  The present Lecture Notes have been prepared to back up a series of a few
seminars given by the author at the Albert Einstein Institute in Potsdam. These
Notes aim at reviewing a research project conducted over the last six years
about a quite interesting and challenging topic, namely the use of the
generalized Kronheimer construction and the generalized McKay correspondence
for the crepant resolution $Y^\Gamma_{[3]} \to \frac{\mathbb{C}^3}{\Gamma}$ of
orbifold singularities, $\Gamma$ being s finite subgroup of $\mathrm{SU(3)}$,
as a strategic tool to construct holographic dual pairs of $\mathcal{N}=1$
gauge theories in $D=4$ and D3-brane solutions of type IIB supergravity. The
project, developed by Ugo Bruzzo with the present author, in various
co-authorships with Massimo Bianchi, Anna Fino, Pietro Antonio Grassi, Dimitry
Markushevich, Dario Martelli, Mario Trigiante and Umar Shahzad, is still going
on and there are a lot of interesting unanswered questions that are illustrated
throughout these Lecture Notes and particularly emphasized in its last chapter.
The mathematical roots of this project are very robust and ramified in
differential geometry, group theory and algebraic geometry. A particularly
challenging issue is that of constructing Ricci flat metrics on the total space
of line-bundles over compact K\"ahler manifolds of real dimension four. In that
direction the symplectic action/angle formalism turns out to be a winning
weapon. The ultimate goal is that of deducing all the aspects of holographic
duality from finite group theory or, to be more precise, SU(3) finite subgroups
