D-Branes on Vanishing del Pezzo Surfaces

We analyze in detail the case of a marginally stable D-Brane on a collapseddel Pezzo surface in a Calabi-Yau threefold using the derived category ofquiver representations and the idea of aligned gradings. We show how thederived category approach to D-branes provides a straight-forward and rigorousconstruction of quiver gauge theories associated to such singularities. Ourmethod shows that a procedure involving exceptional collections used elsewherein the literature is only valid if some tachyon-inducing Ext3 groups are zero.We then analyze in generality a large class of Seiberg dualities which arisefrom tilting equivalences. It follows that some (but not all) mutations ofexceptional collections induce Seiberg duality in this context. The sametilting equivalence can also be used to remove unwanted Ext3 groups and convertan unphysical quiver into a physical one.Comment: 32 pages, several xy-pic figures, typos and minor errors fixed, section 4.3 removed due to possible error