Anti-Quill Part 1: Compactifications

This is the $CPT$-conjugated version of Aayush's very good Quill series. My series is conjugated in the sense that I won't be as mathematical or elaborate or excessively (and frankly concerningly) obsessed with algebraic geometry.

In string theory, compactifications reduce to the usual 4D Einsteinian gravity in the low energy limit, with  an ``external" (although usually this is considered the internal space) manifold $K$ with the extra dimensions surpassed into a small scale similar to the $S^{1}$ with small $r$ in Kaluza-Klein compactifications. In our case, 10D string theory would be $M_{4}\times K^{6}$ with Ricci flatness and a more special condition, that the first Chern class $c_{1}=0$. These lead us to Kahler manifolds and more specifically, Calabi-Yau manifolds. In general, what comprise compactifications are the following: moduli of massless scalar fields, a moduli space of geometries admissible under these moduli, addition of fluxes $\mathcal{F}$ from the theory, D-branes contributing with positive $T$ and orientifold planes contributing with negative $T$. The last three are a messy subject in an EFTs sense because of the tadpole cancellation conjecture in the swampland. A very simple example of stringy compactifications (although not very realistic here because we ignore perturbative corrections) is that of Freund-Rubin compactifications, where you can take magnetic fluxes across $S^{2}$ and write the effective potential in terms of the addition of these fluxes. This is usually an order $\sim 1/R^{6}$ (where $R$ is the volume modulus) against the genus $g$, but the negative curvature contributions are cancelled out when the number of fluxes $N$ is large. When you have large fluxes, these do much to ``stabilize" the moduli against the negative curvature contributions and in more complicated theories, these fluxes being used for moduli stabilization tells you a lot about the EFT and vacua. A similar philosophy applies to more realistic stringy scenarios. In type IIA superstring theory, you add in O-planes and D-branes as well, but with caution since you have to worry about tadpole cancellation. This is typically a problem in F-theory where Calabi-Yau 4-folds are stabilised, but with fluxes adding to the D3-brane tadpole.