\Psi_{k}(P+b_{k})-\Psi_{k}(P)=-4{\rm{i}}\pi\psi_{q^{k}}(P)-4\pi^{2}\frac{d\lambda(P)}{\lambda(P)},&\hbox{if}\quad\psi=\psi_{q^{k}};\\\
\\\
\Psi_{k}(P+b_{j})-\Psi_{k}(P)=-2{\rm{i}}\pi\psi_{q^{k}}(P),&\hbox{if}\quad\psi=\psi_{q^{j}}\quad\text{and}\quad
j\neq k;\\\ \\\ \Psi_{k}(P+b_{j})-\Psi_{k}(P)=0,&\hbox{if}\quad\text{$\psi$ is
Abelian and $j\neq k$}.\end{array}$

To summarize, the jumps along the cycles $b_{j}$ can be rewritten as

$\displaystyle\Psi_{k}(P+b_{j})-\Psi_{k}(P)$
$\displaystyle=\big{(}1-\delta_{\psi,\psi_{q^{k}}}\big{)}\big{(}-2{\rm{i}}\pi\psi(P)\big{)}+\delta_{\psi,\psi_{q^{k}}}\Big{(}-4{\rm{i}}\pi\psi_{q^{k}}(P)-4\pi^{2}\frac{d\lambda(P)}{\lambda(P)}\Big{)}$
$\displaystyle\quad+(1-\delta_{jk})\delta_{\psi,\psi_{q^{j}}}\big{(}-2{\rm{i}}\pi\psi_{q^{k}}(P)\big{)}.$

On the other hand, observing that the multivalued differentials
$\psi_{q^{1}},\dots,\psi_{q^{g}}$ are all single valued and holomorphic inside
the fundamental polygon $F_{g}$ associated with $C_{g}$, we deduce that the
differential $\Psi_{k}$ becomes meromorphic inside $F_{g}$ and the
ramification points $P_{1},\dots,P_{N}$ are its unique simple poles belonging
to $F_{g}$.
Therefore, from (3.53) we have

$\displaystyle E.q^{k}(\psi)$
$\displaystyle=\sum_{s=1}^{N}\underset{P_{s}}{{\rm
res}}\Psi_{k}(P)=\frac{1}{2{\rm{i}}\pi}\oint_{\partial{F_{g}}}\Psi_{k}(P)$
$\displaystyle=\frac{1}{2{\rm{i}}\pi}\sum_{j=1}^{g}\bigg{(}\oint_{a_{j}}\Psi_{k}(P)-\oint_{a_{j}}\Psi_{k}(P+b_{j})\bigg{)}+\frac{1}{2{\rm{i}}\pi}\sum_{j=1}^{g}\bigg{(}\oint_{b_{j}}\Psi_{k}(P)-\oint_{b_{j}}\Psi_{k}(P+a_{j}^{-1})\bigg{)}$
$\displaystyle=\bigg{(}\sum_{j=1,j\neq
k}^{g}\delta_{\psi,\psi_{q^{j}}}\oint_{a_{j}}\psi_{q^{k}}\bigg{)}+\big{(}1-\delta_{\psi,\psi_{q^{k}}}\big{)}\oint_{a_{k}}\psi+\delta_{\psi,\psi_{q^{k}}}\oint_{a_{k}}\bigg{(}2\psi_{q^{k}(P)}-2{\rm{i}}\pi\frac{d\lambda(P)}{\lambda(P)}\bigg{)}$
$\displaystyle=\big{(}1-\delta_{\psi,\psi_{q^{k}}}\big{)}\delta_{\psi,\psi_{\rho^{k}}}=\delta_{\psi,\psi_{\rho^{k}}},$

with $\delta_{\psi,\psi_{\rho^{k}}}$ being the Kronecker delta. This finishes
the proof of the theorem.

$\square$

###### Corollary 3.2

The nonzero entries of constant matrix of the intersection form
$\mathbf{g}(\omega)$ are given by

$\begin{split}&\mathbf{g}(\omega)\big{(}\partial_{s^{i}(\omega)},\partial_{s^{j}(\omega)}\big{)}=(n_{i}+1)\delta_{ij}-\frac{(n_{i}+1)(n_{j}+1)}{L};\\\
&\mathbf{g}(\omega)\big{(}\partial_{y^{i}(\omega)},\partial_{y^{j}(\omega)}\big{)}=-(m_{i}+1)\delta_{ij}+\frac{(m_{i}+1)(m_{j}+1)}{L};\\\
&\mathbf{g}(\omega)\big{(}\partial_{\rho^{i}(\omega)},\partial_{q^{j}(\omega)}\big{)}=\delta_{ij}.\end{split}$
(3.54)

Here $L$ denotes the common number of sheets of the coverings
$[(C_{g},\lambda)]\in\widehat{\mathcal{H}}_{g,L}(\mathbf{n};\mathbf{m})$.

_Proof:_ Let $\mathbf{A}=(\mathbf{A}_{ij})$ be a $m\times m$ square matrix of
the form

$\mathbf{A}_{ij}=\alpha_{0}+\delta_{ij}\alpha_{i},$

where $\alpha_{0},\alpha_{1},\dots,\alpha_{m}$ are all nonzero complex
numbers. Then one can check that the matrix $\mathbf{A}$ is invertible and the
entries of its inverse are:

$\mathbf{A}^{ij}=\frac{1}{\alpha_{i}}\delta_{ij}-\frac{1}{\alpha_{i}\alpha_{j}\sigma},\quad\quad\text{with}\quad\sigma:=\sum_{k=0}^{m}\frac{1}{\alpha_{k}}.$

When $\alpha_{0},\dots,\alpha_{m}$ are the positive real numbers:

$\alpha_{i}=\frac{1}{n_{i}+1},\quad\forall\ i=0,1,\dots,m,$

we have

$\sigma=\sum_{k=0}^{m}\frac{1}{\alpha_{k}}=L.$

This and (3.48) imply (3.54).

$\square$

###### Corollary 3.3

Let
$u^{A}(\omega)\in\Big{\\{}s^{i}(\omega),y^{j}(\omega),\rho^{k}(\omega),q^{k}(\omega)\Big{\\}}$.
Then the partial derivative $\partial_{u^{A}(\omega)}$ has the following form

$\partial_{u^{A}(\omega)}=\sum_{j}\bigg{(}\frac{\lambda_{j}}{\omega(P_{j})}\sum_{\alpha}\textsf{G}_{A\alpha}\psi_{u^{\alpha}}(P_{j})\bigg{)}\partial_{\lambda_{j}},$
(3.55)

where
$\textsf{G}_{A\alpha}=\mathbf{g}(\omega)\big{(}\partial_{u^{A}(\omega)},\partial_{u^{\alpha}(\omega)}\big{)}$
are the entries (3.54) of the constant matrix of the metric
$\mathbf{g}(\omega)$ and $\psi_{u^{A}}$ is among the differentials (3.50)
corresponding to the function $u^{A}$.

_Proof:_ By arguing as in the proof of (3.42), we can check that the partial
derivatives of $\lambda_{j}$ with respect to flat coordinates
$\\{u^{A}(\omega)\\}_{A}$ are given by

$\partial_{u^{A}(\omega)}\lambda_{j}=\frac{\lambda_{j}}{\omega(P_{j})}\sum_{\alpha}\textsf{G}_{A\alpha}\psi_{u^{\alpha}}(P_{j}).$

This shows that (3.55) holds true.

$\square$

At the end of this subsection, we shall describe the matrix of the
intersection form $\mathbf{g}^{*}(\omega)$ in flat coordinates (3.21)-(3.22)
of the Darboux-Egoroff metric $\eta(\omega)$ defined by (3.20). Then we shall
view $\mathbf{g}(\omega)$ as a metric on an open subset of the particular
double Hurwitz space $\widehat{\mathcal{H}}_{g,L}(\mathbf{n};\mathbf{0})$. The
latter is an open subset of the simple Hurwitz space
$\widehat{\mathcal{H}}_{g,L}(n_{0},\dots,n_{m})$ determined by the conditions
$\lambda_{j}\neq 0$, for all $j=1,\dots,N$.
From [14] (see also next section), it is known that the vector field
$E=\sum_{j}\lambda_{j}\partial_{\lambda_{j}}$ plays the role of the Euler
vector field of the semi-simple Hurwitz-Frobenius manifold structures induced
by Dubrovin’s primary differentials (3.23). Thus the obtained matrix of
$\mathbf{g}^{*}(\omega)$ describes, through formula (3.56) below, the quasi-
homogeneity properties of flat coordinates of the $N$ Darboux-Egoroff metrics
$\big{\\{}\eta(\phi_{t^{A}})\big{\\}}_{A}$ of Dubrovin.

###### Theorem 3.6

Let $\omega(P)=\int_{\gamma}h(\lambda(Q))W(P,Q)$ be a differential on a
surface $C_{g}$ satisfying (3.1) and (3.3), $\eta(\omega)$ be the
corresponding Darboux-Egoroff metric (3.20) and $E$ be the vector field
$E=\sum_{j=1}^{N}\lambda_{j}\partial_{\lambda_{j}}$.
Then, in the flat coordinates $t^{A}(\omega)\in\mathcal{S}(\omega)$ described
in (3.21)-(3.22), the entries of the matrix of the intersection form
$\mathbf{g}^{*}(\omega)$ are:

$\mathbf{g}^{*}(\omega)\left(dt^{A}(\omega),dt^{B}(\omega)\right)=E.t^{A}(\phi_{t^{B}})=(d_{A}+d_{B})t^{A}(\phi_{t^{B}})+r_{A,B},$
(3.56)

where $r_{A,B}$ is the constant

$r_{A,B}=\sum_{i,j=1}^{m}\delta_{t^{A},s^{i}}\delta_{t^{B},s^{j}}\Big{(}\frac{1}{n_{0}+1}+\delta_{ij}\frac{1}{n_{i}+1}\Big{)}=r_{B,A}$
(3.57)

and $d_{A}$ (resp. $d_{B}$) is the nonnegative constant appearing in the
integral representation (3.24) of the differential $\phi_{t^{A}}$ (resp.
$\phi_{t^{B}}$).

_Proof:_ According to Theorem 3.2 and formulas (3.18)-(3.19), the following
two equalities are valid:

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

This shows the first equality in (3.56).
Let $\phi$ be one of the differentials belonging to the set (3.23). If
$\phi\neq\phi_{u^{k}}$ (i.e. $\phi$ has no jumps), then the point $\infty^{i}$
is a pole of the meromorphic function $\frac{\lambda(P)\phi(P)}{d\lambda(P)}$
if and only if $\phi$ is of type $\phi=\phi_{t^{i,\alpha}}$ or
$\phi=\phi_{v^{i}}$. Moreover, according to behaviors (3.25) and (3.26) of the
Abelian differentials $\phi_{t^{i,\alpha}}$ and $\phi_{v^{i}}$ near the point
$\infty^{i}$, we have

$\displaystyle\frac{\lambda(P)\phi(P)}{d\lambda(P)}$
$\displaystyle\underset{P\sim\infty^{i}}{=}-\delta_{\phi,\phi_{t^{i,\alpha}}}\Big{(}\frac{z_{i}(P)^{-\alpha}}{\sqrt{n_{i}+1}}+\text{reg}\Big{)}-\delta_{\phi,\phi_{v^{i}}}\Big{(}z_{i}(P)^{-n_{i}-1}+\text{reg}\Big{)}$
$\displaystyle\underset{P\sim\infty^{i}}{=}-\delta_{\phi,\phi_{t^{i,\alpha}}}\Big{(}\frac{1}{\sqrt{n_{i}+1}}\lambda(P)^{\frac{\alpha}{n_{i}+1}}+\text{reg}\Big{)}-\delta_{\phi,\phi_{v^{i}}}\Big{(}\lambda(P)+\text{reg}\Big{)}.$

Moreover, when $\phi=\phi_{u^{k}}$, $k=1,\dots,g$, then the restriction of the
function $\frac{\lambda(P)\phi(P)}{d\lambda(P)}$ to the fundamental polygon
$F_{g}$ associated with the surface $C_{g}$ is holomorphic.
Therefore, the following useful identity holds true

$\underset{\infty^{i}}{{\rm
res}}\bigg{(}\frac{\lambda(P)\phi(P)}{d\lambda(P)}\phi_{0}(P)\bigg{)}=-\delta_{\phi,\phi_{t^{i,\alpha}}}\frac{\alpha}{n_{i}+1}t^{i,\alpha}(\phi_{0})-\delta_{\phi,\phi_{v^{i}}}v^{i}(\phi_{0})$
(3.58)

holds true whenever the differential $\phi_{0}$ is holomorphic near
$\infty^{i}$ and $\phi$ is among the differentials listed in (3.23).
Now, in order to prove the second equality in (3.56), we distinguish five
cases depending on the five types of operations $t^{i,\alpha},\ v^{i},\
s^{i},\ \rho^{k},\ u^{k}$.
First case: $t^{A}=u^{k}$. The equality
$E.u^{k}(\phi_{t^{B}})=(1+d_{B})u^{k}(\phi_{t^{B}})$ is a particular case of
formula (8.12) established in Appendix 2.
Second case: $t^{A}=\rho^{k}$. Assume that
$\phi\in\big{\\{}\phi_{t^{i,\alpha}},\phi_{v^{i}},\phi_{s^{i}},\phi_{\rho^{k}}=\omega_{k}\big{\\}}$,
i.e. $\phi\neq\phi_{u^{k}}$ for all $k=1,\dots,g$. Using the residue theorem
and applying (3.58) to $\phi_{0}=\phi_{\rho^{k}}=\omega_{k}$ (holomorphic
differential), we deduce that

$\displaystyle E.\rho^{k}(\phi)$
$\displaystyle=-\sum_{i=0}^{m}\sum_{\alpha=1}^{n_{i}}\delta_{\phi,\phi_{t^{i,\alpha}}}\underset{\infty^{i}}{{\rm
res}}\
\bigg{(}\frac{\lambda(P)\phi(P)}{d\lambda(P)}\omega_{k}(P)\bigg{)}-\sum_{i=1}^{m}\delta_{\phi,\phi_{v^{i}}}\underset{\infty^{i}}{{\rm
res}}\ \bigg{(}\frac{\lambda(P)\phi(P)}{d\lambda(P)}\omega_{k}(P)\bigg{)}$
$\displaystyle=\sum_{i=0}^{m}\sum_{\alpha=1}^{n_{i}}\delta_{\phi,\phi_{t^{i,\alpha}}}\frac{\alpha}{n_{i}+1}t^{i,\alpha}\big{(}\omega_{k}\big{)}+\sum_{i=1}^{m}\delta_{\phi,\phi_{v^{i}}}v^{i}\big{(}\omega_{k}\big{)}$
$\displaystyle=\sum_{i=0}^{m}\sum_{\alpha=1}^{n_{i}}\delta_{\phi,\phi_{t^{i,\alpha}}}\frac{\alpha}{n_{i}+1}\rho^{k}\big{(}\phi_{t^{i,\alpha}}\big{)}+\sum_{i=1}^{m}\delta_{\phi,\phi_{v^{i}}}\rho^{k}\big{(}\phi_{v^{i}}\big{)}.$

When $\phi=\phi_{u^{k}}$, then the symmetry property (3.36) and the first case
imply that

$E.\rho^{k}(\phi_{u^{k}})=E.u^{k}(\phi_{\rho^{k}})=u^{k}(\phi_{\rho^{k}})=\rho^{k}(\phi_{u^{k}}).$

Thus the desired formula (3.56) is valid in this case.
Third case: $t^{A}=v^{j}$. By (3.36), we know that
$v^{j}(\phi_{u^{k}})=u^{k}(\phi_{v^{j}})$ and
$v^{j}(\phi_{\rho^{k}})=\rho^{k}(\phi_{v^{j}})$ for all $k=1,\dots,g$. Hence
the action of the vector field $E$ on the functions $v^{j}(\phi_{u^{k}})$ and
$v^{j}(\phi_{\rho^{k}})$ is studied in the first and second cases. Thus, it
suffices to consider the cases $E.v^{j}(\phi)$, where
$\phi\in\big{\\{}\phi_{t^{i,\alpha}},\phi_{v^{i}},\phi_{s^{i}}\big{\\}}$.
Given that the differential $\phi_{0}(P):=\phi_{v^{j}}(P)+d\lambda(P)$ is
holomorphic near $\infty^{j}$ (by (3.26), we see that the left hand side of
(3.58) applied to this $\phi_{0}$ can be rewritten as

$\underset{\infty^{j}}{{\rm
res}}\bigg{(}\frac{\lambda(P)\phi(P)}{d\lambda(P)}\phi_{v^{j}}(P)\bigg{)}+v^{j}(\phi)$

while its right hand side becomes

$\displaystyle-\delta_{\phi,\phi_{t^{j,\alpha}}}\frac{\alpha}{n_{j}+1}\Big{(}t^{j,\alpha}\big{(}\phi_{v^{j}}\big{)}+t^{j,\alpha}\big{(}d\lambda(P)\big{)}\Big{)}-\delta_{\phi,\phi_{v^{j}}}\Big{(}v^{j}\big{(}\phi_{v^{j}}\big{)}+v^{j}\big{(}d\lambda(P)\big{)}\Big{)}$
$\displaystyle=-\delta_{\phi,\phi_{t^{j,\alpha}}}\frac{\alpha}{n_{j}+1}t^{j,\alpha}\big{(}\phi_{v^{j}}\big{)}-\delta_{\phi,\phi_{v^{j}}}v^{j}\big{(}\phi_{v^{j}}\big{)}.$

This amounts to the following equality:

$\underset{\infty^{j}}{{\rm res}}\
\bigg{(}\frac{\lambda(P)\phi(P)}{d\lambda(P)}\phi_{v^{j}}(P)\bigg{)}=-v^{j}(\phi)-\delta_{\phi,\phi_{t^{j,\alpha}}}\frac{\alpha}{n_{j}+1}t^{j,\alpha}\big{(}\phi_{v^{j}}\big{)}-\delta_{\phi,\phi_{v^{j}}}v^{j}\big{(}\phi_{v^{j}}\big{)}.$

This and the key identity (3.58) lead to:

$\displaystyle E.v^{j}(\phi)=\sum_{k=1}^{N}\underset{P_{k}}{{\rm res}}\
\bigg{(}\frac{\lambda(P)\phi(P)}{d\lambda(P)}\phi_{v^{j}}(P)\bigg{)}$
$\displaystyle=-\underset{\infty^{j}}{{\rm res}}\
\bigg{(}\frac{\lambda(P)\phi(P)}{d\lambda(P)}\phi_{v^{j}}(P)\bigg{)}-\sum_{i=0,i\neq
j}^{m}\sum_{\alpha=1}^{n_{i}}\delta_{\phi,\phi_{t^{i,\alpha}}}\underset{\infty^{i}}{{\rm
res}}\
\bigg{(}\frac{\lambda(P)\phi(P)}{d\lambda(P)}\phi_{v^{j}}(P)\bigg{)}-\sum_{i=1,i\neq
j}^{m}\delta_{\phi,\phi_{v^{i}}}\underset{\infty^{i}}{{\rm res}}\
\bigg{(}\frac{\lambda(P)\phi(P)}{d\lambda(P)}\phi_{v^{j}}(P)\bigg{)}$
$\displaystyle=v^{j}(\phi)+\delta_{\phi,\phi_{t^{j,\alpha}}}\frac{\alpha}{n_{j}+1}t^{j,\alpha}\big{(}\phi_{v^{j}}\big{)}+\delta_{\phi,\phi_{v^{j}}}v^{j}\big{(}\phi_{v^{j}}\big{)}+\sum_{i=0,i\neq
j}^{m}\sum_{\alpha=1}^{n_{i}}\delta_{\phi,\phi_{t^{i,\alpha}}}\frac{\alpha}{n_{i}+1}t^{i,\alpha}\big{(}\phi_{v^{j}}\big{)}+\sum_{i=1,i\neq
j}^{m}\delta_{\phi,\phi_{v^{i}}}v^{i}\big{(}\phi_{v^{j}}\big{)}$
$\displaystyle=v^{j}(\phi)+\sum_{i=0}^{m}\sum_{\alpha=1}^{n_{i}}\delta_{\phi,\phi_{t^{i,\alpha}}}\frac{\alpha}{n_{i}+1}v^{j}\big{(}\phi_{t^{i,\alpha}}\big{)}+\sum_{i=1}^{m}\delta_{\phi,\phi_{v^{i}}}v^{j}\big{(}\phi_{v^{i}}\big{)},$

with the last equality being a direct consequence of the following particular
cases of (3.36):

$t^{i,\alpha}\big{(}\phi_{v^{j}}\big{)}=v^{j}\big{(}\phi_{t^{i,\alpha}}\big{)},\quad\quad
v^{i}\big{(}\phi_{v^{j}}\big{)}=v^{j}\big{(}\phi_{v^{i}}\big{)}.$

Forth case: $t^{A}=t^{j,\beta}$. Assume that
$\phi\in\big{\\{}\phi_{t^{i,\alpha}},\phi_{s^{i}}\big{\\}}$. Then by similar
arguments we obtain

$\displaystyle E.t^{j,\beta}(\phi)$ $\displaystyle=-\underset{\infty^{j}}{{\rm
res}}\
\bigg{(}\frac{\lambda(P)\phi(P)}{d\lambda(P)}\phi_{t^{j,\beta}}(P)\bigg{)}-\sum_{i=0,i\neq
j}^{m}\sum_{\alpha=1}^{n_{i}}\delta_{\phi,\phi_{t^{i,\alpha}}}\underset{\infty^{i}}{{\rm
res}}\
\bigg{(}\frac{\lambda(P)\phi(P)}{d\lambda(P)}\phi_{t^{j,\beta}}(P)\bigg{)}$
$\displaystyle=\frac{\beta}{n_{j}+1}t^{j,\beta}(\phi)+\sum_{i=0}^{m}\sum_{\alpha=1}^{n_{i}}\delta_{\phi,\phi_{t^{i,\alpha}}}\frac{\alpha}{n_{i}+1}t^{j,\beta}\big{(}\phi_{t^{i,\alpha}}\big{)}.$

Fifth case: $t^{A}=s^{j}$. By symmetry, it suffices to treat the case
$\phi=\phi_{s^{i}}=\Omega_{\infty^{0}\infty^{i}}$. The action of the vector
field $E$ on $s^{j}(\phi_{s^{i}})=s_{ij}$ was already described in (3.48).

$\square$

##  4 Hurwitz-Frobenius manifold structures

###  4.1 Frobenius manifold structures on simple Hurwitz spaces

The goal of this subsection is to build some Frobenius manifold structures on
simple Hurwitz spaces $\widehat{\mathcal{H}}_{g,L}(n_{0},\dots,n_{m})$ induced
by quasi-homogeneous differentials (in the sense of Definition 4.2 below) and
construct the corresponding prepotentials via a new approach. As we will see
below, Dubrovin’s primary differentials in (3.23) are all quasi-homogeneous
differentials. Thus our construction includes semi-simple Hurwitz-Frobenius
manifold of Dubrovin as particular cases.
As pointed out in the introduction, the prepotentials of the obtained
Frobenius manifold structures provide quasi-homogeneous solutions to the
generalized WDVV equations (1.6).

