sum_{j}\partial_{\lambda_{j}}t^{A}(\omega)d\lambda_{j}=\frac{1}{2}\sum_{j}\omega(P_{j})\phi_{t^{A}}(P_{j})d\lambda_{j}.$

On the other hand, using twice the fact that $\eta(\omega)$ induces a linear
isomorphism from the tangent space onto the cotangent space, namely regarding
$(\Lambda_{AB})$ and $(\eta_{jj})$ as the change of basis matrices, we can
write

$\displaystyle dt^{A}(\omega)$
$\displaystyle=\sum_{B}\Lambda^{AB}\eta(\omega)\Big{(}\partial_{t^{B}(\omega)},\cdot\Big{)}$
$\displaystyle=\sum_{B}\Lambda^{AB}\eta(\omega)\Big{(}\sum_{j}\big{(}\partial_{t^{B}(\omega)}\lambda_{j}\big{)}\partial_{\lambda_{j}},\cdot\Big{)}$
$\displaystyle=\sum_{j}\sum_{B}\Lambda^{AB}\big{(}\partial_{t^{B}(\omega)}\lambda_{j}\big{)}\eta(\omega)(\partial_{\lambda_{j}},\cdot)$
$\displaystyle=\sum_{j}\bigg{(}\eta_{jj}\sum_{B}\Lambda^{AB}\big{(}\partial_{t^{B}(\omega)}\lambda_{j}\big{)}\bigg{)}d\lambda_{j}.$

Thus we must have

$\sum_{B}\Lambda^{AB}\big{(}\partial_{t^{B}(\omega)}\lambda_{j}\big{)}=\frac{1}{2}\eta^{jj}\omega(P_{j})\phi_{t^{A}}(P_{j})=\frac{\phi_{t^{A}}(P_{j})}{\omega(P_{j})}.$

Since
$\Lambda^{AB}=\eta^{*}(\omega)\big{(}dt^{A}(\omega),dt^{B}(\omega)\big{)}=\delta_{t^{B},t^{A^{\prime}}}$,
we deduce that

$\sum_{B}\Lambda^{AB}\big{(}\partial_{t^{B}(\omega)}\lambda_{j}\big{)}=\partial_{t^{A^{\prime}}(\omega)}\lambda_{j}.$

This establishes the desired formula (3.42).

$\square$

###### Corollary 3.1

The following assertions hold true:

$\displaystyle\begin{split}&\partial_{t^{A}(\omega)}=\sum_{j}\frac{\phi_{t^{A^{\prime}}}(P_{j})}{\omega(P_{j})}\partial_{\lambda_{j}};\end{split}$
(3.43)
$\displaystyle\begin{split}&\eta(\omega)\left(\partial_{t^{A}(\omega)},\partial_{t^{B}(\omega)}\right)=\frac{1}{2}\sum_{j}\phi_{t^{A^{\prime}}}(P_{j})\phi_{t^{B^{\prime}}}(P_{j})=\eta^{*}(\omega)\left(dt^{A^{\prime}}(\omega),t^{B^{\prime}}(\omega)\right).\end{split}$
(3.44)

In particular, the flat metric $\eta(\omega)$ can be written as

$\eta(\omega)=\textstyle\sum_{A}dt^{A}(\omega)\otimes
dt^{A^{\prime}}(\omega).$ (3.45)

_Proof:_ The relation (LABEL:flat-canonical_relation) is a direct consequence
of (3.42) combined with the chain rule formula. The first equality in
(LABEL:Entries-eta2) follows from (LABEL:flat-canonical_relation) and (3.20)
and the second one from (3.17). Lastly, in view of (LABEL:Entries-eta2) and
(3.37), we obtain the stated expression for the metric $\eta(\omega)$.

$\square$

###### Remark 3.6

In the case where $\omega=\phi$ is a primary differential, Dubrovin’s method
to calculate the constant matrix of the flat metric $\eta(\phi)$ in flat
coordinates uses his bilinear pairing (see formula (5.60) in [14]). In [41],
p. 562-564, Shramchenko employed the same method and gave a detailed
computation of constant matrices of the Daroubx-Egoroff metrics in the real
double setting of Frobenius manifolds.

