14.3 One-loop amplitudes

As an illustration of the formalism developed in the previous section, we compute the one-loop vacuum-to-vacuum amplitude of the closed orientable bosonic string theory. Since ${\dim }_{\mathbb{C}}\mathrm{Teich}({\Sigma }_1)=1$ and ${\dim }_{\mathbb{C}}\ker {\nabla }_z^{(-1)}=1$ , we have

$$Z_1={\int }_{\text{M o d}}\frac{\mathrm{d}\tau }{V(\mathrm{CKV})}\frac{|(\mu ,\varphi ){|}^2}{(\varphi ,\varphi )\cdot (\Phi ,\Phi )}\stackrel{\prime }{\det }{\Delta }_{(1)}^{+}{\left(\frac{{\det }^{\prime }\Delta }{\int {\mathrm{d}}^2\xi \sqrt{\gamma }}\right)}^{-13}.\text{\hspace{1em}(14.121)}$$

To evaluate (14.121) we need to take several steps.

14.3.1 Moduli spaces, CKV, Beltrami and quadratic differentials

In example 8.2, we have shown that the complex structure, namely the conformal structure, of the torus is specified by a complex parameter $\tau$ ( $\mathrm{Im}\tau >0$ ). Figure 8.3 shows the moduli space

$$\mathrm{Mod}\left({\Sigma }_g\right)={\mathcal{M}}_1/G=\mathrm{Teich}\left({\Sigma }_g\right)/\mathrm{SL}(2,\mathbb{Z})=H/\mathrm{SL}(2,\mathbb{Z})$$

where $H$ is the upper half-plane.

Take the torus $T_{\tau }$ specified by the Teichmüller parameter $\tau ={\tau }_1+\mathrm{i}{\tau }_2$ ( ${\tau }_2>0$ ). As a representative, we take a torus in figure 14.8. The metric in $\mathbb{C}$ naturally induces a flat metric (as guaranteed by the uniformization theorem)

Figure 14.8. The parallelogram whose complex structure is parametrized by $\tau$ .

$$\gamma =\frac{1}{2}[\mathrm{d}z\otimes \mathrm{d}\bar{z}+\mathrm{d}\bar{z}\otimes \mathrm{d}z].\text{\hspace{1em}(14.122)}$$

The CKV are globally defined holomorphic vectors. We take $\Phi =\alpha \partial /\partial z$ as the normalized basis of the CKV. The condition $(\Phi ,\Phi )=1$ yields $\int {\mathrm{d}}^{2}z|\alpha {|}^2={\tau }_2|\alpha {|}^2=1$ , that is $\alpha ={\tau }_2^{-1/2}$ (we have dropped the phase). The vector $\Phi$ generates translations in the complex plane,

$$z\to z^{\prime }=z+{\tau }_2^{-1/2}\left(v^1+\mathrm{i}{v}^2\right).\text{\hspace{1em}(14.123)}$$

We must note, however, that the translation is defined modulo the lattice; ${\tau }_2^{-1/2}(v^1+\mathrm{i}{v}^2)$ and ${\tau }_2^{-1/2}(v^1+\mathrm{i}{v}^2)+(m+\tau n)$ yield the identical translation. This forces ${\tau }_2^{-1/2}(v_1+\mathrm{i}{v}^2)$ to lie within the parallelogram of figure 14.8. Since

$${\tau }_2=\int {\mathrm{d}}^{2}z={\tau }_2^{-1}\int {\mathrm{d}}^{2}v$$

$V(\mathrm{CKV})$ is found to be

$$V(\mathrm{CKV})=\int {\mathrm{d}}^{2}v={\tau }_2^2.\text{\hspace{1em}(14.124)}$$

Our next task is to evaluate the Weil-Petersson measure. On the torus there is one quadratic differential $\varphi$ . Since $\varphi \in {\mathfrak{T}}^{-2}$ is a globally defined holomorphic differential, it must be of the form,

$$\varphi =a\mathrm{d}z\otimes \mathrm{d}za\in \mathbb{C}.\text{\hspace{1em}(14.125)}$$

To find the Beltrami differential, we evaluate the change of the metric under a small variation of $\tau$ . For this purpose, it is convenient to introduce the ${\xi }^{\alpha }$ -coordinate system in figure 14.8. The point A corresponds to $(1,0)$ and B to $(0,1)$ . Accordingly, we have $z={\xi }^1+\tau {\xi }^2$ . Under a small change $\delta \tau$ of the Teichmüller parameter, we have, up to a conformal factor,

$$\begin{array}{rl}& |\mathrm{d}z{|}^2\to |\mathrm{d}{\xi }^1+(\tau +\delta \tau )\mathrm{d}{\xi }^2{|}^2=|\mathrm{d}z+\delta \tau \mathrm{d}{\xi }^2{|}^2\\ & ={\left|\mathrm{d}z+\mathrm{d}\tau \frac{\mathrm{d}z-\mathrm{d}\bar{z}}{2\mathrm{i}{\tau }_2}\right|}^2=\left|\mathrm{d}z+\delta \tau \frac{\mathrm{id}\bar{z}}{2{\tau }_2}\right|.\end{array}$$

