TITLE: Topologically distinct Calabi-Yau threefolds
QUESTION [28 upvotes]: In dimensions 1 and 2 there is only one, respectively 2, compact Kaehler manifolds with zero first Chern class, up to diffeomorphism. However, it is an open problem whether or not the number of topological types of such manifolds of dimension 3 (Calabi-Yau threefolds) is bounded. I would like to ask what is known in this direction. In particular, is it known that the Euler characteristic or the total Betti number of Calabi-Yau threefolds can't be arbitrarily large? Are there any mathematical (or physical?) reasons to expect either answer?
As a side question: I remember having heard several times something like "Calabi-Yau 3-folds parametrize (some kind of) vacua in string theory" but was never able to make precise sense of this. So any comments on this point or references accessible to mathematicians would be very welcome.

REPLY [10 votes]: I am going to give a rough account of the physical interpretation of Calabi-Yau manifolds in string theory.  
We are studying particles moving around in 10 spacetime dimensions.  We want to pick these 10 dimensions so that spacetime looks realistic, so we choose our Lorentzian 10-manifold to be $R^{1,3}\times X$ where $X$ is a compact 6-manifold.  If we choose $X$ to be sufficiently small then at low energies $X$ itself is not directly observable.  Thus to an observer who studies the world with low energy probes spacetime will have just the familiar 4 dimensions.
Nevertheless, one cannot chose $X$ in a completely arbitrary way.  Einstein's theory of general relativity tells us that the metric on $X$ carries energy measured by its Ricci curvature.  In the absence of matter, that is $in  \ the  \ vacuum,$ the spacetime must be in a lowest energy Ricci flat configuration.  So $X$ must be such that it admits a metric of vanishing Ricci curvature.  
To get to the Calabi-Yau condition we need to add one more physical criterion and this is called supersymmetry. Supersymmetry is an extremely attractive conjectural symmetry of nature that says that bosons (e.g. photons), and fermions (e.g. electrons) are paired together and related.  Mathematically fermions are described by spinor valued fields on spacetime, while bosons are described by functions or one-forms on spacetime.  To relate these objects, and thus have a theory with supersymmetry, what is required is a covariantly constant spinor field on $X$.  This constraint on $X$ reduces its holonomy from that of a general Riemanian 6-manifold, $SO(6)$, to $SU(3)$.  Indeed $SO(6)$ is locally isomorphic to $SU(4)$ and the spinor representation is $4 \oplus \bar{4}$ where 4 denotes the fundamental representation of $SU(4)$.  Since $X$ admits a covairiantly constant spinor its holonomy must be contained in the subgroup of $SU(4)$ that stabilizes a spinor, and it is clear form the decomposition above that this subgroup is $SU(3)$.  Hence for supersymmetry, $X$ must be Calabi-Yau.
Thus we can now clearly state exacty what it is that Calabi-Yau threefolds describe physically.  They are lowest energy vacuum configurations of string theory where the physics is supersymmetric.  For more details one might want to take a look at the book "Mirror Symmetry" by Vafa and Zaslow.