§ INTRODUCTION

One of the main powers of quantum field theory is that it comes with  a set of relatively simple rules for calculating measurable quantities.  For instance, it allows us to compute the quantities of interest, such as the scattering amplitudes, as a  perturbative expansion in the coupling parameters of the theory by  evaluating the Feynman diagrams using  the Feynman rules that are derived from the Lagrangian of the theory. The Feynman rules expresses the amplitudes with arbitrary external states and loops as well-defined integrals. Therefore, we can consider Feynman prescription as a calculable formulation of the perturbative quantum field theory.

Similarly, in the conventional formulation of string theory,  scattering amplitudes are obtained  by evaluating the string diagrams which are the string theory analogue of the Feynman diagrams. Unlike in  quantum field theory, the rules for evaluating the string diagrams are  not derived from a Lagrangian. Instead the rules are given without referring to an underlying Lagrangian  [1, 2, 3].   By exploring the conformal structure of string theory, powerful covariant methods to compute  amplitudes in perturbative string theory were introduced in [3]. Some aspects of the formalism that  remained slightly opaque also got cleared due to the recent studies and the revival of old results [10, 8, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. Therefore, we can safely say that  perturbative string theory provides a well-defined formal procedure for computing the amplitudes as an expansion in the string coupling.  However, we can consider this as a calculable formulation of the string theory, only if it comes with a set of well defined practical  rules for  evaluating  the amplitudes with any number of arbitrary external states and loops.  Unfortunately, such a set of practical rules for computing the amplitudes  is not readily available.In this paper, we describe a practical procedure for performing computations in perturbative string theory by using hyperbolic geometry. Below, we spell out the main objectives of this paper followed by a brief summary of the results.

§.§ Objectives

The path integral formulation of string theory requires summing over all possible worldsheets traced by the relativistic strings in spacetime with contribution from each worldsheet weighted by a unimodular complex number whose phase is the area of the corresponding worldsheet. In his seminal papers [1, 2], Polyakov showed that this summation over the two dimensional surfaces reduces to two dimensional exactly solvable conformal field theory on Riemann surfaces. This connection was established by introducing a metric on each of the worldsheets. The net effect of this is to introduce an unphysical degree of freedom, Liouville field, into the problem that disappears from the worldsheet action due to its Weyl invariance. Demanding the Weyl invariance at the quantum level makes sure that the Liouville field will not reappear in the action due to the quantum correction. Moreover, if we assume that the external states involved in the scattering process satisfy the classical on-shell condition, then the amplitudes are also independent of the specific choice of the metric on the world-sheet. Since the classical on-shell condition is assumed for the external states, we shall call the amplitude computed  this way as the on-shell string amplitudes.

§.§.§ Bosonic String Amplitudes and Moduli Space of Riemann Surfaces

The conventional formulation of the bosonic string theory  defines the $g$-loop contribution to the amplitude for the scattering of  $n$ asymptotic closed string states as a path integral over the embedding $X^{\mu}(z,\bar z)$ of the string worldsheet, which is a Riemann surfaces with $n$ boundaries, in spacetime and all inequivalent metrics on such Riemann surfaces. Here $z$ denotes the complex coordinate on  the worldsheet.  Using  the  state-operator correspondence, it is possible to replace the boundary conditions with  vertex operators, in an appropriate conformal field theory on the genus $g$ Riemann surface. Let us denote the vertex operator representing a state carrying  quantum numbers $a_i$ and momentum $k^{\mu}_i$ by $V_{a_i,k_i}(z_i,\bar z_i)$. Then the momentum space expression for the $g$-loop string amplitude for $n$ closed-string states having quantum numbers $a_i$ and momentum $k_i^{\mu}$ with $i$ running from $1$ to $n$ is given by,

\begin{equation}\label{nloop}
A^{(n)}_g(a_1,k_1;...;a_n,k_n)=\int\mathop{\underbrace{\Big[ dg_{ab}~dX^{\mu}\Big]_{g,n}}_{\mathrm{On~genus}~ g~\mathrm{Riemann ~surface}}}_{\mathrm{with~}n~\mathrm{punctures} }e^{-S_p[ g_{ab},X^{\mu}]}\prod_{i=1}^n\int d^2z_i\sqrt{-g(z_i)}~V_{a_i,k_i}(z_i),\nonumber\\
\end{equation}

where $S_p$ is the Polyakov action for a free relativistic string propagating through the space time.

The path integral <ref> can be made well-defined by carrying out the gauge fixing by introducing the Fadeev-Popov  ghosts. Redundant transformations that must be fixed  are  the Weyl transformations and the reparametrizations that keep the locations of punctures unaffected. This can be achieved by restricting the path integral over all possible metrics on the Riemann surface to a set of metrics which are not related to each other by the above-mentioned gauge transformations. The elements in this set of  metrics are the representatives of each conformal classes. Different conformal classes correspond to Riemann surfaces with different complex structures. As it is shown by Riemann himself and can be seen via index theorem, the number of parameters specifying each class is finite. These parameters form a non-compact space called the moduli space of Riemann surfaces. Therefore, the gauge-fixed path integral over the metrics on the Riemann surface is reduced to an integration over the moduli space of Riemann surfaces. Hence the $g$-loop on-shell amplitudes in bosonic string theory  with $n$ external closed-string states can be considered  as the integral of a suitable differential form of degree $6g-6+2n$ over the compactified  moduli space of genus-$g$ Riemann surfaces with $n$ punctures $\overline{\mathcal{M}}_{g,n}$ :

\begin{equation}
A^{(n)}_g(a_1,k_1;\cdots;a_n,k_n)=\int_{\overline{\mathcal{M}}_{g,n}}\Omega_{6g-g+n}. \label{amplitude as an integral of form over moduli space}
\end{equation}

The compactified moduli space can be obtained by adding the so-called Deligne-Mumford compactification locus, associated to the Riemann surfaces with nodes, to $\mathcal{M}_{g,n}$ [40].

Therefore, in order to obtain  a practical prescription for evaluating amplitudes in string theory, we need to understand the geometry of the moduli space of Riemann surfaces and  specify the rules for  performing integration over it.  Unfortunately, there is no known simple description of the moduli space. This is mainly due to the following fact. Characterizing the space of  Riemann surfaces require varying the complex structure by infinitesimal deformation of the metric on the Riemann surface.  But the infinitesimal deformations only lead us to the Teichmüller space of  Riemann surfaces  instead of  the moduli space.  The Teichmüller space is the space of Riemann surfaces with metrics  which are  not  related by  diffeomorphisms that  can be continuously connected to the identity transformation. The moduli space of  genus-$g$ Riemann surfaces with $n$ distinguished punctures is  obtained by identifying the points in the Teichmüller space $\mathcal{T}_{g,n}$ of the genus-$g$ Riemann surface with $n$ distinguished punctures that are related by the action of the  mapping class group Mod$_{g,n}$:

\begin{equation}
\mathcal{M}_{g,n}\equiv\frac{\mathcal{T}_{g,n}}{\mathrm{Mod}_{g,n}}.
\end{equation}

Mod$_{g,n}$ is  the group of diffeomorphims on the Riemann surface that can not be continuously connected to the identity transformation.  The action of mapping class group on Teichmüller space is very complicated and as a result  the geometry and the topology of the moduli space is also very complicated. Therefore, finding an explicit fundamental domain of the mapping class group in the Teichmüller space is a highly nontrivial task. This makes explicit integration of a function over the moduli space  of an arbitrary Riemann surface is almost impossible.The only known explicit computation of an integral over the moduli space $\mathcal{M}_{g,n}$, for generic values of $g$ and $n$, is the computation of the Weil-Petersson (WP) volumes of the moduli space of the bordered  Riemann surfaces, parametrized using the Fenchel-Nielsen coordinates, due to Mirzakhani [36]. This outstanding progress seems to suggest that the Fenchel-Nielsen parametrization of the moduli space of Riemann surfaces has the potential to lead us to a calculable formulation of string theory.

Objective 1: Express the bosonic string amplitudes as an integral over an explicit region in Teichmüller space of hyperbolic Riemann surfaces parameterized using the Fenchel-Nielsen coordinates.

§.§.§ Superstring Amplitudes and Picture-Changing Operators

The amplitudes in  the superstring theory  are the super analog of the  amplitudes in the bosonic string theory. Instead of all possible Riemann surfaces with  punctures, we need to sum over all possible super-Riemann surfaces with Neveu-Schwarz (NS) and/or Ramond (R) punctures. More precisely,  $g$-loop closed superstring amplitude of  $n_R$ asymptotic closed superstring  states from the R sector and $n_{NS}$ asymptotic closed superstring states from the NS sector is defined as the path integral over the embedding $X^{\mu}(z,\theta;\bar z,\bar\theta)$ of the superstring worldsheet, i.e. a super-Riemann surface with  $n_R$ boundaries having R boundary conditions and $n_{NS}$ boundaries having  NS boundary conditions, in spacetime and all metrics on such super-Riemann surfaces. Here $(z,\theta)$ is the complex coordinate on  the super-Riemann surface. Gauge redundancies of this path integral are the super-reparametrizations that do not move the punctures and the super-Weyl transformations. Like the bosonic string theory, we must gauge fix the path integral to get rid of these gauge redundancies. The gauge-fixing  restricts the structures on the super-Riemann surface to certain classes. Each of these conjugacy classes defines the superconfromal structure or the super-complex structure on the super-Riemann surface. Again using the index theorem, it can be shown that the number of parameters that describes each class is finite. The parameters that describe these classes involve both commuting and  anti-commuting.  Therefore, the gauge-fixed path integral reduces to an integration over the super-moduli space of  super-Riemann surfaces [11].

Alternatively, the picture-changing formalism of the superstring theory in the RNS framework gives an equivalent prescription for writing down the superstring amplitudes as an integral over the moduli space of ordinary Riemann surfaces [3]. It tells us to integrate out odd super moduli at the cost of introducing the so called picture-changing operators (PCO) on the ordinary Riemann surface. The naive  picture-changing prescription assumes that the positions where we are placing the PCOs do not matter as long as we impose the following condition on their distribution on the Riemann surfaces belonging to the boundary of the moduli space.  For a genus-$g$ Riemann surface with  $n_{NS}$ NS punctures and $2n_R$ R punctures, denoted by $\mathcal{R}_{g,n_{NS},2n_{R}}$, we must insert $2g-2+n_{NS}+n_R$ number of PCO's on arbitrary locations on the Riemann surface. Consider such a surface with one degenerating cycle. Assume that cutting along this cycle separates the surface into two  surfaces $\mathcal{R}_{g_1,n^1_{NS},2n^1_{R}}$ and $\mathcal{R}_{g_2,n^2_{NS},2n^2_{R}}$. Then  the PCOs distribution on the degenerate surface must be such that each component surface have $2g_i-2+n^i_{NS}+n^i_R,~i=1,2$ number of PCOs.  If there are more than one degenerating curves then similar condition must be true with respect to all of the  subsurfaces. Then the $g$-loop superstring amplitude of $n_{NS}$ NS states and $2n_{R}$ R states can be defined as an integral over an appropriately-constructed form on the moduli space of genus-$g$ Riemann surface with $n=n_{NS}+2n_R$ punctures:

\begin{equation} \label{superstring amplitude as an integral of form over section}
A^{(n_{NS},n_R)}_g(a^{NS}_i,k^{NS}_i;a^{R}_j,k^{R}_j)=\int_{\overline{\mathcal{M}}_{g,n}}\Omega_{6g-g+2n} \Big(z_1(m),\cdots,z_{2g-2+n_{NS}+n_R}(m)\Big),\nonumber\\
\end{equation}

where $z_i(m)$ denotes the position of $i^{th}$ PCO on the Riemann surface corresponding to the point in the moduli space  represented by the coordinate $m$. The set $\Big(z_1(m),\cdots,z_{2g-2+n_{NS}+n_R}(m)\Big)$ provides a distribution of PCOs over the entire region of moduli space that satisfy the above-mentioned property. $a_i^{NS},k_i^{NS}$ denote the quantum numbers and momentum vector of the $i^{th}$ NS external state and  $a_j^{R},k_j^{R}$ denote the quantum numbers and momentum vector of the $j^{th}$ R external state.

The superstring amplitudes defined using the naive picture-changing formalism is not well defined due to the presence of  the unphysical singularities. The origin of these singularities can be identified with the fact that replacing the odd coordinates of the supermoduli space with PCOs is not possible in a smooth way over the entire supermoduli space. The choice of the locations of PCOs corresponds to a particular choice of a gauge for the worldsheet gravitino field [28]. It is known that in general, choosing a globally-valid gauge for the gravitino field is impossible [11, 12]. In the picture-changing formalism, this breakdown of the global choice of the gauge for the  gravitino field shows up as the spurious singularities of the integration measure of the superstring amplitudes. The presence of the spurious singularities means that the  computation of amplitudes in the  superstring theory  using PCOs has to be based on piecing together local descriptions. The vertical integration prescription introduced in [15] and systematically developed in [16] provides an efficient method to piece together the local descriptions. However, in order to implement the vertical integration prescription it is essential to know the explicit region over which superstring measure is supposed to be integrated.

Objective 2:  Express the superstring amplitudes as an integration over an explicit region in Teichmüller space of hyperbolic Riemann surfaces in terms of Fenchel-Nielsen coordinates and describe a distribution of PCOs on the worldsheets that respects the above-mentioned factorization property.  This makes implementing the vertical integration possible. The resulting superstring amplitudes are free from spurious poles.

§.§.§ Off-Shell Superstring Amplitudes and 1PI Approach

The conventional formulation of string perturbation theory requires imposing the classical on-shell condition on the external states and  Weyl invariance.  Demanding the classical on-shell condition for external states and the Weyl invariance at the quantum level have significant consequences which we explain below.

Mass Renormalization and Classical On-Shell Condition: Consider a string theory amplitude corresponding to the scattering of $n$-external states representing particles carrying momenta $k_1, \cdots, k_n$ and other discrete quantum numbers $a_1, \cdots, a_n$ with tree level masses $m_{a_1} , \cdots, m_{a_n}$. The momenta $k_i$ are required to satisfy the tree-level on-shell condition $k^2_i = -m^2_{a_i}$. Conventional formulation of string theory yields the result for what in a quantum field theory can be called `truncated Green's function on classical mass shell':

≡lim_k_i^2 →-m_a_i^2 F^(n)_a_1⋯a_n (k_1, ⋯k_n),

F^(n)_a_1⋯a_n  (k_1, ⋯k_n)   ≡G^(n)_a_1⋯a_n (k_1,⋯k_n)∏_i=1^n (k_i^2 + m_a_i^2),

where $G^{(n)}_{a_1\cdots a_n} (k_1,\cdots k_n)$ corresponds to the momentum-space Green's function in the quantum field theory.  This expression is similar to the expression for the S-matrix elements in a quantum field theory but it has significant differences.

To define S-matrix elements in quantum field theory, we need to first consider the two-point function $G^{(2)}_{ab}(k, k')$  for all set of fields whose tree-level masses are all equal to $m$ described by the matrix

\begin{equation} \label{ek6}
G^{(2)}_{ab}(k, k') = (2\pi)^{D+1} \delta^{(D+1)}(k+k') \, Z^{1/2}(k)_{ac} (k^2 + M_p^2)_{cd}^{-1}
\end{equation}

where $M_p^2$ is the mass$^2$ matrix and $Z^{1/2}(k)$ is the wave-function renormalization matrix, the latter being free from poles near $k^2+m^2\simeq 0$.  The sum over $c,d$ are restricted to states which have the same tree-level mass $m$ as the states labeled by the indices $a,b$. $D+1$ is the total number of non-compact space-time dimensions. We can diagonalize $M_p^2$ and absorb the diagonalizing  matrices into the wave-function renormalization factor $Z^{1/2}(k)$ to express  $M_p^2$ as a diagonal matrix.  These eigenvalues, which we shall denote by $m_{a,p}^2$, are the squares of the physical masses.  Then S-matrix elements are defined by

\begin{equation} \label{ek5}
S^{(n)}_{a_1\cdots a_n}(k_1,\cdots k_n) =
\lim_{k_i^2 \to -m_{a_i,p}^2} G^{(n)}_{b_1\cdots b_n}
(k_1,\cdots k_n)\prod_{i=1}^n \left\{Z_i^{-1/2}(k_i)_{a_i, b_i} (k_i^2 + m_{a_i,p}^2)\right\},
\end{equation}

where $m_{a_i,p}$ is the physical mass of the $i$-th particle, defined as the location of the pole in the two-point Green's function $G^{(2)}$ as a function of $-k^2$ and $Z_i(k_i)_{ a_i,b_i}$ is the residue at this pole.

