PhD defence: Asymptotic Hodge Theory in String Compactifications and Integrable Systems

String theory is a proposal for a description of Nature at the tiniest length scales, where both quantum mechanics and gravity are expected to play an essential role. A curious feature of the theory is that its fundamental building blocks – one-dimensional strings – behave as if they are moving in a nine-dimensional space, as opposed to a three-dimensional one. Importantly, what goes on in the remaining 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.

In a beautiful interplay of physics and mathematics, these features are determined by the geometric properties of the internal six-dimensional space. It is the main objective of this thesis to study the latter using the sophisticated framework of asymptotic Hodge theory. One of the central outcomes is the fact that one can obtain a good approximation to the physical observables by studying the allowed singularities of the six-dimensional internal space. In accordance with some foundational theorems in the field, these can be classified in great generality and reveal intriguing underlying structures. Notably, from these abstract considerations we obtain a very general, yet completely algorithmic, procedure for computing physical observables in compactifications of string theory.  At the same time, we observe that the same mathematical structures arise in a rather different corner of physics, namely the study of integrable non-linear sigma models. In particular, the same methods can be used to find previously unknown solutions to certain classes of such models.