Proposition 1 in park2015potentials  for the case of punctures. According to
Section 3.1, Lemma 3.3 and Eq.(109), for a given Riemann orbisurface
$O\cong[\mathbb{H}/\Gamma]$ one can write:

  * •

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

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

In the above equation, we have used the notation
$(\varepsilon\mu)_{jk}=\varepsilon_{j}\mu_{j}+\varepsilon_{k}\mu_{k}$. Using
the commutative diagram (87), one has

$(F^{\varepsilon_{j}\mu_{j}+\varepsilon_{k}\mu_{k}})^{\ast}\Big{(}e^{\varphi^{\varepsilon_{j}\mu_{j}+\varepsilon_{k}\mu_{k}}}\Big{)}=(J^{-1})^{\ast}(f^{\varepsilon_{j}\mu_{j}+\varepsilon_{k}\mu_{k}})^{\ast}(\rho).$

Taking the logarithm of the above formula, we get

$\varphi^{\varepsilon_{j}\mu_{j}+\varepsilon_{k}\mu_{k}}\circ
F^{\varepsilon_{j}\mu_{j}+\varepsilon_{k}\mu_{k}}+\log|\partial_{w}F^{\varepsilon_{j}\mu_{j}+\varepsilon_{k}\mu_{k}}|^{2}=\log((f^{\varepsilon_{j}\mu_{j}+\varepsilon_{k}\mu_{k}})^{\ast}(\rho))\circ
J^{-1}+\log|(J^{-1})^{\prime}|^{2}.$

Then, using the above equations together with Ahlfors formulae (104) and
Wolpert’s formula (49), we have

$\begin{split}-\left.\left(\partialderivative{\log\mathsf{h}_{i}^{\varepsilon_{j}\mu_{j}+\varepsilon_{k}\mu_{k}}}{\varepsilon_{j}}{\bar{\varepsilon}_{k}}\right)\right|_{\varepsilon_{j}=\varepsilon_{k}=0}&=\lim_{w\to
w_{i}}\left.\left(\partialderivative{}{\varepsilon_{j}}{\bar{\varepsilon}_{k}}\right)\right|_{\varepsilon_{j}=\varepsilon_{k}=0}\log((f^{\varepsilon_{j}\mu_{j}+\varepsilon_{k}\mu_{k}})^{\ast}(\rho))\circ
J^{-1}\\\ &\overset{\eqref{WolpertSecVar}}{=}\frac{1}{2}\lim_{w\to
w_{i}}f_{\mu_{j}\bar{\mu}_{k}}\circ
J^{-1}(w)=\frac{1}{2}\left\langle\partialderivative{w_{j}},\partialderivative{w_{k}}\right\rangle_{\text{TZ},i}^{\text{ell}}.\end{split}$

  * •

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

$\begin{split}-\partialderivative{\log\mathsf{h}_{i}}{w_{j}}{\bar{w}_{k}}&=-\left.\left(\partialderivative{\log\mathsf{h}_{i}^{\varepsilon_{j}\mu_{j}+\varepsilon_{k}\mu_{k}}}{\varepsilon_{j}}{\bar{\varepsilon}_{k}}\right)\right|_{\varepsilon_{j}=\varepsilon_{k}=0}\\\
&=2\lim_{w\to
w_{i}}\left\\{\frac{1}{|w-w_{i}|}\left.\partialderivative{}{\varepsilon_{j}}{\bar{\varepsilon}_{k}}\right|_{\varepsilon_{j}=\varepsilon_{k}=0}\left(F^{(\varepsilon\mu)_{jk}}\right)^{\ast}\left(e^{-\frac{1}{2}\varphi^{(\varepsilon\mu)_{jk}}(w)}\right)\right.\\\
&\hskip
28.45274pt+\left.e^{-\frac{1}{2}\varphi(w)}\left.\partialderivative{\varepsilon_{j}}\right|_{\varepsilon_{j}=0}\left(\frac{\partial_{w}F^{\varepsilon_{j}\mu_{j}}(w)}{F^{\varepsilon_{j}\mu_{j}}(w-w_{i})}\right)^{\frac{1}{2}}\left.\partialderivative{\bar{\varepsilon}_{k}}\right|_{\varepsilon_{k}=0}\left(\frac{\overline{\partial_{w}F^{\varepsilon_{k}\mu_{k}}(w)}}{\overline{F^{\varepsilon_{k}\mu_{k}}(w-w_{i})}}\right)^{\frac{1}{2}}\right\\}\\\
&\overset{\eqref{expphivariations}}{=}\lim_{w\to
w_{i}}\left\\{-\frac{1}{2}\log|w-w_{i}|f_{\mu_{j}\bar{\mu}_{k}}\circ
J^{-1}(w)+\frac{1}{2}e^{-\frac{1}{2}\varphi(w)}\,\times\right.\\\
&\left.\frac{\left((w-w_{i})\partial_{w}\dot{F}^{j}(w)-\dot{F}^{j}(w)+\dot{F}^{j}(w_{i})\right)}{|w-w_{i}|(\bar{w}-\bar{w}_{i})^{\frac{1}{2}}}\frac{\left(\overline{(w-w_{i})\partial_{w}\dot{F}^{k}(w)-\dot{F}^{k}(w)+\dot{F}^{k}(w_{i})}\right)}{|w-w_{i}|(w-w_{i})^{\frac{1}{2}}}\right\\}\\\
&\overset{\eqref{diagJexpansion}}{=}-\frac{1}{2}\lim_{w\to\infty}\frac{\log|w-w_{i}|}{\imaginary(\varsigma_{i}^{-1}\circ
J^{-1}(w))}(\imaginary\varsigma_{i}z)f_{\mu_{j}\bar{\mu}_{k}}(\varsigma_{i}z)\\\
&=\pi\lim_{w\to\infty}(\imaginary\varsigma_{i}z)f_{\mu_{j}\bar{\mu}_{k}}(\varsigma_{i}z)\overset{\eqref{constcuspmetric}}{=}\frac{4\pi}{3}\left\langle\partialderivative{w_{j}},\partialderivative{w_{k}}\right\rangle_{TZ,i}^{cusp},\end{split}$

where in the last line, we have used the fact that
$\log|w-w_{i}|/\imaginary(\varsigma_{i}^{-1}\circ J^{-1}(w))=-2\pi$ as $w\to
w_{i}$.

  * •

_$i=n$ case:_

$\begin{split}\partialderivative{\log\mathsf{h}_{1}}{w_{j}}{\bar{w}_{k}}&=\left.\left(\partialderivative{\log\mathsf{h}_{1}^{\varepsilon_{j}\mu_{j}+\varepsilon_{k}\mu_{k}}}{\varepsilon_{j}}{\bar{\varepsilon}_{k}}\right)\right|_{\varepsilon_{j}=\varepsilon_{k}=0}\\\
&=\lim_{w\to\infty}\left\\{\left.\partialderivative{}{\varepsilon_{j}}{\bar{\varepsilon}_{k}}\right|_{\varepsilon_{j}=\varepsilon_{k}=0}\left(\log\left|F^{(\varepsilon\mu)_{jk}}\right|^{2}-2\left(F^{(\varepsilon\mu)_{jk}}\right)^{\ast}\left(e^{-\frac{1}{2}\varphi^{(\varepsilon\mu)_{jk}}}\right)\left|\frac{\partial_{w}F^{(\varepsilon\mu)_{jk}}}{F^{(\varepsilon\mu)_{jk}}}\right|\right)(w)\right\\}\\\
&=-2\lim_{w\to\infty}\left\\{\frac{1}{|w|}\left.\partialderivative{}{\varepsilon_{j}}{\bar{\varepsilon}_{k}}\right|_{\varepsilon_{j}=\varepsilon_{k}=0}\left(F^{(\varepsilon\mu)_{jk}}\right)^{\ast}\left(e^{-\frac{1}{2}\varphi^{(\varepsilon\mu)_{jk}}}\right)\right.\\\
&\hskip
42.67912pt+\left.e^{-\frac{1}{2}\varphi(w)}\left.\partialderivative{\varepsilon_{j}}\right|_{\varepsilon_{j}=0}\left(\frac{\partial_{w}F^{\varepsilon_{j}\mu_{j}}(w)}{F^{\varepsilon_{j}\mu_{j}}(w)}\right)^{\frac{1}{2}}\left.\partialderivative{\bar{\varepsilon}_{k}}\right|_{\varepsilon_{k}=0}\left(\frac{\overline{\partial_{w}F^{\varepsilon_{k}\mu_{k}}(w)}}{\overline{F^{\varepsilon_{k}\mu_{k}}(w)}}\right)^{\frac{1}{2}}\right\\}\\\
&\overset{\eqref{expphivariations}}{=}\lim_{w\to\infty}\left\\{\frac{e^{-\frac{1}{2}\varphi(w)}}{2|w|}f_{\mu_{j}\bar{\mu}_{k}}\circ
J^{-1}-\frac{e^{-\frac{1}{2}\varphi(w)}}{2}\frac{\left(w\partial_{w}\dot{F}^{j}(w)-\dot{F}^{j}(w)\right)}{|w|\bar{w}^{\frac{1}{2}}}\frac{\left(\overline{w\partial_{w}\dot{F}^{k}(w)-\dot{F}^{k}(w)}\right)}{|w|w^{\frac{1}{2}}}\right\\}\\\
&=\lim_{w\to\infty}\left\\{\frac{|w|\log|w|}{2|w|}f_{\mu_{j}\bar{\mu}_{k}}\left(J^{-1}(w)\right)\right\\}\overset{\eqref{diagJexpansion}}{=}\frac{1}{2}\lim_{w\to\infty}\frac{\log|w|}{\imaginary(J^{-1}(w))}(\imaginary
z)f_{\mu_{j}\bar{\mu}_{k}}(z)\\\ &=\pi\lim_{w\to\infty}(\imaginary
z)f_{\mu_{j}\bar{\mu}_{k}}(z)\overset{\eqref{constcuspmetric}}{=}\frac{4\pi}{3}\left\langle\partialderivative{w_{j}},\partialderivative{w_{k}}\right\rangle_{\text{TZ},n}^{\text{cusp}}.\end{split}$

In the above equation, we have used the fact that
$\log|w|/\imaginary(J^{-1}(w))=2\pi$ as $w\to\infty$.

∎

Using Lemma 3.2 and 5.2, we have

######  Corollary 5.2.1.

The function $-\log\mathsf{H}=-m_{1}h_{1}\log\mathsf{h}_{1}\dots-
m_{n-1}h_{n-1}\log\mathsf{h}_{n-1}+\log\mathsf{h}_{n}$ is a potential for the
combination
$\frac{4\pi}{3}\omega_{TZ}^{cusp}+\frac{1}{2}\sum_{j=1}^{n_{e}}m_{j}h_{j}\hskip
1.42262pt\omega_{\text{TZ},j}^{\text{ell}}$ on
$\mathcal{M}_{0,\boldsymbol{m}}(\boldsymbol{m})$. The first Chern form of the
Hermitian line bundle $(\lambda_{0,\boldsymbol{m}},\mathsf{H})$ over
$\mathfrak{M}_{0,n}(\boldsymbol{m})$ is given by

$\mathsf{c}_{1}\left(\lambda_{0,\boldsymbol{m}},\mathsf{H}\right)=\frac{4\pi}{3}\omega_{\text{TZ}}^{\text{cusp}}+\frac{1}{2}\sum_{j=1}^{n_{e}}m_{j}h_{j}\hskip
1.42262pt\omega_{\text{TZ},j}^{\text{ell}}.$ (204)

Let us remind that $r(z)$ is the projection of the regular automorphic form
$-\text{Sch}\left(J;z\right)$ of weight 4 to the subspace of cusp forms for
each marked Fuchsian group of first kind $\Gamma$. Its decomposition can be
written as follows

$r(z)=\sum_{i=1}^{n-3}\mathrm{a}_{i}\,r_{i}(z),$ (205)

where

$\mathrm{a}_{i}=\iint_{\mathcal{F}}\text{Sch}\left(J;z\right)\mu_{i}(z)\differential[2]{z}.$
(206)

According to Section 3.2, for varying $\Gamma$, the $r(z)$ determines a
$(1,0)$-form $\mathrm{r}$ on $\mathcal{T}_{0,n}$ and the latter one is
corresponding to the $(1,0)$-form $\vartheta$,565656See Lemma 3.3.

$\vartheta=\sum_{i=1}^{n-3}\mathrm{a}_{i}\differential{w_{i}},$

on $\mathcal{M}_{0,n}$. Therefore, we also have

######  Corollary 5.2.2.

The function $\mathscr{S}_{\boldsymbol{m}}:\mathcal{M}_{0,n}\to\mathbb{R}$
satisfies

$\partial\mathscr{S}_{\boldsymbol{m}}=2\vartheta$ (207)

and

$\bar{\partial}\partial\mathscr{S}_{\boldsymbol{m}}=-2\sqrt{-1}\left[\omega_{\text{WP}}-\frac{4\pi^{2}}{3}\omega_{\text{TZ}}^{\text{cusp}}-\frac{\pi}{2}\sum_{j=1}^{n_{e}}m_{j}h_{j}\hskip
1.42262pt\omega^{\text{ell}}_{\text{TZ},j}\right].$ (208)

To see the (207), put
$\mathscr{S}_{\boldsymbol{m}}=S_{\boldsymbol{m}}-\pi\log\mathsf{H}$ and
combine Lemma 5.1 with the proof of Theorem 4.1 which it gives

$\begin{split}\partial\mathscr{S}_{\boldsymbol{m}}&=\partial
S_{\boldsymbol{m}}-\pi\sum_{j=1}^{n}m_{j}h_{j}\hskip
1.42262pt\partial\log\mathsf{h}_{j}=\sum_{i=1}^{n-3}\left(\frac{\partial
S_{\boldsymbol{m}}}{\partial{w_{i}}}\right)dw_{i}-\pi\sum_{j=1}^{n}\sum_{i=1}^{n-3}m_{j}h_{j}\left(\frac{\partial}{\partial
w_{i}}\log\mathsf{h}_{j}\right)dw_{i}\\\
&=-2\pi\sum_{i=1}^{n-3}c_{i}dw_{i}-\pi\sum_{i=1}^{n-3}\sum_{j=1}^{n}h_{j}\dot{F}^{i}_{w}(w_{j})dw_{i}=2\sum_{i=1}^{n-3}\left(-\pi
c_{i}+\sum_{j=1}^{n}h_{j}(\mathscr{E}_{j},M_{i})\right)dw_{i}\\\
&\overset{\ref{EnergyMomentumEi}}{=}2\sum_{i=1}^{n-3}\mathrm{a}_{i}dw_{i}=2\vartheta\end{split}$

In the second line of the above equation, we noted that the Eq.(93) implies
that

$\dot{F}^{i}_{w}(w)=-\frac{1}{\pi}\iint_{\mathbb{C}}M_{i}(w^{\prime})\hskip
1.42262pt\partial_{w}R(w^{\prime},w)d^{2}w^{\prime}=-\frac{1}{\pi}\iint_{\mathbb{C}}M_{i}(w^{\prime})\left(\frac{1}{(w^{\prime}-w)^{2}}-\frac{1}{w^{\prime}(w^{\prime}-1)}\right)d^{2}w^{\prime},$

which at the point $w=w_{j}$ is simplified to

$\begin{split}\dot{F}^{i}_{w}(w_{j})&=-\frac{1}{\pi}\iint_{\mathbb{C}}M_{i}(w)\left(\frac{1}{(w-w_{j})^{2}}-\frac{1}{w_{j}(w_{j}-1)}\right)d^{2}w=-\frac{2}{\pi}\iint_{\mathbb{C}}M_{i}(w)\mathscr{E}_{j}(w)d^{2}w\\\
&=-\frac{2}{\pi}\iint_{\mathbb{C}}(\mathscr{E}_{j},M_{i}).\end{split}$

The Eq.(208) can be obtained easily by using Theorem 4.2 as well as Lemma 5.2.

######  Remark 5.1.

According to the Lemmas 3.2 and 4.3, the functions $\mathsf{H}$ and
$\exp[S_{\boldsymbol{m}}/\pi]$ are Hermitian metrics in the line bundle
$\lambda_{0,\boldsymbol{m}}$ over $\mathfrak{M}_{0,n}(\boldsymbol{m})$. This
implies that
$\mathscr{S}_{\boldsymbol{m}}=S_{\boldsymbol{m}}-\pi\log\mathsf{H}$ determines
a function on $\mathfrak{M}_{0,n}(\boldsymbol{m})$. Interestingly, the
combination
$\omega_{{}_{\text{WP}}}-\tfrac{4\pi^{2}}{3}\omega_{{}_{\text{TZ}}}^{\text{cusp}}-\tfrac{\pi}{2}\sum_{j=1}^{n_{e}}m_{j}h_{j}\hskip
1.42262pt\omega^{\text{ell}}_{\text{TZ},j}$ in Eq.(208) with overall factor of
$\tfrac{1}{12\pi}$ appears also in the local index theorem for orbifold
Riemann surfaces for $k=0,1$. This is the sign that the function
$\mathscr{S}_{\boldsymbol{m}}$ can play the role of the _Quillen’s metric_ in
the Hodge line bundle $\lambda_{1}$, defined in ZT_2018 . We will explain a
little bit more about this observation in conclusion.

###  5.2 Chern Forms and Potentials on Schottky Space
$\mathfrak{S}_{g,n}(\boldsymbol{m})$

Let
$O=\overset{{}_{\curlywedge}}{\Omega}\backslash\Sigma=[\mathbb{H}\backslash\Gamma]$
and $\Gamma$ be respectively an orbifold Riemann surface and Fuchsian group of
signature $(g;m_{1},\dots,m_{n_{e}};n_{p})$ and
$J:\mathbb{H}^{\ast}\to\overset{{}_{\curlywedge}}{\Omega}$ be the
corresponding branched covering map. As we have explained in Section 3.3, the
automorphic form $\text{Sch}\left(J^{-1};w\right)$ of weight 4 for the
$\Sigma$ can be projected to the subspace
$\mathcal{H}^{2,0}(\overset{{}_{\curlywedge}}{\Omega},\Sigma)\cong
T_{[0]}^{\ast}\mathfrak{S}_{g,n}(\boldsymbol{m})$,

$\mathsf{R}(w)=\sum_{i=1}^{3g-3+n}\mathrm{b}_{i}P_{i}(w)=\sum_{i=1}^{3g-3+n}\left(\text{Sch}\left(J^{-1}\right),M_{i}\right)P_{i}(w),$
(209)

where $P_{i},i=1,\dots,3g-3+n$ are given by (127) and (129) and by using
Eq.(131) we get

$\mathsf{R}(w)=\sum_{i=1}^{3g-3+n}\left(-\pi
c_{i}+\sum_{j=1}^{n}h_{j}\left(\mathscr{E}_{j},M_{i}\right)\right)P_{i}(w)\equiv\pi\mathsf{R}_{0}(w)+\sum_{j=1}^{n}\mathsf{R}_{j}(w).$
(210)

The definition of $\mathscr{E}_{j}$ is provided by (132). The $\mathsf{R}(w)$
coincides with a $(1,0)$-form $\mathscr{Q}$ on the Schottky pace
$\mathfrak{S}_{g,n}(\boldsymbol{m})$ 575757Actually, $dw_{i}$ in (211)
coincides with $dt_{i}$ in Eq.(24). To be more precise,
$dt_{1},\dots,dt_{3g-3}\equiv
d\lambda_{1},\dots,d\lambda_{g},da_{3},\dots,da_{g},db_{2},\dots,db_{g}$ and
$dt_{3g-3+1},\dots,dt_{3g-3+n}=dw_{1},\dots,dw_{n}$.

$\begin{split}\mathscr{Q}=&\sum_{i=1}^{3g-3+n}\mathrm{b}_{i}dw_{i}=\mathrm{b}_{1}d\lambda_{1}+\dots+\mathrm{b}_{g}d\lambda_{g}+\mathrm{b}_{g+1}da_{3}+\dots+\mathrm{b}_{2g-3}da_{g}\\\
&+\mathrm{b}_{2g-2}db_{2}+\dots+\mathrm{b}_{3g-3}db_{g}+\mathrm{b}_{3g-2}dw_{1}+\dots+\mathrm{b}_{3g-3+n}dw_{n}.\end{split}$
(211)

In the following two theorems, we will explicitly describe canonical
connections and curvatures on the Hermitian line bundle $\mathscr{L}_{i}$ and
$\mathbb{Q}$-line bundle
$\mathscr{L}=\bigotimes_{i=1}^{n}\mathscr{L}_{i}^{\mathsf{h}_{i}}$.

######  Theorem 5.3.

Let $\partial$ and $\bar{\partial}$ be $(1,0)$ and $(0,1)$ components of the
de Rham differential on Schottky space $\mathfrak{S}_{g,n}(\boldsymbol{m})$.
The following statements are true.

