Topological string theory and applications

This thesis focuses on various applications of topological string theory based on different types of Calabi-Yau (CY) manifolds. The first type considered is the toric CY manifold, which is intimately related to spectral problems of difference operators. The particular example considered in the thesis closely resembles the Harper-Hofstadter model in condensed matter physics. We first study the non-perturbative sectors in this model, and then propose a new way to compute them using topological string theory. In the second part of the thesis, we consider partition functions on elliptically fibered CY manifolds. These exhibit interesting modular behavior. We show that for geometries which do not lead to non-Abelian gauge symmetries, the topological string partition functions can be reconstructed based solely on genus zero Gromov-Witten invariants. Finally, we discuss ongoing work regarding the relation of the topological string partition functions on the so-called Higgsing trees in F-theory. (author)