Let us continue the summary following Denef’s notes. arXiv:0803.1194 [hep-th]

All the discussion so far is based on a simple analysis with fluxes turned off. (Flux has the same definition as in Gauss’s Law in E & M but with more generalized potential field, e.g. n dimensional antisymmetric tensor).

When we turn the flux on, they give a nice tree level potential for the moduli field, so we can currently evade from the non-computable corrections problem. However, it’s not nice enough, since for a Ricci-flat (remember we start with a CY) internal space, we can always rescale the metric to make the potential depend on r^{-a} (r characterizes the rescale of the metric and a is some positive number that we don’t care much now). So the flux potential looks like r^{-a}\int_X F^2(X is the internal manifold and F is the field strength of the potential field), which means it’s positive definite and we have a runaway solution at r\rightarrow \infty.

The idea is that adding flux potential is going to deform our original Ricci-flat internal space to be nonflat anymore. The Einstein-Hilbert action scales differently from the flux potential and can compete with it(if it has the correct sign) to generate a minimum for the moduli at some finite field range. The problem with this is that for p\geq 2 form fluxes we’ll end up in an Ads space at the KK scale, which means we’ll need to find quantum corrections to lift the Ads to ds space and the quantum corrections are (again!) of the order of the potential we constructed with flux and geometry! Something interesting happens when we allow F_1\neq 0 or F_0\neg 0. I’ll refer the reader back to the Denef’s notes.

Now we’ll have to add a lot of other stuff (like D-branes, O-planes) in order to generate some vacua with general fluxes, so we’ll have a lot of terms in the potential, and intuitively we’ll get a minimum much easier. For example, for type IIB, we can use orientifold, O3 and O7 planes, D3 and D7 branes and NSNS and RR 3-form fluxes to stabilize all the complex structure (shape) moduli, and use non-perturbative effects to stabilize the Kahler(size) moduli. In conclusion, we might be able to produce some reasonably controlled vacua, but it’s not the simple smooth exact classical solutions we were hoping for anymore.

Hence, we are living in a dirty world.