Exceptional Collections and del Pezzo Gauge Theories
Christopher P. Herzog
TL;DR
This work develops an explicit bridge between algebraic geometry and ${\cal N}=1$ quiver gauge theories by using exceptional collections on del Pezzo surfaces. It provides a concrete procedure to extract quivers, gauge-group ranks, and R-charges from a given exceptional collection and its dual, yielding two key results: the node ordering is fixed up to cyclic permutation, and a simple conformal-rank formula in terms of bifundamental data. A thorough analysis of four-node quivers reveals that only $A$- and $F$-type orderings survive consistency constraints, with Seiberg duality corresponding to mutations for $A$-type but not for $F$-type quivers, and introduces the notion of well-split quivers. These insights illuminate how geometric data controls the gauge theory dynamics and may guide generalizations of KS-like RG flows in non-toric del Pezzo geometries, linking mutation theory, dualities, and anomaly constraints within a unified framework.
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.