On disc-with-hole and disc-with-handle partition functions in bosonic string theory

To parametrise the annulus we use a strip in the complex plane:

$$\{\xi^{i}\},\hskip 14.22636pt\xi^{1},\xi^{2}\in\mathbb{R},\hskip 14.22636pt0\leq\xi^{2}\leq 1.$$

(4)

This parametrises the annulus in the complex plane given by

$$q\leq|z|\leq 1,\hskip 14.22636pt0<q<1\hskip 14.22636ptz\in\mathbb{C}$$

(5)

using the map

$$z=\exp(2\pi i(\xi^{1}+i\tau\xi^{2})),\hskip 14.22636pt\tau\equiv-\frac{1}{2\pi}\ln q.$$

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Under this map, which is periodic with period 1 in the $\xi^{1}$ direction, the top boundary ($\xi^{2}=1$) of the strip is mapped to the inner boundary of the annulus, while the bottom boundary ($\xi^{2}=0$) is mapped to the outer boundary of the annulus. Since we are interested in the path integral over the annulus as a one-loop correction to a phenomenological model of the Wilson loop, we will use ‘modified Dirichlet’ Alvarez:1982zi conditions for the embedding functions $X^{\mu}$ on the outer boundary. After gauge fixing, these become Dirichlet boundary conditions. On the other hand, the inner boundary is not mapped to a fixed curve, and we simply choose Neumann boundary conditions on it. Next, the $X^{\mu}$ and $g$ are obviously required to be periodic in the $\xi^{1}$ direction:

		$\displaystyle X^{\mu}(\xi^{1}+1,\xi^{2})=X^{\mu}(\xi^{1},\xi^{2}),$		(7)
		$\displaystyle g_{ab}(\xi^{1}+1,\xi^{2})=g_{ab}(\xi^{1},\xi^{2}).$

On the upper boundary and lower boundaries $g_{ab}$ must satisfy Alvarez:1982zi

$$g_{ab}=\partial_{a}X^{\mu}\partial_{b}X_{\mu},$$

(8)

The boundary conditions needed to compute the determinant of the Laplacian are discussed below.

To perform the $X^{\mu}$ integration one redefines the integration variable as an expansion about the classical solution:

$$X^{\mu}=X^{\mu}_{cl}+y^{\mu}.$$

(9)

Here $X^{\mu}_{cl}$ is the solution to the classical equation of motion obtained by varying the action. Note that this solution depends on $g$, thus the answer will be a function of the metric and remains inside the integral when we integrate over the metric. Then, the $X^{\mu}$ integration gives²²2We set the string tension $T$ to 1 below as it can be scaled out of the partition function.

$\displaystyle W_{g}$

$\displaystyle\equiv\int DX\ e^{-I[X,g]}=e^{-I[X_{cl},g]}\left(\mathrm{det}(\Delta_{g})\right)^{-13},$

(10)

where $\Delta_{g}$ is the ordinary Laplacian differential operator, i.e.

$$\Delta_{g}\equiv-\frac{1}{\sqrt{g}}\partial_{a}\sqrt{g}g^{ab}\partial_{b}.$$

(11)

The determinant of $\Delta_{g}$ is taken with Dirichlet boundary conditions. Note that $X_{cl}$ depends on $g$ only through the moduli variables since $I$ is invariant under diffeomorphisms and conformal transformations. Our final result for the partition function will be in the form of an integral over these moduli. To perform the integration over $g$ we first note that any metric on the annulus can be taken via a combined general coordinate and Weyl transformation (respecting the above boundary conditions) to a metric of the form (see, e.g., Rodrigues:1986us )

$$g_{ab}=(d\xi^{1})^{2}+(\tau d\xi^{2})^{2}.$$

(12)

Here $\tau=-\frac{1}{2\pi}\ln q$ is the Teichmuller parameter. The integral over metrics then reduces (in $D=26$) to the integral over $\tau$

