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Various topological properties of D-branes in the type--IIA theory arecaptured by the topologically twisted B-model, treating D-branes as objects inthe bounded derived category of coherent sheaves on the compact part of thetarget space. The set of basic D-branes wrapped on the homology cycles of thecompact space are taken to reside in the heart of t-structures of the derivedcategory of coherent sheaves on the space at any point in the K\"ahler modulispace. The stability data entails specifying a t-structure along with a gradefor sorting the branes. Considering an example of a degenerate Calabi-Yauspace, obtained via geometric engineering, that retains but a projective curveas the sole non-compact part, we identify the regions in the K\"ahler modulispace of the curve that pertain to the different t-structures of the boundedderived category of coherent sheaves on the curve corresponding to thedifferent phases of the topological branes.

Branes in hearts with perverse sheaves