####  4.1.1 Frobenius algebra structures

Let
$\eta(\omega):=\frac{1}{2}\sum_{j}\big{(}\omega(P_{j})\big{)}^{2}(d\lambda_{j})^{2}$
be the Darboux-Egoroff metric on the manifold
$\mathcal{M}_{\omega}\subset\widehat{\mathcal{H}}_{g,L}(n_{0},\dots,n_{m})$
determined by

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

Here, as above, the differential $\omega$ is of the form
$\omega(P)=\int_{\gamma}h(\lambda(Q))W(P,Q)$ such that the contour $\gamma$
and the function $h$ satisfy respectively the conditions C1) and C2) in (3.1).
On each tangent space $T_{p}\mathcal{M}_{\omega}$, we define a commutative and
associative multiplication by

$\partial_{\lambda_{i}}\circ\partial_{\lambda_{j}}=\delta_{ij}\partial_{\lambda_{j}}.$
(4.1)

Particularly, we easily see that the vector field
$e=\sum_{j}\partial_{\lambda_{j}}$ is the unit vector field of the
multiplication (4.1).
For $x=\sum_{j}x_{j}\partial_{\lambda_{j}}$,
$y=\sum_{j}y_{j}\partial_{\lambda_{j}}$ and
$z=\sum_{j}z_{j}\partial_{\lambda_{j}}\in T_{p}\mathcal{M}_{\omega}$, consider
the $(0,3)$-tensor

$\mathbf{c}_{\omega}(x,y,z):=\eta(\omega)(x\circ
y,z)=\frac{1}{2}\sum_{j}x_{j}y_{j}z_{j}\omega(P_{j})^{2}.$ (4.2)

Since the tensor $\mathbf{c}_{\omega}(x,y,z)$ is symmetric, the quadruplet
$\big{(}T_{p}\mathcal{M}_{\omega},\circ,e,\eta(\omega)\big{)}$ is a
commutative and associative Frobenius algebra with unity (according to
Definition 2.1).
Note that formula (LABEL:flat-canonical_relation) allows us to describe the
Frobenius algebra structure of $T_{p}\mathcal{M}_{\omega}$ by means of the new
system of flat coordinates
$\mathcal{S}(\omega)=\big{\\{}t^{i,\alpha}(\omega),v^{i}(\omega),s^{i}(\omega),\rho^{k}(\omega),u^{k}(\omega)\big{\\}}$
of the metric $\eta(\omega)$. The upcoming obtained expressions (4.3)-(4.5)
involve the $\eta(\omega)$-duality relations between flat coordinates that
were introduced in (3.40) and (3.41):

  * a)

The multiplication “$\circ$” in (4.1) takes the following form in flat
coordinates:

$\displaystyle\begin{split}\partial_{t^{A}(\omega)}\circ\partial_{t^{B}(\omega)}&=\sum_{j=1}^{N}\frac{\phi_{t^{A^{\prime}}}(P_{j})\phi_{t^{B^{\prime}}}(P_{j})}{\omega(P_{j})^{2}}\partial_{\lambda_{j}}\\\
&=\sum_{C}\bigg{(}\frac{1}{2}\sum_{j=1}^{N}\frac{\phi_{t^{A^{\prime}}}(P_{j})\phi_{t^{B^{\prime}}}(P_{j})\phi_{t^{C}}(P_{j})}{\omega(P_{j})}\bigg{)}\partial_{t^{C}(\omega)}.\end{split}$
(4.3)

  * b)

The unit vector field $e=\sum_{j=1}^{N}\partial_{\lambda_{j}}$ is represented
by:

$e=\sum_{A}\bigg{(}\frac{1}{2}\sum_{j=1}^{N}\phi_{t^{A}}(P_{j})\omega(P_{j})\bigg{)}\partial_{t^{A}(\omega)}.$
(4.4)

  * c)

The symmetric 3-tensor $\mathbf{c}_{\omega}$ defined by (4.2) has the form:

$\mathbf{c}_{\omega}\big{(}\partial_{t^{A}(\omega)},\partial_{t^{B}(\omega)},\partial_{t^{C}(\omega)}\big{)}=\frac{1}{2}\sum_{j=1}^{N}\frac{\phi_{t^{A^{\prime}}}(P_{j})\phi_{t^{B^{\prime}}}(P_{j})\phi_{t^{C^{\prime}}}(P_{j})}{\omega(P_{j})}.$
(4.5)

This can be rewritten as a residue:

$\mathbf{c}_{\omega}\big{(}\partial_{t^{A}(\omega)},\partial_{t^{B}(\omega)},\partial_{t^{C}(\omega)}\big{)}=\sum_{j=1}^{N}\underset{P_{j}}{\text{res}}\frac{\phi_{t^{A^{\prime}}}(P)\phi_{t^{B^{\prime}}}(P)\phi_{t^{C^{\prime}}}(P)}{d\lambda(P)\omega(P)}.$

Note that expressions (4.3) and (4.5) imply that

$\partial_{t^{A}(\omega)}\circ\partial_{t^{B}(\omega)}=\textstyle\sum_{C}\mathbf{c}_{\omega}\big{(}\partial_{t^{A}(\omega)},\partial_{t^{B}(\omega)},\partial_{t^{C^{\prime}}(\omega)}\big{)}\partial_{t^{C}(\omega)}$

and thus the structure constants of the algebra multiplication “$\circ$” are

$\mathbf{c}_{\omega}\big{(}\partial_{t^{A}(\omega)},\partial_{t^{B}(\omega)},\partial_{t^{C^{\prime}}(\omega)}\big{)}=\sum_{\alpha=1}^{N}\eta^{C\alpha}\mathbf{c}_{\omega}\big{(}\partial_{t^{A}(\omega)},\partial_{t^{B}(\omega)},\partial_{t^{\alpha}(\omega)}\big{)},$

with $\eta^{AB}=\delta_{B,A^{\prime}}$ being the constant matrix (3.37).

####  4.1.2 Primary differentials

###### Definition 4.1

Let $\omega(P)=\int_{\gamma}h(\lambda(Q))W(Q,P)$ be a fixed differential as in
(3.1) and (3.3) and $\eta(\omega)$ be the corresponding Darboux-Egoroff metric
(3.20).
We shall call the differential $\omega$ primary with respect to unit vector
field $e=\sum_{j}\partial_{\lambda_{j}}$ ($e$-primary) if

$\nabla^{\eta(\omega)}.e=0,$

where $\nabla^{\eta(\omega)}$ is the Levi-Civita connection of the metric
$\eta(\omega)$.

In other words, $e$-primary differentials are exactly those that ensure the
flatness of the unit vector field $e$. In this case, the components $e^{A}(t)$
of $e$ in flat coordinates $\\{t^{A}(\omega)\\}$ of the metric $\eta(\omega)$
are constant functions.

###### Proposition 4.1

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

$\nabla^{\eta(\omega)}_{\partial_{\lambda_{j}}}.e=\frac{e.\omega(P_{j})}{\omega(P_{j})}\partial_{\lambda_{j}}.$
(4.6)

In particular, the following statements are equivalent:

1.

The differential $\omega$ is primary with respect to $e$;

2.

For all $j=1,\dots,N$,

$e.\omega(P_{j})=0;$ (4.7)

3.

For all $j=1,\dots,N$,

$\int_{\gamma}h^{\prime}(\lambda(Q))W(Q,P_{j})=0.$ (4.8)

_Proof:_ Bearing in mind the Christoffel coefficients of the diagonal metric
$\eta(\omega)$ given by (3.6) and the form $e=\sum_{j}\partial_{\lambda_{j}}$
of the unit vector field $e$, we obtain

$\displaystyle\nabla^{\eta(\omega)}_{\partial_{\lambda_{i}}}e$
$\displaystyle=\nabla^{\eta(\omega)}_{\partial_{\lambda_{i}}}\Big{(}\sum_{j=1}\partial_{\lambda_{j}}\Big{)}=\sum_{j}\sum_{k}\Gamma_{ij}^{k}(\eta)\partial_{\lambda_{k}}=\sum_{j\neq
i}\sum_{k}\Gamma_{ij}^{k}(\eta)\partial_{\lambda_{k}}+\sum_{k}\Gamma_{ii}^{k}(\eta)\partial_{\lambda_{k}}$
$\displaystyle=\Big{(}\sum_{j\neq
i}\Gamma_{ij}^{i}(\eta)\Big{)}\partial_{\lambda_{i}}+\sum_{j\neq
i}\Gamma_{ij}^{j}(\eta)\partial_{\lambda_{j}}+\sum_{k\neq
i}\Gamma_{ii}^{k}(\eta)\partial_{\lambda_{k}}+\Gamma_{ii}^{i}(\eta)\partial_{\lambda_{i}}$
$\displaystyle=\frac{1}{\omega(P_{i})}\Big{(}\sum_{j\neq
i}\partial_{\lambda_{j}}\omega(P_{i})\Big{)}\partial_{\lambda_{i}}+\frac{\partial_{\lambda_{i}}\omega(P_{i})}{\omega(P_{i})}\partial_{\lambda_{i}}$
$\displaystyle=\frac{e.\omega(P_{i})}{\omega(P_{i})}\partial_{\lambda_{i}}.$

This proves (4.6).
The equivalence between the three assertions follows immediately from (4.6)
and the action (3.8) of the vector field $e$ on the functions $\omega(P_{i})$.

$\square$

As a corollary, the characterization (4.8) and the properties of the symmetric
bidifferential yield the following.

###### Corollary 4.1

Consider the $N$ differentials
$\phi_{t^{A}}\in\Big{\\{}\phi_{t^{i,\alpha}},\phi_{v^{i}},\phi_{s^{i}},\phi_{\rho^{k}},\phi_{u^{k}}\Big{\\}}$
associated with the $\eta(\omega)$-flat coordinates and given by (3.23), with
$N=2g+2m+\textstyle\sum_{i=0}^{m}n_{i}$.
Then all the $N$ differentials $\phi_{t^{A}}$ are primary with respect to the
unit vector field $e=\sum_{j=1}^{N}\partial_{\lambda_{j}}$.

These primary differentials coincide (up to multiplicative constants) with the
Dubrovin primary differentials [14, 41].

###### Proposition 4.2

Assume that $\omega_{0}:=\phi_{t^{A_{0}}}$ is a fixed primary differential
belonging to the family (3.23).
Then the unit vector field $e=\sum_{j}\partial_{\lambda_{j}}$ of the Frobenius
algebra associated with $\omega_{0}$ reduces to

$e=\partial_{t^{A_{0}^{\prime}}(\omega_{0})}.$ (4.9)

_Proof:_ In view of (4.4), (3.17) with $(\beta_{1},\beta_{2})=(0,1)$ and
(3.40), we obtain

$\displaystyle e$
$\displaystyle=\sum_{B}\bigg{(}\frac{1}{2}\sum_{j}\phi_{t^{B}}(P_{j})\phi_{t^{A_{0}}}(P_{j})\bigg{)}\partial_{t^{B}(\omega_{0})}$
$\displaystyle=\sum_{B}\bigg{(}\eta^{*}(\omega_{0})\big{(}dt^{A_{0}}(\omega_{0}),dt^{B}(\omega_{0})\big{)}\bigg{)}\partial_{t^{B}(\omega_{0})}$
$\displaystyle=\sum_{B}\delta_{B,A_{0}^{\prime}}\partial_{t^{B}(\omega_{0})}=\partial_{t^{A_{0}^{\prime}}(\omega_{0})}.$

Alternatively, the result can be also checked by applying (LABEL:flat-
canonical_relation) to the partial derivative
$\partial_{t^{A_{0}^{\prime}}(\omega_{0})}$ and $\omega_{0}=\phi_{t^{A_{0}}}$.

$\square$

####  4.1.3 Euler vector field and quasi-homogeneous differentials

###### Definition 4.2

Let $\omega_{0}$ be a differential as in (3.1) and (3.3). We shall call
$\omega_{0}$ quasi-homogenous of degree $d_{0}\in\mathbb{R}$ if there is a
nonzero constant $c_{0}\in\mathbb{C}$ such that

$\omega_{0}(P)=c_{0}\int_{\ell_{0}}\big{(}\lambda(Q)\big{)}^{d_{0}}W(P,Q).$
(4.10)

More generally, a differential is called quasi-homogeneous of degree $d_{0}$
if it can be written as linear combination of differentials of the form
$\int_{\ell_{0}}\big{(}\lambda(Q)\big{)}^{d_{0}}W(P,Q)$.

From their integral representation (3.24) with respect to the kernel $W(P,Q)$,
primary differentials given in (3.23) are quasi-homogeneous and their degrees
satisfy (4.12). Note that duality relation (4.12) will be important for our
construction of prepotentials associated with a quasi-homogeneous differential
(see next subsection).

###### Proposition 4.3

The $N$ primary differentials listed in (3.23) are all quasi-homogeneous with
degrees

$\begin{array}[]{ccccc}&{\rm{deg}}\left(\phi_{t^{i,\alpha}}\right)=\frac{\alpha}{n_{i}+1},&\hskip
28.45274pti=0,\dots,m,&\hskip 28.45274pt\alpha=1,\dots,n_{i};\\\
&{\rm{deg}}\left(\phi_{v^{i}}\right)=deg\left(\phi_{u^{k}}\right)=1,&\hskip
28.45274pti=1,\dots,m,&\hskip 28.45274ptk=1,\dots,g;\\\
&{\rm{deg}}\left(\phi_{s^{i}}\right)=deg\left(\phi_{\rho^{k}}\right)=0,&\hskip
28.45274pti=1,\dots,m,&\hskip 28.45274ptk=1,\dots,g.\end{array}$ (4.11)

Furthermore, their degrees satisfy the following duality relation:

${\rm{deg}}\big{(}\phi_{t^{A}}\big{)}+{\rm{deg}}\big{(}\phi_{t^{A^{\prime}}}\big{)}=1,\quad\text{for
all A}.$ (4.12)

Here $t^{A^{\prime}}$ is the $\eta$-dual flat coordinate of $t^{A}$ in the
sense of (3.40) and (3.41).

_Proof:_ Using (3.41) and (4.11), we see that relation (4.12) holds.

$\square$

We are going to consider the following infinite family of quasi-homogenous
differentials, none of which is primary with respect the unit vector field
$e$. Each quasi-homogenous differential can give a non-normalized (i.e. with
non-flat unity) Frobenius manifold structure (see Definition 2.2).

$\bullet$ For $i=0,\dots,m$ and $\beta\in\mathbb{N}$, $\beta\geq 1$,

$\phi_{t^{i,n_{i}+\beta+1}}(P):=\underset{\infty^{i}}{\rm res}\
\lambda(Q)^{1+\frac{\beta}{n_{i}+1}}W(P,Q)$

is an Abelian differential of the second kind with unique pole of order
$n_{i}+\beta+2$ at $\infty^{i}$. The differential $\phi_{t^{i,n_{i}+\beta+1}}$
has the following behavior

$\phi_{t^{i,n_{i}+\beta+1}}(P)\underset{P\sim\infty^{i}}{=}\frac{n_{i}+\beta+1}{z_{i}(P)^{n_{i}+\beta+2}}dz_{i}(P)+\
\text{holomorphic},\quad\quad\lambda(P)=z_{i}(P)^{-n_{i}-1}.$

Let us point out that a formula analogous to (3.38) and the residue theorem
show that the action of the unit vector field
$e=\sum_{j}\partial_{\lambda_{j}}$ on the function
$t^{A}\big{(}\phi_{t^{i,n_{i}+\beta+1}}\big{)}$ is determined by

$\displaystyle e.t^{A}\big{(}\phi_{t^{i,n_{i}+\beta+1}}\big{)}$
$\displaystyle=\sum_{r=1}^{N}\underset{P_{r}}{{\rm
res}}\frac{\phi_{t^{A}}(P)\phi_{t^{i,n_{i}+\beta+1}}(P)}{d\lambda(P)}$
$\displaystyle=-\underset{\infty^{i}}{{\rm
res}}\frac{\phi_{t^{A}}(P)\phi_{t^{i,n_{i}+\beta+1}}(P)}{d\lambda(P)}$
$\displaystyle=\frac{n_{i}+\beta+1}{n_{i}+1}\underset{\infty^{i}}{{\rm
res}}\lambda(P)^{\frac{\beta}{n_{i}+1}}\phi_{t^{A}}(P).$

Here $t^{A}$ is one of the operations
$\big{\\{}t^{j,\alpha},v^{j},s^{j},\rho^{k},u^{k}\big{\\}}$ introduced in
(3.21) and $\phi_{t^{A}}$ is the corresponding differential.
In particular

$\begin{array}[]{ll}\displaystyle\partial_{\lambda_{j}}\Big{(}e.t^{A}\big{(}\phi_{t^{i,n_{i}+\beta+1}}\big{)}\Big{)}=\frac{n_{i}+\beta+1}{2(n_{i}+1)}\phi_{t^{i,\beta}}(P_{j})\phi_{t^{A}}(P_{j}),&\hbox{if}\quad
j=1,\dots,N;\\\ \\\ \displaystyle
e.t^{A}\big{(}\phi_{t^{i,n_{i}+\beta+1}}\big{)}=2v^{i}(\phi_{t^{A}}),&\hbox{if}\quad\beta=n_{i}+1;\\\
\\\ \displaystyle
e.t^{A}\big{(}\phi_{t^{i,n_{i}+\beta+1}}\big{)}=\frac{\beta(n_{i}+\beta+1)}{(n_{i}+1)^{3/2}}t^{i,\beta}(\phi_{t^{A}}),&\hbox{if}\quad\beta=1,\dots,n_{i}.\end{array}$

$\bullet$ Assume that the covering $(C_{g},\lambda)$ belongs to the particular
double Hurwitz space $\widehat{\mathcal{H}}_{g,L}(\mathbf{n};\mathbf{0})$,
with $\mathbf{n}=(n_{0}+1,\dots,n_{m}+1)$. As indicated before, this is an
open subset of the simple Hurwitz space
$\widehat{\mathcal{H}}_{g,L}(n_{0},\dots,n_{m})$ determined by the conditions
$\lambda_{j}\neq 0$, for all $j=1,\dots,N$. Let
$\mathbf{z}^{0},\dots,\mathbf{z}^{L-1}$ be the zeros of the meromorphic
function $\lambda$. Then the following (normalized) Abelian differential of
the third kind:

$\Omega_{\mathbf{z}^{0}\mathbf{z}^{j}}(P)=\int_{\mathbf{z}^{0}}^{\mathbf{z}^{j}}W(P,Q),\quad\Omega_{\mathbf{z}^{0}\infty^{0}}(P)=\int_{\mathbf{z}^{0}}^{\infty^{0}}W(P,Q),\quad
j=1,\dots,L-1,$

are quasi-homogeneous differentials of degree 0.
In this case, for all
$t^{A}\in\big{\\{}t^{j,\alpha},v^{j},s^{j},\rho^{k},u^{k}\big{\\}}$ we have

$\displaystyle e.t^{A}\big{(}\Omega_{\mathbf{z}^{0}\mathbf{z}^{j}}\big{)}$
$\displaystyle=\sum_{r=1}^{N}\underset{P_{r}}{{\rm
res}}\frac{\phi_{t^{A}}(P)\Omega_{\mathbf{z}^{0}\mathbf{z}^{j}}(P)}{d\lambda(P)}$
$\displaystyle=-\underset{\mathbf{z}^{j}}{{\rm
res}}\frac{\phi_{t^{A}}(P)\Omega_{\mathbf{z}^{0}\mathbf{z}^{j}}(P)}{d\lambda(P)}-\underset{\mathbf{z}^{0}}{{\rm
res}}\frac{\phi_{t^{A}}(P)\Omega_{\mathbf{z}^{0}\mathbf{z}^{j}}(P)}{d\lambda(P)}$
$\displaystyle=\phi_{t^{A}}(\mathbf{z}^{0})-\phi_{t^{A}}(\mathbf{z}^{j});$
$\displaystyle e.t^{A}\big{(}\Omega_{\mathbf{z}^{0}\infty^{0}}\big{)}$
$\displaystyle=-\underset{\mathbf{z}^{0}}{{\rm
res}}\frac{\phi_{t^{A}}(P)\Omega_{\mathbf{z}^{0}\infty^{0}}(P)}{d\lambda(P)}=\phi_{t^{A}}(\mathbf{z}^{0}),$

where $\phi_{t^{A}}(\mathbf{z}^{j})$ is the evaluation (2.7) of the
differential $\phi_{t^{A}}$ at the zero $\mathbf{z}^{j}$.

$\bullet$ Let $k=1,\dots,g$ and $\sigma,\nu$ be two positive integers. Then
the following differentials

$\phi_{u^{k,\sigma}}(P):=\oint_{a_{k}}\big{(}\lambda(Q)\big{)}^{\sigma}W(P,Q)\quad\quad\text{and}\quad\quad\phi_{\widetilde{u}^{k,\nu}}(P):=\oint_{b_{k}}\big{(}\lambda(Q)\big{)}^{\nu}W(P,Q)$

are quasi-homogeneous of degrees $\sigma$ and $\nu$, respectively.
The differentials $\phi_{u^{k,\sigma}}$ and $\phi_{\widetilde{u}^{k,\nu}}$ are
multivalued and their jumps are studied in (8.8) (see Appendix 2 below).
Furthermore, if $t^{A}$ denotes one of the operations
$t^{i,\alpha},v^{i},s^{i},\rho^{k},u^{k}$ listed in (3.21)-(3.22), then the
action of the unit vector field $e=\sum_{j}\partial_{\lambda_{j}}$ on the
functions $t^{A}\big{(}\phi_{u^{k,\sigma}}\big{)}$ and
$t^{A}\big{(}\phi_{\widetilde{u}^{k,\nu}}\big{)}$ is also described in
Appendix 2 (formula (8.10)).

Let us introduce a notation for the set of the quasi-homogeneous differentials
mentioned so far.

$\mathrm{QH}(E):=\Big{\\{}\text{$e$-primary differentials
(\ref{Primary-e})}\Big{\\}}\bigcup\Big{\\{}\phi_{t^{i,n_{i}+\beta+1}},\Omega_{\mathbf{z}^{0}\mathbf{z}^{j}},\Omega_{\mathbf{z}^{0}\infty^{0}},\phi_{u^{k,\sigma}},\phi_{\widetilde{u}^{k,\nu}}\Big{\\}}.$
(4.13)

The above quasi-homogeneity terminology is justified by the following
property.

###### Proposition 4.4

Let $(C_{g},\lambda)$ be a point in the simple Hurwitz space
$\widehat{\mathcal{H}}_{g,L}(n_{0},\dots,n_{m})$. Let $D_{E}$ be the vector
field defined by:

$D_{E}:=\lambda(P)\partial_{\lambda(P)}+\textstyle\sum_{j}\lambda_{j}\partial_{\lambda_{j}}=\lambda(P)\partial_{\lambda(P)}+E.$

If $\omega_{0}$ is a quasi-homogeneous differential of degree $d_{0}\geq 0$
(4.10), then

$D_{E}.\omega_{0}(P)=d_{0}\omega_{0}(P).$ (4.14)

_Proof:_ Given a branched covering $(C_{g},\lambda)$ of $\mathbb{P}^{1}$,
consider the family of branched coverings
$\big{\\{}(C_{g},\lambda_{\varepsilon})\big{\\}}_{\varepsilon}$ defined by

$\lambda_{\varepsilon}(P)=e^{\varepsilon}\lambda(P),$

where $\varepsilon\in\mathbb{C}$ and $|\varepsilon|$ is sufficiently small.
This is nothing but the particular case of (3.10) where
$(\beta_{1},\beta_{2})=(1,0)$.
Under this transformation, the differential $\omega_{0}$ changes as follows:

$\displaystyle\omega_{0}^{\varepsilon}(P)=c_{0}\int_{\ell_{0}}\big{(}\lambda_{\varepsilon}(Q)\big{)}^{d_{0}}W(P,Q)=e^{d_{0}\varepsilon}\omega_{0}(P).$

Therefore

$D_{E}\omega_{0}(P)=\frac{d}{d{\varepsilon}}\omega_{0}^{\varepsilon}(P)\Big{|}_{\varepsilon=0}=d_{0}\omega_{0}(P).$

$\square$

###### Remark 4.1

Formula (4.14) allows us to consider quasi-homogeneous differentials of degree
$d_{0}\geq 0$ as the differentials belonging to the vector space
$Ker\big{(}D_{E}-d_{0}\big{)}$.
For example, we can view a linear combination of holomorphic differentials as
a quasi-homogeneous differential of degree 0.

