Witten's topological twisting

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 :

1. 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.

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

3. 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.

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First, a historical note: the twisting procedure for $N=2$ SCFTs is due to Eguchi and Yang; although a twisting of sorts had already appeared in Witten's Topological Quantum Field Theory paper of 1988.

3. Any $N=2$ SCFT can be twisted in this way, but more generally one can twist also other conformal field theories. For example, one can twist string theories or also theories based on a Kazama algebra, such as the $G/G$ gauged Wess-Zumino-Witten model. More generally still, it used to be the case that all known two-dimensional topological conformal field theories are "cohomologically equivalent" to a twisted $N=2$ SCFT. (My information is probably out of date, since I last looked at this topic 15 years ago!)