Mathematics

We advocate that a generalized Kronheimer construction of the Kählerquotient crepant resolution M_z \rightarrow C3/Gamma of an orbifold singularity, where \Gamma\subsetSU(3)} is a finite subgroup, naturally defines the field content andinteraction structure of a superconformal Chern-Simons Gauge Theory. This issupposedly the dual of an M2-brane solution of D=11 supergravity with C x M_z  as transverse space. We illustrate anddiscuss many aspects of this construction emphasizing that the equationp \wedge p = 0 which provides the Kähler analogue of theholomorphic sector in the hyperK\"ahler moment map equations canonicallydefines the structure of a universal superpotential in the CS theory. Thekernel of the above equation can be described as the orbit with respect to a quiver Lie group G_Gamma of a locus  L_Gamma \subset Hom_\Gamma(QxR,R)that has also a universal definition. We discuss the relation between the coset manifoldG_Gamma/\F_Gamma, the gauge group F_Gammabeing the maximal compact subgroup of the quiver group, the moment mapequations and the first Chern classes of the tautological vector bundles thatare in a one-to-one correspondence with the nontrivial irreps of Gamma.These first Chern classes provide a basis for the cohomology groupH2(M_z). We discuss the relation with the conjugacy classes ofGamma and provide the explicit construction of several examples, emphasizingthe role of a generalized McKay correspondence. The case of the ALE manifoldresolution of C2/Gamma singularities is utilized as a comparisonterm and new formulae related with the complex presentation of Gibbons-Hawkingmetrics are exhibited. Joint work with U. Bruzzo and A. Fino.