Easy question on CFTs in string theory

In summary, a CFT (Conformal Field Theory) is a type of quantum field theory that describes the physics of strings in a particular background geometry. It is an integral part of string theory and can be seen as a generalization of CFTs. CFTs have properties such as conformal invariance, a stress-energy tensor, and a central charge. They are studied using analytical and numerical methods, and have applications in understanding black holes, the AdS/CFT correspondence, and condensed matter physics.
  • #1
physicus
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Hi, I have a question concerning simple bosonic string theory governed by the Polyakov action. Under which assumptions/requirements is the corresponding worldsheet theory a conformal field theory? Is this always true or does it depend on the topology of the worldsheet?

Conformal symmetry of the worldsheet theory arises from diffeomorphism and Weyl invariance after gauge fixing of the worldsheet metric. (Is that true? Why can the metric be gauge fixed? Why can one not fix "more", s.t. also the conformal invariance vanishes?)
I think my confusion arises from the fact, that in the lecture we studied the corresponding conformal field theory on the complex plane. I understand, that there is a conformal map, for example, from the cylinder to the complex plane. I do not see how one would obtain such a map if the topology of the worldsheet is more complicated. But this does not mean, that the worldsheet theory is not a conformal field theory, right?

Thank you very much for your help.
physicus
 
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  • #2


Dear physicus,

Thank you for your question. The conformal symmetry of the worldsheet theory in bosonic string theory is indeed a result of diffeomorphism and Weyl invariance. This means that the action remains invariant under reparametrization of the worldsheet coordinates and rescaling of the worldsheet metric. This is why the metric can be gauge fixed, as it does not affect the overall action.

It is important to note that the conformal symmetry of the worldsheet theory does not depend on the topology of the worldsheet. As long as the worldsheet action is diffeomorphism and Weyl invariant, it will have conformal symmetry. This is true regardless of whether the worldsheet is a simple plane, a cylinder, or a more complicated surface.

The reason we often study the conformal field theory on a complex plane is because it is a convenient choice for calculations and visualization. However, as you mentioned, there are conformal maps that can be used to map other topologies to the complex plane. This does not change the fact that the worldsheet theory is conformal.

I hope this helps clarify your confusion. Please let me know if you have any further questions.
 

Related to Easy question on CFTs in string theory

1. What is a CFT in string theory?

A CFT (Conformal Field Theory) is a type of quantum field theory that describes the physics of strings in a particular background geometry. It is a powerful tool for studying the dynamics of strings and their interactions in a specific spacetime.

2. How is a CFT related to string theory?

CFTs are an integral part of string theory, as they provide a framework for understanding the behavior of strings in different backgrounds. In fact, string theory can be seen as a generalization of CFTs, incorporating the dynamics of strings in a wider range of spacetimes.

3. What are some properties of CFTs?

CFTs have a number of key properties, including conformal invariance, which means that they are invariant under conformal transformations. They also have a stress-energy tensor, which describes the energy, momentum, and other properties of the system. Additionally, CFTs have a central charge, which is a measure of the symmetry of the theory.

4. How are CFTs studied in string theory?

CFTs in string theory are typically studied using a combination of analytical and numerical methods. This can include techniques such as perturbation theory, conformal bootstrap, and numerical simulations. Additionally, many researchers use computer programs and tools to aid in their study of CFTs.

5. What are some applications of CFTs in string theory?

CFTs have a wide range of applications in string theory, including their use in understanding the behavior of black holes, the AdS/CFT correspondence, and the holographic principle. They also have applications in condensed matter physics, where they can be used to describe the behavior of certain materials at a critical point.

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