We close this subsection by mentioning the following duality-exchange
relation:

###### Proposition 3.7

Let $\omega$ be a given differential satisfying (3.1) and (3.3). Let
$t^{A}(\omega),t^{B}(\omega)$ be two $\eta(\omega)$-flat coordinates from the
set (3.21)-(3.22) and $\phi_{t^{A}}$, $\phi_{t^{B}}$ be the corresponding
differentials belonging to the list (3.23).
Then the following holds:

$\frac{\partial\phi_{t^{B}}(P)}{\partial{t^{A}(\omega)}}=\frac{\partial\phi_{t^{A^{\prime}}}(P)}{\partial{t^{B^{\prime}}(\omega)}}.$

_Proof:_ We have

$\displaystyle\partial_{t^{A}(\omega)}\phi_{t^{B}}(P)$
$\displaystyle=\sum_{j}\frac{\phi_{t^{A^{\prime}}}(P_{j})}{\omega(P_{j})}\partial_{\lambda_{j}}\phi_{t^{B}}(P)=\frac{1}{2}\sum_{j}\frac{\phi_{t^{A^{\prime}}}(P_{j})}{\omega(P_{j})}\phi_{t^{B}}(P_{j})W(P,P_{j})$
$\displaystyle=\sum_{j}\frac{\phi_{t^{B}}(P_{j})}{\omega(P_{j})}\partial_{\lambda_{j}}\phi_{t^{A^{\prime}}}(P)=\partial_{t^{B^{\prime}}(\omega)}\phi_{t^{A^{\prime}}}(P),$

where the first and fourth equalities follow from (LABEL:flat-
canonical_relation), while the second and third ones are a direct consequence
of Rauch’s formulas (3.2).

$\square$

###  3.4 Flat coordinates of intersection forms

According to Dubrovin [14], given a semi-simple Frobenius manifold
$(\mathcal{M},\mathop{\langle\cdot,\cdot\rangle})$, there is a second flat
metric $\mathfrak{g}$, called the intersection form, defined on an open subset
of $\mathcal{M}$. The metric $\mathfrak{g}$ and the initial metric
$\mathop{\langle\cdot,\cdot\rangle}$ give rise to the so-called flat pencil.
In the case of a semi-simple Hurwitz-Frobenius manifolds, the (covariant
metric induced by the) intersection form is defined by:

$\mathbf{g}(\omega):=\mathbf{ds^{2}_{1,0}}(\omega)=\sum_{j=1}^{N}\frac{\omega(P_{j})^{2}}{2\lambda_{j}}(d\lambda_{j})^{2},$
(3.46)

where, as above, $\omega$ denotes a differential satisfying (3.1) and (3.3).
In this subsection, we focus on flat coordinates of the intersection form
(3.46) considered as a flat metric on a connected open subset
$\widehat{\mathcal{H}}_{g,L}(\mathbf{n};\mathbf{m})$ of the double Hurwitz
space. Recently, Romano used a Dubrovin type bilinear pairing and gave a set
of flat coordinates of the metric $\mathbf{g}(\omega)$ [34]. However, he did
not calculate the corresponding constant matrix.
In our case, by employing the language of the symmetric bidifferential
$W(P,Q)$, we show that the approach used in Section 3.3 can be adapted to
study of the metric $\mathbf{g}(\omega)$.

###### Theorem 3.5

Let $\big{\\{}\big{(}C_{g},\lambda\big{)},\\{a_{k},b_{k}\\}\big{\\}}$ be a
point of the double Hurwitz space
$\widehat{\mathcal{H}}_{g,L}(\mathbf{n};\mathbf{m})$ and Let $P_{0}$ be a
marked point on $C_{g}$ such that $\lambda(P_{0})=1$. Assume that all the
cycles $\\{a_{k},b_{k}\\}$ start at the point $P_{0}$ and the differential
$\omega$ has zero $a$-periods.
Then the following $N=2g+m+n$ functions provide a system of flat coordinates
of the metric $\mathbf{g}(\omega)$:

$\begin{array}[]{llll}s^{i}(\omega):=\displaystyle
p.v.\int_{\infty^{0}}^{\infty^{i}}\omega(P),&\quad i=1,\dots,m;\\\ \\\
y^{j}(\omega):=\displaystyle
p.v.\int_{\mathbf{z}^{0}}^{\mathbf{z}^{j}}\omega(P),&\quad j=1,\dots,n;\\\ \\\
\rho^{k}(\omega):=\displaystyle\frac{1}{2{\rm{i}}\pi}\oint_{b_{k}}\omega(P),&\quad
k=1,\dots,g;\\\ \\\
q^{k}(\omega):=\displaystyle\oint_{a_{k}}\log\big{(}\lambda(P)\big{)}\omega(P),&\quad
k=1,\dots,g.\end{array}$ (3.47)