At the tree level $Z=1$, $M_p^2=m^2 \, I$, where $I$ is the identity matrix, and hence $R^{(n)}$ and $S^{(n)}$  agree.  However, in general $R^{(n)}$ and $S^{(n)}$ are different. While $S^{(n)}$ defined in <ref> is the physically-relevant quantity, the conventional formulation of string theory directly computes $R^{(n)}$ defined in <ref>. This is a serious trouble for states receiving mass renormalization. Assume that an external state with quantum number $a_i$ and tree-level mass $m_{a_i}$ undergoes mass renormalization. Then beyond one loop, the radiative corrections introduce a series of $\frac{1}{k_i^2+m_{a_i}^2}$. At the same time string theory requires $k_i^2+m_{a_i}^2=0$. This makes the on-shell amplitudes involving  the states receiving mass renormalization ill-defined beyond tree level.

Dynamical Shift of Vacuum and Conformal Invariance : The background spacetimes that avoid the Weyl anomaly satisfy a set of classical equations. However, there are  situations in which  the quantum corrections modify such backgrounds.  Consider an $\mathcal{N}=1$ supersymmetric compactification of string theory down to 3+1 dimensions, where we have $U(1)$ gauge fields with Fayet-Iliopoulos (FI) terms generated at one  loop [4, 5, 6, 7].  It is possible to ensure that only one gauge field has FI term by choosing suitable linear combination of these gauge fields. Typically there are massless scalars $\phi_i$ charged under $U(1)$ gauge fields. The FI term generates a term in the potential of the form

\begin{equation} \label{e1}
{1\over g_s^2}\, \left( \sum_i q_i \phi_i^* \phi_i - C\, g_s^2\right)^2,
\end{equation}

where $q_i$ is the charge carried by $\phi_i$, $C$ is a numerical constant that determines the coefficient of the FI term, and $g_s$ is the string coupling constant. $C$ could be positive or negative and $q_i$s for different fields could have different signs. If we expand the potential in powers of $\phi_i$ around the perturbative vacuum $\phi_i=0$, it is clear that some of these scalars can become tachyonic. The form of effective potential suggests that the correct procedure to compute physical quantities is to shift the corresponding fields so that we have a new vacuum where $\sum_i q_i \langle\phi_i^*\rangle \langle\phi_i\rangle = C\, g_s^2$, and quantize string theory around this new background. However since classically the $C\, g_s^2$ term is absent from this potential <ref>, this new vacuum is not a solution to the classical equations of motion. As a result, on-shell  methods [8, 9, 11], which require that we begin with a conformally-invariant worldsheet theory, is not suitable for carrying out a systematic perturbation expansion around this new background.

It is possible to cure the singularities associated to mass renormalization and dynamical modification of vacuum by defining appropriate off-shell string amplitudes [13, 14, 17, 15, 18, 20, 19, 21, 24]. The off-shell amplitudes are defined by relaxing the classical on-shell condition on external states. The classical on-shell condition, in the worldsheet conformal field theory language, is equivalent to choosing integrated vertex operators having conformal dimensions $(1,1)$. Therefore, in order to construct off-shell amplitudes we must use vertex operators with arbitrary conformal dimensions. Consider a reference coordinate system $z$ on a Riemann surface. Let $z_i$ denote the location of the $i$-th puncture in the $z$-coordinate system and $w_i$ denote the local coordinate system around the $i$-th puncture, related to $z$ by some functional relation $z=f_i(w_i)$ such that the $w_i=0$ maps to $z=z_i$: $f_i(0)=z_i$. Then the contribution to the $n$-point off-shell amplitude from the genus-$g$  Riemann surfaces can be expressed as

\begin{equation}
\int_{\mathcal{M}_{g,n}}  \left\langle \prod_{i=1}^n f_i\circ V_i(0) \, \times \,
\hbox{ghost insertions}\right\rangle\,,
\end{equation}

where $f\circ V(0)$ denotes the conformal transformation of the vertex operator $V$ by the function $f(w)$, the correlator $\langle\cdots\rangle$ is evaluated in the reference
$z$-coordinate system and $\int_{\mathcal{M}_{g,n}}$ denotes the integration over the moduli space of Riemann surfaces of genus $g$ with $n$ punctures. A detailed description of how to construct the integration measure for a given choice of local coordinate system will be explained later. The off-shell amplitudes defined this way depend on the choice of local coordinate system $w_i$ but  are independent of the choice of the reference coordinate system $z$.

It is important to  make sure that physical quantities extracted from the off-shell amplitudes defined this way are independent of the choice of local coordinates. It is showed in [13, 14, 15] that the physical quantities like the renormalized masses of physical states and S-matrix elements are independent of the choice of local coordinates,  if we choose to work with a class of local coordinates satisfying the following  properties:

* The local coordinate system is  symmetric in all of the puncture, i.e. the function $f_i(w)$ should depend on $i$ only via the location $z_i$ of the puncture.

* The choice of local coordinates is continuous over the entire region of moduli space.

* Riemann surfaces that can be obtained by gluing two or more Riemann surfaces using the plumbing fixture procedure are declared to  contribute to the 1PR amplitudes. The gluing relation is

\begin{equation} \label{egluing}
w_1 w_2 = e^{-s+i\theta}, \, \quad 0\le \theta < 2\pi, \quad 0\le s<\infty\,,
\end{equation}

where  $w_1$ and $w_2$ are the local coordinates at the punctures used for gluing. We shall call the corresponding
Riemann surfaces the 1PR Riemann surfaces. The rest of the Riemann surfaces are declared as 1PI Riemann surfaces. The choice of local coordinates on a 1PR surface must be the one induced from the local coordinates of the glued 1PI surfaces. However, the choice of local coordinates on the 1PI surfaces are arbitrary.

We shall call local coordinates satisfying the above criteria
gluing-compatible local coordinates. In the case of the superstring amplitudes, we also need to impose similar restriction on the distribution of PCOs.  On 1PI Riemann surfaces, we are allowed to choose arbitrary  distribution of PCOs. But on 1PR Riemann surfaces, we need to choose  distribution of PCOs that is induced from the surfaces that are being glued using plumbing fixture construction.

Note that the off-shell string amplitudes $\Gamma_{a_1\cdots a_n}^{(n)}(k_1,\cdots,k_n)$ for $n$ external states with quantum numbers $a_i$, momentum  $k_i$, and tree level mass $m_i$, do not compute the off-shell Green's function  $G_{a_1\cdots a_n}^{(n)}(k_1,\cdots,k_n)$ of the string theory. They are however related as

\begin{align}\label{offgreen}
\Gamma_{a_1\cdots a_n}^{(n)}(k_1,\cdots,k_n)=G_{a_1\cdots a_n}^{(n)}(k_1,\cdots,k_n)\prod_{i=1}^n(k_i^2+m_i^2).
\end{align}

Scattering amplitudes defined following LSZ-prescription using  off-shell amplitudes defined this way  are unitary and satisfy cutting rules [22, 23, 24, 25].

Objective 3:  Describe an explicit construction of the 1PI region inside the moduli space of  Riemann surfaces in terms of Fenchel-Nielsen coordinates and also describe a gluing compatible choice of local coordinates around the punctures. Also provide a systematic method for distributing PCOs over the entire region of moduli space in a gluing compatible fashion.  Then construct the superstring measure using the Beltrami differentials associated with the Fenchel-Nielsen coordinate vector fields.

§.§ Summary of the Results

The  uniformization theorem of Poincaré and Koebe [59] asserts that every genus-$g$ Riemann surface $\mathcal{R}_{g,n}$ with $n$ distinguished punctures subject to the constraint $2g+n\ge 3$, can be obtained by the proper discontinuous action of a Fuchsian group $\Gamma$ on the Poincaré upper half plane $\mathbb{H}$:

\begin{equation}
\mathcal{R}_{g,n}\simeq \frac{\mathbb{H}}{\Gamma} \label{Uniformization Theorem}.
\end{equation}

The Fuchsian group $\Gamma$ is a subgroup of the automorphic group of the Poincaré upper half plane. Riemann surfaces obtained this way are hyperbolic and have constant curvature $-1$ everywhere.  A Riemann surface is naturally distinguished by its complex structure. This implies the following one-to-one correspondence:

\begin{equation}
\text{Hyperbolic Structure} \qquad \Longleftrightarrow \qquad  \text{Complex Structure}.
\end{equation}

Therefore, the moduli space of a Riemann surface which is essentially the classifying space for the conjugacy classes of the distinguished complex structures on it can naturally be identified  with the moduli space of hyperbolic Riemann surfaces.

The Fenchel-Nielsen coordinates provide a convenient parametrization of the Teichmüller space of hyperbolic Riemann surfaces [50].  The basic idea behind the Fenchel-Nielsen  parametrization is that every compact hyperbolic Riemann surface of genus $g$  with  $n$ punctures  can be obtained by gluing $2g-2+n$ pairs of pants. By varying the parameters of this construction, every compact hyperbolic metric can be obtained. At each gluing site there are two parameters. At the gluing site, where we glue the boundary $b_1$ of the pair of pants $P$ with boundary $\beta_1$ of the pair of pants $Q$, these parameters are the  length $l(b_1)=l(\beta_1)=\ell$ of the boundaries $b_1$ and $\beta_1$, and the twist parameter $\tau$. Since there are $3g-3+n$ gluing sites for  a Riemann surface of genus $g$  with  $n$ punctures, the  Fenchel-Nielsen coordinates  of the corresponding Teichmüller space $\mathcal{T}_{g,n}$ is given by $(\tau_i,\ell_i),~i=1,\cdots,3g-3+n$.

A specific value of the Fenchel-Nielsen coordinates corresponds to a particular Riemann surface $\mcal{R}$. As we mentioned above, it is also possible to construct $\mcal{R}$ by considering the quotient of a fundamental domain in the upper half-plane by the action of a Fuchsian group $\Gamma$. It turns out that the generators of $\Gamma$ can be expressed in terms of the associated Fenchel-Nielsen coordinates [32]. For more details, see appendix <ref>.

§.§.§ Bosonic-String Measure and the Fenchel-Nielsen Coordinates

Assume that the Fenchel-Nielsen  coordinates $(\tau_i,\ell_i),~i=1,\cdots,3g-3+n$ are defined with respect to the pair of pants decomposition $\{C_i\},~i=1,\cdots,3g-3$, where $C_i$ denote a simple closed geodesic[A simple geodesic is a geodesic that does not intersect itself.] on the hyperbolic surface. The tangent space at a point in the Teichmüller space is spanned by the Fenchel-Nielsen  coordinate vector fields

\begin{equation*}
\left\{\frac{\partial}{\partial \tau_i},\frac{\partial }{\partial \ell_i}\right\},\qquad i=1,\cdots,3g-3+n.
\end{equation*}

The Fenchel-Nielsen  coordinate vector field $\frac{\partial}{\partial \tau_i}$ is  the  twist vector field $t_{C_i}$ associated with the curve $C_i$.  The twist field $t_{\alpha}$, where $\alpha$ is a simple closed geodesic,  generates a flow in $\mathcal{T}_{g,n}$. This flow can be understood as the operation of  cutting the hyperbolic surface along $\alpha$ and attaching the boundaries after rotating one boundary relative to the other by some amount $\delta$. The magnitude $\delta$ parametrizes the flow in  $\mathcal{T}_{g,n}$. The vector field $\frac{\partial}{\partial \ell_i}$ is the dual to $\frac{\partial}{\partial \tau_i}$ with respect to the Weil-Petersson symplectic form on the Teichmüller space.

Using the Beltrami differentials that represent the Fenchel-Nielsen  coordinate vector fields $\{\frac{\partial}{\partial \tau_i},\frac{\partial }{\partial \ell_i}\},~i=1,\cdots,3g-3+n$ we can write the off-shell bosonic string measure as

\begin{equation}\label{pformpgnbos1}
\Omega_{6g-6+2n}^{(g,n)}(|\Phi\rangle)=(2\pi \mathrm{i})^{-(3g-3+n)}\langle\mathcal{R}|B_{6g-6+2n}|\Phi\rangle,
\end{equation}

\begin{equation*}\label{opevormbos1}
B_{6g-6+2n}\left[\frac{\partial}{\partial \ell_1},\frac{\partial}{\partial \tau_1},\cdots,\frac{\partial}{\partial \ell_{3g-3+n}},\frac{\partial}{\partial \tau_{3g-3+n}}\right]\equiv\prod\limits_{a=1}^{3g-3+n}b(t_{\tau_a})\delta\tau_a\,b(t_{\ell_a})\delta\ell_a,
\end{equation*}

\begin{equation*}\label{bv1}
b(t_{\tau_i})\equiv \int_{\mathcal{F}} d^2z\left(b_{zz}\mathbf{t}_{\tau_i}+b_{\bar z\bar z}\bar{\mathbf{t}}_{\tau_i}\right) \qquad  b(t_{\ell_i})\equiv\int_{\mathcal{F}} d^2z\left(b_{zz}\mathbf{t}_{\ell_i}+b_{\bar z\bar z}\bar{\mathbf{t}}_{\ell_i}\right).
\end{equation*}

Here $\mathcal{F}$ denotes the fundamental region that represents the Riemann surface $\mathcal{R}$ in the upper half plane. $\mathbf{t}_{\tau_a}$ is the representative of the vector field $\frac{\partial}{\partial \tau_a}$ and $\mathbf{t}_{\ell_a}$ is the representative of the vector field $\frac{\partial}{\partial \ell_a}$. $\mathbf{t}_{\tau_a}$ deforms the hyperbolic structure of the surface by infinitesimal amount $\delta\tau_a$. Similarly, $\mathbf{t}_{\ell_a}$ deforms the hyperbolic structure of the surface by infinitesimal amount $\delta\ell_a$. $\langle \mathcal{R}|$ is the surface state associated with the surface $\mathcal{R}$  which describes the state that is created on the boundaries of the   disc $D_i, ~i=1,\cdots,n$ around the  punctures having unit radius measured in the local coordinates around the punctures  by performing the functional integral over the fields of the conformal field theory on $\mathcal{R}-\sum_iD_i$.  The state
$$|\Phi\rangle=|\Psi_1\rangle\otimes\cdots\otimes |\Psi_n\rangle\in\mathcal{H}^{\otimes n}$$
where $\mathcal{H}$ is the Hilbert space of the worldsheet conformal field theory.  Then $\langle\mathcal{R}|B_{6g-6+2n}|\Phi\rangle$ describes the $n$-point correlation function on $\mathcal{R}$ with the vertex operator for $|\Psi_i\rangle$ inserted at the $i^{th}$ puncture using the gluing compatible-local coordinate system  around that puncture.

§.§.§ Gluing-Compatible Local Coordinates and Hyperbolic Metric

The gluing-compatibility requirement demands that the choice of local coordinates on a 1PR Riemann surface must be the same as the local coordinates induced on it from the component surfaces. At first sight, the local coordinates around the punctures that are induced from the hyperbolic metric on the surface seem to be a gluing-compatible choice. This is true  for surfaces with nodes, constructed by the plumbing fixture of hyperbolic surfaces. However, it turns out that  when we construct the plumbing family of the hyperbolic Riemann surfaces without nodes, the resulting surfaces do not have constant curvature $-1$ everywhere. This implies that the induced local coordinates on the glued surface are not the local coordinates defined using the hyperbolic metric on it. Therefore, the induced local coordinate system from the hyperbolic metric is not gluing-compatible.

