# Generators of 2D global conformal group in terms of differential operators?

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