http://www.math.wisc.edu/~shi/topological_structures/McKay_correspondences.htm Workshop Schedule Monday (March 20, 2006) 9:00-9:15 Welcome 9:15-10:15 Paul Aspinwall (Duke University) Title: D-Branes, Mukai and McKay Tea Break 11:00-12:00 Tom Bridgeland (University of Edinburgh) Title: From categories to geometry : stability conditions and Kleinian singularities 2:15-3:15 Andrei Caldararu (University of Wisconsin) Title: The Hopf algebra governing orbifold Hochschild cohomology Tea Break 3:45-4:45 Kentaro Hori (University of Toronto) Title: Matrix factorizations and complexes of vector bundle ---- an approach from 2d QFT with boundary Tuesday 9:30-10:30 Alexey Bondal (Steklov Mathematical Institute) Title: Integrable systems related to triangulated categories Tea Break 11:00-12:00 Dmitry Kaledin (Steklov Institute) Title: McKay and generalizations in the symplectic case 1:30-2:30 Miles Reid (University of Warwick) Title: Orbifold Riemann--Roch and plurigenera 2:45-3:45 Wei-Ping Li (Hong Kong Univ. of Science & Technology) Title: Integral cohomology of the Hilbert schemes of points on surfaces Tea Break 4:00-5:00 Bohui Chen (Sichuan University) Title: DeRham model of Chen-Ruan orbifold cohomology ring on abelian orbifolds Wednesday 9:00-10:00 Lev Borisov (UW-Madison) Title: McKay correspondence for elliptic genera Tea Break 10:30-11:30 Fabio Perroni (University of Zürich) Title: The cohomological crepant resolution conjecture for orbifold with transversal A_n-singularities 1:00-2:00 Weiqiang Wang (University of Virginia) Title: The cohomology rings of Hilbert schemes of points and McKay-Ruan correspondence 2:15-3:15 Yasuyuki Kawahigashi (Univ. of Tokyo) Title: Conformal Field Theory and Operator Algebras Tea Break 3:45-4:45 Adrian Ocneanu (Pennsylvania State University) Title: Quantum Subgroups and higher quantum McKay correspondences Thursday 9:00-10:00 Raphael Rouquier (CNRS) Title: McKay's correspondence and modular representations of finite groups Tea Break 10:30-11:30 Chongying Dong (UC Santa Cruz) Title: Representation theory for vertex operator algebras 1:00-2:00 Victor Ginzburg (University of Chicago) Title: Noncommutative geometry and Calabi-Yau algebras 2:15-3:15 Naihuan Jing (North Carolina State University) Title: Vertex operators and quantum cohomology Tea Break Friday 9:30-10:30 Alexander Kirillov, jr. (SUNY at Stony Brook) Title: McKay correspondence and equivariant sheaves on P^1 Tea Break 11:00-12:00 Viktor Ostrik (University of Oregon) Title: Quantum versions of McKay correspondence The original McKay correspondence related finite subgroups of Sl(2) and Dynkin diagrams of type ADE; the latter occur as intersection pairings in the cohomology of crepant resolutions of C2/ . Mirror symmetry inspired a hope for similar relations in higher dimensions. This question has attracted an increasing number of physicists and mathematicians, and has grown well beyond the scope of McKay’s original correspondence. The broader or generalized McKay correspondence can be understood as a duality between the algebra of finite groups and the geometry of crepant resolutions. There are two basic invariants of a finite group: its representation ring and the center of its group algebra. They are commutative rings, of the same rank, but with distinct product structures; the first leads naturally to K-theory and more generally to the derived category. The K-theoretic McKay correspondence can be viewed as an equivalence of the derived category of equivariant coherent sheaves on X, or of coherent sheaves on the orbifold X/G, with the derived category of coherent sheaves on its crepant resolution. These equivalences have been extensively studied in two cases : when the dimension of X is three, and when is a symmetric group. One of the most surprising achievements is a theorem of Bridgeland-King-Reid and Haiman: there is a natural isomorphism between the derived categories of representations of and of coherent sheaves on the crepant resolution. This isomorphism is given by a so-called Fourier-Mukai transform. In many areas of mathematics, an equivalence of derived categories often defines striking correspondences between apparently di erent things; the famous solution of the Kazhdan-Lusztig conjecture in representation theory is one such example. The Fourier-Mukai transform is expected to play an important role in many other situations. Derived categories are central to the theory of D-branes, so this can be interpreted as an open string version of McKay’s correspondence. The center of a group algebra generalizes to the Chen-Ruan orbifold co-homology of X/G. This version of the correspondence