###### Proposition 4.5

Let $\omega_{0}$ be a quasi-homogeneous differential of degree $d_{0}\geq 0$
belonging to the set $\mathrm{QH}(E)$ (4.13) and $t^{A}(\omega_{0})$ be one of
the $\eta(\omega_{0})$-flat coordinates (3.21)-(3.22).
Then the vector field $E:=\sum_{j}\lambda_{j}\partial_{\lambda_{j}}$ acts on
the function $t^{A}(\omega_{0})$ as follows:

$E.t^{A}(\omega_{0})=(d_{A}+d_{0})t^{A}(\omega_{0})+r_{\omega_{0},\phi_{t^{A}}},$
(4.15)

where $d_{A}$ is the quasi-homogeneous degree of the primary differential
$\phi_{t^{A}}$ (see relations (4.10) and (4.11)) and
$r_{\omega_{0},\phi_{t^{A}}}$ is the constant defined by

$r_{\omega_{0},\phi_{t^{A}}}=\left\\{\begin{array}[]{ll}\displaystyle
r_{B,A}=r_{A,B}&\hbox{if}\quad\text{$\omega_{0}=\phi_{t^{B}}$ is one of the
primary differentials (\ref{Primary-e})};\\\ \\\
\displaystyle-\frac{1}{n_{0}+1}\sum_{i=1}^{m}\delta_{\phi_{t^{A}},\phi_{s^{i}}},&\hbox{if}\quad\omega_{0}=\Omega_{\mathbf{z}^{0}\infty^{0}};\\\
\\\ 0,&\hbox{}\quad\text{otherwise}.\end{array}\right.$ (4.16)

Here $r_{A,B}$ are given by (3.57).
Moreover, in flat coordinates (3.21)-(3.22) of the metric $\eta(\omega_{0})$
(3.20), the vector field $E$ admits the following representation

$E=\sum_{A}\Big{(}(d_{A}+d_{0})t^{A}(\omega_{0})+r_{\omega_{0},\phi_{t^{A}}}\Big{)}\partial_{t^{A}(\omega_{0})}.$
(4.17)

_Proof:_ As in Section 3.4, in order to calculate the action of the vector
field $E$ on the functions $t^{A}(\omega_{0})$, we shall use the following
identity:

$E.t^{A}(\omega_{0}):=\sum_{j}\lambda_{j}\partial_{\lambda_{j}}t^{A}(\omega_{0})=\sum_{j}\underset{P_{j}}{\text{res}}\frac{\lambda(P)\phi_{t^{A}}(P)\omega_{0}(P)}{d\lambda(P)}.$

Here the second equality follows directly from (3.18) and (3.19) where we take
$(\beta_{1},\beta_{2})=(1,0)$.
If $\omega_{0}=\phi_{t^{B}}$ is among the $e$-primary differentials (3.23),
then (4.15) is nothing but the second equality in (3.56).
In the case where $\omega_{0}$ is one of the multivalued differentials
$\big{\\{}\phi_{u^{k,\sigma}},\phi_{\widetilde{u}^{k,\nu}}\big{\\}}$, we give
a detailed calculation that proves (4.15) in Appendix 2 (formulas (8.12) and
(8.13)). With the help of the residue theorem and identity (3.58), we can
straightforwardly obtain the remaining cases.
Finally, we have

$E=\sum_{j}\lambda_{j}\partial_{\lambda_{j}}=\sum_{j}\lambda_{j}\bigg{(}\sum_{A}\Big{(}\partial_{\lambda_{j}}t^{A}(\omega_{0})\Big{)}\partial_{t^{A}(\omega_{0})}\bigg{)}=\sum_{A}\Big{(}E.t^{A}(\omega_{0})\Big{)}\partial_{t^{A}(\omega_{0})}.$

Thus representation (4.17) of the vector field $E$ is an immediate consequence
of (4.15).

$\square$

###### Proposition 4.6

Let $\omega_{0}(P):=c_{0}\int_{\ell_{0}}\left(\lambda(Q)\right)^{d_{0}}W(P,Q)$
be a fixed quasi-homogeneous differential of degree $d_{0}\geq 0$.
Then the vector field $E=\sum_{j}\lambda_{j}\partial_{\lambda_{j}}$ satisfies:

$\displaystyle\begin{array}[]{llllll}&[E,e]=-e;\\\ \\\ &[E,x\circ
y]-[E,x]\circ y-x\circ[E,y]=x\circ y;\\\ \\\
&E.\omega_{0}(P_{j})=(d_{0}-1/2)\omega_{0}(P_{j}),&\hbox{}\quad\quad
j=1,\dots,N;\\\ \\\
&\big{(}Lie_{E}.\eta(\omega_{0})\big{)}\big{(}\partial_{\lambda_{j}},\partial_{\lambda_{j}}\big{)}=(2d_{0}-3)\eta(\omega_{0})\big{(}\partial_{\lambda_{j}},\partial_{\lambda_{j}}\big{)},&\hbox{}\quad\quad
j=1,\dots,N.\end{array}$

_Proof:_ By simple calculation, we can check that the first two relations hold
true. The third one is a particular case of (3.8) where we take
$h(\lambda)=c_{0}\lambda^{d_{0}}$ and $(\beta_{1},\beta_{2})=(1,0)$.
Finally, the action of the Lie derivative along the vector field $E$ on the
diagonal metric
$\eta(\omega_{0})=\frac{1}{2}\sum_{j}\omega_{0}(P_{j})^{2}(d\lambda_{j})^{2}$
is as follows

$\displaystyle\big{(}Lie_{E}.\eta(\omega_{0})\big{)}\big{(}\partial_{\lambda_{j}},\partial_{\lambda_{j}}\big{)}$
$\displaystyle:=E.\Big{(}\eta(\omega_{0})\big{(}\partial_{\lambda_{j}},\partial_{\lambda_{j}}\big{)}\Big{)}-2\eta(\omega_{0})\big{(}[E,\partial_{\lambda_{j}}],\partial_{\lambda_{j}}\big{)}$
$\displaystyle=\big{(}E.\omega_{0}(P_{j})\big{)}\omega_{0}(P_{j})+\omega_{0}(P_{j})^{2}$
$\displaystyle=(2d_{0}-3)\frac{\omega_{0}(P_{j})^{2}}{2}=(2d_{0}-3)\eta(\omega_{0})\big{(}\partial_{\lambda_{j}},\partial_{\lambda_{j}}\big{)}.$

$\square$

Summarizing the above results, Propositions 4.5 and 4.6 tell us that the
vector field $E=\sum_{j=1}^{N}\lambda_{j}\partial_{\lambda_{j}}$ is
covariantly linear and obeys the conditions (LABEL:Euler_vector_def) and thus
it is an Euler vector field.

####  4.1.4 A new formula for the prepotential

A prepotential of a Frobenius manifold structure
$(\mathcal{M},\circ,\mathop{\langle\cdot,\cdot\rangle})$ is a function
$\mathrm{F}$ of flat coordinates $\\{t^{A}\\}_{A}$ of the corresponding flat
metric $\mathop{\langle\cdot,\cdot\rangle}$ such that its third derivatives
are given by the symmetric 3-tensor $\mathrm{c}$ from the Definition 2.2 of a
Frobenius manifold:

$\displaystyle\partial_{t^{A}}\partial_{t^{B}}\partial_{t^{C}}\mathrm{F}=\mathrm{c}\left(\partial_{t^{A}},\partial_{t^{B}},\partial_{t^{C}}\right)=\mathop{\langle\partial_{t^{A}}\circ\partial_{t^{B}},\partial_{t^{C}}\rangle}.$

Let us start with the following crucial observation concerning the symmetric
3-tensor (4.5):

###### Proposition 4.7

Let $\omega_{0}$ be a differential satisfying (3.1) and (3.3) and
$\eta(\omega_{0})$ be the corresponding Darboux-Egoroff metric (3.20). Then
the symmetric 3-tensor $\mathbf{c}_{\omega_{0}}$ (4.5) associated with
$\omega_{0}$ enjoys the following formula:

$\mathbf{c}_{\omega_{0}}\big{(}\partial_{t^{A}(\omega_{0})},\partial_{t^{B}(\omega_{0})},\partial_{t^{C}(\omega_{0})}\big{)}=\partial_{t^{C}(\omega_{0})}t^{A^{\prime}}\left(\phi_{t^{B^{\prime}}}\right)=\partial_{t^{C}(\omega_{0})}t^{B^{\prime}}\left(\phi_{t^{A^{\prime}}}\right).$
(4.18)

_Proof:_ From (LABEL:flat-canonical_relation) we know that

$\partial_{t^{C}(\omega_{0})}=\sum_{j}\frac{\phi_{t^{C^{\prime}}}(P_{j})}{\omega_{0}(P_{j})}\partial_{\lambda_{j}}.$

This relation, Rauch’s formula (3.15) and expression (4.5) for the tensor
$\mathbf{c}_{\omega_{0}}$ imply that the desired result (4.18) holds true.

$\square$

In view of (4.18), given that one can permute the order of differentiation, we
see that the 4-tensor

$\partial_{t^{D}(\omega_{0})}\mathbf{c}_{\omega_{0}}\big{(}\partial_{t^{A}(\omega_{0})},\partial_{t^{B}(\omega_{0})},\partial_{t^{C}(\omega_{0})}\big{)}$

is totally symmetric in the four components. As a consequence, according to
the above results and Definitions 2.2 and 2.3, we reach the following
conclusions.

###### Corollary 4.2

Assume that $\omega_{0}$ is a quasi-homogeneous differential (4.10) of degree
$d_{0}\geq 0$. Let $\mathcal{M}_{\omega_{0}}$ be the open subset of the simple
Hurwitz space $\widehat{\mathcal{H}}_{g,L}(n_{0},\dots,n_{m})$ determined by
the conditions:

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

Then the open set $\mathcal{M}_{\omega_{0}}$ carries a Frobenius manifold
structure given by the flat metric $\eta(\omega_{0})$ (3.20), multiplication
law (4.1) and Euler vector field (4.17).
If the differential $\omega_{0}$ is additionally chosen among primary
differentials (3.23) with respect to the unit vector field $e$, then the above
Frobenius manifold structure becomes normalized in the sense of Definition
2.3.

The following theorem is the central result of this work. It establishes a new
formula for prepotentials associated with the obtained Frobenius manifold
structures that are induced by quasi-homogenous differentials and described in
the above corollary.

###### Theorem 4.1

Let $\omega_{0}$ be a given quasi-homogeneous differential (4.10) of degree
$d_{0}\geq 0$ satisfying (3.3). Let
$\big{(}\mathcal{M}_{\omega_{0}},\eta(\omega_{0}),\circ,e,E\big{)}$ be the
Frobenius manifold structure from Corollary 4.2. Let also

i)

$E.t^{A}(\omega_{0})$ be the affine function
$E.t^{A}(\omega_{0}):=(d_{0}+d_{A})t^{A}(\omega_{0})+r_{\omega_{0},\phi_{t^{A}}}$,
for all $A$ (see (4.15));

ii)

$t^{A^{\prime}}$ denote the $\eta$-dual flat coordinate of $t^{A}$ introduced
in (3.40)-(3.41);

iii)

$\big{\\{}\phi_{t^{A}}\big{\\}}_{A}$ be the family of the $N$ primary
differentials (3.23) of Dubrovin;

iv)

$r_{A,B}$ be the constants given by (3.57).

Then the following function is a prepotential of the Frobenius manifold
$\big{(}\mathcal{M}_{\omega_{0}},\eta(\omega_{0}),\circ,e,E\big{)}$:

$\begin{split}\mathbf{F}_{\omega_{0}}&=\frac{1}{4(1+d_{0})}\sum_{A,B}\frac{\left(E.t^{A}(\omega_{0})\right)\left(E.t^{B}(\omega_{0})\right)}{1+d_{0}+d_{A^{\prime}}}\Big{(}t^{A^{\prime}}(\phi_{t^{B^{\prime}}})+t^{B^{\prime}}(\phi_{t^{A^{\prime}}})\Big{)}\\\
&\quad-\frac{1}{4(1+d_{0})}\sum_{A,B}\bigg{(}2\frac{E.t^{A}(\omega_{0})}{1+d_{0}+d_{A^{\prime}}}+t^{A}(\omega_{0})\bigg{)}r_{A^{\prime},B^{\prime}}t^{B}(\omega_{0}).\end{split}$
(4.19)

Moreover, the Hessian matrix of the function $\mathbf{F}_{\omega_{0}}$ does
not depend on the chosen quasi-homogeneous differential $\omega_{0}$ and its
entries are given by

$\partial_{t^{A}(\omega_{0})}\partial_{t^{B}(\omega_{0})}\mathbf{F}_{\omega_{0}}=\frac{1}{2}\left(t^{A^{\prime}}\left(\phi_{t^{B^{\prime}}}\right)+t^{B^{\prime}}\left(\phi_{t^{A^{\prime}}}\right)\right).$
(4.20)

The proof of this theorem is a direct consequence of relation (4.18) combined
with upcoming results (4.22) and (4.25).

###### Proposition 4.8

Consider the $N$ functions $\big{\\{}\mathbf{F}_{A,\omega_{0}}\big{\\}}_{A}$,
with

$\mathbf{F}_{A,\omega_{0}}=\frac{1}{2\big{(}1+d_{0}+d_{A^{\prime}}\big{)}}\sum_{B}\bigg{(}\left(E.t^{B}(\omega_{0})\right)\Big{(}t^{A^{\prime}}\left(\phi_{t^{B^{\prime}}}\right)+t^{B^{\prime}}\left(\phi_{t^{A^{\prime}}}\right)\Big{)}-2r_{A^{\prime},B^{\prime}}t^{B}(\omega_{0})\bigg{)}.$
(4.21)

Then the following statements hold:

1.

For any $B$, we have

$\partial_{t^{B}(\omega_{0})}\mathbf{F}_{A,\omega_{0}}=\frac{1}{2}\Big{(}t^{A^{\prime}}\left(\phi_{t^{B^{\prime}}}\right)+t^{B^{\prime}}\left(\phi_{t^{A^{\prime}}}\right)\Big{)}=\partial_{t^{A}(\omega_{0})}\mathbf{F}_{B,\omega_{0}}.$
(4.22)

2.

The Euler vector field (4.17) acts on the function $\mathbf{F}_{A,\omega_{0}}$
as follows:

$E.\mathbf{F}_{A,\omega_{0}}=\left(1+d_{0}+d_{A^{\prime}}\right)\mathbf{F}_{A,\omega_{0}}+\sum_{B}r_{A^{\prime},B^{\prime}}t^{B}(\omega_{0}).$
(4.23)

3.

The function $\mathbf{F}_{\omega_{0}}$ defined by (4.19) admits the following
representation involving the functions $\mathbf{F}_{A,\omega_{0}}$:

$\mathbf{F}_{\omega_{0}}=\frac{1}{2(1+d_{0})}\sum_{A}\big{(}E.t^{A}(\omega_{0})\big{)}\mathbf{F}_{A,\omega_{0}}-\frac{1}{4(1+d_{0})}\sum_{A,B}r_{A^{\prime},B^{\prime}}t^{A}(\omega_{0})t^{B}(\omega_{0}).$
(4.24)

4.

The partial derivatives of $\mathbf{F}_{\omega_{0}}$ are:

$\partial_{t^{A}(\omega_{0})}\mathbf{F}_{\omega_{0}}=\mathbf{F}_{A,\omega_{0}}.$
(4.25)

_Proof:_ Using the symmetry property of the tensor $\mathbf{c}_{\omega_{0}}$
following from expression (4.5) together with the quasi-homogeneity property
(3.56) of the functions $t^{A}(\phi_{t^{B}})$, i.e.
$E.t^{A}(\phi_{t^{B}})=(d_{A}+d_{B})t^{A}(\phi_{t^{B}})+r_{A,B}$ for the Euler
vector field $E$, we deduce that

$\displaystyle
2\big{(}1+d_{0}+d_{A^{\prime}}\big{)}\partial_{t^{B}(\omega_{0})}\mathbf{F}_{A,\omega_{0}}$
$\displaystyle=(d_{B}+d_{0})\Big{(}t^{A^{\prime}}\left(\phi_{t^{B^{\prime}}}\right)+t^{B^{\prime}}\left(\phi_{t^{A^{\prime}}}\right)\Big{)}-2r_{A^{\prime},B^{\prime}}+\sum_{D}\left(E.t^{D}(\omega_{0})\right)\partial_{t^{B}(\omega_{0})}\Big{(}t^{A^{\prime}}\left(\phi_{t^{D^{\prime}}}\right)+t^{D^{\prime}}\left(\phi_{t^{A^{\prime}}}\right)\Big{)}$
$\displaystyle=(d_{B}+d_{0})\Big{(}t^{A^{\prime}}\left(\phi_{t^{B^{\prime}}}\right)+t^{B^{\prime}}\left(\phi_{t^{A^{\prime}}}\right)\Big{)}-2r_{A^{\prime},B^{\prime}}+\sum_{D}\left(E.t^{D}(\omega_{0})\right)\partial_{t^{D}(\omega_{0})}\Big{(}t^{A^{\prime}}\left(\phi_{t^{B^{\prime}}}\right)+t^{B^{\prime}}\left(\phi_{t^{A^{\prime}}}\right)\Big{)}$
$\displaystyle=(d_{B}+d_{0})\Big{(}t^{A^{\prime}}\left(\phi_{t^{B^{\prime}}}\right)+t^{B^{\prime}}\left(\phi_{t^{A^{\prime}}}\right)\Big{)}-2r_{A^{\prime},B^{\prime}}+E.t^{A^{\prime}}\left(\phi_{t^{B^{\prime}}}\right)+E.t^{B^{\prime}}\left(\phi_{t^{A^{\prime}}}\right)$
$\displaystyle=\big{(}d_{B}+d_{0}+d_{B^{\prime}}+d_{A^{\prime}}\big{)}\Big{(}t^{A^{\prime}}\left(\phi_{t^{B^{\prime}}}\right)+t^{B^{\prime}}\left(\phi_{t^{A^{\prime}}}\right)\Big{)}$
$\displaystyle=\big{(}1+d_{0}+d_{A^{\prime}}\big{)}\Big{(}t^{A^{\prime}}\left(\phi_{t^{B^{\prime}}}\right)+t^{B^{\prime}}\left(\phi_{t^{A^{\prime}}}\right)\Big{)},$

where we have used the key duality formula $d_{B}+d_{B^{\prime}}=1$ in the
last line.
Now, by employing (4.21) and (4.22) and expression (4.17) for the Euler vector
field, we directly arrive at (4.23). Furthermore, replacing the function
$\mathbf{F}_{A,\omega_{0}}$ by its expression (4.21) in the left hand side of
formula (4.24) and comparing the resulting with (4.19), we conclude that
(4.24) holds.
Lastly, we are going to look at (4.25). Differentiating in (4.24) with respect
to $t^{A}(\omega_{0})$, we obtain

$\displaystyle\partial_{t^{A}(\omega_{0})}\mathbf{F}_{\omega_{0}}$
$\displaystyle=\frac{d_{A}+d_{0}}{2(1+d_{0})}\mathbf{F}_{A,\omega_{0}}+\frac{1}{2(1+d_{0})}\sum_{B}\big{(}E.t^{B}(\omega_{0})\big{)}\partial_{t^{A}(\omega_{0})}\mathbf{F}_{B,\omega_{0}}-\frac{1}{2(1+d_{0})}\sum_{B}r_{A^{\prime},B^{\prime}}t^{B}(\omega_{0})$
$\displaystyle=\frac{d_{A}+d_{0}}{2(1+d_{0})}\mathbf{F}_{A,\omega_{0}}+\frac{1}{2(1+d_{0})}\sum_{B}\big{(}E.t^{B}(\omega_{0})\big{)}\partial_{t^{B}(\omega_{0})}\mathbf{F}_{A,\omega_{0}}-\frac{1}{2(1+d_{0})}\sum_{B}r_{A^{\prime},B^{\prime}}t^{B}(\omega_{0})$
$\displaystyle=\frac{d_{A}+d_{0}}{2(1+d_{0})}\mathbf{F}_{A,\omega_{0}}+\frac{1}{2(1+d_{0})}E.\mathbf{F}_{A,\omega_{0}}-\frac{1}{2(1+d_{0})}\sum_{B}r_{A^{\prime},B^{\prime}}t^{B}(\omega_{0})$
$\displaystyle=\frac{d_{A}+d_{A^{\prime}}+2d_{0}+1}{2(1+d_{0})}\mathbf{F}_{A,\omega_{0}}$
$\displaystyle=\mathbf{F}_{A,\omega_{0}};$

where we have used relation (4.22) in the second equality, (4.17) in the third
equality, (4.23) in forth one and once again the duality relation
$d_{A}+d_{A^{\prime}}=1$ in the last equality. The proof of the proposition is
now complete.

$\square$

_Proof of Theorem 4.1:_ Differentiating three times the function
$\mathbf{F}_{\omega_{0}}$ given by (4.19) and using successively (4.25),
(4.22) and (4.18), we obtain

$\displaystyle\partial_{t^{C}(\omega_{0})}\partial_{t^{B}(\omega_{0})}\partial_{t^{A}(\omega_{0})}\mathbf{F}_{\omega_{0}}$
$\displaystyle=\partial_{t^{C}(\omega_{0})}\partial_{t^{B}(\omega_{0})}\mathbf{F}_{A,\omega_{0}}=\frac{1}{2}\partial_{t^{C}(\omega_{0})}\Big{(}t^{A^{\prime}}\left(\phi_{t^{B^{\prime}}}\right)+t^{B^{\prime}}\left(\phi_{t^{A^{\prime}}}\right)\Big{)}$
$\displaystyle=\mathbf{c}_{\omega_{0}}\big{(}\partial_{t^{A}(\omega_{0})},\partial_{t^{B}(\omega_{0})},\partial_{t^{C}(\omega_{0})}\big{)}.$

$\square$

###### Corollary 4.3

The prepotential $\mathbf{F}_{\omega_{0}}$ in (4.19) as well as its first,
second and third partial derivatives are quasi-homogeneous functions, with
respect to the Euler vector field $E$ given by (4.17), in the sense of (2.4).
More precisely, for the quantities

$\mathbf{F}_{A,\omega_{0}}:=\partial_{t^{A}(\omega_{0})}\mathbf{F}_{\omega_{0}},\quad\quad\mathbf{F}_{AB,\omega_{0}}:=\partial_{t^{A}(\omega_{0})}\partial_{t^{B}(\omega_{0})}\mathbf{F}_{\omega_{0}},\quad\quad\mathbf{F}_{ABC,\omega_{0}}:=\partial_{t^{A}(\omega_{0})}\partial_{t^{B}(\omega_{0})}\partial_{t^{C}(\omega_{0})}\mathbf{F}_{\omega_{0}},$

we have

$\begin{split}&E.\mathbf{F}_{\omega_{0}}=\textstyle
2(1+d_{0})\mathbf{F}_{\omega_{0}}+\frac{1}{2}\sum_{A,B}r_{A^{\prime},B^{\prime}}t^{A}(\omega_{0})t^{B}(\omega_{0});\\\
&E.\mathbf{F}_{A,\omega_{0}}=\textstyle\left(1+d_{0}+d_{A^{\prime}}\right)\mathbf{F}_{A,\omega_{0}}+\sum_{B}r_{A^{\prime},B^{\prime}}t^{B}(\omega_{0});\\\
&E.\mathbf{F}_{AB,\omega_{0}}=\left(d_{A^{\prime}}+d_{B^{\prime}}\right)\mathbf{F}_{AB,\omega_{0}}+r_{A^{\prime},B^{\prime}};\\\
&E.\mathbf{F}_{ABC,\omega_{0}}=\big{(}2-d_{0}-d_{A}-d_{B}-d_{C}\big{)}\mathbf{F}_{ABC,\omega_{0}}.\end{split}$
(4.26)

Here $r_{A,B}$ are the constants given by (3.57).

