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Why do Calabi-Yau manifolds crop up in String Theory? From reading "The Shape of Inner Space", I gather one reason is of course that Calabi-Yaus are vacuum solutions of the GR equations. But are there any other reasons?
Also, given those reasons, what tend to be the most physically suggestive and useful or amenable forms among the widely different expressions obtainable by birational transformations and other kinds?
Actually, just some examples of C-Ys, and what they are supposed to represent, would be very interesting.