The K\"ahler Quotient Resolution of ${\mathbb{C}}^3\mathrm{/}\mathrm{\Gamma }$ singularities, the McKay correspondence and D=3 $\mathcal{N}=2$ Chern-Simons gauge theories

We advocate that a generalized Kronheimer construction of the K"ahler quotient crepant resolution $$\mathcal{M}$\zeta \longrightarrow $\mathbb{C}$^3/\Gamma$ofanorbifoldsingularitywhere$\Gamma\subset $\mathrm{S}\mathrm{U}(3)$$ is a finite subgroup naturally defines the field content and interaction structure of a superconformal Chern-Simons Gauge Theory. This is supposedly the dual of an M2-brane solution of $D=11$ supergravity with $\mathbb{C}\times {\mathcal{M}}_{\zeta }$ as transverse space. We illustrate and discuss many aspects of this of constructions emphasizing that the equation $p\wedge p=0$ which provides the K"ahler analogue of the holomorphic sector in the hyperK"ahler moment map equations canonically defines the structure of a universal superpotential in the CS theory. The kernel of the above equation can be described as the orbit with respect to a quiver Lie group ${\mathcal{G}}_{\mathrm{\Gamma }}$ of a locus $L\Gamma \subset $\mathrm{H}\mathrm{o}\mathrm{m}$_\Gamma($\mathcal{Q}$\otimes R,R)$ that has also a universal definition. We discuss the relation between the coset manifold ${\mathcal{G}}_{\mathrm{\Gamma }}\mathrm{/}{\mathcal{F}}_{\mathrm{\Gamma }}$, the gauge group ${\mathcal{F}}_{\mathrm{\Gamma }}$ being the maximal compact subgroup of the quiver group, the moment map equations and the first Chern classes of the tautological vector bundles that are in a one-to-one correspondence with the nontrivial irreps of $\mathrm{\Gamma }$. These first Chern classes provide a basis for the cohomology group $H^2({\mathcal{M}}_{\zeta })$. We discuss the relation with conjugacy classes of $\mathrm{\Gamma }$ and provide the explicit construction of several examples emphasizing the role of a generalized McKay correspondence. The case of the ALE manifold resolution of ${\mathbb{C}}^2\mathrm{/}\mathrm{\Gamma }$ singularities is utilized as a comparison term and new formulae related with the complex presentation of Gibbons-Hawking metrics are exhibited.