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 singu
larity. We deduce the field content and Lagrangian of such theories and pre
sent a systematic method for mapping the moment map levels characterizing t
he partial resolutions of the singularity to the Fayet-Iliopoulos parameter
s of the D-brane world-volume theory. As opposed to the simpler cases studi
ed before, we find a complex web of partial resolutions and associated fiel
d-theoretic Fayet-Iliopoulos deformations. The analysis is performed by tor
ic methods, leading to a structure which can be efficiently described in th
e 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-p
roduct, we present a systematic way of extracting the birational geometry o
f the classical moduli spaces, thus generalizing previous work on resolutio
n of singularities by D-branes. (C) 2000 Elsevier Science B.V. All rights r
eserved.