Operator-state correpondence in CFT

  • Thread starter physicus
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In summary, the operator-state correspondence in CFT is specifically true for conformal field theories. This is because the map from the cylinder to the complex plane is conformal, meaning that conformally invariant theories are not altered by the mapping. It is not applicable to general 2D QFTs on the cylinder. The correspondence may also be valid for boundary CFTs, but this is not certain.
  • #1
physicus
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I am having trouble to understand a conceptionally important points of the operator-state corredondence in CFT. I am using David Tong's script on string theory, chapter 4, to learn CFT. My questions are the following:
1. Why is the state-operator map only true for conformal field theories. If I consider a general 2D QFT on the cylinder, why can't I use the same procedure to map it to the complex plane and then identify states in the far past to operator insertions at the origin of the complex plane? Is this connected to the fact that the map from the cylinder to the complex plane is conformal and, therefore, conformally invariant theories are not altered by the mapping but general QFTs are?
2. Is the correspondence only true for CFTs on the cylinder or is it also valid for boundary CFTs?

Thank you very much for your help.

physicus
 
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  • #2
It's been a while since I thought about CFT, but from what I remember you're right about point 1. Afraid I can't help you with point 2. because I don't know what a "boundary CFT" is.
 

What is operator-state correspondence in CFT?

Operator-state correspondence in CFT refers to the relationship between operators and states in a conformal field theory. It states that a primary operator in a CFT corresponds to a primary state, and the operator's conformal dimensions determine the state's scaling dimensions.

What is the significance of operator-state correspondence in CFT?

Operator-state correspondence is significant because it allows us to study the properties and behavior of a CFT through its operators. It also provides a mathematical framework for understanding the duality between local operators and bulk states in the AdS/CFT correspondence.

How is operator-state correspondence used in CFT calculations?

Operator-state correspondence is used in CFT calculations to relate operators to their corresponding states and to determine the scaling dimensions of states from the conformal dimensions of operators. This allows us to make predictions about the behavior of the theory and its correlation functions.

What are the implications of operator-state correspondence for holography?

The implications of operator-state correspondence for holography are significant. It provides a bridge between the local operators in a CFT and the bulk states in the AdS space, allowing us to understand the relationship between the two and make predictions about the behavior of the bulk theory.

Are there any limitations or exceptions to operator-state correspondence in CFT?

There are some limitations and exceptions to operator-state correspondence in CFT. For example, it does not apply to non-unitary CFTs, and there may be cases where a single operator corresponds to multiple states. Additionally, there are situations where operators have non-vanishing two-point functions with states that are not primary.

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