Why do we regard $z$ and $\bar z$ as independent in CFT? [duplicate]

I have been studying String Theory and CFT for a while, and I am sad to say I do not know why we treat $$z$$ and $$\bar z$$ as independent variables, and why we go on to consider the algebra $$Vir\oplus\bar {Vir}$$ instead of simply $$Vir$$. The literature I followed never really explained this, just simply did it. Is there a clear explanation of this fact?

For simplicity let us consider global conformal symmetry rather than the full Virasoro symmetry. Global symmetry corresponds to the group $$SL(2,\mathbb{C})$$ acting on $$z$$ as $$z\to \frac{az+b}{cz+d}$$. So far $$\bar z$$ is the complex conjugate of $$z$$, not independent.
In a quantum theory we want the symmetry algebra to act on a complex space of states, not just on the geometry. So we have to complexify the symmetry algebra. In two dimensions, it turns out that the complexified symmetry algebra factorizes, $$sl(2,\mathbb{C})^\mathbb{C} = sl(2,\mathbb{C}) \oplus sl(2,\mathbb{C})$$ The two factors lead to Ward identities that involve $$z$$ and $$\bar z$$ independently: this can be interpreted as $$z$$ and $$\bar z$$ becoming independent. But this independence only holds at the level of Ward identities. Physical quantities (such as correlation functions) are functions of $$z\in \mathbb{C}$$, not $$(z,\bar z)\in\mathbb{C}^2$$.
• Note that it is only in Euclidean signature that $z=\bar z^*$. For example, in Lorentzian signature $z$ and $\bar z$ are real and independent. More generally, correlators are analytic functions of independent complex $z$ and $\bar z$ and interpolate between Euclidean and Lorentzian correlators, although they are only known to be analytic in certain subsets of $\mathbb{C}^2$. Jun 14, 2019 at 1:22