&\dot{f}_{-}^{\mu}=-\frac{1}{\pi}\iint_{\mathbb{H}}\overline{\mu(z^{\prime})}\,R(\overline{z^{\prime}},z)\differential[2]{z^{\prime}},\end{split}\right.$
(91)

where

$R(z^{\prime},z)\overset{\text{\tiny\it
def.}}{\equiv}\frac{1}{z^{\prime}-z}+\frac{z-1}{z^{\prime}}-\frac{z}{z^{\prime}-1}=\frac{z(z-1)}{(z^{\prime}-z)z^{\prime}(z^{\prime}-1)}.$
(92)

In turn, the function $F^{\mu}(w)$ is holomorphic with respect to
$\varepsilon$ for each particular $w\in\mathbb{C}$, and

$\dot{F}^{\mu}(w)=-\frac{1}{\pi}\iint_{\mathbb{C}}M(w^{\prime})\,R(w^{\prime},w)\differential[2]{w^{\prime}},$
(93)

where $\dot{F}^{\mu}$ is understood to be given by $(\partial
F^{\varepsilon\mu}/\partial\varepsilon)|_{\varepsilon=0}$, and $M$ was defined
in Eq.(89).313131According to (88), $\partial_{\bar{w}}\dot{F}=M$. The Green-
function equation and solution for this equation are given by
$\partial_{\bar{w}}R(w^{\prime},w)=-\pi\delta(w^{\prime},w)$ and (93),
respectively. Therefore, the kernel $R(w^{\prime},w)$, roughly speaking,
inverts the action of $\bar{\partial}$-operator on Beltrami differentials on
$\hat{\mathbb{C}}$. The precise statement (see (Takhtajan:2001uj, , Lemma 5)
and (Ahlfors_quasiconformal_06, , Section V.C)) is essentially a version of
the Pompeiu formula. Proof of these assertions can be found in
Ahlfors_quasiconformal_06 ; Ahlfors1960RIEMANNSMT ; Bers_1960 .

Moreover, let

$R_{i}(w)=-\frac{1}{\pi}R(w,w_{i})=-\frac{w_{i}(w_{i}-1)}{\pi(w-w_{i})w(w-1)},\qquad\qquad
i=1,\dots,n-3;$ (94)

where $R_{i}$s are linearly independent and generate the space
$\mathcal{H}^{2,0}(O)$. Denote by $Q_{i}$, $i=1,\dots,n-3$, the basis in
$\mathcal{H}^{2,0}(O)$ biorthogonal to $R_{i}$ in the sense of the Petersson
inner product323232See (319).–i.e. $\langle R_{i},Q_{j}\rangle=\delta_{ij}$
where $\delta_{ij}$ is the Kronecker delta. The desired basis in
$\mathcal{H}^{-1,1}(\mathbb{H},\Gamma)\cong
T_{[0]}\mathcal{T}_{0,n}(\boldsymbol{m})$ has the form

$\mu_{i}(z)=\rho(z)^{-1}\overline{q_{i}(z)},$ (95)

where $q_{i}(z)=Q_{i}\circ J\,J^{\prime 2}$ for $i=1,\dots,n-3$ form a basis
for $\mathcal{H}^{2,0}(\mathbb{H},\Gamma)\cong
T_{[0]}^{\ast}\mathcal{T}_{0,n}(\boldsymbol{m})$ and are biorthogonal to
$\mathcal{H}^{2,0}(\mathbb{H},\Gamma)\ni r_{i}(z)=R_{i}\circ J\,J^{\prime 2}$
with respect to the Petersson inner product.333333In the space
$\mathcal{H}{{}^{2,0}}(\mathbb{H},\Gamma)$, it is defined as $\langle
q_{1},q_{2}\rangle_{\text{WP}}=\iint_{\mathcal{F}}q_{1}(z)\,\overline{q_{2}(z)}\,\rho^{-1}(z)\differential[2]{z}.$
The basis
$q^{(\tilde{\mu})}_{i}(z)\in\mathcal{H}^{2,0}(\mathbb{H},\Gamma^{\tilde{\mu}})$
and
$\mu_{i}^{(\tilde{\mu})}\in\mathcal{H}^{-1,1}(\mathbb{H},\Gamma^{\tilde{\mu}})$
for $[\tilde{\mu}]\in\mathcal{T}_{0,n}(\boldsymbol{m})$ can also be defined in
a similar way. Then, the following lemma connects the motion of punctures and
conical singularities on $\hat{\mathbb{C}}$ with the geometry of Teichmüller
space:

######  Lemma 3.3.

For The mapping $\Psi:\mathcal{T}_{0,n}(\boldsymbol{m})\to\mathcal{M}_{0,n}$
is a complex-analytic covering, and we have

$\differential{\Psi_{[\tilde{\mu}]}}(\mu_{i}^{(\tilde{\mu})})=\partialderivative{w_{i}}\quad\text{and}\quad\Psi_{[\tilde{\mu}]}^{\ast}(\differential{w_{i}})=\mathrm{r}_{i}^{(\tilde{\mu})},\qquad
i=1,\dots,n-3.$ (96)

###### Proof.

Let us start with proving the statement
$\differential{\Psi_{[\tilde{\mu}]}}(\mu_{i}^{(\tilde{\mu})})=\partialderivative{w_{i}}$
which is actually repeating the proof of Lemma 3 in Zograf1988ONLE ; Remark
3.1 implies that it is sufficient to verify this statement at the point $[0]$.
In a neighborhood of $[0]\in\mathcal{T}_{0,n}(\boldsymbol{m})$ the Bers’
coefficients $(t_{1},\dots,t_{n-3})\in\mathbb{C}^{n-3}$ are determined in the
basis $\mu_{1},\dots,\mu_{n-3}\in\mathcal{H}^{-1,1}(\mathbb{H},\Gamma)$ from
the expansion

$\mu=\sum_{i=1}^{n-3}t_{i}\mu_{i},$ (97)

where $\mu\in\mathcal{H}^{-1,1}(\mathbb{H},\Gamma)\cap\mathfrak{D}(\Gamma)$.
In these coordinates, the mapping $\Psi$ is given by

$(t_{1},\dots,t_{n-3})\overset{\Psi}{\longmapsto}\left(F^{\mu}(w_{1}),\dots,F^{\mu}(w_{n-3})\right),$
(98)

where $(w_{1},\dots,w_{n-3})=\Psi([0])\in\mathcal{M}_{0,n}$. Since $F^{\mu}$
depends complex-analytically on $t_{1},\dots,t_{n-3}$, the mapping $\Psi$ is
also complex-analytic. We now compute its differential $\differential{\Psi}$
at the point $[0]\in\mathcal{T}_{0,n}(\boldsymbol{m})$: It follows from the
definitions of $\mu_{i}$ and $M_{i}$, equations (95) and (89), as well as
Eq.(80) that

$M_{i}=\mu_{i}\circ
J^{-1}\frac{\overline{(J^{-1})^{\prime}}}{(J^{-1})^{\prime}}=\left(\imaginary
J^{-1}\right)^{2}\overline{q_{i}\circ
J^{-1}}\frac{\overline{(J^{-1})^{\prime}}}{(J^{-1})^{\prime}}=e^{-\varphi}\overline{Q_{i}},$
(99)

where $Q_{i}=q_{i}\circ J^{-1}(J^{-1})^{\prime 2}$. Then, using the above
equation, the definition of $R_{i}$s in (94), and Eq.(93), we get that

$\begin{split}\dot{F}^{\mu_{i}}(w_{j})&=\iint_{\mathbb{C}}M_{i}(w)\,R_{j}(w)\differential[2]{w}\overset{\eqref{MiQi}}{=}\iint_{\mathbb{C}}e^{-\varphi}\overline{Q_{i}}\,R_{j}(w)\differential[2]{w}\overset{\eqref{Qinnerprod}}{=}\langle
R_{j},Q_{i}\rangle=\delta_{ij}.\end{split}$ (100)

This precisely means that in these coordinates the differential
$\differential{\Psi}\Big{|}_{[0]\in\mathcal{T}(\Gamma)}$ is given by the
identity matrix – i.e.
$\differential{\Psi}_{[0]}(\mu_{i})=\partialderivative{w_{i}}$. The second
statement,
$\Psi_{[\tilde{\mu}]}^{\ast}(\differential{w_{i}})=\mathrm{r}_{i}^{(\tilde{\mu})}$
is effectively the inverse of the first statement. Accordingly, it follows
from Eq.(96) that the mapping $\Psi$ is a local diffeomorphism. We now show
that $\Psi$ is a covering. For $\eta\in\operatorname{Aut}_{\ast}(\Gamma)$,
denote by $z_{i}^{\eta}$, $(i=1,\dots,n_{e})$ the fixed-points of elliptic
element $\eta(\tau_{i})\in\Gamma$ of order $m_{i}$ and by $z_{n_{e}+j}^{\eta}$
the fixed-points of the parabolic elements $\eta(\kappa_{j})\in\Gamma$ for
$j=1,\dots,n_{p}$. Let
$\mathsf{s}_{\eta}(w_{1},\dots,w_{n})\overset{\text{\tiny\it
def.}}{\equiv}\left(J(z_{1}^{\eta}),\dots,J(z_{n}^{\eta})\right)$; the
correspondence $\eta\mapsto\mathsf{s}_{\eta}$ determines an epimorphism (i.e.
a surjective group homomorphism) of the group $\operatorname{MCG}(\Gamma)$
onto the product symmetric group
$\operatorname{Symm}\left(\boldsymbol{\mathsf{s}}\right)$.343434As a concrete
example, consider the case of $n$-punctured sphere. In that case, the mapping
class group $\operatorname{MCG}_{0,n}$ is given by the braid group $B_{n}$
quotiented by its center farb2011primer . The kernel of this epimorphism is
$\operatorname{Aut}(\Psi)$ Zograf1988ONLE . The mapping $\Psi$ is invariant
under $\operatorname{Aut}(\Psi)$ and it is clear that
$\mathcal{M}(\Gamma)=\mathcal{T}(\Gamma)/\operatorname{Aut}(\Psi)$. Hence,
$\Psi$ is a covering with automorphism group $\operatorname{Aut}(\Psi)$. ∎

######  Remark 3.5.

Each elliptic and cuspidal metric,
$\langle\cdot,\cdot\rangle^{\text{ell}}_{\text{TZ},i}$ for $i=1,\dots,n_{e}$
and $\langle\cdot,\cdot\rangle^{\text{cusp}}_{\text{TZ},n_{e}+j}$ for
$j=1,\dots,n_{p}$, on $\mathcal{T}_{0,n}(\boldsymbol{m})$ is invariant with
respect to the automorphism group of the covering
$\Psi:\mathcal{T}_{0,n}(\boldsymbol{m})\to\mathcal{M}_{0,n}$, i.e. the pure
mapping classes $\operatorname{PMCG}$, and determines a Kähler metric on
$\mathcal{M}_{0,n}$.

######  Remark 3.6.

We can also rewrite the expression for Energy-Momentum Tensor (82) using
$R_{i}(w)$,

$\text{Sch}\left(J^{-1};w\right)=\sum_{i=1}^{n}h_{i}\varE_{i}(w)-\pi\sum_{i=1}^{n-3}c_{i}R_{i}(w),$
(101)

where

$\left\\{\begin{split}&\varE_{i}(w)=\frac{1}{2(w-w_{i})^{2}}-\frac{1}{2w(w-1)},\quad&i=1,\dots,n-1,\\\
&\varE_{i}(w)=\frac{1}{2w(w-1)},\quad&i=n,\end{split}\right.$ (102)

and $h_{i}=1$ for $i=n_{e}+1,\dots,n$. The corresponding functions
$e_{i}(z)=\varE_{i}\circ J\,J^{\prime 2}$ on $\mathbb{H}$ are automorphic
forms of weight 4 for $\Gamma$ with the non-zero constant term at the cusp or
conical singularities. Let us obtain the equation (101) for the simplest case
with one ramification point of order $m$ at $w_{1}$ and three punctures at
$w_{2}=0,w_{3}=1,$ and $w_{4}\rightarrow\infty$, respectively. Solving (85) in
favor of $c_{3}$ and $c_{2}$ gives

$c_{3}=-(c_{1}+c_{2}),\hskip
14.22636ptc_{2}=-\frac{1+h_{1}+2c_{1}(w_{1}-w_{3})}{2(w_{2}-w_{3})}.$

By substituting the above relations in (82) and noting that $w_{2}=0,w_{3}=1$,
one obtains

$\displaystyle
T_{\varphi}=h_{1}\left(\frac{1}{2(w-w_{1})^{2}}-\frac{1}{2w(w-1)}\right)+\frac{1}{2w^{2}}+\frac{1}{2(w-1)^{2}}-\frac{1}{2w(w-1)}$
$\displaystyle\hskip
28.45274pt+c_{1}\left(\frac{1}{w-w_{1}}+\frac{w_{1}-1}{w}-\frac{w_{1}}{w-1}\right),$

which by using (102) and (94) is equal to (101). The desired general case can
be obtained in the same way.

######  Corollary 3.3.1.

The following statements are true:

  1. (i)

$\dot{F}^{i}(0)=\dot{F}^{i}(1)=0$ and
$\dot{F}^{i}(w)=\text{\scriptsize{$\mathcal{O}$}}\left(|w|^{2}\right)\quad\text{as}\quad
w\to\infty$.

  2. (ii)

$\partial_{\bar{w}}\dot{F}^{i}=M_{i}=e^{-\varphi}\overline{Q_{i}}$.

  3. (iii)

The functions $\dot{F}^{i}(w)$ has the following asymptotics

$\dot{F}^{i}(w)=\left\\{\begin{split}&\delta_{ij}+(w-w_{j})\partial_{w}\dot{F}^{i}(w_{j})+\text{\scriptsize{$\mathcal{O}$}}\left(\frac{|w-w_{j}|}{\log|w-w_{j}|}\right)\hskip
2.84544pt(j=1,\dots,n_{e})\hskip 2.84544pt\text{as}\hskip 5.69046ptw\to
w_{j},\\\
&\delta_{ij}+(w-w_{j})\partial_{w}\dot{F}^{i}(w_{j})+\text{\scriptsize{$\mathcal{O}$}}\left(\frac{|w-w_{j}|}{\log|w-w_{j}|}\right)\hskip
2.84544pt(j=n_{e}+1,\dots,n-1)\hskip 2.84544pt\text{as}\hskip 5.69046ptw\to
w_{j},\\\
&w\partial_{w}\dot{F}^{i}(\infty)+\text{\scriptsize{$\mathcal{O}$}}\left(\frac{|w|}{\log|w|}\right)\hskip
56.9055pt\hskip 2.84544pt\text{as}\hskip
5.69046ptw\to\infty.\end{split}\right.$

###### Proof.

The part _(i)_ can be proved by using the equation (93) and noting to the
expression for $R(w,w_{i})$ in (94). The part _(ii)_ can also be proved by
using (93) and noting to (99). The general idea of proof the part _(iii)_ can
also be found in (park2015potentials, , Remark 3). ∎

In section 4, we will need the derivatives of $\varphi$ with respect to the
variables $w_{1},\dots,w_{n-3}$ and the following lemma will enable us to give
a geometric description of them:

######  Lemma 3.4.

The Liouville filed $\varphi$ is a continuously differentiable function on
$\mathcal{M}_{0,n}$, and

$\partial_{w_{i}}\varphi+\dot{F}^{i}\partial_{w}\varphi+\partial_{w}\dot{F}^{i}=0,\quad\text{for}\quad
i=1,\dots,n-3,$ (103)

###### Proof.

It can be proved exactly in the same way as Lemma 4 in Zograf1988ONLE . Let
$\Gamma$ be a Fuchsian group of the first kind that uniformizes the orbifold
Riemann surface $O=(\hat{\mathbb{C}},\mathscr{D})$ and let $f^{\mu}(z)$ be the
unique solution of the Beltrami equation (34) with
$\mathcal{H}^{-1,1}(\mathbb{H},\Gamma)\ni\mu=t_{1}\,\mu_{1}+\dotsm+t_{n-3}\,\mu_{n-3}$
that fixes the points $z_{n-2}=0$, $z_{n-1}=1$, and $z_{n}=\infty$. The
function $F^{\mu}(w)=(J_{\mu}\circ f^{\mu}\circ J^{-1})(w)$ is differentiable
with respect to $w$ on $O=(\hat{\mathbb{C}},\mathscr{D})$ and depends
analytically on the Bers’ coordinates $t_{1},\dots,t_{n-3}$. It follows from
the commutative diagram (87) that $J_{\mu}$ and $J^{\prime}_{\mu}$ are
continuously differentiable with respect to the Bers’ coordinates
$t_{1},\dots,t_{n-3}$ and that suitable branches of the functions
$J_{\mu}^{-1}$ and $(J_{\mu}^{-1})^{\prime}$ have this property locally
outside the set of singular points. The continuous differentiability of
$\varphi$ on $\mathcal{M}_{0,n}$ now follows from Eq.(80) and Lemma 3.3. As
for equation (103), it is a reformulation of lemma 3.1 due to Ahlfors
Ahlfors1961SomeRO  on the vanishing of the first variation of the area element
in the Poincaré metric on $O=(\hat{\mathbb{C}},\mathscr{D})$ under quasi-
conformal mappings corresponding to harmonic Beltrami differentials: for any
$\mu\in\mathcal{H}^{-1,1}(\mathbb{H},\Gamma)$

$\mathcal{L}_{\mu}\rho=\partialderivative{\varepsilon}\Bigg{|}_{\varepsilon=0}(f^{\varepsilon\mu})^{\ast}(\rho)=\partialderivative{\varepsilon}\frac{|\partial_{z}f^{\varepsilon\mu}|^{2}}{\left(\imaginary
f^{\varepsilon\mu}\right)^{2}}\Bigg{|}_{\varepsilon=0}=0.$ (104)

Let $e^{\varphi^{\mu}(w)}\,|\differential{w}|^{2}$ be the hyperbolic metric on
the Riemann orbisurface $O^{\mu}$ where
$\varphi^{\mu}(w)=\varphi\left(w;F^{\mu}(w_{1}),\dots,F^{\mu}(w_{n-3})\right)$.
It follows from the commutative diagram (87) that

$(f^{\mu})^{\ast}(\rho)=(f^{\mu})^{\ast}\underbrace{(J_{\mu})^{\ast}(e^{\varphi^{\mu}})}_{\rho};$

then, using Eq.(46) as well as $F^{\mu}\circ J=J_{\mu}\circ f^{\mu}$, one gets

$\frac{|\partial_{z}f^{\mu}|^{2}}{\left(\imaginary
f^{\mu}\right)^{2}}=\exp(\varphi^{\mu}\circ J_{\mu}\circ
f^{\mu})\left|\partial_{z}(J_{\mu}\circ
f^{\mu})\right|^{2}=\exp(\varphi^{\mu}\circ F^{\mu}\circ
J)\left|\partial_{w}F^{\mu}\circ J\right|^{2}|J^{\prime}|^{2}.$ (105)

Finally, it follows from (104) that

$\partialderivative{\varepsilon}\Bigg{|}_{\varepsilon=0}e^{\varphi^{\varepsilon\mu}\circ
F^{\varepsilon\mu}\circ J}\left|\partial_{w}F^{\varepsilon\mu}\circ
J\right|^{2}|J^{\prime}|^{2}=\partialderivative{\varphi^{\varepsilon\mu}}{\varepsilon}\Bigg{|}_{\varepsilon=0}+\partial_{w}\varphi\dot{F}^{\mu}+\partial_{w}\dot{F}^{\mu}=0,$
(106)

which by setting $\mu=\mu_{i}$ and recalling lemma 3.3 (i.e.
$d\Psi_{[0]}(\mu_{i})=\partial/\partial_{w_{i}}$) gives us our desired result
(103). ∎

######  Corollary 3.4.1.

According to Lemma C.1, the solution $\varphi$ to the Liouville equation (81)
with singularities has the following asymptotic expansion near the singular
points

$\partial_{w_{i}}\varphi(w)=\left\\{\begin{split}&-\delta_{ij}\partial_{w}\varphi(w)-\big{(}(w-w_{j})\partial_{w}\varphi(w)+1\big{)}\partial_{w}\dot{F}^{i}(w)+\text{\scriptsize{$\mathcal{O}$}}\left(1\right)\quad&\text{as}\quad&w\to
w_{j},\\\
&-\big{(}w\partial_{w}\varphi(w)+1\big{)}\partial_{w}\dot{F}^{i}(w)+\text{\scriptsize{$\mathcal{O}$}}\left(1\right)\quad&\text{as}\quad&w\to\infty,\end{split}\right.$

where $j=1,\dots,n-1$.

