What is the motivation for using Calabi-Yau manifolds in string theory?

Question

I have just begin to study Calabi-Yau compactification. Looking in many book I found that, if we start with a critical superstring theory in $D=10$, we are in search of a compact $D=6$ Calabi-Yau manifold, i.e. a manifold with a spinor that is transported parallely. We do this because we want to conserve some supersymmetry. The thing that I did not understand is why we search only this kind of manifold (which, I read, exists in every even dimension) for the compactified, six-dimensional part of the space time. Why we do not care of the supersymmetry of the remaining four dimensional, "physical" non-compact, spacetime?

Answer 1

The reason is clearly given in the famous paper "Vacuum configurations for superstrings" - http://www.sciencedirect.com/science/article/pii/0550321385906029 -. Here, I am just copying the introduction of that paper. I cannot tell the reason why with better words.

Recently, the discovery [6] of anomaly cancellation in a modified version of d = 10 supergravity and superstring theory with gauge group $O(32)$ or $E_8 \times E_8$ has opened the possibility that these theories might be phenomenologically realistic as well as mathematically consistent. A new string theory with $E_8 \times E_8$ gauge group has recently been constructed [7] along with a second $O(32)$ theory.

For these theories to be realistic, it is necessary that the vacuum state be of the form $M_4 \times K$, where $M_4$ is four-dimensional Minkowski space and K is some compact six-dimensional manifold. (Indeed, Kaluza-Klein theory- with its now widely accepted interpretation that all dimensions are on the same logical footing - was first proposed [8] in an effort to make sense out of higher-dimensional string theories). Quantum numbers of quarks and leptons are then determined by topological invariants of K and of an $O(32)$ or $E_8 \times E_8$ gauge field defined on K [9]. Such considerations, however, are far from uniquely determining K.

In this paper, we will discuss some considerations, which, if valid, come very close to determining K uniquely. We require

(i) The geometry to be of the form $H_4 \times K$, where $H_4$ is a maximally symmetric spacetime.

(ii) There should be an unbroken N = 1 supersymmetry in four dimensions. General arguments [10] and explicit demonstrations [11] have shown that supersymmetry may play an essential role in resolving the gauge hierarchy or Dirac large numbers problem. These arguments require that supersymmetry is unbroken at the Planck (or compactification) scale.

(iii) The gauge group and fermion spectrum should be realistic.

These requirements turn out to be extremely restrictive. In previous ten-dimensional supergravity theories, supersymmetric configurations have never given rise to chiral fermions- let alone to a realistic spectrum. However, the modification introduced by Green and Schwarz to produce an anomaly-free field theory also makes it possible to satisfy these requirements. We will see that unbroken N = 1 supersymmetry requires that K have, for perturbatively accessible configurations, $SU(3)$ holonomy* and that the four-dimensional cosmological constant vanish. The existence of spaces with $SU(3)$ holonomy was conjectured by Calabi [12] and proved by Yau [13].

Answer 2

The root of it the so-called anomaly, which makes superstring theories mathematically inconsistent. The only dimension where it may vanish is the "critical" one (10), but that in itself is not enough if the global topology is non-trivial. And it has to be non-trivial to compactify the extra dimensions. Since the physical spacetime is topologically trivial the issue reduces to the compactified part only, and implies that its first Chern class must vanish, making it Calabi-Yau.