In all of the sources I could find, the hermitian conjugate of a quasi-primary operator O with weights (h,ˉh) in (2D Euclidean) CFT is given as
O(z,ˉz)†=ˉz−2hz−2ˉhO(1ˉz,1z)
The justification is then, that we require operators to be self-adjoint in Lorentzian signature. The Euclidean time x0=it is then inverted by complex conjugation, such that the coordinate z=ex0+ix1 is effectively inverted: z↦1/ˉz. But how does this justify the prefactors?
Since inversion is not an orientation preserving transformation (meaning ∂z(1/ˉz)=0), the usual formula for conformal transformation of a quasi primary field
O(z,ˉz)↦(∂f(z)∂z)h(∂ˉf(ˉz)∂ˉz)ˉhO(f(z),ˉf(ˉz))
does not apply directly. However I would guess that one could compose such a transformation with a parity transformation (which I will take as complex conjugation (?)) to get the same effect. That is, first transform z↦1/z and then perform an overall complex conjugation.
The first part would look like
O(z,ˉz)↦(−1z2)h(−1ˉz2)ˉhO(1z,1ˉz)
and performing an overall complex conjugation would then result in
O(z,ˉz)↦(−1ˉz2)h(−1z2)ˉhO(1ˉz,1z).
Evidently this is still not the result that we want. What happens to the sign in the bracket? Why does no one even bother to mention it (from what I've been able to find)? Was my assumption, that we just need to conjugate the whole expression in the last step false?