###### Proof.

It follows from (103) and from the asymptotics of $\dot{F}^{i}$ (see Corollary
3.3.1). ∎

######  Remark 3.7.

One can also use the results of Ahlfors and Wolpert mentioned in subsection
3.1.2, i.e. lemma 3.1 and Eq.(49), to calculate the first and second
variations of the family of hyperbolic metrics
$\exp(\varphi^{\varepsilon\mu}(w))|\differential{w}|^{2}$ on the orbifold
Riemann surfaces $O^{\varepsilon\mu}=F^{\varepsilon\mu}(O)$. To do so, let us
use the commutative diagram (87) once again to write

$(F^{\mu})^{\ast}(e^{\varphi^{\mu}})=(J^{-1})^{\ast}(f^{\mu})^{\ast}(\rho).$
(107)

Then, it is easy to show that the first and second variations of the family of
hyperbolic metrics on Riemann orbisurfaces $O^{\varepsilon\mu}$ are given by
the same formulas in subsection 3.1.2 but with $\rho$ replaced by
$e^{\varphi}$ and $f_{\mu\overline{\tilde{\mu}}}$ replaced by
$(J^{-1})^{\ast}(f_{\mu\overline{\tilde{\mu}}})=f_{\mu\overline{\tilde{\mu}}}\circ
J^{-1}$. Moreover, for any $\mathtt{a}\in\mathbb{R}$ we have

$(F^{\mu})^{\ast}(e^{\mathtt{a}\varphi^{\mu}})=\left((F^{\mu})^{\ast}(e^{\varphi^{\mu}})\right)^{\mathtt{a}}.$
(108)

Therefore, we get the following formulas for the first and second variations
of $\exp(\varphi^{\varepsilon\mu}(w))$

$\left\\{\begin{split}&\partialderivative{\varepsilon}\Bigg{|}_{\varepsilon=0}(F^{\varepsilon\mu})^{\ast}(e^{\mathtt{a}\varphi^{\varepsilon\mu}})=0,\\\
\\\
&\partialderivative{}{\varepsilon_{1}}{\bar{\varepsilon_{2}}}\Bigg{|}_{\varepsilon_{1}=\varepsilon_{2}=0}(F^{\varepsilon_{1}\mu+\varepsilon_{2}\tilde{\mu}})^{\ast}(e^{\mathtt{a}\varphi^{\varepsilon\mu}})=\frac{\mathtt{a}}{2}e^{\mathtt{a}\varphi}f_{\mu\overline{\tilde{\mu}}}\circ
J^{-1}.\end{split}\right.$ (109)

###  3.3 Schottky Space $\mathfrak{S}_{g,n}(\boldsymbol{m})$

Let us start this subsection by recalling with more detail how a compact
Riemann surface $X$ of genus $g\geq 2$ is uniformized by a Schottky group. We
begin with a few definitions. Schottky groups are an important class of
_Kleinian groups_ : discrete subgroups of Möbius group
$\operatorname{PSL}(2,\mathbb{C})$ that act properly discontinuous on some
domain (called region of discontinuity) of the Riemann sphere
$\hat{\mathbb{C}}$. A _Schottky group_ $\Sigma$ is a free finitely generated
strictly loxodromic Kleinian group Maskit_1967 . Its limit set $\Lambda$ is a
Cantor set353535For more details on geometry of limit sets see Ref. Seade_2015
. and the _region of discontinuity_ $\Omega=\hat{\mathbb{C}}\backslash\Lambda$
is connected. A Schottky group $\Sigma$ of rank $g$ with a relation-free
system of generators $L_{1},\dots,L_{g}\in\operatorname{PSL}(2,\mathbb{C})$ is
called _marked_. There is a notion of equivalence between two marked Schottky
groups: $(\Sigma;L_{1},\dots,L_{g})$ is equivalent to
$(\tilde{\Sigma};\tilde{L}_{1},\dots,\tilde{L}_{g})$ if there exists a Möbius
transformation $\varsigma\in\operatorname{PSL}(2,\mathbb{C})$ such that
$\tilde{L}_{i}=\varsigma L_{i}\varsigma^{-1}$ for all $i=1,\dots,g$. The set
of equivalence classes of marked Schottky groups of genus $g$ is called the
_Schottky space_ of genus $g$ and is denoted by $\mathfrak{S}_{g}$. Similar to
Fuchsian groups, Schottky groups can be used to construct surfaces since the
action of $\Sigma$ on $\Omega$ produces a _compact_ Riemann surface
$\Omega/\Sigma$. An important result is that for every marked Schottky group
$(\Sigma;L_{1},\dots,L_{g})$ there is a _fundamental domain_
$\mathcal{D}$363636However, this fundamental domain $\mathcal{D}$ is _not
uniquely determined_ by the choice of marking for the Schottky group $\Sigma$.
for $\Sigma$ in $\Omega$ which is a (connected) region in $\hat{\mathbb{C}}$
bounded by $2g$ disjoint Jordan curves
$C_{1},\dots,C_{g},C^{\prime}_{1},\dots,C^{\prime}_{g}$ with
$C^{\prime}_{i}=-L_{i}(C_{i})$, $i=1,\dots,g$. Here, $C_{i}$ and
$C^{\prime}_{i}$ are oriented as components of $\partial\mathcal{D}$, and the
minus sign means the reverse orientation. Each element $L_{i}$ can be
represented in the _normal form_

$\frac{L_{i}(w)-a_{i}}{L_{i}(w)-b_{i}}=\lambda_{i}\frac{w-a_{i}}{w-b_{i}},\qquad
w\in\hat{\mathbb{C}},$ (110)

where $a_{i}$ and $b_{i}$ are the respective _attracting_ and _repelling_
fixed points of the loxodromic element $L_{i}$ and $0<|\lambda_{i}|<1$ is the
corresponding multiplier. Given this normal form, one can explicitly construct
the fundamental domain $\mathcal{D}$ in the following way:373737For more
details, see (Gaberdiel:2010jf, , Appendix C). Let us define the Möbius
transformations

$\varsigma_{a_{i},b_{i}}(w)=\frac{b_{i}w+a_{i}}{w+1},$ (111)

satisfying $\varsigma_{a_{i},b_{i}}(0)=a_{i}$ and
$\varsigma_{a_{i},b_{i}}(\infty)=b_{i}$, so that the generators $L_{i}$ of the
marked Schottky group $(\Sigma;L_{1},\dots,L_{g})$ can be written as

$L_{i}=\varsigma_{a_{i},b_{i}}\hskip 2.84526pt\varsigma_{\lambda_{i}}\hskip
2.84526pt\varsigma_{a_{i},b_{i}}^{-1}\quad\text{for}\quad i=1,\dots,g.$ (112)

In the above equation, the Möbius transformation $\varsigma_{\lambda_{i}}$ is
defined by $\varsigma_{\lambda_{i}}(w)=\lambda_{i}w$. Then, a fundamental
domain for $(\Sigma;L_{1},\dots,L_{g})$ is given by

$\mathcal{D}\overset{\text{\tiny\it
def.}}{\equiv}\hat{\mathbb{C}}\Big{\backslash}\bigcup_{i=1}^{g}(\mathrm{D}_{i}\cup\mathrm{D}_{-i}),$
(113)

where

$\left\\{\begin{split}&\mathrm{D}_{i}=\left\\{w\in\mathbb{C}\,\Bigg{|}\,\frac{|w-a_{i}|}{|w-b_{i}|}<|\mathrm{R}_{i}|\right\\}=\varsigma_{a_{i},b_{i}}\hskip
2.84526pt\varsigma_{\mathrm{R}_{i}}(\mathrm{D}),\\\ \\\
&\mathrm{D}_{-i}=\left\\{w\in\mathbb{C}\,\Bigg{|}\,\frac{|w-b_{i}|}{|w-a_{i}|}<|\mathrm{R}_{-i}|\right\\}=\varsigma_{a_{i},b_{i}}\hskip
2.84526pt\varsigma_{inv}\hskip
2.84526pt\varsigma_{\mathrm{R}_{-i}}(\mathrm{D}),\end{split}\right.$ (114)

$\varsigma_{inv}$ is defined by $\varsigma_{inv}(w)=-1/w$, and $\mathrm{D}$ is
the unit disk,

$\mathrm{D}=\left\\{w\in\mathbb{C}\,\Big{|}\,|w|<1\right\\}.$ (115)

Here $\mathrm{R}_{i}$ and $\mathrm{R}_{-i}$ represent the radii of disks
$\mathrm{D}_{i}$ and $\mathrm{D}_{-i}$ respectively and satisfy383838Equation
(116) makes clear the fact that, as mentioned in footnote 36, the fundamental
domain $\mathcal{D}$ cannot be uniquely determined by a choice of marking for
the Schottky group $\Sigma$.

$\mathrm{R}_{i}\mathrm{R}_{-i}=\lambda_{i}\quad\text{for}\quad i=1,\dots,g.$
(116)

The boundary $\partial\mathcal{D}=\bigcup_{i}C_{i}\cup C^{\prime}_{i}$ has
components

$C_{i}=\varsigma_{a_{i},b_{i}}\hskip
2.84526pt\varsigma_{\mathrm{R}_{i}}(C)\quad\text{and}\quad
C^{\prime}_{i}=\varsigma_{a_{i},b_{i}}\hskip 2.84526pt\varsigma_{inv}\hskip
2.84526pt\varsigma_{\mathrm{R}_{-i}}(C),$ (117)

where $C=\partial\mathrm{D}$ is th unit circle. In the remainder of this
paper, we will always assume that a marked Schottky group is _normalized_ ,
that is, $a_{1}=0$, $b_{1}=\infty$, and $a_{2}=1$. In particular, this means
that $\infty\notin\mathcal{D}$.

Under the canonical holomorphic map $\Omega\to\Omega/\Sigma$, the boundary
curves of a standard fundamental domain described above are mapped onto smooth
non-intersecting simple closed curves $\mathsf{a}_{1},\dots,\mathsf{a}_{g}$ on
the Riemann surface. This motivates the following terminology introduced by
Bers Bers1961UniformizationBB : A complete set of _retrosections_ on a Riemann
surface of genus $g$ is a choice of $g$ smooth simple non-intersecting,
homologically independent, closed curves
$\mathsf{a}_{1},\dots,\mathsf{a}_{g}$. We therefore see that a marked Schottky
group, together with the choice of a standard fundamental domain, determines a
Riemann surface with a complete set of retrosections. A much more profound
statement is that _every_ compact Riemann surface can be obtained in this way,
which is the content of the classical Koebe’s retrosection theorem Kra_1972 :

######  Theorem 3.5 (Koebe’s retrosection theorem).

For every compact Riemann surface $X$ of genus $g$ with a complete set of
retrosections $(\mathsf{a}_{1},\dots,\mathsf{a}_{g})$, there exists a marked
Schottky group of genus $g$, $(\Sigma;L_{1},\dots,L_{g})$ , and a fundamental
domain $\mathcal{D}\subset\Omega$ for $\Sigma$ with $2g$ boundary curves
$C_{1},\dots,C_{g},C^{\prime}_{1},\dots,C^{\prime}_{g}$ such that
$X=\Omega/\Sigma$ and the map $\Omega\to X$ sends both $C_{i}$ and
$C^{\prime}_{i}$ to $\mathsf{a}_{i}$. Moreover, the marked Schottky group is
unique up to equivalence
$(\Sigma;L_{1},\dots,L_{g})\sim(\tilde{\Sigma};\tilde{L}_{1},\dots,\tilde{L}_{g})$
defined before as well as $L_{i}\mapsto L_{i}^{-1}$.

######  Remark 3.8.

The theorem 3.5 implies that given a compact Riemann surface $X$ uniformized
by the marked Schottky group $(\Sigma;L_{1},\dots,L_{g})$, we can take the
homology classes of $C_{1},\dots,C_{g}$ as the generators
$[\mathsf{a}_{1}],\dots,[\mathsf{a}_{g}]$ in the symplectic basis of the first
homology group $H_{1}(X,\mathbb{Z})$. However, determining a canonical basis
for $H_{1}(X,\mathbb{Z})$, i.e. a symplectic basis
$\left\\{[\mathsf{a}_{1}],\dots,[\mathsf{a}_{g}],[\mathsf{b}_{1}],\dots,[\mathsf{b}_{g}]\right\\}$
with intersection pairings given by
$\\#\left([\mathsf{a}_{i}],[\mathsf{a}_{j}]\right)=0=\\#\left([\mathsf{b}_{i}],[\mathsf{b}_{j}]\right)$
and $\\#\left([\mathsf{a}_{i}],[\mathsf{b}_{j}]\right)=\delta_{ij}$, depends
also on the choice of $\mathsf{b}$-cycles on the Riemann surface $X$.
Therefore, we can choose the elements
$[\mathsf{b}_{1}],\dots,[\mathsf{b}_{g}]$ in the canonical basis of
$H_{1}(X,\mathbb{Z})$ such that the projections of their representative curves
onto the marked Schottky group $\Sigma$ are precisely the marked generators
$L_{1},\dots,L_{g}$.393939The choice of a canonical basis for the first
homology group $H_{1}(X_{O},\mathbb{Z})$ is usually called a _Torelli marking_
for the compact Riemann surface $X_{O}$. The set of isomorphic Riemann
surfaces marked in the Torelli sense is called _Torelli space_
$\mathfrak{T}_{g}$. The Torelli space is a covering of the Schottky space, by
forgetting half of the Torelli marking, but also up to the outer automorphisms
of the Schottky group.

######  Remark 3.9.

It is clear that there is a one-to-one correspondence between the set of
normalized marked Schottky groups and the Schottky space
$\mathfrak{S}_{g}\subset\mathbb{C}^{3g-3}$, i.e.404040The space
$\mathfrak{S}_{g}$ is a finite covering of the moduli space $\mathcal{M}_{g}$
of compact Riemann surfaces.

$(\Sigma;L_{1},\dots,L_{g})\mapsto(a_{3},\dots,a_{g},b_{2},\dots,b_{g},\lambda_{1},\dots,\lambda_{g})\in\mathbb{C}^{3g-3}$
(118)

For our purposes in the following sections, it will be crucial to give another
(equivalent) definition for the Schottky space $\mathfrak{S}_{g}$. Let
$\mathcal{A}^{-1,1}(\Omega,\Sigma)$ be the complex Banach space of Beltrami
differentials for $\Sigma$ and, in analogy with the Teichmüller case, let us
define the _deformation space_ $\mathfrak{D}(\Sigma)$ to be the open ball of
radius $1$ (in the sense of $L^{\infty}$-norm) in
$\mathcal{A}^{-1,1}(\Omega,\Sigma)$:

$\mathfrak{D}(\Sigma)=\left\\{\mu\in\mathcal{A}^{-1,1}(\Omega,\Sigma)\,\big{|}\,\norm{\mu}_{\infty}<1\right\\}.$
(119)

A homeomorphism $F$ of a plane domain $\Omega$ onto another plane domain
$\tilde{\Omega}$ is said to be _quasi-conformal_ if it satisfies the Beltrami
equation at each point in $\Omega$. For each $\mu\in\mathfrak{D}(\Sigma)$, let
$F^{\mu}$ be the unique normalized (i.e. $F^{\mu}(0)=0$ and $F^{\mu}(1)=1$)
solution of the corresponding Beltrami equation on $\mathbb{C}$ that gives a
quasi-conformal homeomorphism of $\mathbb{C}$ onto itself. Then, the
restriction of $F^{\mu}$ to the region of discontinuity
$\Omega\subset\mathbb{C}$ gives the desired quasi-conformal homeomorphism
$F^{\mu}:\Omega\to\Omega^{\mu}$ and each element $\mu\in\mathfrak{D}(\Sigma)$
gives a faithful representation $\varrho_{\mu}$ of $\Sigma$ in
$\operatorname{PSL}(2,\mathbb{C})$ according to the formula $\sigma\mapsto
F^{\mu}\circ\sigma\circ(F^{\mu})^{-1}$, $\sigma\in\Sigma$. As mentioned
before, two representations $\varrho_{\mu_{1}}$ and $\varrho_{\mu_{2}}$ are
equivalent if they differ by an _inner automorphism_ of
$\operatorname{PSL}(2,\mathbb{C})$, i.e., if
$\varrho_{\mu_{2}}=\varsigma\varrho_{\mu_{1}}\varsigma^{-1}$,
$\varsigma\in\operatorname{PSL}(2,\mathbb{C})$. Accordingly, the Schottky
space $\mathfrak{S}_{g}$ is defined to be the set of equivalence classes of
representations
$\left[\varrho_{\mu}\right]:\Sigma\to\operatorname{PSL}(2,\mathbb{C})$,
$\mu\in\mathfrak{D}(\Sigma)$. In other words,

$\mathfrak{S}_{g}\cong\mathfrak{D}(\Sigma)/\sim,$ (120)

where $\mu_{1}\sim\mu_{2}$ if and only if
$F^{\mu_{1}}\circ\sigma\circ(F^{\mu_{1}})^{-1}=F^{\mu_{2}}\circ\sigma\circ(F^{\mu_{2}})^{-1}$
for all $\sigma\in\Sigma$ (or equivalently,
$F^{\mu_{1}}\big{|}_{\Lambda}=F^{\mu_{2}}\big{|}_{\Lambda}$). The group
$\Sigma$ corresponds to $\mu=0$ and is the base point of
$\mathfrak{D}(\Sigma)$.

The above alternative definition of Schottky space $\mathfrak{S}_{g}$ gives
this opportunity to us to define also the Schottky space
$\mathfrak{S}_{g,n}(\boldsymbol{m})$ of signature
$(g;m_{1},\dots,m_{n_{e}};n_{p})$ Riemann orbisurfaces. Let us consider the
configuration spaces
$\mathscr{F}_{n}\left(\Omega^{\mu}/\Sigma^{\mu}\right)=\mathscr{F}_{n}\left(\mathcal{D}^{\mu}\right)$
with $\Sigma^{\mu}=F^{\mu}\circ\Sigma\circ(F^{\mu})^{-1},\hskip
2.84526pt\Omega^{\mu}=F^{\mu}(\Omega)$ and the deformation space of a marked
Schottky group $(\Sigma;L_{1},\dots,L_{g})$ together with a point
$(w_{1},\dots,w_{n_{e}},w_{n_{e}+1},\dots,w_{n})\in\mathscr{F}_{n}\left(\mathcal{D}\right)$,

$\mathfrak{D}(\Sigma;L_{1},\dots,L_{g};w_{1},\dots,w_{n_{e}};w_{n_{e}+1},\dots,w_{n})\\\
=\left\\{(\mu;w_{1}^{\mu},\dots,w_{n_{e}}^{\mu};w_{n_{e}+1}^{\mu},\dots,w_{n}^{\mu})\in\mathcal{A}^{-1,1}(\Omega,\Sigma)\times\mathscr{F}_{n}\left(\mathcal{D}^{\mu}\right)\,\Big{|}\,\norm{\mu}_{\infty}<1\right\\},$
(121)

where $w^{\mu}_{i}=F^{\mu}(w_{i})$. Just as in the case of $\mathfrak{S}_{g}$,
each element $\mu\in\mathfrak{D}(\Sigma;L_{1},\dots,L_{g};w_{1},\dots,w_{n})$
gives a faithful representation $\varrho_{\mu}$ of
$(\Sigma;L_{1},\dots,L_{g};w_{1},\dots,w_{n})$ in
$\operatorname{PSL}(2,\mathbb{C})\times\mathscr{F}_{n}\left(\mathcal{D}^{\mu}\right)$
according to the formula $L_{i}\mapsto F^{\mu}\circ L_{i}\circ(F^{\mu})^{-1}$
for all marked generators $L_{i}\in\Sigma$, $i=1,\dots,g$ and $w_{j}\mapsto
F^{\mu}(w_{j})$ for $j=1,\dots,n$. Two representations are then equivalent,
$\varrho_{\mu_{1}}\sim\varrho_{\mu_{2}}$, if $F^{\mu_{1}}\circ
L_{i}\circ(F^{\mu_{1}})^{-1}=F^{\mu_{2}}\circ L_{i}\circ(F^{\mu_{2}})^{-1}$
for all $i=1,\dots,g$ as well as $w_{j}^{\mu_{1}}=w_{j}^{\mu_{2}}$ for all
$j=1,\dots,n$ and the Schottky space $\mathfrak{S}_{g,n}(\boldsymbol{m})$ is
defined to be the set of equivalence classes of representations
$\left[\varrho_{\mu}\right]$

$\mathfrak{S}_{g,n}(\boldsymbol{m})\cong\mathfrak{D}(\Sigma;L_{1},\dots,L_{g};w_{1},\dots,w_{n_{e}};w_{n_{e}+1},\dots,w_{n})/\sim.$
(122)

Let us remind that the Schottky uniformization of an orbisurface $O$ is
connected with the Fuchsian uniformization of it by the commutative diagram
(12). Accordingly, each marked Fuchsian group $\Gamma$ with signature
$(g;m_{1},\dots,m_{n_{e}},n_{p})$ is correspond to the unique marked
normalized Schottky group $\Sigma\simeq\Gamma/N$ with the domain of
discontinuity $\overset{{}_{\curlywedge}}{\Omega}$ such that
$\mathbb{H}/\Gamma\cong\overset{{}_{\curlywedge}}{\Omega}/\Sigma$ and this
correspondence determines the following map

$\pi:\mathcal{T}_{g,n}(\boldsymbol{m})\to\mathfrak{S}_{g,n}(\boldsymbol{m})$
(123)

by putting $w_{i}=J(z_{i})$ for $i=1,\dots,n$.414141This map has the same role
as the covering map $\Psi$ in lemma 3.3 and is a complex covering map.
Actually, we can use this map to understand the tangent and cotangent space to
the Schottky space $\mathfrak{S}_{g,n}(\boldsymbol{m})$ as follows. The map
$J$ in diagram (12) push-forwards the vector space
$\mathcal{H}^{2,0}(\mathbb{H},\Gamma)$ to a vector space
$\mathcal{H}^{2,0}(\overset{{}_{\curlywedge}}{\Omega},\Sigma)$ of meromorphic
functions on $\overset{{}_{\curlywedge}}{\Omega}$. This latter vector space
contains automorphic forms of weight 4 for the Schottky group $\Sigma$, which
admit a meromorphic extension to $\overset{{}_{\curlywedge}}{\Omega}$ with at
most simple poles at $\operatorname{Supp}\widetilde{\mathscr{D}}$. Each
$Q\in\mathcal{H}^{2,0}(\overset{{}_{\curlywedge}}{\Omega},\Sigma)$ has the
form

$Q(w)=(q\circ J^{-1})(w)\left(J^{-1}(w)^{\prime}\right)^{2},\qquad
q(z)\in\mathcal{H}^{2,0}(\mathbb{H},\Gamma).$ (124)

The vector space
$\mathcal{H}^{2,0}(\overset{{}_{\curlywedge}}{\Omega},\Sigma)$ itself
coincides with the holomorphic cotangent space
$T_{[0]}^{\ast}\mathfrak{S}_{g,n}(\boldsymbol{m})$ to
$\mathfrak{S}_{g,n}(\boldsymbol{m})$ at the origin. This implies that the
holomorphic tangent space $T_{[0]}\mathfrak{S}_{g,n}(\boldsymbol{m})$ is
identified with the complex vector space
$\mathcal{H}^{-1,1}(\overset{{}_{\curlywedge}}{\Omega},\Sigma)$ of harmonic
Beltrami differentials (harmonic with respect to the hyperbolic metric on
$\overset{{}_{\curlywedge}}{\Omega}$). Each
$M\in\mathcal{H}^{-1,1}(\overset{{}_{\curlywedge}}{\Omega},\Sigma)$ has the
form

$M(w)=e^{-\varphi(w)}\overline{Q(w)},\qquad
Q\in\mathcal{H}^{2,0}(\overset{{}_{\curlywedge}}{\Omega},\Sigma).$ (125)

From Eq.(121), which implies that it exists a fibration
$\jmath:\mathfrak{S}_{g,n}(\boldsymbol{m})\to\mathfrak{S}_{g}$ whose fibers
over the points $[\mu]\in\mathfrak{S}_{g}$ are
$\mathscr{F}_{n}\left(\Omega^{\mu}/\Sigma^{\mu}\right)$, it follows that
$T_{[0]}^{\ast}\mathfrak{S}_{g,n}(\boldsymbol{m})$ has a subspace
$\jmath^{\ast}(T_{[0]}^{\ast}\mathfrak{S}_{g})\cong\mathcal{H}^{2,0}(\Omega,\Sigma)$.
The natural basis in this subspace of $\mathcal{H}^{2,0}(\Omega,\Sigma)$ are
given by

