Operator-state correpondence in CFT

[physicus]

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

 

[muppet]

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.