Anti-Quill Part 3: Calabi-Yau 1

 In Anti-Quill Part 2, I mentioned how we add in D-branes and O-planes into the theory and not just the moduli and the fluxes. The way we gauge whether or not a given theory has viable compactifications is by looking at the effective potential for phenomenological reasons. As an example, the Freund-Rubin effective potential is 

\[ V \sim \frac{1}{R^{4}} \left( \frac{2g-2}{R^{2}}+\frac{N^{2}}{R^{2}} \right)\;, \]

where $N$ are the number of fluxes. However, Freund-Rubin are not complete or realistic compactifications (even if you add in perturbative corrections). One way in which you could see this is to note that $g$ is not zero for compactifications we wish to pursue (Candelas et al). We instead wish to work with Calabi-Yau manifolds, which are Kahler manifolds with $c_{1}(K_{M})=0$, where $K_{M}$ is the complex line bundle. Very quickly, we will notice two interesting features. To start, see that $T^{2}$ can be decomposed into to two $S^{1}$  with radii $(r_{1}, r_{2})$, and the moduli space for this would be a 2-dimensional manifold $M(r_{1}, r_{2})$. Defining $\tau = iT$, where $T=R_{2}/R_{1}$, notice that there is a $T$-duality, which more generally is a mirror symmetry for reasons I will defer from here. In string theory, one usually also adds in a $B$-field, but the point here is that Calabi-Yau manifolds have this $T$-duality. For general moduli spaces $M_{D}$ of Calabi-Yau manifolds, we define a Weil-Petersson metric $g_{WP}$, which is a Kahlerian metric on $\mathrm{Teich}(n, g)$. An interesting associated result is that the Weil-Petersson volume $V_{WP}(M)$ is finite, and that $R(g_{WP})$ for $M(CY3)$ is either positive or negative, unlike for general $M(K3)$ (something I am still trying to understand). It is then of interest how the curvature and the Weil-Petersson metric affect the stringy moduli space, which typically decomposes into $M=M_{c}\times M_{K}$.