McKay correspondence

Abstract

This is a rough write-up of my lecture at Kinosaki and two lectures at RIMS workshops in Dec 1996, on work in progress that has not yet reached any really worthwhile conclusion, but contains lots of fun calculations. History of Vafa's formula, how the McKay correspondence for finite subgroups of SL(n,C) relates to mirror symmetry. The main aim is to give numerical examples of how the 2 McKay correspondences (1) representations of G <--> cohomology of resolution (2) conjugacy classes of G <--> homology must work, and to restate my 1992 Conjecture as a tautology, like cohomology or K-theory of projective space. Another aim is to give an introduction to Nakamura's results on the Hilbert scheme of G-clusters, following his preprints and his many helpful explanations. This is partly based on joint work with Y. Ito, and has benefited from encouragement and invaluable suggestions of S. Mukai.

V2 cured 2 misguided crossreferences and some errors of punctuation. This v3 gives references sent in by listeners to this network, and centres the graphics, a triumph of mind over computer manual!

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• Boundary Rings and N=2 Coset Models
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• Higher dimensional Automorphic Lie Algebras
• Stability of Quiver Representations and Topology Change
• Lectures on D-branes, Gauge Theories and Calabi-Yau Singularities
• Derived McKay correspondence via pure-sheaf transforms
• The ABCDEFG of Little Strings
• Brane Cube Realization of Three-dimensional Nonabelian Orbifolds
• Brane Configurations for Three-dimensional Nonabelian Orbifolds
• Little String Defects and Bala-Carter Theory
• Anticoherent Subspaces
• Geometrical McKay Correspondence for Isolated Singularities
• The Geometry of Noncommutative Singularity Resolutions
• Limit theories and continuous orbifolds
• Multiplicative McKay correspondence in the symplectic case
• Fractional two-branes, toric orbifolds and the quantum McKay correspondence
• Invariants of Automorphic Lie Algebras
• Crepant Resolutions of ${\mathbb{C}}^3/{\mathbb{Z}}_4$ and the Generalized Kronheimer Construction (in view of the Gauge/Gravity Correspondence)
• McKay correspondence and Hilbert schemes in dimension three
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• ADE subalgebras of the triplet vertex algebra W(p): A-series
• Geometric Reid's recipe for dimer models
• An explicit construction of the McKay correspondence for A-Hilb C^3
• Moduli of McKay quiver representations I: the coherent component
• Danilov resolution and representations of McKay quiver
• Gale duality and the linearisation map for noncommutative crepant resolutions