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 dierent 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.