2D CFT decoupling of holomorphic and anti-holomorphic parts

Question

I saw this post but it didn't really help me Decoupling of Holomorphic and Anti-holomorphic parts in 2D CFT

I am trying to fully understand as to why the holomorpic and anti-holomorphic part decouple. I feel like in a lots of lecture notes or books they just say at some point "we only look at holomorphic part because they decouple" but I dont see why this happens. As in at the level of the conformal group I indeed noticed that one always has $z \to w(z)$ and $\bar{z} \to \bar{w}(\bar{z})$ but I don't see how it "decouples" and what exactly decouples and how to put it back, as sometimes they say something like

$\phi(z) = ...$ then how would one find $\phi(z,\bar{z}) = ...$ or something like that.

Answer 1

This is in response to the question posed by OP at the end. I think the OP conflates two different "decouplings" here.

Generally speaking, a primary field depends on both $z$ and $\bar{z}$, and the correlation function factorizes, e.g.: $$ \langle \Phi_1(z, \bar{z}) \Phi_2(0,0) \rangle = G_{12}(z) \bar{G}_{12}(\bar{z}) $$ where $G$ ($\bar{G}$) is (anti)-holomorphic. This is true for all CFTs.

On the other hand, the free boson and fermion are special cases where the primary fields are themselves (anti)-holomorphic. It is a special feture of these theories that they can be written as the sum of purely holomorphic and and antiholomorphic parts. This doesn't necessarily happens in every CFT.