Topics in Gromov-Witten theory

This thesis explores different aspects of Gromov–Witten theory and is divided into two parts.
The first investigates conjectures of Bousseau, Brini and van Garrel relating three a priori very different curve counts: Logarithmic Gromov–Witten theory of Looijenga pairs (certain logarithmic Calabi–Yau surfaces), open Gromov–Witten theory of toric Calabi–Yau threefolds and local Gromov–Witten theory of higher dimensional Calabi–Yau varieties. We concentrate on the case where the logarithmic boundary of the initial surface geometry has two components. First we establish the logarithmic-open correspondence in an explicit example where the Looijenga pair is a del Pezzo surface of degree six. The proof relies on a direct calculation using quantum scattering diagrams and involves an intricate identity of q-hypergeometric functions. After this case study we proceed with a more general, geometric approach and ultimately establish the logarithmic-local and all-genus logarithmic-open correspondence for all Looijenga pairs with two boundary components. The proof of the correspondences involves a delicate application the degeneration formula and torus localisation.
In the second part we propose a mathematical interpretation of the so called refined topological string on a Calabi–Yau threefold in terms of equivariant Gromov–Witten theory of an extended Calabi–Yau fivefold geometry. We perform initial checks which indicate that our proposal meets several expectations formulated in the physics literature. We state a refined BPS integrality conjecture and provide evidence in case the threefold is the resolved conifold or a local del Pezzo surface. In the latter case we do so by identifying the Nekrasov–Shatashvili limit with the relative Gromov–Witten theory of the surface relative a smooth anticanonical curve.