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In Section 11.2.2 of the book on Conformal Field Theory by di Francesco, Mathieu, and Senechal (page 417), the two point function on the Upper Half Plane is written as being equal to the four point function in the CFT on the full complex plane:

$$G_{s}(y_1, y_2, \rho) \equiv \langle \sigma(z_1, \bar{\sigma}_1)\sigma(z_2, \bar{z}_2)\rangle_{UHP} = \langle \sigma(z_1)\sigma(z_2)\sigma(z_1^*)\sigma(z_2^*)\rangle$$

I have a number of questions about this:

  1. Is this doubling trick even applicable to the 2D Ising Model?

  2. When do operators factorize into their chiral and anti-chiral parts in general? It is clear that this should happen in free-field theories, but in interacting systems (or minimal models for that matter), when can this happen?

Also, the spin operator has conformal dimension $h = 1/16$, but this is presumably $\sigma(z)$. How does equating a two point function to a four point function respect conformal weights? Is it just because $\sigma(z_1)$ has half the conformal weight that $\sigma(z_1, \bar{z}_1)$ does?

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The short answer is that the doubling trick applies to correlators at the level of conformal blocks. It is not true that the two point function is the "same" as the four point function. Rather, one must look at the equality in a formal way as emphasizing the equivalence at a block-by-block level.

Thanks to Prof. John Cardy for a discussion.

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