# Riemann surface of four closed string interaction (Zwiebach section26.5)

1. Feb 27, 2010

### ismaili

There are three kinds of light-cone string diagrams for four closed string interactions. As displayed by fig. 26.10, 26.11 and 26.12 of section 26.5 of Zwiebach's book.

For each light-cone string diagram, it is characterized by two parameters, the time difference of the two interaction points, $$\Delta T$$ and the twist angle $$\theta$$.

As for the corresponding Riemann surfaces of the three diagrams, it turns out that the three Riemann surfaces comprise a Riemann sphere with four punctures.

The subtlety I don't understand is the resulting moduli space. It turns out to be $$M_{0,4}$$, the Riemann sphere with three punctures. The three kinds of light-cone string diagrams contribute three disks which are fully cover $$M_{0,4}$$. What I don't understand is, for the diagram, fig. 26.10, the corresponding moduli is a deformed disk. The last paragraph of this section mentioned that "...the one around $$\lambda = 0$$ has an unusual shape. You may take it as a good challenge to explain the features of Figure 26.13 by looking carefully at the $$\theta$$ dependence of each of the three string diagrams in the limit that the intermediate time goes to zero."

Thanks for any further or detail explanations.