# Calculation of conformal block coefficients

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I am reading Di Franceso et Al Chapter 6. He defines the conformal blocks as

$$\mathcal{F}^{21}_{34}(p|x)=x^{h_p-h_3-h_4} \sum_{k} \beta ^{p\{ k_i\}}_{34} x^{K} \frac{\langle h_1| \phi_2 (1) L_{-k_1}\cdot \cdot L_{-k_N} |h_p \rangle}{ \langle h_1 | \phi_2(1) |h_p \rangle}\\= x^{h_p-h_3-h_4} \sum_{K=0}^{\infty} \mathcal{F}_Kx^{K}$$

These are the conformal blocks and I wan't to calculate the co-efficients $\mathcal{F}_K$ of its power series. The expectation value is taken between asymptotic states. The book quotes for for $\mathcal{F}_1$

$$\mathcal{F}_1=\frac{(h_p+h_2-h_1) (h_p+h_3-h_4)}{2h_p}$$ and I know the value of $\beta^{p\{1\}}_34=1/2$. I have no idea how to do this simple computation. I want to calculate

$$\frac{\langle h_1 | \phi_2(1) L_{-1}|h_p \rangle}{\langle h_1| \phi_2(1) |h_p \rangle}= \frac{\langle 0| \phi_1(x) \partial\phi_2(1) \phi_p(0 )|0 \rangle}{\langle 0| \phi_1(x) \phi_p(0 )|0 \rangle}$$

I got the second step after using the commutation relation $[L_{n},\phi(w, \bar{w})]=h(n+1)w^n \phi(w ,\bar{w})+w^{n+1} \partial \phi (w, \bar{w})$ and writing the asymptotic states in terms of their operator fields.

How do I proceed from here? I have no idea how to calculate the correlator in the numerator and denominator resp. I just know the OPE for product of two fields of the **same** conformal dimension, but how do I do this

asked Jun 2, 2014
reshown Aug 1, 2014

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