	$\displaystyle W=$	$\displaystyle\frac{1}{2}\int_{0}^{\infty}d\tau\frac{1}{\sqrt{\tau}}\left(\frac{\det^{2}<T,\psi>}{\det<\psi,\psi>}\right)^{\frac{1}{2}}\left(\mathrm{det}^{\prime}P^{\dagger}_{1}P_{1}\right)^{\frac{1}{2}}$		(13)
		$\displaystyle\times\left(\mathrm{det}(\Delta_{g})\right)^{-13}e^{-I[X_{cl},g]}.$

where we have used the following definitions:

	$\displaystyle(T_{ab})$	$\displaystyle\equiv(\frac{\partial g_{ab}}{\partial\tau}-\frac{1}{2}g_{ab}g^{cd}\frac{\partial g_{cd}}{\partial\tau})$		(14)
	$\displaystyle(P_{1}\delta v)_{ab}$	$\displaystyle\equiv\nabla_{a}\delta v_{b}+\nabla_{b}\delta v_{a}-\nabla_{c}\delta v^{c}g_{ab}$
	$\displaystyle(P_{1}^{\dagger}\alpha)_{n}$	$\displaystyle\equiv-2\nabla^{m}\alpha_{mn}\ .$

$P_{1}$ acts on two-dimensional vectors (diffeomorphism generators) $\delta v$ and $P_{1}^{\dagger}$ acts on rank-2 traceless tensors $\alpha$. The norm on rank-2 traceless tensors is defined by:

$$<A,B>_{g}=\int_{\Sigma}d\sigma^{1}d\sigma^{2}\sqrt{g(\sigma)}\ g^{mi}(\sigma)g^{nj}(\sigma)\ A_{mn}(\sigma)B_{ij}(\sigma)$$

(15)

and $\psi$ is any element in $\mathrm{Ker}(P_{1}^{\dagger})$. The Riemann-Roch theorem implies Alvarez:1982zi :

$$\mathrm{dim}\ \mathrm{Ker}(P_{1})-\mathrm{dim}\ \mathrm{Ker}(P_{1}^{\dagger})=3\chi(\Sigma)=0.$$

(16)

From this we see that the conformal Killing vectors (CKV), i.e. vectors in in $\mathrm{Ker}(P_{1})$, on the annulus form a one-dimensional real vector space (corresponding to rigid translation in the $\xi^{1}$ direction). Also note that the mapping class group is isomorphic to $\mathbb{Z}_{2}$ and is given by the coordinate transformation Rodrigues:1986us

$$\xi^{1}\rightarrow-\xi^{1},\hskip 14.22636pt\xi^{2}\rightarrow 1-\xi^{2}.$$

(17)

The volume of the mapping class group is therefore $2$, and this is where the factor of $\frac{1}{2}$ in front of the path integral comes from.

We can calculate the Weil-Petersson measure directly. We have

$$(T_{ab})\equiv(\frac{\partial g_{ab}}{\partial\tau}-\frac{1}{2}g_{ab}g^{cd}\frac{\partial g_{cd}}{\partial\tau})=\frac{1}{\tau}\begin{pmatrix}-1&0\\ 0&\tau^{2}\end{pmatrix}.$$

(18)

The boundary conditions and the symmetric and traceless requirement on matrices in $\mathrm{Ker}(P_{1}^{\dagger}$) mean we must choose

$$\psi=\begin{pmatrix}1&0\\ 0&-\tau^{2}\end{pmatrix}.$$

(19)

A direct calculation then gives

$$\left(\frac{\det^{2}<T,\psi>}{\det<\psi,\psi>}\right)^{\frac{1}{2}}=\sqrt{\frac{2}{\tau}}.$$

(20)

All that is left is to calculate the determinants of the Laplacians. Firstly,

$\displaystyle(P_{1}^{\dagger}P_{1}\delta v)_{a}$

$\displaystyle=-2\begin{pmatrix}\nabla_{g}^{2}&0\\ 0&\nabla_{g}^{2}\end{pmatrix}\begin{pmatrix}\delta v_{1}\\ \delta v_{2}\end{pmatrix}.$

(21)

We have chosen boundary conditions such that changes in the metric do not change the normal direction at the boundary. This imposes Neumann boundary conditions on $\delta v_{1}$ and Dirichlet on $\delta v_{2}$ Rodrigues:1986us . Hence,

$$\mathrm{det}^{\prime}(P^{\dagger}_{1}P_{1})=\mathrm{det}^{\prime}_{N}(-2\nabla_{g}^{2})\ \mathrm{det}^{\prime}_{D}(-2\nabla_{g}^{2}).$$

