This is a summary of Denef’s argument about the moduli in string theory in his paper/lecture notes arXiv:0803.1194[hep-th]. First I’ll give the list of moduli and then I’ll argue why they are always a problem.

List of moduli:

1 We always have the overall volume modulus, since g_{\mu \nu}\rightarrow r g_{\mu \nu} is a symmetry of vacuum Einstein equations.

2 The dilaton e^{\phi}, for all string theories. This is the string coupling g_s.

3 Axions. These comes from p-form potentials C_p integrated over nontrivial p-cycles \Sigma_p.

4 Metric moduli: complex structure(shape) and Kahler(size) moduli.

5 Brane deformation and/or vector bundle moduli. (Who wants to explain this please do so.)

There are thousands of these moduli and they are no good. Even if these light scalars only couple with matter with gravitational strength, they’ll induce a “long range fifth force” which is not observed. Also, it’s difficult to allow light scalars while maintaining the predictions of standard cosmology.

Dine and Seiberg have a famous argument about the universal problem of theoretical physics,

When the corrections are computable, they are not important; when they are important, they are not computable.

In terms of moduli stabilization, an example is given as follows.

Suppose we have the property that when the corresponding modulus \rho\rightarrow \infty, our string theory is weakly coupled. Then the quantum corrections will generate a potential for \rho, which will satisfy lim_{\rho \rightarrow \infty} V(\rho)=0. Now we have two choices for this potential. If it’s positive, then we’ll have our modulus runaway to infinity. If it’s negative, then our modulus will be trapped at strong coupling where we can’t trust our calculation. We can add higher order corrections to deform the potential so that we can stabilize the modulus at a finite nonzero regime. But it’s usually the case that the corrections we add are so important that we can’t ignore the even higher order corrections anymore, i.e. we are in the strong coupling regime again. Thus Dine and Seiberg concluded: we might as live in a strong coupling string vacuum now.