_Proof:_ In view of (4.24) and (4.25), we obtain

$\displaystyle E.\mathbf{F}_{\omega_{0}}$
$\displaystyle=\sum_{A}\big{(}E.t^{A}(\omega_{0})\big{)}\partial_{t^{A}(\omega_{0})}\mathbf{F}_{\omega_{0}}=\sum_{A}\big{(}E.t^{A}(\omega_{0})\big{)}\mathbf{F}_{A,\omega_{0}}$
$\displaystyle=2(1+d_{0})\mathbf{F}_{\omega_{0}}+\frac{1}{2}\sum_{A,B}r_{A^{\prime},B^{\prime}}t^{A}(\omega_{0})t^{B}(\omega_{0}).$

The second relation in (4.26) is nothing but (4.23). In addition, using (4.20)
and (3.56), we see that the second partial derivatives
$\mathbf{F}_{AB,\omega_{0}}$ of $\mathbf{F}_{\omega_{0}}$ are also quasi-
homogeneous functions.
Now let us deal with the last equality in (4.26). By employing Theorem 4.1,
relation (4.18), the duality formula $d_{A}+d_{A^{\prime}}=1$ as well as
expression (4.17) for the Euler vector field, we can write

$\displaystyle E.\mathbf{F}_{ABC,\omega_{0}}$
$\displaystyle=E.\mathbf{c}_{\omega_{0}}\big{(}\partial_{t^{A}(\omega_{0})},\partial_{t^{B}(\omega_{0})},\partial_{t^{C}(\omega_{0})}\big{)}=E.\partial_{t^{C}(\omega_{0})}t^{A^{\prime}}\left(\phi_{t^{B^{\prime}}}\right)$
$\displaystyle=\partial_{t^{C}(\omega_{0})}\Big{(}E.t^{A^{\prime}}\left(\phi_{t^{B^{\prime}}}\right)\Big{)}+[E,\partial_{t^{C}(\omega_{0})}].t^{A^{\prime}}\left(\phi_{t^{B^{\prime}}}\right)$
$\displaystyle=(d_{A^{\prime}}+d_{B^{\prime}})\partial_{t^{C}(\omega_{0})}t^{A^{\prime}}\left(\phi_{t^{B^{\prime}}}\right)-(d_{0}+d_{C})\partial_{t^{C}(\omega_{0})}t^{A^{\prime}}\left(\phi_{t^{B^{\prime}}}\right)$
$\displaystyle=\big{(}2-d_{0}-d_{A}-d_{B}-d_{C}\big{)}\mathbf{F}_{ABC,\omega_{0}}.$

$\square$

The aim of the upcoming result is to improve formula (4.19) by providing
further precisions for some terms.

###### Proposition 4.9

The prepotential $\mathbf{F}_{\omega_{0}}$ in (4.19) can be written in the
following form:

$\begin{split}&\mathbf{F}_{\omega_{0}}=\frac{1}{2(1+d_{0})}\sum_{A,B}\frac{\big{(}(d_{0}+d_{A})t^{A}(\omega_{0})+r_{\omega_{0},\phi_{t^{A}}}\big{)}\big{(}(d_{0}+d_{B})t^{B}(\omega_{0})+r_{\omega_{0},\phi_{t^{B}}}\big{)}}{1+d_{0}+d_{A^{\prime}}}t^{A^{\prime}}(\phi_{t^{B^{\prime}}})\\\
&\quad+\frac{\log(-1)}{4}\sum_{i,j=1,i\neq
j}^{m}v^{i}(\omega_{0})v^{j}(\omega_{0})-\frac{3}{4(1+d_{0})}\sum_{i,j=1}^{m}\bigg{(}\frac{1}{n_{0}+1}+\delta_{ij}\frac{1}{n_{i}+1}\bigg{)}v^{i}(\omega_{0})v^{j}(\omega_{0}),\end{split}$
(4.27)

where $v^{i}(\omega_{0})$ is the function
$v^{i}(\omega_{0}):=\underset{\infty^{i}}{{\rm res}\
}\lambda(P)\omega_{0}(P)$. Moreover, the first relation in (4.26) becomes

$E.\mathbf{F}_{\omega_{0}}=2(1+d_{0})\mathbf{F}_{\omega_{0}}+\frac{1}{2}\sum_{i,j=1}^{m}\bigg{(}\frac{1}{n_{0}+1}+\delta_{ij}\frac{1}{n_{i}+1}\bigg{)}v^{i}(\omega_{0})v^{j}(\omega_{0}).$
(4.28)

_Proof:_ The proof of formula (4.27) is based on the symmetry property (3.36)
of the functions $t^{A}(\phi_{t^{B}})$ and the fact that the constants
$r_{A,B}$, given by (3.57), are such that $r_{A,B}=0$ whenever at least one of
the operations $t^{A}$ and $t^{B}$ is not of type $s^{i}$. Here $t^{A}$,
$t^{B}$ and $s^{i}$ are among the five types of operations given by (3.21) and
(3.22) with $s^{i}:=p.v.\int_{\infty^{0}}^{\infty^{i}}$.
Denote by $\widetilde{F}_{\omega_{0}}$ (resp. $\widehat{F}_{\omega_{0}}$ ) the
first (resp. second) sum in expression (4.19) for the prepotential
$\mathbf{F}_{\omega_{0}}$. That is we can write
$\mathbf{F}_{\omega_{0}}=\widetilde{F}_{\omega_{0}}-\widehat{F}_{\omega_{0}}$.
Let us first mention the following remarks:

  1. 1.

By (3.41) we have $\delta_{t^{A^{\prime}},s^{i}}=\delta_{t^{A},v^{i}}$ (the
Kronecker delta symbol).

  2. 2.

Let $d_{A}:={\rm{deg}}\big{(}\phi_{t^{A}}\big{)}$ be the degree of the quasi-
homogeneous (and primary) differential $\phi_{t^{A}}$ defined by (4.10).
According to (4.11), we have

$\delta_{t^{A},v^{i}}d_{A^{\prime}}={\rm{deg}}\big{(}\phi_{{v^{i}}^{\prime}}\big{)}={\rm{deg}}\big{(}\phi_{s^{i}}\big{)}=0.$

  3. 3.

$v^{i}(\omega_{0})$ is an eigenfunction of the Euler vector field $E$ with
$E.v^{i}(\omega_{0})=(d_{0}+1)v^{i}(\omega_{0})$.

Now, using these remarks as well as the symmetry relation (3.36), we deduce
that

$\displaystyle\widetilde{F}_{\omega_{0}}-\frac{1}{2(1+d_{0})}\sum_{A,B}\frac{\left(E.t^{A}(\omega_{0})\right)\left(E.t^{B}(\omega_{0})\right)}{1+d_{0}+d_{A^{\prime}}}t^{A^{\prime}}(\phi_{t^{B^{\prime}}})$
$\displaystyle=\frac{1}{4(1+d_{0})}\sum_{A,B}\frac{\left(E.t^{A}(\omega_{0})\right)\left(E.t^{B}(\omega_{0})\right)}{1+d_{0}+d_{A^{\prime}}}\bigg{(}\delta_{t^{A^{\prime}},s^{i}}\delta_{t^{B^{\prime}},s^{j}}(1-\delta_{ij})\log(-1)\bigg{)}$
$\displaystyle=\frac{\log(-1)}{4(1+d_{0})}\sum_{i,j=1}^{m}\big{(}1-\delta_{ij}\big{)}\frac{\left(E.v^{i}(\omega_{0})\right)\left(E.v^{j}(\omega_{0})\right)}{1+d_{0}+deg(\phi_{{v^{i}}^{\prime}})}$
$\displaystyle=\frac{\log(-1)}{4}\sum_{i,j=1}^{m}\big{(}1-\delta_{ij}\big{)}v^{i}(\omega_{0})v^{j}(\omega_{0}).$

On the other hand, by employing once again the aforementioned remarks and the
precise values of the constants $r_{A^{\prime},B^{\prime}}$ (3.57):

$r_{A^{\prime},B^{\prime}}=\sum_{i,j=1}^{m}\delta_{t^{A^{\prime}},s^{i}}\delta_{t^{B^{\prime}},s^{j}}\Big{(}\frac{1}{n_{0}+1}+\delta_{ij}\frac{1}{n_{i}+1}\Big{)}=\sum_{i,j=1}^{m}\delta_{t^{A},v^{i}}\delta_{t^{B},v^{j}}\Big{(}\frac{1}{n_{0}+1}+\delta_{ij}\frac{1}{n_{i}+1}\Big{)},$

we arrive at

$\displaystyle\widehat{F}_{\omega_{0}}$
$\displaystyle:=\frac{1}{4(1+d_{0})}\sum_{A,B}\bigg{(}2\frac{E.t^{A}(\omega_{0})}{1+d_{0}+d_{A^{\prime}}}+t^{A}(\omega_{0})\bigg{)}r_{A^{\prime},B^{\prime}}t^{B}(\omega_{0})$
$\displaystyle=\frac{3}{4(1+d_{0})}\sum_{i,j=1}^{m}\bigg{(}\frac{1}{n_{0}+1}+\delta_{ij}\frac{1}{n_{i}+1}\bigg{)}v^{i}(\omega_{0})v^{j}(\omega_{0}).$

Thus, we obtain (4.27). On the other hand, by using the first relation in
(4.26) and the mentioned values of the constants $r_{A^{\prime},B^{\prime}}$,
we conclude that the function $\mathbf{F}_{\omega_{0}}$ satisfies the quasi-
homogeneity property (4.28).

$\square$

###### Remark 4.2

$\bullet$ According to Theorem 4.1 and relation (4.28), for each quasi-
homogeneous differential $\omega_{0}$ of degree $d_{0}\geq 0$, the function
$\mathbf{F}_{\omega_{0}}$ (4.19) is a $(2+2d_{0})$-quasi-homogeneous solution
to the generalized WDVV equations (1.6).

$\bullet$ By using (4.27) and omitting the quadratic terms, we deduce that

$\mathcal{F}_{\omega_{0}}=\frac{1}{2(1+d_{0})}\sum_{A,B}\frac{\big{(}(d_{0}+d_{A})t^{A}(\omega_{0})+r_{\omega_{0},\phi_{t^{A}}}\big{)}\big{(}(d_{0}+d_{B})t^{B}(\omega_{0})+r_{\omega_{0},\phi_{t^{B}}}\big{)}}{1+d_{0}+d_{A^{\prime}}}t^{A^{\prime}}(\phi_{t^{B^{\prime}}})$
(4.29)

is also a prepotential of the described Frobenius manifold determined by the
quasi-homogeneous differential $\omega_{0}$.

$\bullet$ In formulas (4.19) and (4.27), we have chosen to add quadratic terms
in order to construct prepotentials that share the Hessian matrix (4.20) when
they correspond to Frobenius manifold structures of the same family.
This property could be useful in various situation. We refer to Section 7 for
some examples of applications.

$\bullet$ According to the symmetry relations (3.36) and the relation
$\delta_{t^{A^{\prime}},s^{i}}=\delta_{t^{A},v^{i}}$, formula (4.20) can be
rewritten as follows:

$t^{A^{\prime}}\left(\phi_{t^{B^{\prime}}}\right)=\partial_{t^{A}(\omega_{0})}\partial_{t^{B}(\omega_{0})}\mathbf{F}_{\omega_{0}}-\frac{\log(-1)}{2}\sum_{i,j=1}^{m}(1-\delta_{ij})\delta_{t^{A},v^{i}}\delta_{t^{B},v^{j}}.$
(4.30)

$\bullet$ Assume that the simple Hurwitz space is of type
$\mathcal{H}_{g,L}(n_{0})$, i.e. all the considered coverings $\lambda$ have a
unique pole $\infty^{0}$ of order $n_{0}+1=L$ ($L$ is the number of sheets).

  * i)

The list of primary differentials (3.23) becomes
$\big{\\{}\phi_{t^{0,\alpha}},\phi_{\rho^{k}}=\omega_{k},\phi_{u^{k}}\big{\\}}$.

  * ii)

We only have three types of flat coordinates among those in (3.21)-(3.22)
determined by the operations:

$t^{0,\alpha},\quad\rho^{k},\quad u^{k},\quad\quad\alpha=1,\dots,n_{0},\ \
k=1,\dots,g.$

  * iii)

The set $\mathrm{QH}(E)$ (4.13) consisting of quasi-homogeneous differential
reduces to

$\Big{\\{}\phi_{t^{0,\alpha}},\quad\phi_{\rho^{k}}=\omega_{k},\quad\phi_{u^{k}},\quad\phi_{t^{0,n_{0}+\beta+1}},\quad\Omega_{\mathbf{z}^{0}\mathbf{z}^{j}},\quad\phi_{u^{k,\sigma}},\quad\phi_{\widetilde{u}^{k,\nu}}\Big{\\}}.$

  * iv)

When $\omega_{0}$ is among these quasi-homogeneous differentials, then, by
(4.15) and (4.16), the flat functions $t^{A}(\omega_{0})$ are eigenfunctions
of the Euler vector field $E$ with eigenvalues $d_{0}+d_{A}$.
As a consequence, formulas (4.27) and (4.29) coincide and reduce to the
following

$\mathbf{F}_{\omega_{0}}=\mathcal{F}_{\omega_{0}}=\frac{1}{2(1+d_{0})}\sum_{A,B}\frac{(d_{0}+d_{A})(d_{0}+d_{B})}{1+d_{0}+d_{A^{\prime}}}t^{A}(\omega_{0})t^{B}(\omega_{0})t^{A^{\prime}}(\phi_{t^{B^{\prime}}}).$
(4.31)

Note that the described situation in the last point is realized by the genus
zero Hurwitz space $\mathcal{H}_{0,n+1}(n)$ of polynomial functions

$\lambda(P)=z_{P}^{n+1}+a_{n}z_{P}^{n-1}+\dots+a_{1},$

where $a_{1},\dots,a_{n}$, $n\geq 3$, are complex numbers chosen in such a way
the $n$ zeros $\lambda^{\prime}(P)$ are distinct.

$\bullet$ Assume that $\omega_{0}$ is a linear combination of quasi-
homogeneous differentials of the same degree $d_{0}\geq 0$ such that the
differential $\omega_{0}$ fulfills the conditions in (3.3). According to
Remark 4.1, $\omega_{0}$ is again a quasi-homogeneous differential of degree
$d_{0}$. Let $t^{A}(\omega_{0})$ be function from the set
$\Big{\\{}t^{i,\alpha}(\omega_{0}),v^{i}(\omega_{0}),s^{i}(\omega_{0}),\rho^{k}(\omega_{0}),u^{k}(\omega_{0})\Big{\\}}$
defined by (3.21)-(3.22). Since the Euler vector field
$E=\sum_{j}\lambda_{j}\partial_{\lambda_{j}}$ acts on $t^{A}(\omega_{0})$ in
analogous way as in (4.15), it follows that the quasi-homogeneous differential
$\omega_{0}$ gives a Frobenius manifold structure.
As example, we can produce a quasi-homogeneous differential $\omega_{0}$ of
\- degree 0 by taking $\omega_{0}$ as a linear combination of holomorphic
differentials $\omega_{k}=\phi_{\rho^{k}}$ and Abelian differentials of the
third kind $\phi_{s^{i}}=\Omega_{\infty^{0}\infty^{i}}$;
-degree 1 by taking $\omega_{0}$ in the vector space generating by the differentials $\phi_{v^{i}},\phi_{u^{k}}$ and $\phi_{\widetilde{u}^{k}}(P):=\oint_{b_{k}}\lambda(Q)W(P,Q)$.

####  4.1.5 Prepotentials associated with holomorphic differentials: examples
in high dimension

In this subsection we further investigate the quasi-homogeneous solutions
(4.19)-(4.27) to the WDVV equations induced by normalized holomorphic
differentials $\omega_{1},\dots,\omega_{g}$, $g\geq 1$.
Note that, by Corollary 4.1 and Proposition 4.11, normalized holomorphic
differentials are all primary with respect to the unit vector field
$e=\sum_{j=1}^{N}\partial_{\lambda_{j}}$ and quasi-homogeneous of degree 0.
Let us begin with the following result which gives the general explicit form
of the prepotential $\mathbf{F}_{\omega_{0}}$ (4.19)-(4.27) corresponding to
the differential $\omega_{0}\in\big{\\{}\omega_{1},\dots,\omega_{g}\big{\\}}$.

###### Theorem 4.2

Assume that $g\geq 1$ and $\omega_{0}$ is one of the normalized holomorphic
differentials. Then the prepotential associated with $\omega_{0}$ is given by:

$\begin{split}\mathbf{F}_{\omega_{0}}&=\frac{1}{2}\sum_{i,j=0}^{m}\sum_{\alpha=1}^{n_{i}}\sum_{\beta=1}^{n_{j}}\frac{\alpha\beta}{(2n_{i}+2-\alpha)(n_{j}+1)}t^{i,\alpha}(\omega_{0})t^{j,\beta}(\omega_{0})t^{i,n_{i}+1-\alpha}(\phi_{t^{j,n_{j}+1-\beta}})\\\
&\quad+\frac{1}{2}\sum_{i=0}^{m}\sum_{\alpha=1}^{n_{i}}\sum_{j=1}^{m}\Big{(}\frac{\alpha}{2n_{i}+2-\alpha}+\frac{\alpha}{n_{i}+1}\Big{)}t^{i,\alpha}(\omega_{0})v^{j}(\omega_{0})t^{i,n_{i}+1-\alpha}(\phi_{s^{j}})\\\
&\quad+\frac{1}{2}\sum_{i=0}^{m}\sum_{\alpha=1}^{n_{i}}\sum_{k=1}^{g}\Big{(}\frac{\alpha}{2n_{i}+2-\alpha}+\frac{\alpha}{n_{i}+1}\Big{)}t^{i,\alpha}(\omega_{0})u^{k}(\omega_{0})t^{i,n_{i}+1-\alpha}(\phi_{\rho^{k}})\\\
&\quad+\frac{1}{2}\sum_{i,j=1}^{m}v^{i}(\omega_{0})v^{j}(\omega_{0})s^{i}(\phi_{s^{j}})+\sum_{i=1}^{m}\sum_{k=1}^{g}v^{i}(\omega_{0})u^{k}(\omega_{0})s^{i}(\phi_{\rho^{k}})+\frac{1}{2}\sum_{k,l=1}^{g}u^{k}(\omega_{0})u^{l}(\omega_{0})\rho^{k}(\phi_{\rho^{l}})\\\
&\quad+\frac{\log(-1)}{4}\sum_{i,j=1,i\neq
j}^{m}v^{i}(\omega_{0})v^{j}(\omega_{0})-\frac{3}{4}\sum_{i,j=1}^{m}\bigg{(}\frac{1}{n_{0}+1}+\delta_{ij}\frac{1}{n_{i}+1}\bigg{)}v^{i}(\omega_{0})v^{j}(\omega_{0}).\end{split}$
(4.32)

In addition, this prepotential satisfies

$E_{\omega_{0}}.\mathbf{F}_{\omega_{0}}=2\mathbf{F}_{\omega_{0}}+\frac{1}{2}\sum_{i,j=1}^{m}\Big{(}\frac{1}{n_{0}+1}+\delta_{ij}\frac{1}{n_{i}+1}\Big{)}v^{i}(\omega_{0})v^{j}(\omega_{0}),$

where $E_{\omega_{0}}$ is the following Euler vector field:

$E_{\omega_{0}}=\sum_{i=0}^{m}\sum_{\alpha=1}^{n_{i}}\frac{\alpha}{n_{i}+1}t^{i,\alpha}(\omega_{0})\partial_{t^{i,\alpha}(\omega_{0})}+\sum_{i=1}^{m}v^{i}(\omega_{0})\partial_{v^{i}(\omega_{0})}+\sum_{k=1}^{g}u^{k}(\omega_{0})\partial_{u^{k}(\omega_{0})}.$
(4.33)

_Proof:_ Since the holomorphic differential $\omega_{0}$ is of degree
$d_{0}=0$, formulas (3.56) and (4.15) tell us that

$E.t^{A}(\omega_{0})=d_{A}t^{A}(\omega_{0}),\quad\quad\text{for all $A$}.$

Accordingly, formula (4.27) reads as:

$\displaystyle\mathbf{F}_{\omega_{0}}-\frac{\log(-1)}{4}\sum_{i,j=1,i\neq
j}^{m}v^{i}(\omega_{0})v^{j}(\omega_{0})+\frac{3}{4}\sum_{i,j=1}^{m}\bigg{(}\frac{1}{n_{0}+1}+\delta_{ij}\frac{1}{n_{i}+1}\bigg{)}v^{i}(\omega_{0})v^{j}(\omega_{0})$
$\displaystyle=\frac{1}{2}\sum_{t^{A},t^{B}}\frac{d_{A}d_{B}}{2-d_{A}}t^{A}(\omega_{0})t^{B}(\omega_{0})t^{A^{\prime}}(\phi_{t^{B^{\prime}}})$
$\displaystyle=\frac{1}{2}\sum_{t^{A},t^{B}\in\\{t^{i,\alpha},v^{i},u^{k}\\}}\frac{d_{A}d_{B}}{2-d_{A}}t^{A}(\omega_{0})t^{B}(\omega_{0})t^{A^{\prime}}(\phi_{t^{B^{\prime}}}),$

where we have used the relation $d_{A^{\prime}}=1-d_{A}$ in the first equality
and in the second equality we used the fact that $d_{A}\neq 0$ only when
$t^{A}$ is one of the three operations $t^{i,\alpha},v^{i},u^{k}$ belonging to
the list (3.21)-(3.22).
Thus we arrive at (4.32) by considering the degrees of the differentials
$\phi_{t^{i,\alpha}}$, $\phi_{v^{i}}$ and $\phi_{u^{k}}$ in (4.11) and duality
relations (3.41).
On the other hand, expression (4.33) of the Euler vector field is nothing but
the particular case of (4.17) where $d_{0}=0$ and
$r_{\omega_{0},\phi_{t^{A}}}=0$ for all $A$. In addition, the stated quasi-
homogeneity property of the prepotential (4.32) is an immediate consequence of
(4.28).

$\square$

###### Remark 4.3

For the forth line of the prepotential (4.32), we mention the following:

$\begin{split}&\rho^{k}(\phi_{\rho^{l}}):=\frac{1}{2{\rm
i}\pi}\oint_{b_{k}}\omega_{l}=\frac{\mathbb{B}_{kl}}{2{\rm{i}}\pi};\\\
&s^{i}(\phi_{\rho^{k}})=s^{i}(\omega_{k}):=\int_{\infty^{0}}^{\infty^{i}}\omega_{k};\\\
&s^{i}(\phi_{s^{j}})=s_{ji}=\text{the functions given by (\ref{s-ij
Formula})}.\end{split}$ (4.34)

Now, by considering some specific cases of the combinatorial parameters
$(g,L,n_{0},\dots,n_{m})$ of the simple Hurwitz space
$\widehat{\mathcal{H}}_{g,L}(n_{0},\dots,n_{m})$, we shall discuss some
consequences of formula (4.32).

