Graduate Program in Physics.Kozçaz, Can.Domurcukgül, Tolga.2025-04-142025-04-142023Graduate Program in Physics. SCED 2023 O48 (Thes MIS 2023 B37 PhDhttps://digitalarchive.library.bogazici.edu.tr/handle/123456789/21558This thesis covers topological string theory, Seiberg-Witten theory, and instanton counting which provide insights into the interplay between geometry, topology, and quantum field theory. The target space interpretation of topological strings is explored, including its connections to physical string theory and geometric engineering. Along with that, M-theory compactification on a circle provides a target space interpretation for topological strings, leading to the discovery of new integer invariants known as Gopakumar-Vafa invariants. Seiberg-Witten theory plays a central role in understanding the dynamics of supersymmetric gauge theories. The moduli space of vacua offers a comprehensive description of the possible vacuum states of these theories, and the effective field theory approach enables the derivation of the quantum prepotential. The Seiberg-Witten solution provides exact results for various quantities in these theories and is examined in the context of elliptic curves. Then the Nekrasov partition function is used to study instantons in N = 2 super Yang-Mills theory and the resolution of singularities and equivariant cohomology techniques are used to address non- compactness issues and subtleties arising from singularities. The role of Nekrasov factors in capturing contributions from different sectors of the instanton moduli space has been explored. The five-dimensional lift of Nekrasov partition function is introduced and studied through the connection to the Gopakumar-Vafa expansion of topological string theory.String models.Geometry, Algebraic.viii, 133 leaves