Exceptional collections and D-branes probing toric singularities

TL;DR

This paper shows how strongly exceptional collections on singular toric surfaces can be used to derive gauge theories for D3-branes probing Calabi-Yau singularities, providing a systematic approach for various spaces.

Contribution

It introduces a method to derive gauge theories from strongly exceptional collections on singular toric surfaces, including new proofs for specific Y^{p,q} cases.

Findings

Derived gauge theories for weighted projective spaces

Proved strong exceptionality for Y^{p,p-1} and Y^{p,p-2r} cases

Established a basis of D-branes using exceptional collections

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.