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

Abstract

We advocate that a generalized Kronheimer construction of the Kähler quotient crepant resolution $\mathcal{M}_ζ\longrightarrow \mathbb{C}^3/Γ$ of an orbifold singularity where $Γ\subset \mathrm{SU(3)}$ 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 $\mathbb{C}\times\mathcal{M}_ζ$ 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 Kähler analogue of the holomorphic sector in the hyperKähler 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 $\mathcal{G}_Γ$ of a locus $L_Γ\subset \mathrm{Hom}_Γ(\mathcal{Q}\otimes R,R)$ that has also a universal definition. We discuss the relation between the coset manifold $\mathcal{G}_Γ/\mathcal{F}_Γ$, the gauge group $\mathcal{F}_Γ$ 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 $Γ$. These first Chern classes provide a basis for the cohomology group $H^2(\mathcal{M}_ζ)$. We discuss the relation with conjugacy classes of $Γ$ and provide the explicit construction of several examples emphasizing the role of a generalized McKay correspondence. The case of the ALE manifold resolution of $\mathbb{C}^2/Γ$ singularities is utilized as a comparison term and new formulae related with the complex presentation of Gibbons-Hawking metrics are exhibited.