We have then succeeded in constructing 4D superstring models from toroidal compactifications and understand the full class of these models given by the moduli space . Unfortunately, all of these models have supersymmetry and therefore they are not interesting for phenomenology, because they are not chiral. To obtain a chiral model we should construct models with at most supersymmetry. If we still want to use the benefits of free 2D theories, we should construct models from flat space and modify only the boundary conditions. We have already considered identifications by shift symmetries of a lattice defining the tori. We still have the option to also use rotations and consider `twisted' boundary conditions [11]. As an example let us start with the torus discussed before. If we make the identification we are constructing the orbifold , shown in figure 2, where the twist is rotation by . This space is not a manifold because it is singular at the points left fixed by the rotation . Notice that, for instance, the point is fixed because it is transformed to which is identical to the original point after a lattice shift. In general, the discrete group of rotations defining the orbifold is called the point group , whereas the nonabelian group including the rotations and also the translations of the lattice , is the space group . So usually a torus is defined as and an orbifold .

We can easily construct 4D strings from orbifold compactifications in which the 10D spacetime of the heterotic string is the product of 4D flat spacetime and a sixdimensional orbifold . The heterotic string is particularly interesting because we can extend the action of the point group to the 16D lattice of the gauge group by embedding the action of the orbifold twist in the gauge degrees of freedom defined by the lattice, say. This can easily be done in two ways:
These embeddings on the gauge degrees of freedom allow us to break the gauge group, reduce the number of supersymmetries and generate chiral models in 4D as desired. The reason for this is the following: using the embedding by a shift , we start with the spectrum of the toroidal compactification and have to project out all the states that are not invariant by the orbifold twist. For the gauge group, only the elements satisfying remain, where , breaking the gauge group to a subgroup of the same rank. The four gravitinos of the toroidal compactification also transform and depending on the orbifold twist they are reduced to only one or two invariant states, indicating that there is only or supersymmetry. Actually there are only four twists leading to (for ) and some twenty or twists leading to supersymmetry [12], which are the phenomenologically interesting ones. For each of these twists we can have several () different embeddings on the gauge degrees of freedom. One of these embeddings is called the standard embedding because it acts identically in the gauge degrees of freedom as in the 6D space, this embedding also describes compactifications of the type II strings and is distinguished because in the 2D worldsheet, the corresponding model has two supersymmetries on the leftmovers and two supersymmetries on the rightmovers, the corresponding models are called models. All other embeddings do not have supersymmetry in the left moving side and are called models.
On top of all these embeddings we can also add Wilson lines [13] , by embedding the shifts of the lattice defining the 6D compactified torus, on the gauge degrees of freedom in terms of further shifts of the 16D gauge lattice, which will further break the gauge group. This increases the number of possible consistent models by a large amount, which we can only estimate between millions and billions because some may turn out to be actually equivalent. The Wilson lines can also be interpreted as the full embedding of the space group in the gauge degrees of freedom. In this case, using the two alternative embeddings mentioned above will give a completely different result because in the first case, both and the Wilson lines will act as shifts and so the embedding is abelian, whereas in the second option we will have both shifts and twists so the embedding is nonabelian. This possibility allows for two important properties: the Wilson lines are continuous rather than quantized and the rank of the gauge group can be reduced. In the absence of Wilson lines both embeddings are equivalent. Both classes of embeddings can be obtained by starting with the Narain lattice of toroidal compactifications and twist it in a consistent manner. This already takes into account the discrete and continuous Wilson lines (which were already present, parametrizing ) and also allows for the possibility of performing leftright asymmetric twists, the socalled asymmetric orbifolds [14]. This extra degree of freedom increases the number of possible models.
We can see now how a vast amount of heterotic string orbifold models can be generated. There are many (billions?) classes of models; different classes differentiated by the choice of original 6D toroidal lattice, the orbifold point group, the embeddings and the discrete Wilson lines. But each of these discrete choices allows for a variation of different continuous parameters such as the moduli fields (like ), the continuous Wilson lines (that correspond to charged untwisted sector moduli fields) and there are also charged twistedsector moduli fields[15]. All the continuous parameters can be seen as flat potentials for fields in the effective field theoretical effective action.
Only a few of these classes of models
includes quasirealistic models. As an example
[16], the
model based on the orbifold with embedding
and nonvanishing Wilson lines given by:
(12) 