On the geometry of quiver gauge theories (stacking exceptional collections)
Topological D-branes are represented as objects in the derived category. In this context fractional branes provide ways to determine the low energy gauge theory description D-branes probing a Calabi-Yau singularity. This paper advances the program of using exceptional collections to understand the gauge theory description of a D-brane probing a Calabi-Yau singularity. In many cases they contain only line bundles supported on the exceptional divisor that partially resolves the Calabi-Yau singularity. The exceptional collections were usuful for studying a large class of singularities, such as toric varieties, and also Fano with the help of smooth stacks. In this paper, proved is vanishing of tachyons for these Calabi-Yau singularities, and the authors give a strong exceptional collection for all \(Y^{p,q}\) spaces with \(p-q-2 > 0\). The spaces \(Y^{p,p}\), \(Y^{p,p-1}\) and \(Y^{p,p-2}\) are already shown in the previous paper of the authors [J. High Energy Phys. 2006, No. 02, 061 (2006); \url{arXiv:hep-th/0507175}].
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