The Kähler Quotient Resolution of $\mathbb{C}^3/Γ$ singularities, the McKay correspondence and D=3 $\mathcal{N}=2$ Chern-Simons gauge theories (1710.01046v4)

Abstract: We advocate that a generalized Kronheimer construction of the K\"ahler quotient crepant resolution $\mathcal{M}\zeta \longrightarrow \mathbb{C}^3/\Gamma$ of an orbifold singularity where $\Gamma\subset \mathrm{SU(3)}$ 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 $\mathbb{C}\times\mathcal{M}\zeta$ as transverse space. We illustrate and discuss many aspects of this of constructions emphasizing that the equation $\pmb{p}\wedge\pmb{p}=0$ 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 $\mathcal{G}\Gamma$ 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.