  1. (i)

In a local holomorphic frame, the canonical connection on the Hermitian line
bundle $(\mathscr{L}_{i},\mathsf{h}_{i})$ is given by

$\partial\log\mathsf{h}_{i}=\left\\{\begin{split}&-\frac{2}{\pi
m_{i}h_{i}}\mathsf{R}_{i}\quad&\text{for}\quad&i=1,\dots,n_{e},\\\
&-\frac{2}{\pi}\mathsf{R}_{i}&\text{for}\quad&i=n_{e}+1,\dots,n.\end{split}\right.$

  2. (ii)

In a local holomorphic frame, the canonical connection on the Hermitian
$\mathbb{Q}$-line bundle $(\mathscr{L},\exp\\{S_{\boldsymbol{m}}/\pi\\})$ is
given by

$\frac{1}{\pi}\partial S_{\boldsymbol{m}}=2\mathsf{R}_{0}.$

  3. (iii)

The function
$\mathscr{S}_{\boldsymbol{m}}:\mathfrak{S}_{g,n}(\boldsymbol{m})\to\mathbb{R}$
given by Eq.(200) satisfies

$\partial\mathscr{S}_{\boldsymbol{m}}=2\mathscr{Q}.$

###### Proof.

We will prove each statement separately:

  * •

In order to prove part $(i)$, it is sufficient to show that

$\left.\left(\partialderivative{\log\mathsf{h}_{i}^{\varepsilon\mu_{j}}}{\varepsilon}\right)\right|_{\varepsilon=0}=\left\\{\begin{split}&-\frac{2}{\pi
m_{i}}\left(\mathscr{E}_{i},M_{j}\right)\quad&\text{for}\quad&i=1,\dots,n_{e},\\\
&-\frac{2}{\pi}\left(\mathscr{E}_{i},M_{j}\right)&\text{for}\quad&i=n_{e}+1,\dots,n.\end{split}\right.$

Using Lemma 5.1, we have

$\left.\left(\partialderivative{\log\mathsf{h}_{i}^{\varepsilon\mu_{j}}}{\varepsilon}\right)\right|_{\varepsilon=0}=\left\\{\begin{split}&\frac{1}{m_{i}}\partial_{w}\dot{F}^{j}(w_{i})\quad&\text{for}\quad&i=1,\dots,n_{e},\\\
&\partial_{w}\dot{F}^{j}(w_{i})&\text{for}\quad&i=n_{e}+1,\dots,n.\end{split}\right.$
(212)

In another side, according to (93) and (132), one see

$\pi\partial_{w}\dot{F}^{j}(w_{i})=-\iint_{\overset{{}_{\curlywedge}}{\mathcal{D}}}M_{j}(w)\left(\frac{1}{(w-w_{i})^{2}}-\frac{1}{w(w-1)}\right)\differential[2]{w}=-2\left(\mathscr{E}_{i},M_{j}\right),$

which by substituting in (212) gives the desired result.

  * •

To prove part $(ii)$, we need to show that

$\left.\partialderivative{\varepsilon}\right|_{\varepsilon=0}S_{\boldsymbol{m}}([L_{1}^{\varepsilon\mu_{i}},\dots,L_{g}^{\varepsilon\mu_{i}}];w_{1}^{\varepsilon\mu_{i}},\dots,w_{n}^{\varepsilon\mu_{i}})=-2\pi
c_{i}\quad\text{for}\quad i=1,\dots,3g-3+n.$

We have

$\begin{split}&\mathcal{L}_{\mu_{i}}S_{\boldsymbol{m}}=\left.\partialderivative{\varepsilon}\right|_{\varepsilon=0}S_{\boldsymbol{m}}\left([L_{1}^{\varepsilon\mu_{i}},\dots,L_{g}^{\varepsilon\mu_{i}}];w_{1}^{\varepsilon\mu_{i}},\dots,w_{n}^{\varepsilon\mu_{i}}\right)\\\
&\hskip 31.2982pt=\frac{\sqrt{-1}}{2}\lim_{\epsilon\to
0}\left.\partialderivative{\varepsilon}\right|_{\varepsilon=0}I_{\epsilon}(\varepsilon)+\lim_{\epsilon\to
0}\frac{\partial\tilde{S}^{\text{(ct)}\epsilon}}{\partial w_{i}},\end{split}$
(213)

with $\tilde{S}^{\text{(ct)}\epsilon}$ is given in equation (148) and

$\begin{split}&I_{\epsilon}(\varepsilon)=\iint_{F^{\varepsilon\mu_{i}}(\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon})}|\partial_{w}\varphi^{\varepsilon\mu_{i}}|^{2}\differential{w}\wedge\differential{\bar{w}}+\sum_{k=2}^{g}\oint_{F^{\varepsilon\mu_{i}}(C_{k})}\theta_{(L_{k}^{\varepsilon\mu_{i}})^{-1}}(\varphi^{\varepsilon\mu_{i}})\\\
&\hskip
28.45274pt+\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{F^{\varepsilon\mu_{i}}(C_{j}^{\epsilon})}\varphi^{\varepsilon\mu_{i}}\left(\frac{\differential{\bar{w}}}{\bar{w}-\overline{w_{j}^{\varepsilon\mu_{i}}}}-\frac{\differential{w}}{w-w_{j}^{\varepsilon\mu_{i}}}\right),\end{split}$

where once again, we have used the Gauss-Bonnet formula for Riemann
orbisurfaces Troyanov1991PrescribingCO ; Li_orbifold_2018

$\frac{\sqrt{-1}}{2}\iint_{F^{\varepsilon\mu_{i}}(\overset{{}_{\curlywedge}}{\mathcal{D}})}e^{\varphi^{\varepsilon\mu_{i}}}\differential{w}\wedge\differential{\bar{w}}=2\pi\left(2g+\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)+n_{p}-2\right)=-2\pi\chi(X),$

to conclude that

$\frac{\sqrt{-1}}{2}\mathcal{L}_{\mu_{i}}\iint_{\overset{{}_{\curlywedge}}{\mathcal{D}}}e^{\varphi}\differential{w}\wedge\differential{\bar{w}}=0.$

The calculation of $\mathcal{L}_{\mu_{i}}S_{\boldsymbol{m}}$ closely follows
the corresponding computation in the proof of Theorem 1 in 1988SbMat..60..297Z
, where regularization at the punctures can be found in the proof of Theorem 1
in Zograf1988ONLE  and for ramification points in the proof of Theorem 4.1.
More explicitly, using the commutative diagram 133 and the change of variable
formula
$\int_{F(\overset{{}_{\curlywedge}}{\mathcal{D}})}\omega=\int_{\overset{{}_{\curlywedge}}{\mathcal{D}}}F^{\ast}(\omega)$,
we get

$\begin{split}I_{\epsilon}(\varepsilon)&=\iint_{\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}(\varepsilon)}(F^{\varepsilon\mu_{i}})^{\ast}\Big{(}|\partial_{w}\varphi^{\varepsilon\mu_{i}}|^{2}\differential{w}\wedge\differential{\bar{w}}\Big{)}+\sum_{k=2}^{g}\oint_{C_{k}}(F^{\varepsilon\mu_{i}})^{\ast}\left(\theta_{(L_{k}^{\varepsilon\mu_{i}})^{-1}}(\varphi^{\varepsilon\mu_{i}})\right)\\\
&+\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_{\overset{{}_{\curlywedge}}{\mathcal{D}}_{\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_{k=2}^{g}\oint_{C_{k}}\left(\varphi^{\varepsilon\mu_{i}}\circ
F^{\varepsilon\mu_{i}}-\frac{1}{2}\log|(L_{k}^{\varepsilon\mu_{i}})^{\prime}\circ
F^{\varepsilon\mu_{i}}|^{2}-\log|l_{k}^{\varepsilon\mu_{i}}|^{2}\right)\\\
&\hskip
79.6678pt\times\left(\frac{(L_{k}^{\varepsilon\mu_{i}})^{\prime\prime}\circ
F^{\varepsilon\mu_{i}}}{(L_{k}^{\varepsilon\mu_{i}})^{\prime}\circ
F^{\varepsilon\mu_{i}}}\differential{F^{\varepsilon\mu_{i}}(w)}-\frac{\overline{(L_{k}^{\varepsilon\mu_{i}})^{\prime\prime}\circ
F^{\varepsilon\mu_{i}}}}{\overline{(L_{k}^{\varepsilon\mu_{i}})^{\prime}\circ
F^{\varepsilon\mu_{i}}}}\differential{\overline{F^{\varepsilon\mu_{i}}(w)}}\right)\\\
&+\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).\end{split}$

which by noting that
$\differential{F^{\varepsilon\mu_{i}}(w)}=\partial_{w}F^{\varepsilon\mu_{i}}(dw+\varepsilon
M_{i}\hskip 1.42262ptd\bar{w})$ and
$\differential{\overline{F^{\varepsilon\mu_{i}}(w)}}=\overline{\partial_{w}F^{\varepsilon\mu_{i}}}(\overline{\varepsilon}\overline{M_{i}}dw+d\bar{w})$,
it turns to

$\begin{split}I_{\epsilon}(\varepsilon)&=\iint_{\overset{{}_{\curlywedge}}{\mathcal{D}}_{\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}}\\\
&-2\sum_{k=2}^{g}\oint_{C_{k}}\varphi^{\varepsilon\mu_{i}}\circ
F^{\varepsilon\mu_{i}}\,\frac{\overline{(L_{k}^{\varepsilon\mu_{i}})^{\prime\prime}\circ
F^{\varepsilon\mu_{i}}}}{\overline{(L_{k}^{\varepsilon\mu_{i}})^{\prime}\circ
F^{\varepsilon\mu_{i}}}}\,\overline{\partial_{w}F^{\varepsilon\mu_{i}}}(\bar{\varepsilon}\overline{M_{i}}\differential{w}+\differential{\bar{w}})\\\
&-\sum_{k=2}^{g}\oint_{C_{k}}\partial_{w}\varphi^{\varepsilon\mu_{i}}\circ
F^{\varepsilon\mu_{i}}\,\log|(L_{k}^{\varepsilon\mu_{i}})^{\prime}\circ
F^{\varepsilon\mu_{i}}|^{2}\,\partial_{w}F^{\varepsilon\mu_{i}}(\differential{w}+\varepsilon
M_{i}\differential{\bar{w}})\\\
&-\sum_{k=2}^{g}\oint_{C_{k}}\partial_{\bar{w}}\varphi^{\varepsilon\mu_{i}}\circ
F^{\varepsilon\mu_{i}}\,\log|(L_{k}^{\varepsilon\mu_{i}})^{\prime}\circ
F^{\varepsilon\mu_{i}}|^{2}\,\overline{\partial_{w}F^{\varepsilon\mu_{i}}}(\bar{\varepsilon}\overline{M_{i}}\differential{w}+\differential{\bar{w}})\\\
&+\sum_{k=2}^{g}\oint_{C_{k}}\log|(L_{k}^{\varepsilon\mu_{i}})^{\prime}\circ
F^{\varepsilon\mu_{i}}|^{2}\,\frac{\overline{(L_{k}^{\varepsilon\mu_{i}})^{\prime\prime}\circ
F^{\varepsilon\mu_{i}}}}{\overline{(L_{k}^{\varepsilon\mu_{i}})^{\prime}\circ
F^{\varepsilon\mu_{i}}}}\,\overline{\partial_{w}F^{\varepsilon\mu_{i}}}(\bar{\varepsilon}\overline{M_{i}}\differential{w}+\differential{\bar{w}})\\\
&+8\pi\sqrt{-1}\sum_{k=2}^{g}\log|l_{k}^{\varepsilon\mu_{i}}|^{2}\\\
&+\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 $C_{k}$s are the usual components of
$\partial\overset{{}_{\curlywedge}}{\mathcal{D}}$ and

$\left\\{\begin{split}&\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}(\varepsilon)=\overset{{}_{\curlywedge}}{\mathcal{D}}\Big{\backslash}\bigcup_{i=1}^{n}D_{i}^{\epsilon}(\varepsilon),\\\
&D_{i}^{\epsilon}(\varepsilon)=\left\\{w\in\mathcal{D}\,\Big{|}\,|w^{\varepsilon\mu_{i}}-w_{i}^{\varepsilon\mu_{i}}|<\epsilon\right\\},\\\
&C_{j}^{\epsilon}(\varepsilon)=\partial
D_{j}^{\epsilon}(\varepsilon)=\left\\{w\in\mathcal{D}\,\Big{|}\,|w^{\varepsilon\mu_{i}}-w_{j}^{\varepsilon\mu_{i}}|=\epsilon\right\\}.\end{split}\right.$

As in the proof of Theorem 4.1, in order to compute $\dot{I}^{i}_{\epsilon}$,
we need to differentiate under the integral sign as well as over the variable
integration domain
$\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}(\varepsilon)$. Using
Eq.(155), the second contribution gives

$\left.\partialderivative{\varepsilon}\right|_{\varepsilon=0}\iint_{\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}(\varepsilon)}|\partial_{w}\varphi|^{2}\differential{w}\wedge\differential{\bar{w}}=-\sum_{j=1}^{n}\oint_{\partial
D_{j}^{\epsilon}(\varepsilon)}|\partial_{w}\varphi|^{2}\left(\dot{F}^{i}(w)-\dot{F}^{i}(w_{j})\right)\differential{\bar{w}},$
(214)

where $\partial D_{j}^{\epsilon}(\varepsilon)$ are oriented as a boundary of
$D_{j}^{\epsilon}(\varepsilon)$ which is opposite to the orientation of
$\partial\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}$. Moreover, the
contribution from the variation of integral domain $C_{j}^{\varepsilon}$
vanishes since $\partial C_{j}^{\epsilon}=\emptyset$. The differentiation
under the integral sign repeats the calculations done in the proof of
(1988SbMat..60..297Z, , Thoerem 1) almost word-for-word. The only change is
that the integration domain has now changed to $\mathcal{D}_{\epsilon}$.
Accordingly,

$\begin{split}\dot{I}^{i}_{\epsilon}&=\iint_{\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}}\left[(\partial_{w}\dot{\varphi}^{i}+\partial_{w}^{2}\varphi\dot{F}^{i})\partial_{\bar{w}}\varphi+(\partial_{\bar{w}}\dot{\varphi}^{i}+\partial_{w}\partial_{\bar{w}}\varphi\dot{F}^{i})\partial_{w}\varphi+|\partial_{w}\varphi|^{2}\partial_{w}\dot{F}^{i}\right]\differential{w}\wedge\differential{\bar{w}}\\\
&-\sum_{j=1}^{n}\oint_{\partial
D_{j}^{\epsilon}}|\partial_{w}\varphi|^{2}\left(\dot{F}^{i}(w)-\dot{F}^{i}(w_{j})\right)\differential{\bar{w}}-2\sum_{k=2}^{g}\oint_{C_{k}}(\dot{\varphi}^{i}+\partial_{w}\varphi\dot{F}^{i})\frac{\overline{L_{k}^{\prime\prime}}}{\overline{L_{k}^{\prime}}}\differential{\bar{w}}\\\
&-\sum_{k=2}^{g}\oint_{C_{k}}\left[(\partial_{w}\dot{\varphi}^{i}+\partial_{w}^{2}\varphi\dot{F}^{i})\log|L_{k}^{\prime}|^{2}\differential{w}+\partial_{w}\varphi\frac{(\dot{L}_{k}^{i})^{\prime}+L_{k}^{\prime\prime}\dot{F}^{i}}{L^{\prime}_{k}}\differential{w}\right.\\\
&\hskip
85.35826pt+\partial_{w}\varphi\log|L^{\prime}_{k}|^{2}\partial_{w}\dot{F}^{i}\differential{w}+\partial_{w}\varphi\log|L^{\prime}_{k}|^{2}M_{i}\differential{\bar{w}}\Bigg{]}\\\
&-\sum_{k=2}^{g}\oint_{C_{k}}\left[(\partial_{\bar{w}}\dot{\varphi}^{i}+\partial_{w}\partial_{\bar{w}}\varphi\dot{F}^{i})\log|L_{k}^{\prime}|^{2}+\partial_{w}\varphi\frac{(\dot{L}_{k}^{i})^{\prime}+L_{k}^{\prime\prime}\dot{F}^{i}}{L^{\prime}_{k}}\right]\differential{\bar{w}}\\\
&+\sum_{k=2}^{g}\oint_{C_{k}}\frac{(\dot{L}_{k}^{i})^{\prime}+L_{k}^{\prime\prime}\dot{F}^{i}}{L^{\prime}_{k}}\frac{\overline{L_{k}^{\prime\prime}}}{\overline{L_{k}^{\prime}}}\differential{\bar{w}}+8\pi\sqrt{-1}\sum_{k=2}^{g}\frac{\dot{l}_{k}^{i}}{l_{k}}\\\
&-\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{C_{j}^{\epsilon}}\partial_{w}\dot{F}^{i}\left(\frac{d\bar{w}}{\bar{w}-\bar{w}_{j}}-\frac{dw}{w-w_{j}}\right)+\text{\scriptsize{$\mathcal{O}$}}\left(1\right),\end{split}$

where we have used the Eq.(157) and the fact that $F^{\varepsilon\mu_{i}}$ and
$L_{k}^{\varepsilon\mu_{i}}$ are holomorphic in the variable
$\varepsilon\in\mathbb{C}$. As in the proof of Theorem 4.1, we can use Lemma
3.4 and the equality $\partial_{\bar{w}}\dot{F}^{i}=M_{i}$ to rewrite the
above equation as

$\begin{split}\dot{I}^{i}_{\epsilon}&=\iint_{\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}}\left[(-\partial_{w}\varphi\partial_{w}\dot{F}^{i}-\partial_{w}^{2}\dot{F}^{i})\partial_{\bar{w}}\varphi+(-\partial_{w}\varphi\partial_{\bar{w}}\dot{F}^{i}-\partial_{w}\partial_{\bar{w}}\dot{F}^{i})\partial_{w}\varphi+|\partial_{w}\varphi|^{2}\partial_{w}\dot{F}^{i}\right]\differential{w}\wedge\differential{\bar{w}}\\\
&-\sum_{j=1}^{n}\oint_{\partial
D_{j}^{\epsilon}}|\partial_{w}\varphi|^{2}\left(\dot{F}^{i}(w)-\dot{F}^{i}(w_{j})\right)\differential{\bar{w}}+2\sum_{k=2}^{g}\oint_{C_{k}}\partial_{w}\dot{F}^{i}\frac{\overline{L_{k}^{\prime\prime}}}{\overline{L_{k}^{\prime}}}\differential{\bar{w}}\\\
&-\sum_{k=2}^{g}\oint_{C_{k}}\left[(-\partial_{w}\varphi\partial_{w}\dot{F}^{i}-\partial_{w}^{2}\dot{F}^{i})\log|L_{k}^{\prime}|^{2}\differential{w}+\partial_{w}\varphi\frac{(\dot{L}_{k}^{i})^{\prime}+L_{k}^{\prime\prime}\dot{F}^{i}}{L^{\prime}_{k}}\differential{w}\right.\\\
&\hskip
105.2751pt+\partial_{w}\varphi\log|L^{\prime}_{k}|^{2}\partial_{w}\dot{F}^{i}\differential{w}+\partial_{w}\partial_{\bar{w}}\dot{F}^{i}\log|L^{\prime}_{k}|^{2}\differential{\bar{w}}\Bigg{]}\\\
&-\sum_{k=2}^{g}\oint_{C_{k}}\left(\partial_{w}\varphi-\frac{\overline{L_{k}^{\prime\prime}}}{\overline{L_{k}^{\prime}}}\right)\frac{(\dot{L}_{k}^{i})^{\prime}+L_{k}^{\prime\prime}\dot{F}^{i}}{L^{\prime}_{k}}\differential{\bar{w}}+8\pi\sqrt{-1}\sum_{k=2}^{g}\frac{\dot{l}_{k}^{i}}{l_{k}}\\\
&-\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{C_{j}^{\epsilon}}\partial_{w}\dot{F}^{i}\left(\frac{d\bar{w}}{\bar{w}-\bar{w}_{j}}-\frac{dw}{w-w_{j}}\right)+\text{\scriptsize{$\mathcal{O}$}}\left(1\right).\end{split}$
(215)

Let us first compute the integral over
$\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}$:

