Performing Integrations over Moduli Space for String Theory

[Ichigo449]

To prepare for a meeting I'm having with a prof in 2 weeks I've been told to compute things in string perturbation theory. During this process I have come to performing the calculation of the vacuum amplitude at one loop directly from Polyakov's action, as performed by Polchinski in 1986. I can understand the reduction to an integral over the fundamental domain of the moduli space of a torus, but have no idea how to proceed further. If you could offer any insights that would be fantastic, thank you.

 

[jvicens]

Hello,

It's great to hear that you are working on computing the vacuum amplitude at one loop using string perturbation theory. This is an important topic in theoretical physics and I am glad to see that you are taking on this challenge.

To start, it's important to understand the concept of string perturbation theory. This is a method used to calculate the interactions between strings, which are considered to be the fundamental building blocks of the universe. In this theory, the interactions are described by a series of corrections, known as perturbative corrections, which are calculated using mathematical techniques.

Now, let's focus on your specific question about computing the vacuum amplitude at one loop directly from Polyakov's action. The first step is to understand the concept of the Polyakov action. This is a mathematical expression that describes the dynamics of a string in a particular space-time. It is given by the sum of two terms: the Nambu-Goto action, which describes the motion of the string, and the Weyl term, which accounts for the string's intrinsic curvature.

To proceed further, you will need to use the technique of integration over the fundamental domain of the moduli space of a torus. This involves integrating over all possible shapes and sizes of the torus, which is the surface on which the string is propagating. This is a crucial step in calculating the vacuum amplitude at one loop, as it allows us to sum over all possible interactions between the strings.

To perform this integration, you will need to use the Riemann theta function, which is a mathematical tool used to evaluate integrals over the moduli space of a torus. This function is closely related to the Jacobian theta function, which you may have encountered in your studies.

I would also recommend consulting Polchinski's paper from 1986, as it provides a detailed explanation of the steps involved in computing the vacuum amplitude at one loop from Polyakov's action.

I hope this helps to provide some insights into your calculations. If you have any further questions, please don't hesitate to ask. Good luck with your meeting with the professor in two weeks!