$P_{1}(w),\dots,P_{3g-3}(w)\in\mathcal{H}^{2,0}(\Omega,\Sigma),$ (126)

which are holomorphic automorphic forms of weight 4 for $\Sigma$ and they
coincide with the cotangent vectors

$\differential{\lambda_{1}},\dots,\differential{\lambda_{g}},\differential{a_{3}},\dots,\differential{a_{g}},\differential{b_{2}},\dots,\differential{b_{g}}\in\jmath^{\ast}(T_{[0]}^{\ast}\mathfrak{S}_{g}).$
(127)

The subspace that is the complement of
$\jmath^{\ast}(T_{[0]}^{\ast}\mathfrak{S}_{g})$ in
$T_{[0]}^{\ast}\mathfrak{S}_{g,n}(\boldsymbol{m})$ is isomorphic to the
subspace $T_{[0]}^{\ast}\mathscr{F}_{n}\left(\mathcal{D}\right)$. The subspace
$T_{[0]}^{\ast}\mathscr{F}_{n}\left(\mathcal{D}\right)$ actually is the
cotangent space to the configuration space at the base point
$(w_{1},\dots,w_{n})$. In other words, we have

$T_{[0]}^{\ast}\mathfrak{S}_{g,n}(\boldsymbol{m})\cong\jmath^{\ast}(T_{[0]}^{\ast}\mathfrak{S}_{g})\oplus
T_{[0]}^{\ast}\mathscr{F}_{n}\left(\mathcal{D}\right).$ (128)

It follows from Eq.(93) that a natural basis for
$T_{[0]}^{\ast}\mathscr{F}_{n}\left(\mathcal{D}\right)$ is given by the
following meromorphic automorphic forms of weight 4,

$P_{3g-3+i}(w)=-\frac{1}{\pi}\sum_{\sigma\in\Sigma}R(\sigma
w,w_{i})\sigma^{\prime}(w)^{2},\qquad w\in\overset{{}_{\curlywedge}}{\Omega},$
(129)

which represent $\differential{w_{i}}$ for $i=1,\dots,n$. According to the
following pairing, which actually is analog of pairing (40),

$\left(Q,M\right)=\iint_{\mathcal{D}}Q(w)M(w)\differential[2]{w},$ (130)

we can obtain the dual basis for $P_{1}(w),\dots P_{3g-3+n}(w)$, i.e. the
basis $M_{1}(w),\dots M_{3g-3+n}(w)$ in
$\mathcal{H}^{-1,1}(\overset{{}_{\curlywedge}}{\Omega},\Sigma)$ which they
coincide with the tangent vectors
$\partialderivative{w_{1}},\dots,\partialderivative{w_{n}}\in
T_{[0]}\mathfrak{S}_{g,n}$. Similarly, the corresponding bases in the tangent
and cotangent spaces to $\mathfrak{S}_{g,n}$ at arbitrary point
$([L_{1}^{\mu},\dots,L_{g}^{\mu}];w_{1}^{\mu},\dots,w_{n}^{\mu})$, i.e. in
$T_{[\mu]}\mathfrak{S}_{g,n}(\boldsymbol{m})$ and
$T_{[\mu]}^{\ast}\mathfrak{S}_{g,n}(\boldsymbol{m})$, can also be defined.
This implies that
$\text{Sch}\left(J^{-1};w\right)=\partial_{w}^{2}\varphi(w)-\frac{1}{2}\left(\partial_{w}\varphi(w)\right)^{2}$
(note to the asymptotic behavior of $\varphi(w)$ and its derivatives as $w\to
w_{i}$ (see lemma C.1)) can be decomposed as following

$\text{Sch}\left(J^{-1};w\right)=\sum_{i=1}^{n}h_{i}\mathscr{E}_{i}(w)-\pi\sum_{i=1}^{3g-3+n}c_{i}P_{i}(w),$
(131)

where

$\mathscr{E}_{i}(w)=\frac{1}{2}\sum_{\sigma\in\Sigma}\left(\frac{1}{(\sigma
w-w_{i})^{2}}-\frac{1}{\sigma w(\sigma
w-1)}\right)\sigma^{\prime}(w)^{2},\quad\text{for}\quad i=1,\dots,n,$ (132)

are meromorphic automorphic forms of weight 4 for $\Sigma$ with the second
order poles at $\Sigma\cdot w_{i}$ and $c_{1},\dots,c_{3g-3+n}$ are accessory
parameters. Moreover, for the first and second variations of the family of
hyperbolic metrics on the Schottky domains
$\overset{{}_{\curlywedge}}{\Omega}$μ, we have the same formulas introduced in
subsection 3.1.2 and 3.2.1. Furthermore, the formula (103) is also valid in
this case. Finally, each cuspidal
$\langle\cdot,\cdot\rangle_{\text{TZ},i}^{\text{cusp}}$ and elliptic
$\langle\cdot,\cdot\rangle_{\text{TZ},i}^{\text{ell}}$ metric on
$\mathcal{T}_{g,n}(\boldsymbol{m})$ is invariant with respect to the
automorphism group of the covering
$\pi:\mathcal{T}_{g,n}(\boldsymbol{m})\to\mathfrak{S}_{g,n}(\boldsymbol{m})$
and determines a Kähler metric on $\mathfrak{S}_{g,n}(\boldsymbol{m})$. All
these also imply that in analogy with the commutative diagram (87), we have

$\begin{CD}\mathbb{H}@>{f^{\mu}}>{}>\mathbb{H}\\\ @V{J}V{}V@V{}V{J_{\mu}}V\\\
\overset{{}_{\curlywedge}}{\Omega}@>{F^{\mu}}>{}>\overset{{}_{\curlywedge}}{\Omega}\hskip
1.13809pt^{\mu}\end{CD}$ (133)

where

$\partial_{\bar{w}}F^{\mu}=M(w)\hskip
2.84526pt\partial_{w}F^{\mu}\qquad\text{for}\quad
w\in\overset{{}_{\curlywedge}}{\Omega},$ (134)

and $F^{\varepsilon\mu}$ is complex-analytic in $\varepsilon$ and
$\dot{F}^{\mu}$ is given by similar equation with Eq.(93) where
$w\in\overset{{}_{\curlywedge}}{\Omega}$.

Before closing this section let also mention that the mapping $J$ in diagram
(12) has the following expansions near cusps and ramification points of
$\Gamma$ (see Appendix C)

$J(z)=\left\\{\begin{split}&w_{i}+\sum_{k=1}^{\infty}J^{(i)}_{k}\left(\frac{z-z_{i}}{z-\bar{z}_{i}}\right)^{km_{i}}\hskip
95.3169pt(i=1,\dots,n_{e}),\hskip 5.69046ptz\to z_{i},\\\ \\\
&w_{i}+\sum_{k=1}^{\infty}J^{(i)}_{k}\exp(-\frac{2\pi\hskip
2.84526pt\sqrt{-1}k}{|\delta_{i}|(z-z_{i})})\hskip
28.45274pt(i=n_{e}+1,\dots,n-1),\hskip 5.69046ptz\to z_{i},\end{split}\right.$
(135)

where $w_{i}=J(z_{i})$ for $i=1,\dots,n-1$.424242Note that since $\Sigma$ is
normalized, $\infty\notin\Omega$.. If $e^{\varphi(w)}|\differential{w}|^{2}$
would be the push-forward of the hyperbolic metric on $\mathbb{H}$ by the map
$J$, then the density of hyperbolic metric on
$\overset{{}_{\curlywedge}}{\Omega}$ (i.e. $\rho(w)=e^{\varphi(w)}$) is once
again given by Eq.(80), where $\varphi(w)$ is smooth on $\Omega^{\text{reg}}$.
The function $\varphi(w)$ satisfies434343This equation follows from the
invariance of hyperbolic metric on $\Omega^{\text{reg}}$ under the action of
$\Sigma$ – i.e.
$e^{\varphi(w)}\differential{w}\differential{\bar{w}}=e^{\varphi(\sigma
w)}\differential{\sigma(w)}\differential{\overline{\sigma(w)}}$ for all
$\sigma\in\Sigma$.

$\varphi(\sigma
w)=\varphi(w)-\log|\sigma^{\prime}(w)|^{2}\quad\text{for}\quad\forall\sigma\in\Sigma,\,w\in\Omega^{\text{reg}},$
(136)

and has the same asymptotic as in Lemma C.1,

$\varphi(w)=\left\\{\begin{split}&-2(1-\frac{1}{m_{i}})\log|w-w_{i}|+\log\frac{4|J^{(i)}_{1}|^{-\frac{2}{m_{i}}}}{m_{i}^{2}}+\text{\scriptsize{$\mathcal{O}$}}\left(1\right)\hskip
36.98866ptw\to w_{i},\\\
&-2\log|w-w_{j}|-2\log\left|\log\left|\frac{w-w_{j}}{J^{(i)}_{1}}\right|\right|+\text{\scriptsize{$\mathcal{O}$}}\left(1\right)\hskip
56.9055ptw\to w_{j},\\\
&-2\log|w|-2\log\log\left|\frac{w}{J^{(n)}_{-1}}\right|+\order{|w|^{-1}},\hskip
79.6678ptw\to\infty,\end{split}\right.$ (137)

with $i=1,\dots,n_{e}$ and $j=n_{e}+1,\dots,n-1$.

######  Remark 3.10.

It follows from the above asymptotics that (see statement 3 of lemma C.1 for
more details)

$\left\\{\begin{split}&\log\mathsf{h}_{i}=-2\log m_{i}+2\log 2-\lim_{w\to
w_{i}}\left(\varphi(w)+\left(1-\frac{1}{m_{i}}\right)\log|w-w_{i}|^{2}\right),\hskip
5.69046pti=1,\dots,n_{e},\\\ \\\ &\log\mathsf{h}_{i}=\lim_{w\to
w_{i}}\left(\log|w-w_{i}|^{2}-\frac{2e^{-\frac{\varphi(w)}{2}}}{|w-w_{i}|}\right),\hskip
56.9055pti=n_{e}+1,\dots,n-1,\\\
&\log\mathsf{h}_{n}=\lim_{w\to\infty}\left(\log|w|^{2}-\frac{2e^{-\frac{\varphi(w)}{2}}}{|w|}\right).\end{split}\right.$
(138)

with $\mathsf{h}_{i}=\left|J^{(i)}_{1}\right|^{\frac{2}{m_{i}}}$ for
$i=1,\dots,n_{e}$, and $\mathsf{h}_{i}=\left|J^{(i)}_{1}\right|^{2}$ for
$i=n_{e}+1,\dots,n$.

Last but not least, let $\mathscr{L}_{i}$ be the $i$-th _relative dualizing
sheaf_ on $\mathfrak{S}_{g,n}(\boldsymbol{m})$ that is a holomorphic line
bundle dual to the vertical tangent bundle on
$\mathfrak{S}_{g,n}(\boldsymbol{m})$ along the fibers of the projection
$p_{i}:\mathfrak{S}_{g,n}(\boldsymbol{m})\to\mathfrak{S}_{g,n-1}(\boldsymbol{m})$
which “forgets” the marked point $w_{i}$, $i=1,\dots,n$. The bundles
$\mathscr{L}_{i}$, also called _tautological line bundles_ , are characterized
by the property that the fibers of $\mathscr{L}_{i}$ over a point
$([L_{1},\dots,L_{g}];w_{1},\dots,w_{n})\in\mathfrak{S}_{g,n}$ is the
cotangent space $T^{\ast}_{w_{i}}(\overset{{}_{\curlywedge}}{\Omega}/\Sigma)$.
Since $J\circ\beta_{k}=L_{k}\circ J$, the points
$w_{1},\dots,L_{k}w_{i},\dots,w_{n}$ correspond to the ramification points and
cusps $z_{1},\dots,\beta_{k}z_{i},\dots,z_{n}$, and the first coefficient of
the expansion (135) of $J(z)$ near the equivalent ramification and cusp point
$\beta_{k}z_{i}$, is given by $L^{\prime}_{k}(w_{i})J_{1}^{(i)}$. As a result,
the smooth positive functions
$\mathsf{h}_{i}=\left|J^{(i)}_{1}\right|^{\frac{2}{m_{i}}}$ for
$i=1,\dots,n_{e}$, and $\mathsf{h}_{i}=\left|J^{(i)}_{1}\right|^{2}$ for
$i=n_{e}+1,\dots,n$ get replaced by
$\mathsf{h}_{i}\left|L^{\prime}_{k}(w_{i})\right|^{\frac{2}{m_{i}}}$ and
$\mathsf{h}_{i}\left|L^{\prime}_{k}(w_{i})\right|^{2}$ respectively. Using the
above interpretation of the line bundles $\mathscr{L}_{i}$, we arrive at the
following statement.

######  Lemma 3.6.

The quantities $\mathsf{h}_{i}^{m_{i}}$ determine Hermitian metrics in the
holomorphic line bundles $\mathscr{L}_{i}$ for $i=1,\dots,n_{e}$.

###### Proof.

To prove this lemma for the ramification points, we use the transformation of
$\mathsf{h}_{i}$’s under the action of the generators $L_{k}$ of the Schottky
group. As we explained above sending $w_{1},\dots,w_{i},\dots,w_{n_{e}}$ to
$w_{1},\dots,L_{k}w_{i},\dots,w_{n_{e}}$ will transform $\mathsf{h}_{i}$ to
$\mathsf{h}_{i}\left|L^{\prime}_{k}(w_{i})\right|^{\frac{2}{m_{i}}}$. Thus, we
have:

$\Delta\log\mathsf{h}_{i}=\frac{1}{m_{i}}\log\left|L^{\prime}_{k}(w_{i})\right|^{2}.$
(139)

This means that $\mathsf{h}_{i}^{m_{i}}$ is a Hermitian metric in the line
bundle $\mathscr{L}_{i}$. For the case of cusp, a similar proof can be
applicable, and $\mathsf{h}_{i}$ will be a Hermitian metric on the line bundle
$\mathscr{L}_{i}$. ∎

##  4 Classical Liouville Action

In this section, we study the classical Liouville action with details for the
Riemann orbisurfaces with the genus $g=0$ and $g\neq 0$, separately. Before
proceeding, we should mention here that all the proofs in sections 4 and 5 are
previously provided for the case of Riemann surfaces with just punctures (and
arbitrary genus) in Zograf1988ONLE  and park2015potentials . For calculations
related to the punctures, we could only refer to those articles, but for the
convenience of the reader and to better recognize differences in the presence
of conical singularities, we thought it appropriate to display them all next
to each other.

###  4.1 Riemann Orbisurfaces of Genus 0

Let $O=(\hat{\mathbb{C}},\mathscr{D})$ be a marked Riemann orbisurface with
signature $(0;m_{1},\dots,m_{n_{e}},n_{p})$ and let $\boldsymbol{m}$ denotes
the ordered set of ramification indices $(m_{1},\dots,m_{n})$. The regularized
action functional for the Liouville equation (81) and in presence of conical
singularities with conical angles $2\pi/m_{1},\dots,2\pi/m_{n_{e}}$ at
$w_{1},\dots,w_{n_{e}}$ together with punctures at
$w_{n_{e}+1},\dots,w_{n-2}=0,w_{n-1}=1,w_{n}=\infty$ is defined as follows
(see Zograf1988ONLE ,Takhtajan:2001uj ,Takhtajan:1995fd )

$S_{\boldsymbol{m}}[\varphi]=\\\ \lim_{\epsilon\to
0^{+}}\left(\iint_{O_{\epsilon}}(|\partial_{w}\varphi|^{2}+e^{2\varphi})\differential[2]{w}+\frac{\sqrt{-1}}{2}\sum_{i=1}^{n_{e}}\left(1-\frac{1}{m_{i}}\right)\oint_{C_{i}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{i}}-\frac{\differential{w}}{w-w_{i}}\right)\right.\\\
\left.-2\pi\sum_{i=1}^{n_{e}}\left(1-\frac{1}{m_{i}}\right)^{2}\log\epsilon+2\pi
n_{p}\log\epsilon+4\pi(n_{p}-2)\log|\log\epsilon|\right),$ (140)

where444444Note that $\lim_{\epsilon\rightarrow
0}O_{\epsilon}=O^{\text{reg}}=X_{O}^{\text{reg}}$.

$O_{\epsilon}=\mathbb{C}\Big{\backslash}\bigcup_{i=1}^{n-1}\left\\{w\,\Big{|}\,|w-w_{i}|<\epsilon\right\\}\cup\left\\{w\,\Big{|}\,|w|>\epsilon^{-1}\right\\},$

and the circles

$C_{i}^{\epsilon}=\left\\{w\,\Big{|}\,|w-w_{i}|=\epsilon\right\\},$

are oriented as a component of the boundary $\partial O_{\epsilon}$ of
$O_{\epsilon}$.

######  Remark 4.1.

When $n_{p}=0$, the appropriate classical Liouville action is given by

$S_{\boldsymbol{m}}[\varphi]=\\\ \lim_{\epsilon\to
0^{+}}\Bigg{(}\iint_{O_{\epsilon}}(|\partial_{w}\varphi|^{2}+e^{2\varphi})\differential[2]{w}+\frac{\sqrt{-1}}{2}\sum_{i=1}^{n-1}\left(1-\frac{1}{m_{i}}\right)\oint_{C_{i}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{i}}-\frac{\differential{w}}{w-w_{i}}\right)\\\
+\frac{\sqrt{-1}}{2}\left(1+\frac{1}{m_{n}}\right)\oint_{C_{n}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}}-\frac{\differential{w}}{w}\right)-2\pi\sum_{i=1}^{n-1}\left(1-\frac{1}{m_{i}}\right)^{2}\log\epsilon-2\pi\left(1+\frac{1}{m_{n}}\right)^{2}\log\epsilon\Bigg{)}.$
(141)

######  Remark 4.2.

It is worth to provide some explanation regarding the action (140) and why the
terms associated with conical points appear different from those related to
punctures. The contour integral in the first line of (140) is added to ensure
a well-defined variational principle around the conical
singularities.454545Alternatively, the line integrals are necessary in order
to ensure the proper asymptotic behavior (137). See the explanation following
(Takhtajan:1995fd, , Eq. (8)) for more details. The variation of this integral
cancels the boundary term arising from the variation of the bulk term. To
ensure a well-defined variational principle around the punctures, additional
contour integrals must be considered, which are given by (Takhtajan:1995fd, ,
§2)

$\frac{\sqrt{-1}}{2}\sum_{j=n_{e}+1}^{n-1}\oint_{C_{j}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{j}}-\text{c.c}\right)+\frac{\sqrt{-1}}{2}\sum_{j=n_{e}+1}^{n-1}\oint_{C_{j}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{(\bar{w}-\bar{w}_{j})\log|w-w_{j}|}-\text{c.c.}\right)\\\
-\frac{\sqrt{-1}}{2}\oint_{C_{n}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}}-\text{c.c}\right)-\frac{\sqrt{-1}}{2}\oint_{C_{n}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}\log|w|}-\text{c.c.}\right).$
(142)

More concretely, the classical Liouville action (140) in the presence of these
additional contour integrals takes the form (Takhtajan:1995fd, , Eq. (8))

$S_{\boldsymbol{m}}[\varphi]=\\\ \lim_{\epsilon\to
0^{+}}\left(\iint_{O_{\epsilon}}(|\partial_{w}\varphi|^{2}+e^{2\varphi})\differential[2]{w}+\frac{\sqrt{-1}}{2}\sum_{i=1}^{n-1}\left(1-\frac{1}{m_{i}}\right)\oint_{C_{i}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{i}}-\frac{\differential{w}}{w-w_{i}}\right)\right.\\\
+\frac{\sqrt{-1}}{2}\sum_{j=n_{e}+1}^{n-1}\oint_{C_{j}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{(\bar{w}-\bar{w}_{j})\log|w-w_{j}|}-\frac{\differential{w}}{(w-w_{j})\log|w-w_{j}|}\right)-\frac{\sqrt{-1}}{2}\oint_{C_{n}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}}-\frac{\differential{w}}{w}\right)\\\
\left.-\frac{\sqrt{-1}}{2}\oint_{C_{n}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}\log|w|}-\frac{\differential{w}}{w\log|w|}\right)-2\pi\sum_{i=1}^{n}\left(1-\frac{1}{m_{i}}\right)^{2}\log\epsilon\right),$
(143)

where $m_{i}=\infty$ for $i=n_{e}+1,\dots,n$. By substituting the asymptotic
form (137) of the Liouville field $\varphi$ near punctures in the contour
integrals of (143), up to $\text{\scriptsize{$\mathcal{O}$}}\left(1\right)$
terms, one gets

$\begin{split}&\frac{\sqrt{-1}}{2}\sum_{j=n_{e}+1}^{n-1}\oint_{C_{j}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{j}}-\text{c.c}\right)+\frac{\sqrt{-1}}{2}\sum_{j=n_{e}+1}^{n-1}\oint_{C_{j}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{(\bar{w}-\bar{w}_{j})\log|w-w_{j}|}-\text{c.c.}\right)\\\
&-\frac{\sqrt{-1}}{2}\oint_{C_{n}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}}-\text{c.c}\right)-\frac{\sqrt{-1}}{2}\oint_{C_{n}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}\log|w|}-\text{c.c.}\right)=4\pi
n_{p}\log\epsilon+4\pi(n_{p}-2)\log|\log\epsilon|.\end{split}$ (144)

Since the contour integral (142) evaluated on-shell is merely a divergent
term, we can add it to the counterterm $-2\pi n_{p}\log\epsilon$ in (143) to
get the Liouville action (140). The resulting regulating terms will have the
opposite sign (i.e., plus sign instead of minus sign) and correctly cancel the
divergence coming from the bulk term $\lim_{\epsilon\to
0^{+}}\iint_{O_{\epsilon}}|\partial_{w}\varphi|^{2}\differential[2]{w}$.

Performing a similar calculation for the case of conical points, i.e.,
substituting the asymptotic form (137) of the Liouville field $\varphi$ near
the conical points in the contour integral of (140), one observes that the
result is given by

$\frac{\sqrt{-1}}{2}\sum_{i=1}^{n-1}\left(1-\frac{1}{m_{i}}\right)\oint_{C_{i}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{i}}-\frac{\differential{w}}{w-w_{i}}\right)=4\pi\sum_{i=1}^{n_{e}}\big{[}(1-\tfrac{1}{m_{i}})^{2}\log\epsilon+\tfrac{1}{2}(1-\tfrac{1}{m_{i}})\log\mathsf{h}_{i}\big{]}+\text{\scriptsize{$\mathcal{O}$}}\left(1\right).$
(145)

Notice that, apart from the divergent part, the above expression contains a
finite part (i.e., $2\pi\sum_{i}(1-\tfrac{1}{m_{i}})\log\mathsf{h}_{i}$) which
behaves non-trivially under quasi-conformal transformations and therefore
cannot be ignored (see lemma 5.1 for more details).464646For both punctures
and conical points, there are finite constant terms in the calculation of the
contour integrals which can be safely ignored. These terms are not written in
equations (144) and (145). We have decided to keep the line integrals around
conical points in their integral form since this form is more familiar in the
literature on Liouville CFT and can also be used when the conical points are
of a more general type (see e.g., Takhtajan:2001uj ). Therefore, one arrives
at the classical Liouville action (140).

