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

We advocate that a generalized Kronheimer construction of the Kahler quotient crepant resolution $mathcal{M}_zeta longrightarrow mathbb{C}^3/Gamma$ of an orbifold singularity where $Gammasubset mathrm{SU(3)}$ 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 $mathbb{C}timesmathcal{M}_zeta$ as transverse space. We illustrate and discuss many aspects of this of constructions emphasizing that the equation $pmb{p}wedgepmb{p}=0$ which provides the Kahler analogue of the holomorphic sector in the hyperKahler 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 $mathcal{G}_Gamma$ of a locus $L_Gamma subset mathrm{Hom}_Gamma(mathcal{Q}otimes R,R)$ that has also a universal definition. We discuss the relation between the coset manifold $mathcal{G}_Gamma/mathcal{F}_Gamma$, the gauge group $mathcal{F}_Gamma$ 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 $Gamma$. These first Chern classes provide a basis for the cohomology group $H^2(mathcal{M}_zeta)$. We discuss the relation with conjugacy classes of $Gamma$ and provide the explicit construction of several examples emphasizing the role of a generalized McKay correspondence. The case of the ALE manifold resolution of $mathbb{C}^2/Gamma$ singularities is utilized as a comparison term and new formulae related with the complex presentation of Gibbons-Hawking metrics are exhibited.