Comparing this with (14.110), we find that

$${\mu }_{zz}=\mathrm{i}/2{\tau }_2.\text{\hspace{1em}(14.126)}$$

Here $(\delta \tau )\mu$ is the complex conjugate of $(\delta \tau )\mu$ in (14.110). Of course, this is a reparametrization of the Teichmüller space and does not affect the results. If the reader feels awkward with this, s/he may choose $\bar{\tau }$ as the Teichmüller parameter. From (14.125) and (14.126), we have, up to irrelevant constants,

$$(\mu ,\varphi )=\int {\mathrm{d}}^{2}z\overline{{\mu }^{zz}}{\varphi }_{zz}=\frac{\mathrm{i}}{2{\tau }_2}a{\tau }_2\propto a$$

$$(\varphi ,\varphi )=\int {\mathrm{d}}^{2}z\overline{{\varphi }^{zz}}{\varphi }_{zz}=a^2{\tau }_2.$$

Finally, we have obtained

$$\frac{{\left|(\mu ,\varphi )\right|}^2}{(\varphi ,\varphi )}={\tau }_2^{-1}.\text{\hspace{1em}(14.127)}$$

14.3.2 The evaluation of determinants

We first consider ${\det }^{\prime }{P}_1^{†}{P}_1={\det }^{\prime }{\Delta }_{(1)}^{+}$ . Since we take a flat metric, the Laplacian takes quite a simple form,

$${\Delta }_{(1)}^{+}=-2{\partial }_z{\partial }_{\bar{z}}=\Delta \text{\hspace{1em}(14.128)}$$

where $\Delta$ is the Laplacian defined by (14.87b). Since

$$\int {\mathrm{d}}^2\xi \sqrt{\gamma }=\int {\mathrm{d}}^{2}z={\tau }_2$$

the amplitude (14.121) reduces to

$$Z_1={\int }_{\text{M o d}}\frac{\mathrm{d}\tau }{{\tau }_2^2}\frac{{\det }^{\prime }\Delta }{{\tau }_2}{\left(\frac{{\det }^{\prime }\Delta }{{\tau }_2}\right)}^{-13}\text{\hspace{1em}(14.129)}$$

$$\begin{array}{ccc}\uparrow & \uparrow & \uparrow \\ V(\mathsf{CKV})& \mathsf{W}-\mathsf{P}& \int {\mathsf{d}}^{2}z\end{array}$$

where we have used (14.124) and (14.127). We have factorized the integrand so that the modular invariance is manifest, see exercise 14.7.

Let us compute the spectrum of $\Delta$ . It is convenient to express the Laplacian in ${\xi }^{\alpha }$ -coordinates. From

$${\xi }^1=\mathrm{i}(\bar{\tau }z-\tau \bar{z})/2{\tau }_2{\xi }^2=(z-\bar{z})/2\mathrm{i}{\tau }_2\text{\hspace{1em}(14.130)}$$

we readily find that

$$\Delta =-\frac{1}{2{\tau }_2^2}\left[|\tau {|}^2{\left({\partial }_1\right)}^2-2{\tau }_1{\partial }_1{\partial }_2+{\left({\partial }_2\right)}^2\right]\text{\hspace{1em}(14.131)}$$

where ${\partial }_1=\partial /\partial {\xi }^1$ etc. The eigenfunction satisfying the periodic boundary condition on the torus is

$${\psi }_{m,n}(\xi )=\exp [2\pi \mathrm{i}(n{\xi }^1+m{\xi }^2)](m,n)\in {\mathbb{Z}}^2.\text{\hspace{1em}(14.132)}$$

Substituting this into (14.131), we find the eigenvalue

$${\lambda }_{m,n}=\frac{2{\pi }^2}{{\tau }_2^2}(m-\tau n)(m-\bar{\tau }n).\text{\hspace{1em}(14.133)}$$

The determinant is expressed as an infinite product:

$$\stackrel{\prime }{\det }\Delta =\prod _{m,n}^{\prime }\frac{2{\pi }^2}{{\tau }_2^2}|m+\tau n{|}^2\text{\hspace{1em}(14.134)}$$

the product being taken for all integers $(m,n)\ne (0,0)$

Clearly ${\det }^{\prime }\Delta$ is ill defined and needs to be regularized. Let us introduce the Eisenstein series (Siegel 1980, Lang 1987) defined by

$$E(\tau ,s)\equiv \sum _{m,n}^{\prime }\frac{{\tau }_2^s}{|m+\tau n{|}^{2s}}\text{\hspace{1em}(14.135)}$$

the summation being taken for all integers $(m,n)\ne (0,0)$ . This series converges for $\mathrm{Re}s>1$ and can be analytically continued to the complex $s$ -plane. The series $E(\tau ,s)$ has a simple pole at $s=1$ where we have a Laurent expansion,

$$E(\tau ,s)=\frac{\pi }{s-1}+2\pi [\gamma -\ln 2-\ln \left(\sqrt{{\tau }_2}|\eta (\tau ){|}^2\right)]+\mathcal{O}(s-1).\text{\hspace{1em}(14.136)}$$

