The Kähler quotient resolution of C3/Γ singularities, the McKay correspondence and D=3 N = 2 Chern-Simons gauge theories

We advocate that the generalized Kronheimer construction of the Ka ̈hler quotient crepant resolution Mζ −→ C3/Γ of an orbifold singularity where Γ ⊂ SU(3) is a finite subgroup naturally defines the field content and the interaction structure of a superconformal Chern-Simons Gauge Theory. This latter is suppos- edlythedualofanM2-branesolutionofD=11supergravitywithC×Mζ astransversespace.Weillustrate and discuss many aspects of this type of constructions emphasizing that the equation p ∧ p = 0 which provides the Ka ̈hler analogue of the holomorphic sector in the hyperKa ̈hler moment map equations canonically defines the structure of a universal superpotential in the CS theory. Furthermore the kernel DΓ of the above equation can be described as the orbit with respect to a quiver Lie group GΓ of a special locus LΓ ⊂ HomΓ (Q ⊗ R, R) that has also a universal definition. We provide an extensive discussion of the relation between the coset manifold GΓ/FΓ, the gauge group FΓ being the maximal compact subgroup of the quiver group, the moment map equations and the first Chern classes of the so named tautological vector bundles that are in one-to-one correspondence with the nontrivial irreps of Γ. These first Chern classes are represented by (1,1)-forms on Mζ and provide a basis for the cohomology group H2(Mζ ). We also discuss the relation with conjugacy classes of Γ and we provide the explicit construction of several examples emphasizing the role of a general- ized McKay correspondence. The case of the ALE manifold resolution of C2/Γ singularities is utilized as a comparison term and new formulae related with the complex presentation of Gibbons-Hawking metrics are exhibited.