I am reading the Witten's topological twisting for $N = 2$ Superconformal Field Theory(SCFT) http://arxiv.org/abs/hep-th/9112056 In this paper Witten constructed 2 TQFTs i.e. A-model and B-model from an $N=2$ SCFT with a Kahler target manifold. My queries are the following :

When we do the twist, we're actually changing the interpretation of the fermionic fields e.g. we're taking the field $\psi_{+}^{i}$ to be a section of $\Phi^{\star}(T^{0,1}X)$ instead of a section of $K^{\frac{1}{2}}$ tensor $\Phi^{\star}(T^{0,1}X)$.Mathematically, this seems OK. But on the physics side, are we changing any physics.

Are the topological A and B models still supersymmetric. Does the twisting preserves any supersymmetric.

And is this type of twisting always possible, for any SCFT.

I am sorry for the question is not very clear, feel free to modify.