I'm looking for a reference that lists generators of two dimensional global conformal group on a complex sphere in terms of differential operators that may act on quasi primary fields $\phi(z,\bar z)$. E.g. dilatation operator acts as:

$$D\phi(z,\bar z)=\left(-z\frac{\partial}{\partial z}+\Delta\right)\phi(z,\bar z)~~~,~~~\bar D \phi(z,\bar z)=\left(-\bar z\frac{\partial}{\partial \bar z}+\bar\Delta\right)\phi(z,\bar z)$$

with left moving and right moving dimensions $\Delta,\bar \Delta$.

How do the rest of the generators $P,J,K$ act?

This should be pretty standard stuff, but I've been googling a while now and it seems to be very elusive.


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    $\begingroup$ This is somewhere in the first few chapters of Francesco et al. $\endgroup$ – AccidentalFourierTransform Aug 3 at 13:08

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