In the simplest form of a degeneration, two parts of a degenerate Riemann surface are connected by a single long cylinder. Assume that we are gluing surfaces with hyperbolic metric on them. In this case, the cylinder connecting the component surfaces has a curve on it where the curvature is accumulated. This implies that the glued surface is not hyperbolic. This is the consequence of the following facts: 1) in the plumbing fixture construction, one glue two surfaces by identifying curves around the punctures, 2) the glued surface is hyperbolic only if the component surfaces are glued through a geodesic, 3) however there are no geodesic curves in the neighborhood of punctures. According to the Uniformization Theorem, any surface subject to the condition $2g+n\ge 3$ can be made hyperbolic. In fact, we can make it hyperbolic by performing a Weyl transformation. The correct Weyl factor can be obtained by solving the so-called curvature correction equation [41, 42, 43].  To describe this equation, consider a compact Riemann surface having  metric $ds^2$ with Gauss curvature $\mathbf{C}$. The conformally-equivalent  metric $e^{2f}ds^2$ has constant curvature $-1$, if $f$ satisfy the curvature correction equation

\begin{equation}\label{constantcurvatureq1}
\end{equation}

where $D$ denotes the Laplace-Beltrami operator on the surface. The analysis can be generalized to a surface with arbitrary number of nodes. The result provides the relation between induced local coordinates on the glued surface and the local coordinates defined using the hyperbolic metric. We can then find a gluing-compatible choice of local coordinates. Before specifying this choice, it is important to describe 1PI and 1PR regions of the moduli space described in section <ref>. The 1PI region inside the moduli space consists of surfaces which there is no closed geodesics on them with length less than or equal to $c_*$, the collar constant. The 1PR region inside the moduli space consists of surfaces which there is at least one closed geodesics on them with length less than $c_*$. The length of the collar constant $c_*$ is assumed be infinitesimal compared to any geodesic on the surface having finite length.We can now specify a gluing-compatible choice of local coordinates found by solving the curvature-correction equation to second-order in $c_*$. For this, let us introduce another infinitesimal parameter $\epsilon$. In order to define local coordinates in the 1PI region, we divide it into subregions. Let us denote the subregion in the 1PI region consists of surfaces with $m$ simple closed geodesics of length between $c_*$ and $(1+\epsilon)c_*$ by $\mathbf{R}_m$. For surfaces belong to the subregion $\mathbf{R}_0$, we choose the local coordinate around the $i^{th}$ puncture to be $e^{\frac{\pi^2}{c_*}}w_i$. In terms of $w_i$, the hyperbolic metric in the neighborhood of the puncture takes the following form

\begin{equation}
d s^2_{hyp}=\left(\frac{|dw_i|}{|w_i|\ln|w_i|}\right)^2, \qquad  i=1,\cdots,n.
\end{equation}

For surfaces belong to the region $\mathbf{R}_{m}$ with $m\ne 0$, we choose the local coordinates around the $i^{th}$ puncture to be  $e^{\frac{\pi^2}{c_*}}\widetilde w_{i,m}$, where $\widetilde w_{i,m}$ is obtained by solving the following equation

\begin{equation}
\left(\frac{|d\widetilde{w}_{i,m}|}{|\widetilde{w}_{i,m}|\mathrm{ln}|\widetilde{w}_{i,m}|}\right)^2=\left(\frac{|dw_i|}{|w_i|\mathrm{ln}|w_i|}\right)^2\left(1-\sum_{j=1}^mf(l_j)E^0_j(w_i)\right).
\end{equation}

$f(x)$ is a smooth function of $x$ such that $f(c_*)=\frac{c_*^2}{3}$ and $f((1+\epsilon)c_*)=0$. $E^0_{j}(w_i)$ is the leading term in the Eisenstein series defined with respect to the $j^{th}$ node on the surface. We shall describe this notion in section <ref>. The 1PR region consists of surfaces with   $m$ simple closed geodesics of length  $0\le l_j<c_* ,~j=1,\cdots,m$. We choose the local  coordinates around the $i^{th}$ puncture to be $e^{\frac{\pi^2}{c_*}}\widehat{w}_{i,m}$ where $\widehat{w}_{i,m}$ is obtained by solving the following equation

\begin{equation}
\left(\frac{|d\widehat{w}_{i,m}|}{|\widehat{w}_{i,m}|\mathrm{ln}|\widehat{w}_{i,m}|}\right)^2=\left(\frac{|dw_i|}{|w_i|\mathrm{ln}|w_i|}\right)^2\left(1-\sum_{j=1}^m\frac{l_j^2}{3}E^0_j(w_i)\right).
\end{equation}

Note that different choice of function $f$ and different values for the parameters $c_*$ and  $\epsilon$ give different choices of gluing-compatible local coordinates. It is shown in [13, 14, 15]  that all such choices of local coordinates give the same value for the measurable quantities. Therefore, the function $f$ and the arbitrary parameters $c_*$ and $\epsilon$ will not appear in the measurable quantities as longs as  we restrict our computations to the second order in $c_*$.

§.§.§ Integration of the Bosonic-String Measure

The off-shell bosonic-string measure that we constructed using the Fenchel-Nielsen coordinates of Teichmüller space is an object which lives in the moduli space of hyperbolic Riemann surfaces, because it is invariant under mapping class group (MCG) transformations. The action of MCG on the Teichmüller space is highly nontrivial and, as a result, the geometry of moduli space is considerably more difficult than the corresponding Teichmüller space. This means that in order to evaluate the string amplitudes we need a special method for integrating an MCG-invariant function over the moduli space. Below, we describe such a method.

Consider a hyperbolic Riemann surface with $n$ borders having hyperbolic lengths $L_i,~i=1,\cdots,n$. Assume that we have the following identity,

\begin{equation}\label{mcshane1}
\sum_{\alpha\in\overline{\text{MCG}}\cdot \gamma} f_i(l_{\alpha})=L_i,
\end{equation}

where $f_i$ are real functions of the hyperbolic length $l_{\alpha}$ of the curve $\alpha$ on hyperbolic Riemann surface, which is an MCG image of curve $\gamma=\sum_{i=1}^k\gamma_i$. Here, $\gamma_i$ are simple closed curves on the hyperbolic bordered Riemann surface $\mathcal{R}$ with hyperbolic length $l_{\gamma_i}$. $\overline{\text{MCG}}\cdot \gamma$ denotes the set of inequivalent MCG images of the curve $\gamma$. We also assume that cutting along curves $\gamma$ provides $s$ disconnected bordered Riemann surfaces $\mathcal{R}^j(\gamma),~j=1,\cdots,s$. The Mirzakhani-McShane identity for bordered hyperbolic surfaces [35, 36] is an example of identities of the form  (<ref>). We will explain the details in the section <ref>. Consider an MCG-invariant function $H(l_i,\tau_i)$ defined on the bordered surface $\mathcal{R}$ with borders having lengths $\mathbf{L}=(L_1,\cdots,L_n)$. $(l_i,\tau_i)$ denotes the Fenchel-Nielsen coordinates defined with respect to a pair of pants decomposition of $\mathcal{R}$. Since $H(l_i,\tau_i)$ is invariant under MCG transformations, we are free to choose any pair of pants decomposition of the surface in order to define Fenchel-Nielsen coordinates without changing the form of it. Interestingly, the geometry of the moduli space and the structure of the identity (<ref>) allows us to reduce the integration of $H(l_i,\tau_i)$ over the moduli space to the integration over the lower-dimensional moduli spaces as follows:

∫_ℳ(ℛ)dV H(l,τ)   =1/L_i∑_α∈MCG·γ∫_ℳ(ℛ)dV f_i(ℓ_α)H(ℓ,τ)

=1/L_i|Sym(γ)|∫_ℳ(ℛ)^γdV f_i(ℓ_γ)H(ℓ,τ)

=1/L_i|Sym(γ)|∫_ℝ_+^kdℓ_γ_1⋯dℓ_γ_k ∫_0^2^-M_γ_1ℓ_γ_1⋯∫_0^2^-M_γ_kℓ_γ_k dτ_γ_1⋯dτ_γ_k

×∫_ℳ(ℛ^1(γ))dV(ℛ^1(γ))⋯∫_ℳ(ℛ^s(γ))dV(ℛ^s(γ)) f_i(ℓ_γ)H(ℓ,τ).

Here $\tau_{\gamma_i}$ is the twist along the curve $\gamma_i$ and  $M_{\gamma_i}=1$ if $\gamma_i$ bounds a torus with one boundary component otherwise  $M_{\gamma_i}=0$. In the second step, we absorbed the MCG images of curve $\gamma$ to the integration by enlarging the region of integration from the moduli space $\mathcal{M}(\mathcal{R})$ to the covering space $\mathcal{M}(\mathcal{R})^{\gamma}$. The covering space $\mathcal{M}(\mathcal{R})^{\gamma}$ can be obtained from the Teichmüller space by taking the quotient with the subgroup of the full mapping class group that do not act on the curve $\gamma$. In the third step, we used the fact that the space $\mathcal{M}(\mathcal{R})^{\gamma}$ can be obtained from the moduli spaces of component surfaces, obtained by cutting the surface $\mathcal{R}$ along the curves $\gamma_i, ~i=1,\cdots,k$, and $k$ number of infinite cones formed by $(\ell_i,\tau_i)$ for $i=1,\cdots,k$. For more details, see section <ref>.  Now, we can repeat the same procedure for integration over $\mathcal{M(R}^j(\gamma))$. In this way, the integral can be written over an explicit domain.

§.§.§ Superstring Amplitudes and the PCOs

Off-shell amplitudes in the superstring theory can be obtained by constructing the off-shell superstring measure  and then integrating it over the moduli space of hyperbolic Riemann surfaces. Like in the bosonic-string theory, using the Beltrami differentials associated with Fenchel-Nielsen coordinate vector fields in the tangent space of Teichmüller space and  a choice of gluing compatible local coordinates around the punctures, we construct the off-shell superstring measure. But the  construction of the off-shell superstring measure is more involved than the bosonic string theory due to the presence of PCOs. For obtaining consistent off-shell amplitudes, we need to impose gluing-compatibility requirement for the  PCOs distribution as well as for the local coordinates. A gluing-compatible choice of PCOs distribution can be found using the Mirzakhani-McShane identity as described in [62]. It will be explained in section <ref>. Although the gluing-compatible choice of PCOs distribution provides off-shell superstring amplitudes that is consistent with off-shell factorization properties still it might be ill-defined due to the presence of unphysical singularities. These unwanted singularities appear when the PCOs distribution satisfy certain global conditions on the moduli space. It is not clear that the proposed distribution of PCOs avoids these conditions. In the case that it satisfies, one can follows the vertical integration prescription, proposed in [15] by Sen and later elaborated in [16] by Sen and Witten, to avoid such singularities[There is a generalization of the vertical integration procedure by Erler and Konopka developed in [64].].

§.§ Organization of the Paper

This paper is organized as follows. In section <ref>, we discuss the formal construction of off-shell amplitudes in bosonic string theory. In section <ref>, we describe the explicit construction of off-shell amplitudes in bosonic string theory using hyperbolic Riemann surfaces. In section <ref>, we explain the general construction of off-shell amplitudes in superstring theory that are free from infrared and spurious divergences.  In section <ref>, we describe an explicit construction of superstring amplitudes with off-shell external states in terms of hyperbolic Riemann surfaces and the Fenchel-Nielsen coordinates of the Teichmüller space. We end the main part of the paper in section <ref> by mentioning some interesting directions that require further study. In appendix <ref>, we discuss basic facts about hyperbolic Riemann surfaces and their Teichmüller space. Appendix <ref> is devoted to the description of a systematic algorithm for expressing the generators of a Fuchsian group in terms of the Fenchel-Nielsen coordinates of the associated surface.

§ THE OFF-SHELL BOSONIC-STRING AMPLITUDES

In this section, we shall  review the general construction of off-shell amplitudes in bosonic-string theory.

§.§ The Worldsheet Theory

The bosonic string theory is formulated in terms of a conformal field theory (CFT) defined on a Riemann surface. This conformal field theory has two sectors: matter sector and ghost sector. The matter CFT has  central charge $(26,26)$ and the central charge of the reparametrization ghosts CFT is $(-26,-26)$. The ghost system is composed of the anti-commuting fields $b,c,\bar b,\bar c$.  The total CFT has central charge $(0,0)$. The  Hilbert space of CFT is denoted by $\mathcal{H}$. We denote by $\mathcal{H}_0$ a  subspace of $\mathcal{H}$ defined as

\begin{equation}\label{h0bosonic}
|\Psi\rangle\in \mathcal{H}_0,\qquad\mathrm{if}\qquad(b_0-\bar b_0)|\Psi\rangle=0,\quad(L_0-\bar L_0)|\Psi\rangle =0,
\end{equation}

where $\bar L_n$ and $L_n$ denote the total Virasoro generators in the left and the right-moving sectors of the worldsheet theory.  Physical states that appear as external states in the S-matrix elements computation belong to the subspace  $\mathcal{H}_1$  of $\mathcal{H}_0$ satisfying the extra condition

\begin{equation}\label{h1bosonic}
|\Psi\rangle\in \mathcal{H}_1,\qquad\mathrm{if}\qquad |\Psi\rangle\in \mathcal{H}_0,\quad (b_0+\bar b_0)|\Psi\rangle=0,\quad\mathrm{ghost ~number}(|\Psi\rangle) =2.
\end{equation}

We shall denote the BPZ-conjugate of the state $|\phi_i\rangle\in \mathcal{H}_0$  by $\langle \phi_i^c|$ satisfying

\begin{equation}\label{BPZconjugateb}
\langle\phi_i^c|\phi_j\rangle=\delta_{ij},\quad\langle\phi_j|\phi_i^c\rangle=(-1)^{n_{\phi_i}}\delta_{ij},\quad\sum_i|\phi_i\rangle\langle \phi^c_i|=(-1)^{n_{\phi_i}}\sum_i|\phi_j^c\rangle\langle\phi_j|=1,
\end{equation}

where $n_{\phi_i}$ is the ghost number of the state $|\phi_i\rangle$.

§.§ Off-Shell Bosonic-String Amplitudes

The $g$-loop on-shell amplitudes with $n$ external states in bosonic-string theory are obtained by  integrating an appropriate $6g-6+2n$ real dimensional differential form over  $\mathcal{M}_{g,n}$, the moduli space of Riemann surfaces. This differential form on  $\mathcal{M}_{g,n}$ is constructed by computing the CFT correlator of unintegrated vertex operators having conformal dimension $(1,1)$ corresponding to the external states satisfying the tree-level on shell condition.  However, generic states of string theory undergo mass  renormalization. States having masses different from the tree level masses are mapped to vertex operators having conformal dimensions different from $(1,1)$. Therefore,  for generic states in string theory on-shell amplitudes defined using  vertex operators with conformal dimension $(1,1)$ do not compute S-matrix elements beyond tree level. This forces us to consider the off-shell amplitudes constructed using vertex operators of arbitrary conformal dimensions.

Since the  on-shell amplitudes are defined using vertex operators of conformal dimension $(1,1)$, they do not depend on spurious data like the choice of local coordinates around the punctures at which vertex operators are inserted. This means that the integration measure of an on-shell amplitude is a genuine differential form on $\mathcal{M}_{g,n}$. But off-shell amplitudes defined using vertex operators with arbitrary conformal dimensions do depend on the choice of local coordinates around the punctures. Therefore, we can not consider the integration measure of an off-shell amplitude as a genuine differential form on $\mathcal{M}_{g,n}$. Instead, we need to  think of them as differential form defined on a section of a larger space $\mathcal{P}_{g,n}$. This space is  defined as a fiber bundle over $\mathcal{M}_{g,n}$. The fiber direction of  $\mathcal{P}_{g,n}$ corresponds to possible choices  of the local coordinates around the $n$ punctures of a genus-$g$ Riemann surface. If we restrict ourselves to the states that belongs to the Hilbert space $\mathcal{H}_0$, then we can consider the differential form of our interest as defined on a section of space $\widehat{\mathcal{P}}_{g,n}$. This space is smaller compared to $\mathcal{P}_{g,n}$[We mode out the phases of local coordinates. Thus we essentially consider the quotient of $\mathcal{P}_{g,n}$ by the group generated out of rotations of the local coordinates. Therefore, the resulting space, $\widehat{\mathcal{P}}_{g,n}$, is smaller that $\mathcal{P}_{g,n}$.]. We can understand  $\widehat{\mathcal{P}}_{g,n}$ as a base space of the fiber bundle $\mathcal{P}_{g,n}$ with the fiber direction corresponds to the phases of local coordinates.

§.§.§ Tangent Vectors of $\mathcal{P}_{g,n}$

In order to construct a differential form on a space, we need  to first study its tangent space at a generic point. Since we are interested in constructing a differential form on a section of  $\mathcal{P}_{g,n}$, we need to study the  tangent space of $\mathcal{P}_{g,n}$ associated with deformations of the punctured Riemann surface and/or the choice of local coordinates around the punctures. We shall do this using the idea of Schiffer variation [37, 38, 29, 15].

