D-Branes on Vanishing del Pezzo Surfaces

TL;DR

This paper uses derived categories and tilting equivalences to analyze D-branes on collapsed del Pezzo surfaces, clarifying conditions for quiver gauge theories and Seiberg dualities in Calabi-Yau threefolds.

Contribution

It provides a rigorous derived category framework for D-branes on singularities, clarifies when exceptional collections yield valid quiver theories, and explores Seiberg dualities via tilting equivalences.

Findings

Derived category approach constructs quiver gauge theories for singularities.

Exceptional collection procedures are valid only if certain Ext3 groups vanish.

Tilting equivalences can induce Seiberg dualities and remove unphysical quivers.

Abstract

We analyze in detail the case of a marginally stable D-Brane on a collapsed del Pezzo surface in a Calabi-Yau threefold using the derived category of quiver representations and the idea of aligned gradings. We show how the derived category approach to D-branes provides a straight-forward and rigorous construction of quiver gauge theories associated to such singularities. Our method shows that a procedure involving exceptional collections used elsewhere in 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 arise from tilting equivalences. It follows that some (but not all) mutations of exceptional collections induce Seiberg duality in this context. The same tilting equivalence can also be used to remove unwanted Ext3 groups and convert an unphysical quiver into a physical one.