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 ChernSimons Gauge Theory. This issupposedly the dual of an M2brane 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 onetoone 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 GibbonsHawkingmetrics are exhibited. Joint work with U. Bruzzo and A. Fino.
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The Kähler Quotient Resolution of C3/Gamma singularities, the McKay correspondence and D=3 N=2 ChernSimons gauge theories
Research Group:
Professor
Pietro Fré
Institution:
Università di Torino
Location:
A004
Schedule:
Thursday, October 19, 2017  14:30
Abstract:
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