[black,   thick] (0,2) to[curve through=(1,2.2)..(3,3)..(8,0) .. (3,-3) ..(0,-2)  .. (-3,-3)..(-8,0)..(-3,3)..(-1,2.2)] (0,2);
[black,  thick] (2.5,0) to[curve through=(3.5,1)..(5,1)] (6,0);
[black,  thick] (2.3,.2) to[curve through=(3.5,-.75)..(5,-.75)] (6.2,.2);
[black,  thick] (-2.5,0) to[curve through=(-3.5,1)..(-5,1)] (-6,0);
[black,  thick] (-2.3,.2) to[curve through=(-3.5,-.75)..(-5,-.75)] (-6.2,.2);
[black,  very thick] (-5,2) ellipse (.06 and .06);
[black,  very thick] (-5,-2) ellipse (.06 and .06);
[black,  very thick] (7,0) ellipse (.06 and .06);
[black,  very thick] (5,2.1) ellipse (.5 and 1);
[black,  very thick] (5,-1.95) ellipse (.5 and 1.1);
[black,  very thick] (-3.2,1.95) ellipse (.5 and 1);
[black,  very thick] (-3.2,-1.85) ellipse (.5 and 1.1);
[black,  very thick] (0,0) ellipse (.5 and 2);
[black,  very thick] (-7,0) ellipse (1 and .75);
[gray,  very thick] (-5,2.05) ellipse (.7 and .7);
[gray,  very thick] (-5,-1.95) ellipse (.7 and .7);
[gray,  very thick] (7,0) ellipse (.7 and .7);
node[above] at (-5,3) $D_1$;
node[below] at (-5,-3) $D_2$;
node[right] at (8,0) $D_3$;
node[below right] at (-7.5,2.3) $P_1$;
node[above right] at (-7.5,-2.3) $P_2$;
node[above ] at (1.5,-.7) $P_4$;
node[above ] at (-1.5,-.7) $P_3$;
node[below ] at (6.5,-.7) $P_5$;

A pair of pants decomposition of genus-2 surface with 3 punctures. $P_i$ denotes a pair of pants with local coordinate $z_i$ inside $P_i$ for $i=1,\cdots,5$. $D_i$ denotes the disc around the $i^{th}$ puncture with unit radius with respect to the local coordinate  $w_i$ defined around the $i^{th}$ puncture for $i=1,2,3$.

To elucidate the idea of Schiffer variation consider a Riemann surface $\mathcal{R}\in\mathcal{P}_{g,n}$. This means that $\mathcal{R}$ is a genus-$g$ Riemann surface with $n$ punctures and a specific choice of local coordinates around the punctures. We shall denote the local coordinate around the $i^{th}$ puncture by $w_i$  and the disc  around the $i^{th}$ puncture by $D_i$ for $i=1,\cdots,n$. $D_i$ is defined by all $w_i$s that satisfy $|w_i|\le 1$. Now, consider a pair of pants decomposition of $\mathcal{R}-\sum_i D_i$  by choosing $3g-3+n$ homotopically non-trivial disjoint curves on it. This gives $2g-2+n$ number of pairs of pants denoted by $P_i,~i=1,\cdots,2g-2+n$, see figure <ref>. Denote the coordinate inside $P_i$ by $z_i$. Assume that the $i^{th}$ disc $D_i$ shares its boundary $|w_i|=1$ with the $j^{th}$ pair of pants $P_j$. Also, assume that the $k^{th}$ pair of pants $P_k$ shares a boundary with the $m^{th}$ pair of pants $P_m$. Then we have the following transition functions

\begin{align}\label{transitionf1}
z_j&=f_{i}(w_i),\quad\text{analytic for}\quad P_j\cap{D}_i,\quad\text{can have  singularities elsewhere},\nonumber\\
z_k&=f_{km}(z_m),\quad\text{analytic for}\quad P_k\cap{P}_m,\quad\text{can have singularities elsewhere}.
\end{align}

The Schiffer variation generates all deformations of $\mathcal{P}_{g,n}$ by varying the transition functions associated with the discs around the punctures $f_i(w_i),~i=1,\cdots,n$ by keeping all other transition functions $f_{km}(z_m)$ fixed. We can generate such variations by keeping the coordinates $z_k$ inside the pair of pants $P_k,~k=1,\cdots,2g-2+n$ fixed and changing the coordinates inside the disc $D_i$ from $w_i\to w^{\epsilon}_i$ for $i=1,\cdots,n$. This change of coordinates deforms the transition function associated with the disc $D_i$ around the $i^{th}$ puncture as follows

\begin{equation}\label{deformfi}
f_i^{\epsilon}(w_i)=f_i(w_i)-\epsilon v^{(i)}(z_j),~v^{(i)}(z_j)=f'_i(w_i)v^{(i)}(w_i).
\end{equation}

Here, we assumed that the boundary of $D_i$ is shared with the pair of pants $P_j$. The form of $v^{(i)}(w_i)$ can be obtained from the fact that $f^{\epsilon}_i(w_i^{\epsilon})=z_k=f_i(w_i)$. Then, the tangent vector of $\mathcal{P}_{g,n}$ is given by

\begin{equation}\label{tangentvpgn}
\vec{v}(z)=(v^{(1)}(z),\cdots,v^{(n)}(z)).
\end{equation}

$z$ denotes the coordinate on $\mathcal{R}-\sum_i D_i$. The behavior of $v^{(i)}(z)$ on $\mathcal{R}$ determines the kind of deformations it induces on $\mathcal{P}_{g,n}$:

* $\vec{v}(z)$ is a null vector: if it is holomorphic everywhere except possibly at the punctures.

* $\vec{v}(z)$ deforms the local coordinates around punctures: if it is holomorphic inside $D_i,~i=1,\cdots,n$ which vanishes at the punctures and it does not holomorphically extends into $\mathcal{R}-\sum_i D_i$.

* $\vec{v}(z)$ moves the punctures: if it is holomorphic inside $D_i,~i=1,\cdots,n$ and is nonvanishing at the punctures and also it does not holomorphically extends into $\mathcal{R}-\sum_i D_i$.

* $\vec{v}(z)$ generates deformations in the intrinsic moduli of $\mathcal{R}$ which is not associated with punctures: if it has poles at one or more punctures and also it does not holomorphically extends into $\mathcal{R}-\sum_i D_i$. The set of $3g-3$ of such vector fields with poles of order $q=1,\cdots,3g-3$ at any of the punctures generate the complete set of deformations of $\mathcal{M}_g$.

§.§.§ Differential Forms on $\widehat{\mathcal{P}}_{g,n}$

Consider $p$ tangent vectors $V_1,\cdots,V_p$ of $\mathcal{P}_{g,n}$ and let $\vec{v}_1,\cdots,\vec{v}_p$ be the corresponding $n$-tuple vector fields. We can construct an operator-valued $p$-form $B_p$, whose contraction with the tangent vectors $V_1,\cdots,V_p$ is given by

\begin{equation}\label{opevormbos}
B_p[V_1,\cdots,V_p]=b(\vec{v}_1)\cdots b(\vec{v}_p),
\end{equation}

where $b(\vec{v})$ is defined as

\begin{equation}\label{bv}
b(\vec{v})=\sum_i\oint dw_iv^{(i)}(w_i)b^{(i)}(w_i)+\sum_i\oint d\bar w_i\bar v^{(i)}(\bar w_i)\bar b^{(i)}(\bar w_i).
\end{equation}

$b,\bar b$ denote the anti-ghost fields. We define the $p$-form on $\mathcal{P}_{g,n}$ as

\begin{equation}\label{pformpgnbos}
\Omega_p^{(g,n)}(|\Phi\rangle)=(2\pi \mathrm{i})^{-(3g-3+n)}\langle\mathcal{R}|B_p|\Phi\rangle.
\end{equation}

Here, $|\Phi\rangle$ is some element of $\mathcal{H}^{\otimes n}$ with ghost number

$\langle \mathcal{R}|$ is the surface-state associated with the surface $\mathcal{R}$, which describes the state that is created on the boundaries of $D_i$ by performing a functional integral over the fields of the CFT on $\mathcal{R}-\sum_iD_i$.  Inner product between $\langle\mathcal{R}|$ and a state $|\Psi_1\rangle\otimes\cdots\otimes |\Psi_n\rangle\in\mathcal{H}^{\otimes n}$

\begin{equation}\label{innerprcft}
\langle\mcal{R}|(|\Psi_1\rangle\otimes\cdots\otimes |\Psi_n\rangle),
\end{equation}

describes the $n$-point correlation function on $\mathcal{R}$ with the vertex operator for $|\Psi_i\rangle$ inserted at the $i^{th}$ puncture using the local coordinate system $w_i$ around that puncture. It is clear that $\Omega_p^{(g,n)}(|\Phi\rangle)$ is a $p$-form on $\mathcal{P}_{g,n}$. Remember that a $p$-form on a space generates a number when contracted with $p$ tangent vectors of this space and this number is antisymmetric under the exchange of any pair of tangent vectors. Since anti-ghost fields $b,\bar b$ are anti-commuting, $\Omega_p^{(g,n)}(|\Phi\rangle)$ also has this property and, hence, $\Omega^{(g,n)}_p$ is a $p$-form on $\mathcal{P}_{g,n}$.It is argued in [65] that constructing consistent off-shell string measure requires us to restrict $|\Phi\rangle$ to be an element of $\mathcal{H}_0^{\otimes n}$. In other words, the $|\Phi\rangle$ has to satisfy the following conditions

\begin{equation}\label{eq:conditions on off-shell states b_0 L_0}
(b_0-\bar b_0)|\Phi\rangle=(L_0-\bar L_0)|\Phi\rangle=0.
\end{equation}

These conditions mean that the state $|\Phi\rangle$ is insensitive to the phase of the chosen local coordinates. As a result, the $p$-form $\Omega^{(g,n)}_p$ is independent of the phase of local coordinates around the punctures. In order to check this claim, note that a tangent vector that generates a phase rotation for the local coordinates has the following nonvanishing components, $v^{(i)}(w_i)=w_i$ and $\bar v^{(i)}(\bar w_i)=-\bar w_i$. It is clear that such vectors do not change $\Omega_p^{(g,n)}(|\Phi\rangle)$ if $|\Phi\rangle\in \mathcal{H}_0$. This is because for  $\vec{v}=(0,\cdots,w_i,\cdots,0)$, $b(\vec{v})=b^{(i)}_0-\bar b^{(i)}_0$ and $T(\vec{v})=L^{(i)}_0-\bar L^{(i)}_0$, where the superscripts denote that the modes  $b_0,L_0$ and their complex conjugates are defined with respect to the local coordinates inside the $i^{th}$ puncture. Here

\begin{equation}\label{tv}
T(\vec{v})=\sum_i\oint dw_iv^{(i)}(w_i)T^{(i)}(w_i)+\sum_i\oint d\bar w_i\bar v^{(i)}(\bar w_i)\bar T^{(i)}(\bar w_i).
\end{equation}

Since $(b_0-\bar b_0)|\Phi\rangle=(L_0-\bar L_0)|\Phi\rangle=0$, the  change in  $\Omega_p^{(g,n)}(|\Phi\rangle)$ due to the phase rotation of local coordinates vanishes. Therefore, consistent off-shell string measure are $p$-forms on $\widehat{\mathcal{P}}_{g,n}$, where $\widehat{\mathcal{P}}_{g,n}$ is the space obtained by identifying points in ${\mathcal{P}}_{g,n}$ that are related by phase rotations of local coordinates.

§.§.§ Differential Form on Sections of $\widehat{\mathcal{P}}_{g,n}$

Let us discuss the construction of tangent vectors on a section of $\widehat{\mathcal{P}}_{g,n}$ where choosing a section corresponds to a specific choice of local coordinates. We can move along a section by changing the moduli parameters. Therefore, in order to construct a $p$-form on a section of $\widehat{\mathcal{P}}_{g,n}$, we need to consider only those tangent vectors that give rise to variations of moduli parameters.

Consider a genus-$g$ Riemann surface $\mcal{R}$ with $n$ punctures. Let us divide $\mcal{R}$ into $2g-2+2n$ patches, where $n$ patches are obtained by considering small disks $D_i$s around the $n$ punctures, and $2g-2+n$ remaining patches are pairs of pants obtained by pair-of-pants decomposition of $\mcal{R}-\sum_iD_i$. We denote the local coordinate on the mth patch by $z_m$. Since there are $3g-3+2n$ number of interfaces between these patches ($n$ interfaces between the disks and the remaining pairs of pants, and $3g-3+n$ interfaces between the pairs of pants). We thus have $3g-3+2n$ number of transition functions. As a result, tangent vectors of $\pcab$ constructed using the idea of Schiffer variation will have $3g-3+2n$ number of components.

Denote the real coordinates of the moduli space of genus-$g$ Riemann surface with $n$ punctures by $(\mfk{t}^{(1)},\cdots,\mfk{t}^{(6g-6+2n)})$. Assume that the $m$th and $n$th coordinate patches have a non-empty intersection along the contour $C_{mn}$ runs between them. For the $(3g-3+2n)$-tuple vector field  $\vec{v}_{k}=\left(\cdots,\frac{\partial z_m}{\partial \mfk{t}^{(k)}} \Big|_{z_n},\cdots\right)$ corresponding to the variation in the $\mfk{t}^{(k)}$ moduli, $b(\vec{v}_k)$ defined in (<ref>) is given by

\begin{equation}\label{bvgen}
b(\vec{v}_{k})=\sum_{(mn)}\oint_{C_{mn}}\left(dz_m\left.\frac{\partial z_m}{\partial \mfk{t}^{(k)}}\right|_{z_n}b_{z_mz_m}-d\bar z_{m}\left.\frac{\partial \bar z_m}{\mfk{t}^{(k)}}\right|_{z_n}b_{\bar z_m\bar z_m}\right),
\end{equation}

where the sum over $(mn)$ denotes the summation over the number of overlaps between coordinate patches. Then, the change in the local coordinates $z_m$ and $z_n$ under a change in the moduli $\mfk{t}^{(k)}$ are related as follows [30]

\begin{equation}\label{modtvar}
\left.\frac{\partial z_m}{\partial \mfk{t}^{(k)}} \right|_{z_n}=v_{km}^{z_m}-\left.\frac{\partial z_m}{\partial z_n}\right|_{\mfk{t}}v_{kn}^{z_n}=v_{km}^{z_m}-v_{kn}^{z_m},
\end{equation}

where $v_{km}^{z_m}=\frac{d z_m}{d \mfk{t}^{(k)}}$. From (<ref>), we see that

\begin{equation}\label{bvgen1}
b(\vec{v}_{k})=\sum_{m=1}^{2g-2+2n}\oint_{C_{m}}\left(dz_mv^{z_m}_{km}b_{z_mz_m}-d\bar z_{m}v^{\bar z_m}_{km}b_{\bar z_m\bar z_m}\right).
\end{equation}

Using Stokes Theorem we get

\begin{equation}\label{bvgen2}
b(\vec{v}_{k})=\int d^2z\Big(b_{zz}\mu_{k\bar z}^z+b_{\bar z\bar z}\mu_{kz}^{\bar z}\Big).
\end{equation}

Here $\mu_k$ denotes the Beltrami differential associated with the moduli $t^{(k)}$ which is related to the coordinate changes via the infinitesimal version of the Beltrami equation

μ_kz_m^z̅_m   =∂_z_mv_km^z̅_m,

μ_kz̅_m^z_m   =∂_z̅_mv_km^ z_m.

