On Correspondences Between Toric Singularities and (p,q)-webs

TL;DR

This paper clarifies the relationship between toric singularities and (p,q)-webs in gauge theories, distinguishing two types of webs to resolve existing paradoxes and analyzing their implications for gauge theory phases.

Contribution

It introduces and differentiates between toric and quiver (p,q)-webs, resolving ambiguities and enhancing understanding of gauge theories from toric Calabi-Yau threefolds.

Findings

Toric (p,q)-webs correspond uniquely to toric diagrams.

Quiver (p,q)-webs can represent multiple gauge theories.

Analysis of Higgsing, blowups, and brane splittings reveals complex interrelations.

Abstract

We study four-dimensional N=1 gauge theories which arise from D3-brane probes of toric Calabi-Yau threefolds. There are some standing paradoxes in the literature regarding relations among (p,q)-webs, toric diagrams and various phases of the gauge theories, we resolve them by proposing and carefully distinguishing between two kinds of (p,q)-webs: toric and quiver (p,q)-webs. The former has a one to one correspondence with the toric diagram while the latter can correspond to multiple gauge theories. The key reason for this ambiguity is that a given quiver (p,q)-web can not capture non-chiral matter fields in the gauge theory. To support our claim we analyse families of theories emerging from partial resolution of Abelian orbifolds using the Inverse Algorithm of hep-th/0003085 as well as (p,q)-web techniques. We present complex inter-relations among these theories by Higgsing, blowups and brane splittings. We also point out subtleties involved in the ordering of legs in the (p,q) diagram.