D3-branes on partial resolutions of abelian quotient singularities of Calabi-Yau threefolds

We investigate field theories on the world-volume of a D3-brane transverse to partial resolutions of a Z(3) x Z(3) Calabi-Yau threefold quotient singularity. We deduce the field content and Lagrangian of such theories and present a systematic method for mapping the moment map levels characterizing the partial resolutions of the singularity to the Fayet-Iliopoulos parameters of the D-brane world-volume theory. As opposed to the simpler cases studied before, we find a complex web of partial resolutions and associated field-theoretic Fayet-Iliopoulos deformations. The analysis is performed by toric methods, leading to a structure which can be efficiently described in the language of convex geometry. For the world-volume theory, the analysis of the moduli space has an elegant description in terms of quivers. As a by-product, we present a systematic way of extracting the birational geometry of the classical moduli spaces, thus generalizing previous work on resolution of singularities by D-branes. (C) 2000 Elsevier Science B.V. All rights reserved.