String Theory and M-Theory

458 Flux compactifications
This happens in M-theory, for example, due to higher-order quantum gravity
corrections to the D = 11 supergravity action, as is explained in Section 10.5.
Flux compactifications
Let us begin by considering compactifications of M-theory on manifolds that
are conformally Calabi–Yau four-folds. For these compactifications, the metric
differs from a Calabi–Yau metric by a conformal factor. Even though
these models are phenomenologically unrealistic, since they lead to threedimensional
Minkowski space-time, in some cases they are related to N = 1
theories in four dimensions. This relatively simple class of models illustrates
many of the main features of flux compactifications. More complicated examples,
such as type IIB and heterotic flux compactifications, are discussed
next. In the latter case nonzero fluxes require that the internal compactification
manifolds are non-Kähler but still complex. It is convenient to describe
them using a connection with torsion.
The dilaton and the radial modulus
Two examples of moduli are the dilaton, whose value determines the string
coupling constant, and the radial modulus, whose value determines the size
of the internal manifold. Classical analysis that neglects string loop and instanton
corrections is justified when the coupling constant is small enough.
Similarly, a supergravity approximation to string theory is justified when
the size of the internal manifold is large compared to the string scale. When
there is no potential that fixes these two moduli, as is the case in the absence
of fluxes, these moduli can be tuned so that these approximations are
arbitrarily good. Therefore, even though compactifications without fluxes
are unrealistic, at least one can be confident that the formulas have a regime
of validity. This is less obvious for flux compactifications with a stabilized
dilaton and radial modulus, but it will be shown that the supergravity approximation
has a regime of validity for flux compactifications of M-theory
on manifolds that are conformally Calabi–Yau four-folds.
More generally, moduli fields are stabilized dynamically in flux compactifications.
While this is certainly what one wants, it also raises new challenges.
How can one be sure that a classical supergravity approximation has any
validity at all, once the value of the radial modulus and the dilaton are stabilized?
There is generally a trade-off between the number of moduli that
are stabilized and the amount of mathematical control that one has. This
poses a challenge, since in a realistic model all moduli should be stabilized.