Abstract:
We investigate field theories on the worldvolume of a D3-
brane transverse to partial resolutions of a $\Z_3\times\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
worldvolume 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 worldvolume 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.