Tame Geometry and the String Landscape

Abstract

The idea of tame geometry, made precise by the notion of an o-minimal structure, has recently provided many new insights into mathematics and physics. On the mathematical side it has led to promising new results in Hodge theory, and on the physical side it has inspired a deeper understanding of the landscape of string theory vacua. In this thesis we review these exciting developments, focusing on the tameness of the period map in Hodge theory and on the proposal of a new conjectured aspect of low-energy effective the- ories arising from quantum gravity, called the Tameness Conjecture. We provide further evidence for the Tameness Conjecture in the setting of higher supergravity theories. In particular, we show that in such theories the geometry of the scalar field space is tame, and we argue for the tameness of gauge couplings functions. We point out several sub- tleties arising from the interaction between tameness and arithmetic aspects.