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

[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.

 

[eljose79]

Thank you for bringing up this interesting topic about light-cone string diagrams and their corresponding Riemann surfaces. Let me try to provide some further explanation and details to help clarify the subtlety you mentioned.

First, the three kinds of light-cone string diagrams for four closed string interactions are indeed characterized by two parameters, the time difference \Delta T and the twist angle \theta. These parameters determine the shape and position of the strings in the diagrams, which in turn affect the resulting Riemann surfaces.

As you mentioned, the three Riemann surfaces formed by these diagrams comprise a Riemann sphere with four punctures. This is because each of the three diagrams contributes a disk to the moduli space M_{0,4}, which is essentially the space of all possible Riemann surfaces with four punctures. However, the subtlety arises when we look at the shape of these disks in more detail.

The diagram in fig. 26.10 corresponds to a deformed disk in the moduli space. This is due to the fact that as the intermediate time goes to zero, the shape and position of the strings in the diagram also change significantly. In other words, the twist angle \theta plays a crucial role in determining the shape of the resulting Riemann surface. This is why the last paragraph of section 26.5 mentions the unusual shape of the disk around \lambda = 0 and encourages further investigation into the \theta dependence of each diagram in this limit.

To better understand this, we can think of the \theta dependence as a way to continuously deform the disk in the moduli space. As \theta changes, the shape and position of the strings in the diagram also change, resulting in a different Riemann surface. This is why the moduli space M_{0,4} is not just a simple union of three disks, but rather a complex space with continuously varying shapes and positions of the disks.

I hope this explanation helps to clarify the subtlety you mentioned. It is indeed a challenging topic, but by carefully studying the \theta dependence of each diagram, we can gain a better understanding of the features shown in Figure 26.13. Thank you for your interest in this topic and for bringing it up for further discussion.