ACT A - What's in a Moduli?
In a previous post, we discussed the Kaluza-Klein (KK) compactifications and extended the KK mechanism to string compactifications. Now, we do not really get a unique solution for these compactifications as one can really scale the metric to obtain some other solution. We can thus make a family of these solutions which are called moduli. We will take them on manifold \mathcal{M} with singularities. The metric, let us call it g, that we usually consider are Ricci-flat and they are never unique. If a Calabi-Yau manifold has a Ricc(g) = 0, then there can always be perturbations of the kind that can still make the Ricci curvature vanish (see this). One can always construct a cohomology class (with some restrictions of the perturbation) of the perturbations so that the class of non-trivial metric perturbations can be studied which preserves the Ricci flatness conditions.A moduli space in algebraic geometry, differential geometry, and algebraic topology is about the classifications of objects which are equivalent and those which are not equivalent. (In topology, it is usually called a classifying space.) In essence, it is a geometric interpretation of the solutions. For example, a moduli space of Riemman surface, a moduli space of Elliptic curves, and so on.
Coming back to the compactifications, here, the moduli space is of Ricci flat metrics. The moduli of the compactification consist of the scalar fields (the massless modes) which are the solutions of the equations that are written for motion. These scalar fields correspond to an infinite tower of modes, also called Kaluza-Klein modes. One can take a family of Ricci flat metric \mathcal{G} and a perturbation around it (as we discussed above). We can then show that some perturbations \delta \mathcal{G} will correspond to these varying of \mathcal{G} (which is equivalent to saying that the perturbations would not hurt the conditions of K that we started with). These will still solve the equations of motion. And each variation (in \delta \mathcal{G}) will give us a massless scalar field in our compactified 4-manifold. As is remarked in this paper, there is no need for symmetry for these variations (like one needs for the description of Goldstone bosons).
One then is left with different values, which only differ by the varying scalar fields. We can then write a family of the vacuas and call it the moduli space \mathcal{M}. Now, this is not very exciting because it predicts a large number (a very big number) of vacua for describing our four-dimensional world. This is why moduli stabilization is considered. Moduli is lifted to make the massless modest pick up some mass. (There are non-supersymmetric and supersymmetric realizations of this though.)
ACT \alpha - A Note on the Cohomology of Quasi-Coherent Sheaves
On this blog, we have mentioned about quasi-coherent sheaves and coherent sheaves in multiple places. I wrote a (very) short note on the cohomology of quasi-coherent sheaves. A subject like this is at the heart of algebraic geometry and stems from the ideas of homological algebra (see Tohuku by Grothendieck) and sheaf theory (coherent sheaves, see FAC by Serre). We discuss, in the note, Serre's vanishing theorem, Čech cohomology, and the vanishing of higher cohomology.