§.§.§ Gluing-Compatible Integration Cycle

Off-shell amplitudes in bosonic string theory are given by integrating top-degree  differential forms on a section of $\widehat{\mathcal{P}}_{g,n}$. These forms are built using states $|\Phi\rangle\in\mathcal{H}_1^{\otimes n}$, defined in (<ref>), representing the off-shell external states. In particular the ghost number of $|\Phi\rangle$ is $2n$. So the degree of relevant differential form is $p=6g-6+2n$. This matches with the dimension of moduli space of the genus-$g$ Riemann surface $\mathcal{R}$ with $n$ punctures. Although the degree of $\Omega_{6g-6+2n}^{(g,n)}(|\Phi\rangle)$ is equal to the dimension of $\mathcal{M}_{g,n}$, we can not regard it as a genuine top-form on $\mathcal{M}_{g,n}$. This reason is that $\Omega_{6g-6+2n}^{(g,n)}(|\Phi\rangle)$ depends on the choice of local coordinates around the punctures and is non-zero when a tangent vector $\vec{v}$ generates deformations of local coordinate without varying the surface $\mathcal{R}$.  Therefore, $\Omega_{6g-6+2n}^{(g,n)}(|\Phi\rangle)$ is sensitive to the choice of section of $\pcab$

However, the physical quantities that can be extracted from off-shell amplitudes like the renormalized masses and the S-matrix elements must be independent of the choice of section of $\pcab$ [15]. This is true only if we  impose a condition on the choice of this section, known as gluing-compatibility [14, 13]. To describe this condition, consider two Riemann surfaces $\mathcal{R}_1$ and $\mathcal{R}_2$. $\mathcal{R}_1$ is a genus-$g_1$ surface with $n_1$ punctures and $\mathcal{R}_2$ is a genus $g_2$ surface with $n_2$ punctures. Denote the collection of pairs of pants in the pair-of-pants decomposition of $\mathcal{R}_1$ by $\{P^{(1)}_k\}$ and the discs around punctures by $D_1^{(1)},\cdots,D_{n_1}^{(1)}$. Similarly denote the collection of pairs of pants in the pair-of-pants $\mathcal{R}_2$ by $\{P^{(2)}_k\}$ and the discs around punctures by $D_1^{(2)},\cdots,D_{n_2}^{(2)}$. We can glue discs $D_i^{(1)}$ and  $D_j^{(2)}$ using the plumbing-fixture relation:

\begin{equation}
w_i^{(1)}w_j^{(2)}=e^{-s+\mathrm{i}\theta},\qquad 0\leq s<\infty,\quad 0\leq\theta<2\pi. \label{plumb}
\end{equation}

This will produce another a surface $\mathcal{R}$ with genus $g=g_1+g_2$ and $n=n_1+n_2-2$ punctures. Surfaces that can be constructed this way belong to the region near the boundary of $\mathcal{M}_{g,n}$. This part of $\mathcal{M}_{g,n}$ can be parametrized by the coordinates of $\mathcal{M}_{g_1,n_1}$, $\mathcal{M}_{g_2,n_2}$ and $(s,\theta)$. The gluing-compatibility condition requires that the section in fiber bundle $\widehat {\mathcal{P}}_{g,n}$ over this region of moduli space be chosen such that the relationship between coordinates of $\{P_k^{(1)}\}$ and $D_1^{(1)},\cdots,\cancel{D_i^{(1)}},\cdots, D^{(1)}_{n_1}$ depends only on the moduli of $\mathcal{M}_{g_1,n_1}$ and not on the moduli of $\mathcal{M}_{g_2,n_2}$ and $(s,\theta)$. Similarly  the relation between coordinates of $\{P_k^{(2)}\}$ and $D_1^{(2)},\cdots,\cancel{D_j^{(2)}},\cdots, D^{(2)}_{n_2}$ depends only on the moduli of $\mathcal{M}_{g_2,n_2}$ and not on the moduli of $\mathcal{M}_{g_1,n_1}$ and $(s,\theta)$. Also, the dependence of these relations on the moduli of $\mathcal{M}_{g_l,n_l}$ must be the one induced from the choice of section $\widehat{\mathcal{P}}_{g_l,n_l}$ for $l=1,2$.

§ HYPERBOLIC GEOMETRY AND OFF-SHELL BOSONIC-STRING AMPLITUDES

In this section, we shall explicitly construct off-shell  amplitudes in bosonic-string theory using hyperbolic Riemann surfaces following the general construction described in section <ref>. For a quick introduction into the theory of hyperbolic Riemann surfaces and their moduli space see appendix <ref>.

§.§ A Gluing-Compatible Section of $\widehat{\mathcal{P}}_{g,n}$

A Riemann surface is called hyperbolic if it is equipped with a hyperbolic metric, i.e. a metric with constant curvature $-1$ everywhere on the surface.
One important advantage of using hyperbolic metric is that Riemann surfaces with nodes obtained by the plumbing fixture of hyperbolic surfaces are again hyperbolic.  This suggests that if we choose local coordinates around the punctures as the one induced from the hyperbolic metric on the surface, at least on the  complete degeneration limit, where the plumbing parameter vanishes and a node is developed, this choice will match with local coordinates induced from the component surfaces. This is an essential constraint satisfied by a gluing-compatible section of $\widehat{\mathcal{P}}_{g,n}$.  We shall argue in the rest of this section that  away from the degeneration locus also we can  define  a consistent choice of local coordinate that leads to a gluing-compatible section of $\widehat{\mathcal{P}}_{g,n}$.

A hyperbolic Riemann surface can be represented  as a quotient of the upper half-plane $\mathbb{H} $ by a Fuchsian group.  A puncture on a hyperbolic Riemann surface corresponds to the fixed point of a parabolic element of the Fuchsian group acting on the upper half-plane $\mathbb{H} $. For a puncture $p$, there is a natural local coordinate $w$ with $w(p) = 0$ and the hyperbolic metric around the puncture is locally given by [44]

\begin{equation}
\end{equation}

Let $z$ be the coordinate on upper half-plane. Then, the distinguished local coordinate is given as

\begin{equation}
w = e^{2\pi\mathrm{i}z}, \label{local coordinate for the cusp at infinity}
\end{equation}

for the puncture that corresponding parabolic element whose fixed point is at infinity $z_{\infty}=i\infty$ on the upper half-plane $\mathbb{H}$. As required this choice of local coordinate  is invariant under the translation, $z \to z + 1$, which represents  the action of the generator of the corresponding parabolic element.  In terms of coordinate $z$, the metric around the puncture takes the form

\begin{equation}
\end{equation}

which is the  hyperbolic metric for the Poincaré upper half-plane $\mathbb{H}$, as it should be.  The local canonical coordinate around the puncture $w$ given by (<ref>) is unique modulo a phase factor.  If the fixed point of the parabolic element corresponding to the cusp is at a finite point $x$  on the real axis of upper-half plane, then the local coordinate is given by

\begin{equation}
w = e^{-\frac{2\pi\mathrm{i}}{z-x}}. \label{local coordinate for cusp at finite real value $x$}
\end{equation}

Before using the proposed choice of local coordinates for constructing the off-shell amplitudes, it is important to ensure that it satisfies the gluing-compatibility  requirement.  For this, we should first answer the following question: What are the local coordinates induced around the punctures on the surface obtained via plumbing fixture of hyperbolic Riemann surfaces? To proceed, let us analyze the metric on Riemann surfaces obtained via the plumbing fixture of surfaces which are hyperbolic.

§.§.§ Plumbing Fixture and Hyperbolic Riemann Surfaces

The degenerating families of Riemann surfaces are readily given by cut-and-paste constructions in hyperbolic geometry, following Fenchel and Nielsen. Below, we shall discuss the relation between the cut-and-paste construction and the plumbing-fixture construction of degenerating families of hyperbolic Riemann surfaces. For more details on the cut-and-paste construction and Fenchel-Nielsen coordinates, see section <ref> in the appendices.

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[line width=1pt,color=black] (0.35,-0.5) ellipse (.085 and .115)node  [below ] $\gamma_2$;
[line width=1pt] (-1,0) .. controls(-.7,-.25) ..(1,-0.5)node[above] $p$;
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[line width=1pt] (4,0) .. controls(3.5,-.4) ..(2,-0.5)node[above] $q$;
[line width=1pt] (4,-1) .. controls(3.5,-.65)  ..(2,-0.5);
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(-3.75,.5) node[above ] $\beta_2$  (-3.75,-2) node[above] $\beta_1$ (8.2,-.2)node [below right ] $\beta_3$;
[->,line width =1pt] (9.5,-.5)–(10.5,-.5);

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(12.15,.5) node[above ] $\beta_2$  (12.15,-2) node[above ] $\beta_1$ (21.2,-.2)node [below right ] $\beta_3$;

The plumbing fixture of two surfaces at punctures $p$ and $q$.

The Plumbing-Fixture Construction: Consider $\overline{\mathcal{M}}_g$, the Deligne-Mumford stable curve compactification of the  moduli space of genus-$g$ Riemann surfaces [40]. Let us denote the compactification divisor of $\overline{\mathcal{M}}_g$ by $\mathcal{D}$. A point of $\mathcal{D}$ represents a Riemann surface $\mathcal{R}$ with nodes. By definition, a neighborhood of a node $\mfk{a}$ of $\mathcal{R}$ is complex isomorphic to either $\{|w^{(1)}|<\epsilon\}$ or

\begin{equation}
\end{equation}

where $w^{(1)}$ and $w^{(2)}$ are the local coordinates around the  two sides of $\mfk{a}$. To move away from the compactification divisor, let us consider the following family of surfaces fibered over a disk with complex coordinate $t$

\begin{equation}
\left\{w^{(1)}w^{(2)}=t|~|w^{(1)}|,|w^{(2)}|<\epsilon,|t|<\epsilon\right\}.
\end{equation}

We can identify $U$ with the fiber at $t=0$. A deformation of $\mathcal{R}\in \overline{\mathcal{M}}_g$ which opens the node is given by varying the parameter $t$. Consider a Riemann surface $\mathcal{R}_0\in \mathcal{D}\subset \overline{\mathcal{M}}_g$ with $m$ nodes denoted by $\mfk{a}_1,\cdots,\mfk{a}_m$. For the node $\mfk{a}_i$, the punctures $p_i$ and $q_i$ of $\mathcal{R}_0-\{\mfk{a}_1,\cdots,\mfk{a}_m\}$ are paired. Let

\begin{equation}
U_i^1=\left\{|w_i^1|<1\right\}, \quad U_i^2=\left\{|w_i^2|<1\right\};~i=1,\cdots, m,
\end{equation}

be the disjoint neighborhoods of the punctures $p_i$ and $q_i$, respectively. Here, $w^{(1)}_i$ and $w^{(2)}_i$ with $w_i^{(1)}(p_i)=0$ and $w_i^{(2)}(q_i)=0$ are the local coordinates around the two sides of the node $\mfk{a}_i$. Consider an open set $\mathcal{V}\subset \mathcal{R}_0$ disjoint from the set $U_i^1,U_i^2$ which support the Beltrami differentials $\{\mu_a\}$. The Beltrami differentials span the tangent space of the Teichmüller space of $\mathcal{R}_0-\{\mfk{a}_1,\cdots,\mfk{a}_m\}$. The dimension of this space is $3g-3-m$. Given
$$s=(s_1,\cdots,s_{3g-3-m})\in \mathbb{C}^{3g-3-m},$$
for a neighborhood of the origin, the sum  $\mu(s)=\sum_js_j\mu_j$  is a solution $\omega^{\mu(s)}$ of the Beltrami equation. Assume that the surface $\omega^{\mu(s)}(\mathcal{R}_0)=\mathcal{R}_s$ is a quasiconformal deformation of $\mathcal{R}_0$.  Then, we shall parametrize the opening of nodes as follows. The map $\omega^{\mu(s)}$ is conformal on $U_i^1$ and $U_i^2$ and therefore $w_i^{(1)}$ and $w_i^{(2)}$ serve as local coordinates for $\omega^{\mu(s)}(U_i^1),~\omega^{\mu(s)}(U_i^2)\subset \omega^{\mu(s)}(\mathcal{R}_0)$. Given
$$ t=(t_1,\cdots,t_m)\in \mathbb{C}^m,~|t_i|<1,$$
we construct the family of surface $\mathcal{R}_{t,s}$, parametrized by $s$ and $t$ as follows. We first remove the discs $\{0<|w_i^{(1)}|\leq |t_i|\}$ and $\{0<|w^{(2)}_i|\leq |t_i|\}$ from $\mathcal{R}_s$ (see figure <ref>), and then attach $\{|t_i|<|w^{(1)}_i|< 1\}$ to $\{|t_i|<|w^{(2)}_i|< 1\}$ by identifying $w^{(1)}_i$ and $\frac{t_i}{w^{(2)}_i}$. The construction is complex, and the tuple $(t,s)$ parametrizing $\mathcal{R}_{t,s}$ provides a local complex coordinate chart.

[thick,color=black] (3,-2.5) ellipse (2 and 2)(3,-1)node$D_2$;
[thick,color=gray] (3,-2.5) ellipse (1 and 1);

[thick,color=gray] (-3,-2.5) ellipse (2 and 2);
[thick,color=black] (-3,-2.5) ellipse (1 and 1);

Two annuli with inner radius $|t|$ and outer radius 1 obtained by removing a disc of radius $|t|$ from $D_1$ and $D_2$ where $t$ is a complex parameter.

Plumbing Fixture of Hyperbolic Riemann Surfaces: Now we shall discuss the plumbing fixture of Riemann surfaces with hyperbolic metric. For a geodesic $\alpha$ on the hyperbolic surface $\mathcal{R}$ of length $l_{\alpha}$, a neighborhood with area $2l_{\alpha}\cot\frac{l_{\alpha}}{2}$ is called the collar around $\alpha$.  The standard collar around the geodesic $\alpha$  is the  collection of points $p$ whose hyperbolic distance from $\alpha$ is less than  $w(\alpha)$ given by

\begin{equation}
\sinh w(\alpha)\sinh\frac{l_{\alpha}}{2}=1.
\end{equation}

The standard collar can be described as a quotient of the upper half-plane $\mathbb{H}$. To describe this quotient space, consider the transformation $z\to e^{l_{\alpha}}z$ which is represented by the following matrix in PSL$(2,\mathbb{R})$

\begin{equation}
M=\left(\begin{array}{cc}e^{\frac{l_{\alpha}}{2}} & 0 \\0 & e^{-\frac{l_{\alpha}}{2}}\end{array}\right).
\end{equation}

It is clear from this form that it generates a cyclic subgroup of PSL$(2,\mathbb{R})$. We shall denote this subgroup by $\Gamma_{\alpha}$. The fundamental domain is given by a strip in $\mathbb{H}$ (see figure <ref>). If we quotient $\mathbb{H}$ with $z\to e^{l_{\alpha}}z$ relation, we identify the two sides of the strip. This gives a cylinder which is topologically  an annulus which has an induced hyperbolic structure from $\mathbb{H}$. The simple closed geodesic of this hyperbolic annulus has hyperbolic length $l_{\alpha}$. The standard collar approximately the  quotient of the wedge $\left\{\frac{l_{\alpha}}{2}<\mathrm{arg} z<\pi-\frac{l_{\alpha}}{2}\right\}$ by $\Gamma_{\alpha}$ (see figure <ref>).

[<->,line width=.25pt](-3,0)–(11,0);
[->,line width=.25pt](4,0)–(4,5);

[line width=1pt] (1,0) .. controls (2,3) and (6,3)  ..(7,0);
[line width=1pt] (-1,0) .. controls (.5,5) and (7.5,5)  ..(9,0);

[line width=1pt, color=gray] (1.25,3)node[above left]$\pi-l$–(4,0) – (6.75,3)node[above right]$l$;

The fundamental domain of a collar is the region between the annulus and the wedge.

The standard collar or  hyperbolic annulus can be constructed via the plumbing fixture of two discs $D_1=\{|y_1|<1\}$ and $D_2=\{|y_2|<1\}$ and by endowing a hyperbolic metric on it.   The  plumbing fixture locus

\begin{equation}\label{plumbingfiber}
\mathcal{F}=\{y_1y_2=t\Big|~|y_1|,|y_2|,|t|<1\}
\end{equation}

is a complex manifold fibered  over the disk $D=\{|t|<1\}$. The $t=0$ fibre is singular. It is the union of the discs $D_1$ and $D_2$ joined at the origin. To obtain a hyperbolic metric, we need to remove the origin from $D_1$ and $D_2$. Each of the punctured disk has a complete hyperbolic metric given by

\begin{equation}\label{hypemetricd}
ds^2_0=\left(\frac{|dy_1|}{|y_1|\ln|y_1|}\right)^2\qquad \{0<|y_1|<1\}\cup \{0<|y_2|<1\}.
\end{equation}

The  $t\neq 0$ fibres are the annuli $\{|t|<|y_1|<1\}$ with complete hyperbolic metric,

\begin{equation}\label{hype}
ds_t^2=\text{sinc}^{-2}\left(\frac{\pi\ln|y_1|}{\ln|t_j|} \right)ds_0^2,
\end{equation}

in the $j^{th}$ collar of $\mathcal{R}_t$, where sinc is the normalized sinc function given by $\text{sinc}(x)=\frac{\text{sin}(\pi x)}{\pi x}$. For small $|t|$, we have the following expansion of the hyperbolic metric on the punctured disc

\begin{equation}
ds^2_t= \left(1+\frac{1}{3}\Theta^2+\frac{1}{15}\Theta^4+\cdots\right)ds^2_0, \qquad \{|t|<|y_1|<1\},
\end{equation}

where $\Theta\equiv\pi\frac{\ln |y_1|}{\ln|t|}$. It is clear form this expansion that the limit $t\longrightarrow 0$ of $ds_t^2$ is real analytic in $(\ln\frac{1}{|t|})^{-1}$ and uniform for $|y_1|,|y_2|>\epsilon$.

