Let us consider first the couplings generated at string tree-level
and also tree-level in the sigma-model expansion [34,35].
Besides the 4D Poincaré symmetry, supersymmetry and
gauge symmetries which determine the Cremmer *et al* Lagrangian,
we also can use the `axionic' symmetry:
This is a symmetry which for the 4D fields would
imply that can be shifted by an arbitrary imaginary constant.
There are also two scaling properties of the 4D Lagrangian
,
for which the Lagrangian
scales as
.
Also, given a scale , define
where gives Newton's constant in 10D. The transformations
,
with similar transformations for the other fields, imply that the
Lagrangian should scale as
. These scaling properties are not symmetries of the
Lagrangian but of the classical field equations and so
they can be used to restrict the form of the
*tree-level* effective action only.

Using these symmetries we can extract the full dependence of the
effective action on the dilaton field , which is the most
generic field in all compactifications. We conclude that at tree-level
in both expansions [35]:

With still undetermined. This is however a very crude approximation. What we really want is to know these functions at tree-level in the string expansion but

(19) |

Second, and more important, the superpotential above does not depend on which is the string loop-counting parameter, and therefore does not get renormalized in string perturbation theory! [39]. This means that we only need to compute at the tree level and it will not be changed by radiative corrections. This is the string version of the standard non-renormalization theorems of supersymmetric theories. Also for the superpotential vanishes, independent of the values of ( )! There are not self couplings among the `moduli' fields and therefore they represent flat directions in field space (see for instance [40] ). Notice that due to the non-renormalization theorems, this result is exact in string perturbation theory!. The only possibilty we have to lift this vacuum degeneracy is by nonperturbative string effects.

The quantity we have less information on, even at tree-level, is the
Kahler potential . It has been computed only for several
simple cases. For instance in the simplest possible Calabi Yau
compactification (
) a consistent
truncation from the 10D action gives [34]:

(20) |

(21) |

and compute the moduli dependent quantities . This has been done explicitly for some orbifold compactifications. For instance for factorized orbifolds, that is orbifolds of a 6D torus which is the product of three 2D tori , the dependence on the corresponding moduli fields is given by

and . Giving rise to the Kahler potential:

Where the fractional numbers are the `modular weights' of the fields with respect to the duality symmetries related to the moduli or . For instance, under duality, the fields transform as:

(24) |

Furthermore, for (Calabi-Yau) models, there is
a very interesting observation
[43]. Since these compactifications are also
compactifications of type II strings, and in that case
the corresponding 4D theory has supersymmetry, the moduli
dependent part of the Kähler potential, has to have the
same dependence as for the models which are much more restrictive.
This underlying structure has been very fruitful to extract
information on models and comes with the name of
`special geometry'.
This restricts the function which gives the metric in the moduli
space. First, the moduli space
of the forms and the forms
factorizes and so:

(25) |

(26) |

(27) |

(28) |

Since and are holomorphic, they may get similar constraints as the superpotential above. In particular, since counts sigma model loops, then is not renormalized and so the dependent part of the Kähler potential is given exactly by the tree-level result!. Similarly, the Yukawa couplings are exact at tree-level. Here is where the mirror symmetry explained in the previous section plays an important role. Since by mirror symmetry we understand that the compactification on the original manifold and its mirror represent the same CFT, and so the same string model, in the version with the manifold , the roles of and are interchanged, therefore, computing the dependent part of the Kähler potential in (which is exact at tree level) gives the dependent part of the Kähler potential in ! This fact has been used to compute explicitly the moduli dependent part of the Kähler potential in some examples. This overcomes the Calabi-Yau problem of not knowing the exact CFT behind the compactification, at least for these couplings.