Asymptotic Hodge Theory and String Theory: An application to the swampland program

Abstract

The thesis applies asymptotic Hodge theory of the internal Calabi-Yau space in string compactification to study the four-dimensional physics near various corners of the field space. This approach sets up some universal understandings of the swampland in quantum gravity. The first chapter introduces the basic concepts used in this thesis, which are string compactification, swampland, and Hodge theory. The second chapter uses Hodge theory to study the swampland distance conjecture. The setting is type IIB string theory on Calabi-Yau threefolds, and we examine the distance conjecture in the complex structure moduli space of the Calabi-Yau's. With Hodge theory, we equip every limit in the moduli space with an algebraic structure, and classify all possible structures that can appear within a Calabi-Yau. When the Calabi-Yau admits multiple-stage degenerations, all possible transitions between the corresponding algebraic structures are classified. Thus, an intricate network of the algebraic structures associated with infinite distance limits is discovered. Using this network of algebraic structures, we find a light tower of states in the distance conjecture in most infinite distance limits. The approach is exemplified in two-parameter models. The third chapter uses Hodge theory to study the swampland de Sitter conjecture near the corners in the scalar field space. The setting is F-theory on Calabi-Yau fourfolds with four-form flux, and we work in the complex structure moduli space of the fourfolds. There are again algebraic structures associated to every asymptotic limit in the complex structure moduli space, and these algebraic structures can be classified. Using the classification, we further exhaust the possible forms of the scalar potentials in the asymptotic limits. We carry out this program for two-parameter families of Calabi-Yau fourfolds, and study the vacuum structure of these theories near the limits. We confirm the de Sitter conjecture near the limits. Via dualities, these F-theoretical conclusions generalise the known results in type IIA and IIB orientifold settings. In the end, we use the same approach to write down possible asymptotic forms of the axion potentials in the axion monodromy inflation models, and study the backreaction of axions on saxions. We discover that the previously found linear backreaction behaviour of axions on saxions in examples are intimately related to Hodge theory in the field space. The fourth chapter inquires further the topic at the end of chapter three. We examine more closely the implication of Hodge theory on axion monodromy inflation models. We work again in the complex structure moduli space of F-theory compactified on Calabi-Yau fourfolds. The problem is approached differently than in chapter three, using no classification of the algebraic structures associated to the limits of the field space. Thus, the asymptotic form of the axion monodromy potential is generally studied. The backreaction of axions on saxions is attacked using the Puiseux series solution of the vacuum equations, and the method is exemplified in two-parameter models. We confirm that, besides some exceptional cases, the linear backreaction of axion on saxion found previously in several examples are universal in axion monodromy models.