Exceptional collections and D-branes probing toric singularities

TL;DR

This paper presents a constructive link between strongly exceptional collections on singular toric surfaces and the gauge theories on D3-branes probing Calabi-Yau toric singularities. The authors develop an algorithm to derive quiver gauge theories from toric data by constructing strong exceptional collections of line bundles on the base surface (or stack) and computing Ext groups, aided by computer algebra tools. They provide explicit collections and quivers for a wide range of examples, including weighted projective spaces and the $Y^{p,q}$, $X^{p,q}$, and $L^{p,q,r}$ families, and prove strong exceptionality for $Y^{p,p-1}$ and $Y^{p,p-2r}$ under gcd conditions. The results establish a principled, computationally implementable route from toric geometry to 4d ${\mathcal N}=1$ quiver gauge theories, with Seiberg duality realized via mutations and a stated conjecture for four-ray toric surfaces.

Abstract

We demonstrate that a strongly exceptional collection on a singular toric surface can be used to derive the gauge theory on a stack of D3-branes probing the Calabi-Yau singularity caused by the surface shrinking to zero size. A strongly exceptional collection, i.e., an ordered set of sheaves satisfying special mapping properties, gives a convenient basis of D-branes. We find such collections and analyze the gauge theories for weighted projective spaces, and many of the Y^{p,q} and L^{p,q,r} spaces. In particular, we prove the strong exceptionality for all p in the Y^{p,p-1} case, and similarly for the Y^{p,p-2r} case.

Theorems & Definitions (14)

• Proposition 6.1
• proof
• Lemma 6.2
• proof : Proof of the Lemma
• Proposition 6.3
• proof
• Lemma 6.4
• proof : Proof of the Lemma
• Proposition 6.5
• proof