######  Theorem 4.1.

For any fixed ordered set of ramification indices
$\boldsymbol{m}=(m_{1},\dots,m_{n})$ such that $2\leq m_{1}\leq
m_{2}\leq\dots\leq m_{n}$ and
$\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)+n_{p}>2$, the function
$S_{\boldsymbol{m}}:\mathcal{M}_{0,n}\to\mathbb{R}$ is differentiable and

$c_{i}=-\frac{1}{2\pi}\partialderivative{S_{\boldsymbol{m}}}{w_{i}}\quad\text{for}\quad
i=1,\dots,n-3,$ (146)

where $c_{i}$ are the accessory parameters defined by (84).

###### Proof.

Let

$\tilde{S}_{\boldsymbol{m}}^{\epsilon}(w_{1},\dots,w_{n-3})=\tilde{S}_{\boldsymbol{m}}^{(B)\epsilon}(w_{1},\dots,w_{n-3})+\tilde{S}^{\text{(ct)}\epsilon}-2\pi\chi(X),$
(147)

with474747In view of the Gauss-Bonnet formula for Riemann orbisurfaces
Troyanov1991PrescribingCO ; Li_orbifold_2018  $\\\
\frac{\sqrt{-1}}{2}\iint_{O}e^{\varphi}\differential{w}\wedge\differential{\bar{w}}=2\pi\left(\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)+n_{p}-2\right)=-2\pi\chi(O).$

$\tilde{S}_{\boldsymbol{m}}^{(B)\epsilon}(w_{1},\dots,w_{n-3})=\iint_{O_{\epsilon}}|\partial_{w}\varphi|^{2}\differential[2]{w}+\frac{\sqrt{-1}}{2}\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{C_{i}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{j}}-\frac{\differential{w}}{w-w_{j}}\right)\\\
\tilde{S}^{(\text{ct})\epsilon}=-2\pi\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)^{2}\log\epsilon+2\pi(n-n_{e})\log\epsilon+4\pi(n-n_{e}-2)\log|\log\epsilon|.$
(148)

For any $\epsilon>0$, the function $\tilde{S}_{\boldsymbol{m}}^{\epsilon}$ is
continuously differentiable on $\mathcal{M}_{0,n}$. To prove the theorem it
suffices to show that
$\mathcal{L}_{\mu_{i}}\tilde{S}_{\boldsymbol{m}}^{\epsilon}$ converges
uniformly to $-2\pi c_{i}$ as $\epsilon\to 0$ in a neighborhood of any point
of the moduli space $\mathcal{M}_{0,n}$. More explicitly, one needs to show
that

$\mathcal{L}_{\mu_{i}}\tilde{S}_{\boldsymbol{m}}^{\epsilon}=\partialderivative{\tilde{S}_{\boldsymbol{m}}^{\epsilon}}{w_{i}}\overset{\text{\tiny\it
def.}}{\equiv}\partialderivative{\varepsilon}\Bigg{|}_{\varepsilon=0}\tilde{S}_{\boldsymbol{m}}^{\epsilon}(w_{1}^{\varepsilon\mu_{i}},\dots,w_{n-3}^{\varepsilon\mu_{i}})=-2\pi
c_{i}\quad\text{for}\quad i=1,\dots,n-3,$ (149)

pointwise on the moduli space $\mathcal{M}_{0,n}$.484848We remind the reader
that the basis $\\{\mu_{i}\\}$ for $T_{[0]}\mathcal{T}_{0,n}$ has been defined
in Eq.(95). Firstly, we can write

$\partialderivative{\tilde{S}_{\boldsymbol{m}}^{(B)\epsilon}}{w_{i}}=\frac{\sqrt{-1}}{2}\partialderivative{\varepsilon}\Bigg{|}_{\varepsilon=0}I_{\epsilon}(\varepsilon),$
(150)

where

$\begin{split}&I_{\epsilon}(\varepsilon)=\left(I_{\epsilon}[\varphi]\circ\Psi\circ\Phi\right)(\varepsilon\mu_{i}),\\\
&\varphi^{\varepsilon\mu_{i}}(w)=\varphi\Big{(}w;(\Psi\circ\Phi)(\varepsilon\mu_{i})\Big{)}=\varphi(w;\underbrace{F^{\varepsilon\mu_{i}}(w_{1})}_{w_{1}^{\varepsilon\mu_{i}}},\dots,\underbrace{F^{\varepsilon\mu_{i}}(w_{n-3})}_{w_{n-3}^{\varepsilon\mu_{i}}}),\end{split}$

and

$I_{\epsilon}[\varphi]=\iint_{O_{\epsilon}}|\partial_{w}\varphi|^{2}\differential{w}\wedge\differential{\bar{w}}+\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{C_{i}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{j}}-\frac{\differential{w}}{w-w_{j}}\right).$

Accordingly, one gets

$\begin{split}I_{\epsilon}(\varepsilon)=&\iint_{F^{\varepsilon\mu_{i}}(O_{\epsilon})}|\partial_{w}\varphi^{\varepsilon\mu_{i}}(w)|^{2}\differential{w}\wedge\differential{\bar{w}}\\\
&+\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{F^{\varepsilon\mu_{i}}(C_{j}^{\epsilon})}\varphi^{\varepsilon\mu_{i}}(w)\left(\frac{\differential{\bar{w}}}{\bar{w}-\overline{w_{j}^{\varepsilon\mu_{i}}}}-\frac{\differential{w}}{w-w_{j}^{\varepsilon\mu_{i}}}\right),\end{split}$
(151)

with

$\left\\{\begin{split}&F^{\varepsilon\mu_{i}}(O_{\epsilon})=\mathbb{C}\Big{\backslash}\bigcup_{k=1}^{n-1}\left\\{w\,\Big{|}\,|w-w_{k}^{\varepsilon\mu_{i}}|<\epsilon\right\\}\cup\left\\{w\,\Big{|}\,|w|>\epsilon^{-1}\right\\},\\\
&F^{\varepsilon\mu_{i}}(C_{j}^{\epsilon})=\left\\{w\,\Big{|}\,|w-w_{j}^{\varepsilon\mu_{i}}|=\epsilon\right\\}.\end{split}\right.$
(152)

The calculation of (150) can be done by almost verbatim repeating the
corresponding computations in the proof of Theorem 1 in Zograf1988ONLE .
Accordingly, let us now use the _change of variable formula_ for differential
forms, $\int_{F(X)}\omega=\int_{X}F^{\ast}(\omega)$, and the commutative
diagram (87) to write

$\begin{split}I_{\epsilon}(\varepsilon)&=\iint_{O_{\epsilon}(\varepsilon)}(F^{\varepsilon\mu_{i}})^{\ast}\Big{(}|\partial_{w}\varphi^{\varepsilon\mu_{i}}|^{2}\differential{w}\wedge\differential{\bar{w}}\Big{)}\\\
&+\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{C_{j}^{\epsilon}(\varepsilon)}(F^{\varepsilon\mu_{i}})^{\ast}\left(\varphi^{\varepsilon\mu_{i}}\left(\frac{\differential{\bar{w}}}{\bar{w}-\overline{w_{j}^{\varepsilon\mu_{i}}}}-\frac{\differential{w}}{w-w_{j}^{\varepsilon\mu_{i}}}\right)\right)\\\
&=\iint_{O_{\epsilon}(\varepsilon)}|\partial_{w}\varphi^{\varepsilon\mu_{i}}\circ
F^{\varepsilon\mu_{i}}|^{2}\differential{F^{\varepsilon\mu_{i}}(w)}\wedge\differential{\overline{F^{\varepsilon\mu_{i}}(w)}}\\\
&+\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{C_{j}^{\epsilon}(\varepsilon)}(\varphi^{\varepsilon\mu_{i}}\circ
F^{\varepsilon\mu_{i}})\left(\frac{\differential{\overline{F^{\varepsilon\mu_{i}}(w)}}}{\overline{F^{\varepsilon\mu_{i}}(w)}-\overline{w_{j}^{\varepsilon\mu_{i}}}}-\frac{\differential{F^{\varepsilon\mu_{i}}(w)}}{F^{\varepsilon\mu_{i}}(w)-w_{j}^{\varepsilon\mu_{i}}}\right)\\\
&=\iint_{O_{\epsilon}(\varepsilon)}|\partial_{w}\varphi^{\varepsilon\mu_{i}}\circ
F^{\varepsilon\mu_{i}}|^{2}\,|\partial_{w}F^{\varepsilon\mu_{i}}|^{2}(1-|\varepsilon
M_{i}|^{2})\differential{w}\wedge\differential{\bar{w}}\\\
&+\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{C_{j}^{\epsilon}(\varepsilon)}(\varphi^{\varepsilon\mu_{i}}\circ
F^{\varepsilon\mu_{i}})\left(\frac{\overline{\partial_{w}F^{\varepsilon\mu_{i}}}(\bar{\varepsilon}\overline{M_{i}}\differential{w}+\differential{\bar{w}})}{\overline{w^{\varepsilon\mu_{i}}}-\overline{w_{j}^{\varepsilon\mu_{i}}}}-\frac{\partial_{w}F^{\varepsilon\mu_{i}}(\differential{w}+\varepsilon
M_{i}\differential{\bar{w}})}{w^{\varepsilon\mu_{i}}-w_{j}^{\varepsilon\mu_{i}}}\right),\end{split}$

where

$\left\\{\begin{split}&O_{\epsilon}(\varepsilon)=\mathbb{C}\Big{\backslash}\bigcup_{k=1}^{n-1}\left\\{w\,\Big{|}\,|w^{\varepsilon\mu_{i}}-w_{i}^{\varepsilon\mu_{i}}|<\epsilon\right\\}\cup\left\\{w\,\Big{|}\,|w^{\varepsilon\mu_{i}}|>\epsilon^{-1}\right\\},\\\
&C_{j}^{\epsilon}(\varepsilon)=\left\\{w\,\Big{|}\,|w^{\varepsilon\mu_{i}}-w_{j}^{\varepsilon\mu_{i}}|=\epsilon\right\\}.\end{split}\right.$
(153)

To compute $\partial
I_{\epsilon}(\varepsilon)/\partial\varepsilon|_{\varepsilon=0}$, we need to
differentiate under the integral sign as well as over the variable integration
domains $O_{\epsilon}(\varepsilon)$ and $C_{j}^{\epsilon}(\varepsilon)$:

$\begin{split}\partialderivative{\varepsilon}&\Bigg{|}_{\varepsilon=0}I_{\epsilon}(\varepsilon)=\iint_{O_{\epsilon}}\partialderivative{\varepsilon}\Bigg{|}_{\varepsilon=0}|\partial_{w}\varphi^{\varepsilon\mu_{i}}\circ
F^{\varepsilon\mu_{i}}|^{2}\,|\partial_{w}F^{\varepsilon\mu_{i}}|^{2}(1-|\varepsilon
M_{i}|^{2})\differential{w}\wedge\differential{\bar{w}}\\\
&+\partialderivative{\varepsilon}\Bigg{|}_{\varepsilon=0}\iint_{O_{\epsilon}(\varepsilon)}|\partial_{w}\varphi|^{2}\differential{w}\wedge\differential{\bar{w}}\\\
&+\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{C_{j}^{\epsilon}}\partialderivative{\varepsilon}\Bigg{|}_{\varepsilon=0}(\varphi^{\varepsilon\mu_{i}}\circ
F^{\varepsilon\mu_{i}})\left(\frac{\overline{\partial_{w}F^{\varepsilon\mu_{i}}}(\bar{\varepsilon}\overline{M_{i}}\differential{w}+\differential{\bar{w}})}{\overline{w^{\varepsilon\mu_{i}}}-\overline{w_{j}^{\varepsilon\mu_{i}}}}-\frac{\partial_{w}F^{\varepsilon\mu_{i}}(\differential{w}+\varepsilon
M_{i}\differential{\bar{w}})}{w^{\varepsilon\mu_{i}}-w_{j}^{\varepsilon\mu_{i}}}\right)\\\
&+\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\partialderivative{\varepsilon}\Bigg{|}_{\varepsilon=0}\oint_{C_{j}^{\epsilon}(\varepsilon)}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{j}}-\frac{\differential{w}}{w-w_{j}}\right).\end{split}$
(154)

The second and fourth terms in Eq.(154) can be computed using the formula for
differentiating a given $k$-form $\omega$ over a smooth family of variable
domains $O(\varepsilon)$

$\partialderivative{\varepsilon}\Bigg{|}_{\varepsilon=0}\underbrace{\int\dotsm\int_{O_{\epsilon}(\varepsilon)}}_{k}\omega=\underbrace{\int\dotsm\int_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\partial
O_{\epsilon}}}}_{k-1}i_{V}(\omega),$ (155)

where $i_{V}(\omega)$ denotes the interior product of the $k$-form $\omega$
with vector field $V$ which is the vector field along $\partial O_{\epsilon}$
corresponding to the family of curves $\partial O_{\epsilon}(\varepsilon)$. As
a result we get

$\begin{split}\partialderivative{I_{\epsilon}}{w_{i}}&=\iint_{O_{\epsilon}}\left[(\partial_{w_{i}}\partial_{w}\varphi+\partial_{w}^{2}\varphi\dot{F}^{i})\partial_{\bar{w}}\varphi+(\partial_{w_{i}}\partial_{\bar{w}}\varphi+\partial_{w}\partial_{\bar{w}}\varphi\dot{F}^{i})\partial_{w}\varphi+|\partial_{w}\varphi|^{2}\partial_{w}\dot{F}^{i}\right]\differential{w}\wedge\differential{\bar{w}}\\\
&-\sum_{k=1}^{n}\int_{\partial
O^{\epsilon}_{k}}|\partial_{w}\varphi|^{2}\Big{(}\dot{F}^{i}(w)-\dot{F}^{i}(w_{k})\Big{)}\differential{\bar{w}}\\\
&+\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{C_{j}^{\epsilon}}\partialderivative{\varepsilon}\Bigg{|}_{\varepsilon=0}(\varphi^{\varepsilon\mu_{i}}\circ
F^{\varepsilon\mu_{i}})\left(\frac{\overline{\partial_{w}F^{\varepsilon\mu_{i}}}(\bar{\varepsilon}\overline{M_{i}}\differential{w}+\differential{\bar{w}})}{\overline{w^{\varepsilon\mu_{i}}}-\overline{w_{j}^{\varepsilon\mu_{i}}}}-\frac{\partial_{w}F^{\varepsilon\mu_{i}}(\differential{w}+\varepsilon
M_{i}\differential{\bar{w}})}{w^{\varepsilon\mu_{i}}-w_{j}^{\varepsilon\mu_{i}}}\right),\end{split}$
(156)

where the last term in Eq.(154) has vanished due to the fact that $\partial
C_{j}^{\epsilon}=\emptyset$. By noting that

$\begin{split}&\oint_{C_{j}^{\epsilon}}\varphi\left(\frac{d\bar{w}}{\bar{w}-\bar{w}_{j}}-\frac{dw}{w-w_{j}}\right)=\\\
&\hskip
14.22636pt-2\oint_{C_{j}^{\epsilon}}\varphi\left(\frac{dw}{w-w_{j}}\right)-\oint_{C_{j}^{\epsilon}}\partial_{w}\varphi\log|w-w_{j}|^{2}dw-\oint_{C_{j}^{\epsilon}}\partial_{\bar{w}}\varphi\log|w-w_{j}|^{2}d\bar{w},\end{split}$

we have

$\begin{split}&\oint_{C_{j}^{\epsilon}}(\varphi^{\varepsilon\mu_{i}}\circ
F^{\varepsilon\mu_{i}})\left(\frac{\overline{\partial_{w}F^{\varepsilon\mu_{i}}}(\bar{\varepsilon}\overline{M_{i}}\differential{w}+\differential{\bar{w}})}{\overline{w^{\varepsilon\mu_{i}}}-\overline{w_{j}^{\varepsilon\mu_{i}}}}-\frac{\partial_{w}F^{\varepsilon\mu_{i}}(\differential{w}+\varepsilon
M_{i}\differential{\bar{w}})}{w^{\varepsilon\mu_{i}}-w_{j}^{\varepsilon\mu_{i}}}\right)\\\
&=\underbrace{-2\oint_{C_{j}^{\epsilon}}(\varphi^{\varepsilon\mu_{i}}\circ
F^{\varepsilon\mu_{i}})\left(\frac{\partial_{w}F^{\varepsilon\mu_{i}}(dw+\varepsilon
M_{i}d\bar{w})}{F^{\varepsilon\mu_{i}}(w)-F^{\varepsilon\mu_{i}}(w_{j})}\right)}_{B_{1}}\\\
&\underbrace{-\oint_{C_{j}^{\epsilon}}\left(\partial_{w}\varphi^{\varepsilon\mu_{i}}\circ
F^{\varepsilon\mu_{i}}\right)\log|F^{\varepsilon\mu_{i}}(w)-F^{\varepsilon\mu_{i}}(w_{j})|^{2}\hskip
2.84526pt\partial_{w}F^{\varepsilon\mu_{i}}\left(dw+\varepsilon
M_{i}d\bar{w}\right)}_{B_{2}}\\\
&\underbrace{-\oint_{C_{j}^{\epsilon}}\left(\partial_{\bar{w}}\varphi^{\varepsilon\mu_{i}}\circ
F^{\varepsilon\mu_{i}}\right)\log|F^{\varepsilon\mu_{i}}(w)-F^{\varepsilon\mu_{i}}(w_{j})|^{2}\hskip
2.84526pt\overline{\partial_{w}F^{\varepsilon\mu_{i}}}\left(\bar{\varepsilon}\overline{M_{i}}dw+d\bar{w}\right)}_{B_{3}}.\end{split}$

After simple calculations and using the Lemma 3.4 one can see as $\epsilon\to
0$,

$\begin{split}&\partialderivative{\varepsilon}\Bigg{|}_{\varepsilon=0}B_{1}=2\oint_{C_{j}^{\epsilon}}\partial_{w}\dot{F}^{i}(w)\left(\frac{dw}{w-w_{j}}\right)-2\oint_{C_{j}^{\epsilon}}\varphi\left(\frac{M_{i}\hskip
2.84526ptd\bar{w}}{w-w_{j}}\right)+\text{\scriptsize{$\mathcal{O}$}}\left(1\right)\\\
&\partialderivative{\varepsilon}\Bigg{|}_{\varepsilon=0}B_{2}=\oint_{C_{j}^{\epsilon}}\partial^{2}_{w}\dot{F}^{i}\log|w-w_{j}|^{2}dw-\oint_{C_{j}^{\epsilon}}\partial_{w}\varphi\partial_{w}\dot{F}^{i}(w_{j})dw\\\
&\hskip 56.9055pt-\oint_{C_{j}^{\epsilon}}\partial_{w}\varphi\hskip
2.84526ptM_{i}\log|w-w_{j}|^{2}d\bar{w}+\text{\scriptsize{$\mathcal{O}$}}\left(1\right),\\\
&\partialderivative{\varepsilon}\Bigg{|}_{\varepsilon=0}B_{3}=\oint_{C_{j}^{\epsilon}}\partial_{w}\varphi\hskip
2.84526ptM_{i}\log|w-w_{j}|^{2}d\bar{w}+\oint_{C_{j}^{\epsilon}}\partial_{\bar{w}}\partial_{w}\dot{F}^{i}\log|w-w_{j}|^{2}d\bar{w}\\\
&\hskip
56.9055pt-\oint_{C_{j}^{\epsilon}}\partial_{\bar{w}}\varphi\partial_{w}\dot{F}^{i}(w_{j})d\bar{w}+\text{\scriptsize{$\mathcal{O}$}}\left(1\right).\end{split}$

which it implies that,

$\begin{split}&\partialderivative{\varepsilon}\Bigg{|}_{\varepsilon=0}\left(B_{1}+B_{2}+B_{3}\right)\\\
&\hskip
28.45274pt=-\oint_{C_{j}^{\epsilon}}\partial_{w}\dot{F}^{i}\left(\frac{d\bar{w}}{\bar{w}-\bar{w}_{j}}-\frac{dw}{w-w_{j}}\right)-\oint_{C_{j}^{\epsilon}}\partial_{w}\dot{F}^{i}(w_{j})d\varphi-2\oint_{C_{j}^{\epsilon}}\varphi\left(\frac{M_{i}\hskip
2.84526ptd\bar{w}}{w-w_{j}}\right)+\text{\scriptsize{$\mathcal{O}$}}\left(1\right)\\\
&\hskip
28.45274pt\overset{\eqref{MAsymptotic}}{=}-\oint_{C_{j}^{\epsilon}}\partial_{w}\dot{F}^{i}\left(\frac{d\bar{w}}{\bar{w}-\bar{w}_{j}}-\frac{dw}{w-w_{j}}\right)+\text{\scriptsize{$\mathcal{O}$}}\left(1\right)\end{split}$

Therefore, according to the above result, for the third term in (156) we get

$\begin{split}&\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{C_{j}^{\epsilon}}\partialderivative{\varepsilon}\Bigg{|}_{\varepsilon=0}(\varphi^{\varepsilon\mu_{i}}\circ
F^{\varepsilon\mu_{i}})\left(\frac{\overline{\partial_{w}F^{\varepsilon\mu_{i}}}(\bar{\varepsilon}\overline{M_{i}}\differential{w}+\differential{\bar{w}})}{\overline{w^{\varepsilon\mu_{i}}}-\overline{w_{j}^{\varepsilon\mu_{i}}}}-\frac{\partial_{w}F^{\varepsilon\mu_{i}}(\differential{w}+\varepsilon
M_{i}\differential{\bar{w}})}{w^{\varepsilon\mu_{i}}-w_{j}^{\varepsilon\mu_{i}}}\right)\\\
&\hskip
28.45274pt=-\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{C_{j}^{\epsilon}}\partial_{w}\dot{F}^{i}\left(\frac{d\bar{w}}{\bar{w}-\bar{w}_{j}}-\frac{dw}{w-w_{j}}\right)+\text{\scriptsize{$\mathcal{O}$}}\left(1\right)\end{split}$
(157)

