Asymptotic Hodge Theory in String Compactifications and Integrable Systems

Abstract

String theory is a proposal for a description of nature on the smallest length scales, where both quantum mechanics and gravity are expected to play an essential role. A curious feature of the theory is that its basic building blocks – one-dimensional strings – behave as if they were moving in nine-dimensional space, as opposed to the three-dimensional space we are familiar with. String theory proposes that the six additional dimensions are curled up into extremely small sizes, in a process called “compactification”. Importantly, what happens in these hidden six dimensions has an enormous influence on the physics we observe at larger length scales, including the strength of the interactions between elementary particles and the value of the cosmological constant.

An intriguing aspect of string theory is that there are countless possible ways to compactify these extra dimensions, resulting in a vast "landscape" of potential universes, each with distinct physical properties. In a beautiful interplay of physics and mathematics, the physical characteristics of these universes are intricately linked to the geometric properties of the internal six-dimensional space.

The main aim of this thesis is to study the latter using the sophisticated mathematical framework of asymptotic Hodge theory. One of the central results is that one can obtain a good approximation of numerous physical observables by studying the allowed singularities of the six-dimensional internal space. In accordance with some fundamental theorems in the field of algebraic geometry, these can be classified in great generality and reveal intriguing underlying structures. In particular, this abstract approach leads to a comprehensive, algorithmic method for calculating physical observables in string compactifications. In this way, we obtain a very general characterization of universes that can be constructed in string theory and find that they are in fact finite in number. Additionally, based on recent advances in the field of tame geometry it is suggested that this number might be much smaller than previously expected.

Finally, we observe that the same mathematical structures applied above in the context of string compactifications arise in a completely different corner of physics: the study of integrable non-linear sigma models. In particular, the same computational methods used to compute physical observables in string theory can be used to find previously unknown solutions for certain classes of such models. Our results open up a new avenue of research, bridging two distinct areas of theoretical physics.