Phase Structure of D-brane Gauge Theories and Toric Duality

TL;DR

This paper investigates toric duality for D-brane gauge theories by analyzing how different toric diagrams related by unimodular transformations can yield inequivalent gauge theories with the same toric moduli space. It formalizes the notion of toric isomorphisms and dissects the forward and reverse toric algorithms to isolate degrees of freedom that leave physics invariant, defining phases of partial resolution. Through detailed application to ${ m Z}_3 imes{ m Z}_3$ resolutions and toric del Pezzo surfaces (including $F_0$), it shows that several unimodular representations collapse to the same physical theory, while others produce distinct phases. The work clarifies when toric dual theories are physically distinct and discusses implications for AdS/CFT, brane webs, and the broader classification of toric phases, outlining future directions for a formal phase taxonomy. Overall, it advances understanding of how geometric equivalences translate into gauge-theory dualities and provides a framework to catalog phases arising from toric data.

Abstract

Harnessing the unimodular degree of freedom in the definition of any toric diagram, we present a method of constructing inequivalent gauge theories which are world-volume theories of D-branes probing the same toric singularity. These theories are various phases in partial resolution of Abelian orbifolds. As examples, two phases are constructed for both the zeroth Hirzebruch and the second del Pezzo surfaces. We show that such a phenomenon is a special case of ``Toric Duality'' proposed in hep-th/0003085. Furthermore, we investigate the general conditions that distinguish these different gauge theories with the same (toric) moduli space.