Spanning Class in the Category of Branes

Abstract: Given a generic anticanonical hypersurface $Y$ of a toric variety determined by a reflexive polytope, we define a line bundle ${\mathcal L}$ on $Y$ that generates a spanning class in the bounded derivative category $D^b(Y)$. From this fact, we deduce properties of some spaces of strings related with the brane ${\mathcal L}$. We prove a vanishing theorem for the vertex operators associated to strings stretching from branes of the form ${\mathcal L}^{\otimes i}$ to nonzero objects in $D^b(Y)$. We also define a gauge field on ${\mathcal L}$ which minimizes the corresponding Yang-Mills functional.