###### Proposition 4.10

Assume that the Hurwitz space is $\widehat{\mathcal{H}}_{g,n+1}(n)$, i.e. the
considered meromorphic functions have a unique pole $\infty^{0}$ of order
$n+1\geq 2$. Then the prepotential induced by a normalized holomorphic
differential $\omega_{0}$ is of the following form:

$\begin{split}\mathbf{F}_{\omega_{0}}&=\frac{1}{2(n+1)}\sum_{\alpha,\beta=1}^{n}\frac{\alpha\beta}{(2n+2-\alpha)}t^{\alpha}(\omega_{0})t^{\beta}(\omega_{0})t^{n+1-\alpha}(\phi_{t^{n+1-\beta}})\\\
&\quad+\frac{1}{2}\sum_{\alpha=1}^{n}\sum_{k=1}^{g}\Big{(}\frac{\alpha}{2n+2-\alpha}+\frac{\alpha}{n+1}\Big{)}t^{\alpha}(\omega_{0})u^{k}(\omega_{0})t^{n+1-\alpha}(\phi_{\rho^{k}})\\\
&\quad+\frac{1}{4{\rm{i}}\pi}\sum_{k,l=1}^{g}u^{k}(\omega_{0})u^{l}(\omega_{0})\mathbb{B}_{kl}.\end{split}$
(4.35)

_Proof:_ Formula (4.35) follows from (4.32) and (4.34).

$\square$

In the following corollary, we are going to rewrite (4.35) when $n=1$, i.e.,
the considered Hurwitz space is that of hyperelliptic curves

$\Big{\\{}(\mu,\lambda)\in\mathbb{C}^{2}:\quad\mu^{2}=\prod_{k=1}^{2g+1}(\lambda-\lambda_{i}),\quad\text{$\lambda_{i}\neq\lambda_{j}$
for $i\neq j$}\Big{\\}}.$ (4.36)

###### Corollary 4.4

Assume that the Hurwitz space is $\mathcal{H}_{g,2}(1)$. Then the prepotential
in (4.35) reduces to:

$\begin{split}\mathbf{F}_{\omega_{0}}&=\frac{1}{12}\big{(}t(\omega_{0})\big{)}^{2}t(\phi_{t})+\frac{5}{12}t(\omega_{0})\sum_{k=1}^{g}u^{k}(\omega_{0})t(\omega_{k})+\frac{1}{4{\rm{i}}\pi}\sum_{k,l=1}^{g}u^{k}(\omega_{0})u^{l}(\omega_{0})\mathbb{B}_{kl}.\end{split}$
(4.37)

Here $\phi_{t}(P)=\sqrt{2}W(P,\infty^{0})$ is the Abelian differential of the
second kind and $t(\phi)$ is the flat function
$t(\phi):=\sqrt{2}\underset{\infty^{0}}{{\rm res}\
}\sqrt{\lambda(P)}\phi(P)=\sqrt{2}\phi(\infty^{0})$, with
$\phi\in\big{\\{}\phi_{t},\omega_{1},\dots,\omega_{g}\big{\\}}$.

As another corollary, we shall analyse formula (4.35) in the genus one case.
For this, let us denote the chosen flat coordinates (3.21)-(3.22) associated
with the unique normalized holomorphic differential $\omega_{0}$ as follows:

$\displaystyle t_{0}=\tau/(2{\rm{i}}\pi),\quad\text{with}\quad\Im(\tau)>0;$
$\displaystyle t_{\alpha}=t^{\alpha}(\omega_{0}):=\underset{\infty^{0}}{\rm
res}\ \lambda(P)^{\frac{\alpha}{n+1}}\omega_{0}(P),\quad\alpha=1,\dots,n;$
$\displaystyle t_{n+1}:=u(\omega_{0}):=\oint_{a}\lambda(P)\omega_{0}(P).$

With respect to this system of flat coordinates, the Dubrovin flat metric
$\eta(\omega_{0})$ (3.20) becomes anti-diagonal. Moreover, (4.9) yields that
the unit vector $e=\sum_{j=1}^{n+2}\partial_{\lambda_{j}}$ is represented by
$e=\partial_{t_{n+1}}$.

###### Corollary 4.5

With the above notation, the genus one case of the prepotential (4.35) is as
follows:

$\begin{split}\mathbf{F}_{\omega_{0}}(t_{0},t_{1},\dots,t_{n+1})&=\frac{t_{0}}{2}(t_{n+1})^{2}+\frac{t_{n+1}}{2}\sum_{\alpha=1}^{n}\Big{(}\frac{\alpha}{2n+2-\alpha}+\frac{\alpha}{n+1}\Big{)}t_{\alpha}t_{n+1-\alpha}\\\
&\quad+\frac{1}{2(n+1)}\sum_{\alpha,\beta=1}^{n}\frac{\alpha\beta}{(2n+2-\alpha)}t_{\alpha}t_{\beta}t^{n+1-\alpha}(\phi_{t^{n+1-\beta}}).\end{split}$
(4.38)

Note that the unknown functions $\big{\\{}t^{\alpha}(\phi_{t^{\beta}}):\quad
1\leq\alpha,\beta\leq n\big{\\}}$ in (4.38) are functions of
$t_{0},t_{1},\dots,t_{n},t_{n+1}$ such that

$\partial_{t_{n+1}}t^{\alpha}(\phi_{t^{\beta}})=e.t^{\alpha}(\phi_{t^{\beta}})=\delta_{n+1,\alpha+\beta}.$

Moreover, since the Euler vector field (4.33) is represented by
$E=\sum_{k=1}^{n+1}\frac{k}{n+1}\partial_{t_{k}}$, it follows that the
function $t^{\alpha}(\phi_{t^{\beta}})$ satisfies the following partial
differential equation:

$\sum_{k=1}^{n}k\frac{\partial}{\partial{t_{k}}}t^{\alpha}(\phi_{t^{\beta}})=(\alpha+\beta)t^{\alpha}(\phi_{t^{\beta}})-(n+1)\delta_{n+1,\alpha+\beta}.$

On the other hand, according to (3.36), the functions
$\big{\\{}t^{\alpha}(\phi_{t^{\beta}})\big{\\}}$ satisfy:
$t^{\alpha}(\phi_{t^{\beta}})=t^{\beta}(\phi_{t^{\alpha}})$.

###### Proposition 4.11

Assume that the simple Hurwitz space is of type
$\widehat{\mathcal{H}}_{g,m+1}(0,\dots,0)$, that is the meromorphic functions
have $m+1$ simples poles $\infty^{0},\dots,\infty^{m}$, with $m\geq 1$.
Then the prepotential (4.32) associated with the holomorphic differential
$\omega_{0}\in\big{\\{}\omega_{1},\dots,\omega_{g}\big{\\}}$ takes the
following form:

$\begin{split}&\mathbf{F}_{\omega_{0}}=\frac{1}{2}\sum_{i=1}^{m}\big{(}v^{i}(\omega_{0})\big{)}^{2}\log\bigg{(}\sum_{k=1}^{g}v^{i}(\omega_{k})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{|}\mathbb{B}\big{)}\bigg{)}\\\
&\quad-\bigg{(}\sum_{i=1}^{m}\big{(}v^{i}(\omega_{0})\big{)}^{2}\bigg{)}\log\Big{(}\Theta_{\Delta}\Big{(}s^{i}(\omega_{1}),\dots,s^{i}(\omega_{g})\big{|}\mathbb{B}\Big{)}\Big{)}\\\
&\quad+\frac{1}{2}\bigg{(}\sum_{i,j=1}^{m}v^{i}(\omega_{0})v^{j}(\omega_{0})\bigg{)}\log\left(\sum_{r=1}^{m}\sum_{k=1}^{g}v^{r}(\omega_{k})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{|}\mathbb{B}\big{)}\right)\\\
&\quad+\frac{1}{2}\sum_{i,j=1,i\neq
j}^{m}v^{i}(\omega_{0})v^{j}(\omega_{0})\log\left(\frac{\Theta_{\Delta}\Big{(}s^{i}(\omega_{1})-s^{j}(\omega_{1}),\dots,s^{i}(\omega_{g})-s^{j}(\omega_{g})\big{|}\mathbb{B}\Big{)}}{\Theta_{\Delta}\Big{(}s^{i}(\omega_{1}),\dots,s^{i}(\omega_{g})\big{|}\mathbb{B}\Big{)}\Theta_{\Delta}\Big{(}s^{j}(\omega_{1}),\dots,s^{j}(\omega_{g})\big{|}\mathbb{B}\Big{)}}\right)\\\
&\quad+\sum_{i=1}^{m}\sum_{k=1}^{g}v^{i}(\omega_{0})u^{k}(\omega_{0})s^{i}(\omega_{k})+\frac{1}{2}\sum_{k,l=1}^{g}u^{k}(\omega_{0})u^{l}(\omega_{0})\Big{(}\mathbb{B}_{kl}/(2{\rm{i}}\pi)\Big{)}\\\
&\quad+\frac{\log(-1)}{4}\sum_{i,j=1,i\neq
j}^{m}v^{i}(\omega_{0})v^{j}(\omega_{0})-\frac{3}{4}\sum_{i,j=1}^{m}\big{(}1+\delta_{ij}\big{)}v^{i}(\omega_{0})v^{j}(\omega_{0}).\end{split}$
(4.39)

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

_Proof:_ Due to (4.32), (4.34) and the fact that the poles of $\lambda$ are
all simple, we have

$\displaystyle\mathbf{F}_{\omega_{0}}$
$\displaystyle=\frac{1}{2}\sum_{i,j=1}^{m}v^{i}(\omega_{0})v^{j}(\omega_{0})s^{i}(\phi_{s^{j}})+\sum_{i=1}^{m}\sum_{k=1}^{g}v^{i}(\omega_{0})u^{k}(\omega_{0})s^{i}(\omega_{k})+\frac{1}{2}\sum_{k,l=1}^{g}u^{k}(\omega_{0})u^{l}(\omega_{0})\Big{(}\mathbb{B}_{kl}/(2{\rm{i}}\pi)\Big{)}$
$\displaystyle\quad+\frac{\log(-1)}{4}\sum_{i,j=1,i\neq
j}^{m}v^{i}(\omega_{0})v^{j}(\omega_{0})-\frac{3}{4}\sum_{i,j=1}^{m}\big{(}1+\delta_{ij}\big{)}v^{i}(\omega_{0})v^{j}(\omega_{0}).$

In order to establish the desired formula, it suffices to express the
functions $s^{i}(\phi_{s^{j}})=s_{ji}$, obtained in (3.32), by means of flat
coordinates $v^{i}(\omega_{k}),s^{i}(\omega_{k}),\mathbb{B}_{kl}$
corresponding to the normalized holomorphic differentials
$\omega_{1},\dots,\omega_{g}$.
Let us first mention that by notation (2.7) and the residue theorem, we can
write

$\omega_{k}(\infty^{i})=\underset{\infty^{i}}{{\rm res}}\
\lambda(P)\omega_{k}(P)=v^{i}(\omega_{k})$

and

$\omega_{k}(\infty^{0})=\underset{\infty^{0}}{{\rm res}}\
\lambda(P)\omega_{k}(P)=-\sum_{i=1}^{m}\underset{\infty^{i}}{{\rm res}}\
\lambda(P)\omega_{k}(P)=-\sum_{i=1}^{m}v^{i}(\omega_{k}).$

Therefore, by substituting this into formula (3.32), we get

$\displaystyle s_{ii}$
$\displaystyle=\log\left(\sum_{k=1}^{g}\omega_{k}(\infty^{i})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{)}\right)+\log\left(\sum_{k=1}^{g}\omega_{k}(\infty^{0})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{)}\right)-2\log\left(\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{i})-\mathcal{A}(\infty^{0})\big{)}\right)-\log(-1)$
$\displaystyle=\log\left(\sum_{k=1}^{g}v^{i}(\omega_{k})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{|}\mathbb{B}\big{)}\right)+\log\left(\sum_{r=1}^{m}\sum_{k=1}^{g}v^{r}(\omega_{k})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{|}\mathbb{B}\big{)}\right)-2\log\Big{(}\Theta_{\Delta}\Big{(}s^{i}(\omega_{1}),\dots,s^{i}(\omega_{g})\big{|}\mathbb{B}\Big{)}\Big{)}$

and when $i\neq j$,

$\displaystyle s^{i}(\phi_{s^{j}})$
$\displaystyle=s_{ji}=\log\left(\frac{\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{i})-\mathcal{A}(\infty^{j})\big{)}}{\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{i})-\mathcal{A}(\infty^{0})\big{)}\Theta_{\Delta}\big{(}\mathcal{A}(\infty^{j})-\mathcal{A}(\infty^{0})\big{)}}\right)+\log\left(\sum_{k=1}^{g}\omega_{k}(\infty^{0})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{)}\right)-\log(-1)$
$\displaystyle=\log\left(\frac{\Theta_{\Delta}\Big{(}s^{i}(\omega_{1})-s^{j}(\omega_{1}),\dots,s^{i}(\omega_{g})-s^{j}(\omega_{g})\big{|}\mathbb{B}\Big{)}}{\Theta_{\Delta}\Big{(}s^{i}(\omega_{1}),\dots,s^{i}(\omega_{g})\big{|}\mathbb{B}\Big{)}\Theta_{\Delta}\Big{(}s^{j}(\omega_{1}),\dots,s^{j}(\omega_{g})\big{|}\mathbb{B}\Big{)}}\right)+\log\left(\sum_{r=1}^{m}\sum_{k=1}^{g}v^{r}(\omega_{k})\partial_{z_{k}}\Theta_{\Delta}\big{(}0\big{|}\mathbb{B}\big{)}\right).$

This completes the proof of (4.39).

$\square$

Note that we have obtained the prepotential (4.39) by simple specialization of
the general formula (4.19) (see also (4.27)) to the case of the Hurwitz space
$\widehat{\mathcal{H}}_{g,m+1}(0,\dots,0)$ and the holomorphic primary
differential $\omega_{0}$. Now we show how formula (4.39) leads directly to an
explicit solution of the (non-generalized) WDVV equations if we specialize it
to the genus one case. In other words, we obtain an explicit solution
depending on $2m+2$ variables without much calculation but simply by
specializing the general formula to the case of the Hurwitz space
$\widehat{\mathcal{H}}_{1,m+1}(0,\dots,0)$ and a holomorphic differential
$\omega_{0}$.
Consider the lattice $\mathbb{L}:=\mathbb{Z}+\tau\mathbb{Z}$, with
$\Im(\tau)>0$. Then the Riemann theta function $\Theta_{\Delta}(\cdot|\tau)$
(2.8), corresponding to the odd and non-singular half integer characteristic
$\Delta=[1/2,1/2]$, is closely related to the $\theta_{1}$-Jacobi function

$\Theta_{[\frac{1}{2},\frac{1}{2}]}(u|\tau):=i\sum_{n\in\mathbb{Z}}(-1)^{n}e^{{\rm{i}}\pi\tau(n+1/2)^{2}+2{\rm{i}}\pi(n+1/2)u}=-\theta_{1}(u|\theta),\quad\quad\Im(\tau)>0.$
(4.40)

We refer to [8] (Section 13.20) and [2, 6, 43] for details on the Jacobi theta
functions.
The dimension of the Frobenius manifold associated with the normalized
holomorphic differential $\omega_{0}$ is $N=2m+2$, with $m+1$ being the common
degree of the coverings belonging to the Hurwitz space
$\mathcal{H}_{1,m+1}(0,\dots,0)$.
Moreover, according to Theorem 3.4, the following functions provide a system
of flat coordinates of the flat metric $\eta(\omega_{0})$ (3.20):

$\begin{split}&t_{0}:=\tau/{2{\rm i}\pi}=\frac{1}{2{\rm
i}\pi}\oint_{b}\omega_{0};\\\
&t_{k}:=s^{k}(\omega_{0})=\int_{\infty^{0}}^{\infty^{k}}\omega_{0},\quad
k=1,\dots,m;\\\ &t_{2m+1-k}:=v^{k}(\omega_{0})=\underset{\infty^{k}}{{\rm
res}}\ \lambda(P)\omega_{0}(P),\quad k=1,\dots,m;\\\
&t_{2m+1}:=u(\omega_{0})=\oint_{a}\lambda(P)\omega_{0}(P).\end{split}$ (4.41)

According to (3.37) and (3.45), the constant matrix of the metric
$\eta(\omega_{0})$, with respect to flat coordinates (4.41), is anti-diagonal
:

$\eta_{\alpha\beta}=\delta_{2m+1,\alpha+\beta}=\eta^{\alpha\beta},\quad\alpha,\beta=0,\dots,2m+1.$

###### Corollary 4.6

Assume that the Hurwitz space is $\mathcal{H}_{1,m+1}(0,\dots,0)$. With the
preceding notation, the prepotential
$\mathbf{F}_{\omega_{0}}=\mathbf{F}_{\omega_{0}}(t_{0},t_{1},\dots,t_{2m+1})$
induced by the normalized holomorphic differential $\omega_{0}$ is given by:

$\begin{split}\mathbf{F}_{\omega_{0}}&=\frac{(t_{2m+1})^{2}}{2}t_{0}+t_{2m+1}\sum_{k=1}^{m}t_{k}t_{2m+1-k}+\frac{1}{2}\sum_{k=1}^{m}\big{(}t_{2m+1-k}\big{)}^{2}\log(t_{2m+1-k})\\\
&\quad+\frac{1}{2}\bigg{(}\sum_{k=1}^{m}t_{2m+1-k}\bigg{)}^{2}\log\bigg{(}\sum_{j=1}^{m}t_{2m+1-j}\bigg{)}-\sum_{k=1}^{m}\big{(}t_{2m+1-k}\big{)}^{2}\log\left(\frac{\theta_{1}\big{(}t_{k}\big{|}{2{\rm
i}\pi}t_{0}\big{)}}{\theta_{1}^{\prime}\big{(}0\big{|}2{\rm i}\pi
t_{0}\big{)}}\right)\\\ &\quad+\frac{1}{2}\sum_{j,k=1,j\neq
k}^{m}t_{2m+1-j}t_{2m+1-k}\log\bigg{(}\frac{\theta_{1}^{\prime}\big{(}0\big{|}{2{\rm
i}\pi}t_{0}\big{)}\theta_{1}\big{(}t_{j}-t_{k}\big{|}{2{\rm
i}\pi}t_{0}\big{)}}{\theta_{1}\big{(}t_{j}\big{|}{2{\rm
i}\pi}t_{0}\big{)}\theta_{1}\big{(}t_{k}\big{|}{2{\rm
i}\pi}t_{0}\big{)}}\bigg{)}\\\ &\quad+\frac{\log(-1)}{4}\sum_{j,k=1,j\neq
k}^{m}t_{2m+1-j}t_{2m+1-k}-\frac{3}{4}\sum_{j,k=1}^{m}(1+\delta_{jk})t_{2m+1-j}t_{2m+1-k}.\end{split}$
(4.42)

In addition, the function $\mathbf{F}_{\omega_{0}}$ is quasi-homogeneous of
degree $2$ with respect to the Euler vector field
$E_{\omega_{0}}=\sum_{k=0}^{m}t_{2n+1-k}\partial_{t_{2n+1-k}}$ and

$E_{\omega_{0}}\mathbf{F}_{\omega_{0}}=2\mathbf{F}_{\omega_{0}}+\frac{1}{2}\sum_{k=1}^{n}(t_{2n+1-k})^{2}+\frac{1}{2}\left(\sum_{k=1}^{n}t_{2n+1-k}\right)^{2}.$

_Proof:_ Formula (4.39) implies that

$\displaystyle\mathbf{F}_{\omega_{0}}=\frac{1}{2}\sum_{i=1}^{m}(v^{i})^{2}\log\Big{(}v^{i}\theta_{[\frac{1}{2},\frac{1}{2}]}^{\prime}\big{(}0\big{|}\tau\big{)}\Big{)}-\bigg{(}\sum_{i=1}^{m}(v^{i})^{2}\bigg{)}\log\Big{(}\theta_{1}\big{(}s^{i}\big{|}\tau\big{)}\Big{)}$
$\displaystyle\quad+\frac{1}{2}\bigg{(}\sum_{i,j=1}^{m}v^{i}v^{j}\bigg{)}\log\bigg{(}\sum_{r=1}^{m}v^{r}\theta_{[\frac{1}{2},\frac{1}{2}]}^{\prime}\big{(}0\big{|}\tau\big{)}\bigg{)}+\frac{1}{2}\sum_{i,j=1,i\neq
j}^{m}v^{i}v^{j}\log\bigg{(}\frac{\theta_{[\frac{1}{2},\frac{1}{2}]}\big{(}s^{i}-s^{j}\big{|}\tau\big{)}}{\theta_{[\frac{1}{2},\frac{1}{2}]}\big{(}s^{i}\big{|}\tau\big{)}\theta_{[\frac{1}{2},\frac{1}{2}]}\big{(}s^{j}\big{|}\tau\big{)}}\bigg{)}$
$\displaystyle\quad+u\sum_{i=1}^{m}v^{i}s^{i}+\frac{(u)^{2}}{4{\rm
i}\pi}\tau+\frac{\log(-1)}{4}\sum_{i,j=1,i\neq
j}^{m}v^{i}v^{j}-\frac{3}{4}\sum_{i,j=1}^{m}\big{(}1+\delta_{ij}\big{)}v^{i}v^{j}$
$\displaystyle=\frac{1}{2}\sum_{i=1}^{m}(v^{i})^{2}\log(v^{i})+\bigg{(}\sum_{i=1}^{m}(v^{i})^{2}\bigg{)}\log\Big{(}\theta_{[\frac{1}{2},\frac{1}{2}]}^{\prime}\big{(}0\big{|}2{\rm
i}\pi
t_{0}\big{)}\Big{)}-\bigg{(}\sum_{i=1}^{m}(v^{i})^{2}\bigg{)}\log\Big{(}\theta_{1}\big{(}s^{i}\big{|}\tau\big{)}\Big{)}$
$\displaystyle\quad+\frac{1}{2}\bigg{(}\sum_{i,j=1}^{m}v^{i}v^{j}\bigg{)}\log\bigg{(}\sum_{r=1}^{m}v^{r}\bigg{)}+\frac{1}{2}\sum_{i,j=1,i\neq
j}^{m}v^{i}v^{j}\log\bigg{(}\frac{\theta_{[\frac{1}{2},\frac{1}{2}]}^{\prime}\big{(}0\big{|}\tau\big{)}\theta_{[\frac{1}{2},\frac{1}{2}]}\big{(}s^{i}-s^{j}\big{|}\tau\big{)}}{\theta_{[\frac{1}{2},\frac{1}{2}]}\big{(}s^{i}\big{|}\tau\big{)}\theta_{[\frac{1}{2},\frac{1}{2}]}\big{(}s^{j}\big{|}\tau\big{)}}\bigg{)}$
$\displaystyle\quad+u\sum_{i=1}^{m}v^{i}s^{i}+\frac{(u)^{2}}{4{\rm
i}\pi}\tau+\frac{\log(-1)}{4}\sum_{i,j=1,i\neq
j}^{m}v^{i}v^{j}-\frac{3}{4}\sum_{i,j=1}^{m}\big{(}1+\delta_{ij}\big{)}v^{i}v^{j}.$

Thus we obtain (4.42) by using (4.41) and (4.40).

$\square$

###### Remark 4.4

When $m=1$, (4.42) becomes

$\mathbf{F}_{\omega_{0}}=\frac{(t_{3})^{2}}{2}t_{0}+t_{1}t_{2}t_{3}+(t_{2})^{2}\log(t_{2})-(t_{2})^{2}\log\bigg{(}\frac{\theta_{1}(t_{1}|2\rm{i}\pi
t_{0})}{\theta^{\prime}_{1}\big{(}0|2{\rm{i}}\pi
t_{0}\big{)}}\bigg{)}-\frac{3}{2}(t_{2})^{2}.$ (4.43)

Note that, up to the change of variables
$(t_{0},t_{1},t_{2},t_{3})\longleftrightarrow(u_{1},u_{2},u_{3},u_{4})=(-t_{3},t_{2},t_{1},t_{0})$
and up to the quadratic term $-\frac{3}{2}t_{2}^{2}$, our prepotential (4.43)
coincides with the one appearing in [19, 11] and obtained using a long
computation based on Dubrovin’s bilinear pairing method. The mentioned change
of variables does not affect the WDVV equations.