(22)

Furthermore, the factor $(\mathrm{det}^{\prime}(\Delta_{g}))^{-13}$ which arises due to the $X$-integration must be taken with Dirichlet boundary conditions at the lower edge ($\xi^{2}=0$) and Neumann at the upper edge ($\xi^{2}=1$). Now we calculate each determinant individually. First, the operator in the determinant

$$\mathrm{det}^{\prime}_{N}(-2\nabla_{g}^{2})=\mathrm{det}^{\prime}_{N}(-2(\partial_{1}^{2}+\frac{1}{\tau^{2}}\partial_{2}^{2}))$$

(23)

has eigenvectors

$$\cos(2\pi m\xi^{1})\cos(\pi n\xi^{2}),\hskip 14.22636ptm,n\in\mathbb{Z}\hskip 14.22636ptm,n\geq 0,\hskip 14.22636pt(m,n)\neq(0,0)$$

(24)

and the corresponding eigenvalues

$\displaystyle\lambda_{mn}$

$\displaystyle=\frac{8\pi^{2}}{\tau^{\prime 2}}|\tau^{\prime\prime}m+n|^{2},\hskip 14.22636pt\tau^{\prime}\equiv 2\tau,\tau^{\prime\prime}\equiv i\tau^{\prime}.$

(25)

These are identical to the eigenvalues encountered in the one-loop closed string (torus) calculation in the case that the torus Teichmuller parameter is purely imaginary. Such a determinant is calculated, for example, in Nakahara:2003nw using the Eisenstein series. This method can be immediately adapted to obtain:

$$\mathrm{det}^{\prime}_{N}(-2\nabla_{g}^{2})=\frac{1}{\sqrt{2}}2\tau|\eta(2i\tau)|^{2}=\sqrt{2}\tau|\eta(2i\tau)|^{2}.$$

(26)

Next, the operator in the determinant

$$\mathrm{det}^{\prime}_{D}(-2\nabla_{g}^{2})=\mathrm{det}^{\prime}_{D}(-2(\partial_{1}^{2}+\frac{1}{\tau^{2}}\partial_{2}^{2}))$$

(27)

has eigenvectors

$$\cos(2\pi m\xi^{1})\sin(\pi n\xi^{2}),\hskip 14.22636ptm,n\in\mathbb{Z},\hskip 14.22636ptn>0,\hskip 14.22636ptm\geq 0,$$

(28)

and identical eigenvalues

$\displaystyle\lambda_{mn}$

$\displaystyle=\frac{8\pi^{2}}{\tau^{\prime 2}}|\tau^{\prime\prime}m+n|^{2},\hskip 14.22636pt\tau^{\prime\prime}\equiv i2\tau.$

(29)

The only difference is that we allow $n=0$ so long as $m\neq 0$ in the Neumann case. Hence we obtain

$$\mathrm{det}^{\prime}_{N}(-2\nabla_{g}^{2})=\mathrm{det}^{\prime}_{D}(-2\nabla_{g}^{2})\prod_{m\neq 0}(2\pi^{2}m^{2}).$$

(30)

We may regularize this factor as follows:

$$\prod_{m\geq 0}(2\pi^{2}m^{2})=\exp(-\zeta^{\prime}_{\Delta}(0)),$$

(31)

where

$$\zeta_{\Delta}(s)\equiv\sum_{m\geq 1}(2\pi m)^{-2s}=\frac{1}{(2\pi)^{-2s}}\zeta_{R}(2s).$$

(32)

Then

$$-\zeta^{\prime}_{\Delta}(0)=(2\zeta^{\prime}_{R}(0)-2\log(2\pi)\zeta_{R}(0))=0,$$

(33)

and hence the factor regularizes to $1$.