In addition, in these coordinates, the nonzero entries of the contravariant
metric $\mathbf{g}^{*}(\omega)$ are

$\begin{split}&\mathbf{g}^{*}(\omega)\big{(}ds^{i}(\omega),ds^{j}(\omega)\big{)}=\frac{1}{n_{0}+1}+\delta_{ij}\frac{1}{n_{i}+1};\\\
&\mathbf{g}^{*}(\omega)\big{(}dy^{i}(\omega),dy^{j}(\omega)\big{)}=-\frac{1}{m_{0}+1}-\delta_{ij}\frac{1}{m_{i}+1};\\\
&\mathbf{g}^{*}(\omega)\big{(}d\rho^{i}(\omega),dq^{j}(\omega)\big{)}=\delta_{ij}.\end{split}$
(3.48)

Here the positive integers $n_{i}+1$ and $m_{j}+1$ denote respectively the
multiplicities of the prescribed ramifications points $\infty^{i}$ and
$\mathbf{z}^{j}$, with $i=0,\dots,m$ and $j=0,\dots,n$.

Before we start to prove this theorem, let us mention some remarks and
preliminary results.

$\bullet$ Similarly to the function $s^{i}(\omega)$, the principal value in
the formula of $y^{j}(\omega)$ is defined by subtraction of the divergent part
of the integral as a function of the local parameters
$\mathfrak{z}_{0}(P):=\lambda(P)^{\frac{1}{m_{0}+1}}$ and
$\mathfrak{z}_{i}(P):=\lambda(P)^{\frac{1}{m_{i}+1}}$ near the points
$\mathbf{z}^{0}$ and $\mathbf{z}^{i}$, respectively.

$\bullet$ Let us denote by $\Omega_{\mathbf{z}^{0}\mathbf{z}^{i}}(P)$ the
normalized Abelian differential of the third kind having simple poles at the
points $\mathbf{z}^{0}$ and $\mathbf{z}^{i}$, i.e.

$\underset{\mathbf{z}^{i}}{{\rm res}}\
\Omega_{\mathbf{z}^{0}\mathbf{z}^{i}}=1,\quad\underset{\mathbf{z}^{0}}{{\rm
res}}\
\Omega_{\mathbf{z}^{0}\mathbf{z}^{i}}=-1,\quad\oint_{a_{k}}\Omega_{\mathbf{z}^{0}\mathbf{z}^{i}}=0,\quad\forall\
k=1,\dots,g.$

We emphasize that formulas analogous to (3.27) and (3.28) and a direct
adaptation of Theorem 3.3 allow us to calculate the functions

$y_{ij}:=p.v.\int_{\mathbf{z}^{0}}^{\mathbf{z}^{j}}\Omega_{\mathbf{z}^{0}\mathbf{z}^{i}}(P),\quad\quad\forall\
i,j=1,\dots,n.$

More precisely, we obtain the following statements:

1)

Assume that $g=0$ and $\mathbf{z}^{0}\in\mathbb{C}$. For $i=0,\dots,n$,
consider the meromorphic function $\nu_{i}$ defined by

$\nu_{i}(P)=\left\\{\begin{array}[]{ll}(z_{P}-\mathbf{z}^{i})^{-m_{i}-1}\lambda(P),&\quad\hbox{if}\quad\mathbf{z}^{i}\in\mathbb{C},\\\
\\\
z_{P}^{m_{i}+1}\lambda(P),&\quad\hbox{if}\quad\mathbf{z}^{i}=\infty.\end{array}\right.$
(3.49)

  * i)

If $i\neq j$ and $\mathbf{z}^{i},\mathbf{z}^{j}\in\mathbb{C}$, then

$y_{ij}=\log\left(\frac{\mathbf{z}^{j}-\mathbf{z}^{i}}{\big{(}\mathbf{z}^{j}-\mathbf{z}^{0}\big{)}\big{(}\mathbf{z}^{0}-\mathbf{z}^{i}\big{)}}\right)-\frac{\log\big{(}\nu_{0}(\mathbf{z}^{0})\big{)}}{m_{0}+1}.$

When $\mathbf{z}^{r}=\infty$, with $r=i$ or $j$, we deduce the desired formula
from the previous one by letting $\mathbf{z}^{r}\to\infty$.

