We have seen that consistent 4D chiral string models lead
to supersymmetry. For phenomenological purposes,
we are interested in finding the effective action for the
light degrees of freedom, that means we want to integrate out
all the heavy degrees of freedom at the Planck scale
and compute the effective couplings among the light
states (massless at the Planck scale).
This will be
a standard field theoretical action with supersymmetry.
The on-shell massless spectrum of these models
have the graviton-gravitino multiplet
,
the gauge-gaugino multiplets
and the matter and moduli fields which fit
into chiral multiplets of the form
except for the dilaton field which together with the
antisymmetric tensor belong to a linear
multiplet
.
The most general couplings of supergravity to one linear
multiplet and several gauge and chiral multiplets is
*not* yet known, although some progress towards its construction has been done recently [30,31]. Nevertheless, as we mentioned in the
previous chapter, this field can be dualized to construct the
chiral multiplet with
and
. After performing this duality transformation we are
lead with a supergravity theory coupled only to gauge
and chiral multiplets.
The most general such action was constructed more than a decade ago
[32],
and therefore it is more convenient in this sense to work with the dual dilaton rather
than the stringy one . Although, the partial knowledge
about the lagrangian in terms of is enough to understand
most of the results we will mention next [31], we will
only mention the approach with the field which is
the most commonly used.
The general Lagrangian coupling supergravity
to gauge and chiral multiplets
depends on three arbitrary functions of the chiral multiplets:

- (1)The Kähler potential
which is a
*real*function. It determines the kinetic terms of the chiral fields

(14) - (2)The superpotential which is a
*holomorphic*function of the chiral multiplets (it does not depend on )^{}. determines the Yukawa couplings as well as the -term part of the scalar potential (known as -term because it originates after eliminating auxiliary fields associated to the chiral multiplets which are usually called ):

(15) - (3)The gauge kinetic function which
is also
*holomorphic*. It determines the gauge kinetic terms

(16)

(17)

The problem posed in this section is:
given a 4D string model, calculate the functions
. In order to do that let us
separate the fields into the moduli
the dilaton ,
^{} and the matter fields charged under the gauge group .
Most of the structure of the couplings depends
on the model. But there are some couplings which are model independent.
To extract them, the best procedure is to use all the
symmetries at hand. For this let us remark that 4D strings
are controlled by two perturbation expansions. One is the
expansion in the sigma-model (2D worldsheet) which is governed by the
expectation value of a modulus field (the size of the extra dimensions).
Whereas proper string perturbation theory is governed by the dilaton field .