$\begin{split}&\iint_{\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}}\left[(-\partial_{w}\varphi\partial_{w}\dot{F}^{i}-\partial_{w}^{2}\dot{F}^{i})\partial_{\bar{w}}\varphi+(-\partial_{w}\varphi\partial_{\bar{w}}\dot{F}^{i}-\partial_{w}\partial_{\bar{w}}\dot{F}^{i})\partial_{w}\varphi+|\partial_{w}\varphi|^{2}\partial_{w}\dot{F}^{i}\right]\differential{w}\wedge\differential{\bar{w}}\\\
&=\iint_{\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}}\left[\big{(}2\partial_{w}^{2}\varphi-(\partial_{w}\varphi)^{2}\big{)}\partial_{\bar{w}}\dot{F}^{i}-2\partialderivative{w}(\partial_{w}\varphi\partial_{\bar{w}}\dot{F}^{i})\right.\\\
&\hskip
123.76965pt\left.+\partialderivative{\bar{w}}(\partial_{w}\varphi\partial_{w}\dot{F}^{i})-\partialderivative{w}(\partial_{\bar{w}}\varphi\partial_{w}\dot{F}^{i})\right]\differential{w}\wedge\differential{\bar{w}}\\\
&=2\iint_{\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}}T_{\varphi}M_{i}\differential{w}\wedge\differential{\bar{w}}\underbrace{-2\int_{\partial\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}}\partial_{w}\varphi\partial_{\bar{w}}\dot{F}^{i}\differential{\bar{w}}}_{I_{1}}\underbrace{-\int_{\partial\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}}\partial_{w}\varphi\partial_{w}\dot{F}^{i}\differential{w}}_{I_{2}}\underbrace{-\int_{\partial\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}}\partial_{\bar{w}}\varphi\partial_{w}\dot{F}^{i}\differential{\bar{w}}}_{I_{3}},\end{split}$

where in the last line, we have used the definition of the Energy-Momentum
$T_{\varphi}=\text{Sch}\left(J^{-1};w\right)=\partial_{w}^{2}\varphi-\frac{1}{2}(\partial_{w}\varphi)^{2}$.
In order to compute the integrals $I_{1}$, $I_{2}$, and $I_{3}$, we need some
useful identities. From the equality $F^{\varepsilon\mu_{i}}\circ
L_{k}=L_{k}^{\varepsilon\mu_{i}}\circ F^{\varepsilon\mu_{i}}$ for
$k=1,\dots,g$ one can see that

$\left\\{\begin{split}&\dot{F}^{i}\circ
L_{k}=\dot{F}^{i}L^{\prime}_{k}+\dot{L}_{k}^{i},\\\ \\\
&\partial_{w}\dot{F}^{i}\circ
L_{k}\,L^{\prime}_{k}=\partial_{w}\dot{F}^{i}L^{\prime}_{k}+\dot{F}^{i}L^{\prime\prime}_{k}+\dot{L}_{k}^{i},\\\
\\\ &\partial_{\bar{w}}\dot{F}^{i}\circ
L_{k}\,\overline{L^{\prime}_{k}}=\partial_{\bar{w}}\dot{F}^{i}L^{\prime}_{k},\end{split}\right.$

Moreover, from Eq.(136) one has

$\partial_{w}\varphi\circ
L_{k}\,L^{\prime}_{k}+\frac{L^{\prime\prime}_{k}}{L^{\prime}_{k}}=\partial_{w}\varphi,\qquad\partial_{\bar{w}}\varphi\circ
L_{k}\,\overline{L^{\prime}_{k}}+\frac{\overline{L^{\prime\prime}_{k}}}{\overline{L^{\prime}_{k}}}=\partial_{\bar{w}}\varphi.$

Now, recalling that
$\partial\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}=\left[\bigcup\limits_{k=1}^{g}(C_{k}\cup
C^{\prime}_{k})\right]\cup\left[\bigcup\limits_{i=1}^{n+l}C_{i}^{\epsilon}\right]$
where $C^{\prime}_{k}=-L_{k}(C_{k})$, together with implementing some of the
above identities we get

$\begin{split}I_{1}&=-2\int_{\partial\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}}\partial_{w}\varphi\partial_{\bar{w}}\dot{F}^{i}\differential{\bar{w}}\\\
&=-2\sum_{k=2}^{g}\oint_{C_{k}}\partial_{w}\varphi\partial_{\bar{w}}\dot{F}^{i}\differential{\bar{w}}-2\sum_{j=1}^{n_{e}}\oint_{C^{\epsilon}_{j}}\partial_{w}\varphi\partial_{\bar{w}}\dot{F}^{i}\differential{\bar{w}}-2\sum_{j=1}^{n_{p}}\oint_{C^{\epsilon}_{n_{e}+j}}\partial_{w}\varphi\partial_{\bar{w}}\dot{F}^{i}\differential{\bar{w}}\\\
&=-2\sum_{k=2}^{g}\oint_{C_{k}}\partial_{\bar{w}}\dot{F}^{i}\frac{L^{\prime\prime}_{k}}{L^{\prime}_{k}}\differential{\bar{w}}-2\sum_{j=1}^{n_{e}}\oint_{C^{\epsilon}_{j}}\partial_{w}\varphi
M_{i}\differential{\bar{w}}-2\sum_{j=1}^{n_{p}}\oint_{C^{\epsilon}_{n_{e}+j}}\partial_{w}\varphi
M_{i}\differential{\bar{w}},\end{split}$

which it can be simplified more by using the asymptotic expansions (322)

$\begin{split}&I_{1}=2\sum_{k=2}^{g}\oint_{C_{k}}\left[\left(\frac{L^{\prime\prime}_{k}}{L^{\prime}_{k}}\right)^{\prime}\dot{F}^{i}+\frac{L^{\prime\prime}_{k}}{L^{\prime}_{k}}\partial_{w}\dot{F}^{i}\right]\differential{w}\\\
&-2\sum_{j=1}^{n_{e}}\oint_{C^{\epsilon}_{j}}\left(-\frac{1-\frac{1}{m_{j}}}{w-w_{j}}+\frac{c_{j}}{1-\frac{1}{m_{j}}}+\dotsm\right)\left(\frac{\bar{q}^{(j)}_{1}}{4\bar{J}^{(j)}_{1}}\cdot|w-w_{j}|^{1-\frac{2}{m_{j}}}+\dotsm\right)\differential{\bar{w}}\\\
&-2\sum_{j=1}^{n_{p}-1}\oint_{C^{\epsilon}_{n_{e}+j}}\left(-\frac{1}{w-w_{n_{e}+j}}\left(1+\frac{1}{\log\left|\frac{w-w_{n_{e}+j}}{J_{1}^{(n_{e}+j)}}\right|}\right)+\dotsm\right)\times\\\
&\hskip
56.9055pt\times\left(-\frac{|\delta_{n_{e}+j}|^{2}\bar{q}_{1}^{(n_{e}+j)}}{4\pi^{2}\bar{J}_{1}^{(n_{e}+j)}}\cdot|w-w_{n_{e}+j}|\log^{2}|w-w_{n_{e}+j}|+\dotsm\right)\differential{\bar{w}}\\\
&-2\oint_{C_{n}^{\epsilon}}\left(-\frac{1}{w}\left(1+\frac{1}{\log\left|\frac{w}{J_{-1}^{(n)}}\right|}\right)-\frac{c_{n}}{w^{2}}+\dotsm\right)\left(-\frac{|\delta_{n}|^{2}\bar{q}_{1}^{(n)}\bar{J}_{-1}^{(n)}}{4\pi^{2}}\cdot\frac{\log^{2}|w|}{|w|}+\dotsm\right)\differential{\bar{w}}\\\
&=2\sum_{k=2}^{g}\oint_{C_{k}}\left[\left(\frac{L^{\prime\prime}_{k}}{L^{\prime}_{k}}\right)^{\prime}\dot{F}^{i}+\frac{L^{\prime\prime}_{k}}{L^{\prime}_{k}}\partial_{w}\dot{F}^{i}\right]\differential{w}+\text{\scriptsize{$\mathcal{O}$}}\left(1\right);\end{split}$

Moreover,

$\begin{split}I_{2}&=-\int_{\partial\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}}\partial_{w}\varphi\partial_{w}\dot{F}^{i}\differential{w}=-\sum_{k=2}^{g}\oint_{C_{k}}\partial_{w}\varphi\partial_{w}\dot{F}^{i}\differential{w}-\sum_{j=1}^{n}\oint_{C_{j}^{\epsilon}}\partial_{w}\varphi\partial_{w}\dot{F}^{i}\differential{w}\\\
&=-\sum_{k=2}^{g}\oint_{C_{k}}\left[\frac{L^{\prime\prime}_{k}}{L^{\prime}_{k}}\partial_{w}\dot{F}^{i}-\left(\partial_{w}\varphi-\frac{L^{\prime\prime}_{k}}{L^{\prime}_{k}}\right)\frac{(\dot{L}_{k}^{i})^{\prime}+L_{k}^{\prime\prime}\dot{F}^{i}}{L^{\prime}_{k}}\right]\differential{w}-\sum_{j=1}^{n}\oint_{C_{j}^{\epsilon}}\partial_{w}\varphi\partial_{w}\dot{F}^{i}\differential{w};\end{split}$

Furthermore,

$\begin{split}I_{3}&=-\int_{\partial\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}}\partial_{\bar{w}}\varphi\partial_{w}\dot{F}^{i}\differential{\bar{w}}=-\sum_{k=2}^{g}\oint_{C_{k}}\partial_{\bar{w}}\varphi\partial_{w}\dot{F}^{i}\differential{\bar{w}}-\sum_{j=1}^{n}\oint_{C_{j}^{\epsilon}}\partial_{\bar{w}}\varphi\partial_{w}\dot{F}^{i}\differential{\bar{w}}\\\
&=-\sum_{k=2}^{g}\oint_{C_{k}}\left[\frac{\overline{L^{\prime\prime}_{k}}}{\overline{L^{\prime}_{k}}}\partial_{w}\dot{F}^{i}-\left(\partial_{\bar{w}}\varphi-\frac{\overline{L^{\prime\prime}_{k}}}{\overline{L^{\prime}_{k}}}\right)\frac{(\dot{L}_{k}^{i})^{\prime}+L_{k}^{\prime\prime}\dot{F}^{i}}{L^{\prime}_{k}}\right]\differential{\bar{w}}-\sum_{j=1}^{n}\oint_{C_{j}^{\epsilon}}\partial_{\bar{w}}\varphi\partial_{w}\dot{F}^{i}\differential{\bar{w}};\end{split}$

Using the fact that

$-\sum_{j=1}^{n}\oint_{C_{j}^{\epsilon}}\partial_{w}\varphi\partial_{w}\dot{F}^{i}\differential{w}-\sum_{j=1}^{n}\oint_{C_{j}^{\epsilon}}\partial_{\bar{w}}\varphi\partial_{w}\dot{F}^{i}\differential{\bar{w}}=-\sum_{j=1}^{n}\oint_{C_{j}^{\epsilon}}\partial_{w}\dot{F}^{i}\differential{\varphi}=\text{\scriptsize{$\mathcal{O}$}}\left(1\right)\quad\text{as}\quad\epsilon\to
0,$

we get

$\begin{split}I_{1}+I_{2}+I_{3}&=2\sum_{k=2}^{g}\oint_{C_{k}}\left[\left(\frac{L^{\prime\prime}_{k}}{L^{\prime}_{k}}\right)^{\prime}\dot{F}^{i}+\frac{L^{\prime\prime}_{k}}{L^{\prime}_{k}}\partial_{w}\dot{F}^{i}\right]\differential{w}\\\
&-\sum_{k=2}^{g}\oint_{C_{k}}\left[\frac{L^{\prime\prime}_{k}}{L^{\prime}_{k}}\partial_{w}\dot{F}^{i}-\left(\partial_{w}\varphi-\frac{L^{\prime\prime}_{k}}{L^{\prime}_{k}}\right)\frac{(\dot{L}_{k}^{i})^{\prime}+L_{k}^{\prime\prime}\dot{F}^{i}}{L^{\prime}_{k}}\right]\differential{w}\\\
&-\sum_{k=2}^{g}\oint_{C_{k}}\left[\frac{\overline{L^{\prime\prime}_{k}}}{\overline{L^{\prime}_{k}}}\partial_{w}\dot{F}^{i}-\left(\partial_{\bar{w}}\varphi-\frac{\overline{L^{\prime\prime}_{k}}}{\overline{L^{\prime}_{k}}}\right)\frac{(\dot{L}_{k}^{i})^{\prime}+L_{k}^{\prime\prime}\dot{F}^{i}}{L^{\prime}_{k}}\right]\differential{\bar{w}}+\text{\scriptsize{$\mathcal{O}$}}\left(1\right).\end{split}$

Substituting the above results in Eq.(215), we get

$\begin{split}\dot{I}^{i}_{\epsilon}&=2\iint_{\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}}T_{\varphi}M_{i}\differential{w}\wedge\differential{\bar{w}}-\sum_{j=1}^{n}\oint_{C_{j}^{\epsilon}}|\partial_{w}\varphi|^{2}\left(\dot{F}^{i}(w)-\dot{F}^{i}(w_{j})\right)\differential{\bar{w}}\\\
&+\sum_{k=2}^{g}\oint_{C_{k}}\left[\partial_{w}\dot{F}^{i}\left(\frac{L^{\prime\prime}_{k}}{L^{\prime}_{k}}\differential{w}+\frac{\overline{L^{\prime\prime}_{k}}}{\overline{L^{\prime}_{k}}}\differential{\bar{w}}\right)+\log|L^{\prime}_{k}|^{2}\left(\partial_{w}^{2}\dot{F}^{i}\differential{w}+\partial_{\bar{w}}\partial_{w}\dot{F}^{i}\differential{\bar{w}}\right)\right]\\\
&+\sum_{k=2}^{g}\oint_{C_{k}}\dot{F}^{i}\left[2\left(\frac{L^{\prime\prime}_{k}}{L^{\prime}_{k}}\right)^{\prime}-\left(\frac{L^{\prime\prime}_{k}}{L^{\prime}_{k}}\right)^{2}\right]\differential{w}\\\
&+\sum_{k=2}^{g}\oint_{C_{k}}\frac{(\dot{L}_{k}^{i})^{\prime}+L_{k}^{\prime\prime}}{(L^{\prime}_{k})^{2}}\differential{w}+8\pi\sqrt{-1}\sum_{k=2}^{g}\frac{\dot{l}_{k}^{i}}{l_{k}}\\\
&-\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{C_{j}^{\epsilon}}\partial_{w}\dot{F}^{i}\left(\frac{d\bar{w}}{\bar{w}-\bar{w}_{j}}-\frac{dw}{w-w_{j}}\right)+\text{\scriptsize{$\mathcal{O}$}}\left(1\right).\end{split}$
(216)

Using the fact that the second line in (216) can be written as

$\begin{split}&\sum_{k=2}^{g}\oint_{C_{k}}\left[\partial_{w}\dot{F}^{i}\left(\frac{L^{\prime\prime}_{k}}{L^{\prime}_{k}}\differential{w}+\frac{\overline{L^{\prime\prime}_{k}}}{\overline{L^{\prime}_{k}}}\differential{\bar{w}}\right)+\log|L^{\prime}_{k}|^{2}\left(\partial_{w}^{2}\dot{F}^{i}\differential{w}+\partial_{\bar{w}}\partial_{w}\dot{F}^{i}\differential{\bar{w}}\right)\right]\\\
&=\sum_{k=2}^{g}\oint_{C_{k}}\differential{\left(\partial_{w}\dot{F}^{i}\log|L^{\prime}_{k}|^{2}\right)},\end{split}$

one can see that this sum vanishes; the third line in (216) is also equal to
$0$, since

$\sum_{k=2}^{g}\oint_{C_{k}}\dot{F}^{i}\left[2\left(\frac{L^{\prime\prime}_{k}}{L^{\prime}_{k}}\right)^{\prime}-\left(\frac{L^{\prime\prime}_{k}}{L^{\prime}_{k}}\right)^{2}\right]\differential{w}=2\sum_{k=2}^{g}\oint_{C_{k}}\dot{F}^{i}\hskip
2.84526pt\text{Sch}\left(L_{k};w\right)\differential{w}=0.$

As for the fourth line of Eq.(216), since the contour $C_{k}$ encircles a zero
of the function $L^{\prime}_{k}$, i.e. the point $L_{k}^{-1}(\infty)$, we get
from the Cauchy formula that

$\oint_{C_{k}}\frac{(\dot{L}_{k}^{i})^{\prime}+L_{k}^{\prime\prime}}{(L^{\prime}_{k})^{2}}\differential{w}=-8\pi\sqrt{-1}\frac{\dot{l}_{k}^{i}}{l_{k}}\quad\text{for}\quad
k=2,\dots,g;$

therefore, the fourth line of Eq.(216) vanishes as well. Putting everything
together, one has

$\begin{split}\dot{I}^{i}_{\epsilon}&=2\iint_{\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}}T_{\varphi}M_{i}\differential{w}\wedge\differential{\bar{w}}-\sum_{j=1}^{n}\oint_{C_{j}^{\epsilon}}|\partial_{w}\varphi|^{2}\left(\dot{F}^{i}(w)-\dot{F}^{i}(w_{j})\right)\differential{\bar{w}}\\\
&\hskip
28.45274pt-\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\oint_{C_{j}^{\epsilon}}\partial_{w}\dot{F}^{i}\left(\frac{d\bar{w}}{\bar{w}-\bar{w}_{j}}-\frac{dw}{w-w_{j}}\right)+\text{\scriptsize{$\mathcal{O}$}}\left(1\right).\end{split}$
(217)

Using Eq.(213), the limit

$\lim_{\epsilon\to
0}\iint_{\overset{{}_{\curlywedge}}{\mathcal{D}}_{\epsilon}}T_{\varphi}M_{i}\differential{w}\wedge\differential{\bar{w}}\overset{\eqref{Rquaddiffexpansion}}{=}-2\sqrt{-1}\mathrm{b}_{i},$

and

$\begin{split}&\lim_{\epsilon\to
0}\sum_{j=1}^{n}\oint_{C_{j}^{\epsilon}}|\partial_{w}\varphi|^{2}\left(\dot{F}^{i}(w)-\dot{F}^{i}(w_{j})\right)\differential{\bar{w}}\\\
&\overset{\ref{lemma:asymptotics}}{=}\lim_{\epsilon\to
0}\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)^{2}\oint_{C_{j}^{\epsilon}}\frac{\dot{F}^{i}(w)-\dot{F}^{i}(w_{j})}{|w-w_{j}|^{2}}\differential{\bar{w}}\\\
&+\lim_{\epsilon\to
0}\sum_{j=1}^{n_{p}-1}\oint_{C_{j}^{\epsilon}}\left(\frac{\dot{F}^{i}(w)-\dot{F}^{i}(w_{j})}{|w-w_{j}|^{2}}+\frac{2\big{(}\dot{F}^{i}(w)-\dot{F}^{i}(w_{j})\big{)}}{|w-w_{j}|^{2}\log|w-w_{j}|}\right)\differential{\bar{w}}\\\
&+\lim_{\epsilon\to
0}\oint_{C_{n}^{\epsilon}}\left(\frac{\dot{F}^{i}(w)-\dot{F}^{i}(\infty)}{|w|^{2}}+\frac{2\big{(}\dot{F}^{i}(w)-\dot{F}^{i}(\infty)\big{)}}{|w|^{2}\log|w|}\right)\differential{\bar{w}}\\\
&=2\pi\sqrt{-1}\sum_{j=1}^{n}\left(1-\frac{1}{m_{j}}\right)^{2}\partial_{w}\dot{F}^{i}(w_{j})\quad\text{with}\quad
m_{j}=\infty\quad\text{for}\quad j=n_{e}+1,\dots,n,\end{split}$

Accordingly, the Eq.(217) is simplified to

$\begin{split}\dot{I}^{i}_{\epsilon}=-4\sqrt{-1}\mathrm{b}_{i}-2\pi\sqrt{-1}\sum_{j=1}^{n}\left(1-\frac{1}{m_{j}}\right)^{2}\partial_{w}\dot{F}^{i}(w_{j})-4\pi\sqrt{-1}\sum_{j=1}^{n_{e}}\left(1-\frac{1}{m_{j}}\right)\partial_{w}\dot{F}^{i}(w_{j})\end{split}$
(218)

Finally, by putting everything together from (213), (218) and (162), we get

$\begin{split}&\mathcal{L}_{\mu_{i}}S_{\boldsymbol{m}}=2\mathrm{b}_{i}+\pi\sum_{j=1}^{n}h_{j}\partial_{w}\dot{F}^{i}(w_{j})=-2\pi
c_{i}+2\sum_{j=1}^{n}h_{j}\left(\mathscr{E}_{j},M_{i}\right)+\pi\sum_{j=1}^{n}h_{j}\partial_{w}\dot{F}^{i}(w_{j})\\\
&\overset{\eqref{EMinnerprod}}{=}-2\pi c_{i}.\end{split}$

  * •

The proof of part $(iii)$ follows readily from those of parts $(i)$ and
$(ii)$.

∎

######  Theorem 5.4.

The following statements are also true.

