Moduli Stabilization and Stability in Type II/F-theory flux compactifications

In this thesis we study String Theory compactifications to four dimensions focusing on the moduli stabilization process and the associated vacua structure in various frameworks, from Type IIA to F-theory. We interpret the results in the context of the Swampland Program.

More specifically, we generalize the bilinear formalism of the scalar potential to include the contributions of geometric and non-geometric fluxes, which is later used to perform a systematic search of vacua. Using an Ansatz motivated by the goal of achieving stable de Sitter vacua, we study the equations of motion of Type IIA with metric fluxes. We obtain only AdS vacua, both SUSY and non-SUSY, checking their stability and generalizing several results from the literature. We try to find scale separation but fail to do so in the studied solutions. 

We also consider the 10d uplift of AdS4 vacua arising from the 4d massive Type IIA effective theory with only RR and NSNS fluxes. Using the language of SU(3)xSU(3) structures and performing an expansion around the smearing approximation in powers of the string coupling, we study the stability of the supersymmetric solution and its non-supersymmetric partner (associated with the former by a change of sign in the RR 4-form field strength  flux). We contrast the results with the Weak Gravity Conjecture and the AdS instability conjecture in several toroidal orbifold examples and find that some non-supersymmetric cases are in tension with the predictions of those conjectures, hinting at the existence of additional corrections that have not been taken into account.

Concerning F-theory, we study moduli stabilization in the complex structure sector of compactifications over elliptically fibered Calabi-Yau 4-folds in the limit of Large Complex Structure. Using homological mirror symmetry, we are able to replicate the analysis for the Type IIA case and give a bilinear expression for the scalar potential, allowing for a simpler and more detailed study of the vacua structure. In the process, we find two distinct families of flux configurations compatible with the tadpole constraints that allow for full moduli stabilization. The first one requires polynomial corrections to fix all the moduli and the flux contribution to the tadpole scales with the dimension of the moduli space. In contrast, in the second family, polynomial corrections are not needed and only a pair of fluxes enters the tadpole independently of the number of moduli. We thoroughly examine the former in the Type IIB limit, where the superpotential is also quadratic and polynomial corrections can be considered at all orders. We argue that vacua fall into three classes depending on the choice of flux quanta. In particular, we provide analytic expressions for the vacuum expectation values and flux-induced masses of the axio-dilaton and complex structure fields in a large subclass of vacua, independently of the Calabi-Yau and the number of moduli. Finally, we show that at this level of approximation supersymmetric vacua always contain flat directions.

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