defines a conjectural multiplicative equivalence of Chen-Ruan cohomology with a deformation of the cohomology ring of a crepant resolution, involving Gromov-Witten invariants associated to certain exceptional sets. This conjecture has recently checked for certain Hilbert schemes by Fantecchi-G¨ottsche-Uribe, based on earlier work by Lehn-Sorger. No deformation is involved in this case, but in general we need the quantum cohomology of the crepant resolution: This is thus a closed string McKay correspondence. During the attempt to prove McKay correspondences at the level of numerical invariants such as Euler numbers, Batyrev, Denef-Loeser, Kontsevich and others developed the impressive techniques of motivic integration. It is safe to predict that as these results are extended to more general invariants, related important techniques will develop. When is a finite subgroup of Sp(n) (e.g., symmetric groups), the crepant resolution of C2n/ has a hyper-K¨ahler structure. Rotating this structure de-fines deformations of the complex structure of the underlying orbifold, which is best understood in the framework of noncommutative algebraic geometry [Etingof-Ginzburg]. It is also related to Cherednik’s double a ne Hecke algebra. Orbifolds appear not only in geometry. They also naturally appear in the the theory of vertex operator algebras, which are a natural context for the study of tensor structures in conformal field theory. For example, an orbifold construction for VOAs was used in Borcherd’s solution of the famous Moonshine conjecture of McKay and Thompson. These algebraic constructions were motivated by the operator product formalism of string theory. Since the geometric orbifolds mentioned above live in the Lagrangian formalism, the precise mathematical relation between these two aspects of orbifolds are not yet clear. This is certainly an important problem to address. A first hint may be Grojnowski-Nakajima’s geometric construction of a Heisenberg representation on the cohomology of certain Hilbert schemes. This corresponds to the lattice vertex operator algebra, one of the simplest VOAs. Nakajima has also constructed representation of some Lie algebras using the so-called quiver varieties, which are roughly moduli spaces of sheaves on the orbifold C2n/G.

http://www.math.wisc.edu/~shi/topological_structures/McKay_correspondences.htm **Workshop Schedule** **Monday (March 20, 2006)** 9:00-9:15 Welcome 9:15-10:15 Paul Aspinwall (Duke University) Title: D-Branes, Mukai and McKay Tea Break 11:00-12:00 Tom Bridgeland (University of Edinburgh) Title: From categories to geometry : stability conditions and Kleinian singularities 2:15-3:15 Andrei Caldararu (University of Wisconsin) Title: The Hopf algebra governing orbifold Hochschild cohomology Tea Break 3:45-4:45 Kentaro Hori (University of Toronto) Title: Matrix factorizations and complexes of vector bundle ---- an approach from 2d QFT with boundary **Tuesday** 9:30-10:30 Alexey Bondal (Steklov Mathematical Institute) Title: Integrable systems related to triangulated categories Tea Break 11:00-12:00 Dmitry Kaledin (Steklov Institute) Title: McKay and generalizations in the symplectic case 1:30-2:30 Miles Reid (University of Warwick) Title: Orbifold Riemann--Roch and plurigenera 2:45-3:45 Wei-Ping Li (Hong Kong Univ. of Science & Technology) Title: Integral cohomology of the Hilbert schemes of points on surfaces Tea Break 4:00-5:00 Bohui Chen (Sichuan University) Title: DeRham model of Chen-Ruan orbifold cohomology ring on abelian orbifolds **Wednesday** 9:00-10:00 Lev Borisov (UW-Madison) Title: McKay correspondence for elliptic genera Tea Break 10:30-11:30 Fabio Perroni (University of Zürich) Title: The cohomological crepant resolution conjecture for orbifold with transversal A_n-singularities 1:00-2:00 Weiqiang Wang (University of Virginia) Title: The cohomology rings of Hilbert schemes of points and McKay-Ruan correspondence 2:15-3:15 Yasuyuki Kawahigashi (Univ. of Tokyo) Title: Conformal Field Theory and Operator Algebras Tea Break 3:45-4:45 Adrian Ocneanu (Pennsylvania State University) Title: Quantum Subgroups and higher quantum McKay correspondences **Thursday** 9:00-10:00 Raphael Rouquier (CNRS) Title: McKay's correspondence and modular representations of finite groups Tea Break 10:30-11:30 Chongying Dong (UC Santa Cruz) Title: Representation theory for vertex operator algebras 1:00-2:00 Victor Ginzburg (University of Chicago) Title: Noncommutative geometry and Calabi-Yau algebras 2:15-3:15 Naihuan Jing (North Carolina State University) Title: Vertex operators and quantum cohomology Tea Break **Friday** 9:30-10:30 Alexander