  * ii)

If $i=j$ we have

$\begin{array}[]{ll}y_{ii}=\displaystyle-\frac{\log\big{(}\nu_{i}(\mathbf{z}^{i})\big{)}}{m_{i}+1}-\frac{\log\big{(}\nu_{0}(\mathbf{z}^{0})\big{)}}{m_{0}+1}-2\log\big{(}\mathbf{z}^{i}-\mathbf{z}^{0}\big{)}-\log(-1),&\hbox{if}\quad\mathbf{z}^{i}\in\mathbb{C};\\\
\\\
y_{ii}=\displaystyle-\frac{\log\big{(}\nu_{i}(\mathbf{z}^{i})\big{)}}{m_{i}+1}-\frac{\log\big{(}\nu_{0}(\mathbf{z}^{0})\big{)}}{m_{0}+1},&\hbox{if}\quad\mathbf{z}^{i}=\infty.\end{array}$

2)

Assume that $g\geq 1$.

  * i)

When $i\neq j$, we have

$y_{ij}=\log\left(\frac{\Theta_{\Delta}\big{(}\mathcal{A}(\mathbf{z}^{j})-\mathcal{A}(\mathbf{z}^{i})\big{)}}{\Theta_{\Delta}\big{(}\mathcal{A}(\mathbf{z}^{j})-\mathcal{A}(\mathbf{z}^{0})\big{)}\Theta_{\Delta}\big{(}\mathcal{A}(\mathbf{z}^{i})-\mathcal{A}(\mathbf{z}^{0})\big{)}}\right)+\log\left(\sum_{k=1}^{g}\omega_{k}(\mathbf{z}^{0})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{)}\right)-\log(-1).$

  * ii)

If $i=j$ we have

$y_{ii}=\log\left(\sum_{k=1}^{g}\omega_{k}(\mathbf{z}^{i})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{)}\right)+\log\left(\sum_{k=1}^{g}\omega_{k}(\mathbf{z}^{0})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{)}\right)-2\log\left(\Theta_{\Delta}\big{(}\mathcal{A}(\mathbf{z}^{i})-\mathcal{A}(\mathbf{z}^{0})\big{)}\right)-\log(-1).$

$\bullet$ Consider the differentials connected with the four families of flat
functions $s^{i},y^{i},\rho^{k},q^{k}$ (as above, the link here is in the
sense of (3.13) -(3.14)):

$\begin{split}&\psi_{s^{i}}(P)=\int_{\infty^{0}}^{\infty^{i}}W(P,Q)=\Omega_{\infty^{0}\infty^{i}}(P);\\\
&\psi_{y^{i}}(P)=\int_{\mathbf{z}^{0}}^{\mathbf{z}^{i}}W(P,Q)=\Omega_{\mathbf{z}^{0}\mathbf{z}^{i}}(P);\\\
&\psi_{\rho^{k}}(P)=\frac{1}{2{\rm{i}}\pi}\oint_{b_{k}}W(P,Q)=\omega_{k}(P);\\\
&\psi_{q^{k}}(P)=\oint_{a_{k}}\log\big{(}\lambda(Q)\big{)}W(P,Q).\end{split}$
(3.50)

The analytic properties of the Abelian differentials
$\psi_{s^{i}},\psi_{y^{i}},\psi_{\rho^{k}}$ have previously been specified.
Let us deal with the differential $\psi_{q^{k}}$. Since the differential
$W(P,\cdot)$ has vanishing $a$-periods (2.13), the definition of
$\psi_{q^{k}}$ does not depend on the choice of the branch of the complex
logarithm and its $a$-periods are all zero. Now, we claim that
$\psi_{q^{k}}(P)$ is a multivalued differential with jumps along the cycle
$b_{k}$ and

$\psi_{q^{k}}(P+b_{k})-\psi_{q^{k}}(P)=-2{\rm{i}}\pi\frac{d\lambda(P)}{\lambda(P)}.$
(3.51)

Indeed, using (2.11) and integrating by parts, we can write

$\psi_{q^{k}}(P)=-\oint_{a_{k}}d_{P}\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(Q)\big{)}\Big{)}\frac{d\lambda(Q)}{\lambda(Q)}=dF_{q^{k}}(P),$

where $F_{q^{k}}$ is given by

$F_{q^{k}}(P):=\int^{P}\psi_{q^{k}}=-\oint_{a_{k}}\log\Big{(}\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(Q)\big{)}\Big{)}\frac{d\lambda(Q)}{\lambda(Q)}.$