  1. (i)

The first Chern form of the Hermitian line bundle
$(\mathscr{L}_{i},\mathsf{h}_{i})$ is given by

$\mathsf{c}_{1}\left(\mathscr{L}_{i},\mathsf{h}_{i}\right)=\frac{1}{2\pi}\omega^{\text{ell}}_{\text{TZ},i},\qquad
i=1,\dots,n_{e}.$

  2. (ii)

The first Chern form of the Hermitian line bundle
$(\mathscr{L}_{i},\mathsf{h}_{i})$ is given by

$\mathsf{c}_{1}\left(\mathscr{L}_{i},\mathsf{h}_{i}\right)=\frac{4}{3}\omega^{\text{cusp}}_{\text{TZ},i},\qquad
i=n_{e}+1,\dots,n.$

  3. (iii)

The first Chern form of the $\mathbb{Q}$-line bundle
$(\mathscr{L},\exp[S_{\boldsymbol{m}}/\pi])$ is given by

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

  4. (iv)

The function
$\mathscr{S}_{\boldsymbol{m}}=S_{\boldsymbol{m}}-\pi\log\mathsf{H}$ satisfies

$-\hskip
1.42262pt\bar{\partial}\partial\mathscr{S}_{\boldsymbol{m}}=2\sqrt{-1}\left(\omega_{\text{WP}}-\frac{4\pi^{2}}{3}\omega^{\text{\text{cusp}}}_{\text{TZ}}-\frac{\pi}{2}\sum_{j=1}^{n_{e}}m_{j}h_{j}\hskip
1.42262pt\omega^{\text{\text{ell}}}_{\text{TZ},j}\right)$

###### Proof.

Since

$\mathsf{c}_{1}\left(\mathscr{L}_{i},\mathsf{h}_{i}\right)=\frac{\sqrt{-1}}{2\pi}\bar{\partial}\partial\log\mathsf{h}_{i},$

The proof of parts _(i)_ and _(ii)_ are exactly the same as that of Lemma 5.2.
The proof of part _(iii)_ can be obtained by Theorem 5.3 and accordingly
following similar analysis with the Theorem 4.2. The last part immediately
follows from _(i)-(iii)_. ∎

##  6 Discussion and some Future Directions

This paper explores the semi-classical limit of Liouville field theory on
Riemann orbisurfaces with finite type and $g>1$. This is accomplished through
the use of Schottky global coordinates. This study can be seen as an extension
of a previous work by Park, Takhtajan, and Teo park2015potentials  in which
they examined the classical Liouville action on the Schottky space of compact
Riemann surfaces with only punctures. In this paper, we include the
contributions of orbifold points to the Liouville action.

In Section 4.2, we noted that the Liouville action is independent from a
specific choice of fundamental domain. Corollary 4.4.1 already demonstrates
that $\mathscr{S}_{\boldsymbol{m}}=S_{\boldsymbol{m}}-\pi\log\mathsf{H}$ is
not reliant on the choice of a fundamental domain for the Schottky group
$\Sigma$; however, this can also be verified using holography. The holographic
principle for compact Riemann surfaces $X$ posits that the classical Liouville
action on such surfaces is equal to the renormalized volume of a hyperbolic
3-manifold $M$, where that surface is its conformal boundary. witten1998anti ;
Krasnov:2000zq ; Takhtajan:2002cc , extended this principle to punctured (for
both Schottky and quasi-Fuchsian cases), and Teo_2018 ;
chandra2022semiclassical  to orbifold Riemann surfaces (for only the quasi-
Fuchsian case). However, this principle is not yet proven for 3-dimensional
handlebody orbifolds with Riemann orbisurfaces as their conformal boundary.
Thus, in an upcoming paper ANT23 , we aim to demonstrate that the Liouville
action $\mathscr{S}_{\boldsymbol{m}}$ is linked to the renormalized volume of
the corresponding hyperbolic 3-dimensional manifold $M$ up to some constants
that do not rely on moduli parameters. This would also imply that the
Liouville action is independent of the fundamental domain chosen.

Although the presented proofs are only applicable for orbifold Riemann
surfaces with conical points of angle $2\pi/m_{i}$, we believe that most of
our findings can be extended to (hyperbolic) conical Riemann surfaces with
genus $g>1$. Specifically, for weighted punctured Riemann surfaces, the
modified classical Liouville action (with Schottky global coordinates) is
expected to be accurately given by $\mathscr{S}_{\boldsymbol{\alpha}}$ with
$\boldsymbol{\alpha}=(\alpha_{1},...,\alpha_{n})$ and the cone angles
$2\pi(1-\alpha_{i})$ where each factors of $(1-\tfrac{1}{m_{i}})$ in
$\mathscr{S}_{\boldsymbol{m}}$ is replaced by $\alpha_{i}$. Let’s talk about
it in more detail here.

The weighted punctured Riemann surface $(X,\mathscr{D})$ is a compact Riemann
surface $X$ together with an $\mathbb{R}$-divisor
$\mathscr{D}=\sum_{i=1}^{n}\alpha_{i}\,x_{i}$ such that the weights
$0<\alpha_{i}\leq 1$ are associated to each puncture (or marked point)
$x_{i}$. Hyperbolic metrics on weighted punctured Riemann surfaces have, by
definition, conical singularities at the punctures - for this reason, the pair
$(X,\mathscr{D})$ is also called a Riemann cone-surface (or conical Riemann
surface). The existence and uniqueness of a conical hyperbolic metric with
prescribed singularities at a finite number of points on a Riemann surface is
a classical problem that is closely related (and in special cases is,
equivalent) to the famous Uniformization Problem of Klein and Poincaré. Let’s
remember that such metrics have been considered beginning with the work of
Picard picard1893equation . Starting from the classical results by Kazdan-
Warner kazdan1972curvature ; kazdan1974curvature ; kazdan1975direct ,
existence and uniqueness of conical hyperbolic metrics in every conformal
class on a $(X,\mathscr{D})$ were proved by McOwen McOwen-1988  and Troyanov
Troyanov1991PrescribingCO . The necessary and sufficient condition for the
existence of a hyperbolic, conical metric according to McOwen-1988 ;
Troyanov1991PrescribingCO  is that the statement of the Gauss-Bonnet theorem
holds - in other words, the degree of the log-canonical divisor
$\mathcal{K}_{X}+\mathscr{D}$ should be positive, where $\mathcal{K}_{X}$
denotes the canonical divisor of $X$. The positivity of the log-canonical
divisor implies that $\sum_{i=1}^{n}\alpha_{i}>2$. In this case, the unique
hyperbolic metric $ds^{2}$ on $X$ is said to be compatible with the divisor
$\mathscr{D}$ if $ds^{2}$ is a hermitian metric of class
$\mathcal{C}^{\infty}$ on the punctured Riemann surface
$X_{\text{reg}}:X\backslash\text{supp}(\mathscr{D})$ such that if $u_{i}$ is a
holomorphic local coordinate in a neighbourhood $U_{i}$ of $x_{i}$, then there
exists a real-analytic function $\varphi_{i}(u_{i},\bar{u}_{i})$, which is
smooth on $U_{i}\backslash\\{x_{i}\\}$, and such that in $U_{i}$ the metric
$ds^{2}$ is of the form

$ds^{2}=\frac{e^{\varphi_{i}}|du_{i}|^{2}}{|u_{i}-x_{i}|^{2\alpha_{i}}},$

for $0<\alpha_{i}<1$ and

$ds^{2}=\frac{e^{\varphi_{i}}|du_{i}|^{2}}{-2|u_{i}-x_{i}|^{2}\log^{2}|u_{i}-x_{i}|},$

for $\alpha_{i}=1$. Moreover, Area$(X,ds^{2})=\pi\hskip
1.42262pt\text{deg}(\mathcal{K}_{X}+\mathscr{D})=-\pi\chi(X,\mathscr{D})$,
where $\chi(X,\mathscr{D})=\chi(X)-\text{deg}(\mathscr{D})$ is by definition
the Euler-Poincaré characteristic of the Riemann cone-surface
$(X,\mathscr{D})$. The dependence of these hyperbolic cone metrics on the
vector of weights $\boldsymbol{\alpha}=(\alpha_{1},...,\alpha_{n})$ is
characterized by (Schumacher2008WeilPeterssonGF, , Proposition 2.2). The
(unique) conical metrics of constant negative curvature (with fixed weights)
induce new Kähler structures on the Teichmüller spaces of punctured Riemann
surfaces that depend on the cone angles: In Takhtajan:2001uj , Takhtajan and
Zograf introduced a generalized Weil-Petersson metric, parameterized by the
vector of weights $\boldsymbol{\alpha}=(\alpha_{1},...,\alpha_{n})$, on the
moduli spaces of n-pointed rational curves in the context of Liouville
actions. Later, a variation of hyperbolic conical metrics in holomorphic
families were studied by Schumacher and Trapani in Schumacher2005VariationOC ;
Schumacher2008WeilPeterssonGF  (see also mazzeo2017teichmuller ; do2009weil ;
anagnostou2022volumes ). They showed that it is still possible to introduce
"harmonic Beltrami differentials" with respect to a hyperbolic conical metric,
together with a Kodaira-Spencer map needed for the notion of a Weil-Petersson
metric. To be more precise, the generalized Kodaira-Spencer map derived in
Schumacher2005VariationOC ; Schumacher2008WeilPeterssonGF  identifies the
tangent and cotangent spaces to Teichmüller space
$\mathcal{T}_{g,n}(\boldsymbol{\alpha})$585858The
$\mathcal{T}_{g,n}(\boldsymbol{\alpha})$ is complex-analytically isomorphic to
the Teichmüller space of n-pointed stable curves $\mathcal{T}_{g,n}$ (see e.g.
Schumacher2005VariationOC ; Schumacher2008WeilPeterssonGF ). of Riemann cone-
surfaces $(X,\mathscr{D})$ with appropriately defined "harmonic Beltrami
differentials" and "holomorphic quadratic differentials" on the Riemann cone-
surface. It turned out that for $0<\alpha_{i}<1$ the generalized Weil-
Petersson metric depends in a smooth monotone way on the weights $\alpha_{i}$.
In particular, for $\alpha_{i}\rightarrow 0$ one recovers the Weil-Petersson
metric for non-punctured surfaces while for $\alpha_{i}\rightarrow 1$ one gets
the Weil-Petersson metric for Riemann surfaces with cusps. Note that in the
case of general conical singularities, the monodromy group $\Gamma$ of the
Fuchsian differential equation is no longer discrete in P$SL(2,R)$; hence,
$\mathcal{T}_{g,n}(\boldsymbol{\alpha})\neq\mathcal{T}(\Gamma)$ (see
Takhtajan:2001uj ; takhtajan2014real  and (nag1988complex, , §2.2)). Moreover,
one can consider the variation of hyperbolic conical metrics in holomorphic
families, studied in Schumacher2005VariationOC ; Schumacher2008WeilPeterssonGF
, and introduce a generalized Takhtajan-Zograf (TZ) metric. In particular, one
should study the behavior of the integral kernel of the Resolvant
$(\Delta_{0}+1/2)^{-1}$ (see e.g. (garbin2019spectral, , §8)). Although a
conical singularity is simple, its presence has a profound impact on the
Laplace operator. Unlike surfaces with smooth boundaries, the Laplace operator
no longer has a canonical self-adjoint extension. Instead, it has many self-
adjoint extensions and these need not be equivalent loya2007zeta . Probably
the most notable self-adjoint extension of the Laplace-Beltrami operator on a
Riemann surface with conical singularities is given by the so-called
Friedrichs extension (see e.g., kalvin2022determinant ; kalvin2023determinants
). Using the same method as in Schumacher2005VariationOC ;
Schumacher2008WeilPeterssonGF , one can show that this generalized TZ metric
depends in a smooth monotone way on the weights $0<\alpha_{i}<1$ and that, for
$\alpha_{i}=1-1/m_{i}$ with integers $2\leq m_{i}<\infty$, it corresponds to
the elliptic TZ metric introduced in ZT_2018 . One can also construct a Kähler
potential ($\sim\log\mathsf{H}_{\boldsymbol{\alpha}}$) for these generalized
TZ metrics in terms of the solutions of Fuchsian differential equations. To be
more precise, Kähler potentials for generalized TZ metric should be defined in
terms of the coefficients of the expansion (Takhtajan:2001uj, , Eq.(9))
(alternatively, in terms of the expansion of $\varphi$ as in
(kalvin2023determinants, , Eq. (1.2) and (1.5))). This generalizes the results
of park2015potentials  to the case of Riemann cone-surfaces. Lastly,
regularized Liouville action $S_{\boldsymbol{\alpha}}$ should be defined
similar to Takhtajan:2001uj  and the combination
$\mathscr{S}_{\boldsymbol{\boldsymbol{\alpha}}}=S_{\boldsymbol{\alpha}}-\pi\log\mathsf{H}_{\boldsymbol{\alpha}}$
should define a function on $\mathfrak{S}_{g,n}(\boldsymbol{\alpha})$ (and
$\mathfrak{M}_{g,n}(\boldsymbol{\boldsymbol{\alpha}})$) (see also
(kalvin2023determinants, , Theorem 1.1)). Moreover, the different asymptotic
behavior should be derived using the Fuchsian differential equation in analogy
with (Takhtajan:2001uj, , Lemma 4). This asymptotics can be used to derive the
first and second variations of
$\mathscr{S}_{\boldsymbol{\boldsymbol{\alpha}}}=S_{\boldsymbol{\alpha}}-\pi\log\mathsf{H}_{\boldsymbol{\alpha}}$.
It is worth to remind that, when $g>1$, we define $S_{\boldsymbol{\alpha}}$ on
the regularized Schottky fundamental domain; in this case, the fibration
$\jmath:\mathfrak{S}_{g,n}(\boldsymbol{\alpha})\to\mathfrak{S}_{g}$ allows us
to write $T_{[0]}^{\ast}$ as
$\jmath^{\ast}(T_{[0]}^{\ast}\mathfrak{S}_{g})\oplus
T_{[0]}^{\ast}\mathscr{F}_{n}\left(\mathcal{D}\right)$. This decomposition
makes it possible to use the methods of 1988SbMat..60..297Z  on
$\jmath^{\ast}(T_{[0]}^{\ast}\mathfrak{S}_{g})$ while variations w.r.t. cone
points should be carried out using a method similar to Takhtajan:2001uj .
Accordingly, most of our findings can be extended to conical Riemann surfaces
with genus $g>1$.

Finally, we mention some interpretations related to our results and also
future directions:

  * •

For the marked Riemann surface with $g>1$, one can define a normalized basis
$v_{1},\dots,v_{g}$ of the space of holomorphic 1-forms - abelian
differentials of the first kind, and the period matrix
$\boldsymbol{\tau}$,595959The $\boldsymbol{\tau}$ is called period matrix.

$\int_{\alpha_{k}}v_{k^{\prime}}=\delta_{kk^{\prime}},\hskip
28.45274pt\boldsymbol{\tau}_{kk^{\prime}}=\int_{\beta_{k}}v_{k^{\prime}},$

with

$\imaginary\boldsymbol{\tau}_{kk^{\prime}}=\langle
v_{k},v_{k^{\prime}}\rangle=\frac{i}{2}\int_{\mathcal{D}}v_{k}\hskip
1.42262pt\bar{v}_{k^{\prime}}\hskip 2.84526pte^{\phi(w)}dwd\bar{w}.$

By defining also $\det^{\prime}\Box_{0}$ (zeta-function regularized
determinant of the Laplace operator in the hyperbolic metric
$\exp(\varphi)|dw|^{2}$ acting on functions) and
$S_{g}=S_{g,\boldsymbol{m}=0}$ as function on the Schottky space
$\mathfrak{S}_{g}$, Zograf Zograf_1990  shown that there exists a holomorphic
function $\mathfrak{F}_{g}:\mathfrak{S}_{g}\rightarrow\mathbb{C}$, such that

$\frac{\det^{\prime}\Box_{0}}{\det\imaginary\boldsymbol{\tau}}=c_{g}\,e^{-\frac{S_{g}}{12\pi}}\,|\mathfrak{F}_{g}|^{2},$
(219)

with $c_{g}$ as a constant that depends just on the genus $g$ and

$\mathfrak{F_{g}}=\prod_{\\{\gamma\\}}\prod_{k=0}^{\infty}\left(1-\hskip
2.84526ptq_{\gamma}^{1+k}\right),$ (220)

where $\\{\gamma\\}$ runs over all the distinct primitive conjugacy classes in
$\Gamma$ (i.e., $\gamma\in\Gamma$, that cannot be written as the power of any
other element of $\Gamma$)606060Or equivalently, the set of simple closed
geodesics on the surface, and $\log\\{\gamma\\}$ is the corresponding geodesic
length., excluding the identity, and $q_{\gamma}$ is the multiplier of
$\gamma\in\Gamma$ ( $q_{\gamma}+1/q_{\gamma}=|\text{Tr}\gamma|$). Now by
comparing our result in part $(iv)$ of Theorem 5.4,

$\bar{\partial}\partial\mathscr{S}_{\boldsymbol{m}}=-2\sqrt{-1}\left(\omega_{\text{WP}}-\frac{4\pi^{2}}{3}\omega^{\text{\text{cusp}}}_{\text{TZ}}-\frac{\pi}{2}\sum_{j=1}^{n_{e}}m_{j}h_{j}\hskip
1.42262pt\omega^{\text{\text{ell}}}_{\text{TZ},j}\right),$

with ZT_2018 616161In this case, one needs extend the conditions
$\int_{\alpha_{k}}v_{k^{\prime}}=\delta_{kk^{\prime}}$ to
$\int_{\\{\alpha_{k},\kappa_{k},\tau_{k}\\}}v_{k^{\prime}}=\delta_{kk^{\prime}}$.

$\bar{\partial}\partial\hskip
2.84526pt\log\frac{\det\imaginary\boldsymbol{\tau}}{\det^{\prime}\Box_{0}}=-\frac{\sqrt{-1}}{6\pi}\left(\omega_{\text{WP}}-\frac{4\pi^{2}}{3}\omega^{\text{\text{cusp}}}_{\text{TZ}}-\frac{\pi}{2}\sum_{j=1}^{n_{e}}m_{j}h_{j}\hskip
1.42262pt\omega^{\text{\text{ell}}}_{\text{TZ},j}\right),$

we can generalize the Zograf’s formula (219) using a function
$\mathfrak{F}_{g,n}(\boldsymbol{m})$ on $\mathfrak{S}_{g,n}(\boldsymbol{m})$,
i.e.
$\mathfrak{F}_{g,n}(\boldsymbol{m}):\mathfrak{S}_{g,n}(\boldsymbol{m})\rightarrow\mathbb{C}$,
such that

$\frac{\det^{\prime}\Box_{0}}{\det\imaginary\boldsymbol{\tau}}=c_{g,n}(\boldsymbol{m})\hskip
2.84526pte^{-\frac{\mathscr{S}_{\boldsymbol{m}}}{12\pi}}\hskip
2.84526pt\big{|}\mathfrak{F}_{g,n}(\boldsymbol{m})\big{|}^{2},$ (221)

where $\mathscr{S}_{\boldsymbol{m}}=S_{\boldsymbol{m}}-\pi\log\mathsf{H}$ is
the classical generalized Liouville action and $c_{g,n}(\boldsymbol{m})$ is a
constant depending only on $g,n$ and $\boldsymbol{m}$. It would be interesting
to find the exact form of the function $\mathfrak{F}_{g,n}(\boldsymbol{m})$
which its importance, particularly from the perspective of physics, will be
clear in the following items.626262The exact form of this holomorphic anomaly
formula for the determinant of Laplacian has been found for the case of a
sphere with just three conical singularities, see kalvin2023determinants .

