Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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)?

share|improve this question
1  
I have made two small edits, both to aid other readers. First, I have changed two periods into question marks — 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

1 Answer 1

up vote 2 down vote accepted

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.

share|improve this answer
    
@ 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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.