In view of relation (2.9), we obtain

$\displaystyle F_{q^{k}}(P+a_{j})-F_{q^{k}}(P)$
$\displaystyle=\oint_{a_{j}}\psi_{q^{k}}=0;$ $\displaystyle
F_{q^{k}}(P+b_{j})-F_{q^{k}}(P)$
$\displaystyle=\oint_{b_{j}}\psi_{q^{k}}=-\oint_{a_{k}}\log\bigg{(}\frac{\Theta_{\Delta}\big{(}\mathcal{A}(P+b_{j})-\mathcal{A}(Q)\big{)}}{\Theta_{\Delta}\big{(}\mathcal{A}(P)-\mathcal{A}(Q)\big{)}}\bigg{)}\frac{d\lambda(Q)}{\lambda(Q)}$
$\displaystyle=-\oint_{a_{k}}\bigg{(}-{\rm{i}}\pi\mathop{\langle\mathbb{B}e_{j},e_{j}\rangle}-2{\rm{i}}\pi\mathop{\langle\beta,e_{j}\rangle}-2{\rm{i}}\pi\int_{Q}^{P}\omega_{j}\bigg{)}\frac{d\lambda(Q)}{\lambda(Q)}$
$\displaystyle=2{\rm{i}}\pi\oint_{a_{k}}\Big{(}\int_{Q}^{P}\omega_{j}\Big{)}\frac{d\lambda(Q)}{\lambda(Q)}$
$\displaystyle=2{\rm{i}}\pi\oint_{a_{k}}\bigg{(}d_{Q}\Big{(}\log\big{(}\lambda(Q)\big{)}\int_{Q}^{P}\omega_{j}\Big{)}+\log\big{(}\lambda(Q)\big{)}\omega_{j}(Q)\bigg{)}$
$\displaystyle=2{\rm{i}}\pi\bigg{(}\log\big{(}\lambda(Q)\big{)}\int_{Q}^{P}\omega_{j}\bigg{)}\bigg{|}_{Q=Q_{0}}^{Q=Q_{0}+a_{k}}+2{\rm{i}}\pi\oint_{a_{k}}\log\big{(}\lambda(Q)\big{)}\omega_{j}(Q)$
$\displaystyle=-2{\rm{i}}\pi\delta_{jk}\log\big{(}\lambda(Q_{0})\big{)}+2{\rm{i}}\pi
q^{k}(\omega_{j}),$

where $Q_{0}$ is a starting point of the cycle $a_{k}$. Here, we are going to
study general properties of the differential $\psi_{q^{k}}$ and so (contrary
to the assumption of Theorem 3.5) we don’t suppose that the cycles
$\\{a_{k},b_{k}\\}$ start at a particular point $P_{0}$ such that
$\lambda(P_{0})=1$.
From the above calculation, we see that
$\psi_{q^{k}}(P+b_{j})-\psi_{q^{k}}(P)=0$ for all $j\neq k$. Moreover,
regarding $P$ as a starting point of the cycle $b_{k}$ in the expression
$F_{q^{k}}(P+b_{j})-F_{q^{k}}(P)$ and choosing $Q_{0}=P$ to be the
intersection points of $a_{k}$ and $b_{k}$, we arrive at (3.51).

$\bullet$ Let
$u^{A}(\omega),u^{B}(\omega)\in\Big{\\{}s^{i}(\omega),y^{j}(\omega),\rho^{k}(\omega),q^{k}(\omega)\Big{\\}}$.
The above computation shows that

$q^{k}(\psi_{\rho^{j}})=\oint_{a_{k}}\log\big{(}\lambda(Q)\big{)}\omega_{j}(Q)=\frac{1}{2{\rm{i}}\pi}\oint_{b_{j}}\psi_{q^{k}}=\rho^{j}(\psi_{q^{k}}),\quad\quad\forall\
j,k=1,\dots,g;$

provided that the cycles $\\{a_{k},b_{k}\\}$ start at a marked point $P_{0}$
so that $\lambda(P_{0})=1$.
Analogously to the symmetry property (3.36), using the latter assumption as
well as the explicit formulas for the functions $s_{ij}$ and $y_{ij}$, we see
that the interchanging of the order of two operations $u^{A},u^{B}$ among
$\big{\\{}s^{i},y^{j},\rho^{k},q^{k}\big{\\}}$ only results in the addition of
a constant. More concretely, the following holds:

$u^{A}(\psi_{u^{B}})=u^{B}(\psi_{u^{A}})+\log(-1)\sum_{i,j=1}^{m}(1-\delta_{ij})\delta_{s^{i},u^{A}}\delta_{s^{j},u^{B}}+\log(-1)\sum_{i,j=1}^{n}(1-\delta_{ij})\delta_{y^{i},u^{A}}\delta_{y^{j},u^{B}}.$
(3.52)

Now we are in position to prove Theorem 3.5.