Lastly, we need to find the determinant of the Laplacian on the annulus $\mathrm{det}(\Delta_{g})$ where the inner boundary has Neumann boundary conditions, and the outer boundary has Dirichlet boundary conditions. We have

$$\mathrm{det}(\Delta_{g})=\mathrm{det}(-(\partial_{1}^{2}+\frac{1}{\tau^{2}}\partial_{2}^{2})).$$

(34)

This has eigenvectors

$$\cos(2\pi m\xi^{1})\sin(\pi(n+\frac{1}{2})\xi^{2}),\hskip 14.22636ptm,n\in\mathbb{Z},\hskip 14.22636ptm,n\geq 0$$

(35)

with eigenvalues

$\displaystyle\lambda_{mn}$

$\displaystyle=\Big{(}(2\pi m)^{2}+\frac{1}{\tau^{2}}(\pi(n+\frac{1}{2})\Big{)}^{2}.$

(36)

To calculate this determinant we will need to use a different method, relying on contour integrals. Our method is similar to a spectral method used in Kirsten:1999qjn . Consider the initial value problem

$\displaystyle\phi_{m,\mu}^{\prime\prime}(\xi^{2})=-\tau^{2}(\mu^{2}-4\pi^{2}m^{2})\phi_{m,\mu}(\xi^{2}),\hskip 14.22636pt\phi_{m,\mu}(0)=0,\hskip 14.22636pt\phi_{m,\mu}^{\prime}(0)=1.$

(37)

This has the unique solution

$$\phi_{m,\mu}(\xi^{2})=\frac{1}{\tau\sqrt{\mu^{2}-4\pi^{2}m^{2}}}\sin((\tau\sqrt{\mu^{2}-4\pi^{2}m^{2}})\xi^{2}).$$

(38)

In particular, for a given $m$ we will view $\phi^{\prime}_{m,\mu}(1)$ as a meromorphic function of $\mu$:

$$\phi^{\prime}_{m,\mu}(1)=\cos(\tau\sqrt{\mu^{2}-4\pi^{2}m^{2}}).$$

(39)

This function has zeros exactly when

$$\mu^{2}=(4\pi^{2}m^{2}+\frac{1}{\tau^{2}}(n+\frac{1}{2})^{2})\equiv\lambda_{mn},$$

(40)

i.e. exactly when $\mu^{2}$ is an eigenvalue of $\Delta_{g}$. Hence, we will consider the function

$$\frac{d}{d\mu}\log(\phi^{\prime}_{m,\mu}(1))=\frac{\frac{d}{d\mu}\phi^{\prime}_{m,\mu}(1)}{\phi^{\prime}_{m,\mu}(1)}$$

(41)

which clearly has a simple pole at each $\mu^{2}=\lambda_{mn}$ if we consider the expansion of $\phi^{\prime}_{m,\mu}(1)$. Hence, we may write the zeta function as

$$\zeta_{\Delta}(s)=\frac{1}{2\pi i}\int_{\Lambda}d\mu\mu^{-2s}\frac{d}{d\mu}\log(\phi^{\prime}_{0,\mu}(1))+\sum_{m=1}\frac{1}{2\pi i}\int_{\Lambda}d\mu\mu^{-2s}\frac{d}{d\mu}\log(\phi^{\prime}_{m,\mu}(1))$$

(42)

where $\Lambda$ is a contour enclosing all the positive simple poles, which all lie on the real axis. Now we choose the contour $\Lambda$ to be parametrised as

$$\Lambda(k)=\epsilon+ik,\hskip 14.22636pt\epsilon,k\in\mathbb{R}$$

(43)

where $\epsilon$ is a small positive number and $k$ runs from $-\infty$ to $\infty$. Substituting this into the integral and taking the limit of $\epsilon$ going to zero we obtain

	$\displaystyle\zeta_{\Delta}(s)=$	$\displaystyle\frac{\sin(\pi s)}{\pi}\int_{0}^{\infty}dkk^{-2s}\frac{d}{dk}\log(\phi^{\prime}_{0,ik}(1))$		(44)
		$\displaystyle+\sum_{m=1}m^{-2s}\frac{\sin(\pi s)}{\pi}\int_{0}^{\infty}dkk^{-2s}\frac{d}{dk}\log(\phi^{\prime}_{m,imk}(1)),$

where

		$\displaystyle\phi^{\prime}_{0,ik}(1)=\cos(\tau ik)=\cosh(\tau k),$		(45)
		$\displaystyle\phi^{\prime}_{m,imk}(1)=\cos(\tau\sqrt{-m^{2}k^{2}-4\pi^{2}m^{2}})=\cosh(\tau\sqrt{m^{2}k^{2}+4\pi^{2}m^{2}}).$

Now we will regularise these integrals separately.

