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 cannot 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.
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