The closed geodesic in the $t$-fiber is $|y_1|=|t|^{\frac{1}{2}}$ and has the length

\begin{equation}\label{eq:length of the core geodesics}
\end{equation}

It is known that  there exists a positive constant $c_*$ such that if the length $l$ of a geodesic $\gamma$ on a hyperbolic Riemann  surface $\mathcal{R}$ is less than or equal to $c_*$, then the standard collar embeds isometrically about $\gamma$ [49, 48]. This constant $c_*$ is known as the collar constant. We shall call a geodesic whose length is at most $c_*$ a short geodesics. Therefore, whenever the length of a simple geodesic along which the cut-and-paste construction can be done becomes less than the collar constant, we can replace the collar around this short geodesic with the hyperbolic annulus. We can then interpret this collar as a plumbing collar. This allows us to find the needed bridge between the plumbing-fixture and the cut-and-paste construction. The disc $\mathscr{D}=\{|t|<1\}$ can be thought of as the moduli space for the hyperbolic annulus. The Riemannian Weil-Petersson (WP) metric on $\mathscr{D}$ is given by [39]

\begin{equation} \label{WP metric on the moduli space of the Riemann surfaces}
\end{equation}

We can identify the Fenchel-Nielsen coordinates $(\ell,\tau)$ for the moduli space of hyperbolic annulus as follows:

ℓ   =-2π^2/ln|t|,

2πτ/ℓ   =arg t=Im(ln t).

If we use these relations, then (<ref>) can be written as $d\ell\wedge d\tau$. This perfectly agrees with the Wolpert's formula for the WP volume form on the moduli space of hyperbolic Riemann surfaces. This volume form can be obtained via the WP symplectic form [47].This identification can be generalized to the case of a hyperbolic surface with $m$ disjoint short geodesics. The collar neighborhood  of each of the short geodesics can be interpreted as a plumbing collar.  Given an $\epsilon>0$, we have the following estimate for the length $l_i$ of the simple closed geodesic of the $t_i$th annulus [39],

\begin{equation}
\left|\frac{2\pi^2}{l_i}-\ln\frac{1}{|t_i|}\right|<\epsilon,\quad i=1,\cdots,m.
\end{equation}

Therefore,  we see that the degeneration of a hyperbolic metric is associated to the formation of  wide collars about short geodesics. From the above discussion, it is tempting to claim that a hyperbolic Riemann surface near the boundary of the moduli space can be understood as a surface obtained by the plumbing fixture of Riemann surfaces with hyperbolic metric.

§.§.§ A Proposal for a Gluing-Compatible Section of $\pcab$

Before specifying a gluing-compatible section, let us first define the 1PI and 1PR regions of the moduli space:

1PI and 1PR Regions of the Moduli Space: the region inside the moduli space consists of hyperbolic Riemann surfaces having no simple closed geodesics with length less than or equal to $c_*$ is declared as the 1PI region of the moduli space. The region inside the moduli space that is complement to the 1PI region is declared as the 1PR region.

Here, we assume that $c_*$ is a small constant which is much less than the hyperbolic length of any simple closed geodesic on the surface  that is  homotopically-inequivalent to the simple closed geodesic on the plumbing collars. The rationale behind this assumption will be explained in section <ref>. With these definitions of 1PI and 1PR regions of the moduli space, we choose the local coordinates as the one that is induced from the hyperbolic metric on the surface with the scaling factor of $e^{\frac{\pi^2}{c_*}}$. Let us briefly explain the need for the scaling factor. The 1PR region of the moduli space should be constructed using the plumbing fixture in which the plumbing parameter runs between $0$ and $1$. On the other hand, it is clear from the equation (<ref>) that our definition of 1PR region of the moduli space consists of all Riemann surfaces

\begin{equation}
\left\{w_i^{(1)}w_i^{(2)}=t_i~|~|t_i|\leq |w_i^{(1)}|,|w_i^{(2)}|<1,~|t_i|<e^{-\frac{2\pi^2}{c_*}},~i=1,\cdots,m\right\}.
\end{equation}

However we can make the plumbing parameter to run from $0$ to $1$ by rescaling the local coordinates $w_i^{(1)}$ and $w_i^{(2)}$ as follows

\begin{equation}\label{rescaled coordinate}
\widetilde{w}_i^{(1)}\equiv e^{\frac{\pi^2}{c_*}}w_i^{(1)}\qquad \widetilde{ w}_i^{(2)}\equiv e^{\frac{\pi^2}{c_*}}w_i^{(2)}, \qquad i=1,\cdots,m.
\end{equation}

In terms of this rescaled coordinate, the family of surfaces in the 1PR region is constructed by the following plumbing-fixture relation

\begin{equation}
\left\{\widetilde{w}_i^{(1)}\widetilde{w}_i^{(2)}=\widetilde{t}_i~|~|\widetilde {t}_i|\leq|\widetilde{w}^{(1)}_i|,|\widetilde{ w}^{(2)}_i|<e^{\frac{\pi^2}{c_*}},~|\widetilde{t}_i|<1,~i=1,\cdots,m\right\}.
\end{equation}

Note that this definition does not fix the phase of the local coordinates. Such a phase ambiguity will not affect the definition of off-shell amplitudes due to the condition (<ref>) satisfied by the off-shell states. This choice provides a gluing-compatible section of $\pcab$ only if the plumbing-fixture construction applied to one or more hyperbolic Riemann surfaces gives a hyperbolic metric on the resulting surface. Therefore, we must first determine the metric on the plumbing family.

§.§.§ The Hyperbolic Metric on the Plumbing Family

Consider a hyperbolic Riemann surface $\mcal{R}_0$ with $m$ nodes. Using plumbing fixture construction, we can construct a family of non-degenerate Riemann surfaces $\mcal{R}_t$ parametrized by $m$-tuple $t\equiv (t_1,\cdots,t_m)$ from the degenerate hyperbolic Riemann surface $\mcal{R}_0$. By removing the $m$ nodes from $\mcal{R}_0$, we get a hyperbolic surface $\mcal{\widehat{R}}_0$. On $\widehat{\mcal{R}}_0$, we have a pair of punctures $p_i,q_i$  instead of the node $\mfk{a}_i$ for $i=1,\cdots,m$. The surface $\widehat{\mathcal{R}}_0$ is a finite union of Riemann surfaces with punctures. We denote the local coordinates around the punctures $p_i$ and $q_i$ induced from the hyperbolic metric by $w^{(1)}_i$ and $w^{(2)}_i$ with the property that $w^{(1)}_i(p_i)=0$ and $w^{(2)}_i(q_i)=0$.

Let us analyze the metric on $\mcal{R}_t$. In terms of the local coordinates $w^{(1)}_i$ and $w^{(2)}_i$, the hyperbolic metric on $\widehat{\mathcal{R}}_0$ has the following local expression

\begin{equation}\label{eq:hyperbolic metric around the punctures}
ds^2=\left(\frac{|d\zeta|}{|\zeta|\ln|\zeta|}\right)^2,\qquad \zeta=w^{(1)}_i,~\zeta=w^{(2)}_i.
\end{equation}

Using (<ref>), it is straightforward to check that there is no geodesic around the punctures. Applying the plumbing-fixture construction to these punctures identifies the curves $w_i^{(1)}=\sqrt{t}$ and $w_i^{(2)}=\sqrt{t}$ in the neighborhood of the punctures. We can obtain a hyperbolic surface by gluing two hyperbolic surfaces only if the component surfaces glued along a geodesic. However, the curves identified by the plumbing fixture construction are not geodesics. Therefore, the resulting surface $\mcal{R}_t$ cannot be a hyperbolic surface. In fact, one can check that the metric on $\mcal{R}_t$ has constant curvature $-1$ everywhere except along the $w_i^{(1)}=w_i^{(2)}=\sqrt{t}$. Therefore, the plumbing collar does not have hyperbolic metric on it.

We can make the metric on a plumbing collar in $\mcal{R}_t$ hyperbolic by performing a conformal transformation. In terms of local coordinates $w^{(1)}_i$ and $w^{(2)}_i$, the resulting metric on the $i$th plumbing collar takes the following form

\begin{equation}\label{hype1}
ds_t^2=\text{sinc}^{-2}\left(\frac{\pi\ln|\zeta|}{\ln|t_i|} \right)ds_0^2
\qquad~ \zeta=w^{(1)}_i~\text{or}\quad\zeta=w^{(2)}_i.
\end{equation}

As a result of this conformal transformation, the metric on the collars of $\mcal{R}_t$ become hyperbolic. The metric away from the plumbing collars remains hyperbolic as before the transformation. However, these two metrics do not match with each other in the tails of plumbing collars. Therefore, we interpolate between the two choices at the two ends of the plumbing collars.  The result is a smooth grafted metric $ds_{graft}^2$ for $\mathcal{R}_t$, see figure <ref>. The grafted metric has curvature $-1$ everywhere except at the tails of plumbing.

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(16.2,2.5) node[above]the hyperbolic metric on $\mathcal{R}_0$;
(16.2,1.5) node[above]$\substack{\text{\fontsize{12}{12}\selectfont the interpolating} \\ \text{\fontsize{12}{12}\selectfont metric}}$;
(16.2,-3) node[above]the metric on hyperbolic annulus;
(12.15,.5) node[above ] $\beta_2$  (12.15,-2) node[above ] $\beta_1$ (21.2,-.2)node [below right ] $\beta_3$  (16,0.2) node  [below ] $\gamma$;

The grafted metric on the surface obtained via the plumbing fixture.

Since we have found a smooth metric on the glued surface, we can make it hyperbolic by performing an appropriate conformal transformation. The proper conformal transformation that does this job can be found by solving the so-called the curvature correction equation [41, 42, 43]. To describe this equation, consider a compact Riemann surface with the metric $ds^2$ and the Gauss curvature[In two dimension, the Gaussian curvature is half of the Ricci curvature of the surface.] $\mathbf{ C}$.  Then, the  metric $e^{2f}ds^2$ on this surface has constant curvature $-1$ if

\begin{equation}\label{constantcurvatureq}
Df-e^{2f}=\mathbf{ C},
\end{equation}

where $D$ is the Laplace-Beltrami operator on the surface.

In order to find the  hyperbolic metric on $\mathcal{R}_t$, we need to specify the grafted metric properly. For this, let us introduce a positive constant $b_*$ and a negative constant $a_0$. The grafted metric $ds_{graft}^2$ on $\mathcal{R}_t$ is defined as follows:

* In the region complement to the region in $\mathcal{R}_t$ described by

\begin{equation}
\mathcal{F}_{b_*}=\left\{\left(w_i^{(1)},w_i^{(2)},t_i\right)\left|~w_i^{(1)}w_i^{(2)}=t_i,|w_i^{(1)}|,|w_i^{(2)}|<b_*;~i=1,\cdots,m\right\}\right.,
\end{equation}

we use the hyperbolic metric $ds^2$ on $\widehat{\mathcal{R}}_0$ restricted to $\mathcal{R}_{b_*}$. The surface $\mathcal{R}_{b_*}$ is obtained from $\widehat{\mathcal{R}}_0$ by removing the punctured discs $\{0<|w_i^{(1)}|\leq b_*\}$ about $p_i$ and $\{0<|w_i^{(2)}|\leq b_*\}$ about $q_i$ for $i=1,\cdots,m$.

* In the region in  plumbing collar  $ \mathcal{F}_{b_*}$ that is complement to the  collar bands described by the

\begin{equation}
e^{a_0}b_*\leq |w_i^{(1)}|\leq b_*,\quad e^{a_0}b_*\leq |w_i^{(2)}|\leq b_*;~i=1,\cdots,m,
\end{equation}

we use the metric $ds_t^2$  on the hyperbolic annulus given in (<ref>).

* In the collar bands $\{e^{a_0}b_*\leq |w_i^{(j)}|\leq b_*\}$ for $j=1,2$, we use the geometric-interpolation of the two metrics $ds^2$ and $ds_t^2$ given by

\begin{equation}\label{interpmetric}
ds_{graft}^2=(ds^2)^{1-\eta}(ds_t^2)^{\eta} \quad \mathrm{with} \quad \eta=\eta\left(\ln\left(\frac{|w^{(j)}_i|}{b_*}\right)\right).
\end{equation}

Here, $\eta(a)$ is a smooth function which is one for $a\leq a_0<0$ and zero for $a\geq 0$.

Now we shall discuss the leading correction to the grafted metric needed for  making it a hyperbolic metric on the plumbing family [42]. Let us denote the Fuchsian group associated with the Riemann surface $\mcal{R}$ with $\Gamma$. Assume that the $\Gamma$ has the stabilizer of infinity  $\Gamma_{\infty}$ (i.e. the  group generated by the transformation $z\to z+1$). We can then consider the relative Poincaré series

\begin{equation}\label{eisenstein}
E(z;2)=\sum_{A\in \Gamma/\Gamma_{\infty}}(\mathrm{Im}  A (z))^2,
\end{equation}

where $z$ is the coordinate on $\mathbb{H}$. The function $(\mathrm{Im} z)^2$ on $\mathbb{H}$ is an eigenfunction of the hyperbolic Laplacian with eigenvalue 2. This seres, known as the Eisenstein series,  converges locally uniformly on $\mathbb{H}$.  It has the expansion

\begin{equation}\label{eisentseincusp}
E(z;2)=(\mathrm{Im} z)^2+\hat e(z)
\end{equation}

with  $\hat e(z)$ bounded as $\mcal{O}((\mathrm{Im} z)^{-1})$ for large values of $\mathrm{Im} z$. The quotient space $\{\mathrm{Im} (z)>1\}/\Gamma_{\infty}$ embeds in $\mathbb{H}/\Gamma$. This region in the upper half-plane that corresponds to neighborhood of the puncture with hyperbolic area $1$ associated with $\Gamma_{\infty}$ on the hyperbolic Riemann surface $\mathcal{R}=\mathbb{H}/\Gamma$  is known as the cusp region for this puncture in $\mathbb{H}$. Cusp regions for distinct punctures are disjoint. Remember that $\widehat{\mcal{R}}_0$ is a finite union of Riemann surfaces with punctures, and each component has an associated Fuchsian group. We can defined as Eisenstein series with respect to a puncture. This Eisenstein series lives on the component surface that contains the puncture with respect to which we define the Eisenstein series. It is useful to consider a special truncation of these Eisenstein series for a given choice of $\eta$ and parameters $b_*,a_0$ and $t$.

The Truncated Eisenstein Series: The special truncation  $E^{\#}$ of the Eisenstein series is given by the following modification in the cusp regions

* In the cusp region associated with the puncture that is used for the plumbing fixture that produces $\mcal{R}_t$ from $\widehat{\mcal{R}}_0$, and for $\mathrm{Im} z>1$, we define

\begin{equation}\label{trnceisenstein1}
E^{\#}(z;2)\equiv[1-\eta(-2\pi\mathrm{Im}\,z-\mathrm{ln}~b_*)](\mathrm{Im}\,z)^2+\left[1-\eta\left(-2\pi\mathrm{Im}\,z+\mathrm{ln}\left(\frac{b_*}{|t|}\right)+a_0\right)\right]\hat e(z). \nonumber
\end{equation}

* In the cusp region associated with other punctures, and for $\mathrm{Im} z>1$, we define

\begin{equation}\label{trnceisenstein2}
E^{\#}(z;2)\equiv \left[1-\eta(-2\pi\ln\,z+\ln\left(\frac{b_*}{|t|}\right)+a_0)\right]E(z;2). \nonumber
\end{equation}

Let us extend these truncated Eisenstein series to define $E^{\dagger}$, the melding of the Eisenstein series, on $\mcal{R}_t$. For this, we first extend the definition of $E^{\#}$ by zero on the components of $\widehat{\mathcal{R}}_0$ that do not contain the puncture used to define $E$. Then, we define $E^{\dagger}$ on the glued surfaces $\mcal{R}_t$ as follows. Away from the plumbing collars in $\mcal{R}_t$, $E^{\dagger}$ is the same as the non-zero $E^{\#}$ in that region. On the $i$th plumbing collar of $\mcal{R}_t$ on the overlap $\{|t|/b_*<|w^{(1)}|<b_*\}\cap\{|t|/b_*<|w^{(2)}|<b_*\}$, $E^{\dagger}$ is defined as the sum of $E^{\#}$ at $w^{(1)}_i$ and $E^{\#}$ at $w^{(2)}_i=w^{(1)}_i/t$.

