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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!

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  • $\begingroup$ 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 $\endgroup$
    – jdm
    Commented Aug 25, 2011 at 15:28
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    $\begingroup$ 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. $\endgroup$
    – Willie Wong
    Commented Aug 25, 2011 at 17:24
  • $\begingroup$ @Willie, Ahh yes, that does sound very sensible and somehow rings a bell. I should really start reading things more carefully... $\endgroup$
    – Michael
    Commented Aug 25, 2011 at 17:48
  • $\begingroup$ 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. $\endgroup$
    – Willie Wong
    Commented Aug 25, 2011 at 22:26

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Expanding on Willie Wong's comment: to have a geometrical compactification, you must assume that geometry is still approximately valid at the string scale, so that you can use supergravity. The arguments for Calabi-Yau compactification were given by Candelas Horowitz Strominger and Witten in 1985. They worked in the supergravity approximation, but the argument was only based on the supersymmetry of the low energy approximation, so the Calabi-Yaus are expected to lift to exact solutions of string theory.

The stringent condition is that there is a low energy supersymmetry that survives compactification. This means that there is a spinor which is covariantly constant, that is, which is unchanged under parallel transport in the compactification manifold. Parallel transport on a 6 dimensional manifold gives SO(6) rotation for each loop going from point x back to x, and SO(6) is SU(4) (p to a double cover, it's just like SU(2) and SO(3)) and if there is a spinor at x which is constant under these rotations, you can make it (1,0,0,0) by doing an SU(4) rotation, and then the only SU(4) rotations which leave it constant are the SU(3) acting on the last three components (this argument looks like it is a miracle of Lie-group algebra, and restricted to six dimensions, but in any even dimension 2n there is an embedding of SU(n) into SO(2n) which just pairs up the real coordinates into complex coordinates--- this is familiar from the reduction of the SO(10) GUT to the SU(5) GUT).

The definition of Calabi Yaus is that their parallel transport on loops is restricted to an SU(3). So the compactification manifolds which preserve exactly one supersymmetry are classified by Calabi Yaus.

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