This expression is known as the Kronecker first limit formula and is essential for our purposes. In (14.136), $\gamma =0.57721\dots$ is Euler’s constant and $\eta (\tau )$ is the Dedekind $\eta$ -function

$$\eta (\tau )\equiv {\mathrm{e}}^{\mathrm{i}\pi \tau /12}\prod _{n>1}(1-{\mathrm{e}}^{2\mathrm{i}\pi n\tau }).\text{\hspace{1em}(14.137)}$$

Neglecting constant factors, we have

$$\begin{array}{rl}\frac{{\det }^{\prime }\Delta }{{\tau }_2}& =\exp \left(-\ln {\tau }_2+\sum ^{\prime }\ln \frac{|m+\tau n{|}^2}{{\tau }_2^2}\right)\\ & =\exp \left(-\ln {\tau }_2-\frac{\partial }{\partial s}[{\tau }_2^{s}E(\tau ,s)]{|}_{s=0}\right)\\ & =\exp \{-\ln {\tau }_2[1+E(\tau ,0)]-E^{\prime }(\tau ,0)\}.\end{array}\text{\hspace{1em}(14.138)}$$

To evaluate the exponent, we note the functional equation,

$${\pi }^{-s}\Gamma (s)E(\tau ,s)={\pi }^{-(1-s)}\Gamma (1-s)E(\tau ,1-s).\text{\hspace{1em}(14.139)}$$

Taking the limit $s\to 0$ in (14.139), we have

$$\begin{array}{rl}sE(\tau ,1-s)& ={\pi }^{1-2s}\frac{\Gamma (1+s)}{\Gamma (1-s)}E(\tau ,s)\\ & =\pi (1-2s\ln \pi +\dots )\frac{(1-\gamma s+\cdots )}{(1+\gamma s+\cdots )}[E(\tau ,0)+E^{\prime }(\tau ,0)s+\dots ]\\ & =\pi E(\tau ,0)+\left[-2(\ln \pi +\gamma )E(\tau ,0)+E^{\prime }(\tau ,0)\right]\pi s+\dots .\end{array}$$

From (14.136), we also have

$$sE(\tau ,1-s)=-\pi +2\pi s[\gamma -\ln 2-\ln (\sqrt{{\tau }_2}|\eta (\tau ){|}^2)]+\dots .$$

Equating the coefficients of $s^0$ and $s^1$ , we find that

$$E(\tau ,0)=-1\text{\hspace{1em}(14.140a)}$$

$$E^{\prime }(\tau ,0)=-2\left[\ln 2\pi +\ln \left(\sqrt{{\tau }_2}|\eta (\tau ){|}^2\right)\right].\text{\hspace{1em}(14.140b)}$$

Substituting (14.140) into (14.138), we obtain

$$\frac{{\det }^{\prime }\Delta }{{\tau }_2}=\exp \left[-E^{\prime }(\tau ,0)\right]={\tau }_2|\eta (\tau ){|}^4.\text{\hspace{1em}(14.141)}$$

Finally, it follows from (14.129) and (14.141) that

$$Z_1={\int }_{\text{M o d}}\frac{\mathrm{d}\tau }{{\tau }_2^2}{\tau }_2^{-12}|\eta (\tau ){|}^{-48}.\text{\hspace{1em}(14.142)}$$

A neat form of $Z_1$ is obtained if we define the discriminant

$$\Delta (\tau )\equiv (2\pi {)}^{12}\eta (\tau {)}^{24}.\text{\hspace{1em}(14.143)}$$

Up to an irrelevant constant, the one-loop amplitude is

$$Z_1={\int }_{\text{M o d}}\frac{\mathrm{d}\tau }{{\tau }_2^2}{\tau }_2^{-12}|\Delta (\tau ){|}^{-2}.\text{\hspace{1em}(14.144)}$$

$\Delta (\tau )$ is known as the cusp form of weight 12, implying

$$\Delta \left(\frac{a\tau +b}{c\tau +d}\right)=(c\tau +d{)}^{12}\Delta (\tau )\text{\hspace{1em}(14.145)}$$

and $c(0)=0$ , where the $c(n)$ are the Fourier coefficients,

$$\Delta (\tau )=\sum _{n\ge 0}c(n){\mathrm{e}}^{2\pi n\mathrm{i}\tau }.\text{\hspace{1em}(14.146)}$$

Higher genus amplitudes are given by the cusp forms of other weights, see Belavin and Knizhnik (1986), Moore (1986), Gilbert (1986) and Morozov (1987).

Exercise 14.7

Show that

$$\eta (\tau +1)={\mathrm{e}}^{\pi \mathrm{i}/12}\eta (\tau )\eta (-1/\tau )=(-\mathrm{i}\tau {)}^{1/2}\eta (\tau )\text{\hspace{1em}(14.147)}$$

where the branch is chosen so that $\sqrt{z}>0$ if $z>0$ . Use this result to show that $\mathrm{d}\tau /{\tau }_2^2$ and ${\tau }_2^{-12}|\eta (\tau ){|}^{-48}$ are independently invariant under $\tau \to \tau +1$ and $\tau \to -1/\tau$ .

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