Geometry and Infinite Distance Limits in Type IIB Calabi-Yau Compactifications

Sammanfattning

In string theory compactifications, the asymptotic structure of the moduli space plays a key role in determining the validity of the resulting low-energy effective field theories. In Type IIB supergravity compactified on Calabi–Yau threefolds, infinite-distance limits at the boundaries of the complex-structure moduli space correspond to singular geometric degenerations that make an infinite tower of states exponentially light. While recent developments using asymptotic Hodge theory abstractly classify these infinite distance limits into specific degeneration categories (Types II, III, and IV), this thesis investigates whether these boundaries can be explicitly realised and characterised using toric methods. We build the required mathematical frameworks, developing the foundational geometry of Calabi–Yau manifolds, the combinatorial tools of toric geometry and Batyrev’s hypersurface construction, and the formulism of mixed and limiting Hodge structures. By analysing concrete representative examples of Calabi–Yau threefolds, we demonstrate in explicit examples that the combinatorial data of Batyrev's construction reproduces the Type II/III/IV degeneration structures derived from asymptotic Hodge theory. This explicit match validates the abstract mathematical classifications, potentially providing a computationally accessible approach to describing infinite-distance limits within the framework of the Swampland Program.