  * •

Zograf’s formula has an interesting geometric description in the context of
the Quillen metric and local index theorem. The Quillen metric on
$\lambda_{1}$ (the determinant line bundle associated with the Cauchy-Riemann
operator $\bar{\partial}_{1}$, usually called the _Hodge line bundle_) is
defined by

$\left(||v||_{1}^{\text{Quill}}\right)^{2}=\frac{\det\imaginary\boldsymbol{\tau}}{\det^{\prime}\Box_{0}},$

and accordingly, the first Chern form of the Hermition line bundle
$(\lambda_{1},||v||_{1}^{\text{Quill}})$ over $\mathfrak{S}_{g}$ is given by

$c_{1}(\lambda_{1},||v||_{1}^{\text{Quill}})=\frac{1}{12\pi^{2}}\omega_{\text{WP}}.$

This observation provides the existence of an isometry between the line bundle
over $\mathfrak{M}_{g}$ determined by carrying the Hermitian metric
$\exp[S_{g}/12\pi]$ and the line bundle $\lambda_{1}$ with the Quillen metric.
More generally, Takhtajan and Zograf have studied _local index theorem_ for
orbifold Riemann surfaces in ZT_2018 . The main result of that paper (see
(ZT_2018, , Theorem 2)) is the following formula on the moduli space
$\mathfrak{M}_{g,n}(\boldsymbol{m})$ of punctured orbifold Riemann surfaces
$O=[\mathbb{H}/\Gamma]$:

$\mathsf{c}_{1}\left(\lambda_{k},||\cdot||_{k}^{Quill}\right)=\frac{6k^{2}-6k+1}{12\pi^{2}}\omega_{WP}-\frac{1}{9}\omega_{cusp}-\frac{1}{4\pi}\sum_{j=1}^{n_{e}}m_{j}\left[B_{2}\left(\left\\{\frac{k-1}{m_{j}}\right\\}\right)-\frac{1}{6m_{j}^{2}}\right]\omega^{ell}_{j},$
(222)

for $k\geq 1$. Here $\lambda_{k}$ is the determinant line bundle associated
with the Cauchy-Riemann operator $\bar{\partial}_{k}$ and is a holomorphic
$\operatorname{MCG}(\Gamma)$-invariant line bundle on $\mathcal{T}(\Gamma)$
whose fibers are given by the determinant lines
$\bigwedge^{\max}\ker\bar{\partial}_{k}\otimes\left(\bigwedge^{\max}\operatorname{coker}\bar{\partial}_{k}\right)^{-1}$
while $||\cdot||_{k}^{\text{Quill}}=||\cdot||_{k}/\sqrt{\det\Box_{k}}$ denotes
the _Quillen norm_ in $\lambda_{k}$;636363Hermition line bundles
$\left(\lambda_{k},||\cdot||_{k}^{\text{Quill}}\right)$ and
$\left(\lambda_{1-k},||\cdot||_{1-k}^{\text{Quill}}\right)$ are isometrically
isomorphic. since the determinant line bundle $\lambda_{k}$ is
$\operatorname{MCG}$-invariant, one can alternatively think of $\lambda_{k}$
as a holomorphic $\mathbb{Q}$-line bundle on the moduli space
$\mathfrak{M}(\Gamma)=\mathcal{T}(\Gamma)/\operatorname{MCG}(\Gamma)$.
Finally, $B_{2}(x)=x^{2}-x+1/6$ is the second Bernoulli polynomial, and
$\\{x\\}$ denotes the fractional part of $x\in\mathbb{Q}$. It is clear from
the Eq.(222) that for $k=1$, the first chern form of the determinant line
bundle $\lambda_{1}$ with Quillen norm is given by
$(\omega_{\text{WP}}-\frac{4\pi^{2}}{3}\omega^{\text{\text{cusp}}}_{\text{TZ}}-\frac{\pi}{2}\sum_{j=1}^{n_{e}}m_{j}h_{j}\omega^{\text{\text{ell}}}_{\text{TZ},j})/12\pi^{2}$.
Comparing this observation with our result in part $(iv)$ of Theorem 5.4 and
noting to the Theorem 3.1 in Zograf_1990 , suggests that the following remark
is correct:

######  Remark 6.1.

The function $\mathscr{S}_{\boldsymbol{m}}$ on the Schottky space
$\mathfrak{S}_{g,n}(\boldsymbol{m})$ determines a holomorphic
$\mathbb{Q}$-line bundle $\lambda_{{}_{Sch}}$ on the moduli space
$\mathfrak{M}_{g,n}(\boldsymbol{m})$ with Hermitian metric
$\langle\cdot,\cdot\rangle_{{}_{Sch}}$, where $\langle
1,1\rangle_{{}_{Sch}}=\exp(\mathscr{S}_{\boldsymbol{m}}/12\pi)$ (here $1$ is
understood as the corresponding section of the trivial bundle
$\mathfrak{S}_{g,n}(\boldsymbol{m})\times\mathbb{C}\to\mathfrak{S}_{g,n}(\boldsymbol{m})$).
The Hermitian $\mathbb{Q}$-line bundle
$(\lambda_{{}_{Sch}};\langle\cdot,\cdot\rangle_{{}_{Sch}})$ is isometrically
isomorphic to the Hodge line bundle
$(\lambda_{{}_{Hod}};\langle\cdot,\cdot\rangle_{{}_{Quil}})$ over
$\mathfrak{M}_{g,n}(\boldsymbol{m})$ – i.e. there exists an isomorphism
$\imath:\lambda_{{}_{Sch}}\to\lambda_{{}_{Hod}}$ such that $\langle
s,s\rangle_{{}_{Sch}}=\langle\imath\circ s,\imath\circ s\rangle_{{}_{Quil}}$
for every local section $s$ of the bundle $\lambda_{{}_{Sch}}$.

We do not try to prove the above claim rigorously, but it can be proved by
repeating the computation in (Zograf_1990, , Theorem 3.1.), but with a
modified 1-cocycle $\\{f_{\gamma}\\}$ of modular group
$\gamma\in\operatorname{MCG}(\Gamma)$, i.e.

$f_{\gamma}(t)=\exp(c\int_{t_{\ast}}^{t}\partial(\tilde{\mathscr{S}}_{\boldsymbol{m}}-\tilde{\mathscr{S}}_{\boldsymbol{m}}\circ\gamma)+\frac{c}{2}\left(\tilde{\mathscr{S}}_{\boldsymbol{m}}(t_{\ast})-\tilde{\mathscr{S}}_{\boldsymbol{m}}(\gamma
t_{\ast})+2\pi\sqrt{-1}c_{\gamma}\right)),$

with
$\tilde{\mathscr{S}}_{\boldsymbol{m}}=\mathscr{S}_{\boldsymbol{m}}\circ\Psi$
where
$\Psi:\mathcal{T}_{g,n}(\boldsymbol{m})\to\mathfrak{S}_{g,n}(\boldsymbol{m})$
is a natural holomorphic cover and $t_{\ast}$, $t$ are fixed base point and
any other point in $\mathcal{T}_{g,n}(\boldsymbol{m})$ and $\partial$ denotes
the (1,0)-component of the exterior differentiation operator on
$\mathcal{T}_{g,n}(\boldsymbol{m})$. The constants $c_{\gamma}$ and $c$ can
also be fixed by noting the 1-cocycle nature of
$\\{f_{\gamma}\\}_{\gamma\in\operatorname{MCG}(\Gamma)}$ for the group
$\operatorname{MCG}{(\Gamma)}$ and comparing the curvature form of
$\mathbb{Q}$-line bundle $\lambda_{\emph{Sch}}$ with the curvature form of
Hodge line bundle $\lambda_{\emph{Hod}}$.

It is worth mentioning that the Determinant Line Bundles and Quillen Metrics
in the conical case have been constructed in (Schumacher2008WeilPeterssonGF, ,
§6). The curvature tensor of the Weil-Petersson metric for Teichmüller spaces
of compact (or punctured) Riemann surfaces was computed explicitly by Tromba
tromba1986natural  and Wolpert wolpert1986chern . In
(Schumacher2008WeilPeterssonGF, , §7), the authors show the analogous result
for the weighted punctured case.

  * •

The function $\mathfrak{F}_{g}$ has an important rule to find the physical
wave-function of 3-dimensional pure AdS quantum gravity and define the last
ingredient of canonical quantization, i.e., the scalar product between the
physical wave-functions: the physical norm should be invariant under the
mapping class group transformations. As we will see, it is achievable via the
Quillen norm as follows. Let us recall that according to Eq.(219),

$\frac{\det^{\prime}\Box_{0}}{\det\imaginary\tau}=c_{g}\,e^{-\frac{S_{g}}{12\pi}}\,|\mathfrak{F}_{g}|^{2}.$

The determinant of the operator $\Box_{0}$ is invariant under large
diffeomorphisms, and let’s assume that under $\operatorname{MCG}$
transformation, we have

$\det\imaginary\tau\rightarrow\det\imaginary\tau^{\prime}=|\mathfrak{A}|^{2}\hskip
2.84526pt\det\imaginary\tau,\hskip 28.45274ptS_{g}\rightarrow
S^{\prime}_{g}=S_{g}+\mathfrak{B}+\bar{\mathfrak{B}},$ (223)

which imply that the function $\mathfrak{F}_{g}$ should transforms as follows

$\mathfrak{F}_{g}\rightarrow\mathfrak{F}_{g}^{{}^{\prime}}=\frac{e^{\mathfrak{B}}}{\mathfrak{A}}\hskip
2.84526pt\mathfrak{F}_{g}.$

Now if the matrix $U_{i}^{j}$, under large diffeomorphism, relates wave-
functions defined in two coordinate systems, i.e.
$\Psi^{\prime^{i}}=U^{i}_{j}\hskip 1.42262pte^{-\mathfrak{B}\mathsf{D}}\hskip
1.42262pt\Psi^{j}$, then the functions

$\tilde{\Psi}^{i}=\left(\mathfrak{F}_{g}\right)^{\mathsf{D}}\hskip
0.28453pt\Psi^{i},$ (224)

will transform as

$\tilde{\Psi}^{i}\rightarrow\tilde{\Psi}^{\prime
i}=\mathfrak{A}^{-\mathsf{D}}\hskip 1.42262ptU^{i}_{j}\hskip
1.42262pt\tilde{\Psi}^{j}.$ (225)

By noting to the equations (223), (224) and (225), the MCG-invariant scalar
product of physical wave-functions $\tilde{\Psi}$ can be defined based on
Quillen norm,

$\langle\tilde{\Psi}_{1},\tilde{\Psi}_{2}\rangle=\int_{\mathcal{T}(\Sigma)}\left(\frac{\det^{\prime}\Box_{0}}{\det\imaginary\tau}\right)^{-\mathsf{D}}\hskip
2.84526pt\overline{\tilde{\Psi}}_{1}\wedge\ast\tilde{\Psi}_{2}.$

Accordingly, the function $\mathfrak{F}_{g}$ determines the physical wave-
functions of 3-dimensional pure quantum gravity and their MCG-invariant scalar
product.646464 For more details see kim2015canonical  and maloney2015geometric
. See also eberhardt2022off .

In light of the aforementioned observation and taking into account the remarks
in the Introduction regarding the inclusion of massive particles in the path
integral of 3-dimensional gravity (or considering 3-dimensional Seifert
manifolds whose KK-reductions are related to conical Riemann surfaces), the
function $\mathfrak{F}_{g,n}(\boldsymbol{m})$ plays crucial rule in defining
the physical wave-functions (and their scalar product) of 3-dimensional
quantum gravity in presence of those massive particles (or in presence of
Seifert manifold’s contribution). This itself presents an intriguing problem
warranting further exploration. Apart from that, the function
$\mathfrak{F}_{g}$ plays another significant role in the connection between
the physical wave-function of 3-dimensional pure gravity and dual two-
dimensional Virasoro conformal blocks. According to teschner2003liouville ,
the 3-dimensional physical wave-functions, obeying the Gauss law constraints,
are conformal blocks of quantum Liouville theory. Exploring this proposal in
the presence of conical singularities, based on our results, is also an
interesting problem, i.e., finding the relation between the wave-function of
3-dimensional AdS gravity in the presence of particles with special masses and
quantum Liouville theory with some special vertex operators and even more to
determine whether the partition function of gravity can indeed be decomposed
into Virasoro characters.

  * •

Our results can be applicable to two-dimensional theories of gravity such as
deformed Jackiw-Teitelboim gravity in $AdS_{2}$ witten2020matrix ;
maxfield2021path ; alishahiha2021free ; turiaci20212d ; eberhardt20232d ;
artemev2023p , where the gravitational path integral can be written as an
integral over the moduli space of orbifold (conical) Riemann surfaces.

  * •

Our results, via analytic continuation of the classical Matschull process,
have potential application in the study of black hole production with a non-
trivial topology inside the horizon from the collision of massive point
particles with a certain mass Krasnov:2001ui ; Krasnov:2002rn ;
lindgren2016black .656565Also see Chen:2016dfb ; Chen:2016kyz ; Cho:2017fzo ;
Cardy:2017qhl ; Can:2017ycp ; Collier:2019weq ; Belin:2021ibv  and
(Chang:2016ftb, , §4).

  * •

There is a rich mathematical theory- that of the Selberg trace formula and its
generalizations - where the sum over elements $\gamma\in\Gamma$ is used to
compute the spectrum of differential operators on $\mathbb{H}/\Gamma$. The
Selberg zeta function $Z_{\text{Sz}}(s,\Gamma,\mathfrak{U})$ associated to the
Riemann surface is

$Z_{\text{Sz}}(s,\Gamma,\mathfrak{U})=\prod_{\\{\gamma\\}}\prod_{k=0}^{\infty}\left(1-\mathfrak{U}(\gamma)\hskip
2.84526ptq_{\gamma}^{s+k}\right),$ (226)

where $\mathfrak{U}:\Gamma\rightarrow U(1)$ be a unitary character. For $\real
s>1$, the above product admits a meromorphic continuation to the complex
$s$-plane. According to the Selberg trace formula, for the case where $\Gamma$
has just hyperbolic and parabolic elements, it is shown d1986determinants
that,

$\det^{\prime}\hskip 1.42262pt\Box_{0}\simeq Z_{\text{Sz}}^{\prime}(1),$

where $Z_{\text{Sz}}(s)\equiv Z_{\text{Sz}}(s,\Gamma,\mathfrak{U}=1)$. Hence,
Zograf’s formula (219) also gives a factorization of
$Z_{\text{Sz}}^{\prime}(1)$ as a function on $\mathfrak{S}_{g}$. Accordingly,
it would be intriguing to find a function akin to the Selberg zeta function
$Z_{\text{Sz}}(s)$ acting on $\mathfrak{S}_{g,n}(\boldsymbol{m})$, utilizing
the generalized holomorphic factorization theorem (221). That is important, at
least, for finding the contribution of 1-loop effects to the partition
function of 3-dimensional AdS gravity in the presence of massive particles.
Let us remind that the one-loop partition function of pure three-dimensional
gravity is given by

$Z^{1-\text{loop}}_{\text{gravity}}=\frac{\det\Box^{(1)}}{\sqrt{\det\Box^{(2)}\hskip
2.84526pt\det\Box^{(0)}}},$

where $\Box^{(2)}$ is the kinetic operator for linearized graviton (symmetric
traceless) fluctuations around the chosen background and
$\Box^{(1)},\Box^{(0)}$ are related to kinetic operators of vector ghost and
weyl mode, respectively, since to compute the partition function we must also
include the Fadeev-Popov modes arising due to gauge fixing. By computing each
ones concretely, it is shown that giombi2008one ,

$Z^{1-\text{loop}}_{\text{gravity}}=\prod_{\\{\gamma\\}}\prod_{k=0}^{\infty}\frac{1}{\big{|}1-q_{\gamma}^{2+k}\big{|}}.$
(227)

When $M=\mathbb{H}_{3}/\Gamma$ is a solid torus, $\Gamma=\mathbb{Z}$, and
$\\{\gamma\\}$ consists of the generator of $\Gamma$ and its inverse, the
above formula reduces to

$Z^{1-\text{loop}}_{\text{gravity}}(\tau,\bar{\tau})=\prod_{k=2}^{\infty}\frac{1}{|1-q^{k}|^{2}},$

with $q=e^{2\pi i\tau}$.666666 Unlike the $\mathbb{H}_{3}/\mathbb{Z}$, the
higher-loop contributions to the partition function may not vanish for the
general case $M=\mathbb{H}_{3}/\Gamma$. At least for the handlebody
geometries, there exists a proposal to calculate all-loop expressions
yin2007partition . Comparing (227) with (226) implies that

$Z^{1-\text{loop}}_{\text{gravity}}=Z^{-1}_{\text{Sz}}(2).$

Let us also remind that, according to (220),
$\mathfrak{F_{g}}=Z_{\text{Sz}}(1)$. Hence, with knowledge of the function
$\mathfrak{F}_{g,n}(\boldsymbol{m})$, we can investigate the appropriate
function that, in the presence of conical singularities, assumes the role of
the Selberg zeta function $Z_{\text{Sz}}(s)$. Consequently, this exploration
can also provide insights into the contribution of conical singularities to
the 1-loop gravity partition function. For the special case
$M=\mathbb{H}_{3}/(\mathbb{Z}\times\mathbb{Z}_{m})$, it is shown
benjamin2020pure  that in presence of conical singularities, the 1-loop
partition function again is given by $Z_{\text{Sz}}(s)$ but evaluated at
different point, i.e. $Z^{1-\text{loop}}_{\text{gravity}}\sim
Z^{-1}_{\text{Sz}}(1)$.

  * •

In this paper, we explored further the relation between the modified classical
Liouville action (i.e., the one which its Euler-Lagrange equation admits the
hyperbolic Riemann surfaces with conical singularities as a solution),
uniformization of orbifold Riemann surfaces and complex geometry of moduli
spaces. One knows that, potentially, the quantized Liouville theory can
describe the quantum corrections to those hyperbolic geometries (outside the
singularities). Accordingly, one approach to understanding two-dimensional
quantum gravity through the quantum Liouville theorem is demanding that
quantum Liouville theory take advantage of conformal symmetry akin to its
classical counterpart. If the conformal symmetry is also symmetry of quantum
Liouville theory, it should show itself in the conformal Ward identities
(CWIs) for correlation functions of components of stress-energy tensor with
another operators. An important point to consider is the space on which the
Liouville functional is expressed. This choice significantly impacts the form
of the conformal Ward identities (CWIs) and their implications.

Let’s consider the simplest case, i.e., the Liouville action on the moduli
space $\mathcal{M}_{0,n}$ and identifying $X_{\boldsymbol{m}}$ with
$V_{m_{1}}(x_{1})\dotsm V_{m_{n}}(x_{n})$ and restoring the $\hbar$. Since the
$S_{\boldsymbol{m}}$ in (140) is a well-defined (single-valued) function on
$\mathcal{M}_{0,n}$, the quantum Liouville theorem is defined by

$\expectationvalue{X_{\boldsymbol{m}}}=\int\limits_{\mathscr{CM}_{\boldsymbol{m}}(X)}\mathcal{D}\psi\,\,e^{-\frac{1}{2\pi\hbar}S_{\boldsymbol{m}}[\psi]},$
(228)

where the partition function $\expectationvalue{X_{\boldsymbol{m}}}$ is a
real-valued function on the $\mathcal{M}_{0,n}$ and moreover we have

$\expectationvalue{\boldsymbol{T}(w)X_{\boldsymbol{m}}}=\int\limits_{\mathscr{CM}_{\boldsymbol{m}}(X)}\mathcal{D}\psi\,\,\boldsymbol{T}(\psi)(w)\hskip
2.84526pte^{-\frac{1}{2\pi\hbar}S_{\boldsymbol{m}}[\psi]}.$ (229)

To have conformal symmetry, one is required to prove that

$\frac{1}{\hbar}\expectationvalue{\boldsymbol{T}(w)X_{\boldsymbol{m}}}=\sum_{i=1}^{n}\left(\frac{\boldsymbol{h}_{m_{i}}(\hbar)}{(w-w_{i})^{2}}+\frac{1}{(w-w_{i})}\partial_{w_{i}}\right)\expectationvalue{X_{\boldsymbol{m}}},$
(230)

with $\boldsymbol{h}_{m_{i}}(\hbar)$ are considered as conformal dimensions of
vertex operators $V_{m_{i}}=e^{\frac{\alpha(m_{i})}{\hbar}\psi}$. Note that at
the tree level when $\hbar\rightarrow 0$,

$\begin{split}&\expectationvalue{X_{\boldsymbol{m}}}\sim
e^{-\frac{1}{2\pi\hbar}S_{\boldsymbol{m}}[\varphi]},\\\
&\expectationvalue{\boldsymbol{T}(w)X_{\boldsymbol{m}}}\sim
T_{\varphi}(w)\hskip
2.84526pte^{-\frac{1}{2\pi\hbar}S_{\boldsymbol{m}}[\varphi]},\\\
&\boldsymbol{h}_{m_{i}}(\hbar)\sim\frac{h_{\text{cl}}(m_{i})}{2\hbar}.\end{split}$

By substituting the above relations in (230), one gets Takhtajan:1994vt ;
Takhtajan:1994zi

$\frac{1}{\hbar}\hskip 2.84526ptT_{\varphi}(w)\hskip
2.84526pte^{-\frac{1}{2\pi\hbar}S_{\boldsymbol{\boldsymbol{m}}}[\varphi]}=\sum_{i=1}^{n}\left(\frac{h_{\text{cl}}(m_{i})}{2\hbar(w-w_{i})^{2}}+\frac{1}{(w-w_{i})}\partial_{w_{i}}\right)e^{-\frac{1}{2\pi\hbar}S_{\boldsymbol{\boldsymbol{m}}}[\varphi]},$

which implies that

$T_{\varphi}(w)\hskip
2.84526pt=\sum_{i=1}^{n}\left(\frac{h_{\text{cl}}(m_{i})}{2(w-w_{i})^{2}}-\frac{1}{2\pi}\frac{1}{(w-w_{i})}\partial_{w_{i}}S_{\boldsymbol{\boldsymbol{m}}}[\varphi]\right).$
(231)

Comparing the above result with $T_{\varphi}(w)=\text{Sch}(J^{-1};w)$, where
$\text{Sch}(J^{-1};w)$ is given by (82), implies that

$h_{\text{cl}}(m_{i})=h_{i}=1-\frac{1}{m_{i}^{2}},\hskip
28.45274pt\partial_{w_{i}}S_{\boldsymbol{m}}=-2\pi c_{i},$ (232)

which are in agreement with Theorem.4.1.

