D-branes on Calabi-Yau Spaces

In this thesis the properties of D-branes on Calabi–Yau spaces are investigated. Compactifications of type II string theories on these spaces to which D-branes are added lead to N = 1 supersymmetric gauge theories on the world-volume of these D-branes. Both the Calabi–Yau spaces and the D-branes have in general a moduli space. We examine the dependence of the gauge theory on the choice of the moduli, in particular those of the K¨ahler structure of the Calabi–Yau manifold. For this purpose we choose two points in this moduli space which are distinguished by the fact that there exists an explicit description of the spectrum of the D-branes. One of these points corresponds to a manifold in the large volume limit on which the D-branes are described by classical geometry of vector bundles. At the other points the size of the manifold is smaller than its quantum fluctuations such that the classical geometry looses its meaning and has to be replaced by a conformal field theory. The Witten index in the open string sector is independent of the variation of these moduli and serves, together with mirror symmetry, as a tool to compare the two descriptions. We give an extensive and general presentation of these two descriptions for the class of Fermat hypersurfaces in weighted projective spaces. We explicitly carry out the comparison in many representative examples. Among them are manifolds admitting elliptic and K3-fibrations and manifolds whose moduli space can be embedded into the moduli space of another manifold. One main focus is on D4-branes, in particular on the dimension of their moduli space. Using the methods developed we are able to further confirm with our results the modified geometric hypothesis by Douglas. It essentially states that the properties of these D-branes or of these gauge theories can be determined partly by classical geometry, partly by mirror symmetry. A peculiarity of these gauge theories is the appearance of lines of marginal stability at which BPS states can decay. We show the existence of such lines in the framework of this class of Calabi–Yau spaces in two di®erent ways and discuss the connection to the formation of bound states. Of particular interest is the D0-brane whose appearance in this framework is explained