_Proof of Theorem 3.5:_ By applying (3.16) with $\beta_{1}=1$, $\beta_{2}=0$
and taking into account the properties of the symmetric bidifferential
$W(P,Q)$, we deduce that the functions $s^{i}(\omega)$, $y^{j}(\omega)$,
$\rho^{k}(\omega)$ and $q^{k}(\omega)$ in (3.47) are all flat with respect to
the intersection form $\mathbf{g}(\omega)$.
Now, in order to calculate the entries of the constant matrix of the metric
$\mathbf{g}^{*}(\omega)$, we will adapt the scheme of the proof of Theorem
3.4. More precisely, the crucial ingredient is a formula analogous to (3.38):

$\mathbf{g}^{*}(\omega)\left(du^{A}(\omega),du^{B}(\omega)\right)=E.u^{A}(\psi_{u^{B}})=\sum_{k=1}^{N}\underset{P_{k}}{{\rm
res}}\frac{\lambda(P)\psi_{u^{A}}(P)\psi_{u^{B}}(P)}{d\lambda(P)},$ (3.53)

where $E$ is the vector field
$E:=\sum_{j=1}^{N}\lambda_{j}\partial_{\lambda_{j}}$ and $\psi_{u^{A}}(P)$,
$\psi_{u^{B}}(P)$ are among the differentials in (3.50). Note that (3.53) is
an immediate consequence of (3.18) and (3.19) where we take
$(\beta_{1},\beta_{2})=(1,0)$.
Due to the residue theorem and (3.53) we have

$\displaystyle E.s^{i}(\psi_{s^{j}})$
$\displaystyle=\sum_{k=1}^{N}\underset{P_{k}}{{\rm
res}}\frac{\lambda(P)\Omega_{\infty^{0}\infty^{i}}(P)\Omega_{\infty^{0}\infty^{j}}(P)}{d\lambda(P)}$
$\displaystyle=-\underset{\infty^{0}}{{\rm
res}}\frac{\lambda(P)\Omega_{\infty^{0}\infty^{i}}(P)\Omega_{\infty^{0}\infty^{j}}(P)}{d\lambda(P)}-\delta_{ij}\underset{\infty^{i}}{{\rm
res}}\frac{\lambda(P)\Omega_{\infty^{0}\infty^{i}}(P)\Omega_{\infty^{0}\infty^{j}}(P)}{d\lambda(P)}$
$\displaystyle=\frac{1}{n_{0}+1}+\delta_{ij}\frac{1}{n_{i}+1}.$

Similarly we get

$\displaystyle
E.s^{i}(\psi_{y^{j}})=E.s^{i}(\psi_{\rho^{k}})=E.y^{i}(\psi_{\rho^{k}})=E.\rho^{k}(\psi_{\rho^{j}})=0;$
$\displaystyle
E.y^{i}(\psi_{y^{j}})=-\frac{1}{m_{0}+1}+\delta_{ij}\frac{1}{m_{i}+1}.$

It remains to make precise the action of vector field $E$ on the functions
$q^{k}(\psi)$ with
$\psi\in\big{\\{}\psi_{s^{i}},\psi_{y^{j}},\psi_{\rho^{k}},\psi_{q^{k}}\big{\\}}$.
To this end, let us consider the differential

$\Psi_{k}(P):=\frac{\lambda(P)\psi(P)}{d\lambda(P)}\psi_{q^{k}}(P),\quad
k=1,\dots,g,$

where $\psi$ is one of the differentials belonging to (3.50).
Then $\Psi_{k}$ is a multivalued differential and its jumps are as follows:

$\displaystyle\begin{array}[]{llll}\Psi_{k}(P+a_{j})-\Psi_{k}(P)=0,&\hbox{if}\quad
j=1,\dots,g;\\\ \\\
\Psi_{k}(P+b_{k})-\Psi_{k}(P)=-2{\rm{i}}\pi\psi(P),&\hbox{if}\quad\psi\neq\psi_{q^{k}};\\\
\\\