Why do CalabiYau manifolds crop up in string theory, and what their most useful and suggestive form?
The choice of CalabiYau manifolds was because it was originally believed that the compactified theory would have N=1 supersymmetry. In general compactification on arbitrary manifolds will not preserve the supersymmetry, but compactification on CalabiYau manifolds will. If you want a mathematical explanation of exactly why CY manifolds preserve just the right amount of supersymmetry I'm afraid this is far beyond me!
If the LHC fails to find low energy supersymmetry that puts us in an interesting position because CalabiYau manifolds would no longer be essential. However that's jumping the gun a bit.
Incidentally, I've also read Yau's book "The Shape of Inner Space" and I strongly recommend it for anyone interested in this area. It's targeted at a general audience rather than active researchers or students, but you will need a good grasp of maths to get the most from it. However it's a very clear explanation of what CY manifolds are and why they matter.
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John R Ramsden
Updated on September 15, 2020Comments

John R Ramsden about 3 years
Why do CalabiYau manifolds crop up in String Theory? From reading "The Shape of Inner Space", I gather one reason is of course that CalabiYaus 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 CYs, and what they are supposed to represent, would be very interesting.

Qmechanic over 11 yearsPossible duplicates: physics.stackexchange.com/q/4972/2451 , physics.stackexchange.com/q/13945/2451 and links therein.

Dilaton almost 11 yearsThis is a good, interesting, and important question; I would strongly disagree with this bein closed if it is not a duplicate of another question which already has a nice answer.
