Exceptional Collections and del Pezzo Gauge Theories

TL;DR

This paper uses exceptional collections to analyze gauge theories from D3-branes on del Pezzo surfaces, providing new tools for understanding their structure and dualities in string theory.

Contribution

It introduces methods to determine node ordering and gauge group ranks in quiver theories derived from del Pezzo surfaces, linking mutations to Seiberg duality.

Findings

Node ordering determined up to cyclic permutation

Derived formula for gauge group ranks at conformal point

Analyzed mutations and their relation to Seiberg duality

Abstract

Stacks of D3-branes placed at the tip of a cone over a del Pezzo surface provide a way of geometrically engineering a small but rich class of gauge/gravity dualities. We develop tools for understanding the resulting quiver gauge theories using exceptional collections. We prove two important results for a general quiver gauge theory: 1) we show the ordering of the nodes can be determined up to cyclic permutation and 2) we derive a simple formula for the ranks of the gauge groups (at the conformal point) in terms of the numbers of bifundamentals. We also provide a detailed analysis of four node quivers, examining when precisely mutations of the exceptional collection are related to Seiberg duality.