Anti-Quill Part 4: Calabi-Yau 2

 In Anti-Quill Part 3, we recalled some interesting properties of Calabi-Yau manifolds. Since a Calabi-Yau metric $g$ can be perturbed via some deformation $g+\delta g$ continuously to produce another Ricci flat manifold, this gives us a moduli space of CY manifolds. Using hodge diamond and specifically calculating hodge numbers with quintic hypersurfaces $\Sigma $ in $\mathbb{CP}^{4}$, one can find the dimension of the moduli space. A simple approach is to take the 126 monomials minus 25 degrees of freedom under $\mathbb{GL}(5, \mathbb{C})$ transformations, which gives you 101 dof, times 2 plus 1, which gives you 203 real dimension for the moduli space, which really comes from the calculation of the hodge numbers. In string theory, the actual nature of the moduli space is far more complicated because we want to do EFTs and somehow quantify viable vacua (if they exist!). The swampland distance conjecture tells you that taking two points on the moduli space $M(\text{string theory})$, there is an infinite tower of modes with exponentially vanishing mass: $M \sim m\exp (-\lambda s)$, where $s\to \infty $ is the geodesic distance on $M(\text{string theory})$. We expect usually that the ``landscape" theories lie in a particular part of this moduli space and the UV-complete nature of it, and how EFT corrections contribute in certain limits, like when $g_{\text{string}}$ is weak. There are some subtleties around how you view the converse with the light tower of modes, see van Riet and Zoccarato. However, the full moduli space even geometrically is non-trivial. Addition of fluxes with the split $M=M_{c}\times M_{K}$ from before is done by the Gukov-Vafa-Witten superpotential $V_{GVW}(\Omega _{3}, G_{3})$, where $\Omega _{3}$ is the complex structure moduli and $G_{3}$ is the dilaton.