To evaluate $W$ we need to parametrise the moduli space of $\Sigma$ and evaluate determinants on $\Sigma$. In fact, it will be easier to work with the Schottky double of $\Sigma$, which will be a surface without boundary denoted $\Sigma^{2}$; we will closely follow the approach outlined in BLAU1988285 . $\Sigma^{2}$ is topologically constructed by attaching two copies of $\Sigma$, denoted $\Sigma$ and $\Sigma^{*}$, with opposite orientation along the boundaries. Since $\Sigma$ is a disc-with-handle, $\Sigma^{2}$ is therefore a double-torus. The key point is that the moduli space and determinants on $\Sigma$ can then be related to those on $\Sigma^{2}$ in a manner we will now describe. First, recall that the moduli and Teichmuller spaces on $\Sigma$ can be defined as BLAU1988285

$$\mathscr{M}(\Sigma)\equiv\frac{C(\Sigma)}{\mathrm{Diff}(\Sigma)},\hskip 14.22636pt\mathscr{T}(\Sigma)\equiv\frac{C(\Sigma)}{\mathrm{Diff_{0}}(\Sigma)},$$

(63)

where $\mathrm{Diff(\Sigma)}$ is the group of orientation preserving diffeomorphisms on $\Sigma$, $\mathrm{Diff}_{0}(\Sigma)$ is the subgroup connected to the identity, and $C(\Sigma)$ is the set of almost complex structures on $\Sigma$ (which is in bijection with the set of classes of metrics where metrics in a given class differ only by a Weyl transformation). Now, there exists a natural choice of involution $I$ on the constructed topological space $\Sigma^{2}$ such that

$$I^{2}=1,\hskip 14.22636ptI(\Sigma)=\Sigma^{*},\hskip 14.22636ptI(\Sigma^{*})=\Sigma,\hskip 14.22636ptI|_{\partial\Sigma}=I|_{\partial\Sigma^{*}}=1.$$

(64)

Furthermore, if the surface $\Sigma$ comes with a choice of almost complex structure $J$, then we may define an almost complex structure $\tilde{J}$ on $\Sigma^{2}$ as BLAU1988285

$$\tilde{J}_{p}=J_{p}\ \textrm{if}\ p\in\Sigma,\hskip 14.22636pt\tilde{J}_{p}=-(I_{*}\circ J_{p}\circ I_{*})_{p}\ \textrm{if}\ p\in\Sigma^{*}.$$

(65)

Then, $I$ is in addition anti-conformal. On the other hand, an arbitrary almost complex structure $J$ on $\Sigma^{2}$ may or may not make $\Sigma^{2}$ the double of a surface; it will be considered the double of the surface $\Sigma$ if $J$ is such that the corresponding involution $I$ is anti-conformal. In particular, in BLAU1988285 the following important facts are noted: any metric on $\Sigma$ is equivalent in $\mathscr{T}(\Sigma)$ to a metric which extends smoothly to a metric on $\Sigma^{2}$ which is invariant under $I$, and any metric on $\Sigma^{2}$ invariant under $I$ (up to a diffeomorphism and Weyl transformation) is equivalent in $\mathscr{T}(\Sigma^{2})$ to a metric exactly invariant under $I$. Hence, $\mathscr{T}(\Sigma)$ is a natural subset of $\mathscr{T}(\Sigma^{2})$. In fact, we will therefore be able to write the partition function over $\Sigma$ in terms of objects defined on $\Sigma^{2}$. In particular, the determinants of interest on $\Sigma$ can be expressed in terms of determinants on $\Sigma^{2}$ using the following formulas BLAU1988285 :

