# Are there non-supersymmetric and/or non-Calabi-Yau topological sigma models?

I am reading some aspects of Mirror Symmetry and in mirror symmetry the $N=2$ SCFT on a Calabi Yau Manifold can be divided into two sectors each of which is a topological sigma model, A-Model and B-Model. After some research through some literature about the topological models, it seems that the topological models are constructed only on supersymmetric theory.

Are there any non -Supersymmetric topological sigma models?

Are there some topological models where the target space is not a Calabi-Yau manifold (or in general a Kahler manifold)?

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I have made two small edits, both to aid other readers. First, I have changed two periods into question marks &mdash; this post does include two questions, but a "question mark" really does mark a sentence as a question for people skimming. Second, I have modified the title to encapsulate the thrust of the question, and set it in the form of a question. –  Theo Johnson-Freyd Nov 1 '10 at 18:07
@Theo Thanks for your edits. –  J Verma Nov 1 '10 at 20:28

I believe that A-model does not require a Calabi-Yau target space. In fact, A-model is well-defined on any almost complex manifold, which was Witten's original construction (Comm. Math. Phys. Volume 118, Number 3 (1988), 411-449). On the other hand, B-model can only be defined on a Calabi-Yau manifold, which follows from anomaly cancelation.

In general, topological field theories have many different types (not necessarily supersymmetric). As an example, Chern-Simons theory is topological. Try http://en.wikipedia.org/wiki/Topological_quantum_field_theory for some general discussion.

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@ Moduli thanks for the answer..actually I am reading the Witten's topological twisting paper where he constructs A and B-models from a $N=2$ SCFT on a Kahler manifold, and the topological nature of A-model depends on the Kahler class on the target manifold. Can you suggest some references to look at A and B-models in generality. –  J Verma Nov 13 '10 at 18:56
A-model was discovered ealier than B-model. The original A-model paper (in whihc the name "A-model" has not been invented yet) is the one I gave. I am guessing the paper you are reading is the classic one "Mirror Manifolds And Topological Field Theory" (hep-th/9112056). If so, it was pointed out in that paper that A-model can be defined on almost complex manifolds, while B-model can only be defined on a Calabi-Yau manifold, which follows from anomaly cancelation. –  Moduli Nov 13 '10 at 21:08
@ Moduli ya you are right, I am reading the same paper you mentioned. I am under the impression (correct if I am wrong), that you need Kahler target manifold for $N=2$ Supersymmetry. And the other paper you mentioned is "Topological Sigma model", I always look through this paper on my laptop, But never read it. Do you suggest reading it, for the better understanding of the paper I am reading. I was just wondering,if there is some better,easy to read, review of sigma models in general and topological models in particular. Thanks –  J Verma Nov 14 '10 at 1:41