###  4.2 Almost Frobenius manifold structures on double Hurwitz spaces

In order to simplify formulas, summation over repeated _Greek indices_ will be
assumed throughout this subsection.
Let us start by the following definition of an almost Frobenius structure
which is motivated by Definition 9 in [18].

###### Definition 4.3

Let $\mathcal{M}$ be a manifold of dimension N. An almost Frobenius manifold
structure on $\mathcal{M}$ of charge $D\neq 1$ is the data of
$\big{(}\mathcal{M},\ast,\mathbf{g},E\big{)}$ such that each tangent space
$T_{p}\mathcal{M}$ is a Frobenius algebra varying smoothly or analytically
over $\mathcal{M}$ with the additional properties:

  * _1._

the inner product $\mathbf{g}$ is a flat metric on $\mathcal{M}$;

  * _2._

in flat coordinates $\xi=(\xi^{1},\dots,\xi^{N})$ for the metric $\mathbf{g}$,

    * i)

the structure constants of the multiplication
$\partial_{\xi^{i}}\ast\partial_{\xi^{j}}=\mathbf{c}_{\ast,ij}^{\alpha}(\xi)\partial_{\xi^{\alpha}}$
can be locally represented in the form

$\mathbf{c}_{\ast,ij}^{k}(\xi)=G^{k\alpha}\partial_{\xi^{\alpha}}\partial_{\xi^{i}}\partial_{\xi^{j}}F_{\ast}(\xi)$

for some function $F_{\ast}$, with
$G_{\alpha\beta}=\mathbf{g}\big{(}\partial_{\xi^{\alpha}},\partial_{\xi^{\beta}}\big{)}$
and $\big{(}G^{\alpha\beta}\big{)}$ being the inverse matrix of
$\big{(}G_{\alpha\beta}\big{)}$.

    * ii)

the Euler vector field $E=\frac{(1-D)}{2}\sum_{j}\xi^{j}\partial_{\xi^{j}}$ is
the unity of the Frobenius algebra;

    * iii)

the function $F_{\ast}$ satisfies the homogeneity property:

$E.F_{\ast}=(1-D)F_{\ast}+2G_{\alpha\beta}\xi^{\alpha}\xi^{\beta}.$

####  4.2.1 Frobenius algebras associated with intersection forms

Consider the intersection form
$\mathbf{g}(\omega_{0}):=\frac{1}{2}\sum_{j}\frac{\omega_{0}(P_{j})^{2}}{\lambda_{j}}(d\lambda_{j})^{2}$
(3.46) induced by the differential $\omega_{0}$ satisfying (3.1) and (3.3).
From Appendix 1, we know that $\mathbf{g}(\omega_{0})$ is a flat metric
defined on the open subset $\mathcal{M}_{\omega_{0}}$ of the double Hurwitz
space $\widehat{\mathcal{H}}_{g,L}(\mathbf{n};\mathbf{m})$ determined by the
conditions

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

We have a commutative and associative algebra structure on the tangent spaces
of the complex manifold $\mathcal{M}_{\omega_{0}}$ defined by the
multiplication $``\ast"$:

$\partial_{\lambda_{i}}\ast\partial_{\lambda_{j}}=\frac{\delta_{ij}}{\lambda_{j}}\partial_{\lambda_{j}}.$
(4.44)

Moreover, the multiplication (4.44) admits the vector field
$E=\sum_{j}\lambda_{j}\partial_{\lambda_{j}}$ as the unit vector field.
For $x=\sum_{j}x_{j}\partial_{\lambda_{j}}$,
$y=\sum_{j}y_{j}\partial_{\lambda_{j}}$ and
$z=\sum_{j}z_{j}\partial_{\lambda_{j}}\in T_{p}\mathcal{M}_{\omega_{0}}$,
consider the $(0,3)$-tensor

$\mathbf{c}_{\ast,\omega_{0}}(x,y,z):=\mathbf{g}(\omega_{0})(x\ast
y,z)=\frac{1}{2}\sum_{j}x_{j}y_{j}z_{j}\frac{\omega_{0}(P_{j})^{2}}{\lambda_{j}^{2}}.$
(4.45)

In particular, the multiplication $``\ast"$ is compatible with the metric
$\mathbf{g}(\omega_{0})$ and thus, according to Definition 2.1, the data
$\big{(}\ast,E,\mathbf{g}(\omega_{0})\big{)}$ gives rise to a Frobenius
algebra structure on tangent spaces of $\mathcal{M}_{\omega_{0}}$.

In the following result, we redescribe such Frobenius algebras structure on
tangent spaces by using the already chosen system (3.47) of flat coordinates
$\\{u^{A}(\omega_{0})\\}$ of the metric $\mathbf{g}(\omega_{0})$. The proof is
an immediate consequence of formulas (3.55), (4.44) and (4.45).

###### Proposition 4.12

Let $\big{(}G_{AB}\big{)}$ be the constant matrix (3.54) of the metric
$\mathbf{g}(\omega_{0})$ (3.46) with respect to flat coordinates (3.47). The
following assertions hold:

a)

In $\mathbf{g}(\omega_{0})$-flat coordinates (3.47), the algebra
multiplication “$\ast$” (4.44) is given by:

$\displaystyle\begin{split}\partial_{u^{A}(\omega_{0})}\ast\partial_{u^{B}(\omega_{0})}&=\sum_{j}\bigg{(}\frac{\lambda_{j}}{\omega_{0}(P_{j})^{2}}\textsf{G}_{A\alpha}\phi_{u^{\alpha}}(P_{j})\textsf{G}_{B\beta}\phi_{u^{\beta}}(P_{j})\bigg{)}\partial_{\lambda_{j}}\\\
&=\sum_{C}\bigg{(}\frac{1}{2}\sum_{j}\bigg{(}\frac{\lambda_{j}\phi_{u^{C}}(P_{j})}{\omega_{0}(P_{j})}\textsf{G}_{A\alpha}\phi_{u^{\alpha}}(P_{j})\textsf{G}_{B\beta}\phi_{u^{\beta}}(P_{j})\bigg{)}\bigg{)}\partial_{u^{C}(\omega_{0})}.\end{split}$
(4.46)

b)

In flat coordinates $\\{u^{A}(\omega_{0})\\}_{A}$ (3.47) of the metric
$\mathbf{g}(\omega_{0})$, the unit vector field $E$ takes the following form:

$E=\sum_{A}\bigg{(}\frac{1}{2}\sum_{j=1}^{N}\lambda_{j}\phi_{u^{A}}(P_{j})\omega_{0}(P_{j})\bigg{)}\partial_{u^{A}(\omega_{0})};$
(4.47)

c)

The symmetric 3-tensor $\mathbf{c}_{\ast,\omega}$ (4.45):

$\mathbf{c}_{\ast,\omega_{0}}\big{(}\partial_{u^{A}(\omega_{0})},\partial_{u^{B}(\omega_{0})},\partial_{u^{C}(\omega_{0})}\big{)}=\frac{1}{2}\sum_{j}\frac{\lambda_{j}}{\omega_{0}(P_{j})}\textsf{G}_{A\alpha}\phi_{u^{\alpha}}(P_{j})\textsf{G}_{B\beta}\phi_{u^{\beta}}(P_{j})\textsf{G}_{C\sigma}\phi_{u^{\sigma}}(P_{j}).$
(4.48)

d)

Structure constants of the multiplication “$\ast$” are such that:

$\partial_{u^{A}(\omega_{0})}\ast\partial_{u^{B}(\omega_{0})}=\sum_{C}\Big{(}\textsf{G}^{C\theta}\mathbf{c}_{\ast,\omega_{0}}\big{(}\partial_{u^{A}(\omega_{0})},\partial_{u^{B}(\omega_{0})},\partial_{u^{\theta}(\omega_{0})}\big{)}\Big{)}\partial_{u^{C}(\omega_{0})}.$
(4.49)

###### Definition 4.4

Let $\nabla^{\mathbf{g}(\omega_{0})}$ be the Levi-Civita connection of the
metric $\mathbf{g}(\omega_{0})$. The differential $\omega_{0}$ is called
primary with respect to unit vector field
$E=\sum_{j}\lambda_{j}\partial_{\lambda_{j}}$ if

$\nabla^{\mathbf{g}(\omega_{0})}E=0.$

###### Proposition 4.13

Let $\omega_{0}(P):=\int_{\gamma}h(\lambda(Q))W(P,Q)$.

1.

The following statements are equivalent:

i)

The differential $\omega_{0}$ is primary with respect to
$E=\sum_{j}\lambda_{j}\partial_{\lambda_{j}}$;

ii)

For all $j=1,\dots,N$,

$E.\omega_{0}(P_{j})=-\frac{\omega_{0}(P_{j})}{2};$

iii)

For all $j=1,\dots,N$,

$\int_{\gamma}\lambda(Q)h^{\prime}(\lambda(Q))W(Q,P_{j})=0.$ (4.50)

2.

The $N$ differentials $\psi_{u^{A}}$ in (3.50) are all primary with respect to
the vector field $E$.

3.

Assume that $\omega_{0}=\psi_{u^{A}}$ is one of primary differentials (3.50).
Then, with respect to flat coordinates of the metric $\mathbf{g}(\omega_{0})$,
the unit vector field $E$ is represented by (sum over $\alpha$ is assumed)

$E=\textsf{G}^{A\alpha}\partial_{u^{\alpha}(\omega_{0})},$

with
$\textsf{G}^{A\alpha}:=\mathbf{g}^{*}(\omega_{0})\big{(}du^{A}(\omega_{0}),du^{\alpha}(\omega_{0})\big{)}$
being the entries (3.48) of the intersection form
$\mathbf{g}^{*}(\omega_{0})$.

_Proof:_ 1. By using expressions (3.6) for Christoffel coefficients of the
intersection form $\mathbf{g}(\omega_{0}):=\mathbf{ds^{2}_{1,0}}(\omega_{0})$,
we obtain

$\displaystyle\nabla^{\mathbf{g}(\omega_{0})}_{\partial_{\lambda_{i}}}E$
$\displaystyle=\partial_{\lambda_{i}}+\sum_{j}\lambda_{j}\sum_{k}\Gamma_{ij}^{k}(\mathbf{g})\partial_{\lambda_{k}}=\partial_{\lambda_{i}}+\sum_{j\neq
i}\lambda_{j}\sum_{k}\Gamma_{ij}^{k}(\mathbf{g})\partial_{\lambda_{k}}+\lambda_{i}\sum_{k}\Gamma_{ii}^{k}(\mathbf{g})\partial_{\lambda_{k}}$
$\displaystyle=\partial_{\lambda_{i}}+\Big{(}\sum_{j\neq
i}\lambda_{j}\Gamma_{ij}^{i}(\mathbf{g})\Big{)}\partial_{\lambda_{i}}+\sum_{j\neq
i}\lambda_{j}\Gamma_{ij}^{j}(\mathbf{g})\partial_{\lambda_{j}}+\lambda_{i}\sum_{k\neq
i}\Gamma_{ii}^{k}(\mathbf{g})\partial_{\lambda_{k}}+\lambda_{i}\Gamma_{ii}^{i}(\mathbf{g})\partial_{\lambda_{i}}$
$\displaystyle=\partial_{\lambda_{i}}+\frac{1}{\omega(P_{i})}\Big{(}\sum_{j\neq
i}\lambda_{j}\partial_{\lambda_{j}}\omega(P_{i})\Big{)}\partial_{\lambda_{i}}+\lambda_{i}\Big{(}-\frac{1}{2\lambda_{i}}+\frac{\partial_{\lambda_{i}}\omega(P_{i})}{\omega(P_{i})}\Big{)}\partial_{\lambda_{i}}$
$\displaystyle=\Big{(}\frac{E.\omega(P_{i})}{\omega(P_{i})}+\frac{1}{2}\Big{)}\partial_{\lambda_{i}}.$

Therefore, this and the action (3.8) of the vector field $E$ on the functions
$\omega(P_{i})$ show the equivalence between the three assertions.
2. Since the functions $h(\lambda)$ defining the Abelian differentials $\psi_{s^{i}}=\Omega_{\infty^{0}\infty^{i}}$, $\psi_{y^{j}}=\Omega_{\mathbf{z}^{0}\mathbf{z}^{j}}$ and $\psi_{\rho^{k}}=\omega_{k}$ are constants, they satisfy conditions (4.50). Furthermore, because of (2.13), these conditions are also fulfilled by the multivalued differentials $\psi_{q^{k}}$.
3. Due to formula (3.53) we have

$\textstyle
E=\sum_{j}\lambda_{j}\partial_{\lambda_{j}}=\sum_{B}\Big{(}E.u^{B}(\omega_{0})\Big{)}\partial_{u^{B}(\omega_{0})}=\sum_{B}\Big{(}\mathbf{g}^{*}(\omega_{0})\big{(}du^{A}(\omega_{0}),du^{B}(\omega_{0})\big{)}\Big{)}\partial_{u^{B}(\omega_{0})}.$

$\square$

###### Remark 4.5

The Abelian differential of the third kind
$\omega_{0}(P)=\Omega_{\mathbf{z}^{0}\infty^{0}}(P)=\int_{\mathbf{z}^{0}}^{\infty^{0}}W(P,Q)$
is primary with respect to the vector field $E$, but it does not belong to the
vector space generated by the differentials (3.50).
Moreover, in flat coordinates (3.47) of the metric $\mathbf{g}(\omega_{0})$,
$E$ has the following form:

$\displaystyle
E=\sum_{A}\Big{(}\sum_{k=1}^{N}\underset{P_{k}}{\text{res}}\frac{\lambda(P)\psi_{u^{A}}(P)\omega_{0}(P)}{d\lambda(P)}\Big{)}\partial_{u^{A}(\omega_{0})}=\frac{1}{n_{0}+1}\sum_{i=1}^{m}\partial_{s^{i}(\omega_{0})}-\frac{1}{m_{0}+1}\sum_{i=1}^{n}\partial_{y^{i}(\omega_{0})}.$

####  4.2.2 Prepotentials of almost Frobenius manifolds

Let $\omega_{0}$ be a fixed quasi-homogeneous differential of degree
$d_{0}\neq 0$ with vanishing $a$-periods. Essentially, $\omega_{0}$ will
belong to the following family of Abelian differentials of the second kind:

$\displaystyle\psi_{t^{i,\alpha}}(P):=\underset{\infty^{i}}{\rm
res}\lambda(Q)^{\frac{\alpha}{n_{i}+1}}W(P,Q),\quad\quad\quad\text{$i=1,\dots,m$\quad
and\quad$\alpha\in\mathbb{N}$, $\alpha\geq 1$};$
$\displaystyle\widetilde{\psi}_{t^{j,\beta}}(P):=\underset{\mathbf{z}^{j}}{\rm
res}\lambda(Q)^{-\frac{\beta}{m_{j}+1}}W(P,Q),\quad\quad\text{$j=1,\dots,n$\quad
and\quad$\beta\in\mathbb{N}$, $\beta\geq 1$};$

or the following family of multivalued differentials (studied in Appendix 2):

$\displaystyle\phi_{u^{k,\sigma}}(P):=\oint_{a_{k}}\big{(}\lambda(Q)\big{)}^{\sigma}W(P,Q),\quad\quad\quad\quad\quad\text{$k=1,\dots,g$\quad
and\quad$\sigma\in\mathbb{N}$, $\sigma\geq 1$}.$

Note that the degrees of these differentials are:

$deg\big{(}\psi_{t^{i,\alpha}}\big{)}=\frac{\alpha}{n_{i}+1},\quad\quad
deg\big{(}\widetilde{\psi}_{t^{j,\beta}}\big{)}=-\frac{\beta}{m_{j}+1},\quad\quad
deg\big{(}\phi_{u^{k,\sigma}}\big{)}=\sigma.$

###### Proposition 4.14

Let
$\omega_{0}\in\big{\\{}\psi_{t^{i,\alpha}},\widetilde{\psi}_{t^{j,\beta}},\phi_{u^{k,\sigma}}\big{\\}}$
and $\big{\\{}u^{A}(\omega_{0})\\}_{A}$ be the corresponding system of flat
coordinates (3.47) and $E=\sum_{j}\lambda_{j}\partial_{\lambda_{j}}$ be the
unit vector field. Then

$\begin{split}&E.u^{A}(\omega_{0})=d_{0}u^{A}(\omega_{0}),\quad\text{for all
$A$};\\\
&E=d_{0}\sum_{A}u^{A}(\omega_{0})\partial_{u^{A}(\omega_{0})}.\end{split}$
(4.51)

_Proof:_ The second formula in (4.51) follows from the first one. In order to
prove that $u^{A}(\omega_{0})$ is an eigenfunction of the vector field $E$, we
shall use formula (3.53) where we replace $\psi_{u^{B}}$ by $\omega_{0}$.
Assume first that $\omega_{0}=\psi_{t^{i_{0},\alpha}}(P)$. Similarly to
(3.25), we have

$\psi_{t^{i_{0},\alpha}}(P)\underset{P\sim\infty^{i_{0}}}{=}\frac{\alpha}{z_{i_{0}}(P)^{\alpha+1}}dz_{i_{0}}(P)+\text{holomorphic},\quad\quad\lambda(P)=z_{i_{0}}(P)^{-n_{i_{0}}-1}.$

When $\psi_{u^{A}}$ is among the $E$-primary differentials
$\psi_{s^{i}},\psi_{y^{j}},\psi_{\rho^{k}}$ (see (3.50)), the function
$\frac{\lambda(P)\psi_{u^{A}}(P)}{d\lambda(P)}$ is holomorphic near the point
$\infty^{i_{0}}$. Therefore by writing its local behavior near
$\infty^{i_{0}}$ and using notation (2.7) we deduce that

$\displaystyle E.u^{A}(\omega_{0})$
$\displaystyle=\sum_{k=1}^{N}\underset{P_{k}}{{\rm
res}}\frac{\lambda(P)\psi_{u^{A}}(P)\omega_{0}(P)}{d\lambda(P)}=-\underset{\infty^{i_{0}}}{{\rm
res}}\frac{\lambda(P)\psi_{u^{A}}(P)\omega_{0}(P)}{d\lambda(P)}$
$\displaystyle=\frac{\alpha}{(n_{i_{0}}+1)(\alpha-1)!}\psi_{u^{A}}^{(\alpha-1)}(\infty^{i_{0}})=\frac{\alpha}{n_{i_{0}}+1}\underset{\infty^{i_{0}}}{{\rm
res}}\big{(}\lambda(P)\big{)}^{\frac{\alpha}{n_{i_{0}}+1}}\psi_{u^{A}}(P)$
$\displaystyle=\frac{\alpha}{n_{i_{0}}+1}u^{A}(\omega_{0})\quad\quad(\text{by
changing the order of integration}).$

When $\psi_{u^{A}}$ is among the multivalued differentials
$\psi_{q^{k}}(P):=\oint_{a_{k}}\log(\lambda(Q))W(P,Q)$, the restriction of
function $\frac{\lambda(P)\psi_{q^{k}}(P)}{d\lambda(P)}$ to a chosen
fundamental polygon $F_{g}$ is also holomorphic near the point
$\infty^{i_{0}}$. Accordingly, in view of the jumps (3.51) of $\psi_{q^{k}}$
and the residue theorem, we obtain

$\displaystyle E.q^{k}(\omega_{0})$
$\displaystyle=\sum_{j=1}^{N}\underset{P_{j}}{{\rm
res}}\frac{\lambda(P)\psi_{q^{k}}(P)\omega_{0}(P)}{d\lambda(P)}=\frac{1}{2{\rm{i}}\pi}\oint_{\partial{F_{g}}}\bigg{(}\frac{\lambda(P)\omega_{0}(P)}{d\lambda(P)}\psi_{q^{k}}(P)\bigg{)}-\underset{\infty^{i_{0}}}{{\rm
res}}\frac{\lambda(P)\psi_{q^{k}}(P)\omega_{0}(P)}{d\lambda(P)}$
$\displaystyle=\frac{1}{2{\rm{i}}\pi}\oint_{a_{k}}\bigg{(}\frac{\lambda(P)\omega_{0}(P)}{d\lambda(P)}\Big{(}\psi_{q^{k}}(P)-\psi_{q^{k}}(P+b_{k})\Big{)}\bigg{)}+\frac{\alpha}{(n_{i_{0}}+1)(\alpha-1)!}\psi_{q^{k}}^{(\alpha-1)}(\infty^{i_{0}})$
$\displaystyle=\oint_{a_{k}}\omega_{0}(P)+\frac{\alpha}{n_{i_{0}}+1}\underset{\infty^{i_{0}}}{{\rm
res}}\big{(}\lambda(P)\big{)}^{\frac{\alpha}{n_{i_{0}}+1}}\psi_{q^{k}}(P)$
$\displaystyle=\frac{\alpha}{n_{i_{0}}+1}q^{k}(\omega_{0})\quad\quad(\text{by
changing the order of integration}).$

The case where $\omega_{0}=\widetilde{\psi}_{t^{j,\beta}}$ can be done
similarly. Let us now take $\omega_{0}=\phi_{u^{k_{0},\sigma}}$. Then the
action of $E$ on the functions
$s^{i}(\omega_{0}),\rho^{k}(\omega_{0}),q^{j}(\omega_{0})$ is included in
formulas (8.12) and (8.15). Moreover, the arguments similar to those used in
the preceeding case $E.q^{k}(\omega_{0})$ allow us to arrive at the relation
$E.y^{j}(\omega_{0})=\sigma y^{j}(\omega_{0})$.

$\square$

###### Theorem 4.3

Let
$\omega_{0}\in\big{\\{}\psi_{t^{i,\alpha}},\widetilde{\psi}_{t^{j,\beta}},\phi_{u^{k,\sigma}}\big{\\}}$
be a quasi-homogeneous differential of degree $d_{0}\neq 0$.

1)

Consider the symmetric matrix $(H_{AB})$ defined by

$H_{AB}:=\frac{1}{2}\textsf{G}_{A\alpha}\textsf{G}_{B\beta}\Big{(}u^{\alpha}\big{(}\psi_{u^{\beta}}\big{)}+u^{\beta}\big{(}\psi_{u^{\alpha}}\big{)}\Big{)}.$
(4.52)

Then the 3-tensor (4.48) can be expressed as:

$\begin{split}\partial_{u^{C}(\omega_{0})}H_{AB}=\mathbf{c}_{\ast,\omega_{0}}\big{(}\partial_{u^{A}(\omega_{0})},\partial_{u^{B}(\omega_{0})},\partial_{u^{C}(\omega_{0})}\big{)}=\partial_{u^{B}(\omega_{0})}H_{AC}.\end{split}$
(4.53)

2)

Consider the function

$\mathbf{F}_{\ast,\omega_{0}}=\frac{1}{2}\sum_{A,B}u^{A}(\omega_{0})u^{B}(\omega_{0})\Big{(}H_{AB}-\frac{3}{2d_{0}}\textsf{G}_{AB}\Big{)}.$
(4.54)

Then $\mathbf{F}_{\ast,\omega_{0}}$ satisfies the following properties:

i.

The third order partial derivatives of $\mathbf{F}_{\ast,\omega_{0}}$ are such
that

$\partial_{u^{A}(\omega_{0})}\partial_{u^{B}(\omega_{0})}\partial_{u^{C}(\omega_{0})}\mathbf{F}_{\ast,\omega_{0}}=\mathbf{c}_{\ast,\omega_{0}}\big{(}\partial_{u^{A}(\omega_{0})},\partial_{u^{B}(\omega_{0})},\partial_{u^{C}(\omega_{0})}\big{)}.$

ii.

the function $\mathbf{F}_{\ast,\omega_{0}}$ is quasi-homogeneous w.r.t. the
vector field $E$ in (4.51) with

$E.\mathbf{F}_{\ast,\omega_{0}}=2d_{0}\mathbf{F}_{\ast,\omega_{0}}+\frac{1}{2}\sum_{A,B}\textsf{G}_{AB}u^{A}(\omega_{0})u^{B}(\omega_{0});$

iii.

the matrix $(H_{AB})$ in (4.52), which does not depend on $\omega_{0}$, is the
Hessian matrix of $\mathbf{F}_{\ast,\omega_{0}}$.

