D-brane gauge theories from toric singularities and toric duality

Via partial resolution of Abelian orbifolds we present an algorithm for extracting a consistent set of gauge theory data fur an arbitrary toric variety whose singularity a D-brane probes. As illustrative examples, we tabulate the matter content and super-potential for a D-brane living on the toric del Pezzo surfaces as well as the zeroth Hirzebruch surface. Moreover, we discuss the non-uniqueness of the general problem and present examples of vastly different theories whose moduli spaces are described by the same toric data. Our methods provide new tools for calculating gauge theories which flow to the same universality class in the IR. We shall call it Toric Duality. (C) 2001 Elsevier Science B.V. All rights reserved.