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 $L=\int d^6 x \sqrt{-g} M_6^4 {\cal_R}$ 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 $ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}+R^2\tilde{g_{mn}}dy^m dy^n$, where $R^2$ is the volume of the 2d manifold $M_g$. Then the action can be written as $M_6^4R^2\int d^4x \sqrt{-g}((\int d^2y\sqrt{\tilde{g}}{\cal_R}_2)+R^2{\cal_R}_4)+...$.

We realize that $\int d^2y\sqrt{\tilde{g}}{\cal_R}_2$ is a topology constant which equals $\chi(M_g)=2-2g$ and rescale to Einstein frame(where we have canonical Einstein-Hilbert action $\int \sqrt{-g}{\cal_R}$):$g\rightarrow h=R^2 g$, we find the 4d lagrangian to be $M_4^2\int d^4 x\sqrt{-h}({\cal_R}_h-V(R)$ where $M_4^2=M_6^4 R^2$ and $V(R)\sim (2 g-2)\frac{1}{R^4}$. Apparently this one term is not enough to help us to stabilize the volume modulus $R(x)$.

Let’s add new ingrediants: suppose there is n units of magnetic flux on the 2d internal space $M_g$,i.e. $\int_{M_g} F=n$ then the term in 6d action $\int d^6 x \sqrt{-g}|F|^2$ can give a term proportional to $\frac{1}{R^2}$ given by $\int_{M_g}|F|^2=R^2\times (\frac{n}{R^2})^2$, after rescale to Einstein frame, we obtain $V(R)\sim (2 g-2)\frac{1}{R^4}+\frac{n^2}{R^6}$. Now if $g=0$, 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 $-m\frac{1}{R^4}$, which can stabilize the modulus even when we have $g=1$, 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.