As we mentioned earlier, $(\mathrm{Im}(z))^2$ is the eigenfunction of the hyperbolic Laplacian with eigenvalue $2$. The contribution of the truncation of the Eisenstein series to the hyperbolic metric can be determined by analyzing the quantity $(D_{\mathrm{graft}}-2)E^{\dagger}$ on the collar in which $D_{\mathrm{graft}}$ is the Laplacian in the grafted metric. In the complement of the collar, the grafted metric is the hyperbolic metric and $E^\dagger=E$ and $(D_{\mathrm{graft}}-2)E^{\dagger}$ becomes $(D_{\mathrm{hyp}}-2)E$ for $D_{\mathrm{hyp}}\equiv y^2\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)$ where $z=x+\mathrm{i} y$. By the definition of $E$ given in (<ref>), this quantity is zero. However, this quantity is non-zero on the collar and can be used to determine the contribution to the hyperbolic metric on the collar from the grafted metric defined on the collar band. It can be shown that [42]:

\begin{equation}\label{grafted Laplacian acts on the melting}
(D_{\mathrm{graft}}-2)E^{\dagger}(z)=-\frac{1}{4\pi}\Lambda +\mathcal{O}\left(\left(-\ln|t|\right)^{-1}\right),
\end{equation}

$$\Lambda\equiv\frac{\partial}{\partial a}\left(a^4\frac{\partial}{\partial a}\eta(a)\right), \qquad\mathrm{for},\quad a\equiv\ln|w^{(1)}|\quad\mathrm{or}\quad\ln|w^{(2)}|.$$

Finally we can use this result to find the relation between the hyperbolic metric on the plumbing family and the grafted metric. Assume that $t=(t_1,\cdots,t_m)\in\mathbb{C}^{m}$ and $|t_i|<b_*^4,~i=1,\cdots,m$ are the plumbing fixture parameters for a surface $\mathcal{R}_t$. To find the degenerate expansion of the hyperbolic metric on $\mathcal{R}_t$, one should use curvature-correction equation (<ref>). Using this approach, the hyperbolic metric on the surface $\mathcal{R}_t$ has the following expansion [41, 42]:

\begin{equation}
ds^2_{\mathrm{hyp}}=ds_{graft}^2\left(1+2(D_{\mathrm{graft}}-2)^{-1}(1+{\bf C}_{\mathrm{graft}})+\mathcal{O}\left(\parallel 1+{\bf C}_{\mathrm{graft}}\parallel\right)^2\right) \label{hyperbolicintermsgaussiancurvature},
\end{equation}

In which $\parallel\cdot\parallel$ is an appropriate norm and ${\bf C}_{\mathrm{graft}}$ is the Gaussian curvature of the grafted metric given by [41]:

\begin{equation}
{\bf C}_{\text{graft}}=-1-\frac{\epsilon^2}{6}\Lambda+\mathcal{O}(\epsilon^4), ~ ~ \epsilon\equiv\frac{\pi}{\ln|t|}.
\end{equation}

$\Lambda$ is given in (<ref>). Using this relation, the hyperbolic metric on $\mathcal{R}_t$ can be expanded in terms of the grafted metric (Theorem $4$ of [42]):The Expansion of Hyperbolic Metric on $\mathcal{R}_{t}$: Given a choice of $b_*<1$ and a cut-off function $\eta$ , then for all small $t$ the hyperbolic metric $ds^2_{\mathrm{hyp}}$ of the  Riemann surface $\mathcal{R}_{t}$, obtained by the plumbing fixture of the $m$ pairs of annuli, has the following expansion:

\begin{equation}\label{graftedhyp}
\end{equation}

The functions $E^{\dagger}_{i,1}$ and $E^{\dagger}_{i,2}$ are the melding of the Eisenstein series $E(\cdot;2)$ associated to the pair of cusps plumbed to form the $i$th collar.

This exapnsion for the hyperbolic metric on $\mathcal{R}_{t}$ can be written in terms of the $i$th collar geodesic  $l_{i}=-\frac{2\pi^2}{\ln|t_i|}$ computed in $ds_t^2$ metric:

\begin{equation}\label{graftedhyp2}
\end{equation}

We can use this expansion to compute the lengths of an arbitrary simple closed geodesics on $\mcal{R}_t$

* The hyperbolic length of the geodesic in the $i$th plumbing collar  is given by

\begin{equation}\label{collargeodesic}
\end{equation}

Note that there is no correction of order $l_i^2$ to the length of the $i$th collar geodesic.

* The  length of a simple closed geodesic $\alpha$,  disjoint from the plumbing collars, is given by

\begin{equation}
\widetilde{l}_{\alpha}=l_{\alpha}+\sum_{i=1}^m\frac{l_i^2}{6}\int_{\alpha}ds\,\left(E^\dagger_{i,1}+E^\dagger_{i,2}\right)+\sum_{i=1}^m\mathcal{O}\left(l_i^3\right).
\end{equation}

In this formula, $\widetilde{l}_{\alpha}$ denotes the length of $\alpha$ on $\mcal{R}_t$ and $l_{\alpha}$ denotes the length of $\alpha$ on $\mcal{R}_0$. Away from the collars, $E_{i,1}^\dagger(z,2)=E_{i,1}(z,2)$, so we can write:

\begin{equation}
\widetilde{l}_{\alpha}=l_{\alpha}+\sum_{i=1}^m\frac{l_i^2}{6}\int_{\alpha}ds\,\left(E_{i,1}+E_{i,2}\right)+\sum_{i=1}^m\mathcal{O}\left(l_i^3\right).  \label{lengthawayc}
\end{equation}

§.§.§ A Gluing-Compatible Choice of Local Coordinates

We are in a position to check the validity of the choice of local coordinates proposed in section <ref>. We proposed that the local coordinate around a puncture to be $\widetilde{w}=e^{\frac{\pi^2}{c_*}}w$,  where $w$ is induced from the hyperbolic metric on the surface. In terms of the local coordinate $w$, the hyperbolic metric around the puncture takes the form $\left(\frac{|dw|}{|w|\ln|w|}\right)^2$. One of the main features of a gluing-compatible choice of local coordinates is that each component of the surface in the 1PR region are independent of the plumbing  parameters and moduli parameters of other components. Equation (<ref>) shows that the proposed choice of local coordinates does not satisfy this criteria and as such does not provide a gluing-compatible section of $\pcab$.

Let us elaborate this point. By construction, the local coordinates induced from the grafted metric $ds^2_{graft}$ depend only on the moduli parameters associated with the component surfaces to which the punctures belong. To obtain a hyperbolic metric $ds^2_{hyp}$ from the grafted metric, we should multiply with a Weyl factor that depends on the plumbing parameters. The net effect is that the local coordinate induced from the hyperbolic metric will depend on the plumbing parameter, and hence it will not be a gluing-compatible choice of local coordinates. It is thus clear that we can obtain a choice of local coordinate that does not depend on the plumbing parameters by multiplying the hyperbolic metric with the inverse of the Weyl factor obtained by solving the curvature-correction equation (<ref>), the factor that is multiplied with $ds^2_{graft}$ in (<ref>).

Based on this observation, we can modify the proposed choice of local coordinates and make it gluing-compatible as long as we keep the parameter $c_*$ very small. Let us briefly explain the need for keeping the parameter $c_*$ very small. Exactly at the degeneration locus, the local coordinates on the glued surface match with local coordinates induced from the hyperbolic metric. Otherwise, the difference between them is of order $l_i^2$, where $l_i$ is the length of the simple closed geodesic on the $i$th plumbing collar. When the length $l_i$ is very small compared to lengths of other simple closed geodesics on the surface, we can neglect the higher-order corrections. For a gluing-compatible section of $\pcab$, it is important that the choice of local coordinates in the 1PR region matches with the local coordinates induced from the component surfaces by plumbing fixture. Remember that we defined the 1PR regionby considering the Riemann surfaces that have a simple closed geodesics less than $c_*$. By demanding $c_*$ to be very small, we are making sure that the difference between the induced coordinates induced from the hyperbolic metric and the ones induced from the component surfaces through gluing is very small, or in other words, we can use (<ref>) and ignore all higher-order corrections beyond the order $l_i^2$. Moreover, it is clear from equation (<ref>) that for small $c_*$, the definition of 1PI and 1PR regions given in section <ref> remains the same. Hence at the boundary of 1PI region, $l_i=c_*$.

We shall now describe a choice of local coordinates that is gluing-compatible and is continuous all over the moduli space. As the case of naive local coordinate induced from the hyperbolic metric, this choice of local coordinates is unique up to a phase ambiguity. To specify the choice of local coordinates, let us introduce another infinitesimal parameter $\epsilon$. In order to define local coordinates in the 1PI region, we divide it into subregions. Let us denote the subregion in the 1PI region consists of surfaces with $m$ simple closed geodesics of length between $c_*$ and $(1+\epsilon)c_*$ by $\mathbf{R}_m$. For surfaces belong to the subregion $\mathbf{R}_0$, we choose the local coordinate around the $i^{th}$ puncture to be $e^{\frac{\pi^2}{c_*}}w_i$. In terms of $w_i$, the hyperbolic metric in the neighborhood of the puncture takes the following form

\begin{equation}
d s^2_{hyp}=\left(\frac{|dw_i|}{|w_i|\ln|w_i|}\right)^2, \qquad  i=1,\cdots,n.
\end{equation}

For surfaces belong to the region $\mathbf{R}_{m}$ with $m\ne 0$, we choose the local coordinates around the $i^{th}$ puncture to be  $e^{\frac{\pi^2}{c_*}}\widetilde w_{i,m}$, where $\widetilde w_{i,m}$ is obtained by solving the following equation

\begin{equation}\label{eq:metric on the 1PI and near its boundary}
\left(\frac{|d\widetilde{w}_{i,m}|}{|\widetilde{w}_{i,m}|\mathrm{ln}|\widetilde{w}_{i,m}|}\right)^2=\left(\frac{|dw_i|}{|w_i|\mathrm{ln}|w_i|}\right)^2\left(1-\sum_{j=1}^mf(l_j)E^0_j(w_i)\right).
\end{equation}

$f(x)$ is a smooth function of $x$ such that $f(c_*)=\frac{c_*^2}{3}$ and $f((1+\epsilon)c_*)=0$. $E^0_{j}(w_i)$ is the leading term in the Eisenstein series defined with respect to the $j^{th}$ node on the surface. For more explanation about the expansion of the Eisenstein series around a puncture, see appendix of [62]. The 1PR region consists of surfaces with   $m$ simple closed geodesics of length  $0\le l_j<c_* ,~j=1,\cdots,m$. We choose the local  coordinates around the $i^{th}$ puncture to be $e^{\frac{\pi^2}{c_*}}\widehat{w}_{i,m}$ where $\widehat{w}_{i,m}$ is obtained by solving the following equation

\begin{equation}\label{eq:metric in the 1PR region}
\left(\frac{|d\widehat{w}_{i,m}|}{|\widehat{w}_{i,m}|\mathrm{ln}|\widehat{w}_{i,m}|}\right)^2=\left(\frac{|dw_i|}{|w_i|\mathrm{ln}|w_i|}\right)^2\left(1-\sum_{j=1}^m\frac{l_j^2}{3}E^0_j(w_i)\right).
\end{equation}

Note that different choice of function $f$ and different values for the parameters $c_*$ and  $\epsilon$ give different choices of gluing-compatible local coordinates. It is shown in [13, 14, 15]  that all such choices of local coordinates give the same value for the measurable quantities. Therefore, the function $f$ and the arbitrary parameters $c_*$ and $\epsilon$ will not appear in the measurable quantities as longs as  we restrict our computations to the second order in $c_*$.

We can  generalize the procedure to higher order in $c_*$, say to the $n^{th}$ order in $c_*$.  The idea behind choosing the  local coordinates around the punctures is that it should be same as the coordinate induced from the grafting metric inside the 1PR region of the moduli space.   Therefore, we need to  find the explicit solution for curvature correction equation <ref> to $n^{th}$ order in $c_*$. This will relate the hyperbolic metric with grafting metric that is valid up to $n^{th}$ order in $c_*$. Beyond second order we need to correct both the definition of 1PR region and choice of local coordinates to make it gluing compatible, since the length of the core geodesic can also receive correction beyond the second order. The algorithm for finding a gluing compatible choice of local coordinate around the punctures up to $c_*^n,~n\ge 3$ is as follows:

* We first solve the curvature corrections equation (<ref>) to $n$th order in $c_*$. Using this solution, we can find the correction to the hyperbolic metric $ds^2_{hyp}$ up to order $n$ from (<ref>).

* We then compute the correction to the lengths of the collars using the hyperbolic metric $ds^2_{hyp}$  expressed in terms of grafted metric.

* We next define the 1PI region inside the moduli space as the region consists of hyperbolic surfaces with no simple geodesics of length less than $C_*$. $C_*$ is the length of the geodesic computed using $ds^2_{hyp}$ on the plumbing collars of the family of Riemann surface obtained with plumbing parameters $|t_i|=e^{-\frac{2\pi^2}{c_*}}$. The 1PR region inside the moduli space is the complement of the 1PI region.

* The next step is to define the local coordinates on Riemann surfaces inside the 1PR regions. We choose it to be $e^{\frac{\pi^2}{c_*}}\widehat{w}_{i,m}^{(n)},~i=1,\cdots,n$, where $\widetilde{w}_{i,m}^{(n)}$ is the natural local coordinate around the punctures induced from the grafted metric, represented in terms of the hyperbolic metric.

* In order to define the local coordinates around the punctures on the Riemann surfaces inside the 1PI region,  we divide the 1PI region into subregions. Let us denote the subregion in the 1PI region consists of surfaces with $m$ simple closed geodesics of length between $C_*$ and $(1+\epsilon)C_*$ by $\mathbf{R}^{(n)}_m$. For the surfaces belong to the subregion $\mathbf{R}^{(n)}_0$, we choose the local coordinate around the $i$th puncture to be $e^{\frac{\pi^2}{c_{*}}}w_i$. For the surfaces belong to the subregion $\mathbf{R}^{(n)}_m$ for $m\ne 0$, we choose the local coordinate around the $i$th puncture to be $e^{\frac{\pi^2}{c_*}}\widetilde{w}_{i,m}^{(n)}$. $\widetilde{w}_{i,m}^{(n)}$ is induced from the metric interpolating between the grafted and hyperbolic metrics.

Note that choosing different interpolating functions $f$ and various values for the parameter $c_*$ and  $\epsilon$ give different choices of gluing compatible local coordinates. It is shown in [13, 14, 15]  that all such choices of local coordinates give same value for the measurable quantities. Therefore, the arbitrary parameters $c_*$ and $\epsilon$ will not appear in the measurable quantities as longs as we restrict them to the same order in $c_*$ up to which we restricted the perturbative solution to the curvature correction equation.

§.§ Integration Measure on the Section of $\widehat{\mathcal{P}}_{g,n}$

The construction of string measure on a gluing compatible section of $\widehat{\mathcal{P}}_{g,n}$ requires choosing a parametrization of the moduli space of Riemann surfaces and computing the Beltrami differentials associated with them, as explained in section <ref>. The Fenchel-Nielsen parametrization provides a convenient choice of global coordinates for the Teichmüller space of hyperbolic Riemann surfaces [50]. Although this parameterization  is not well suited for explicitly describing the moduli space, as explained in  section <ref>,  integrations over the moduli space can be efficiently performed using the these coordinates. Therefore, we choose to work with them. Below, we shall discuss the construction of Beltrami differentials associated with the Fenchel-Nielsen coordinates and describe the explicit procedure for writing the integration measure for the off-shell bosonic-string theory.

The basic idea behind the Fenchel-Nielsen  parametrization is that every hyperbolic metric on a Riemann surface can be obtained by piecing together the metric from simple subdomains. The basic building block of a compact hyperbolic Riemann surface is  a  pair of pants. This construction is based on the following observations.

