Is a CFT$_2$ always holomorphically factorizable? I had this idea because that's what we usually see is taken in string theory e.g (taking $z$ and $\bar{z}$ as independent variables). E.g. Ginsparg explains in "Applied Conformal field theory" paper page 7, that due to the emergence of Virasoro symmetry group, it is "useful" to consider them as independent, though he said that we should take $z^*=\bar{z}$ later according to our convenience. But it would be nice to understand what's going on. E.g, does holomorphic factorizability becomes an essential in some CFT$_2$'s? E.g. if we are considering some field theory in AdS$_3$ (no Chern Simons term say), should we expect the boundary CFT to be factorizable (its then not a pure gravity any more)?
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