Accordingly, the Eq.(156) is simplified more to

$\begin{split}\partialderivative{I_{\epsilon}}{w_{i}}&=\iint_{O_{\epsilon}}\left[(\partial_{w_{i}}\partial_{w}\varphi+\partial_{w}^{2}\varphi\dot{F}^{i})\partial_{\bar{w}}\varphi+(\partial_{w_{i}}\partial_{\bar{w}}\varphi+\partial_{w}\partial_{\bar{w}}\varphi\dot{F}^{i})\partial_{w}\varphi+|\partial_{w}\varphi|^{2}\partial_{w}\dot{F}^{i}\right]\differential{w}\wedge\differential{\bar{w}}\\\
&-\sum_{k=1}^{n}\int_{\partial
O^{\epsilon}_{k}}|\partial_{w}\varphi|^{2}\Big{(}\dot{F}^{i}(w)-\dot{F}^{i}(w_{k})\Big{)}\differential{\bar{w}}\\\
&-\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{C_{j}^{\epsilon}}\partial_{w}\dot{F}^{i}\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{j}}-\frac{\differential{w}}{w-w_{j}}\right)+\text{\scriptsize{$\mathcal{O}$}}\left(1\right),\end{split}$
(158)

Using the Lemma 3.4, we get

$\left\\{\begin{split}&\partial_{w_{i}}\partial_{w}\varphi+\partial_{w}^{2}\varphi\,\dot{F}^{i}=-\partial_{w}\varphi\,\partial_{w}\dot{F}^{i}-\partial_{w}^{2}\dot{F}^{i},\\\
\\\
&\partial_{w_{i}}\partial_{\bar{w}}\varphi+\partial_{w}\partial_{\bar{w}}\varphi\dot{F}^{i}=-\partial_{w}\varphi\,\partial_{\bar{w}}\dot{F}^{i}-\partial_{\bar{w}}\partial_{w}\dot{F}^{i},\end{split}\right.$
(159)

which makes it possible to rewrite (158) as

$\begin{split}\partialderivative{I_{\epsilon}}{w_{i}}&=\iint_{O_{\epsilon}}\left[(-\partial_{w}\varphi\,\partial_{w}\dot{F}^{i}-\partial_{w}^{2}\dot{F}^{i})\partial_{\bar{w}}\varphi+(-\partial_{w}\varphi\,\partial_{\bar{w}}\dot{F}^{i}-\partial_{\bar{w}}\partial_{w}\dot{F}^{i})\partial_{w}\varphi+|\partial_{w}\varphi|^{2}\partial_{w}\dot{F}^{i}\right]\differential{w}\wedge\differential{\bar{w}}\\\
&-\sum_{k=1}^{n}\int_{\partial
O^{\epsilon}_{k}}|\partial_{w}\varphi|^{2}\Big{(}\dot{F}^{i}(w)-\dot{F}^{i}(w_{k})\Big{)}\differential{\bar{w}}\\\
&-\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{C_{j}^{\epsilon}}\partial_{w}\dot{F}^{i}\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{j}}-\frac{\differential{w}}{w-w_{j}}\right)+\text{\scriptsize{$\mathcal{O}$}}\left(1\right)\\\
&=\iint_{O_{\epsilon}}\left[\Big{(}2\partial_{w}^{2}\varphi-(\partial_{w}\varphi)^{2}\Big{)}\partial_{\bar{w}}\dot{F}^{i}-2\partialderivative{w}(\partial_{w}\varphi\,\partial_{\bar{w}}\dot{F}^{i})+\partialderivative{\bar{w}}(\partial_{w}\varphi\,\partial_{w}\dot{F}^{i})-\partialderivative{w}(\partial_{\bar{w}}\varphi\,\partial_{w}\dot{F}^{i})\right]\differential{w}\wedge\differential{\bar{w}}\\\
&-\sum_{k=1}^{n}\int_{\partial
O^{\epsilon}_{k}}|\partial_{w}\varphi|^{2}\Big{(}\dot{F}^{i}(w)-\dot{F}^{i}(w_{k})\Big{)}\differential{\bar{w}}\\\
&-\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{C_{j}^{\epsilon}}\partial_{w}\dot{F}^{i}\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{j}}-\frac{\differential{w}}{w-w_{j}}\right)+\text{\scriptsize{$\mathcal{O}$}}\left(1\right)\\\
&=\underbrace{\iint_{O_{\epsilon}}\Big{(}2\partial_{w}^{2}\varphi-(\partial_{w}\varphi)^{2}\Big{)}\partial_{\bar{w}}\dot{F}^{i}\differential{w}\wedge\differential{\bar{w}}}_{I_{1}}\underbrace{-2\int_{\partial
O_{\epsilon}}\partial_{w}\varphi\,\partial_{\bar{w}}\dot{F}^{i}\differential{\bar{w}}}_{I_{2}}\underbrace{-\int_{\partial
O_{\epsilon}}\partial_{w}\varphi\,\partial_{w}\dot{F}^{i}\differential{w}}_{I_{3}}\\\
&\underbrace{-\int_{\partial
O_{\epsilon}}\partial_{\bar{w}}\varphi\,\partial_{w}\dot{F}^{i}\differential{\bar{w}}}_{I_{4}}\underbrace{-\int_{\partial
O_{\epsilon}}|\partial_{w}\varphi|^{2}\dot{F}^{i}\differential{\bar{w}}}_{I_{5}}+\oint_{C^{\epsilon}_{i}}|\partial_{w}\varphi|^{2}\differential{\bar{w}}\\\
&-\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{C_{j}^{\epsilon}}\partial_{w}\dot{F}^{i}\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{j}}-\frac{\differential{w}}{w-w_{j}}\right)+\text{\scriptsize{$\mathcal{O}$}}\left(1\right).\end{split}$
(160)

Let us compute each of the integrals $I_{1},\dots,I_{5}$ separately using
Lemma C.1 and Corollary 3.3.1 as well as equations (82) and (83). We begin
with the integral $I_{1}$:

$\begin{split}I_{1}&=\iint_{O_{\epsilon}}\Big{(}2\partial_{w}^{2}\varphi-(\partial_{w}\varphi)^{2}\Big{)}\partial_{\bar{w}}\dot{F}^{i}\differential{w}\wedge\differential{\bar{w}}=-2\int_{\partial
O_{\epsilon}}T_{\varphi}\dot{F}^{i}\differential{w}\\\
&=-2\sum_{j=1}^{n_{e}}\oint_{C^{\epsilon}_{j}}\left(\frac{h_{j}}{2(w-w_{j})^{2}}+\frac{c_{j}}{w-w_{j}}+\dotsm\right)\left(\delta_{ij}+(w-w_{j})\partial_{w}\dot{F}^{i}(w_{j})+\dotsm\right)\differential{w}\\\
&-2\sum_{j=n_{e}+1}^{n-1}\oint_{C^{\epsilon}_{j}}\left(\frac{1}{2(w-w_{j})^{2}}+\frac{c_{j}}{w-w_{j}}+\dotsm\right)\left(\delta_{ij}+(w-w_{j})\partial_{w}\dot{F}^{i}(w_{j})+\dotsm\right)\differential{w}\\\
&-2\oint_{C_{n}^{\epsilon}}\left(\frac{1}{2w^{2}}+\frac{c_{n}}{w^{3}}+\dotsm\right)\left(w\partial_{w}\dot{F}^{i}(\infty)+\dotsm\right)\differential{w}\\\
&=4\pi\sqrt{-1}c_{i}+2\pi\sqrt{-1}\sum_{k=1}^{n-1}h_{k}\partial_{w}\dot{F}^{i}(w_{k})-2\pi\sqrt{-1}\partial_{w}\dot{F}^{i}(\infty)+\text{\scriptsize{$\mathcal{O}$}}\left(1\right)\quad\text{as}\quad\epsilon\to
0.\end{split}$

In the last line we have used the notation $h_{k}=1-1/m_{k}^{2}$ for
$k=1,\dots,n_{e}$ and $h_{k}=1$ for $k=n_{e}+1,\dots,n-1$ . In addition, we
have

$\begin{split}I_{2}&=-2\int_{\partial
O_{\epsilon}}\partial_{w}\varphi\,\partial_{\bar{w}}\dot{F}^{i}\differential{\bar{w}}=-2\int_{\partial
O_{\epsilon}}\partial_{w}\varphi(w)\,M_{i}(w)\differential{\bar{w}}\\\
&\overset{\eqref{MAsymptotic}}{=}-2\sum_{j=1}^{n_{e}}\oint_{C^{\epsilon}_{j}}\left(-\frac{1-\frac{1}{m_{j}}}{w-w_{j}}+\frac{c_{j}}{1-\frac{1}{m_{j}}}+\dotsm\right)\left(\frac{\bar{q}^{(j)}_{1}}{4\bar{J}^{(j)}_{1}}\cdot|w-w_{j}|^{1-\frac{2}{m_{j}}}+\dotsm\right)\differential{\bar{w}}\\\
&-2\sum_{j=n_{e}+1}^{n-1}\oint_{C^{\epsilon}_{j}}\left(-\frac{1}{w-w_{j}}\left(1+\frac{1}{\log\left|\frac{w-w_{j}}{J_{1}^{(j)}}\right|}\right)+c_{j}+\dotsm\right)\times\\\
&\hskip
76.82234pt\times\left(-\frac{|\delta_{j}|^{2}\bar{q}_{1}^{(j)}}{4\pi^{2}\bar{J}_{1}^{(j)}}\cdot|w-w_{j}|\log^{2}|w-w_{j}|+\dotsm\right)\differential{\bar{w}}\\\
&-2\oint_{C_{n}^{\epsilon}}\left(-\frac{1}{w}\left(1+\frac{1}{\log\left|\frac{w}{J_{-1}^{(n)}}\right|}\right)-\frac{c_{n}}{w^{2}}+\dotsm\right)\left(-\frac{|\delta_{n}|^{2}\bar{q}_{1}^{(n)}\bar{J}_{-1}^{(n)}}{4\pi^{2}}\cdot\frac{\log^{2}|w|}{|w|}+\dotsm\right)\differential{\bar{w}}\\\
&=\text{\scriptsize{$\mathcal{O}$}}\left(1\right),\end{split}$

as $\epsilon\to 0$. Let us now calculate $I_{3}+I_{4}$:

$\begin{split}I_{3}+I_{4}&=-\int_{\partial
O_{\epsilon}}\partial_{w}\varphi\,\partial_{w}\dot{F}^{i}\differential{w}-\int_{\partial
O_{\epsilon}}\partial_{\bar{w}}\varphi\,\partial_{w}\dot{F}^{i}\differential{\bar{w}}=-\int_{\partial
O_{\epsilon}}\partial_{w}\dot{F}^{i}\differential{\varphi}=\text{\scriptsize{$\mathcal{O}$}}\left(1\right),\end{split}$

as $\epsilon\to 0$. Finally, we can turn to calculating $I_{5}$:

$\begin{split}I_{5}&=-\int_{\partial
O_{\epsilon}}|\partial_{w}\varphi|^{2}\dot{F}^{i}\differential{\bar{w}}\\\
&=-\sum_{k=1}^{n-1}\oint_{C^{\epsilon}_{k}}|\partial_{w}\varphi|^{2}\left(\delta_{ik}+(w-w_{k})\partial_{w}\dot{F}^{i}(w_{k})+\dotsm\right)\differential{\bar{w}}\\\
&\hskip
11.38092pt-\oint_{C_{n}^{\epsilon}}\left(\frac{1}{|w|^{2}}+\dotsm\right)\left(w\partial_{w}\dot{F}^{i}(\infty)+\dotsm\right)\differential{\bar{w}}\\\
&=-\oint_{C^{\epsilon}_{i}}|\partial_{w}\varphi|^{2}\differential{\bar{w}}-\sum_{k=1}^{n-1}\oint_{C^{\epsilon}_{k}}\left(\frac{\left(1-\frac{1}{m_{k}}\right)^{2}}{|w-w_{k}|^{2}}+\dotsm\right)\left((w-w_{k})\partial_{w}\dot{F}^{i}(w_{k})+\dotsm\right)\differential{\bar{w}}\\\
&\hskip
11.38092pt-\oint_{C_{n}^{\epsilon}}\left(\frac{1}{|w|^{2}}+\dotsm\right)\left(w\partial_{w}\dot{F}^{i}(\infty)+\dotsm\right)\differential{\bar{w}}\\\
&=-\oint_{C^{\epsilon}_{i}}|\partial_{w}\varphi|^{2}\differential{\bar{w}}-2\pi\sqrt{-1}\sum_{k=1}^{n-1}\left(1-\frac{1}{m_{k}}\right)^{2}\partial_{w}\dot{F}^{i}(w_{k})+2\pi\sqrt{-1}\partial_{w}\dot{F}^{i}(\infty)+\text{\scriptsize{$\mathcal{O}$}}\left(1\right),\end{split}$

as $\epsilon\to 0$ with $m_{k}=\infty$ for $k=n_{e}+1,\dots,n-1$. As a result,
we have

$I_{1}+\dotsm+I_{5}=4\pi\sqrt{-1}c_{i}-\oint_{C^{\epsilon}_{i}}|\partial_{w}\varphi|^{2}\differential{\bar{w}}+2\pi\sqrt{-1}\sum_{j=1}^{n_{e}}\left(-\frac{2}{m_{j}^{2}}+\frac{2}{m_{j}}\right)\partial_{w}\dot{F}^{i}(w_{j})+\text{\scriptsize{$\mathcal{O}$}}\left(1\right),$
(161)

as $\epsilon\to 0$. Moreover, by changing the conformal structure, the
variation of counterterm action $\tilde{S}^{\epsilon}_{\text{ct}}$ gives

$\begin{split}&\partialderivative{\tilde{S}_{\text{ct}}^{\epsilon}}{w_{i}}=-2\pi\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)^{2}\partial_{w}\dot{F}^{i}(w_{j})+\text{\scriptsize{$\mathcal{O}$}}\left(1\right)\hskip
14.22636pt\text{as}\hskip 8.5359pt\epsilon\rightarrow 0\end{split}$ (162)

Notice that we have only taken into account the contributions coming from the
regulating terms for conical points and not the punctures. The reason for this
can be traced back to the equation (142). By doing similar analysis as in
equation (157) for the contour integrals in (142), it is easy to see that the
quasi-conformal transformations of these contour integrals are canceled by
those of the counterterm $-2\pi n_{p}\log\epsilon$. In other words, it is
sufficient to only consider the quasi-conformal transformations of the bulk
term for the case of punctures, which is in agreement with the analysis in
Zograf1988ONLE .

Now, using equations (160), (161), (147), (162), and putting everything
together, we arrive at our desired result

$\begin{split}\partialderivative{\tilde{S}_{\boldsymbol{m}}^{\epsilon}}{w_{i}}&=\frac{\sqrt{-1}}{2}\left(4\pi\sqrt{-1}c_{i}-\oint_{C^{\epsilon}_{i}}|\partial_{w}\varphi|^{2}\differential{\bar{w}}+2\pi\sqrt{-1}\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\left(\frac{2}{m_{j}}\right)\partial_{w}\dot{F}^{i}(w_{j})\right.\\\
&\left.+\oint_{C^{\epsilon}_{i}}|\partial_{w}\varphi|^{2}\differential{\bar{w}}-2\pi\sqrt{-1}\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\left(\frac{2}{m_{j}}\right)\partial_{w}\dot{F}^{i}(w_{j})+\text{\scriptsize{$\mathcal{O}$}}\left(1\right)\right)\\\
&=-2\pi
c_{i}+\text{\scriptsize{$\mathcal{O}$}}\left(1\right)\quad\text{as}\quad\epsilon\to
0.\end{split}$

In order to complete the proof, it remains to show that the remainder in the
above formula can be estimated uniformly in a neighborhood of an arbitrary
point $(w_{1},\dots,w_{n-3})\in\mathcal{M}_{0,n}$. Let $\Gamma$ be a Fuchsian
group uniformizing the orbisurface $O=(\hat{\mathbb{C}},\mathscr{D})$. Then,
the Hauptmodule $J_{\mu}$, where $\mu=t_{1}\mu_{1}+\dotsm+t_{n-3}\hskip
2.84526pt\mu_{n-3}\in\mathcal{H}^{-1,1}(\mathbb{H},\Gamma)$, is continuously
differentiable with respect to the Bers’ coordinates $t_{i}$ and its
coefficients in an expansion similar to Eq.(63) have the same property. From
this, we can conclude with the help of Eq.(80) and Lemma 3.3 that the
remainders in assertions (2) and (4) of Lemma C.1 can be estimated uniformly.
Using the commutative diagram (87), we can conclude that an analogous
assertion is valid also for the remainders in Corollary 3.3.1 for
$\dot{F}^{i}$, and this completes the proof of the theorem. ∎

######  Remark 4.3.

Theorem 4.1 means that
$\sum_{i=1}^{n-3}(c_{i}\differential{w_{i}}+\bar{c}_{i}\differential{\bar{w}_{i}})$
is an exact 1-form on $\mathcal{M}_{0,n}$ with anti-derivative
$-S_{\boldsymbol{m}}/2\pi$.

######  Remark 4.4.

Similar to $\partial S_{\boldsymbol{m}}/\partial w_{i}$ which are proportional
to the accessory parameters, the derivatives $\partial\log\mathsf{h}/\partial
w_{i}$ are defined to be another set of parameters called the _auxiliary
parameters_.

In addition to Theorem 4.1, we can also generalize (Zograf1988ONLE, , Theorem
2) to include ramification points: The Weil-Petersson metric defined on
$\mathcal{T}_{0,n}(\boldsymbol{m})$ in Sec. 3.1.3 can be projected onto
$\mathcal{M}_{0,n}$, since it is invariant under $\operatorname{Aut}(\Psi)$.
We will continue to call this metric obtained on $\mathcal{M}_{0,n}$ as the
Weil-Petersson metric and denote it with the same notation
$\langle\cdot,\cdot\rangle_{{}_{\text{WP}}}$. Then, we have the following
theorem:

######  Theorem 4.2.

For any fixed ordered set of ramification indices
$\boldsymbol{m}=(m_{1},\dots,m_{n})$ such that $2\leq m_{1}\leq
m_{2}\leq\dots\leq m_{n}$ and
$\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)+n_{p}>2$, the function
$-S_{\boldsymbol{m}}$ is a real-analytic Kähler potential for the metric
$\langle\cdot,\cdot\rangle_{{}_{WP}}$ on $\mathcal{M}_{0,n}$

$\bar{\partial}\partial S_{\boldsymbol{m}}=-2\sqrt{-1}\hskip
2.84526pt\omega_{\text{WP}}.$

###### Proof.

In order to prove this theorem, we have to prove that the accessory parameters
$c_{1},\dots,c_{n-3}$ are continuously differentiable on $\mathcal{M}_{0,n}$,
and

$\partialderivative{c_{i}}{\bar{w}_{j}}=\frac{1}{2\pi}\left\langle\partialderivative{w_{i}},\partialderivative{w_{j}}\right\rangle_{{}_{\text{WP}}}\quad\text{for}\quad
i,j=1,\dots,n-3.$ (163)

The proof of continuous differentiability of the functions $c_{i}$ on
$\mathcal{M}_{0,n}$ follows readily from the definition of accessory
parameters (84) and continuous differentiability of the Hauptmodule $J$ with
respect to the Ber’s coordinates. We now turn to proving Eq.(163) in the same
way as for Theorem 2 in Zograf1988ONLE  where no ramification point exists.
Let $(w_{1},\dots,w_{n-3})$ be an arbitrary point in $\mathcal{M}_{0,n}$ and
let $\Gamma$ be a Fuchsian group uniformizing the orbisurface $O$. It follows
from the commutative diagram (87) that

$\text{Sch}\left(J_{\varepsilon\mu_{j}}^{-1}\circ
F^{\varepsilon\mu_{j}};w\right)=\text{Sch}\left(f^{\varepsilon\mu_{j}}\circ
J^{-1};z\right),$

where $\mu_{j}$ is an element of the basis in
$\mathcal{H}^{-1,1}(\mathbb{H},\Gamma)$ given by Eq.(95), and
$\varepsilon\in\mathbb{C}$ is sufficiently small. By using the following well-
known property of _Schwarzian derivative_ ,

$\text{Sch}\left(A\circ B;w\right)=\text{Sch}\left(A;w\right)\circ B\hskip
2.84526ptB^{\prime 2}+\text{Sch}\left(B;w\right)$

we have

$\text{Sch}\left(J_{\varepsilon\mu_{j}}^{-1};w\right)\circ
F^{\varepsilon\mu_{j}}\left(\partial_{w}F^{\varepsilon\mu_{j}}\right)^{2}+\text{Sch}\left(F^{\varepsilon\mu_{j}};w\right)=\text{Sch}\left(f^{\varepsilon\mu_{j}};z\right)\circ
J^{-1}(J^{-1})^{\prime 2}+\text{Sch}\left(J^{-1};z\right).$

We can now differentiate both sides of the above equality with respect to
$\bar{\varepsilon}$ at the point $\varepsilon=0$, using Eq.(82) and the fact
that $F^{\varepsilon\mu_{j}}$, and as a result $w_{i}^{\varepsilon\mu_{j}}$,
are holomorphic functions of $\varepsilon$ at $\varepsilon=0$. The left-hand
side gives

$\begin{split}&\left.\partialderivative{\bar{\varepsilon}}\right|_{\varepsilon=0}\left(\text{Sch}\left(J_{\varepsilon\mu_{j}}^{-1};w\right)\circ
F^{\varepsilon\mu_{j}}\left(\partial_{w}F^{\varepsilon\mu_{j}}\right)^{2}+\text{Sch}\left(F^{\varepsilon\mu_{j}};w\right)\right)\\\
&\hskip
22.76228pt=\left.\partialderivative{T_{\varphi^{\varepsilon\mu_{j}}}}{\bar{\varepsilon}}\right|_{\varepsilon=0}+\partial_{w}T_{\varphi}\left.\partialderivative{F^{\varepsilon\mu_{j}}}{\bar{\varepsilon}}\right|_{\varepsilon=0}+T_{\varphi}\left.\partialderivative{\bar{\varepsilon}}\right|_{\varepsilon=0}\left(\partial_{w}F^{\varepsilon\mu_{j}}\right)^{2}+\left.\partialderivative{\bar{\varepsilon}}\right|_{\varepsilon=0}\text{Sch}\left(F^{\varepsilon\mu_{j}};w\right)\\\
&\hskip
22.76228pt=\left.\partialderivative{\bar{\varepsilon}}\right|_{\varepsilon=0}\sum_{i=1}^{n-1}\left(\frac{h_{i}}{2(w-w_{i}^{\varepsilon\mu_{j}})^{2}}+\frac{c_{i}^{\varepsilon\mu_{j}}}{w-w_{i}^{\varepsilon\mu_{j}}}\right)=\sum_{i=1}^{n-1}\left(\left.\partialderivative{c_{i}^{\varepsilon\mu_{j}}}{\bar{\varepsilon}}\right|_{\varepsilon=0}\right)\frac{1}{w-w_{i}},\end{split}$
(164)

where
$c_{i}^{\varepsilon\mu_{j}}=c_{i}(w_{1}^{\varepsilon\mu_{j}},\dots,w_{n-3}^{\varepsilon\mu_{j}})$.
To compute the right-hand side, we will use the definition of Schwarzian
derivative (15), the fact that $\partial/\partial{\bar{\varepsilon}}$ of the
functions $\partial_{z}f^{\varepsilon\mu_{j}}$ and
$\partial_{z}^{2}f^{\varepsilon\mu_{j}}$ vanishes at $\varepsilon=0$, and the
well-known Ahlfors formula494949The Ahlfors formula can be derived by
comparing the expression for $\dot{f}^{\mu_{j}}_{-}(z)$, given by Eq.(91), and
the equality $\varLambda\varLambda^{\ast}=\operatorname{id}$ discussed in
Section 3.1.1. Ahlfors_quasiconformal_06 ; Ahlfors1961SomeRO

