14 BOSONIC STRING THEORY

In the present chapter, we study the one-loop amplitude of bosonic string theory. Our example is the simplest one: closed, oriented bosonic strings in 26-dimensional Euclidean space. ¹ The action is the Polyakov action

$$S=\frac{1}{2\pi }{\int }_{{\Sigma }_g}{\mathrm{d}}^2\xi \sqrt{\gamma }{\gamma }^{\alpha \beta }{\partial }_{\alpha }{X}^{\mu }{\partial }_{\beta }{X}_{\mu }-\frac{\lambda }{4\pi }{\int }_{{\Sigma }_g}{\mathrm{d}}^2\xi \sqrt{\gamma }\mathcal{R}\text{\hspace{1em}(14.1)}$$

where ${\Sigma }_g$ is a Riemann surface with genus $g$ . The second term is proportional to the Euler characteristic $\chi =2-2g$ and, hence, determines the relative ratio of multi-loop amplitudes; the $g$ -loop amplitude is proportional to $\exp (-\lambda g)$ . We have not written down the possible counter terms explicitly.

In the following sections, we work out the path integral formalism of bosonic strings. We first develop the necessary mathematical tools, namely differential geometry on Riemann surfaces. Then the path integral expression for the vacuum amplitude is written down. As an example, we compute the one-loop vacuum amplitude. Our exposition is based on D’Hoker and Phong (1986), Polchinski (1986) and Moore and Nelson (1986). There are many surveys of these topics, for example, Alvarez-Gaume and Nelson (1986), Bagger (1987), D’Hoker and Phong (1988) and Weinberg (1988).

Endnotes

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正文：英文 · OCR 机器稿 · 待校对

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Footnotes

1. The reason for $D=26$ will be clarified in section 14.2. ↩