Moreover, the conformal symmetry at the quantum level implies that the vertex
operators and components of the energy-momentum tensor satisfy the operator
product expansion (OPE) of Belavin-Polyakon-Zamolodchikov, for example,

$\begin{split}&\frac{1}{\hbar^{2}}\boldsymbol{T}(w)\boldsymbol{T}(w^{\prime})=\frac{\mathsf{c}(\hbar)/2}{(w-w^{\prime})^{4}}+\frac{1}{\hbar}\frac{2\boldsymbol{T}(w^{\prime})}{(w-w^{\prime})^{2}}+\frac{1}{\hbar}\frac{1}{(w-w^{\prime})}\partial_{w^{\prime}}\boldsymbol{T}(w^{\prime})+\text{regular
terms},\\\
&\frac{1}{\hbar^{2}}\boldsymbol{T}(w)\overline{\boldsymbol{T}}(\bar{w}^{\prime})=\text{regular
terms}\end{split}$

which yields the following CWIs,

$\begin{split}&\frac{1}{\hbar^{2}}\expectationvalue{\boldsymbol{T}(w)\boldsymbol{T}(w^{\prime})X_{\boldsymbol{m}}}=\frac{\mathsf{c}(\hbar)/2}{(w-w^{\prime})^{4}}\expectationvalue{X_{\boldsymbol{m}}}+\frac{1}{\hbar}\left(\frac{2}{(w-w^{\prime})^{2}}+\frac{1}{(w-w^{\prime})}\partial_{w^{\prime}}\right)\expectationvalue{\boldsymbol{T}(w^{\prime})X_{\boldsymbol{m}}}\\\
&\hskip
56.9055pt+\frac{1}{\hbar}\sum_{i=1}^{n}\left(\frac{\boldsymbol{h}_{m_{i}}(\hbar)}{(w-w_{i})^{2}}+\frac{1}{(w-w_{i})}\partial_{w_{i}}\right)\expectationvalue{\boldsymbol{T}(w^{\prime})X_{\boldsymbol{m}}},\\\
&\frac{1}{\hbar^{2}}\expectationvalue{\boldsymbol{T}(w)\overline{\boldsymbol{T}}(\bar{w}^{\prime})X_{\boldsymbol{m}}}=\frac{1}{\hbar}\sum_{i=1}^{n}\left(\frac{\boldsymbol{h}_{m_{i}}(\hbar)}{(w-w_{i})^{2}}+\frac{1}{(w-w_{i})}\partial_{w_{i}}\right)\expectationvalue{\overline{\boldsymbol{T}}(\bar{w}^{\prime})X_{\boldsymbol{m}}}.\end{split}$
(233)

Let us show that our results are in agreement also with the second CWI in
(233) at the tree-level. To see that, let us define the normalized connected
two-point function

$\expectationvalue{\boldsymbol{T}(w)\overline{\boldsymbol{T}}(\bar{w}^{\prime})X_{\boldsymbol{m}}}_{\text{nc}}=\frac{\expectationvalue{\boldsymbol{T}(w)\overline{\boldsymbol{T}}(\bar{w}^{\prime})X_{\boldsymbol{m}}}}{\expectationvalue{X_{\boldsymbol{m}}}}-\frac{\expectationvalue{\boldsymbol{T}(w)X_{\boldsymbol{m}}}}{\expectationvalue{X_{\boldsymbol{m}}}}\frac{\expectationvalue{\overline{\boldsymbol{T}}(\bar{w}^{\prime})X_{\boldsymbol{m}}}}{\expectationvalue{X_{\boldsymbol{m}}}}.$
(234)

By using (230) and its similar expression for
$\expectationvalue{\overline{\boldsymbol{T}}(\bar{w}^{\prime})X_{\boldsymbol{m}}}$,

$\frac{1}{\hbar}\expectationvalue{\overline{\boldsymbol{T}}(\bar{w}^{\prime})X_{\boldsymbol{m}}}=\sum_{j=1}^{n}\left(\frac{\bar{\boldsymbol{h}}_{m_{j}}(\hbar)}{(\bar{w}^{\prime}-\bar{w}^{\prime}_{j})^{2}}+\frac{1}{(\bar{w}^{\prime}-\bar{w}^{\prime}_{j})}\partial_{\bar{w}^{\prime}_{j}}\right)\expectationvalue{X_{\boldsymbol{m}}},$

the normalized connected correlation function (234) is simplified to

$\expectationvalue{\boldsymbol{T}(w)\overline{\boldsymbol{T}}(\bar{w}^{\prime})X_{\boldsymbol{m}}}_{\text{nc}}=\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{1}{(w-w_{i})}\frac{1}{(\bar{w}^{\prime}-\bar{w}_{j}^{\prime})}\partial_{w_{i}}\partial_{\bar{w}^{\prime}_{j}}\log\expectationvalue{X_{\boldsymbol{m}}}.$
(235)

At the tree-level, the right-hand side of the above equation has an order of
$\mathcal{O}(\hbar^{-1})$. To find the same order of $\hbar$ on the left-hand
side, one can use the path integral representation of the two-point function,

$\expectationvalue{\boldsymbol{T}(w)\overline{\boldsymbol{T}}(\bar{w}^{\prime})X_{\boldsymbol{m}}}=\int\limits_{\mathscr{CM}_{\boldsymbol{m}}(X)}\mathcal{D}\psi\,\,\boldsymbol{T}(\psi)(w)\overline{\boldsymbol{T}}(\psi)(\bar{w}^{\prime})\hskip
2.84526pte^{-\frac{1}{2\pi\hbar}S_{\boldsymbol{m}}[\psi]}$ (236)

Let us expand the field $\psi$ around its classical value $\varphi$,

$\psi(w)=\varphi(w)+\sqrt{\pi\hbar}\hskip 1.42262pt\delta\psi(w)+...$

which implies that

$\begin{split}&S_{\boldsymbol{m}}[\psi]=S_{\boldsymbol{m}}[\varphi]+\pi\hbar\int_{.}\delta\psi\left(\Box_{0}+\frac{1}{2}\right)\delta\psi\hskip
2.84526pte^{\varphi}d^{2}w+...\\\
&\frac{1}{\hbar}\boldsymbol{T}(w)=\frac{1}{\hbar}T_{\varphi}(w)+\sqrt{\frac{\pi}{\hbar}}\hskip
2.84526pt\mathsf{D}_{w}\delta\psi(w)+...\end{split}$

with
$\mathsf{D}_{w}=\left(\partial_{w}^{2}-(\partial_{w}\varphi)\partial_{w}\right)$.
Now, substituting the above expansion in the right-hand side of (236), the CWI
(235) at the tree-level becomes

$\frac{2\pi}{\hbar}\mathsf{D}_{w}\mathsf{D}_{\bar{w}^{\prime}}\hskip
1.42262ptG(w,w^{\prime})=-\frac{1}{2\pi\hbar}\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{1}{(w-w_{i})}\frac{1}{(\bar{w}^{\prime}-\bar{w}_{j}^{\prime})}\hskip
1.42262pt\partial_{w_{i}}\partial_{\bar{w}^{\prime}_{j}}S_{\boldsymbol{m}}[\varphi],$
(237)

where $G=1/2\hskip 2.84526pt(\Box_{0}+1/2)^{-1}$ is the propagator of quantum
Liouville theorem, $\Box_{0}=-\exp{-\varphi}\hskip
2.84526pt\partial^{2}/\partial_{w}\partial_{\bar{w}}$. Moreover, by using the
overall $SL(2,\mathbb{C})$-symmetry to normalize
$w_{n-2}=0,w_{n-1}=1,w_{n}=\infty$, the right hand side can be written as

$\begin{split}\text{RHS}&=-\frac{1}{2\pi\hbar}\sum_{i=1}^{n-3}\sum_{j=1}^{n-3}R(w,w_{i})R(\bar{w}^{\prime},\bar{w}^{\prime}_{j})\hskip
1.42262pt\partial_{w_{i}}\partial_{\bar{w}^{\prime}_{j}}S_{\boldsymbol{m}}[\varphi],\\\
&=-\frac{\pi}{2\hbar}\sum_{i=1}^{n-3}\sum_{j=1}^{n-3}R_{i}(w)R_{j}(\bar{w}^{\prime})\hskip
1.42262pt\partial_{w_{i}}\partial_{\bar{w}^{\prime}_{j}}S_{\boldsymbol{m}}[\varphi],\end{split}$
(238)

where $R(w,w_{i})$ and $R_{i}$ are defined in (92) and (94), respectively.
Therefore, according to Theorem 5.4, the RHS of (237) is (1,1)-component of
the curvature form for the connection in the holomorphic line bundle over the
moduli. Through rigorous mathematical computations Takhtajan:1994vt ;
Takhtajan:1994zi , it can be demonstrated that the left-hand side of equation
(237) encompasses precisely the identical information. However, a streamlined
approach can be employed to substantiate this claim by using the principles of
Friedan-Shenker modular geometry Friedan:1986ua . Friedan and Shenker
interpret the expectation value $\expectationvalue{X}$ as a Hermitian metric
in a certain holomorphic line bundle over moduli space and the quadratic
differential $\expectationvalue{T(w)X}_{nc}dw^{2}$ as a (1,0)-component of the
canonical metric connection. This interpretation aligns with the tree-level
analysis of the CWI (230) in above. Moreover, they interpret
$\expectationvalue{\boldsymbol{T}(w)\overline{\boldsymbol{T}}(\bar{w}^{\prime})X_{\boldsymbol{m}}}_{\text{nc}}dwd\bar{w}$
as (1,1)-component of the curvature form of that connection which is also in
agreement with the tree level analysis of (235), as illustrated in (238), for
the moduli space $\mathcal{M}_{0,n}$.

Now, One can utilize the aforementioned terminology to find new constraints on
Kähler geometry of moduli space $\mathfrak{M}_{0,n}(\boldsymbol{m})$ by
demanding that the CWIs hold true when they are projected to this space.
Conversly, one can also use the known information about Kähler geometry to
find the "dynamical proof" of the Virasoro symmetry of the Liouville theory.
Let’s find one of those constraints by studying the quantum Liouville theory
and its associated CWIs on the moduli space
$\mathfrak{M}_{0,n}(\boldsymbol{m})$. In comparison to the previous case, two
points should be highlighted. Firstly, to define the path integral
representation of quantum Liouville theory, one needs a Liouville functional
that is a well-defined (single-valued) function on the moduli space
$\mathfrak{M}_{0,n}(\boldsymbol{m})$. It is shown that, unlike the function
$S_{\boldsymbol{m}}$ that is not invariant under the action of
$\operatorname{Symm}\left(\boldsymbol{m}\right)$, at least at the semi-
classical limit, the function $\mathscr{S}_{\boldsymbol{m}}$ has this
property. Accordingly, for the moduli space
$\mathfrak{M}_{0,n}(\boldsymbol{m})$, instead of (228), we have

$\expectationvalue{X_{\boldsymbol{m}}}=\int\limits_{\mathscr{CM}_{\boldsymbol{m}}(X)}\mathcal{D}\psi\,\,e^{-\frac{1}{2\pi\hbar}\mathscr{S}_{\boldsymbol{m}}[\psi]}.$
(239)

Secondly, to check the counterpart of (230) for the moduli space
$\mathfrak{M}_{0,n}(\boldsymbol{m})$, the automorphic form
$\text{Sch}\left(J^{-1};w\right)$ should be projected to the subspace
$T_{[0]}^{\ast}\mathfrak{M}_{0,n}(\boldsymbol{m})$. In order to identify the
appropriate bases for $T_{[0]}^{\ast}\mathfrak{M}_{0,n}(\boldsymbol{m})$, one
can get help from their counterpart bases for
$T_{[0]}^{\ast}\mathcal{M}_{0,n}$. In the semi-classical limit
$\hbar\rightarrow 0$, the projection of $\text{Sch}\left(J^{-1};w\right)$ on
$\mathcal{M}_{0,n}$, according to (101), is given by

$\begin{split}\sum_{i=1}^{n-3}(\text{Sch}\left(J^{-1};w\right),M_{i})dw_{i}&=\sum_{i=1}^{n-3}\bigg{(}-\pi\sum_{j=1}^{n-3}c_{j}R_{j}(w)+\sum_{j=1}^{n}h_{j}\varE_{j}(w),M_{i}\bigg{)}dw_{i}\\\
&=\sum_{i=1}^{n-3}\bigg{(}-\pi
c_{i}+\sum_{j=1}^{n}h_{j}\big{(}\varE_{j}(w),M_{i}\big{)}\bigg{)}dw_{i}\end{split}$
(240)

where $R_{j}(w)$ and $\varE_{j}(w)$ are defined by (94) and (102),
respectively, and none of them are invariant under the action of
$\operatorname{Symm}\left(\boldsymbol{m}\right)$. To find the projection of
$\text{Sch}\left(J^{-1};w\right)$ on the space
$T_{[0]}^{\ast}\mathfrak{M}_{0,n}(\boldsymbol{m})$, one needs the
$\operatorname{Symm}\left(\boldsymbol{m}\right)$-invariant version of
$R_{i}(w)$ and $\varE_{i}(w)$ which are given by

$\begin{split}&\tilde{R}_{i}(w)=-\frac{1}{\pi}\sum_{\gamma_{j,j+1}\in\operatorname{Symm}\left(\boldsymbol{m}\right)}R\left(\gamma_{j,j+1}(w),w_{i}\right)\gamma_{j,j+1}^{\prime}(w)^{2},\\\
&\tilde{\mathscr{E}}_{k}(w)=\frac{1}{2}\sum_{\gamma_{j,j+1}\in\operatorname{Symm}\left(\boldsymbol{m}\right)}\left(\frac{1}{(\gamma_{j,j+1}(w)-w_{k})^{2}}-\frac{1}{\gamma_{j,j+1}(w)(\gamma_{j,j+1}(w)-1)}\right)\gamma_{j,j+1}^{\prime}(w)^{2},\\\
&\tilde{\varE}_{n}(w)=\frac{1}{2}\sum_{\gamma_{j,j+1}\in\operatorname{Symm}\left(\boldsymbol{m}\right)}\frac{1}{\gamma_{j,j+1}(w)(\gamma_{j,j+1}(w)-1)}\hskip
2.84526pt\gamma_{j,j+1}^{\prime}(w)^{2}\end{split}$ (241)

with $k=1,\dots,n-1$ and $\gamma_{j,j+1}(w)$ are defined in Section 3.2.1. In
the above, just for simplicity, we assumed that the signature of $O$ is
$(O;m,...,m)$, but it can be easily extended to the general case. Accordingly,
the projection of $\text{Sch}\left(J^{-1};w\right)$ on moduli space
$\mathfrak{M}_{0,n}(\boldsymbol{m})$ is given by

$\begin{split}\sum_{i=1}^{n-3}\left(\text{Sch}\left(J^{-1};w\right),M_{i}\right)d\tilde{w}_{i}&=\sum_{i=1}^{n-3}\bigg{(}-\pi\sum_{j=1}^{n-3}c_{j}\tilde{R}_{j}(w)+\sum_{j=1}^{n}h_{j}\tilde{\varE}_{j}(w),M_{i}\bigg{)}d\tilde{w}_{i}\\\
&=\sum_{i=1}^{n-3}\bigg{(}-\pi
c_{i}+\big{(}T_{s}(w),M_{i}\big{)}\bigg{)}d\tilde{w}_{i},\end{split}$ (242)

where

$\begin{split}T_{s}(w)&=\frac{1}{2}\sum_{i=1}^{n-1}\sum_{\gamma_{j,j+1}\in\operatorname{Symm}\left(\boldsymbol{m}\right)}\frac{h_{i}}{(\gamma_{j,j+1}(w)-w_{i})^{2}}\gamma_{j,j+1}^{\prime}(w)^{2}\\\
&-\frac{1}{2}\sum_{\gamma_{j,j+1}\in\operatorname{Symm}\left(\boldsymbol{m}\right)}\frac{\sum_{i=1}^{n-1}h_{i}-h_{n}}{\gamma_{j,j+1}(w)(\gamma_{j,j+1}(w)-1)}\gamma_{j,j+1}^{\prime}(w)^{2}.\end{split}$
(243)

In the above, the $d\tilde{w}_{i}$s are $(1,0)$ forms on cotangent space
$T_{[0]}^{\ast}\mathfrak{M}_{0,n}(\boldsymbol{m})$. The counterpart of (230)
for this case is

$T^{\mathfrak{M}_{0,n}(\boldsymbol{m})}_{\varphi}(w)=\sum_{i=1}^{n}\sum_{\gamma_{j,j+1}\in\operatorname{Symm}\left(\boldsymbol{m}\right)}\left(\frac{h_{\text{cl}}(m_{i})}{2(\gamma_{j,j+1}(w)-w_{i})^{2}}-\frac{1}{2\pi}\frac{1}{(\gamma_{j,j+1}(w)-w_{i})}\partial_{\tilde{w}_{i}}\mathscr{S}_{\boldsymbol{m}}[\varphi]\right),$
(244)

which together with the relations (242) and (243) gives information about the
moduli space of Riemann orbisurface $O$, i.e, a relation between accessory
parameters $c_{i}$ in (242) for the Fuchsian uniformization and the first
variation $\partial_{\tilde{w}_{i}}\mathscr{S}_{\boldsymbol{m}}[\varphi]$. It
also provides the opportunity to check the observed relation between the
universal CWIs and Friedan-Shenker modular geometry. It is noteworthy that
using the same methodology, one can probe the Kähler geometry of moduli space
$\mathfrak{M}_{g,n}(\boldsymbol{m})$ base on using the equations
(127),(129),(132) and (241).

Last but not least, it is shown ZT_localindextheorem_1991  that the validity
of CWI (230) at 1-loop approximation for the genus $g=0$ case with just
$n_{p}$ punctures yields a formula for the first derivative of
$Z_{\text{Sz}}(2)$. Moreover, it is shown ZT_localindextheorem_1991  that when
$g\neq 0$ with just $n_{p}$ punctures, the CWI (235) at the 1-loop level is
equivalent to the local index theorem for the families of
$\bar{\partial}$-operators on the punctured Riemann surfaces. Based on the
above-examined cases, it would be interesting to explore more the relation
between CWIs for the correlation functions

$\frac{1}{\hbar^{2}}\expectationvalue{\prod_{k=0}\boldsymbol{T}(w_{k})\prod_{l=0}\overline{\boldsymbol{T}}(\bar{w}_{l}^{\prime})\hskip
2.84526ptX_{\boldsymbol{m}}},$

and uniformization of the Riemann surfaces and complex geometry of Teichmüller
space $\mathcal{T}_{g,n}(\boldsymbol{m})$, Schottky space
$\mathfrak{S}_{g,n}(\boldsymbol{m})$ and Moduli spaces
$\mathcal{M}_{g,n},\mathfrak{M}_{g,n}(\boldsymbol{m})$ and more importantly 2
dimensional quantum gravity. Moreover, the multi-point correlation functions
can provide further evidence in favor of the profound role of Ward identities
in Friedan-Shenker modular geometry.

  * •

Also very recently, it is shown Benjamin:2023uib  that asymptotic series (in
powers of the central charge) for expansion of 2-dimensional conformal blocks,
involving the exchange of identity operator, necessitates the existence of
non-perturbative effects via resurgence analysis. In the dual three-
dimensional theory, this implies that the graviton loop expansion is also an
asymptotic series, and to cure it, one needs to consider new saddle points,
which are particle-like states with large negative mass (non-manifold saddles
with conical excesses). In this paper we focus on geometries with conical
defects (orbifold geometries), but studying the relation between our work and
Benjamin:2023uib  would be interesting.

###### Acknowledgements.

The authors thank Lorenz Eberhardt, Ruben Hidalgo, Kirill Krasnov, Alex
Maloney, Leon Takhtajan, Lee-Peng Teo, Tina Torkaman, and Peter Zograf for
inspiring discussions. We are specially grateful to Peter Zograf for providing
us with an English translation of his paper Zograf_1990  and to Lorenz
Eberhardt and specially Leon Takhtajan for helpful comments on a draft version
of this work. The research of B.T. and A.N. is supported by the Iranian
National Science Foundation (INSF) under Grant No. 4001859.

##  Appendix A Introduction to Orbifold Riemann Surfaces

Orbifolds lie at the intersection of many different areas of mathematics and
Physics, including algebraic and differential geometry, as well as conformal
field theory and string theory. Orbifolds were first introduced into topology
and differential geometry by Satake satake1956generalization , who called them
_V-manifolds_. Satake described them as a generalization of smooth manifolds
that are locally modeled on a quotient of $\mathbb{R}^{\mathfrak{n}}$ by the
action of a finite group and generalized concepts such as de Rham cohomology
and the Gauss-Bonnet theorem to orbifolds. Shortly after the original paper of
Satake, Baily introduced complex V-manifolds and generalized both the Hodge
decomposition theorem baily1956decomposition  and Kodaira’s projective
embedding theorem baily1957imbedding  to V-manifolds. The concept of
V-manifolds was later re-invented by Thurston thurston80  under the name of
“orbifolds” and the notion of fundamental groups was generalized for these
objects. Even though orbifolds were already very important objects in
mathematics, the work of Dixon, Harvey, Vafa, and Witten Dixon:1985jw ;
Dixon:1986jc  as well as the subsequent work of Dixon, Friedan, Martinec, and
Shenker Dixon:1986qv  lead to a dramatic increase of interest in orbifolds
among Physicists. The main objective of this appendix is to compile some basic
facts about orbifolds and fix some notations used throughout this paper.
Although we start with a general setting, the main focus of this appendix is
on complex 1-dimensional orbifolds, called orbifold Riemann surfaces, as they
are the objects of study in the main body of this paper.