Kirillov, jr. (SUNY at Stony Brook) Title: McKay correspondence and equivariant sheaves on P^1 Tea Break 11:00-12:00 Viktor Ostrik (University of Oregon) Title: Quantum versions of McKay correspondence The original McKay correspondence related finite subgroups of Sl(2) and Dynkin diagrams of type ADE; the latter occur as intersection pairings in the cohomology of crepant resolutions of C2/ . Mirror symmetry inspired a hope for similar relations in higher dimensions. This question has attracted an increasing number of physicists and mathematicians, and has grown well beyond the scope of McKay’s original correspondence. The broader or generalized McKay correspondence can be understood as a duality between the algebra of finite groups and the geometry of crepant resolutions. There are two basic invariants of a finite group: its representation ring and the center of its group algebra. They are commutative rings, of the same rank, but with distinct product structures; the first leads naturally to K-theory and more generally to the derived category. The K-theoretic McKay correspondence can be viewed as an equivalence of the derived category of equivariant coherent sheaves on X, or of coherent sheaves on the orbifold X/G, with the derived category of coherent sheaves on its crepant resolution. These equivalences have been extensively studied in two cases : when the dimension of X is three, and when is a symmetric group. One of the most surprising achievements is a theorem of Bridgeland-King-Reid and Haiman: there is a natural isomorphism between the derived categories of representations of and of coherent sheaves on the crepant resolution. This isomorphism is given by a so-called Fourier-Mukai transform. In many areas of mathematics, an equivalence of derived categories often defines striking correspondences between apparently di erent things; the famous solution of the Kazhdan-Lusztig conjecture in representation theory is one such example. The Fourier-Mukai transform is expected to play an important role in many other situations. Derived categories are central to the theory of D-branes, so this can be interpreted as an open string version of McKay’s correspondence. The center of a group algebra generalizes to the Chen-Ruan orbifold co-homology of X/G. This version of the correspondence defines a conjectural multiplicative equivalence of Chen-Ruan cohomology with a deformation of the cohomology ring of a crepant resolution, involving Gromov-Witten invariants associated to certain exceptional sets. This conjecture has recently checked for certain Hilbert schemes by Fantecchi-G¨ottsche-Uribe, based on earlier work by Lehn-Sorger. No deformation is involved in this case, but in general we need the quantum cohomology of the crepant resolution: This is thus a closed string McKay correspondence. During the attempt to prove McKay correspondences at the level of numerical invariants such as Euler numbers, Batyrev, Denef-Loeser, Kontsevich and others developed the impressive techniques of motivic integration. It is safe to predict that as these results are extended to more general invariants, related important techniques will develop. When is a finite subgroup of Sp(n) (e.g., symmetric groups), the crepant resolution of C2n/ has a hyper-K¨ahler structure. Rotating this structure de-fines deformations of the complex structure of the underlying orbifold, which is best understood in the framework of noncommutative algebraic geometry [Etingof-Ginzburg]. It is also related to Cherednik’s double a ne Hecke algebra. Orbifolds appear not only in geometry. They also naturally appear in the the theory of vertex operator algebras, which are a natural context for the study of tensor structures in conformal field theory. For example, an orbifold construction for VOAs was used in Borcherd’s solution of the famous Moonshine conjecture of McKay and Thompson. These algebraic constructions were motivated by the operator product formalism of string theory. Since the geometric orbifolds mentioned above live in the Lagrangian formalism, the precise mathematical relation between these two aspects of orbifolds are not yet clear. This is certainly an important problem to address. A first hint may be Grojnowski-Nakajima’s geometric construction of a Heisenberg representation on the cohomology of certain Hilbert schemes. This corresponds to the lattice vertex operator algebra, one of the simplest VOAs. Nakajima has also constructed representation of some Lie algebras using the so-called quiver varieties, which are roughly moduli spaces of sheaves on the orbifold C2n/G.

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