These are some notes on arXiv:hep-th/0701050 by Denef, Douglas and Kachru.
Flux compactification is an ominous term that often scares people away. Here are some notes I come across can give a simple idea how to do moduli stabilization using flux compactification in 6 dimensions. We choose 6 dimensions because it equals 4(Minkowski spacetime we live in) plus 2(like a torus, the simplest low dimensional Calabi-Yau we can find).
The idea is that we start with Einstein-Hilbert action and dimensional reduce it to a 4d spacetime.
A compact 2d internal space can be characterized by its genus. (g is the number of holes in the “donut”, g=0 is sphere, g=1 is torus…). An ansatz for the 6d metric can be taken to be , where is the volume of the 2d manifold . Then the action can be written as .
We realize that is a topology constant which equals and rescale to Einstein frame(where we have canonical Einstein-Hilbert action ):, we find the 4d lagrangian to be where and . Apparently this one term is not enough to help us to stabilize the volume modulus .
Let’s add new ingrediants: suppose there is n units of magnetic flux on the 2d internal space ,i.e. then the term in 6d action can give a term proportional to given by , after rescale to Einstein frame, we obtain . Now if , it is easy to see that the two terms can compete with each other and stabilize R.
If further more we add m O planes(an ingrediant has negative tension), we have one more term in the potential , which can stabilize the modulus even when we have , i.e. a torus.
In 10 dimensional string theory, we adopt the same idea: inclusion of fluxes and branes and planes will give us a potential that eventually can stabilize all the moduli.