The ‘real’ spacetime in (super)string theories is a derived concept; it is spanned by a part of the field space of the underlying world-sheet field theory. In the simplest version, the fields involved are all bosons and they may be interpreted as mapping the world-sheet into the target spacetime. That is, the effective spacetime, X, is the space in which the fields of the underlying world-sheet are taking values. In a little more complicated cases, additional constraints might be imposed on a selection of the fields, such as periodicity conditions: in effect, the corresponding directions of the spacetime X are made periodic, and so compact (akin to folding up the infinite plain into a cylinder). The uncompactified directions ought to correspond to the observable 3+1-dimensional spacetime, while the compactifid ones ought to be shrunk beyond (current) observation. More complicated models fail to have such a simple interpretation. Nevertheless, a part of the field space ought always to be interpretable as the observable spacetime - if realistic application is desired; the remaining part of the field space remains, in a sense, internal.
This internal (field) space will in general depend on parameters. If the internal space has a geometric interpretation, such as a Calabi-Yau variety, these parameters (modulo certain redundances) are the moduli of the Calabi-Yau variety. The choice of these parameters fixes the internal space, and with it the features of the effective (and observable) spacetime field theory. The possible values of these parameters is determined by the moduli space geometry and parameter space dynamics, and can be studied using the cohomology rings of the internal space, which in turn is simplified by using the mirror and duality transforms.
At any rate, the internal space parameters (moduli) may be allowed to vary over the effective spacetime, whereby the details of the compactification also vary, and so also the features of the effective (and observable) spacetime field theory. Rather generally, this variation will lead to certain critical behavior of the compactifying space at select locations in spacetime, leading therefore to a critical behavior of the effective spacetime field theory. In particular, the effective spacetime metric will also become critical at these select locations, which thereby become recoginzed as macroscopic objects, such as black holes, cosmic strings, cosmic domain walls, etc.
© Tristan Hübsch, 2008