To motivate our interest in orbifold Riemann surfaces, let us recall that
ordinary Riemann surfaces are complex 1-dimensional algebro-geometric objects
with a lot of good properties: Geometrical facts about Riemann surfaces are as
“nice” as possible, and they often provide the intuition and motivation for
generalizations to more complicated manifolds or varieties. The name “surface”
comes from the fact that every Riemann surface is a two-dimensional real
analytic manifold (i.e., a surface), but it contains more structures: In fact,
a Riemann surface is the simplest example of a Kähler manifold which means
that it admits three mutually compatible structures — a complex structure, a
Riemannian structure, and a symplectic structure. In addition, the existence
of non-constant meromorphic functions on these surfaces can be used to show
that any compact Riemann surface is a _projective algebraic curve_ and,
therefore can be given by polynomial equations inside a projective space.
_Orbifold Riemann surfaces_ are the natural generalization of Riemann surfaces
in the orbifold world. Just like their manifold counterparts, orbifold Riemann
surfaces can be viewed both as a complex orbifold of dimension 1 (_complex
analysts viewpoint_) or as a smooth, proper Deligne–Mumford stack676767
_Deligne–Mumford stacks_ (or _DM stacks_) are an algebraic generalization of
orbifolds, and can be roughly thought of as an “orbivariety” or “orbischeme”.
Just as orbifolds are locally the quotient of a manifold by a finite group, a
DM stack can be characterized as being “locally” the quotient of a scheme by a
finite group action. (over $\mathbb{C}$) of dimension 1 (_algebraic geometers
viewpoint_). These orbisurfaces also admit Riemannian metrics and can be
regarded as the simplest examples of Kähler orbifolds. When the emphasis is on
the algebro-geometric viewpoint, orbifold Riemann surfaces are usually called
_orbifold curves_ or _orbicurves_.

The viewpoint that one takes on the singular points of an orbifold depends a
lot on what type of “space” one is working with: When working in the
topological realm, one usually treats the orbifold singularities as an
_additional structure_ – an _orbifold structure_ – on an _underlying
topological space_ in the same way that one thinks of a smooth structure as an
additional structure on a topological manifold (see thurston80 ;
steer1999grothendieck ; Scott1983TheGO ; Cooper_2000 ;
caramello2019introduction ). In particular, a topological space is allowed to
have several different orbifold structures. On the other hand, from an
algebro-geometric viewpoint behrend2006uniformization , it is more convenient
to consider (analytic or algebraic) _stacks_ as the proper notion of space.
Such a stack is then called an orbifold (be it an analytic or an algebraic
one) if it admits a _covering_ by open substacks of the form
$[\tilde{U}/\Gamma]$,686868If $M$ is a complex manifold of dimension
$\mathfrak{n}$ and
$\Gamma\subset\operatorname{Aut}(M)\subset\operatorname{GL}(\mathfrak{n},\mathbb{C})$
is a finite subgroup of holomorphic automorphisms of $M$ which does _not_ act
_freely_ (i.e. has fixed points on $M$), the quotient space $M/\Gamma$ will
have the structure of an analytic stack (or, equivalently, of a complex
orbifold) and we will use the notation $[M/\Gamma]$ to mean $M/\Gamma$ as an
analytic orbifold/stack. The notation $M/\Gamma$ will be reserved for the
_coarse moduli space_ or the _underlying analytic space_ of $[M/\Gamma]$ which
will be a _variety with quotient singularities_. parameterising families of
$\Gamma$-orbits in $\tilde{U}$, where $\tilde{U}$ is the local model for
representable stacks (i.e. manifolds) and $\Gamma$ is a finite subgroup of the
automorphism group of $\tilde{U}$. This second point of view treats an
orbifold singularity as an intrinsic structure of the space.

For us, the appropriate notion of space will be that of _analytic spaces_
Gunning:1969rm , by which we mean a generalization of complex manifolds that
allow the presence of singularities and are locally isomorphic to the common
zero locus of a finite collection of holomorphic functions. Our point of view,
which will be reflected in our introduction to orbifolds, lies somewhere in
between the two extremes mentioned above: We will treat the subclass of
codimension $\geq 2$ orbifold singularities as the intrinsic structure of an
_underlying complex analytic space_ 696969This underlying analytic space is
actually the same as the coarse moduli space of the orbifold regarded as an
analytic stack. while the orbifold singularities of codimension-1 still need
to be treated as additional structures on this analytic space (see Boyer_2008
for more details).

Throughout this paper, we will need to work with different characterizations
of orbifold Riemann surfaces that have appeared in the literature: In order to
introduce orbifold geometric structures in Appendix B, we will need to work
with a definition of complex orbifolds based on _orbifold charts_
kawasaki1979riemann  while a characterization of Riemann orbisurfaces as _log
pairs_ Campana2004  will be more suitable for studying orbifold metrics
McOwen-1988 . A third way of characterizing Riemann orbisurfaces will be as
_Riemann surfaces with signature_ and this viewpoint – which is closely
related to the notion of Riemann orbisurfaces as log pairs – is the one that
we have adapted in the main body of this paper.

Our presentation in this appendix is closer to that of references (Boyer_2008,
, §4), (milnor1990dynamics, , Appx. E), and uludaug2007orbifolds . For more
details about other approaches, the reader is encouraged to consult with
thurston80 ; Scott1983TheGO ; Cooper_2000 ; caramello2019introduction ;
adem2007orbifolds  among others.

###  A.1 Analytic Geometry

We will start our introduction to complex orbifolds with a quick review of
some background information about complex analytic spaces and analytic
mappings — particularly, ramified covering maps of analytic spaces. The reader
is advised to consult with references gunning1970lectures ; Gunning:1969rm ;
Grauert-Remmert ; Riemenschneide_2011 ; namba1991finite ; namba1992finite ;
uludaug2007orbifolds  for more details.

####  A.1.1 Complex Analytic Spaces and Analytic Mappings

Let us start our review of complex analytic spaces with defining a complex
analytic subvariety:

######  Definition A.1 (Analytic subvariety).

Let $U$ be an open subset of $\mathbb{C}^{\mathfrak{n}}$ (or of any complex
analytic manifold $M$) and let $X$ be a subset of $U$. We say that $X$ is an
_analytic subvariety_ in $U$ if, for any point $x$ in $U$, there exist a
neighborhood $V$ of $x$ and a finite number of holomorphic functions
$f_{1},\dots,f_{k}$ on $V$ such that

$X\cap V=\left\\{z\in V\,\Big{|}\,f_{1}(z)=\dotsm=f_{k}(z)=0\right\\}.$

In other words, in some open neighborhood of each point of $U$, an analytic
subvariety $X\subset U$ is the set of common zeros of a finite number of
complex analytic functions. We call $f_{1},\dots,f_{k}$ a system of local
defining functions for $X$ and a non-empty analytic subvariety of $U$ which is
locally defined by a _single_ (not identically zero) holomorphic function will
be called an _analytic hypersurface_ in $U$. Finally, a subvariety $X$ of $U$
is called _irreducible_ if it cannot be written as a union $X=X_{1}\cup X_{2}$
where $X_{i}$ are analytic subvarieties of $U$ properly contained in $X$.

As suggested before, complex analytic subvarieties can be viewed as a
generalization of complex (sub)manifolds which allow for the presence of
singularities:

######  Definition A.2 (Smooth and singular points of subvarieties).

A point $x$ on an analytic subvariety $X$ is said to be _regular_ or _smooth_
if it is possible to choose coordinates $(z_{1},\dots,z_{\mathfrak{n}})$ in an
open neighborhood $V\subset U\subset\mathbb{C}^{\mathfrak{n}}$ of point $x$
such that locally $X$ is a linear subspace $\\{z\in
V\,|\,z_{k+1}=\dotsm=z_{\mathfrak{n}}=0\\}$ — i.e. if $X\cap V$ is a
$k$-dimensional submanifold of $\mathbb{C}^{\mathfrak{n}}$. The set of all
regular points of $X$ is an open dense subset of $X$ and will be denoted by
$\operatorname{Reg}(X)$. The points of a subvariety that are not regular
points are called the _singular points_. The set
$X\backslash\operatorname{Reg}(X)$ of all singular points of $X$ will be
denoted by $\operatorname{Sing}(X)$ and is called the _singular locus_ of $X$.

An analytic subvariety $X$ will be called a _smooth analytic subvariety_ if
$X=\operatorname{Reg}(X)$; evidently, a smooth analytic variety is just a
complex analytic manifold itself. When $X$ is an irreducible analytic
subvariety, the _complex dimension of $X$_ is defined as the dimension of its
smooth part $\operatorname{Reg}(X)$ regarded as a complex manifold. More
generally, if $X$ is reducible, the dimension of $X$ is defined as the maximum
of the dimensions of its irreducible components. A reducible analytic
subvariety $X$ is called _pure dimensional_ if every irreducible component of
$X$ has the same dimension.

It is clear that analytic subvariety $X\subset U$ can be endowed with the
relative topology coming from $U$; however, the main point in the study of
analytic subvarieties is that one should take into account consideration not
only about the topology of these analytic subvarieties but also about their
_function-theoretic properties_ : For simplicity, let us take the open set
$U\subset\mathbb{C}^{\mathfrak{n}}$ to be a sufficiently small open polydisc
$\varDelta(\boldsymbol{\epsilon})$ such that an analytic subvariety $X$ of
$\varDelta(\boldsymbol{\epsilon})$ can be determined as the set of common
zeros of a finite number of functions that are analytic throughout
$\varDelta(\boldsymbol{\epsilon})$. Under the natural addition and
multiplication of complex-valued functions, the set of all holomorphic
functions on $\varDelta(\boldsymbol{\epsilon})$ forms a ring
$\mathfrak{O}_{\varDelta(\boldsymbol{\epsilon})}$ containing the constants
$c\in\mathbb{C}^{\mathfrak{n}}$ — hence in fact a $\mathbb{C}$-algebra. The
set of all analytic functions in $\varDelta(\boldsymbol{\epsilon})$ which
vanish on $X$ form an _ideal_ $\mathfrak{I}(X)$ in the ring
$\mathfrak{O}_{\varDelta(\boldsymbol{\epsilon})}$, called the _ideal of $X$_.
Then, the _ring of holomorphic functions on $X$_ is given by the quotient ring
$\mathfrak{O}_{X}:=\mathfrak{O}_{\varDelta(\boldsymbol{\epsilon})}/\mathfrak{I}(X)$.
It is easy to see that a subvariety $X$ of $\varDelta(\boldsymbol{\epsilon})$
is irreducible precisely when the ideal $\mathfrak{I}(X)$ is a _prime_ ideal
in $\mathfrak{O}_{\varDelta(\boldsymbol{\epsilon})}$, by which we mean
$\mathfrak{I}(X)\neq\mathfrak{O}_{\varDelta(\boldsymbol{\epsilon})}$ and for
any two holomorphic functions
$f,f^{\prime}\in\mathfrak{O}_{\varDelta(\boldsymbol{\epsilon})}$ the statement
$ff^{\prime}\in\mathfrak{I}(X)$ implies $f\in\mathfrak{I}(X)$ or
$f^{\prime}\in\mathfrak{I}(X)$ (or both).

######  Remark A.1.

Let $f_{1},\dots,f_{k}\in\mathfrak{O}_{\varDelta(\boldsymbol{\epsilon})}$ be
the system of defining functions for general (i.e., not necessarily
irreducible) subvariety $X$. Then, the defining functions $f_{1},\dots,f_{k}$
generate an _ideal_ $\mathfrak{I}$ in the
$\mathfrak{O}_{\varDelta(\boldsymbol{\epsilon})}$ which is sometimes called
the _defining ideal_ for $X$. While all holomorphic functions
$f\in\mathfrak{I}$ vanish on $X$, i.e. $\mathfrak{I}\subseteq\mathfrak{I}(X)$,
the inverse statement is _not_ necessarily true — i.e. in general
$\mathfrak{I}(X)\nsubseteq\mathfrak{I}$. However, provided that the polydisc
$\varDelta(\boldsymbol{\epsilon})$ is small enough, an important fact is that
_all_ holomorphic functions $f\in\mathfrak{I}(X)$ have a power which is
contained in $\mathfrak{I}$. This motivates us to define the _radical_ of
$\mathfrak{I}$ as the set

$\sqrt{\mathfrak{I}}:=\left\\{f\in\mathfrak{O}_{\varDelta(\boldsymbol{\epsilon})}\,\Big{|}\,f^{k^{\prime}}\in\mathfrak{I}\text{
for some positive integer }k^{\prime}\right\\}.$

This is again an ideal in $\mathfrak{O}_{\varDelta(\boldsymbol{\epsilon})}$
and we have $\sqrt{\mathfrak{I}}=\mathfrak{I}(X)$. When the defining ideal
$\mathfrak{I}$ is prime in $\mathfrak{O}_{\varDelta(\boldsymbol{\epsilon})}$,
we get $\sqrt{\mathfrak{I}}=\mathfrak{I}$; hence,
$\mathfrak{I}=\mathfrak{I}(X)$ exactly when $X$ is irreducible.

More generally, when the open set $U\subset\mathbb{C}^{\mathfrak{n}}$ is not
restricted to be small (or to be an open polydisc), the above statements hold
true in small neighborhoods of each point $x\in X$. For such local
considerations, it is often convenient to introduce the notion of _germs_ : To
make this precise, consider two pairs $(X,U)$ and $(X^{\prime},U^{\prime})$
where $U,U^{\prime}$ are open neighborhoods of the origin in
$\mathbb{C}^{\mathfrak{n}}$ and $X,X^{\prime}$ are analytic subvarieties of
$U,U^{\prime}$ respectively. The two pairs $(X,U)$ and
$(X^{\prime},U^{\prime})$ are said to define the same _germ of analytic
subvarieties_ at the origin in $\mathbb{C}^{\mathfrak{n}}$ if there exists a
neighborhood $W\subseteq U\cap U^{\prime}$ of 0 such that $X\cap
W=X^{\prime}\cap W$. We will denote the germ of analytic subvariety $X$ at 0
in $\mathbb{C}^{\mathfrak{n}}$ by $[X]_{0}$. Now, let $\mathfrak{O}_{U}$ be
the ring of holomorphic functions in some open subset
$U\subset\mathbb{C}^{\mathfrak{n}}$ containing the origin. Analogously, we can
define an equivalence relation $\sim_{0}$ between two holomorphic functions
$f,f^{\prime}\in\mathfrak{O}_{U}$ where $f\sim_{0}f^{\prime}$ if there exists
a neighborhood $W$ of 0 such that the restrictions of $f$ and $f^{\prime}$ to
$W$ are identical — i.e. $f\big{|}_{{}_{W}}=f^{\prime}\big{|}_{{}_{W}}$. The
equivalence class of a function $f$ is called the _germ of holomorphic
function $f$ at the origin_ and will be denoted by $[f]_{0}$. In addition, the
quotient ring $\mathfrak{O}_{U,0}:=\mathfrak{O}_{U}/\sim_{0}$ will be the
_ring of germs of holomorphic functions at the origin_.707070If we denote by
$\mathbb{C}\\{z_{1},\dots,z_{\mathfrak{n}}\\}$ the set of power series which
converge absolutely in some neighborhood of 0, this set also has the structure
of a ring. Since, as in the one variable case, $f\sim_{0}f^{\prime}$ if and
only if $f$ and $f^{\prime}$ have the same power series expansion, we may
identify $\mathfrak{O}_{U,0}$ with
$\mathbb{C}\\{z_{1},\dots,z_{\mathfrak{n}}\\}$.

Similar to the case of analytic subvariety of a sufficiently small open
polydisc, to each germ $[X]_{0}$ of an analytic subvariety at the origin in
$\mathbb{C}^{\mathfrak{n}}$ there is canonically associated an ideal
$\mathfrak{I}([X]_{0})$ in the local ring717171A commutative ring is called
_local_ if it has only one maximal ideal. The unique maximal ideal of
$\mathfrak{O}_{\mathbb{C}^{\mathfrak{n}},0}$ (equivalently,
$\mathbb{C}\\{z_{1},\dots,z_{\mathfrak{n}}\\}$) corresponds to the set of all
germs of holomorphic functions that vanish at the origin (equivalently, the
set of all power series whose constant term vanishes).
$\mathfrak{O}_{\mathbb{C}^{\mathfrak{n}},0}$ which is defined as the ideal of
germs of all analytic functions vanishing on the subvariety $X$ representing
the germ $[X]_{0}$. In the other direction, to each ideal
$\mathfrak{I}\subseteq\mathfrak{O}_{\mathbb{C}^{\mathfrak{n}},0}$ there is
canonically associated a germ of an analytic subvariety at the origin in
$\mathbb{C}^{\mathfrak{n}}$, called the _locus of the ideal $\mathfrak{I}$_
and denoted by $[X(\mathfrak{I})]_{0}$. The germ $[X(\mathfrak{I})]_{0}$ is
defined as the germ represented by the analytic subvariety $X=\\{z\in
U\,|\,f_{1}(z)=\dotsm=f_{k}(z)=0\\}$ of the open set
$U\subset\mathbb{C}^{\mathfrak{n}}$, where $f_{i}\in\mathfrak{O}_{U}$ are
analytic functions in $U$ whose germs in $\mathfrak{O}_{U,0}$ generate the
ideal $\mathfrak{I}$. Note that, similar to what we saw in Remark A.1,
$\mathfrak{I}\big{(}[X(\mathfrak{I})]_{0}\big{)}=\sqrt{\mathfrak{I}}$ and
$\sqrt{\mathfrak{I}}=\mathfrak{I}$ iff the ideal
$\mathfrak{I}\subseteq\mathfrak{O}_{\mathbb{C}^{\mathfrak{n}},0}$ is prime
(equivalently, if the germ $[X(\mathfrak{I})]_{0}$ is irreducible). Finally,
the residue class ring
$\mathfrak{O}_{\mathbb{C}^{\mathfrak{n}},0}/\mathfrak{I}([X]_{0})$ will now be
denoted by $\mathfrak{O}_{X,0}$ and will be called the _ring of germs of
holomorphic functions on the subvariety $X$_ at the origin in
$\mathbb{C}^{\mathfrak{n}}$.

Having learned how to think about the function-theoretic properties of an
analytic subvariety locally, we are now ready to study those properties from a
global perspective; it is then convenient to think about _sheaves_ of
rings/ideals/modules as we shall now explain: Let us start with the local
rings $\mathfrak{O}_{\mathbb{C}^{\mathfrak{n}},z}$ of germs of holomorphic
functions at any point $z\in\mathbb{C}^{\mathfrak{n}}$. The set of rings
$\mathfrak{O}_{\mathbb{C}^{\mathfrak{n}},z}$ for all points
$z\in\mathbb{C}^{\mathfrak{n}}$ can be taken to form the _sheaf of germs of
holomorphic functions of $\mathfrak{n}$ complex variables_
$\mathscr{O}_{\mathbb{C}^{\mathfrak{n}}}$; the restriction
$\mathscr{O}_{\mathbb{C}^{\mathfrak{n}}}\big{|}_{{}_{U}}$ of the sheaf of
rings $\mathscr{O}_{\mathbb{C}^{\mathfrak{n}}}$ to any open set
$U\subset\mathbb{C}^{\mathfrak{n}}$ will be simply denoted by
$\mathscr{O}_{U}$. Similarly, consider an analytic subvariety $X$ of an open
subset $U\subset\mathbb{C}^{\mathfrak{n}}$ and to each point $z\in U$
associate the ideal
$\mathfrak{I}([X]_{z})\subseteq\mathfrak{O}_{\mathbb{C}^{\mathfrak{n}},z}$ of
the germ of the subvariety $X$ at that point (if $z\notin X$ the ideal
$\mathfrak{I}([X]_{z})$ is of course the trivial ideal
$\mathfrak{O}_{\mathbb{C}^{\mathfrak{n}},z}$). The set of all ideals
$\mathfrak{I}([X]_{z})$ at any point $z\in U$ form an analytic subsheaf
727272An _analytic sheaf_ over an open set $U\subset\mathbb{C}^{\mathfrak{n}}$
is a sheaf of modules over $\mathscr{O}_{U}$. of the sheaf $\mathscr{O}_{U}$
over the set $U$, which will be dented by $\mathscr{I}(X)$ and called the
_sheaf of ideals of the analytic subvariety $X$_. Finally, the restriction to
the subvariety $X$ of the analytic sheaf $\mathscr{O}_{U}/\mathscr{I}(X)$ will
be called the _the sheaf of germs of holomorphic functions on the subvariety
$X$_ and is dented by $\mathscr{O}_{X}$; the local rings
$\mathfrak{O}_{X,x}=\mathfrak{O}_{U,x}/\mathfrak{I}([X]_{x})$ at any point
$x\in X$ can then be viewed as the _stalks_ of $\mathscr{O}_{X}$.

