We consider non-compact Calabi-Yau threefolds that are fibrations over
compact Riemann surfaces, the local curves, and study the dynamics of B-branes
wrapped around the curves. We discuss different but closely related possible
approaches to this problem. In particular, we study the open topological string
field theory of the B-brane and the dimensional reduction of the holomorphic
Chern-Simons functional to the curve. The classical (g_s = 0) limit of these
dynamics for one single brane is given by the deformation theory of the curve
inside the Calabi-Yau threefold; we consider this last approach for the Laufer
curve case.Comment: PhD Thesis, 60 pages, 2 PostScript figures. Some sections reshuffled
and some comments added with respect to the version presented at SISSA on
November 28, 2006, according to the suggestions of the examination committee.
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