$\left.\partialderivative{\bar{\varepsilon}}\partial_{z}^{3}f^{\varepsilon\mu_{j}}\right|_{\varepsilon=0}=-\frac{1}{2}q_{j},$

where $\mu_{j}$ and $q_{j}$ are connected by the relation (95). Then, we have

$\begin{split}&\left.\partialderivative{\bar{\varepsilon}}\right|_{\varepsilon=0}\bigg{(}\text{Sch}\left(f^{\varepsilon\mu_{j}};z\right)\circ
J^{-1}(J^{-1})^{\prime 2}+\text{Sch}\left(J^{-1};z\right)\bigg{)}\\\ &\hskip
34.14322pt=\left.\partialderivative{\bar{\varepsilon}}\left[\partial_{z}^{3}f^{\varepsilon\mu_{j}}\circ
J^{-1}(J^{-1})^{\prime 2}\right]\right|_{\varepsilon=0}=-\frac{1}{2}q_{j}\circ
J^{-1}(J^{-1})^{\prime 2}=-\frac{1}{2}Q_{j}.\end{split}$ (165)

Equating the left-hand side, (164), and the right-hand side, (165), and using
Eq.(85), we get

$\begin{split}&\sum_{i=1}^{n-1}\left(\left.\partialderivative{c_{i}^{\varepsilon\mu_{j}}}{\bar{\varepsilon}}\right|_{\varepsilon=0}\right)\frac{1}{w-w_{i}}\\\
&\hskip
8.5359pt=\sum_{i=1}^{n-3}\left(\left.\partialderivative{c_{i}^{\varepsilon\mu_{j}}}{\bar{\varepsilon}}\right|_{\varepsilon=0}\right)\frac{1}{w-w_{i}}+\left(\left.\partialderivative{\bar{\varepsilon}}c_{n-2}^{\varepsilon\mu_{j}}\right|_{\varepsilon=0}\right)\frac{1}{w}+\left(\left.\partialderivative{\bar{\varepsilon}}c_{n-1}^{\varepsilon\mu_{j}}\right|_{\varepsilon=0}\right)\frac{1}{w-1}\\\
&\hskip
8.5359pt=\sum_{i=1}^{n-3}\left(\left.\partialderivative{c_{i}^{\varepsilon\mu_{j}}}{\bar{\varepsilon}}\right|_{\varepsilon=0}\right)\frac{1}{w-w_{i}}+\left(\left.\partialderivative{\bar{\varepsilon}}\left(-1+\frac{n}{2}+\sum_{i=1}^{n-3}c_{i}^{\varepsilon\mu_{j}}(w_{i}^{\varepsilon\mu_{j}}-1)\right)\right|_{\varepsilon=0}\right)\frac{1}{w}\\\
&\hskip
28.45274pt+\left(\left.\partialderivative{\bar{\varepsilon}}\left(1-\frac{n}{2}-\sum_{i=1}^{n-3}c_{i}^{\varepsilon\mu_{j}}w_{i}^{\varepsilon\mu_{j}}\right)\right|_{\varepsilon=0}\right)\frac{1}{w-1}\\\
&\hskip
8.5359pt=\sum_{i=1}^{n-3}\left(\left.\partialderivative{c_{i}^{\varepsilon\mu_{j}}}{\bar{\varepsilon}}\right|_{\varepsilon=0}\right)\left(\frac{1}{w-w_{i}}+\frac{w_{i}-1}{w}-\frac{w_{i}}{w-1}\right)\overset{\eqref{Ridef}}{=}-\pi\sum_{i=1}^{n-3}\left(\left.\partialderivative{c_{i}^{\varepsilon\mu_{j}}}{\bar{\varepsilon}}\right|_{\varepsilon=0}\right)R_{i}=-\frac{1}{2}Q_{j},\end{split}$

or

$\sum_{i=1}^{n-3}\left(\left.\partialderivative{c_{i}^{\varepsilon\mu_{j}}}{\bar{\varepsilon}}\right|_{\varepsilon=0}\right)R_{i}=\frac{1}{2\pi}Q_{j}.$

It now follows from the biorthogonality of the bases $R_{i}$ and $Q_{j}$ and
Lemma 3.3 that

$\partialderivative{c_{i}}{\bar{w}_{j}}=\frac{1}{2\pi}\langle
Q_{i},Q_{j}\rangle=\frac{1}{2\pi}\left\langle\partialderivative{w_{i}},\partialderivative{w_{j}}\right\rangle_{{}_{\text{WP}}},$

which is what we wanted. Combining Eq.(163) and Theorem 4.1, we can conclude
that the real-valued function $-S_{\boldsymbol{m}}$ is a potential for the
Weil-Petersson metric on $\mathcal{M}_{0,n}$

$-\partialderivative{S_{\boldsymbol{m}}}{w_{i}}{\bar{w}_{j}}=\left\langle\partialderivative{w_{i}},\partialderivative{w_{j}}\right\rangle_{{}_{\text{WP}}},$

while the function $-S_{\boldsymbol{m}}\circ\Psi$ is a potential for the Weil-
Petersson metric on the Teichmüller space $\mathcal{T}_{0,n}(\boldsymbol{m})$.
In addition, the real analyticity of the Weil-Petersson metric (see
Ahlfors1961SomeRO ) implies the real analyticity of the accessory parameters
$c_{1},\dots,c_{n-3}$ and of their generating function
$-S_{\boldsymbol{m}}/2\pi$ on the space $\mathcal{M}_{0,n}$. ∎

######  Remark 4.5.

Theorems 4.1 and 4.2 also give us a proof that the Weil-Petersson metric is a
Kähler metric.

Finally, let
$\mathfrak{M}_{0,n}(\boldsymbol{m})=\mathcal{M}_{0,n}/\operatorname{Symm}\left(\boldsymbol{\mathsf{s}}\right)$
be the moduli space of orbifold Riemann surfaces with signature
$(0;m_{1},\dots,m_{n_{e}};n_{p})$. One can generalize the results of
(Zograf_1990, , Sec.1) to prove the following Lemma:

######  Lemma 4.3.

$\exp[S_{\boldsymbol{m}}/\pi]$ determines a Hermitian metric in a holomorphic
line bundle $\lambda_{0,\boldsymbol{m}}$ over
$\mathfrak{M}_{0,n}(\boldsymbol{m})$ (see Section 3.2), so that

$\mathsf{c}_{1}\left(\lambda_{0,\boldsymbol{m}},\exp[S_{\boldsymbol{m}}/\pi]\right)=\frac{1}{\pi^{2}}\omega_{WP}.$
(166)

###### Proof.

We first need to show that $\exp[S_{\boldsymbol{m}}/\pi]$ is a hermitian
metric on the holomorphic line bundle $\lambda_{0,\boldsymbol{m}}$ defined in
the Section 3.2. To do so we use the same representation of
$\operatorname{Symm}\left(\boldsymbol{\mathsf{s}}\right)$ introduced there.
For our purpose, it suffices to prove this lemma for the case where the
signature of orbifold Riemann surface $O$ is given by
$(0;\underbrace{m,\dots,m}_{s},\underbrace{m^{\prime},\dots,m^{\prime}}_{s^{\prime}})$
with $s\equiv s_{m}$ and $s^{\prime}\equiv s_{m^{\prime}}>3$. Namely, we have
only two kinds of points, $s$ ramification points of order $m$ and
$s^{\prime}$ ramification points of order $m^{\prime}$. Furthermore we fix the
points $0$, $1$, and $\infty$ as the last three points in each
$s^{\prime}$-tuple on the stratum $\operatorname{Sing}_{m^{\prime}}$. Then the
$\mathcal{M}_{0,n}$ would be,

$\displaystyle\\{(w_{1},\dots,w_{s})\in\mathbb{C}^{s}\\}\times$
$\displaystyle\\{(w_{1},w_{2},\dots,w_{s^{\prime}-3},0,1)\in\mathbb{C}^{s^{\prime}-1}\\}$
$\displaystyle=\\{(w_{1},\dots,w_{s},w_{s+1},w_{s+2},\dots,w_{s+s^{\prime}-3},0,1)\\},$

and the generators and 1-cocycles of
$\operatorname{Symm}\left(\boldsymbol{\mathsf{s}}\right)=\operatorname{Symm}\left(s\right)\times\operatorname{Symm}\left(s^{\prime}\right)$
would be the same as the ones we introduced in Section 3.2. Moreover, in our
case, we have

$S_{\boldsymbol{m}}[\varphi]=\\\ \hskip 14.22636pt\lim_{\epsilon\to
0^{+}}\left(\iint_{O_{\epsilon}}(|\partial_{w}\varphi|^{2}+e^{\varphi})\differential[2]{w}+\frac{\sqrt{-1}}{2}\sum_{k=1}^{s+s^{\prime}-1}\left(1-\frac{1}{m_{k}}\right)\oint_{C_{k}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{k}}-\frac{\differential{w}}{w-w_{k}}\right)\right.\\\
\left.+\frac{\sqrt{-1}}{2}\left(1+\frac{1}{m^{\prime}}\right)\oint_{C_{s+s^{\prime}}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}}-\frac{\differential{w}}{w}\right)\right),$
(167)

where

$\left\\{\begin{split}&O_{\epsilon}=\mathbb{C}\Big{\backslash}\bigcup_{k=1}^{s+s^{\prime}-1}\left\\{w\,\Big{|}\,|w-w_{k}|<\epsilon\right\\}\cup\left\\{w\,\Big{|}\,|w|>\epsilon^{-1}\right\\},\\\
&C_{k}^{\epsilon}=\left\\{w\,\Big{|}\,|w-w_{k}|=\epsilon\right\\},\hskip
14.22636ptk=1,2,\dots,s+s^{\prime}-1\\\
&C_{s+s^{\prime}}^{\epsilon}=\left\\{w\,\Big{|}\,|w|=1/\epsilon\right\\},\end{split}\right.$
(168)

and the circles are oriented as a component of the boundary $\partial
O_{\epsilon}$ of $O_{\epsilon}$. Notice that we did not consider the
counterterms in (167). Actually
$\operatorname{Symm}\left(\boldsymbol{\mathsf{s}}\right)$ does not change the
conformal family, and therefore the counterterms will not contribute to the
variation under the action of this group, and therefore writing them would be
redundant. To study the variation of $S_{\boldsymbol{m}}[\varphi]$ under the
action of $\operatorname{Symm}\left(\boldsymbol{\mathsf{s}}\right)$, it
suffices to study its variation under the generators
$\\{(\sigma_{j,j+1},\sigma^{\prime}_{i,i+1})\\}_{j=1,i=1}^{j=s,i=s^{\prime}-1}$
of $\operatorname{Symm}\left(\boldsymbol{\mathsf{s}}\right)$. The effect of
these generators is translated to the induced isomorphism
$\gamma_{j,j+1;i,i+1}$ introduced in Sec.3.2. Readily we have:

$\begin{split}&\Delta
S_{\boldsymbol{m}}[\varphi]=S_{\boldsymbol{m}}[\varphi]\circ(\sigma_{k,k+1},\sigma^{\prime}_{i,i+1})-S_{\boldsymbol{m}}[\varphi]\\\
&\hskip 2.84544pt=\lim_{\epsilon\to
0^{+}}\bigg{(}\frac{\sqrt{-1}}{2}\bigg{\\{}\Delta
S_{\boldsymbol{m}}^{(1)}[\varphi]+\big{(}1-\frac{1}{m}\big{)}\Delta
S_{\boldsymbol{m}}^{(2)}[\varphi]+\big{(}1-\frac{1}{m^{\prime}}\big{)}\Delta
S_{\boldsymbol{m}}^{(3)}[\varphi]+\big{(}1+\frac{1}{m^{\prime}}\big{)}\Delta
S_{\boldsymbol{m}}^{(4)}[\varphi]\bigg{\\}}\bigg{)},\end{split}$ (169)

where

$\begin{split}&\Delta
S_{\boldsymbol{m}}^{(1)}[\varphi]=\iint_{\tilde{O}_{\epsilon}}(|\partial_{\gamma_{j,j+1;i,i+1}w}\tilde{\varphi}|^{2}+e^{\tilde{\varphi}})\differential{(\gamma_{j,j+1;i,i+1}w)}\wedge\differential{(\overline{\gamma_{j,j+1;i,i+1}w})}\\\
&\hskip
56.9055pt-\iint_{O_{\epsilon}}(|\partial_{w}\varphi|^{2}+e^{\varphi})\differential{w}\wedge\differential{\bar{w}},\\\
&\Delta
S_{\boldsymbol{m}}^{(2)}[\varphi]=\sum_{k=1}^{s}\bigg{\\{}\oint_{\tilde{C}_{k}^{\epsilon}}\tilde{\varphi}\left(\frac{\differential{(\overline{\gamma_{j,j+1;i,i+1}w})}}{\overline{\gamma_{j,j+1;i,i+1}w}-\overline{\gamma_{j,j+1;i,i+1}w}_{k}}-\frac{\differential{(\gamma_{j,j+1;i,i+1}w)}}{\gamma_{j,j+1;i,i+1}w-\gamma_{j,j+1;i,i+1}w_{k}}\right)\\\
&\hskip
56.9055pt-\oint_{C_{k}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{k}}-\frac{\differential{w}}{w-w_{k}}\right)\bigg{\\}},\\\
&\Delta
S_{\boldsymbol{m}}^{(3)}[\varphi]=\sum_{k=s+1}^{s+s^{\prime}-1}\bigg{\\{}\oint_{\tilde{C}_{k}^{\epsilon}}\tilde{\varphi}\left(\frac{\differential{(\overline{\gamma_{j,j+1;i,i+1}w})}}{\overline{\gamma_{j,j+1;i,i+1}w}-\overline{\gamma_{j,j+1;i,i+1}w_{k}}}-\frac{\differential{(\gamma_{j,j+1;i,i+1}w)}}{\gamma_{j,j+1;i,i+1}w-\gamma_{j,j+1;i,i+1}w_{k}}\right)\\\
&\hskip
56.9055pt-\oint_{C_{k}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{k}}-\frac{\differential{w}}{w-w_{k}}\right)\bigg{\\}},\\\
&\Delta
S_{\boldsymbol{m}}^{(4)}[\varphi]=\oint_{\tilde{C}_{s+s^{\prime}}^{\epsilon}}\tilde{\varphi}\left(\frac{\differential{(\overline{\gamma_{j,j+1;i,i+1}w})}}{\overline{\gamma_{j,j+1;i,i+1}w}}-\frac{\differential{(\gamma_{j,j+1;i,i+1}w)}}{\gamma_{j,j+1;i,i+1}w}\right)-\oint_{C_{s+s^{\prime}}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}}-\frac{\differential{w}}{w}\right),\end{split}$

and $\tilde{O}_{\epsilon}$, $\tilde{C}_{k}^{\epsilon}$, $\tilde{\varphi}$ are
the transformed orbifold Riemann surface, circles, Liouville field under
$\gamma_{j,j+1;i,i+1}$, respectively, such that
$\tilde{O}_{\epsilon}=\mathbb{C}\backslash\cup_{k=1}^{s+s^{\prime}}\operatorname{Int}\tilde{C}_{k}^{\epsilon}$.
By looking at (67) we see that the index $j$ does not have any non-trivial
effect, and we only have to worry about different values of $i$. For
$i<s^{\prime}-3$ the isomorphism $\gamma_{j,j+1;i,i+1}$ is the identity so for
this cases we have $\Delta S_{\boldsymbol{m}}[\varphi]=0$. Accordingly, the
only non-trivial cases are $i=s^{\prime}-3,s^{\prime}-2,s^{\prime}-1$. Let us
look at the case with $i=s^{\prime}-3$. Here we have the isomorphism
$\gamma_{s^{\prime}-3,s^{\prime}-2}=(w-w_{s+s^{\prime}-3})/(1-w_{s+s^{\prime}-3})$
which for simplicity we call $\gamma$. Let us calculate each contribution in
(169) separately 505050For $\Delta S_{\boldsymbol{m}}^{(1)}[\varphi]$, the
exponential terms give the same constants, and we can safely ignore them in
variation..

$\displaystyle\Delta S_{\boldsymbol{m}}^{(1)}[\varphi]$
$\displaystyle=\iint_{\tilde{O}_{\epsilon}}|\partial_{\gamma
w}\tilde{\varphi}|^{2}\differential{(\gamma
w)}\wedge\differential{(\overline{\gamma
w})}-\iint_{O_{\epsilon}}|\partial_{w}\varphi|^{2}\differential{w}\wedge\differential{\bar{w}}$
$\displaystyle=\iint_{\gamma^{-1}(\tilde{O}_{\epsilon})}|\partial_{w}\tilde{\varphi}\circ\gamma|^{2}|\gamma^{\prime}|^{-2}(|\gamma^{\prime}|^{2}\differential{w}\wedge\differential{\bar{w}})-\iint_{O_{\epsilon}}|\partial_{w}\varphi|^{2}\differential{w}\wedge\differential{\bar{w}}$
$\displaystyle=\iint_{\gamma^{-1}(\tilde{O}_{\epsilon})\backslash
O_{\epsilon}}|\partial_{w}\varphi|^{2}\differential{w}\wedge\differential{\bar{w}},$

where $\gamma^{\prime}=\partial\gamma(w)/\partial w$ and in the last line, we
used the invariance of the hyperbolic metric:

$\displaystyle e^{\tilde{\varphi}\circ\gamma}\differential{(\gamma
w)}\wedge\differential{(\overline{\gamma
w})}=e^{\varphi}\differential{w}\wedge\differential{\bar{w}}\implies\tilde{\varphi}\circ\gamma+\log|\gamma^{\prime}|^{2}=\varphi\implies\partial_{w}\tilde{\varphi}\circ\gamma=\partial_{w}\varphi,$
(170)

since $\gamma^{\prime}=1/(1-w_{s+s^{\prime}-3})$. The region
$\gamma^{-1}(\tilde{O}_{\epsilon})\backslash O_{\epsilon}$ contains a part of
$\mathbb{C}$ bounded by the circles $\tilde{C}_{k}$ and $C_{k}$. Thus, by
integration by parts and using the equations of motion together with taking
into account the orientation of these circles, we have:

$\Delta
S_{\boldsymbol{m}}^{(1)}[\varphi]=\sum_{k=1}^{s+s^{\prime}-1}\left(\oint_{\tilde{C}_{k}^{\epsilon}}\varphi\partial_{\bar{w}}\varphi\differential{\bar{w}}-\oint_{C_{k}^{\epsilon}}\varphi\partial_{\bar{w}}\varphi\differential{\bar{w}}\right)+\left(\oint_{\tilde{C}_{s+s^{\prime}}^{\epsilon}}\varphi\partial_{\bar{w}}\varphi\differential{\bar{w}}-\oint_{C_{s+s^{\prime}}^{\epsilon}}\varphi\partial_{\bar{w}}\varphi\differential{\bar{w}}\right).$
(171)

The the explicit form of the circles $\tilde{C}_{k}^{\epsilon}$, are given by

$\tilde{C}_{k}^{\epsilon}=\left\\{\begin{split}&\big{\\{}w\,\big{|}\,\>|\gamma
w-\gamma
w_{k}|=\epsilon\big{\\}}=\big{\\{}w\,\big{|}\,\>|w-w_{k}|=\epsilon|1-w_{s+s^{\prime}-3}|\big{\\}}&k\leq
s+s^{\prime}-1,\\\ \\\ &\left\\{w\,\big{|}\,\>|\gamma
w|=\tfrac{1}{\epsilon}\right\\}=\big{\\{}w\,\big{|}\,\>|w|=\tfrac{1}{\epsilon}|1-w_{s+s^{\prime}-3}|\big{\\}}&k=s+s^{\prime}.\end{split}\right.$
(172)

Now with the use of (171),(172) and asymptotic form of $\varphi$ from Lemma
C.1, one can find515151Note that the orientation of the contour around the
points at the infinity is opposite of the other points, hence the sign
difference in the last term.

