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

+ 2 like - 0 dislike
222 views

I'm looking for a reference that lists generators of two dimensional 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 $D=z\frac{\partial}{\partial z}$:

$$z\frac{\partial}{\partial z}\phi(z,\bar z)=\Delta\phi(z,\bar z)~~~,~~~\bar z\frac{\partial}{\partial \bar z}\phi(z,\bar z)=\bar\Delta\phi(z,\bar z)$$

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.

asked Aug 3, 2019

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOv$\varnothing$rflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.