The method discussed in the previous section can be generalised to include more dimensions and more complicated fields. We can use it to reduce our ten-dimensional string effective action (3.7) to four dimensions. But let's consider the general case of an effective action in -dimensions that we want to reduce to () dimensions. So in the simplest case we assume that space-time is of the form , where has dimensions and has dimensions and all fields are independent of the coordinates of .
This the simplest way to compactify the extra dimensions and of course there are more interesting ways to do this compactification which have more realistic features, but they are also far more difficult. For the moment we will stick to the case in which the fields are taken to be independent of the extra coordinates.
We will use the following notation: Local coordinates of are ( ), internal coordinates of are , (). The total space has signature () and fields in this -dimensional space are denoted with a hat, as well as their coordinates (, , etc.), ( ). Quantities without a hat are then the ()-dimensional ones.
We begin with the first terms of the low energy string effective action (3.6), the Einstein term coupled to the Dilaton:
It is very useful to do this reduction with the aid of so called vielbeins (vierbeins or tetrads in four dimensions, vielbeins in any other dimension). With the use of these vielbeins we make the connection between curved coordinate systems and local Lorentz (flat) coordinates. See appendix A for a short introduction of these vielbeins.
On the same transformation grounds that we saw in the previous section we can determine the reduction of these vielbeins.
Let again the greek indices denote the curved indices, (
) where labels the internal coordinates.
And the hatted roman indices will denote the Lorentz indices in -dimensions, (
). We assume that the internal space is flat, so there is no difference between the internal 'curved' indices and the internal Lorentz indices. The vielbeins reduce in the following way :
We can put these expressions into the Einstein part of our effective action (3.18) and after a tedious calculation one finds
We also have to reduce the other part of our string effective action involving the anti-symmetric field tensor
We can derive that the following reduction of the -dimensional anti-symmetric field tensor gives us the correct transformation properties in dimensions
is a gauge tensor field:
. This means that the ()-dimensional fields will have the following transformation properties:
We can use the vielbeins to convert the curved indices of to Lorentz indices
and then use the reduced vielbeins ( and ) to convert them back to -dimensional curved indices:
Due to the dimensional reduction there arise extra terms in the definition in the field strength tensor
. These are the so called Abelian Chern-Simons terms.
We can put all these terms back into the action. The total reduced effective action, consisting of the Einstein part as well as the part with the anti-symmetric field tensor, can be written in a very symmetric form . The claim is that there is a global symmetry that keeps this action invariant. (see Appendix B on the symmetry group .)
First we have to introduce the scalar matrix ,
So the reduction of a Low energy String Effective Action in dimensions to dimensions results allways in a action (3.32) which has an symmetry. This dual symmetry is called T-Duality, where the 'T' stands for Target-space. In the case of the string effective action of the bosonic sector of the Heterotic string (3.7) we have to reduce from to four dimensions. This means and the reduced action will be invariant under transformations. This also means there will be 12 vector fields and will be a matrix with 36 independent scalars.
As we mentioned before, we expect that compactification on more complicated manifolds will give more realistic features in four dimensions. In fact there are very many ways to perform this compactification and it is not clear yet which of them should be the correct way.