$\displaystyle\Delta
S_{\boldsymbol{m}}^{(1)}[\varphi]=-4\pi\sqrt{-1}\left(1-\frac{1}{m}\right)^{2}s\log\frac{1}{|1-w_{s+s^{\prime}-3}|}-4\pi\sqrt{-1}\left(1-\frac{1}{m^{\prime}}\right)^{2}(s^{\prime}-1)\log\frac{1}{|1-w_{s+s^{\prime}-3}|}$
(173) $\displaystyle\hskip
49.79231pt+4\pi\sqrt{-1}\left(1+\frac{1}{m^{\prime}}\right)^{2}\log\frac{1}{|1-w_{s+s^{\prime}-3}|}.$

Next for the $\Delta S_{\boldsymbol{m}}^{(2)}[\varphi]$ we have

$\displaystyle\Delta S_{\boldsymbol{m}}^{(2)}[\varphi]$
$\displaystyle=\sum_{k=1}^{s}\left(\oint_{\tilde{C}_{k}^{\epsilon}}\tilde{\varphi}\left(\frac{\differential{(\overline{\gamma
w})}}{\overline{\gamma w}-\overline{\gamma w}_{k}}-\frac{\differential{(\gamma
w)}}{\gamma w-\gamma
w_{k}}\right)\right.\left.-\oint_{C_{k}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{k}}-\frac{\differential{w}}{w-w_{k}}\right)\right)$
$\displaystyle=\sum_{k=1}^{s}\left(\oint_{\tilde{C}_{k}^{\epsilon}}(\varphi-\log|\gamma^{\prime}|^{2})\left(\frac{\bar{\gamma^{\prime}}\differential{\bar{w}}}{\overline{\gamma
w}-\overline{\gamma w}_{k}}-\frac{\gamma^{\prime}\differential{w}}{\gamma
w-\gamma
w_{k}}\right)\right.\left.-\oint_{C_{k}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{k}}-\frac{\differential{w}}{w-w_{k}}\right)\right)$
$\displaystyle=\sum_{k=1}^{s}\left(\oint_{\tilde{C}_{k}^{\epsilon}}(\varphi+2\log|1-w_{s+s^{\prime}-3}|)\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{k}}-\frac{\differential{w}}{w-w_{k}}\right)\right.\left.-\oint_{C_{k}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{k}}-\frac{\differential{w}}{w-w_{k}}\right)\right),$

where we used (170) and the $\gamma^{\prime}=1/(1-w_{s+s^{\prime}-3})$ which
by implementing the asymptotics of $\varphi$ from Lemma C.1 and (172) is
simplified more to

$\displaystyle\Delta S_{\boldsymbol{m}}^{(2)}[\varphi]-8\pi\sqrt{-1}\hskip
2.84526pt\frac{1}{m}s\log\frac{1}{|1-w_{s+s^{\prime}-3}|}.$ (174)

By identical calculation to that of $\Delta S_{\boldsymbol{m}}^{(2)}[\varphi]$
we have:

$\displaystyle\Delta S_{\boldsymbol{m}}^{(3)}[\varphi]=-8\pi\sqrt{-1}\hskip
2.84526pt\frac{1}{m^{\prime}}(s^{\prime}-1)\log\frac{1}{|1-w_{s+s^{\prime}-3}|}.$
(175)

Finally

$\displaystyle\Delta S_{\boldsymbol{m}}^{(4)}[\varphi]$
$\displaystyle=\oint_{\tilde{C}_{s+s^{\prime}}^{\epsilon}}\tilde{\varphi}\left(\frac{\differential{(\overline{\gamma
w})}}{\overline{\gamma w}}-\frac{\differential{(\gamma w)}}{\gamma
w}\right)-\oint_{C_{s+s^{\prime}}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}}-\frac{\differential{w}}{w}\right)$
$\displaystyle=\oint_{\tilde{C}_{s+s^{\prime}}^{\epsilon}}(\varphi-\log|\gamma^{\prime}|^{2})\left(\frac{\bar{\gamma^{\prime}}\differential{\bar{w}}}{\overline{\gamma
w}}-\frac{\gamma^{\prime}\differential{w}}{\gamma
w}\right)-\oint_{C_{s+s^{\prime}}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}}-\frac{\differential{w}}{w}\right)$
$\displaystyle=\oint_{\tilde{C}_{s+s^{\prime}}^{\epsilon}}(\varphi+2\log|1-w_{s+s^{\prime}-3}|)\left(\frac{\differential{\bar{w}}}{\bar{w}}-\frac{\differential{w}}{w}\right)-\oint_{C_{s+s^{\prime}}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}}-\frac{\differential{w}}{w}\right),$

where in the last line we approximated $w-w_{s+s^{\prime}-3}$ with $w$ in the
denominator of the second term due to $w$ having a large absolute value.
Again, using the asymptotics of $\varphi$ from Lemma C.1, Eq.(172) and taking
into account the opposite orientation of contours around points at the
infinity

$\displaystyle\Delta S_{\boldsymbol{m}}^{(4)}[\varphi]=-8\pi\sqrt{-1}\hskip
2.84526pt\frac{1}{m^{\prime}}\log\frac{1}{|1-w_{s+s^{\prime}-3}|}.$ (176)

Putting (169), (173), (174), (175) and (176) together we get

$\Delta
S_{\boldsymbol{m}}[\varphi]=2\pi\left(\frac{1}{m^{2}}-1\right)s\log|1-w_{s+s^{\prime}-3}|+2\pi\left(\frac{1}{m^{\prime
2}}-1\right)(s^{\prime}-2)\log|1-w_{s+s^{\prime}-3}|,$ (177)

which by comparing (177) to (68) we can readily see that for $i=s^{\prime}-3$:

$\Delta
S_{\boldsymbol{m}}[\varphi]=S_{\boldsymbol{m}}[\varphi]\circ(\sigma_{k,k+1},\sigma^{\prime}_{i,i+1})-S_{\boldsymbol{m}}[\varphi]=-2\pi\log|f_{(\sigma_{k,k+1},\sigma^{\prime}_{i,i+1})}|.$
(178)

Doing the analogous calculations for $i=s^{\prime}-2,s^{\prime}-3$ would yield
similar results. One can continue this proof for the case with the direct
product of the symmetric groups of more strata inductively. Furthermore, had
we chosen to fix the points $0,1$ and $\infty$ in the other stratum or decided
to deal with punctures, we would still get the same result with minor changes
in the path of the proof. So in principle, we proved that under the action of
any element $\eta$ of
$\operatorname{Symm}\left(\boldsymbol{\mathsf{s}}\right)$,
$S_{\boldsymbol{m}}$ transforms according to the rule

$\exp[S_{\boldsymbol{m}}\circ\eta/\pi]\hskip
2.84526pt|f_{\eta}|^{2}=\exp[S_{\boldsymbol{m}}/\pi].$ (179)

This means that $\exp[S_{\boldsymbol{m}}/\pi]$ is a Hermitian metric on the
holomorphic line bundle $\lambda_{0,\boldsymbol{m}}$ defined in the Section
3.2. Furthermore, as we mentioned in there, (178) shows that 1-cocycles can be
viewed as the modular anomaly caused by the non-covariance of the action under
the effect of the modular group. To complete the proof, we remind that the
first Chern form
$\mathsf{c}_{1}\left(\lambda_{0,\boldsymbol{m}},\exp[S_{\boldsymbol{m}}/\pi]\right)$
of the metrized line bundle $\lambda_{0,\boldsymbol{m}}$ with a metric
$\exp[S_{\boldsymbol{m}}/\pi]$ is given by

$\mathsf{c}_{1}\left(\lambda_{0,\boldsymbol{m}},\exp[S_{\boldsymbol{m}}/\pi]\right)=-\frac{\sqrt{-1}}{2\pi}\hskip
2.84526pt\partial\bar{\partial}\left(\frac{S_{\boldsymbol{m}}}{\pi}\right).$
(180)

Thus using Theorem 4.2, we find that

$\mathsf{c}_{1}\left(\lambda_{0,\boldsymbol{m}},\exp[S_{\boldsymbol{m}}/\pi]\right)=\frac{1}{\pi^{2}}\omega_{\text{WP}}.$
(181)

This statement completes the proof. ∎

###  4.2 Riemann Orbisurfaces of Genus >1

Let $(\Sigma;L_{1},\dots,L_{g})$ be a marked normalized Schottky group of rank
$g>1$. The classical Liouville action is a critical value of the Liouville
action functional and for the case of a closed Riemann surface with $g>1$ it
was first defined by Zograf and Takhtajan 1988SbMat..60..297Z  (and was later
interpreted by Takhtajan and Teo Takhtajan:2002cc  in cohomological language)
to be given by

$S[\varphi]=\iint_{\mathcal{D}}(|\partial_{w}\varphi|^{2}+e^{\varphi})\differential[2]{w}+\frac{\sqrt{-1}}{2}\sum_{k=2}^{g}\oint_{C_{k}}\theta_{L_{k}^{-1}}(\varphi),$
(182)

where the 1-form $\theta_{L_{k}^{-1}}(\varphi)$ is given by

$\theta_{L_{k}^{-1}}(\varphi)=\left(\varphi-\frac{1}{2}\log|L_{k}^{\prime}|^{2}-\log|l_{k}|^{2}\right)\left(\frac{L_{k}^{\prime\prime}}{L_{k}^{\prime}}\differential{w}-\frac{\overline{L_{k}^{\prime\prime}}}{\overline{L_{k}^{\prime}}}\differential{\bar{w}}\right),\qquad\forall\hskip
1.42262ptL_{k}\in(\Sigma;L_{1},\dots,L_{g}).$ (183)

In the above equations, $\mathcal{D}$ is the fundamental domain of the marked
normalized Schottky group $(\Sigma;L_{1},\dots,L_{g})$,
$\partial\mathcal{D}=\bigcup_{k=1}^{g}(C_{k}\cup C_{k}^{\prime})$, and

$l_{k}=\frac{1-\lambda_{k}}{\sqrt{\lambda_{k}}(a_{k}-b_{k})}$ (184)

is the left-hand lower element in the matrix representation of the generator
$L_{k}\in\operatorname{PSL}(2,\mathbb{C})$ for $k=2,\dots,g$. Notice that
since we have chosen the marked Schottky group to be normalized, in particular
$a_{1}=0$ and $b_{1}=\infty$, $l_{1}=0$ and $\theta_{L_{1}^{-1}}(\varphi)=0$.

######  Remark 4.6.

The addition of the second term in the on-shell Liouville action (182) makes
sure that:

  * •

The variation $\delta S$ of the classical action $S[\varphi]$ has the form

$\delta S[\varphi]\overset{\text{\tiny\it def.}}{\equiv}\lim_{t\to
0}\frac{S[\varphi+t\delta\psi]-S[\varphi]}{t}=\iint_{\mathcal{D}}(-2\partial_{\bar{w}}\partial_{w}\varphi+e^{\varphi})\delta\psi\differential[2]{w},$
(185)

where the variation $\delta\psi\in\mathcal{C}^{\infty}(\Omega,\mathbb{R})$ is
a smooth function on $\Omega$ which is automorphic with respect to $\Sigma$.

  * •

The classical Liouville action $S[\varphi]$ is _independent_ of the specific
choice of a fundamental domain $\mathcal{D}$ for the marked Schottky group
$(\Sigma;L_{1},\dots,L_{g})$525252See footnote 38. and determines a smooth
function $S:\mathfrak{S}_{g}\to\mathbb{R}$.

As in Section 3.3, let us denote by $\partial$ and $\bar{\partial}$ the
$(1,0)$ and $(0,1)$ components of the de Rham differential on
$\mathfrak{S}_{g}$. Zograf and Takhtajan (1988SbMat..60..297Z, , Theorems 1,2)
have proven that

$\partial
S=-2\pi\sum_{i=1}^{3g-3}c_{i}P_{i}\qquad\text{and}\qquad\bar{\partial}\partial
S=-2\sqrt{-1}\omega_{WP},$ (186)

so that $-S$ is a Kähler potential for the (projection of) Weil-Petersson
metric on $\mathfrak{S}_{g}$.

To define the classical Liouville action for the hyperbolic metric on
$\overset{{}_{\curlywedge}}{\Omega}/\Sigma$, one needs to regularize the area
integral in (182) which diverges due to the asymptotic behavior of $\varphi$
as $w\to w_{i}$, see Lemma C.1. We will do so in the same way that we
regulated the Liouville action for the genus $g=0$ case: Let us define
$\overset{{}_{\curlywedge}}{\mathcal{D}}$ as the pair
$(\mathcal{D},\widetilde{\mathscr{D}}\big{|}_{{}_{\mathcal{D}}})$ and suppose
that all $w_{1},\dots,w_{n}\in\operatorname{Int}\mathcal{D}$, the interior of
the fundamental domain $\mathcal{D}$. For sufficiently small $\epsilon>0$
define535353Note that $\lim_{\epsilon\rightarrow
0}\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}=\overset{{}_{\curlywedge}}{\mathcal{D}}_{\text{reg}}$
where
$\overset{{}_{\curlywedge}}{\mathcal{D}}_{\text{reg}}:=\overset{{}_{\curlywedge}}{\mathcal{D}}\backslash(\operatorname{Supp}\widetilde{\mathscr{D}}\cap\mathcal{D})$.

$\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}=\overset{{}_{\curlywedge}}{\mathcal{D}}\big{\backslash}\bigcup_{i=1}^{n}D_{i}^{\epsilon},$
(187)

with $D_{i}^{\epsilon}\overset{\text{\tiny\it
def.}}{\equiv}\left\\{w\,\big{|}\,|w-w_{i}|<\epsilon\right\\}$. It follows
from Lemma C.1 that the following limit exists

$S_{\overset{{}_{\curlywedge}}{\mathcal{D}}_{\text{reg}}}[\varphi]=\\\
\lim_{\epsilon\to
0^{+}}\left(\iint_{\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}}(|\partial_{w}\varphi|^{2}+e^{2\varphi})\differential[2]{w}+\frac{\sqrt{-1}}{2}\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{C_{j}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{j}}-\frac{\differential{w}}{w-w_{j}}\right)\right.\\\
\left.-2\pi\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)^{2}\log\epsilon+2\pi
n_{p}\big{(}\log\epsilon+2\log|\log\epsilon|\big{)}\right).$ (188)

######  Remark 4.7.

When $n_{p}=0$, the appropriate
$S_{\overset{{}_{\curlywedge}}{\mathcal{D}}_{\text{reg}}}[\varphi]$ is given
by

$\lim_{\epsilon\to
0^{+}}\Bigg{(}\iint_{\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}}(|\partial_{w}\varphi|^{2}+e^{2\varphi})\differential[2]{w}+\frac{\sqrt{-1}}{2}\sum_{j=1}^{n}\left(1-\frac{1}{m_{j}}\right)\oint_{C_{j}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{j}}-\frac{\differential{w}}{w-w_{j}}\right)\\\
-2\pi\sum_{i=1}^{n}\left(1-\frac{1}{m_{j}}\right)^{2}\log\epsilon\Bigg{)}.$
(189)

######  Remark 4.8.

Similar to the genus $0$ case, one can derive
$S_{\overset{{}_{\curlywedge}}{\mathcal{D}}_{\text{reg}}}$ by truncating the
hyperbolic metric in $\epsilon$-neighborhood of ramification points and cusps,
and replacing it with a smooth flat metric in these neighborhoods (see Figure
9).

Now, we can define the regularized action as

$\begin{split}&S_{\boldsymbol{m}}[\varphi]=S_{\boldsymbol{m}}(\mathcal{D};w_{1},\dots,w_{n})=S_{\overset{{}_{\curlywedge}}{\mathcal{D}}_{\text{reg}}}[\varphi]+\frac{\sqrt{-1}}{2}\sum_{k=2}^{g}\oint_{C_{k}}\theta_{L_{k}^{-1}}(\varphi).\end{split}$
(190)

This completes the definition of $S_{\boldsymbol{m}}$ provided that all of the
fixed points $w_{1},\dots,w_{n}$ lie in the interior of the fundamental domain
– i.e. $w_{1},\dots,w_{n}\in\operatorname{Int}\mathcal{D}$. The
$S_{\boldsymbol{m}}(\mathcal{D};w_{1},\dots,w_{n})$ depends on the choice of
representatives in $\Sigma\cdot\\{w_{1},\dots,w_{n}\\}$ and _no longer
determines a function on the Schottky space
$\mathfrak{S}_{g,n}(\boldsymbol{m})$_. Note that $w_{i}$ and $L_{k}(w_{i})$
have the same order and are related by the action of
$\operatorname{Symm}\left(\boldsymbol{\mathsf{s}}\right)$ acting on
$\overset{{}_{\curlywedge}}{\Omega}$. The geometric meaning of
$S_{\boldsymbol{m}}$ is given by the following Lemma (also see Lemma 3.6):

######  Lemma 4.4.

The regularized Liouville action determines a Hermitian metric
$\exp[S_{\boldsymbol{m}}/\pi]$ on the holomorphic $\mathbb{Q}$-line bundle
$\mathscr{L}=\bigotimes_{i=1}^{n}\mathscr{L}_{i}^{h_{i}}$ over
$\mathfrak{S}_{g,n}(\boldsymbol{m})$.

###### Proof.

It is sufficient to prove that for $i=1,\dots,n$,

$S_{\boldsymbol{m}}(\tilde{\mathcal{D}};w_{1},\dots,L_{k}w_{i},\dots,w_{n})-S_{\boldsymbol{m}}(\mathcal{D};w_{1},\dots,w_{n})=\pi
h_{i}\hskip 1.42262pt\log|L_{k}^{\prime}(w_{i})|^{2},$ (191)

with $w_{1},\dots,w_{n}\in\operatorname{Int}\mathcal{D}$ and
$w_{1},\dots,L_{k}w_{i},\dots,w_{n}\in\operatorname{Int}\tilde{\mathcal{D}}$.
Furthermore, it is sufficient to consider the case when

$\tilde{\mathcal{D}}=(\mathcal{D}\backslash\mathcal{D}_{0})\cup
L_{k}(\mathcal{D}_{0}),$

and $\mathcal{D}_{0}\subset\mathcal{D}$ is such that
$\partial\mathcal{D}_{0}\cap\partial\mathcal{D}\subset C_{k}$ and
$w_{i}\in\mathcal{D}_{0}$, and all other
$w_{j}\in\mathcal{D}\backslash\mathcal{D}_{0}$ for $j\neq i$. Note that by a
finite combination of such transformations, any choice of a fundamental domain
for $\Sigma$ can be obtained from $\mathcal{D}$. The computation of (191)
closely follows the corresponding computation in the proof of Lemma 3 in
park2015potentials  where no ramification point exists. Let

$\begin{split}I_{\epsilon}(\mathcal{D};w_{1},\dots,w_{n})&=\iint_{\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}}(|\partial_{w}\varphi|^{2}+e^{\varphi})\differential{w}\wedge\differential{\bar{w}}+\sum_{k=2}^{g}\oint_{C_{k}}\theta_{L_{k}^{-1}}(\varphi)\\\
&+\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{C_{j}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{j}}-\frac{\differential{w}}{w-w_{j}}\right),\end{split}$

where we did not include the counterterms in the action due to the fact that
the action of $L_{k}$ does not change the conformal class. Since
$\tilde{C}_{j}=C_{j}$ for $j\neq k$ and
$\tilde{C}_{k}=C_{k}-\partial\mathcal{D}_{0}$, we have

$\begin{split}\Delta
I_{\epsilon}&=I_{\epsilon}(\tilde{\mathcal{D}};w_{1},\dots,L_{k}w_{i},\dots,w_{n})-I_{\epsilon}(\mathcal{D};w_{1},\dots,w_{n})\\\
&=\iint_{L_{k}(\mathcal{D}_{0})\backslash\tilde{D}_{i}^{\epsilon}}(|\partial_{w}\varphi|^{2}+e^{\varphi})\differential{w}\wedge\differential{\bar{w}}-\iint_{\mathcal{D}_{0}\backslash
D_{i}^{\epsilon}}(|\partial_{w}\varphi|^{2}+e^{\varphi})\differential{w}\wedge\differential{\bar{w}}-\oint_{\partial\mathcal{D}_{0}}\theta_{L_{k}^{-1}}(\varphi)\\\
&+\sum_{j=1}^{n_{e}}\delta_{i\,j}\left(1-\frac{1}{m_{j}}\right)\left[\oint_{\tilde{C}_{j}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\overline{L_{k}w_{j}}}-\frac{\differential{w}}{w-L_{k}w_{j}}\right)-\oint_{C_{j}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{j}}-\frac{\differential{w}}{w-w_{j}}\right)\right].\end{split}$

According to Eq.(136)

$\begin{split}L_{k}^{\ast}\left((|\partial_{w}\varphi|^{2}+e^{\varphi})\differential{w}\wedge\differential{\bar{w}}\right)&=\left((|\partial_{w}\varphi|^{2}+e^{\varphi})\differential{w}\wedge\differential{\bar{w}}\right)\circ
L_{k}\,|L_{k}^{\prime}|^{2}\\\
&=(|\partial_{w}\varphi|^{2}+e^{\varphi})\differential{w}\wedge\differential{\bar{w}}+\differential{\theta_{L_{k}^{-1}}(\varphi)},\end{split}$
(192)

which, together with using the Stokes theorem, we get

$\begin{split}\Delta I_{\epsilon}&=\iint_{\mathcal{D}_{0}\backslash
L_{k}^{-1}(\tilde{D}_{i}^{\epsilon})}(|\partial_{w}\varphi|^{2}+e^{\varphi})\differential{w}\wedge\differential{\bar{w}}-\iint_{\mathcal{D}_{0}\backslash
D_{i}^{\epsilon}}(|\partial_{w}\varphi|^{2}+e^{\varphi})\differential{w}\wedge\differential{\bar{w}}-\oint_{\partial
L_{k}^{-1}(\tilde{D}_{i}^{\epsilon})}\theta_{L_{k}^{-1}}(\varphi)\\\
&+\sum_{j=1}^{n_{e}}\delta_{i\,j}\left(1-\frac{1}{m_{j}}\right)\left[\oint_{L_{k}^{-1}(\tilde{C}_{j}^{\epsilon})}(\varphi-\log|L^{\prime}_{k}w|^{2})\left(\frac{\overline{L^{\prime}_{k}w}\differential{\bar{w}}}{\overline{L_{k}w}-\overline{L_{k}w_{j}}}-\frac{L^{\prime}_{k}w\differential{w}}{L_{k}w-L_{k}w_{j}}\right)\right.\\\
&\left.\hskip
56.9055pt-\oint_{C_{j}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{j}}-\frac{\differential{w}}{w-w_{j}}\right)\right].\end{split}$

Since the third term’s integrand does not have a pole, its contribution will
be of $\text{\scriptsize{$\mathcal{O}$}}\left(1\right)$, and it can be safely
omitted. Furthermore, the exponential term in the Liouville term gives the
Euler characteristic, and its contribution will be canceled between related
terms. Now, we can rewrite the above equation as

$\begin{split}\Delta
I_{\epsilon}&=\iint_{D_{i}^{\epsilon}}(|\partial_{w}\varphi|^{2})\differential{w}\wedge\differential{\bar{w}}-\iint_{L_{k}^{-1}(\tilde{D}_{i}^{\epsilon})}(|\partial_{w}\varphi|^{2})\differential{w}\wedge\differential{\bar{w}}\\\
&+\sum_{j=1}^{n_{e}}\delta_{i\,j}\left(1-\frac{1}{m_{j}}\right)\left[\oint_{L_{k}^{-1}(\tilde{C}_{j}^{\epsilon})}(\varphi-\log|L^{\prime}_{k}(w)|^{2})\left(\partial_{\bar{w}}\log|L_{k}w-L_{k}w_{j}|^{2}\right.\right.\\\
&\left.\left.-\partial_{w}\log|L_{k}w-L_{k}w_{j}|^{2}\right)-\oint_{C_{j}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{j}}-\frac{\differential{w}}{w-w_{j}}\right)\right].\end{split}$
(193)

