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.
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- 1.
The alternative operator approach leads to the same results.
- 2.
To have a Lorentzian metric one needs a globally defined vector field which defines the direction of time. Given such a vector field t αand a Riemannian metric g αβwhich always exists on the surfaces of interest, the metric
$$\begin{array}{rcl}{ h}_{\alpha \beta } = {g}_{\alpha \beta } - 2{ {g}_{\alpha \gamma }{g}_{\beta \delta }{t}^{\gamma }{t}^{\delta } \over ({g}_{\gamma \delta }{t}^{\gamma }{t}^{\delta })} & & \\ \end{array}$$is Lorentzian. For a surface without boundary such a vector field only exists on the torus. For surfaces with boundaries a Lorentzian metric exists on the cylinder but not e.g. for the surface shown on the left hand side of Fig. 6.2, if we require that the boundary components are space-like, e.g. time flowing in from the left and out to the right boundary circles. It is clear that the ‘arrow of time’ must merge and split somewhere where it is not well defined.
- 3.
In previous chapters we have seen how the requirement of conformal or BRST invariance puts them on shell.
- 4.
For higher loop amplitudes, e.g. when discussing questions of finiteness of string amplitudes or to prove unitarity of the theory, these tools are necessary. They are fully under control for the bosonic string. For the superstring they are much more subtle and not yet fully worked out in all detail.
- 5.
If the boundary has corners, there is a correction term \({ 1 \over 2\pi } \sum\limits_{i}(\pi - {\theta }_{i})\), where the sum is over all interior angles.
- 6.
For non-orientable surfaces one goes to the orientable double cover (which always exists) with twice as many boundary components; (6.3) then computes twice the Euler number of the non-orientable surface.
- 7.
The choice of a good gauge slice for the world-sheet gravitino of the RNS string is considerably more difficult than for the metric for the bosonic string. We will not discuss these issues.
- 8.
The transformation \(z \rightarrow f(\bar{z}),\,\bar{z} \rightarrow \bar{ w} =\bar{ f}(z)\)reverses the orientation. The determinant of the Jacobian is \(-\vert {\partial }_{z}\bar{f}{\vert }^{2} < 0\).
- 9.
The existence of conformal coordinates in Euclidean signature is guaranteed by the following result: Consider a metric \(d{s}^{2} = {g}_{ij}(x)d{x}^{i}d{x}^{j}\)written in a local coordinate system. We can define local complex coordinates \(z = {x}^{1} + i{x}^{2}\)so that the metric has the general form
$$d{s}^{2} = 2{e}^{2\varphi }\vert dz + \mu \,d\bar{z}{\vert }^{2}.$$(6.17)Via a non-holomorphic coordinate transformation \(z \rightarrow w(z,\bar{z})\)one can bring the metric (6.17) to the form \(d{s}^{2} = 2{e}^{2\tilde{\varphi }}\vert dw{\vert }^{2}\)with \(\tilde{\varphi } = \varphi -\ln \vert {\partial }_{z}w\vert \)provided wis a solution of the Beltrami equation \({\partial }_{\bar{z}}w = \mu {\partial }_{z}w\). The Jacobian of this coordinate transformation is \(\vert \partial w{\vert }^{2} -\vert \overline{\partial }w{\vert }^{2}\)which must be positive to preserve the orientation. Then | μ | 2 < 1. One can show that a solution of the Beltrami equation always exists locally.
- 10.
This is even true for non-orientable world-sheets as each such surface has a double cover which is a Riemann surface.
- 11.
Note that \(\vert \vert {\psi }_{0}\vert {\vert }^{2} = ({\phi }^{i}\vert {\phi }^{j}){\psi }_{0}^{i}\bar{{\psi }}_{0}^{j}\).
- 12.
If we have less then three vertex operator insertions the subgroup of \(PSL(2, \mathbb{C})\)which leaves their positions fixed is non-compact and has infinite volume.
- 13.
With the exception of L − 1 | 0⟩, but this is irrelevant here since we are considering excitations of states | p⟩ and \({L}_{-1}\vert p\rangle eq 0\).
- 14.
Proof: assume \(J \times {\phi }_{1} = {\phi }_{2} + {\phi }_{3} + \ldots \), multiply both sides with J cand obtain \(Vdash \times {\phi }_{1} = {J}^{c} \times {\phi }_{2} + {J}^{c} \times {\phi }_{3} + \ldots \)which is in contradiction with the fusion rule of the identity operator.
- 15.
If the cylinder is mapped to the plane its image is an annulus. The map is \(z =\exp ({ i \over t} (\tau - i\sigma ))\). The modular parameter t′of the annulus is the ratio of the radii of the two boundary circles. Two fundamental regions are \(0 < t \prime \leq 1\)and \(1 \leq t \prime < \infty \)which can be mapped to each other by an Stransformation \(t \prime \rightarrow 1/t \prime \).
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© 2012 Springer-Verlag Berlin Heidelberg
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Blumenhagen, R., Lüst, D., Theisen, S. (2012). String Perturbation Theory and One-Loop Amplitudes. In: Basic Concepts of String Theory. Theoretical and Mathematical Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29497-6_6
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DOI: https://doi.org/10.1007/978-3-642-29497-6_6
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