We saw that the orbifolds obtained from twisting the 6D tori can give rise to chiral models in 4D. Orbifolds are singular objects but they can be smoothed out by blowing-up the singularities at the fixed points. The resulting smooth manifold is a so-called Calabi-Yau manifold [17]. Mathematically, these are 6D complex manifolds with holonomy or equivalently vanishing first Chern class. They were actually the first standard Kaluza-Klein compactification considered in string theory, leading to chiral 4D models and generically gauge group , with a hidden gauge group.

The drawback of compactifications on Calabi Yau manifolds is that they are highly nontrivial spaces and we cannot describe the strings on such manifolds, contrary to what we did in the case of free theories such as tori and orbifolds. In particular we can not compute explicitly the couplings in the effective theory, except for the simplest renormalizable Yukawa couplings.

On the other hand, Calabi-Yau manifolds have been understood
much better during the past few years and have lead to some
beautiful and impressive results. In a way they are more general than orbifolds because an orbifold is only a particular
singular limit of a Calabi-Yau manifold. Also there are
other constructions of these manifolds which are
not related to orbifolds. They can be defined as
hypersurfaces in complex (weighted) projective spaces
where 's are the weights
of the corresponding coordinates for which there is the
identification
. The hypersurface is
defined as the vanishing locus of a polynomial of the
corresponding coordinates. For instance
the surface defined as:

(13) |

- (i) There are classes of moduli fields, generalizing
the fields and of the two-torus mentioned before.
The number of these fields is given by topological numbers
known as Hodge numbers
.
They correspond to the number of
complex harmonic forms that can be defined in the manifold
with holomorphic indices and antiholomorphic
indices. Then the number of complex structure fields (U)
is given by and the number of Kähler structure
fields () is given by .
Many of the forms correspond to the coefficients of
different monomials which can be added to the defining polynomial
that still give rise to the same space, other forms correspond
to the blowing up of possible singularities. Many of the forms
correspond to polynomial deformations of the defining surface,
but others are related with the reparation of singularities.
For
Calabi-Yau manifolds we have , therefore there is
always a special Kähler class deformation which can be thought
as the overall size of the corresponding manifold, it
is usually called, the Kähler form. All the other Hodge numbers
of Calabi-Yau manifolds are fixed (
)
- (ii) The gauge group in 4D is .
The matter fields transform as 's or
's
of . The number of each is also given by the Hodge numbers
and the number of generations is then
topological:
where
is the Euler number of the manifold. This is one of the
most appealing properties of these compactifications since they
imply that topology determines the number of quarks and leptons.
- (iii)Mirror symmetry [18]. Similar to the 2D toroidal compactifications,
it has been found that there is a mirror symmetry in Calabi-Yau
spaces that exchanges the moduli fields and
,
. This means that for every
Calabi-Yau manifold , there exists another
manifold which has the complex and Kähler structure
fields exchanged,
*ie*and opposite Euler number . The mirror symmetry of the two-torus described previously, is only a special example on which the manifold is its own mirror. Mirror symmetry is not only a nontrivial contribution of string theory to modern mathematics, but it has very interesting applications for computing effective Lagrangians as we will see in the next section. It also relates the geometrical modular symmetries associated to the fields of the manifold to generalized, stringy, -duality symmetries for the mirror and viceversa (see for instance [19] ). - (iv) Even though these string models are not completely understood in
terms of 2D CFTs, special points in the moduli space of a given
Calabi-Yau are CFTs, such as the orbifold compactifications mentioned
before. In fact some people believe that there is a one to one
correspondence
between string models with supersymmetry in the worldsheet and
Calabi-Yau manifolds. There is also a description of CFTs in terms of
effective Landau-Ginzburg Lagrangians which are intimately related
to Calabi-Yau compactifications as two phases of the same 2D theory
[20].
In particular the potential of that Landau-Ginzburg theory is
determined by the polynomial defining the Calabi-Yau hypersurface.
- (v) There are also few models with three generations that,
after the symmetry breaking, could lead to quasi-realistic
4D strings. One of these models was analyzed in some detail
[21].
Usually, models also lead to the existence of extra particles
of different kinds. They have been thoroughly studied because of
the potential experimental importance of detecting an
extra massive gauge boson (for a recent discussion see ref. [22] ).
- (vi) Although most of the Calabi-Yau models studied so far correspond to standard embedding in the gauge degrees of freedom ( models), there is also the possibility of constructing models by performing different embeddings, similar to the orbifold case. This increases substantially the number of string models of this construction [23].