		$\displaystyle\det(\Delta)_{g,\Sigma}=\left(\frac{\det^{\prime}(\Delta)_{g,\Sigma^{2}}}{\int_{\Sigma^{2}}\sqrt{g}}\right)^{\frac{1}{2}}(R_{\Sigma^{2},I}(J))^{\frac{1}{2}},$		(66)
		$\displaystyle\det(P_{1}^{\dagger}P_{1})_{g,\Sigma}=(\det(P_{1}^{\dagger}P_{1})_{g,\Sigma^{2}})^{\frac{1}{2}}.$

Here $R_{\Sigma^{2},I}(J)$ is BLAU1988285

$$R_{\Sigma^{2},I}(J)=\det[(1+\Gamma)\textrm{Im}\ \tau+(1-\Gamma)(\textrm{Im}\ \tau)^{-1}]$$

(67)

where $\tau$ and $\Gamma$ are defined as follows. First, we define a canonical homology basis of $A$ cycles and $B$ cycles on our Schottky double $\Sigma^{2}$. In general, the involution $I$ will take $A$ cycles to $A$ cycles and $B$ cycles to $B$ cycles. Then we define $\Gamma$ as the matrix corresponding to how the involution pushes forward A cycles to other A cycles:

$$I_{*}A_{i}=\Gamma_{ij}A_{j}.$$

(68)

In addition, if we let $\{\omega_{i}\}$ be a basis of holomorphic differentials on the Schottky double, normalised to the A-cycles in the sense that

$$\int_{A_{i}}\omega_{j}=\delta_{ij},$$

(69)

then we define the ‘period matrix’ $\tau$ as

$$\int_{B_{i}}\omega_{j}=\tau_{ij}.$$

(70)

Now we turn to the Weil-Petersson measure $d\mu_{\mathrm{WP}}$ on $\Sigma$. Let $\{\mu_{i}\}$ and $\{S_{i}\}$ be bases for the spaces of Beltrami and quadratic differentials on $\Sigma^{2}$ which are even under the involution $I$. Each Beltrami differential $\mu_{i}$ corresponds to a tangent vector in the Teichmuller space which we write as $\frac{d}{dm_{i}}$. Then the Weil-Petersson measure on $\Sigma$ can be written as BLAU1988285

$$d\mu_{\mathrm{WP},\Sigma}=\frac{\det<S_{i},\mu_{j}>}{(\det<S_{i},S_{j}>)^{\frac{1}{2}}}\prod_{i=1}^{3}dm_{i}.$$

(71)

This measure is on a subset of the Teichmuller space of $\Sigma^{2}$ which is isomorphic to the Teichmuller space of $\Sigma$ (in particular, it is on the subset of $\mathscr{T}(\Sigma^{2})$ which is invariant under $I$). Now we compile results, obtaining

$\displaystyle W_{\Sigma}=\int_{\mathscr{F}}d\mu_{\mathrm{WP},\Sigma}(R_{\Sigma^{2},I}(J))^{\frac{-13}{2}}\ \left[(\det(P_{1}^{\dagger}P_{1})_{g,\Sigma^{2}})^{\frac{1}{2}}\left(\frac{\det^{\prime}(\Delta)_{g,\Sigma^{2}}}{\int_{\Sigma^{2}}\sqrt{g}}\right)^{-13}\right]^{\frac{1}{2}}\ e^{-I(X_{cl},g)}.$

(72)

At this stage we review our strategy to obtain an explicit representation of the partition function on $\Sigma$. We wish to write the above integrand in terms of moduli parameters on $\Sigma^{2}$ and restrict the integration to the subset of $\mathscr{T}(\Sigma^{2})$ which is isomorphic to $\mathscr{T}(\Sigma)$ (in fact we restrict further to the moduli spaces). It is in fact easy to carry this out if one notes that the partition function on $\Sigma^{2}$ is given by Alvarez:1982zi

$$W_{\Sigma^{2}}=\int_{\mathscr{F}_{\Sigma^{2}}}d\mu_{\mathrm{WP},\Sigma^{2}}(\det(P_{1}^{\dagger}P_{1})_{g,\Sigma^{2}})^{\frac{1}{2}}\left(\frac{2\pi}{\int_{\Sigma^{2}}\sqrt{g}}\mathrm{det}^{\prime}(\Delta)_{g,\Sigma^{2}}\right)^{-13}\ .$$

(73)

This is the 2-loop closed bosonic string partition function, and its value is well-known from the literature. We see that a detailed analysis of the above integrand in terms of a good set of moduli parameters, combined with knowledge of how the region of integration should be restricted, will allow us to immediately obtain $W_{\Sigma}$.