By noting that

$\displaystyle\partial_{w}\log|L_{k}w-L_{k}w_{j}|^{2}=\partial_{w}\log|w-w_{j}|^{2}+\text{\scriptsize{$\mathcal{O}$}}\left(1\right),$

we can write (193) as follows

$\begin{split}\Delta
I_{\epsilon}&=\iint_{D_{i}^{\epsilon}}(|\partial_{w}\varphi|^{2})\differential{w}\wedge\differential{\bar{w}}-\iint_{L_{k}^{-1}(\tilde{D}_{i}^{\epsilon})}(|\partial_{w}\varphi|^{2})\differential{w}\wedge\differential{\bar{w}}\\\
&+\sum_{j=1}^{n_{e}}\delta_{i\,j}\left(1-\frac{1}{m_{j}}\right)\left[\oint_{L_{k}^{-1}(\tilde{C}_{j}^{\epsilon})}(\varphi-\log|L^{\prime}_{k}(w)|^{2})\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{j}}-\frac{\differential{w}}{w-w_{j}}\right)\right.\\\
&\left.\hskip
56.9055pt-\oint_{C_{j}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{j}}-\frac{\differential{w}}{w-w_{j}}\right)\right].\end{split}$
(194)

Focusing on the first line, by using the integration by parts and equations of
motion together with considering the orientation of the boundary circles:

$\displaystyle\iint_{D_{i}^{\epsilon}}(|\partial_{w}\varphi|^{2})\differential{w}\wedge\differential{\bar{w}}-\iint_{L_{k}^{-1}(\tilde{D}_{i}^{\epsilon})}(|\partial_{w}\varphi|^{2})\differential{w}\wedge\differential{\bar{w}}=\oint_{C_{i}^{\epsilon}}\varphi\partial_{\bar{w}}\varphi\differential\bar{w}-\oint_{\tilde{C}_{i}^{\epsilon}}\varphi\partial_{\bar{w}}\varphi\differential\bar{w},$
(195)

where

$\displaystyle\tilde{C}_{i}^{\epsilon}=\\{w\big{|}\>|L_{k}w-L_{k}w_{i}|=\epsilon\\}\equiv\\{w\big{|}\>|w-w_{i}|=\frac{\epsilon}{L^{\prime}_{k}w_{i}}\\}.$
(196)

To be more precise, the right hand side of (195) had an extra contribution

$\frac{1}{2}\iint_{L_{k}^{-1}(\tilde{D}_{i}^{\epsilon})}\varphi
e^{\varphi}\differential{w}\wedge\differential{\bar{w}}-\frac{1}{2}\iint_{D_{i}^{\epsilon}}\varphi
e^{\varphi}\differential{w}\wedge\differential{\bar{w}},$ (197)

which by defining $y_{i}=|w-w_{i}|,\hskip
1.42262pt\alpha_{i}=1-1/m_{i},[y_{i}]=:\epsilon\leq
y_{i}\leq\epsilon/L^{\prime}_{k}w_{i}$, this contribution for ramification
points and cusps respectively becomes

$\begin{split}&-\frac{2\pi}{2}\iint_{[y_{i}]}\frac{1}{y_{i}^{2\alpha_{i}-1}}\log{y_{i}^{2\alpha_{i}}}dy_{i}=\pi\left(\frac{\alpha_{i}}{2(1-\alpha_{i})^{2}}y_{i}^{2-2\alpha_{i}}-\frac{\alpha_{i}}{(1-\alpha_{i})}y_{i}^{2-2\alpha_{i}}\log
y_{i}\right)\bigg{|}^{\epsilon/L^{\prime}_{k}w_{i}}_{\epsilon}\\\
&-\frac{2\pi}{2}\iint_{[y_{i}]}\frac{1}{y_{i}\log^{2}y_{i}}\log\left({y_{i}^{2}}\log^{2}y_{i}\right)dy_{i}=\pi\left(2+\frac{2}{\log
y_{i}}-2\log|\log y_{i}|+\frac{2}{\log y_{i}}\log|\log
y_{i}|\right)\bigg{|}^{\epsilon/L^{\prime}_{k}w_{i}}_{\epsilon}\end{split}$

but both of them, and accordingly (197), vanish in the limit
$\epsilon\rightarrow 0$. Now, by using the asymptotic form of $\varphi$ from
the Lemma C.1 together with the relation (196), the Eq.(195) is simplified
to545454The $D_{i}^{\epsilon}$’s boundaries have the opposite orientation of
the boundary of the fundamental domain.

$\displaystyle\iint_{D_{i}^{\epsilon}}(|\partial_{w}\varphi|^{2})\differential{w}\wedge\differential{\bar{w}}-\iint_{L_{k}^{-1}(\tilde{D}_{i}^{\epsilon})}(|\partial_{w}\varphi|^{2})\differential{w}\wedge\differential{\bar{w}}=-4\pi\sqrt{-1}\left(1-\frac{1}{m_{i}}\right)^{2}\log|L^{\prime}_{k}w_{i}|.$
(198)

For the rest of the integrals in (194), again by using the asymptotics of
$\varphi$ from Lemma C.1 and (196) we have

$\displaystyle\sum_{j=1}^{n_{e}}\delta_{ij}\left(1-\frac{1}{m_{j}}\right)\left[\oint_{L_{k}^{-1}(\tilde{C}_{j}^{\epsilon})}(\varphi-\log|L^{\prime}_{k}(w)|^{2})\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{j}}-\frac{\differential{w}}{w-w_{j}}\right)\right.$
(199) $\displaystyle\left.\hskip
36.98866pt-\oint_{C_{j}^{\epsilon}}\varphi\left(\frac{\differential{\bar{w}}}{\bar{w}-\bar{w}_{j}}-\frac{\differential{w}}{w-w_{j}}\right)\right]=8\pi\sqrt{-1}\left(\frac{1}{m_{i}^{2}}-\frac{1}{m_{i}}\right)\log|L^{\prime}_{k}w_{i}|.$

Now combining (198) and (LABEL:schottkyboundary) gives

$\Delta I_{\epsilon}=-4\pi\sqrt{-1}\hskip
2.84526pth_{i}\log|L^{\prime}_{k}w_{i}|.$

So, in principle, we proved that

$\displaystyle\exp[S_{\boldsymbol{m}}(\tilde{\mathcal{D}};w_{1},\dots,L_{k}w_{i},\dots,w_{n})/(\pi
h_{i})]=\exp[S_{\boldsymbol{m}}(\mathcal{D};w_{1},\dots,w_{n})/(\pi
h_{i})]\hskip 2.84526pt|L^{\prime}_{k}w_{i}|^{2},$

which means that $\exp[S_{\boldsymbol{m}}/\pi]$ is a Hermitian metric on
$\mathscr{L}=\bigotimes_{i=1}^{n_{p}+n_{e}}\mathscr{L}_{i}^{h_{i}}$. ∎

Combining Lemmas 4.4 and 3.6, we can deduce the following statement

######  Corollary 4.4.1.

Put
$\mathsf{H}=\mathsf{h}_{1}^{m_{1}h_{1}}\dotsm\mathsf{h}_{n_{e}}^{m_{n_{e}}h_{n_{e}}}\mathsf{h}_{n_{e}+1}\dotsm\mathsf{h}_{n}$.
Then

$\mathscr{S}_{\boldsymbol{m}}=S_{\boldsymbol{m}}-\pi\log\mathsf{H},$ (200)

determines a smooth real-valued function on
$\mathfrak{S}_{g,n}(\boldsymbol{m})$.

By demanding a physical condition, the form of the function
$\mathscr{S}_{\boldsymbol{m}}$ may be easy to understand. Let us remember that
by acting the $L_{k}$ on each $w_{i}$ in a particular fundamental domain
$\mathcal{D}$, we find the same order singularity in another fundamental
domain. Effectively, this means that the action of the Schottky group is the
same as the action of $\operatorname{Symm}\left(\boldsymbol{m}\right)$ group
acting on $\overset{{}_{\curlywedge}}{\Omega}$. According to the Section
3.2.1, the group $\operatorname{Symm}\left(\boldsymbol{m}\right)$ will be
generated by the transpositions
$\\{\sigma_{i,i+1}\\}_{i=1}^{\boldsymbol{m}-1}$and the action of
transpositions will be followed by a $\operatorname{PSL}(2,\mathbb{C})$
transformation to ensure that the last three coordinates in the Teichmüller
space remain $0,1$ and $\infty$. This last transformation causes the
regularization radius $\epsilon_{i}$ changes for some of the circles around
the singularities. This means that for each fundamental domain, one should
find a new counterterm action and, accordingly, a new action. But since we
would like to have an action which is independent from the choice of
fundamental domain, we should add a new term to the Liouville action
$S_{\boldsymbol{m}}$ that can compensate that changing of regularization radii
$\epsilon_{i}$s and this new term is exactly $\pi\log\mathsf{H}$.

##  5 Potentials for Weil–Petersson and Takhtajan–Zograf Metrics

In this section, following park2015potentials ; ZT_2018 , we construct a
global potential for the cuspidal and elliptic metrics on $\mathcal{M}_{0,n}$
(see Section 3.1.3). We will also prove that the first Chern forms of the line
bundles $\mathscr{L}_{i}$ over the Schottky space
$\mathfrak{S}_{g,n}(\boldsymbol{m})$ with Hermitian metrics $\mathsf{h}_{i}$
are given by $\tfrac{1}{2\pi}\omega^{\text{ell}}_{\text{TZ},i}$ for
$i=1,\dots,n_{e}$ and $\tfrac{4}{3}\omega^{\text{cusp}}_{\text{TZ},i}$ for
$i=n_{e}+1,\dots,n$. In addition, we will show that
$\tfrac{1}{\pi^{2}}\omega_{{}_{\text{WP}}}$ is the first Chern form of the
$\mathbb{Q}$-line bundle
$\mathscr{L}=\bigotimes_{i=1}^{n}\mathscr{L}_{i}^{h_{i}}$ with Hermitian
metric $\exp[S_{\boldsymbol{m}}/\pi]$, where the regularized classical
Liouville action $S_{\boldsymbol{m}}$ is given by Eq.(190). Then, it follows
readily from these two results that the specific combination
$\omega_{{}_{\text{WP}}}-\tfrac{4\pi^{2}}{3}\omega_{{}_{\text{TZ}}}^{\text{cusp}}-\tfrac{\pi}{2}\sum_{j=1}^{n_{e}}m_{j}h_{j}\hskip
1.42262pt\omega^{\text{ell}}_{\text{TZ},j}$ of Weil-Petersson metric as well
as cuspidal and elliptic Takhtajan-Zograf metrics has a _global_ Kähler
potential on $\mathfrak{S}_{g,n}(\boldsymbol{m})$.

###  5.1 Potentials for Cuspidal and Elliptic TZ Metrics on
$\mathcal{M}_{0,n}$

As in Section 3.2, let $\Gamma$ be a marked normalized Fuchsian group with
signature $(0;m_{1},\dots,m_{n_{e}},n_{p})$ that uniformizes the orbifold
Riemann surface $O$ and let $J:\mathbb{H}\to O$ be the Klien’s Hauptmodule. In
addition, let $\mathsf{h}_{i}=\left|J^{(i)}_{1}\right|^{\frac{2}{m_{i}}}$ for
$i=1,\dots,n_{e}$, $\mathsf{h}_{i}=\left|J^{(i)}_{1}\right|^{2}$ for
$i=n_{e}+1,\dots,n-1$, and $\mathsf{h}_{n}=\left|J^{(n)}_{-1}\right|^{2}$ be
smooth positive functions on $\mathcal{M}_{0,n}$.555555See Eq.(64). Now,
according to the expressions for $\log\mathsf{h}_{i}$ in Remark 3.10, we prove
the following lemma:

######  Lemma 5.1.

For all $k=1,\dots,n-3$, we have

$\begin{split}&\partialderivative{w_{k}}\log\mathsf{h}_{i}=\frac{1}{m_{i}}\partial_{w}\dot{F}^{k}(w_{i}),\hskip
28.45274pti=1,\dots,n_{e},\\\ \\\
&\partialderivative{w_{k}}\log\mathsf{h}_{i}=\partial_{w}\dot{F}^{k}(w_{i}),\hskip
42.67912pti=n_{e}+1,\dots,n.\end{split}$ (201)

###### Proof.

Consider the orbifold Riemann surface $O\cong[\mathbb{H}/\Gamma]$. Using Lemma
3.3, it is sufficient to show that

$\left.\left(\partialderivative{\log\mathsf{h}_{i}^{\varepsilon\mu_{k}}}{\varepsilon}\right)\right|_{\varepsilon=0}=\left\\{\begin{split}&\frac{1}{m_{i}}\partial_{w}\dot{F}^{k}(w_{i})&\text{for}\quad&i=1,\dots,n_{e},\\\
\\\
&\partial_{w}\dot{F}^{k}(w_{i})\quad&\text{for}\quad&i=n_{e}+1,\dots,n,\end{split}\right.$
(202)

and for all $k=1,\dots,n-3$. The following proof repeats verbatim the proof of
Lemma 4 in park2015potentials  for the case of punctures. Using the fact that
$F^{\varepsilon\mu_{k}}$ is holomorphic in $\varepsilon$ at $\varepsilon=0$,
Corollary 3.3.1 and formulas in Remark 3.10, Eq.(109) we get:

  * •

_$i=1,\dots,n_{e}$ case:_

$\begin{split}\left.\left(\partialderivative{\log\mathsf{h}_{i}^{\varepsilon\mu_{k}}}{\varepsilon}\right)\right|_{\varepsilon=0}&=-\left.\left(\partialderivative{\varepsilon}\right)\right|_{\varepsilon=0}\lim_{w\to
w_{i}}\ \left(\varphi^{\varepsilon\mu_{k}}\circ
F^{\varepsilon\mu_{k}}+\left(1-\frac{1}{m_{i}}\right)\log|F^{\varepsilon\mu_{k}}(w-w_{i})|^{2}\right)\\\
&=-\lim_{w\to
w_{i}}\left\\{\left.\partialderivative{\varepsilon}\right|_{\varepsilon=0}\left(\varphi^{\varepsilon\mu_{k}}\circ
F^{\varepsilon\mu_{k}}+\left(1-\frac{1}{m_{i}}\right)\log|F^{\varepsilon\mu_{k}}(w-w_{i})|^{2}\right)\right\\}\\\
&=-\lim_{w\to
w_{i}}\left(\partial_{w_{k}}\varphi+\partial_{w}\varphi\dot{F}^{k}(w)+\left(1-\frac{1}{m_{i}}\right)\frac{\dot{F}^{k}(w)-\dot{F}^{k}(w_{i})}{w-w_{i}}\right)\\\
&\overset{\eqref{widerivative}}{=}-\lim_{w\to
w_{i}}\left(-\partial_{w}\varphi\dot{F}^{k}(w)-\partial_{w}\dot{F}^{k}(w)+\partial_{w}\varphi\dot{F}^{k}(w)+\left(1-\frac{1}{m_{i}}\right)\partial_{w}\dot{F}^{k}(w)\right)\\\
&=\frac{1}{m_{i}}\partial_{w}\dot{F}^{k}(w_{i}).\end{split}$

In going from the first line to the second line, we have interchanged the
order of the limit $w\to w_{i}$ and the differentiation. This is allowed due
to convergence in the above formula and the fact that the definition of
$\mathsf{h}_{i}$ is uniform in a neighborhood of an arbitrary point
$(w_{1},\dots,w_{n-3})\in\mathcal{M}_{0,n}$.

  * •

_$i=n_{e}+1,\dots,n-1$ case:_

$\begin{split}\left.\left(\partialderivative{\log\mathsf{h}_{i}^{\varepsilon\mu_{k}}}{\varepsilon}\right)\right|_{\varepsilon=0}&=\left.\left(\partialderivative{\varepsilon}\right)\right|_{\varepsilon=0}\lim_{w\to
w_{i}}\left(\log|F^{\varepsilon\mu_{k}}(w-w_{i})|^{2}-\frac{2e^{-\frac{\varphi^{\varepsilon\mu_{k}}\circ
F^{\varepsilon\mu_{k}}(w)}{2}}}{|F^{\varepsilon\mu_{k}}(w-w_{i})|}\right)\\\
&=\lim_{w\to
w_{i}}\left\\{\left.\partialderivative{\varepsilon}\right|_{\varepsilon=0}\left(\log|F^{\varepsilon\mu_{k}}(w-w_{i})|^{2}-2(F^{\varepsilon\mu_{k}})^{\ast}\left(e^{-\frac{1}{2}\varphi^{\varepsilon\mu_{k}}}\right)\left|\frac{\partial_{w}F^{\varepsilon\mu_{k}}(w)}{F^{\varepsilon\mu_{k}}(w-w_{i})}\right|\right)\right\\}\\\
&=\lim_{w\to
w_{i}}\left\\{\left.\partialderivative{\varepsilon}\right|_{\varepsilon=0}\log(F^{\varepsilon\mu_{k}}(w-w_{i}))-2\frac{e^{-\frac{1}{2}\varphi}}{(\bar{w}-\bar{w}_{i})^{\frac{1}{2}}}\left.\partialderivative{\varepsilon}\right|_{\varepsilon=0}\left(\frac{\partial_{w}F^{\varepsilon\mu_{k}}(w)}{F^{\varepsilon\mu_{k}}(w-w_{i})}\right)^{\frac{1}{2}}\right\\}\\\
&=\lim_{w\to
w_{i}}\left(\frac{\dot{F}^{k}(w)-\dot{F}^{k}(w_{i})}{w-w_{i}}-\frac{e^{-\frac{1}{2}\varphi(w)}\left((w-w_{i})\partial_{w}\dot{F}^{k}(w)-\dot{F}^{k}(w)+\dot{F}^{k}(w_{i})\right)}{(w-w_{i})|w-w_{i}|}\right)\\\
&=\partial_{w}\dot{F}^{k}(w_{i}).\end{split}$

  * •

_$i=n$ case:_

$\begin{split}\left.\left(\partialderivative{\log\mathsf{h}_{n}^{\varepsilon\mu_{k}}}{\varepsilon}\right)\right|_{\varepsilon=0}&=\left.\left(\partialderivative{\varepsilon}\right)\right|_{\varepsilon=0}\lim_{w\to\infty}\left(\log|F^{\varepsilon\mu_{k}}(w)|^{2}-\frac{2e^{-\frac{\varphi^{\varepsilon\mu_{k}}\circ
F^{\varepsilon\mu_{k}}(w)}{2}}}{|F^{\varepsilon\mu_{k}}(w)|}\right)\\\
&=\lim_{w\to\infty}\left\\{\left.\partialderivative{\varepsilon}\right|_{\varepsilon=0}\left(\log|F^{\varepsilon\mu_{k}}|^{2}-2(F^{\varepsilon\mu_{k}})^{\ast}\left(e^{-\frac{1}{2}\varphi^{\varepsilon\mu_{k}}}\right)\left|\frac{\partial_{w}F^{\varepsilon\mu_{k}}}{F^{\varepsilon\mu_{k}}}\right|\right)(w)\right\\}\\\
&=\lim_{w\to\infty}\left\\{\left.\partialderivative{\varepsilon}\right|_{\varepsilon=0}\log(F^{\varepsilon\mu_{k}}(w)\overline{F^{\varepsilon\mu_{k}}(w)})-2\frac{e^{-\frac{1}{2}\varphi}}{\bar{w}^{\frac{1}{2}}}\left.\partialderivative{\varepsilon}\right|_{\varepsilon=0}\left(\frac{\partial_{w}F^{\varepsilon\mu_{k}}(w)}{F^{\varepsilon\mu_{k}}(w)}\right)^{\frac{1}{2}}\right\\}\\\
&=\lim_{w\to\infty}\left(\frac{\dot{F}^{k}(w)}{w}-\frac{e^{-\frac{1}{2}\varphi(w)}(w\partial_{w}\dot{F}^{k}(w)-\dot{F}^{k}(w))|w|}{w^{2}\bar{w}}\right)\\\
&=\partial_{w}\dot{F}^{k}(\infty).\end{split}$

∎

As before, let $\partial$ and $\bar{\partial}$ be the $(1,0)$ and $(0,1)$
components of de Rham differential $d=\partial+\bar{\partial}$ on
$\mathcal{M}_{0,n}$. We have

######  Lemma 5.2.

The functions
$-\log\mathsf{h}_{i},-\log\mathsf{h}_{j},\log\mathsf{h}_{n},:\mathcal{M}_{0,n}\to\mathbb{R}_{>0}$
for $i=1,\dots,n_{e}$ and $j=n_{e}+1,\dots,n-1$ are Kähler potential for TZ
metrics $\frac{1}{2}\langle\cdot,\cdot\rangle_{\text{TZ},i}^{\text{ell}}$
,$\frac{4\pi}{3}\langle\cdot,\cdot\rangle_{\text{TZ},j}^{\text{cusp}}$,
$-\frac{4\pi}{3}\langle\cdot,\cdot\rangle_{\text{TZ},n}^{\text{cusp}}$,
respectively:

$\bar{\partial}\partial\log\mathsf{h}_{i}=-\sqrt{-1}\hskip
1.42262pt\omega_{\text{TZ},i}^{\text{ell}},\qquad\bar{\partial}\partial\log\mathsf{h}_{j}=-\frac{8\pi\sqrt{-1}}{3}\hskip
1.42262pt\omega_{\text{TZ},j}^{\text{cusp}},\qquad\bar{\partial}\partial\log\mathsf{h}_{n}=\frac{8\pi\sqrt{-1}}{3}\hskip
1.42262pt\omega_{\text{TZ},n}^{\text{cusp}}.$ (203)

###### Proof.

We need to prove that for all $j,k=1,\dots,n-3$,

$-\partialderivative{\log\mathsf{h}_{i}}{w_{j}}{\bar{w}_{k}}=\left\\{\begin{split}&\frac{1}{2}\left\langle\partialderivative{w_{j}},\partialderivative{w_{k}}\right\rangle_{\text{TZ},i}^{\text{ell}}&\text{for}\quad&i=1,\dots,n_{e},\\\
\\\
&\frac{4\pi}{3}\left\langle\partialderivative{w_{j}},\partialderivative{w_{k}}\right\rangle_{\text{TZ},i}^{\text{cusp}}&\text{for}\quad&i=n_{e}+1,\dots,n-1,\\\
\\\
&-\frac{4\pi}{3}\left\langle\partialderivative{w_{j}},\partialderivative{w_{k}}\right\rangle_{\text{TZ},i}^{\text{cusp}}\quad&\text{for}\quad&i=n.\end{split}\right.$

Let us consider the three cases $i=1,...,n_{e}$, $i=n_{e}+1,\dots,n-1$, and
$i=n$ separately. The following proof repeats verbatim the proof of