##### Abstract

<p> String theory is a a physical field theory in which point particles are replaced by 1-manifolds propagating in time, called strings. The 2-manifold representing the time evolution of a string is called the string worldsheet. Strings can be either closed (meaning their worldsheets are closed surfaces) or open (meaning their worldsheets have boundary). A D-brane is a submanifold of the spacetime manifold on which string endpoints are constrained to lie.
<p> There are five different string theories that have supersymmetry, and they are all related by various dualities. This dissertation will review how D-branes are classified by K-theory. We will then explore the $K$-theoretic aspects of a hypothesized duality between the type I theory compactified on a 4-torus and the type IIA theory compactified on a K3 surface, by looking at a certain blow down of the singular limit of K3. This dissertation concludes by classifying D-branes on the type II orientifold <bold>T</bold><sup>4</sup>/<bold>Z</bold><sub>2</sub> when the <bold>Z</bold><sub>2</sub> action is multiplication by -1 and the H-flux is trivial. We find that classifying D-branes on the singular limit of K3, <bold>T</bold><sup>4</sup>/<bold>Z</bold><sub>2</sub> by equivariant K-theory agrees with the classification of D-branes on a smooth K3 surface by ordinary K-theory.