Locally, a germ of an analytic subvariety determines a germ of a topological
space and this space further possess a distinguished subring of the ring of
germs of continuous complex-valued functions, namely the ring of germs of
holomorphic functions on the subvariety. This observation suggests that the
correct way of characterizing an analytic subvariety $X$ is as a _$\mathbb{C}$
-ringed space_:

######  Definition A.3.

A _ringed space_ $X$ is a pair $(|X|,\mathscr{O}_{X})$ consisting of a
Hausdorff topological space $|X|$ and a sheaf of rings $\mathscr{O}_{X}$ on
$|X|$, called the _structure sheaf_ of $X$. It is called a _locally ringed
space_ when, for every $x\in|X|$, the stalk $\mathscr{O}_{X,x}$ is a local
ring. Its maximal ideal is denoted by $\mathfrak{m}_{X,x}$. A locally ringed
space is called a _$\mathbb{C}$ -ringed space_ when furthermore
$\mathscr{O}_{X}$ is a sheaf of $\mathbb{C}$-algebras and, for every
$x\in|X|$, there is an isomorphism
$\mathscr{O}_{X,x}/\mathfrak{m}_{X,x}\cong\mathbb{C}$ of
$\mathbb{C}$-algebras.

It is clear that for analytic subvarieties, the role of structure sheaf is
played by the _sheaf of germs of holomorphic functions_ on the subvariety.

The notion of an analytic subvariety as defined in A.1 depends quite
essentially on a particular embedding in the ambient space
$\mathbb{C}^{\mathfrak{n}}$. For example, the germ of an analytic subvariety
at the origin in $\mathbb{C}^{\mathfrak{n}}$ can also be viewed as the germ of
an analytic subvariety at the origin in $\mathbb{C}^{\mathfrak{n}+1}$ through
the canonical embedding
$\mathbb{C}^{\mathfrak{n}}\hookrightarrow\mathbb{C}^{\mathfrak{n}+1}$ but
these will be inequivalent germs of analytic subvarieties. It is thus evident
that there is a point to introducing an equivalence relation among analytic
subvarieties in order to investigate those properties, which are to some
extent independent of the embeddings of these subvarieties in their ambient
complex number spaces.

Once again, for the sake of simplicity, let us consider sufficiently small
open polydiscs
$\varDelta^{\mathfrak{n}_{1}}(\boldsymbol{\epsilon}_{1})\subset\mathbb{C}^{\mathfrak{n}_{1}}$
and
$\varDelta^{\mathfrak{n}_{2}}(\boldsymbol{\epsilon}_{2})\subset\mathbb{C}^{\mathfrak{n}_{2}}$
such that analytic subvarieties
$X_{1}\subset\varDelta^{\mathfrak{n}_{1}}(\boldsymbol{\epsilon}_{1})$ and
$X_{2}\subset\varDelta^{\mathfrak{n}_{2}}(\boldsymbol{\epsilon}_{2})$ can be
determined as the set of common zeros of a finite number of functions that are
analytic throughout $\varDelta^{\mathfrak{n}_{1}}(\boldsymbol{\epsilon}_{1})$
and $\varDelta^{\mathfrak{n}_{2}}(\boldsymbol{\epsilon}_{2})$, respectively. A
continuous mapping between two such analytic subvarieties
$X_{1}\subset\varDelta^{\mathfrak{n}_{1}}(\boldsymbol{\epsilon}_{1})$ and
$X_{2}\subset\varDelta^{\mathfrak{n}_{2}}(\boldsymbol{\epsilon}_{2})$ is said
to be a _complex analytic mapping_ $f:X_{1}\to X_{2}$ between these two
subvarieties if there is a holomorphic mapping
$F:\varDelta^{\mathfrak{n}_{1}}(\boldsymbol{\epsilon}_{1})\to\varDelta^{\mathfrak{n}_{2}}(\boldsymbol{\epsilon}_{2})$
such that the restriction of $F$ to the subvariety
$X_{1}\subset\varDelta^{\mathfrak{n}_{1}}(\boldsymbol{\epsilon}_{1})$ is just
$f$ — i.e. $F\big{|}_{X_{1}}=f$. Additionally, two analytic subvarieties
$X_{1}\subset\varDelta^{\mathfrak{n}_{1}}(\boldsymbol{\epsilon}_{1})$ and
$X_{2}\subset\varDelta^{\mathfrak{n}_{2}}(\boldsymbol{\epsilon}_{2})$ are said
to be _analytically equivalent_ if there are complex analytic mappings
$f:X_{1}\to X_{2}$ and $g:X_{2}\to X_{1}$ such that the compositions $f\circ
g$ and $g\circ f$ are the appropriate identity mappings. This notion of
equivalence thus allows one to speak of analytic subvarieties without
reference to the spaces in which they are embedded; an equivalence class is
called an _analytic variety_ , and a space which has locally the structure of
an analytic variety is called an _analytic space_.

For global considerations, it is more convenient to think about an analytic
subvariety $X$ as a $\mathbb{C}$-ringed space $(|X|,\mathscr{O}_{X})$. Complex
analytic mappings $f:X_{1}\to X_{2}$ between two subvarieties $X_{1}$ and
$X_{2}$ are then viewed as _morphisms of $\mathbb{C}$-ringed spaces_ between
$(|X_{1}|,\mathscr{O}_{X_{1}})$ and $(|X_{2}|,\mathscr{O}_{X_{2}})$:

######  Definition A.4 (Morphism of $\mathbb{C}$-ringed spaces).

A _morphism_ $f:X_{1}\to X_{2}$ of ringed spaces
$(|X_{1}|,\mathscr{O}_{X_{1}})$ and $(|X_{2}|,\mathscr{O}_{X_{2}})$ is a pair
$f=(|f|,f^{\ast})$ consisting of a continuous map

$|f|:|X_{1}|\to|X_{2}|$

and a homomorphism

$f^{\ast}:\mathscr{O}_{X_{2}}\to|f|_{\ast}(\mathscr{O}_{X_{1}})$

of sheaves of rings on $X_{2}$. For any point $x\in X_{1}$, we think of
$f^{\ast}_{x}$ as the ring homomorphism

$f_{x}^{\ast}:\mathscr{O}_{X_{2},f(x)}\to\mathscr{O}_{X_{1},x}$

defined as the composition of the canonical homomorphisms

$\mathscr{O}_{X_{2},f(x)}\to\Big{(}|f|_{\ast}(\mathscr{O}_{X_{1}})\Big{)}_{f(x)}\to\mathscr{O}_{X_{1},x}.$

In case $X_{1}$ and $X_{2}$ are locally ringed spaces, a morphism by
definition has to be _local_ , that is, satisfy

$f_{x}^{\ast}\big{(}\mathfrak{m}_{X_{2},f(x)}\big{)}\subset\mathfrak{m}_{X_{1},x}$

for every $x\in X_{1}$. A _morphism of $\mathbb{C}$-ringed spaces_ $X_{1}$ and
$X_{2}$ is a morphism of ringed spaces, where $f^{\ast}$ is furthermore a
homomorphism of sheaves of $\mathbb{C}$-algebras. In this case, $f_{x}^{\ast}$
is automatically local for every $x\in X_{1}$.

It is an immediate consequence of this definition that two analytic
subvarieties $X_{1}$ and $X_{2}$ determine _equivalent varieties_ if and only
if there is a _topological homeomorphism_ $|f|:|X_{1}|\to|X_{2}|$ inducing an
isomorphism
$f^{\ast}:\mathscr{O}_{X_{2}}\xrightarrow{\cong}|f|_{\ast}(\mathscr{O}_{X_{1}})$
between the sheaves of $\mathbb{C}$-algebras. Thus, the coherent analytic
sheaf $\mathscr{O}_{X}$ on an analytic subvariety $X$ is the complete
invariant determining equivalence as varieties — hence the name _structure
sheaf_.

To build up complex analytic spaces, we construct local models as follows: Let
$U\subset\mathbb{C}^{\mathfrak{n}}$ be an open subset, and assume that
$\mathscr{I}$ is a coherent sheaf of $\mathscr{O}_{U}$-ideals. Then

$\operatorname{Supp}\big{(}\mathscr{O}_{U}/\mathscr{I}\big{)}=\Big{\\{}x\in
U\,\Big{|}\,\big{(}\mathscr{O}_{U}/\mathscr{I}\big{)}_{x}\neq 0\Big{\\}}$

is an analytic subset of $U$ which we will denote by $\tilde{A}$. The pair
$\big{(}\tilde{A},(\mathscr{O}_{U}/\mathscr{I})\big{|}_{\tilde{A}}\big{)}$ is
a $\mathbb{C}$-ringed space, which is called a _local model of an analytic
space_.

######  Definition A.5 (Complex analytic spaces and analytic mappings).

A _complex analytic space_ , or an _analytic space_ for short, is a
$\mathbb{C}$-ringed space $(|X|,\mathscr{O}_{X})$ satisfying the following
conditions:

  1. (i)

$|X|$ is Hausdorff,

  2. (ii)

For every $x\in X$, there is an open neighbourhood $V_{x}$ of $x$ such that
$(V_{x},\mathscr{O}_{X}\big{|}_{V_{x}})$ is isomorphic (as $\mathbb{C}$-ringed
space) to some local model.

If $X=(|X|,\mathscr{O}_{X})$ and $Y=(|Y|,\mathscr{O}_{Y})$ are complex
analytic spaces, then any morphism

$(|f|,f^{\ast}):(|X|,\mathscr{O}_{X})\to(|Y|,\mathscr{O}_{Y}),$

of $\mathbb{C}$-ringed spaces is called an _analytic map_ (or _holomorphic
map_).

######  Remark A.2.

Note that any complex manifold $M$ can be considered as an analytic space
$(|M|,\mathscr{O}_{M})$ and holomorphic maps $f:M\to N$ between complex
manifolds can be extended to analytic mappings
$(|f|,f^{\ast}):(|M|,\mathscr{O}_{M})\to(|N|,\mathscr{O}_{N})$.

Let $|X|$ be a topological space. Any analytic variety
$(|U|,\mathscr{O}_{U})$, where $|U|$ is an open set in $|X|$, is called an
_analytic chart_ on $|X|$. A family
$\big{\\{}(|U_{a}|,\mathscr{O}_{a}),g^{\ast}_{ab}\big{\\}}_{a,b\in A}$
consisting of complex charts on $|X|$ and of $\mathbb{C}$-algebra isomorphisms

$g^{\ast}_{ab}:\mathscr{O}_{b}\big{|}_{{}_{|U_{a}|\cap|U_{b}|}}\to\mathscr{O}_{a}\big{|}_{{}_{|U_{a}|\cap|U_{b}|}},$

is called an _analytic atlas_ on $X$, if $\big{\\{}|U_{a}|\big{\\}}_{a\in A}$
is an open covering of $|X|$, and if furthermore

$g^{\ast}_{ab}\circ g^{\ast}_{bc}=g^{\ast}_{ac}\quad\text{for all}\quad
a,b,c\in A\qquad\text{(coycle condition)};$

the maps $g^{\ast}_{ab}$ are the _gluing isomorphisms_ of the atlas.
Furthermore, we have the following lemma (see e.g. (Grauert-Remmert, , §1.7)
for the proof):

######  Lemma A.1 (Gluing Lemma).

Let $\big{\\{}(|V_{i}|,\mathscr{O}_{i}),g^{\ast}_{ij}\big{\\}}_{i,j\in I}$ be
an analytic atlas on a Hausdorff topological space $|X|$. Then, there exists a
unique (up to isomorphism) complex analytic space $(|X|,\mathscr{O}_{X})$ and
$\mathbb{C}$-algebra isomorphisms
$f^{\ast}_{i}:\mathscr{O}_{X}\big{|}_{{}_{|V_{i}|}}\to\mathscr{O}_{i}$ for all
$i\in I$, such that

$g^{\ast}_{ij}=f^{\ast}_{i}\circ(f^{\ast}_{j})^{-1}$

on every intersection $|V_{i}|\cap|V_{j}|$.

A complex analytic space $Y$ is called an _open complex analytic subspace_ of
$X$, if $|Y|$ is an open subset of $|X|$, and
$\mathscr{O}_{Y}=\mathscr{O}_{X}\big{|}_{{}_{Y}}$. In addition, $Y$ is called
a _closed complex analytic subspace_ of $X$, if there is a coherent ideal
$\mathscr{J}\subset\mathscr{O}_{X}$ such that
$|Y|=\operatorname{Supp}(\mathscr{O}_{X}/\mathscr{J})$ and
$\mathscr{O}_{Y}=(\mathscr{O}_{X}/\mathscr{J})\big{|}_{Y}$. In this case,
there is a canonical analytic map determined by the injection, which we denote
by $Y\hookrightarrow X$. A subset $A$ of a complex analytic space $X$ is
called _analytic_ when there is a coherent ideal
$\mathscr{J}\subset\mathscr{O}_{X}$ such that
$A=\operatorname{Supp}(\mathscr{O}_{X}/\mathscr{J})$.

We now discuss several possibilities of how good (respectively, how bad) a
given analytic space $X=(|X|,\mathscr{O}_{X})$ may behave at a point $x\in X$.
The situation is optimal if $x$ is a smooth point of $X$:

######  Definition A.6 (Smooth and singular points of analytic spaces).

A point $x$ in an analytic space $X=(|X|,\mathscr{O}_{X})$ is called _smooth_
or _regular_ , if there exists a neighborhood $V_{x}$ of $x$ in $|X|$ and an
open set $U$ in some complex number space $\mathbb{C}^{\mathfrak{n}}$ such
that $(V_{x},\mathscr{O}_{X}\big{|}_{{}_{V_{x}}})$ and $(U,\mathscr{O}_{U})$
are analytically isomorphic — i.e. if there exist analytic mappings

$(|f|,f^{\ast}):(V_{x},\mathscr{O}_{X}\big{|}_{{}_{V_{x}}})\to(U,\mathscr{O}_{U})\quad\text{and}\quad(|g|,g^{\ast}):(U,\mathscr{O}_{U})\to(V_{x},\mathscr{O}_{X}\big{|}_{{}_{V_{x}}}),$

such that the compositions $|f|\circ|g|$, $|g|\circ|f|$, $f^{\ast}\circ
g^{\ast}$, and $g^{\ast}\circ f^{\ast}$ are the appropriate identity mappings.
In other words, a point $x\in X$ is smooth if and only if
$\mathscr{O}_{X,x}\cong\mathscr{O}_{\mathbb{C}^{\mathfrak{n}},x}$. Of course,
the singular locus is defined to be the set of all non-smooth points:
$\operatorname{Sing}(X)=X\backslash\operatorname{Reg}(X)$.

When $X$ is not necessarily smooth (i.e., has a non-empty singular locus), we
define the following notions regarding the behavior of an analytic space $X$
at a point $x\in X$:

  * •

The analytic space $X$ is called _irreducible at point $x$_ if the stalk
$\mathscr{O}_{X,x}$ is an integral domain, otherwise $X$ is called reducible
at $x$. All smooth points are irreducible points, since for such a point $x$,
the stalk $\mathscr{O}_{X,x}$ is isomorphic to the ring of convergent power
series $\mathbb{C}\\{z_{1},\dots,z_{\mathfrak{n}}\\}$. The analytic space $X$
will be called _locally irreducible_ if all points of $X$ are irreducible; in
particular, complex manifolds are locally irreducible.

  * •

The complex analytic space $X$ is called _reduced at $x$_ if the stalk
$\mathscr{O}_{X,x}$ is a reduced ring — i.e. does not contain nilpotent
elements. All irreducible points are reduced points of $X$. We call $X$ a
_reduced analytic space_ if $X$ is reduced at all its points; this happens
when every local model for the space is defined by a radical sheaf of ideals.
An analytic space $X$, which isn’t reduced has a reduction $X_{\text{red}}$,
which is a reduced analytic space with the same underlying topological space.
There exists a canonical embedding $\iota:X_{\text{red}}\hookrightarrow X$ and
every morphism from $X$ to a reduced analytic space factors through $\iota$.
In particular, every analytic mapping $f:Y\to X$ of complex analytic spaces
induces a canonical analytic morphism $f_{\text{red}}:Y_{\text{red}}\to
X_{\text{red}}$ of their reductions such that $f\circ\iota_{Y}=\iota_{X}\circ
f_{\text{red}}$ where $\iota_{X}:X_{\text{red}}\hookrightarrow X$ and
$\iota_{Y}:Y_{red}\hookrightarrow Y$ are the canonical embeddings. When
$Y=Y_{\text{red}}$, the sheaf homomorphism component, $f^{\ast}$, of an
analytic map $(|f|,f^{\ast}):(|Y|,\mathscr{O}_{Y})\to(|X|,\mathscr{O}_{X})$ is
_uniquely_ determined by its continuous mapping component $|f|$.

  * •

A reduced point $x\in X$ will be called a _normal point_ of $X$, if the stalk
$\mathscr{O}_{X,x}$ is integrally closed in its quotient ring. Smooth points
are normal, and $X$ is irreducible at every normal point. An analytic space
$X$ will be called _normal_ if every point $x\in X$ is a normal point; in a
normal analytic space, the _singular locus has codimension at least two_. Once
again, all non-normal analytic spaces can be smoothed out into normal spaces
in a canonical way; this construction is called the _normalization_.

In the rest of this appendix, we will mainly focus on _normal analytic
spaces_.

####  A.1.2 An Intermezzo on Line Bundles and Divisors

In this subsection, we will introduce the basic definitions concerning line
bundles and divisors. Although we are mainly interested in complex analytic
spaces, we will mostly focus our attention to complex manifolds (i.e., smooth
analytic spaces) to avoid complication; we will comment on some of the
subtleties of generalizing the introduced notion to singular analytic spaces.
We start with reviewing the notions of connection, curvature, and Chern
classes for complex vector bundles (see Boyer_2008 ; Grauert-Remmert ;
suwa2007introduction ; akhiezer2012several  for more details).

#### Complex Vector Bundles

Let $M$ be a complex manifold of dimension $\mathfrak{n}$. A _complex vector
bundle_ of _rank_ $r$ over $M$ is a smooth manifold $E$ together with a
continuous map $\pi:E\to M$ such that there exists an open covering
$\mathcal{V}=\\{V_{\alpha}\\}_{\alpha\in A}$ of $M$ with the following
properties:

  1. (i)

for each $\alpha\in A$, there is a homeomorphism

$\psi_{\alpha}:\pi^{-1}(V_{\alpha})\xrightarrow{\sim}V_{\alpha}\times\mathbb{C}^{r}$

with $\operatorname{pr}\circ\psi_{\alpha}=\pi$ where $\operatorname{pr}$
denotes the projection $V_{\alpha}\times\mathbb{C}^{r}\to V_{\alpha}$;

  2. (ii)

for each pair $(\alpha,\beta)\in A\times A$, there is a $\mathcal{C}^{\infty}$
map

$g^{\alpha\beta}:V_{\alpha}\cap V_{\beta}\to\operatorname{GL}(r,\mathbb{C})$

with

$\psi_{\alpha}\circ\psi_{\beta}^{-1}(p,\zeta)=\big{(}p,g^{\alpha\beta}(p)\zeta\big{)}\qquad\text{for}\qquad(p,\zeta)\in
V_{\alpha}\cap V_{\beta}\times\mathbb{C}^{r}.$

We call $\psi_{\alpha}$ a _trivialization_ of $E$ on $V_{\alpha}$. We also
call $g^{\alpha\beta}$ the _transition matrix_ of $E$ on $V_{\alpha}\cap
V_{\beta}$ and the collection $\\{g^{\alpha\beta}\\}_{(\alpha,\beta)\in
A\times A}$ the _system of transition matrices_ of $E$. For each point $p$ in
$V_{\alpha}\cap V_{\beta}\cap V_{\gamma}$ we have the identity

$g^{\alpha\beta}(p)\,g^{\beta\gamma}(p)=g^{\alpha\gamma}(p)\qquad\qquad\text{(cocycle
condition)}.$ (245)

Thus, in particular, $g^{\alpha\alpha}(p)=\mathbbm{1}$ (the identity matrix)
and $g^{\beta\alpha}(p)=\big{(}g^{\alpha\beta}(p)\big{)}^{-1}$. We may think
of the system
$\big{\\{}(V_{\alpha},\psi_{\alpha},g^{\alpha\beta})\big{\\}}_{(\alpha,\beta)\in
A\times A}$ as defining a _vector bundle structure_ on $E$.

Conversely, if we are given an open covering
$\mathcal{V}=\\{V_{\alpha}\\}_{\alpha\in A}$ of $M$ and a collection
$\\{g^{\alpha\beta}\\}_{(\alpha,\beta)\in A\times A}$ of
$\mathcal{C}^{\infty}$ maps

$g^{\alpha\beta}:V_{\alpha}\cap V_{\beta}\to\operatorname{GL}(r,\mathbb{C}),$

satisfying the cocycle condition (245) for $p\in V_{\alpha}\cap V_{\beta}\cap
V_{\gamma}$, we may construct a vector bundle as follows: For