Refined topological vertex and supersymmetric gauge theories

This thesis focuses on open topological string theory on Calabi-Yau threefolds without compact four-cycles. We first review relevant aspects of topological string theories and gauge theories. Then we use the refined topological vertex formalism to compute refined open topological string amplitudes. We find that the refined geometric transition, or in other words, Higgsing, is the key step in this computation. We find there are different types of refined open topological branes, and the corresponding refined Ooguri-Vafa formulas encode refined open BPS invariants. Moreover, we discuss a quiver structure of refined open topological string amplitudes on toric Calabi-Yau threefolds. In order to physically understand this quiver structure, we study the corresponding 3d N=2 gauge theories and find quivers encode actually effective mixed Chern-Simons levels of the mirror dual theories. Along the way, we discuss brane webs for 3d N=2 theories and find quiver structure also for nonabelian theories.