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why in homological mirror symmetry, we restrict us to a projective variety (calabi-yau)? Because in physics we don't need this condition. What's the general picture for general calabi-yau?

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I think that this is much more likely to get answers if posted on mathoverflow. –  Matt E Oct 11 '10 at 4:29

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Yau gives some quick explanation of this in his scholaropedia page for CY manifolds (which is really a great read if you're interested in such things). In particular, speaking about spacetime manifolds $\mathbb{R}^{3,1}\times X$ (with standard Minkowski metric on the first part), he says "for the most basic product models $N=1$ supersymmetry, the space $X$ must be a Calabi-Yau manifold of complex dimension $3$".

He references a 1985 paper of Candelas, Horowitz, Strominger and Witten.

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