Various topological properties of D-branes in the type--IIA theory are captured by the topologically twisted B-model, treating D-branes as objects in the bounded derived category of coherent sheaves on the compact part of the target space. The set of
basic D-branes wrapped on the homology cycles of the compact space are taken to reside in the heart of t-structures of the derived category of coherent sheaves on the space at any point in the Kahler moduli space. The stability data entails specifying a t-structure along with a grade for sorting the branes. Considering an example of a degenerate Calabi-Yau space, obtained via geometric engineering, that retains but a projective curve as the sole non-compact part, we identify the regions in the Kahler moduli space of the curve that pertain to the different t-structures of the bounded derived category of coherent sheaves on the curve corresponding to the different phases of the topological branes.
Problems of stabilizing moduli of the type--IIB string theory on toroidal orientifolds $T^6/Z_2$, in presence of worldvolume fluxes on various D-branes, are considered. For $Z_2$ actions, introducing either O9 or O3 planes, we rule out the possibilit
y of moduli stabilization in a wide class of models with $mathcal{N}=1$ supersymmetry, characterized by the type of fluxes turned on along D-brane worldvolume. Our results, in particular, imply that Abelian worldvolume fluxes can not by themselves stabilize closed string moduli, in a consistent supersymmtric model, for above orientifold compactifications. We also discuss other $Z_2$ orientifolds of $T^6$ and show that certain other brane wrappings are also ruled out by similar consistency requirements. In specific setups we consider examples with D9-branes wrapping on a complex three-torus with its world-volume fluxes taken to be semi-homogeneous bundles and D7-branes wrapping holomorphic four-cycles of the complex three-torus carrying world-volume fluxes.
