Why (in relatively nontechnical terms) are CalabiYau manifolds favored for compactified dimensions in string theory?
Solution 1
There is one simple reason: in such scenario the physics at the string scale has supersymmetry. Supersymmetry (more technically $N=1$ supersymmetry) has some nice phenomenological features that make it an attractive bridge between low energy physics and string theory. The existence of this symmetry translates directly to the requirement that the compactification manifold is CalabiYau.
Solution 2
Since the word "supersymmetry" did not appear in your list of forbidden words let me give you this answer:
Because CalabiYau manifolds leave unbroken some part of the original supersymmetry, which is advantageous for model building.
But there are alternatives to CalabiYaus, like flux compactififcations or large extra dimensions.
Solution 3
We can have compactifications over 7D manifolds with a $G_2$ holonomy, or an 8D manifold with an $SO(7)$ holonomy. We can have orbifolds, or flux compactifications. We can have warped compactifications like $AdS_5 \times S^5$.
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Gordon
Updated on October 19, 2020Comments

Gordon about 3 years
I was hoping for an answer in general terms avoiding things like holonomy, Chern classes, Kahler manifolds, fibre bundles and terms of similar ilk. Simply, what are the compelling reasons for restricting the landscape to admittedly bizarre CalabiYau manifolds? I have Yau's semipopular book but haven't read it yet, nor, obviously, String Theory Demystified :)

Urs Schreiber about 10 yearsSee also on the nLab for more: ncatlab.org/nlab/show/supersymmetry+and+CalabiYau+manifolds


Gordon over 12 yearsWell,yes I realise that superstring theory means supersymmetric string theory, but was unaware that restricted the manifolds of the extra dimensions in that way.

stupidity over 12 yearsThere is extended supersymmetry at very short distances, but then part of it is broken by what manifold you choose for compactification. For CalabiYau manifold the remaining amount is what is most attractive for phenomenology, the minimal amount.

lurscher over 12 yearshow can large extra dimensions be consistent with everyday experience?

Daniel Grumiller over 12 years"Large" means "much larger than the Planck scale", but it still can be tiny. Interestingly, experiments do not rule out large extra dimensions of submillimeter size. See, for instance, arxiv.org/abs/hepph/0011014