Equivalence of D-brane categories

Abstract

In this thesis certain aspects of D-branes and their role in a deeper understanding of gauged linear sigma models and quiver gauge theories are investigated, including the relevant categories in the mathematical setup. Our applications include the development of a McKay correspondence for Landau-Ginzburg models and the construction of noncommutative resolutions of ADE fibered Calabi-Yau threefolds. A proof of the Homological Mirror Symmetry conjecture for toric Del Pezzo surfaces will be also discussed.