Lectures on resolutions à la Kronheimer of orbifold singularities, McKay quivers for Gauge Theories on D3 branes, and the issue of Ricci flat metrics on the resolved three-folds

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 $Y_{[3]}^{\mathrm{\Gamma }}\to \frac{{\mathbb{C}}^3}{\mathrm{\Gamma }}$ of orbifold singularities, $\mathrm{\Gamma }$ being s finite subgroup of $\mathrm{S}\mathrm{U}(3)$, 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 Kähler 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