# Why is Compactification restricted to Toroids, Calabi-Yau et al?

I think I've missed this point somehow. I've just started with Compactification and so far, I don't really see why it is restricted to the above mentioned types of manifolds?

I have to admit, when studying T-Duality, I simply took Toroidal compactification as a kind of "why not"-thing.

Could someone point me in the right direction? Been skipping through my books to look for some explanation of this, but couldn't find anything. Thanks!

• I don't know how it is in string theory, so I'm not posting this as an answer, but in general you can have compactifications over many kinds of manifolds. The simplist case is $S^1$ (e.g. Kaluza-Klein). However, if you want to have chiral Fermions, you'd have to use the orbifold $S^1/Z_2$ (UED). I can imagine that the reasons for using Calabi-Yau manifolds are similar, to get the Fermions / the supersymmetry right. Possibly related: physics.stackexchange.com/q/4972
– jdm
Aug 25, 2011 at 15:28
• Not an expert, but my understand is that it is not the compactification that restricts the manifold type, but the requirement of supersymmetry. In principle you can take the product manifold between any compact Riemannian manifold and a space-time manifold, and by inserting in an appropriate fact you can make the extra dimensions "small". It is the supersymmetry requirement that leads to the compactified dimensions needing there holonomy groups be restricted. Aug 25, 2011 at 17:24
• @Willie, Ahh yes, that does sound very sensible and somehow rings a bell. I should really start reading things more carefully... Aug 25, 2011 at 17:48
• Egads, I must be really hungry when I typed that last comment. So many typos. "my understandING" in the first sentence. "an appropriate factorOR" in the third line, and "thEIR holonomy groups" in the last sentence. Sorry for any confusion. Aug 25, 2011 at 22:26