_Proof:_ 1) From (3.55) we know that:

$\partial_{u^{C}(\omega_{0})}=\textstyle\sum_{j}\bigg{(}\frac{\lambda_{j}}{\omega_{0}(P_{j})}\textsf{G}_{C\sigma}\psi_{u^{\sigma}}(P_{j})\bigg{)}\partial_{\lambda_{j}}.$

Using this, the Rauch formula and (4.48) we deduce that

$\displaystyle\partial_{u^{C}(\omega_{0})}H_{AB}$
$\displaystyle=\frac{1}{2}\sum_{j}\frac{\lambda_{j}}{\omega_{0}(P_{j})}\textsf{G}_{C\sigma}\psi_{u^{\sigma}}(P_{j})\textsf{G}_{A\alpha}\textsf{G}_{B\beta}\psi_{u^{\alpha}}(P_{j})\psi_{u^{\beta}}(P_{j})$
$\displaystyle=\mathbf{c}_{\ast,\omega_{0}}\big{(}\partial_{u^{A}(\omega_{0})},\partial_{u^{B}(\omega_{0})},\partial_{u^{C}(\omega_{0})}\big{)},$

which gives (4.53).
2) Using the first equality in (3.53), we deduce that the action of the vector
field $E$ on the functions $H_{AB}$ is determined by

$\displaystyle
E.H_{AB}=\frac{1}{2}\textsf{G}_{A\alpha}\textsf{G}_{B\beta}\Big{(}E.u^{\alpha}\big{(}\psi_{u^{\beta}}\big{)}+E.u^{\alpha}\big{(}\psi_{u^{\beta}}\big{)}\Big{)}$
$\displaystyle=\textsf{G}_{A\alpha}\textsf{G}_{B\beta}G^{\alpha\beta}=G_{AB}.$

This and (4.51) lead to:

$\displaystyle\partial_{u^{A}(\omega_{0})}\mathbf{F}_{\ast,\omega_{0}}$
$\displaystyle=\sum_{C}u^{C}(\omega_{0})\Big{(}H_{AC}-\frac{3}{2d_{0}}\textsf{G}_{AC}\Big{)}+\frac{1}{2}\sum_{C}u^{C}(\omega_{0})\left(\sum_{D}u^{D}(\omega_{0})\partial_{u^{A}(\omega_{0})}H_{CD}\right)$
$\displaystyle=\sum_{C}u^{C}(\omega_{0})\Big{(}H_{AC}-\frac{3}{2d_{0}}\textsf{G}_{AC}\Big{)}+\frac{1}{2}\sum_{C}u^{C}(\omega_{0})\Big{(}\sum_{D}u^{D}(\omega_{0})\partial_{u^{D}(\omega_{0})}H_{AC}\Big{)}$
$\displaystyle=\sum_{C}u^{C}(\omega_{0})\Big{(}H_{AC}-\frac{3}{2d_{0}}\textsf{G}_{AC}\Big{)}+\frac{1}{2d_{0}}\sum_{C}u^{C}(\omega_{0})E.H_{AC}$
$\displaystyle=\sum_{C}u^{C}(\omega_{0})H_{AC}-\frac{1}{d_{0}}\sum_{C}\textsf{G}_{AC}u^{C}(\omega_{0}).$

In particular

$E.\mathbf{F}_{\ast,\omega_{0}}=d_{0}\sum_{A}u^{A}(\omega_{0})\partial_{u^{A}(\omega_{0})}\mathbf{F}_{\ast,\omega_{0}}=2d_{0}\mathbf{F}_{\ast,\omega_{0}}+\frac{1}{2}\sum_{A,B}\textsf{G}_{AB}u^{A}(\omega_{0})u^{B}(\omega_{0}).$

Moreover, like before we have

$\displaystyle\partial_{u^{B}(\omega_{0})}\partial_{u^{A}(\omega_{0})}\mathbf{F}_{\ast,\omega_{0}}$
$\displaystyle=H_{AB}+\sum_{C}u^{C}(\omega_{0})\partial_{u^{B}(\omega_{0})}H_{AC}-\frac{1}{d_{0}}\textsf{G}_{AB}$
$\displaystyle=H_{AB}+\frac{1}{d_{0}}E.H_{AB}-\frac{1}{d_{0}}\textsf{G}_{AB}$
$\displaystyle=H_{AB}.$

Finally, by relation (4.53) we conclude that

$\partial_{u^{C}(\omega_{0})}\partial_{u^{B}(\omega_{0})}\partial_{u^{A}(\omega_{0})}\mathbf{F}_{\ast,\omega_{0}}=\partial_{u^{C}(\omega_{0})}H_{AB}=\mathbf{c}_{\ast,\omega_{0}}\big{(}\partial_{u^{A}(\omega_{0})},\partial_{u^{B}(\omega_{0})},\partial_{u^{C}(\omega_{0})}\big{)}.$

$\square$

Theorem 4.3 and Definition 4.3 enable us to conclude the following.

###### Corollary 4.7

Let
$\omega_{0}\in\big{\\{}\psi_{t^{i,\alpha}},\widetilde{\psi}_{t^{j,\beta}},\phi_{u^{k,\sigma}}\big{\\}}$
be a quasi-homogeneous differential of degree $d_{0}\neq 0$. Then the open set
$\mathcal{M}_{\omega_{0}}\subset\widehat{\mathcal{H}}_{g,L}(\mathbf{n};\mathbf{m})$
carries an almost Frobenius structure determined by the intersection form
$\mathbf{g}(\omega_{0})$ (3.46), the multiplication $``\ast"$ (4.46), the
unit=Euler vector field $E$ (4.51) and the prepotential
$\mathbf{F}_{\ast,\omega_{0}}$ (4.54). The charge is given by $D=1-2d_{0}\neq
1$.

##  5 Prepotentials of deformed Hurwitz-Frobenius manifold structures

In this section, our goal is to extend formula (4.19)-(4.27) to quasi-
homogeneous solutions associated with deformations of Frobenius manifold
structures on the simple Hurwitz spaces
$\widehat{\mathcal{H}}_{g,L}(n_{0},\dots,n_{m})$, with $g\geq 1$. As already
indicated in the introduction, the deformed Frobenius structures on
$\widehat{\mathcal{H}}_{g,L}(n_{0},\dots,n_{m})$ depending $g(g+1)/2$ complex
parameters were studied by Shramchenko in [42].
In order to achieve our goal, we follow the pattern of the Sections 3.3 and
4.1 and use the results obtained there.

###  5.1 Deformations of Frobenius manifold structures on simple Hurwitz
spaces

Let $\left\\{(C_{g},\lambda),\\{a_{k},b_{k}\\}\right\\}$ be a point in
$\widehat{\mathcal{H}}_{g,L}(n_{0},\dots,n_{m})$ and
$\omega_{1},\dots,\omega_{g}$ be the basis of normalized holomorphic
differential on the surface $C_{g}$, with
$\oint_{a_{l}}\omega_{k}=\delta_{kl}$. Here, the cycles $a_{k},b_{k}$ are
assumed to start at a chosen marked point $P_{0}\in C_{g}$ so that
$\lambda(P_{0})=0$.
We will use repeatedly the following preliminary result about the Riemann
matrix of $b$-periods $\mathbb{B}=(\mathbb{B}_{kl})$.

###### Lemma 5.1

We have

$e.\mathbb{B}_{kl}=0=E.\mathbb{B}_{kl},\quad\quad\forall\ k,l=1,\dots,g,$
(5.1)

where $e=\sum_{j}\partial_{\lambda_{j}}$ and
$E=\sum_{j}\lambda_{j}\partial_{\lambda_{j}}$.

_Proof:_ Using notation (3.21), we have $\mathbb{B}_{kl}={2{\rm
i}\pi}\rho^{k}(\phi_{\rho^{l}})$. Thus (5.1) follows from (3.38) and (3.56)
and the fact that normalized holomorphic differentials are quasi-homogeneous
differentials of degree 0 (as we know from (4.11)).

$\square$

$\bullet$

Darboux-Egoroff metrics in terms of the $\mathbf{q}$-bidifferentials

Let $\mathfrak{p}=\big{(}\mathfrak{p}_{kl}\big{)}$ and
$\mathfrak{q}=\big{(}\mathfrak{q}_{kl}\big{)}$ be two $g\times g$ matrices
such that

i)

the matrix $\mathfrak{q}$ is nondegenerate and the matrix
$\mathbf{q}:=\mathfrak{q}^{-1}\mathfrak{p}=\big{(}\mathbf{q}_{kl}\big{)}$ is
symmetric;

ii)

the matrix $I+\mathbf{q}\mathbb{B}$ is nondegenerate (here $\mathbb{B}$ is the
Riemann matrix and $I$ is the identity matrix);

iii)

the matrix $\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}$ is symmetric;

iv)

the parameters $\mathbf{q}_{kl}$ are constants with respect to the branch
points $\\{\lambda_{j}\\}$ of $(C_{g},\lambda)$ and $\lambda$.

Note that item i) tells us that the following $2g\times 2g$ matrix is
symplectic:

$\mathcal{S}:=\begin{pmatrix}\left(\mathfrak{q}^{-1}\right)^{T}&0\\\
\mathfrak{p}&\mathfrak{q}\end{pmatrix},$ (5.2)

that is $\mathcal{S}^{T}\mathcal{J}\mathcal{S}=\mathcal{J}$, with
$\mathcal{J}=\begin{pmatrix}0&-I\\\ I&0\end{pmatrix}$.
Moreover, when the matrix $\mathbf{q}$ is nondegenerate (and symmetric),
$\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}=\big{(}\mathbb{B}+\mathbf{q}^{-1}\big{)}^{-1}$
is symmetric.
By following [42], let us consider the following $\mathbf{q}$-deformation of
the bidifferential $W(P,Q)$ (2.11):

$\begin{split}W_{\mathbf{q}}(P,Q)&:=W(P,Q)-{2{\rm
i}\pi}\Big{(}\big{(}\mathfrak{q}+\mathfrak{p}\mathbb{B}\big{)}^{-1}\mathfrak{p}\Big{)}_{\alpha\beta}\omega_{\alpha}(P)\omega_{\beta}(Q)\\\
&=W(P,Q)-{2{\rm
i}\pi}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}\omega_{\alpha}(P)\omega_{\beta}(Q),\end{split}$
(5.3)

where $\omega_{1},\dots,\omega_{g}$ are the normalized holomorphic
differentials.
_In formula ( 5.3) as well as during this subsection and the next one,
summation over the repeated Greek indices $\alpha,\beta=1,\dots,g$ is
assumed_.
We shall call $W_{\mathbf{q}}(P,Q)$ the $\mathbf{q}$-bidifferential.
Expression (5.3) for the $\mathbf{q}$-bidifferential differs slightly from the
one used in [42] where the symmetric matrix $\mathbf{q}$ can be degenerate.
From the properties of the symmetric bidifferential $W(P,Q)$ (2.11), the
$\mathbf{q}$-bidifferential $W_{\mathbf{q}}(P,Q)$ (5.3) is also symmetric and
has a pole of second order on the diagonal with biresidue 1. Moreover, for all
$k=1,\dots,g$ we have

$\displaystyle\oint_{Q\in a_{k}}W_{\mathbf{q}}(P,Q)$ $\displaystyle=-{2{\rm
i}\pi}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{k\alpha}\omega_{\alpha}(P);$
$\displaystyle\oint_{Q\in b_{k}}W_{\mathbf{q}}(P,Q)$ $\displaystyle={2{\rm
i}\pi}\omega_{k}(P)-{2{\rm
i}\pi}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}\mathbb{B}_{\beta{k}}\omega_{\alpha}(P)$
$\displaystyle={2{\rm
i}\pi}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\Big{)}_{\alpha{k}}\omega_{\alpha}(P).$

Thus the $\mathbf{q}$-bidifferential $W_{\mathbf{q}}(P,Q)$ fulfills the
normalization condition

$\oint_{Q\in a_{k}}W_{\mathbf{q}}(P,Q)+\mathbf{q}_{k\alpha}\oint_{Q\in
b_{\alpha}}W_{\mathbf{q}}(P,Q)=0,\quad\quad\forall\ k=1,\dots,g,$ (5.4)

with $\mathbf{q}_{\alpha\beta}$ being the coefficient of the matrix
$\mathbf{q}$.
From its expression (5.3), we see that the $\mathbf{q}$-bidifferential
$W_{\mathbf{q}}(P,Q)$ turns into the bidifferential $W(P,Q)$ provided that
$\mathbf{q}$ is the zero matrix.
Before delving more into the properties of the $\mathbf{q}$-bidifferential,
let us emphasize that the deformation (5.3) is motivated by the following
viewpoint. Assume that the symplectic matrix (5.2) belongs to
$Sp(2g,\mathbb{Z})$. Then the cycles
$\widetilde{\mathbf{a}}:=\big{(}\widetilde{a}_{1},\dots,\widetilde{a}_{g}\big{)}^{T}$
and
$\widetilde{\mathbf{b}}:=\big{(}\widetilde{b}_{1},\dots,\widetilde{b}_{g}\big{)}^{T}$
given by

$\begin{pmatrix}\widetilde{\mathbf{b}}\\\
\widetilde{\mathbf{a}}\end{pmatrix}=\begin{pmatrix}\big{(}\mathfrak{q}^{-1}\big{)}^{T}&0\\\
\mathfrak{p}&\mathfrak{q}\end{pmatrix}\begin{pmatrix}\mathbf{b}\\\
\mathbf{a}\end{pmatrix},\quad\quad\text{with}\quad\mathbf{a}:=\big{(}a_{1},\dots,a_{g}\big{)}^{T},\quad\mathbf{b}:=\big{(}b_{1},\dots,b_{g}\big{)}^{T}$

define a new canonical homology basis on the surface $C_{g}$. Thus the
$\mathbf{q}$-bidifferential (5.3) becomes normalized by

$\displaystyle\oint_{Q\in\widetilde{a}_{k}}W_{\mathbf{q}}(P,Q)$
$\displaystyle=\mathfrak{p}_{k\alpha}\oint_{Q\in
b_{\alpha}}W_{\mathbf{q}}(P,Q)+\mathfrak{q}_{k\beta}\oint_{Q\in
a_{\beta}}W_{\mathbf{q}}(P,Q)$
$\displaystyle=\mathfrak{q}_{k\beta}\left(\mathbf{q}_{\beta\alpha}\oint_{Q\in
b_{\alpha}}W_{\mathbf{q}}(P,Q)+\oint_{Q\in
a_{\beta}}W_{\mathbf{q}}(P,Q)\right)=0,$

where we used the equality $\mathfrak{p}=\mathfrak{q}\mathbf{q}$ and (5.4) in
the second line.
As consequence, the characterization (2.12)-(2.13) allows us to view
$W_{\mathbf{q}}(P,Q)$ (5.3) as the canonical symmetric bidifferential with
respect to the new canonical homology basis
$\\{\widetilde{a}_{k},\widetilde{b}_{k}\\}$ on $C_{g}$ and the corresponding
basis of normalized holomorphic differentials
$\widetilde{\omega}_{1},\dots,\widetilde{\omega}_{g}$, with

$\big{(}\widetilde{\omega}_{1},\dots,\widetilde{\omega}_{g}\big{)}^{T}=\big{(}\mathbb{B}\mathfrak{p}^{T}+\mathfrak{q}^{T}\big{)}^{-1}\big{(}\omega_{1},\dots,\omega_{g}\big{)}^{T}.$

The following result was proven in [42] when the $\mathfrak{p}=I$ and
$\mathbf{q}=\mathfrak{q}^{-1}$ is symmetric and nondegenerate. We include a
proof for the reader’s convenience.

###### Proposition 5.1

The $\mathbf{q}$-bidifferential $W_{\mathbf{q}}(P,Q)$ satisfies the following
variational Rauch formulas:

$\partial_{\lambda_{j}}W_{\mathbf{q}}(P,Q)=\frac{1}{2}W_{\mathbf{q}}(P,P_{j})W_{\mathbf{q}}(P_{j},Q),$
(5.5)

where $W_{\mathbf{q}}(P,P_{j})$ is the evaluation (2.5) of the
$\mathbf{q}$-bidifferential $W_{\mathbf{q}}(P,Q)$ at $Q=P_{j}$.
In particular, the functions

$\kappa_{ij}(\mathbf{q}):=\frac{1}{2}W_{\mathbf{q}}(P_{i},P_{j}):=\frac{1}{2}\frac{W_{\mathbf{q}}(P,Q)}{dx_{i}(P)dx_{j}(Q)},\quad\quad
i\neq j,$

solve the Darboux-Egoroff equations:

$\begin{split}&\partial_{\lambda_{k}}\kappa_{ij}(\mathbf{q})=\kappa_{ik}(\mathbf{q})\kappa_{kj}(\mathbf{q}),\quad\quad\text{$i,j,k$
are distinct};\\\
&e.\kappa_{ij}(\mathbf{q}):=\sum_{j}\partial_{\lambda_{j}}\kappa_{ij}(\mathbf{q})=0.\end{split}$
(5.6)

_Proof:_ Since $\partial_{\lambda_{j}}\mathbb{B}_{kl}={{\rm
i}\pi}\omega_{k}(P_{j})\omega_{l}(P_{j})$ (by Rauch’s formula (2.15)), it
follows that

$\displaystyle\partial_{\lambda_{j}}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}$
$\displaystyle=\partial_{\lambda_{j}}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{\alpha\sigma}\mathbf{q}_{\sigma\beta}\Big{)}$
$\displaystyle=\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{\alpha\mu}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{\nu\sigma}\mathbf{q}_{\sigma\beta}\partial_{\lambda_{j}}\Big{(}\mathbf{q}\mathbb{B}\Big{)}_{\mu\nu}$
$\displaystyle={{\rm
i}\pi}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{\alpha\mu}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{\nu\sigma}\mathbf{q}_{\sigma\beta}\mathbf{q}_{\mu\epsilon}\omega_{\epsilon}(P_{j})\omega_{\nu}(P_{j})$
$\displaystyle={{\rm
i}\pi}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\epsilon}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\nu\beta}\omega_{\epsilon}(P_{j})\omega_{\nu}(P_{j})$

where we have taken the sum over all the repeated Greek indices and denoted by
$\Lambda^{ij}$ the coefficient of a nondegenerate matrix $(\Lambda_{ij})$.
This relation, Rauch’s formulas (2.15) and expression (5.3) for the
$\mathbf{q}$-bidifferential imply that

$\displaystyle\partial_{\lambda_{j}}W_{\mathbf{q}}(P,Q)$
$\displaystyle=\frac{1}{2}W(P,P_{j})W_{\mathbf{q}}(P_{j},Q)-{{\rm
i}\pi}W(P_{j},Q)\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}\omega_{\alpha}(P)\omega_{\beta}(P_{j})$
$\displaystyle\quad-2{{\rm i}\pi}\bigg{(}{{\rm
i}\pi}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\epsilon}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\nu\beta}\omega_{\epsilon}(P_{j})\omega_{\nu}(P_{j})\bigg{)}\omega_{\alpha}(P)\omega_{\beta}(Q)$
$\displaystyle=\frac{1}{2}W_{\mathbf{q}}(P,P_{j})W_{\mathbf{q}}(P_{j},Q).$

The first equation in (5.6) is a direct consequence of Rauch’s variational
formulas (5.5). On the other hand, (5.3) and (5.1) imply that

$e.W_{\mathbf{q}}(P_{i},P_{j})=e.W(P_{i},P_{j})-2{{\rm
i}\pi}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}\Big{(}e.\big{(}\omega_{\alpha}(P_{i})\omega_{\beta}(P_{j})\big{)}\Big{)}.$

Thus the second equation in (5.6) follows from
$e.W(P_{i},P_{j})=0=e.\omega_{k}(P_{j})$, by (3.7) and (3.8).

$\square$

The above result leads to a family of Darboux-Egoroff metrics with respect to
the introduced $\mathbf{q}$-bidifferential $W_{\mathbf{q}}(P,Q)$ (5.3),
similarly to (3.20).

###### Corollary 5.1

Let $\omega_{\mathbf{q}}$ be the $\mathbf{q}$-differential on $C_{g}$ defined
by

$\omega_{\mathbf{q}}(P)=\int_{\gamma}h\big{(}\lambda(Q))W_{\mathbf{q}}(P,Q),$
(5.7)

where the contour $\gamma$ and the function $h$ are as in (3.1). Then the
following formula

$\eta\big{(}\omega_{\mathbf{q}}\big{)}:=\frac{1}{2}\sum_{j=1}^{N}\big{(}\omega_{\mathbf{q}}(P_{j})\big{)}^{2}(d\lambda_{j})^{2}=\sum_{j=1}^{N}\bigg{(}\underset{P_{j}}{{\rm
res}}\frac{\big{(}\omega_{\mathbf{q}}(P)\big{)}^{2}}{d\lambda(P)}\bigg{)}(d\lambda_{j})^{2}.$
(5.8)

defines a Darboux-Egoroff metric on the open subset
$\mathcal{M}_{\omega_{\mathbf{q}}}$ of the Hurwitz space
$\widehat{\mathcal{H}}_{g,L}(n_{0},\dots,n_{m})$ determined by the conditions

$det(I+\mathbf{q}\mathbb{B})\neq
0\quad\quad\text{and}\quad\quad\omega_{\mathbf{q}}(P_{j})\neq 0,\quad\quad
j=1,\dots,N.$ (5.9)

The rotation coefficients of the metric
$\eta\big{(}\omega_{\mathbf{q}}\big{)}$ are given by
$\kappa_{ij}(\mathbf{q}):=W_{\mathbf{q}}(P_{i},P_{j})/2$.

$\bullet$

Normalized holomorphic $\mathbf{q}$-differentials

For $k=1,\dots,g$, let us denote by $\omega_{\mathbf{q},k}$ the holomorphic
$\mathbf{q}$-differentials defined by

$\displaystyle\omega_{\mathbf{q},k}(P)$ $\displaystyle:=\frac{1}{2{\rm
i}\pi}\oint_{b_{k}}W_{\mathbf{q}}(P,Q)=\omega_{k}(P)-\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}\mathbb{B}_{k\beta}\omega_{\alpha}(P).$

Therefore

$\omega_{\mathbf{q},k}(P)=\big{(}I+\mathbb{B}\mathbf{q}\big{)}^{k\alpha}\omega_{\alpha}(P).$
(5.10)

The holomorphic $\mathbf{q}$-differentials $\omega_{\mathbf{q},k}$ enjoy the
following normalization property:

$\oint_{a_{j}}\omega_{\mathbf{q},k}+\mathbf{q}_{j\alpha}\oint_{b_{\alpha}}\omega_{\mathbf{q},k}=\delta_{jk},\quad\quad
j=1,\dots,g.$ (5.11)

Let us now introduce the $\mathbf{q}$-Riemann matrix of $b$-periods of
normalized holomorphic $\mathbf{q}$-differentials (5.10) by

$(\mathbb{B}_{\mathbf{q}})_{kl}:=\oint_{b_{l}}\omega_{\mathbf{q},k}=\big{(}I+\mathbb{B}\mathbf{q}\big{)}^{k\alpha}\mathbb{B}_{\alpha{l}}.$
(5.12)

Note that the normalization property (5.11) specifies the holomorphic
$\mathbf{q}$-differentials $\omega_{\mathbf{q},k}$ uniquely. This follows from
the fact that a holomorphic differential $\phi$ on $C_{g}$ satisfying

$\oint_{a_{j}}\phi+\mathbf{q}_{j\alpha}\oint_{b_{\alpha}}\phi=0,\quad\quad\forall\
j=1,\dots,g,$ (5.13)

must vanish identically. Indeed, if we write
$\phi=c_{1}\omega_{1}+\dots+c_{g}\omega_{g}$, then (5.13) implies that

$\big{(}I+\mathbf{q}\mathbb{B}\big{)}(c_{1},\dots,c_{g})^{T}=0$

and thus we get $c_{1}=\dots=c_{g}=0$.

