String Perturbation Theory and One-Loop Amplitudes
In this chapter we study issues of relevance for the perturbation theory of oriented bosonic strings. After giving a general description of world-sheets of higher genus, we discuss in some detail string one loop diagrams. We first do this for the closed s
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String Perturbation Theory and One-Loop Amplitudes
Abstract In this chapter we study issues of relevance for the perturbation theory of oriented bosonic strings. After giving a general description of world-sheets of higher genus, we discuss in some detail string one loop diagrams. We first do this for the closed string leading to torus diagrams, which we discuss both for the bosonic string and, continuing our presentation from Chap. 4, also for abstract conformal field theories. In this context we also present the simple current method, which provides a powerful tool for generating modular invariant partition functions. We also discuss the one-loop amplitude for open strings. From the one-loop amplitude of an open string stretched between two bosonic Dp-branes we extract the D-brane tension.
6.1 String Perturbation Expansion String world-sheets are two dimensional surfaces. Clearly, fixing the number of inand out-going strings does not yet specify the world-sheet. More complicated worldsheets intuitively correspond to higher orders in perturbation theory. To illustrate this, we look at the closed oriented string whose world-sheets are orientable surfaces. Consider as an example the tree level scattering amplitude of four strings shown in Fig. 6.1. The interactions of strings result from their splitting and joining. The corresponding world-sheet has tubes extending into the past and the future for incoming and outgoing closed strings, respectively. In the Polyakov formulation1 a scattering amplitude is a functional integral over oriented surfaces bounded by the position curves of the initial and final string configurations, weighted with the exponential of the free action (Polyakov action) and integrated with the string wave functions. The key observation is now that conformal invariance allows to consider punctured world-sheets instead of surfaces with boundaries
1
The alternative operator approach leads to the same results.
R. Blumenhagen et al., Basic Concepts of String Theory, Theoretical and Mathematical Physics, DOI 10.1007/978-3-642-29497-6 6, © Springer-Verlag Berlin Heidelberg 2013
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6 String Perturbation Theory and One-Loop Amplitudes
Fig. 6.1 Tree level scattering of four closed strings
Fig. 6.2 Map of asymptotic string states to points on the sphere
corresponding to incoming and outgoing strings. The incoming and outgoing strings are conformally mapped to points (the punctures) of the two-dimensional surface (see Fig. 6.2). Consider, for example, the case of a world-sheet with only one incoming and one outgoing string, described by a cylinder with metric ds 2 D d 2 C d 2 , 1 < < 1, 0 < 2. From here on, unless specified otherwise, we will always work with world-sheets with Euclidean signature metrics. There are several reasons for this. First, it allows us to use techniques of two-dimensional CFT, like the ones we have developed in Chap. 4 and the mathematics of Riemann surfaces. Second, Riemann surfaces generally do not admit non-singular Lorentzian metrics. The only exceptio
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