# 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.

Let me give quick answers to your questions:

1. Twisting per se does not change the physics: it merely allows one to identify a particular subsector of the theory. The topological theory is obtained by restriction to that subsector.

2. The topological A and B models are particular subsectors of the sigma model quantum field theory. Supersymmetries will generally not preserve those subsectors, hence the topological theories are no longer supersymmetric.

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 (My information is probably out of date, since I last looked at this topic 15 years ago!)

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@ Jose thanks for your valuable reply. Since the topological theory is a subsector of the original supersymmetric theory, are we not loosing some information while doing the twist, other than the SUSY. Do the A and B models completely characterize the original theory. –  J Verma Nov 15 '10 at 5:34
@J Verma: one has to distinguish the "twisted theory" from the "topological theory". Twisting reinterprets the fields in a way that it is easy to identify the BRST operator as a particular supersymmetry generator. In the topological theory, you compute correlation functions of fields in the kernel of the BRST operator and these are topological invariants. When you consider target manifolds which are not necessarily Calabi-Yau, the sigma model stops being supersymmetric, but the twisted model with its one "supersymmetry" (i.e., the BRST differential) persists. –  José Figueroa-O'Farrill Nov 15 '10 at 11:31