1)  The lengths $\lambda_1,\lambda_2$ and $\lambda_3$ of alternating sides of a right hexagon in the hyperbolic plane have any value between $(0,\infty)$.  If the hexagon is doubled across the unlabeled sides, then we obtain a pair of pants $P$ with geodesic boundary lengths $l_j=2\lambda_j,~j=1,2,3$.

2)  Given a pair of pants $P$ with boundaries $\alpha_j,~j=1,2,3$ and a pair of pants $Q$ with boundaries $\beta_j,~j=1,2,3$, $P$ can be glued to $Q$ along the boundaries $\alpha_1$ and $\beta_1$ if both of them have the same hyperbolic length. The resulting surface is a geometric sum, that has a hyperbolic metric whose restriction to each piece is the original metric of that piece. The proof of this claim is based on the observation that the geometry in a tubular neighborhood of a simple closed geodesic of length $l$ is completely determined by $l$ [39].

By combining $2g-2+n$ pairs of pants, a compact surface of genus $g$  and $n$ boundaries can be constructed. Assume that the boundary components of the surface are curves with lengths $L_i,~i=1,\cdots,n$. When all $L_i=0,~i=1,\cdots,n$, we obtain a Riemann surface with $n$ punctures. By varying the parameters of this construction every compact hyperbolic metric can be obtained. At each gluing site there are two parameters. At the gluing site where we glue the boundary $\alpha_1$ of the pair of pants $P$ with boundary $\beta_1$ of the pair of pants $Q$, these parameters are the  length $l(\alpha_1)=l(\beta_1)=\ell$ of the boundaries $\alpha_1$ and $\beta_1$ and the twist parameter $\tau$.  The twist parameter is defined as follows. Let $p_1$ be a point on boundary $\alpha_1$ and point $q_1$ be a point on boundary $\beta_1$. Compared to other points on $\alpha_1$, the point $p_1$ is assumed to has the minimum hyperbolic distance from the boundary $\alpha_2$. Similarly  point $q_1$ has the minimum hyperbolic distance from the boundary $\beta_2$. The twist parameter $\tau$ is the distance between $p_1$ and $q_1$ along $\alpha_1\sim\beta_1$.  Then the parameters $(\tau_j,\ell_j),~1\leq j\leq 3g-3+n,~\tau_j\in \mathbb{R},~l_j\in \mathbb{R}^+$ for a fixed pair of pants decomposition $\mathcal{P}$ endows the Teichmüller space $\mathcal{T}_{g,n}$ of the genus-$g$ Riemann surfaces with $n$ boundary components with a global real-analytic coordinates. There is a natural symplectic form known as the Weil-Petersson (WP) symplectic form whose explicit form in terms of the Fenchel-Nielsen coordinates is [47]:

\begin{equation}\label{WolpertMagic}
\omega_{WP}=\sum_{i=1}^{3g-3+n}d\ell_j\wedge d\tau_j.
\end{equation}

Consider the Fenchel-Nielsen  coordinates $(\tau_i,\ell_i),~i=1,\cdots,3g-3+n$  defined with respect to a pants decomposition of a hyperbolic Riemann surface  $\mathcal{R}$  using $3g-3+n$  curves $\{C_i\},~i=1,\cdots,3g-3+n$, where $C_i$ denotes a simple closed geodesic on the  surface. For $i\neq j$, the curves $C_i$ and $C_j$ are disjoint and non-homotopic.  The tangent space at a point in  the Teichmüller space is spanned by the Fenchel-Nielsen  coordinate vector fields $\left\{\frac{\partial}{\partial \tau_i},\frac{\partial }{\partial \ell_i}\right\},~i=1,\cdots,3g-3+n$. The Fenchel-Nielsen  coordinate vector field $\frac{\partial}{\partial \tau_i}$ is  the  twist vector field $t_{C_i}$ associated with the curve $C_i$.  The twist field $t_{\alpha}$, for a simple closed geodesic $\alpha$,  generates a flow on  $\mathcal{T}_{g,n}$. This flow can be understood as the operation of  cutting the hyperbolic surface along $\alpha$ and attaching the boundaries after rotating one boundary relative to the other by some amount $\delta$. The magnitude $\delta$ parametrizes the flow on $\mathcal{T}_{g,n}$.

The universal cover of a hyperbolic Riemann surface $\mathcal{R}$ is the upper half-plane $\mathbb{H}$ endowed with the hyperbolic metric. Assume that the uniformization group is a finitely-generated Fuchsian group $\Gamma$. Suppose that the simple closed geodesic $\alpha$  corresponds to the element

$$A=\left(\begin{array}{cc}a & b \\c & d\end{array}\right),$$
in $ \Gamma$. Let $\langle A\rangle$ denote the infinite cyclic group generated by $A$. Then, the  Beltrami differential corresponding to the twist vector field $t_{\alpha}$ is given by [45]

\begin{equation}\label{beltramitwist}
\mathbf{t }_{\alpha}=\frac{\mathrm{i}}{\pi}(\mathrm{Im}z)^2\overline{ \Theta}_{\alpha},
\end{equation}

where $\Theta_{\alpha}$ is the relative Poincaré series

\begin{equation}\label{poicares}
\Theta_{\alpha}=\sum_{B\in \langle A\rangle \backslash  \Gamma}\omega_{B^{-1}AB}.
\end{equation}

In the above equation, $\omega_A$ is given  by

\begin{equation}
\omega_A=\frac{(a+d)^2-4}{\Big(cz^2+(d-a)z-b\Big)^2}.
\end{equation}

$z$ is the coordinate on the upper half-plane. The Beltrami differential for the Fenchel-Nielsen  coordinate vector field $\frac{\partial}{\partial \tau_i}$ is given by $\mathbf{ t}_{C_i}$. The Beltrami differential for the Fenchel-Nielsen  coordinate vector field $\frac{\partial}{\partial l_i}$ can also be constructed by noting the that with respect to the WP symplectic form $\frac{\partial}{\partial l_i}$ is dual to the twist vector field $\frac{\partial}{\partial \tau_i}$ [46]. We denote the  Beltrami differential for the Fenchel-Nielsen  coordinate vector field $\frac{\partial}{\partial l_i}$ as $\mathbf{ l}_{C_i}$.

Putting these together, the off-shell bosonic-string measure is given by

\begin{equation}\label{eq:the bosonic-string measure}
\Omega_{6g-6+2n}^{(g,n)}(|\Phi\rangle)=(2\pi \mathrm{i})^{-(3g-3+n)}\langle\mathcal{R}|B_{6g-6+2n}|\Phi\rangle,
\end{equation}

\begin{equation}\label{eq:the expression for B}
B_{6g-6+2n}\left[\frac{\partial}{\partial \ell_1},\frac{\partial}{\partial \tau_1},\cdots,\frac{\partial}{\partial \ell_{3g-3+n}},\frac{\partial}{\partial \tau_{3g-3+n}}\right]=b(\mathbf{t}_{C_1})b(\mathbf{l}_{ C_1})\cdots b(\mathbf{t}_{C_{3g-3+n}})b(\mathbf{l}_{ C_{3g-3+n}}),
\end{equation}

b(𝐭_C_i)   =∫_ℱ d^2z(b_zz𝐭_C_i+b_z̅z̅𝐭_C_i),

b(𝐥_C_i)   =∫_ℱ d^2z(b_zz𝐥_C_i+b_z̅z̅𝐥_C_i).

Here $\mathcal{F}$ denotes the fundamental domain in the upper half-plane for the action of the Fuchsian group $\Gamma$ associated with the Riemann surface $\mathcal{R}$.  The state $|\mathcal{R}\rangle$ is the surface state associated with $\mathcal{R}$ and the state  $|\Phi\rangle$ represents the tensor product of the off-shell states inserted at the punctures $|\Psi_1\rangle\otimes\cdots\otimes |\Psi_n\rangle\in\mathcal{H}^{\otimes n}$. The quantity  $\langle\mathcal{R}|B_{6g-6+2n}|\Phi\rangle$ describe the $n$-point correlation function on $\mathcal{R}$ with the off-shell vertex operators for the states $|\Psi_i\rangle, i=1,\cdots,n$, inserted at the punctures using the gluing compatible choice of local coordinates  around the punctures described in  section <ref>.

§.§ Integration over the Moduli Space

In order to obtain the off-shell bosonic string amplitude, we need to integrating the off-shell bosonic-string measure over $\mathcal{M}_{g,n}$, the moduli space  of genus $g$  Riemann surface with $n$ punctures. $\mathcal{M}_{g,n}$  is the quotient of the Teichmüller space  $\mathcal{T}_{g,n}$ with the action of the mapping class group (MCG). Although the Fenchel-Nielsen parameterization provides a simple description of the Teichmüller space of hyperbolic Riemann surfaces, they fails to provide a convenient description of the moduli space.   The reason is that the form of the action of  the mapping class group on  Fenchel-Nielsen coordinates is not known  explicitly  [33, 34].   As a result, integrating a mapping class group invariant function over the moduli space of hyperbolic Riemann surfaces is not a  straightforward. In this section, we shall discuss a way to bypass this difficulty using a prescription introduced by Mirzakhani [36].

The basic idea of [36] is as follows.  The integration over the moduli space can be lifted to an appropriate covering space, and the pull-back of the WP volume form defines a volume form in the covering space. The most important feature of  this covering space is that it can be decomposed  into a product of lower-dimensional moduli spaces  and a set of two dimensional  infinite cones. These infinite cones are parametrized by some pairs $(l_a,\tau_a)$ of the Fenchel-Nielsen coordinates. We can again lift the integration over the lower dimensional moduli spaces to obtain another set of lower dimensional moduli spaces and infinite cones. At the end we will end up with an integration over a set of infinite cones whose explicit domains in the Teichmüller space parametrized using the Fenchel-Nielsen coordinates are known. Below, we shall explain a method to lift an integral over a space to a covering of that space.

Consider the space $M$ with a covering space $N$. The covering map is given by $\pi: N\longrightarrow M$. If $dv_{M}$ is a volume form on $M$, then $dv_N\equiv\pi^{-1}*(dv_M)$ defines a volume form on $N$.  Suppose that $f$ is a smooth function on $N$, then the push forward of $f$ to $M$ using the covering map $\pi$, i.e. $\pi_*f$, is defined by:

\begin{equation}\label{coveri}
(\pi_*f)(x)\equiv\sum_{y\in \pi^{-1}\{x\}}f(y).
\end{equation}

Here, we used the fact that the value of the push-forward of a function $f$ at a point $x$ of $M$ can be obtained by summation over the values of the function $f$ at all points in the fiber of the point $x$ in $N$. This relation defines a smooth function on $M$. Then, the integral of this pullback over $M$ can be lifted to the covering space as follows:

\begin{equation}\label{covint}
\int_{M}dv_M~(\pi_*f)~=\int_{N}dv_N~f.
\end{equation}

Therefore, if we have a covering of the moduli space that has a well-defined region in the Teichmüller space, then  the integration of a function on the moduli space can be performed by expressing this function as a push-forward of a function in the covering space. In the remaining part of this section, we shall explain that it is indeed possible to find such a covering of the moduli space and express the MCG-invariant functions as a push-forward of a function defined in the covering space.

§.§.§ The Covering Space of the Moduli Space

The moduli space of hyperbolic Riemann surfaces with borders can in principle be obtained from the Teichmüller space by stating that different points which are related by the action of all mapping class group elements on  simple closed geodesics on hyperbolic surfaces are equivalent. The study of this action is a complicated. However, the following covering space of the moduli space has a comparatively nicer structure. Assume that $\gamma$ is a simple closed curve on  $\mathcal{R}_{g,n}$ a genus-$g$ Riemann surface with $n$ borders. The borders have hyperbolic lengths $\mathbf{L}\equiv\{L_1,\cdots,L_n\}$. We denote the moduli space of such surfaces as $\mcal{M}_{g,n}(\mathbf{L})$. Consider a space  $\mathcal{M}_{g,n}(\mathbf{L})^{\gamma}$ which is obtained from the Teichmüller space of $\mathcal{R}_{g,n}$ by identifying different points that are related by the subgroup of the mapping class group that keeps $\gamma$ fixed. The space $\mathcal{M}_{g,n}(\mathbf{L})^{\gamma}$ is larger than $\mathcal{M}_{g,n}(\mathbf{L})$ because it is obtained by taking  a quotient  of the Teichmüller space by a smaller group. The most important feature of this space is that it can be thought of as a product of lower dimensional moduli spaces and infinite cones [36] where the infinite cones corresponds to  twists about the curve $\gamma$ from $0$ to $\ell_{\gamma}$ (for generic case)[As we explain below, there is a situation when the range of $\tau$ is between $0$ and $\frac{\ell_{\gamma}}{2}$] and length $\ell_{\gamma}$  of the curve $\gamma$. the lower dimensional moduli spaces are the moduli spaces of surfaces obtained by cutting the original surface along $\gamma$. We shall elaborate this idea. Assume that $\gamma$ is a multi-curve on $\mathcal{R}_{g,n}$

\begin{equation}\label{curvg}
\gamma=\sum_{i=1}^k \gamma_i.
\end{equation}

$\gamma_1,\cdots,\gamma_k$ are disjoint non-homotopic  simple closed curves on $\mathcal{R}_{g,n}$. Denote the mapping class group of the genus $g$ Riemann surface with $n$ borders by $\text{MCG}_{g,n}$. For $h\in \text{MCG}_{g,n}$, let $h\cdot\gamma=\sum_{i=1}^k h\cdot\gamma_i$. Assume that $\mathcal{O}_{\gamma}$ be the set of homotopy classes of elements of the set MCG$_{g,n}\cdot\gamma$

\begin{equation}
\mathcal{O}_{\gamma}\equiv\left\{\left[\alpha\right]\left|~\alpha=h\cdot\gamma,~~[\alpha]\sim[\beta] \iff \alpha\mathrm{ ~and}~ \beta ~\mathrm{are ~homotopic}\right.\right\}.
\end{equation}

Let us consider the covering space $\mathcal{M}_{g,n}(\mathbf{L})^{\gamma}$ defined as follows:

\begin{equation}
\mathcal{M}_{g,n}(\mathbf{L})^{\gamma}\equiv\left\{(\mathcal{R}_{g,n},\eta)| X\in \mathcal{M}_{g,n}(\mathbf{L}),~\eta=\sum_{i=1}^k\eta_i\in \mathcal{O}_{\gamma}, \eta_i\mathrm{ ~ is~a ~closed ~geodesics~on~}\mathcal{R}_{g,n}\right\}.\nonumber \label{mgng}
\end{equation}

For any set $A$ of homotopy classes of simple closed curves on $\mcal{R}_{g,n}$, define the stabilizer of $A$, Stab($A$), by

\begin{equation}\label{stabg}
\mathrm{Stab}(A)\equiv\{h\in \mathrm{MCG}_{g,n}|~h\cdot A=A\}\subset \text{MCG}_{g,n}.
\end{equation}

The symmetry group of $\gamma$, Sym($\gamma$), is defined by

\begin{equation}
\mathrm{Sym}(\gamma)\equiv\mathrm{Stab}(\gamma)/\cap_{i=1}^k\mathrm{Stab}(\gamma_i)\label{symg}.
\end{equation}

Denote the Dehn twist along $\gamma$ by $\phi_{\gamma}$. We can then define the following subgroup of the mapping class group

\begin{equation}\label{ggamma}
G_{\gamma}\equiv\bigcap_{i=1}^k~ \mathrm{Stab}(\gamma_i)\subset \mathrm{MCG}_{g,n}.
\end{equation}

$G_{\gamma}$ is generated by the set of Dehn twists $\phi_{\gamma_i},~i=1,\cdots,k$, and  the elements of the mapping class group of $\mcal{R}_{g,n}(\gamma)$. The surface $\mcal{R}_{g,n}(\gamma)$ denote the surface $\mcal{R}_{g,n}-U_{\gamma}$, where $U_{\gamma}$ is an open set homeomorphic to $\bigcup_{i=1}^k(0,1)\times \gamma_i$ around $\gamma$. Cutting the surface $\mcal{R}_{g,n}$ along each of the curves $\gamma_i$ produces two additional boundary components. Therefore, the surface $\mcal{R}_{g,n}(\gamma)$, which is obtained by cutting the surface along all the curves in $\gamma$, is a possibly disconnected surface with $n+2k$ boundary components and $s$ connected components. Let us denote the connected component of $\mcal{R}_{g,n}({\gamma})$, by $\mathcal{R}^i(\gamma),~i=1,\cdots,s$, see figures (<ref>) and (<ref>).

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