$\bullet$

Quasi-homogeneous $\mathbf{q}$-differentials and primary
$\mathbf{q}$-differentials

We start with the following definition of quasi-homogeneous differentials with
respect to the $\mathbf{q}$-bidifferential (5.3). It is similar to Definition
4.2.

###### Definition 5.1

Let $d_{0}$ be a nonnegative real number. A $\mathbf{q}$-differential
$\omega_{\mathbf{q}}$ is called quasi-homogeneous of degree $d_{0}$ if
$\omega_{\mathbf{q}}$ can be written as a linear combination of
$\mathbf{q}$-differentials of the following form:

$\int_{\ell_{0}}\big{(}\lambda(Q)\big{)}^{d_{0}}W_{\mathbf{q}}(P,Q)$

with $\ell_{0}$ being a contour satisfying condition C1) in (3.1).

We are going to introduce the following family of $N$ typical examples of
quasi-homogeneous $\mathbf{q}$-differentials. They can be regarded as the
$\mathbf{q}$-analogue of the Dubrovin primary differentials (3.23). These
$\mathbf{q}$-differentials and the corresponding Darboux-Egoroff metrics (5.8)
turn out to the so-called $\mathbf{q}$-deformed semi-simple Frobenius manifold
structures, studied in [42].

i) For $t^{A}\in\big{\\{}t^{i,\epsilon},v^{i},s^{i},\rho^{k}\big{\\}}$, with
$t^{i,\epsilon}$, $v^{i}$, $s^{i}$, $\rho^{k}$ being the operations defined by
(3.21), consider the $\mathbf{q}$-differential

$\begin{split}\phi_{\mathbf{q},t^{A}}(P)&:=c_{A}\int_{\ell_{A}}\big{(}\lambda(Q)\big{)}^{d_{A}}W_{\mathbf{q}}(P,Q)=\phi_{t^{A}}(P)-{2{\rm
i}\pi}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}t^{A}(\omega_{\beta})\omega_{\alpha}(P),\end{split}$
(5.14)

where the triplet $(d_{A},c_{A},\ell_{A})$ is determined by integral
representation formulas (3.23) and (3.24) for the Abelian differential
$\phi_{t^{A}}$.
In particular, the $\mathbf{q}$-differential $\phi_{\mathbf{q},t^{A}}$ is a
quasi-homogeneous $\mathbf{q}$-differential of degree $d_{A}$ and has the same
singularity structure as the primary differential $\phi_{t^{A}}$ (3.23)
described in Proposition 3.4. Furthermore, because of the symmetry property
$\oint_{b_{r}}\phi_{t^{A}}={2{\rm
i}\pi}\rho^{r}(\phi_{t^{A}})=t^{A}(\omega_{r})$ (3.36), differential
$\phi_{\mathbf{q},t^{A}}$ has the following property:

$\oint_{a_{j}}\phi_{\mathbf{q},t^{A}}+\mathbf{q}_{j\alpha}\oint_{b_{\alpha}}\phi_{\mathbf{q},t^{A}}=\delta_{t^{A},\rho^{j}},\quad\quad
j=1,\dots,g.$ (5.15)

On the other hand, (5.14) implies that $\phi_{\mathbf{q},t^{A}}$ coincides
with the differential $\phi_{t^{A}}$ provided that the matrix $\mathbf{q}$ is
zero.

ii) For $k=1,\dots,g$, let $\phi_{\mathbf{q},\xi^{k}}$ be the multivalued
$\mathbf{q}$-differential defined by

$\begin{split}\phi_{\mathbf{q},\xi^{k}}(P)&:=\oint_{a_{k}}\lambda(Q)W_{\mathbf{q}}(P,Q)+\mathbf{q}_{k\sigma}\oint_{Q\in
b_{\sigma}}\lambda(Q)W_{\mathbf{q}}(P,Q)\\\ &=\phi_{u^{k}}(P)-{2{\rm
i}\pi}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}u^{k}(\omega_{\beta})\omega_{\alpha}(P)\\\
&\quad+\mathbf{q}_{k\sigma}\Big{(}\phi_{{\widetilde{u}}^{\sigma}}(P)-{2{\rm
i}\pi}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}{\widetilde{u}}^{\sigma}(\omega_{\beta})\omega_{\alpha}(P)\Big{)},\end{split}$
(5.16)

with $\phi_{u^{k}}$ being the primary differential in (3.23), $u^{k}(\omega)$
the function $u^{k}(\omega)=\oint_{a_{k}}\lambda(P)\omega(P)$ and
$\phi_{\widetilde{u}^{r}}$ and $\widetilde{u}^{r}$ being respectively the
differential and the corresponding operation defined by

$\begin{split}&\phi_{{\widetilde{u}}^{r}}(P):=\oint_{b_{r}}\lambda(Q)W(P,Q);\\\
&{\widetilde{u}}^{r}(\omega):=\oint_{b_{r}}\lambda(P)\omega(P),\end{split}\quad\quad
r=1,\dots,g.$ (5.17)

In expression (5.16), we have assumed the sum over the repeated indices
$\alpha,\beta,\sigma$.
The multivalued differential $\phi_{{\widetilde{u}}^{r}}$ is the particular
case $\nu=1$ of the differentials
$\big{\\{}\phi_{{\widetilde{u}}^{k,\nu}}\big{\\}}_{k,\nu}$ studied in Appendix
2.
According to the described jumps (3.29) and (8.8) of the differentials
$\phi_{u^{k}}$ and $\phi_{{\widetilde{u}}^{r}}$, we get

$\begin{split}&\phi_{\mathbf{q},\xi^{k}}(P+a_{j})-\phi_{\mathbf{q},\xi^{k}}(P)={2{\rm
i}\pi}\mathbf{q}_{jk}d\lambda(P);\\\
&\phi_{\mathbf{q},\xi^{k}}(P+b_{j})-\phi_{\mathbf{q},\xi^{k}}(P)=-{2{\rm
i}\pi}\delta_{jk}d\lambda(P).\end{split}$ (5.18)

Moreover, using (8.9) and taking into account the fact that the cycles
$a_{k},b_{k}$ start at a marked point $P_{0}$ such that $\lambda(P_{0})=0$, we
deduce that the multivalued differential $\phi_{\mathbf{q},\xi^{k}}$ (5.16)
satisfies the normalization property (5.15). Note that, the first equality in
(5.16) and Definition 5.1 tell us that $\phi_{\mathbf{q},\xi^{k}}$ is a quasi-
homogeneous $\mathbf{q}$-differential of degree 1. In addition,
$\phi_{\mathbf{q},\xi^{k}}$ turns into the multivalued differential
$\phi_{u^{k}}$ defined in (3.23) when $\mathbf{q}=0$.

###### Proposition 5.2

Assume that $\phi_{\mathbf{q},t^{A}}$ is among the list of
$\mathbf{q}$-differentials

$\Big{\\{}\phi_{\mathbf{q},t^{i,\epsilon}},\phi_{\mathbf{q},v^{i}},\phi_{\mathbf{q},s^{i}},\phi_{\mathbf{q},\rho^{k}},\phi_{\mathbf{q},\xi^{k}}\Big{\\}}$
(5.19)

defined by (5.14) and (5.16).
Then the $\mathbf{q}$-differential $\phi_{\mathbf{q},t^{A}}$ is primary with
respect to vector field $e=\sum_{j=1}^{N}\partial_{\lambda_{j}}$, where by
this we mean that $e$ is covariantly constant with respect to the Levi-Civita
connection of the metric $\eta\big{(}\phi_{{\mathbf{q}},t^{A}}\big{)}$ defined
by (5.8).

_Proof:_ Let $\nabla$ be the Levi-Civita connection of the metric
$\eta\big{(}\phi_{{\mathbf{q}},t^{A}}\big{)}$. Analogously to (4.7), we can
see that the flatness property $\nabla{e}=0$ of the vector field $e$ can be
characterized by the N conditions

$e.\phi_{\mathbf{q},t^{A}}(P_{j})=0,\quad\quad\forall\ j=1,\dots,N.$

Suppose first that
$t^{A}\in\big{\\{}t^{i,\epsilon},v^{i},s^{i},\rho^{k}\big{\\}}$. We have

$\displaystyle e.\phi_{\mathbf{q},t^{A}}(P_{j})$
$\displaystyle=e.\phi_{t^{A}}(P_{j})-{2{\rm
i}\pi}\Big{(}e.\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}\Big{)}t^{A}(\omega_{\beta})\omega_{\alpha}(P_{j})$
$\displaystyle\quad-{2{\rm
i}\pi}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}\Big{(}e.\big{[}t^{A}(\omega_{\beta})\omega_{\alpha}(P_{j})\big{]}\Big{)}$
$\displaystyle=0,$

where we used expression (5.14) for $\phi_{\mathbf{q},t^{A}}$ in the first
equality and in the second equality, we used the following arguments:
1) relations $e.\mathbb{B}=0=e.t^{A}(\omega_{k})$: the first one was observed
in (5.1) and the second follows from (3.37) and (3.38);
2) the differentials $\phi_{t^{i,\epsilon}}$, $\phi_{v^{i}}$, $\phi_{s^{i}}$
and $\phi_{\rho^{k}}=\omega_{k}$ are $e$-primary (by Corollary 4.1) and thus
$e.\phi_{t^{A}}(P_{j})=0$.
Let us now consider the case $t^{A}=\xi^{k}$. We first observe that formula
(3.8) applied to the differential $\omega=\phi_{{\widetilde{u}}^{r}}$ given by
(5.17) and relation (8.10) imply respectively the following:

$e.\phi_{{\widetilde{u}}^{\sigma}}(P_{j})={2{\rm
i}\pi}\omega_{\sigma}(P_{j})\quad\text{and}\quad
e.{\widetilde{u}}^{\sigma}(\omega_{\beta})=\mathbb{B}_{\beta\sigma}.$

These relations, formula (5.16) and the arguments used in the previous case
enable us to arrive at the following:

$\displaystyle e.\phi_{\mathbf{q},\xi^{k}}(P_{j})$
$\displaystyle=e.\phi_{u^{k}}(P_{j})-{2{\rm
i}\pi}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}\big{(}e.u^{k}(\omega_{\beta})\big{)}\omega_{\alpha}(P_{j})$
$\displaystyle\quad+\mathbf{q}_{k\sigma}\bigg{(}e.\phi_{{\widetilde{u}}^{\sigma}}(P_{j})-{2{\rm
i}\pi}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}\big{(}e.{\widetilde{u}}^{\sigma}(\omega_{\beta})\big{)}\omega_{\alpha}(P_{j})\bigg{)}$
$\displaystyle=-{2{\rm
i}\pi}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{k\alpha}\omega_{\alpha}(P_{j})+{2{\rm
i}\pi}\mathbf{q}_{k\sigma}\bigg{(}\omega_{\sigma}(P_{j})-\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}\mathbb{B}_{\beta\sigma}\omega_{\alpha}(P_{j})\bigg{)}$
$\displaystyle=0.$

$\square$

$\bullet$

Flat coordinates of the flat metric $\eta\big{(}\omega_{\mathbf{q}}\big{)}$

Similarly to the link between (3.13) and (3.14), we consider the list of $N$
operations $\big{\\{}t^{i,\epsilon},v^{i},s^{i},\rho^{k},\xi^{k}\big{\\}}$
corresponding to the primary $\mathbf{q}$-differentials (5.14) and (5.16)
where $t^{i,\epsilon},v^{i},s^{i},\rho^{k}$ are the same as in (3.21) and
$\xi^{k}$ is the following operation:

$\xi^{k}(\omega):=\oint_{a_{k}}\lambda(P)\omega(P)+\mathbf{q}_{k\alpha}\oint_{b_{\alpha}}\lambda(P)\omega(P)=u^{k}(\omega)+\mathbf{q}_{k\alpha}\widetilde{u}^{\alpha}(\omega).$
(5.20)

Here $\omega$ is a differential as in (3.1) or a $\mathbf{q}$-differential as
in (5.7) and $u^{k}$ and $\widetilde{u}^{\alpha}$ are respectively the
operations defined in (3.22) and (5.17).

i) Assume that
$t^{A},t^{B}\in\big{\\{}t^{i,\epsilon},v^{i},s^{i},\rho^{k}\big{\\}}$. Then,
due to (5.14) and the symmetry property (3.36), we have

$\begin{split}t^{A}\big{(}\phi_{\mathbf{q},t^{B}}\big{)}&=t^{A}\big{(}\phi_{t^{B}}\big{)}-{2{\rm
i}\pi}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}t^{A}(\omega_{\alpha})t^{B}(\omega_{\beta})\\\
&=t^{B}\big{(}\phi_{\mathbf{q},t^{A}}\big{)}+\log(-1)\sum_{i,j=1}^{m}(1-\delta_{ij})\delta_{t^{A},s^{i}}\delta_{t^{B},s^{j}}.\end{split}$
(5.21)

ii) When $t^{A}\in\big{\\{}t^{i,\epsilon},v^{i},s^{i},\rho^{k}\big{\\}}$ and
$t^{B}=\xi^{k}$, then we have

$\begin{split}t^{A}\big{(}\phi_{\mathbf{q},\xi^{k}}\big{)}&=t^{A}\big{(}\phi_{u^{k}}\big{)}-{2{\rm
i}\pi}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}t^{A}(\omega_{\alpha})u^{k}(\omega_{\beta})\\\
&\quad+\mathbf{q}_{k\sigma}\bigg{(}t^{A}\big{(}\phi_{\widetilde{u}^{\sigma}}\big{)}-{2{\rm
i}\pi}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}t^{A}(\omega_{\alpha}){\widetilde{u}}^{\sigma}(\omega_{\beta})\bigg{)}\\\
&=\xi^{k}\big{(}\phi_{\mathbf{q},t^{A}}\big{)},\end{split}$ (5.22)

where we have used expression (5.16) for the multivalued differential
$\phi_{\mathbf{q},\xi^{k}}$ in the first equality and the symmetry properties

$t^{A}\big{(}\phi_{u^{r}}\big{)}=u^{r}\big{(}\phi_{t^{A}}\big{)}\quad\quad\text{and}\quad\quad
t^{A}\big{(}\phi_{\widetilde{u}^{r}}\big{)}=\widetilde{u}^{r}\big{(}\phi_{t^{A}}\big{)},\quad\quad\forall\
t^{A}=t^{i,\epsilon},v^{i},s^{i},\rho^{k}$

in the second equality. Here, as in (3.30) and (3.36), we have taken into
account the fact that the cycles $a_{k},b_{k}$ start at a marked point
$P_{0}\in C_{g}$ so that $\lambda(P_{0})=0$.

iii) If $t^{A}=\xi^{k}$ and $t^{B}=\xi^{l},$ with $k,l=1,\dots,g$, then by
assuming the sum over the indices $\alpha,\beta,\mu,\nu$, we have

$\begin{split}\xi^{k}\big{(}\phi_{\mathbf{q},\xi^{l}}\big{)}&=u^{k}\big{(}\phi_{u^{l}}\big{)}-{2{\rm
i}\pi}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}u^{k}(\omega_{\alpha})u^{l}(\omega_{\beta})+\mathbf{q}_{k\mu}\widetilde{u}^{\mu}\big{(}\phi_{u^{l}}\big{)}+\mathbf{q}_{l\nu}u^{k}\big{(}\phi_{\widetilde{u}^{\nu}}\big{)}\\\
&\quad-{2{\rm
i}\pi}\mathbf{q}_{k\mu}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}\widetilde{u}^{\mu}(\omega_{\alpha})u^{l}(\omega_{\beta})-{2{\rm
i}\pi}\mathbf{q}_{l\nu}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}u^{k}(\omega_{\alpha})\widetilde{u}^{\nu}(\omega_{\beta})\\\
&\quad+\mathbf{q}_{k\mu}\mathbf{q}_{l\nu}\bigg{(}\widetilde{u}^{\mu}\big{(}\phi_{\widetilde{u}^{\nu}}\big{)}-{2{\rm
i}\pi}\Big{(}\big{(}I+\mathbf{q}\mathbb{B}\big{)}^{-1}\mathbf{q}\Big{)}_{\alpha\beta}\widetilde{u}^{\mu}(\omega_{\alpha})\widetilde{u}^{\nu}(\omega_{\beta})\bigg{)}\\\
&=\xi^{l}\big{(}\phi_{\mathbf{q},\xi^{k}}\big{)}.\end{split}$ (5.23)

Therefore, equalities (5.21), (5.22) and (5.23) yield that

$t^{A}\big{(}\phi_{\mathbf{q},t^{B}}\big{)}=t^{B}\big{(}\phi_{\mathbf{q},t^{A}}\big{)}+\log(-1)\sum_{i,j=1}^{m}(1-\delta_{ij})\delta_{t^{A},s^{i}}\delta_{t^{B},s^{j}},\quad\quad\forall\
t^{A},t^{B}\in\big{\\{}t^{i,\epsilon},v^{i},s^{i},\rho^{k},\xi^{k}\big{\\}}$

provided that the cycles of the canonical homology basis $\\{a_{k},b_{k}\\}$
start at a marked point $P_{0}$ with $\lambda(P_{0})=0$. Furthermore, using
the same relations we see that the functions
$\big{\\{}t^{A}\big{(}\phi_{\mathbf{q},t^{B}}\big{)}\big{\\}}$ satisfy the
following limiting behavior as the matrix $\mathbf{q}$ is the zero matrix:

$t^{A}\big{(}\phi_{\mathbf{q},t^{B}}\big{)}\Big{|}_{\mathbf{q}=0}=\left\\{\begin{array}[]{lll}t^{A}\big{(}\phi_{t^{B}}\big{)},&\hbox{if}\quad
t^{A},t^{B}\in\big{\\{}t^{i,\epsilon},v^{i},s^{i},\rho^{k}\big{\\}};\\\ \\\
t^{A}\big{(}\phi_{u^{k}}\big{)},&\hbox{if}\quad
t^{A}\in\big{\\{}t^{i,\epsilon},v^{i},s^{i},\rho^{k}\big{\\}}\quad\text{and}\quad
t^{B}=\xi^{k};\\\ \\\ u^{k}\big{(}\phi_{u^{l}}\big{)},&\hbox{if}\quad
t^{A}=\xi^{k}\quad\text{and}\quad t^{B}=\xi^{l}.\end{array}\right.$ (5.24)

###### Proposition 5.3

Let $\omega_{\mathbf{q}}$ be the $\mathbf{q}$-differential defined by (5.7)
and $\eta(\omega_{\mathbf{q}}\big{)}$ be the corresponding Darboux-Egoroff
metric (5.8) defined on the open set $\mathcal{M}_{\omega_{\mathbf{q}}}$
determined by conditions (5.9).
Then the following set

$\mathcal{S}(\omega_{\mathbf{q}}):=\Big{\\{}t^{i,\epsilon}\big{(}\omega_{\mathbf{q}}\big{)},v^{i}\big{(}\omega_{\mathbf{q}}\big{)},s^{i}\big{(}\omega_{\mathbf{q}}\big{)},\rho^{k}\big{(}\omega_{\mathbf{q}}\big{)},\xi^{k}\big{(}\omega_{\mathbf{q}}\big{)}\Big{\\}},$
(5.25)

where $t^{i,\epsilon},v^{i},s^{i},\rho^{k}$ are the operations defined in
(3.21) and $\xi^{k}$ is given by (5.20), provides a system of flat coordinates
of the metric $\eta(\omega_{\mathbf{q}}\big{)}$.
In coordinates (5.25), the nonzero entries of the constant matrix of the
contravariant metric $\eta^{*}\big{(}\omega_{\mathbf{q}}\big{)}$ are as
follows:

$\begin{split}&\eta^{*}\big{(}\omega_{\mathbf{q}}\big{)}(dt^{i,\epsilon_{1}}\big{(}\omega_{\mathbf{q}}\big{)},dt^{j,\epsilon_{2}}\big{(}\omega_{\mathbf{q}}\big{)})=\delta_{ij}\delta_{\epsilon_{1}+\epsilon_{2},n_{j}+1};\\\
&\eta^{*}\big{(}\omega_{\mathbf{q}}\big{)}(dv^{i}\big{(}\omega_{\mathbf{q}}\big{)},ds^{j}\big{(}\omega_{\mathbf{q}}\big{)})=\delta_{ij};\\\
&\eta^{*}\big{(}\omega_{\mathbf{q}}\big{)}(d\rho^{k}\big{(}\omega_{\mathbf{q}}\big{)},d\xi^{l}\big{(}\omega_{\mathbf{q}}\big{)})=\delta_{kl}.\end{split}$
(5.26)

_Proof:_ Due to Rauch variational formula (5.5), we have

$\displaystyle\eta^{*}\big{(}\omega_{\mathbf{q}}\big{)}\big{(}dt^{A}\big{(}\omega_{\mathbf{q}}\big{)},dt^{B}\big{(}\omega_{\mathbf{q}}\big{)}\big{)}$
$\displaystyle=2\sum_{j=1}^{N}\frac{\partial_{\lambda_{j}}t^{A}\big{(}\omega_{\mathbf{q}}\big{)}\partial_{\lambda_{j}}t^{B}\big{(}\omega_{\mathbf{q}}\big{)}}{\big{(}\omega_{\mathbf{q}}(P_{j})\big{)}^{2}}=\frac{1}{2}\sum_{j=1}^{N}\phi_{\mathbf{q},t^{A}}(P_{j})\phi_{\mathbf{q},t^{B}}(P_{j})=e.t^{A}\big{(}\phi_{\mathbf{q},t^{B}}\big{)}.$

We consider two cases.
_First case:_
$t^{A},t^{B}\in\big{\\{}t^{i,\epsilon},v^{i},s^{i},\rho^{k}\big{\\}}$. Using
(5.21) and properties (3.37), (3.38) and (3.40) we deduce that

$\displaystyle
e.t^{A}\big{(}\phi_{\mathbf{q},t^{B}}\big{)}=e.t^{A}\big{(}\phi_{t^{B}}\big{)}=\delta_{t^{B},t^{A^{\prime}}},$

with $t^{A^{\prime}}$ being the dual flat coordinate of $t^{A}$ in the sense
of (3.40) and (3.41).
_Second case:_
$t^{A}\in\big{\\{}t^{i,\epsilon},v^{i},s^{i},\rho^{k},\xi^{k}\big{\\}}$ and
$t^{B}=\xi^{l}$. We proceed as in the proof of Theorem 3.4. Consider the
multivalued differential

$\Phi_{\mathbf{q},A,l}(P):=\frac{\phi_{\mathbf{q},t^{A}}(P)\phi_{\mathbf{q},\xi^{l}}(P)}{d\lambda(P)}.$

Relation (5.18) implies that the jumps of $\Phi_{\mathbf{q},A,l}$ are as
follows:

$\displaystyle\Phi_{\mathbf{q},A,l}(P+a_{j})-\Phi_{\mathbf{q},A,l}(P)$
$\displaystyle={2{\rm
i}\pi}\delta_{t^{A},\xi^{k}}\Big{(}\mathbf{q}_{lj}\phi_{\mathbf{q},\xi^{k}}(P)+\mathbf{q}_{kj}\phi_{\mathbf{q},\xi^{l}}(P)+{2{\rm
i}\pi}\mathbf{q}_{kj}\mathbf{q}_{lj}d\lambda(P)\Big{)}$
$\displaystyle\quad+{2{\rm