State operator correpondence and $S^1 \times S^1$ to $R^2$

Question

I want to know the "state operator correspondence" in detail.

What i want to know is the correspondence of S1×S1 to R2 How the states and the operator match to each other?

I know that for S1×S1 via radial quantization one circle plays a role for time, which produce the states.

And for R2, we have operators.

But i don't know how these two (states in S1×S1 and operators in R2) match each other. Can you explain this in detail? (Recommendation of any references will be helpful to me!)

This post is imported from Physics StackExchange at 2016-06-02 04:57 (UTC) posted by SE User phy_math, SE users associated with first three comments below are ACuriousMind, phy_math and ACuriousMind respectively.

Comments

I'm not sure what you are asking. The state-operator correspondence of CFTs means that there is a bijection between states of the theory and operators. What do you mean with the "corresponding of S1×S1 to R2", and how is this supposed to relate to the state-operator correspondence?

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@ACuriousMind, What i want to know how states and operator corresponds. i.e, how we know there is a bijection map between them?

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The proof of the state-operator correspondence should be in every good CFT resource. Please indicate what you don't understand about it. Also, you still have not explained what role S1×S1 or R2 are supposed to play here.