The double torus can be represented as the complex plane (union infinity) with two pairs of non-intersecting circles cut out such that in each pair the two circles are isometric to each other; here, we give a summary of this representation as described, e.g., in TseytlinRG . The handles on the double torus are formed by identification of the boundaries of each pair of circles. To make this identification, we associate with each pair of circles $(I_{1},I^{\prime}_{1})$ and $(I_{2},I^{\prime}_{2})$ a Mobius transformation $T_{i}:\mathbb{C}\rightarrow\mathbb{C}$ where

		$\displaystyle T_{i}(z)\equiv\frac{A_{i}z+B_{i}}{C_{i}z+D_{i}},\hskip 14.22636ptA_{i}D_{i}-B_{i}C_{i}=1,\hskip 14.22636pti\in\{1,2\},$		(74)
		$\displaystyle T_{i}(I_{i})=I^{\prime}_{i},$

i.e. $T_{i}$ maps the circle $I_{i}$ to the isometric circle $I^{\prime}_{i}$. This leads to the fact that the isometric circles are given by

$$I_{i}=\{|C_{i}z+D|=1\},\hskip 14.22636ptI^{\prime}_{i}=\{|C_{i}z-A|=1\}.$$

(75)

The Schottky group $G$ is the group generated by these Mobius transformations. Then, the fundamental region of the double torus is the region of the complex plane exterior to the four circles $I_{1}$, $I^{\prime}_{1}$, $I_{2}$ and $I^{\prime}_{2}$.

Furthermore, it can be shown that any double torus can be represented by the fundamental region of such a Schottky group. Since each of the Mobius transformations $T_{i}$ can be parametrised by three complex numbers $(\xi_{i},\eta_{i},k_{i})$ where $k_{i}$ is the multiplier of $T_{i}$ and $\xi_{i}$ and $\eta_{i}$ are the repulsive and attractive fixed points of $T_{i}$, we have 6 complex parameters. However, if two sets of parameters are related by an overall $\mathrm{SL}(2,\mathbb{C})$ transformation, then the corresponding Riemann surfaces relate to the same point in moduli space.

This reduces the number of complex parameters to 3 as expected. In fact, we will write the integrand in a $\mathrm{SL}(2,\mathbb{C})$ invariant form, and divide out by the volume of $\mathrm{SL}(2,\mathbb{C})$. Now we write down the partition function for the 2-loop closed bosonic string DiVecchia:1987uf

	$\displaystyle W_{\Sigma^{2}}=$	$\displaystyle\frac{1}{\Omega}\int_{\mathrm{mod}(\Sigma^{2})}\prod_{i=1}^{2}\left(\frac{d^{2}k_{i}d^{2}\xi_{i}d^{2}\eta_{i}}{|k_{i}|^{4}|\eta_{i}-\xi_{i}|^{4}}|1-k_{i}|^{4}\right)(\det\ \mathrm{Im}\ \tau)^{-13}$		(76)
		$\displaystyle\ \ \ \ \ \times\prod^{\prime}_{\alpha}\prod_{n=1}^{\infty}|1-k_{\alpha}^{n}|^{-48}\prod^{\prime}_{\alpha}|1-k_{\alpha}|^{-4},$

where $\prod_{\alpha}^{\prime}$ is over all primitive elements of the Schottky group and $\Omega$ is the volume of the $\mathrm{SL}(2,\mathbb{C})$ group. We now assume that the Weil-Petersson measure in this expression is

$$d\mu_{\mathrm{WP},\Sigma^{2}}=\prod_{i=1}^{2}\left(\frac{d^{2}k_{i}d^{2}\xi_{i}d^{2}\eta_{i}}{|k_{i}|^{4}|\eta_{i}-\xi_{i}|^{4}}|1-k_{i}|^{4}\right).$$

(77)