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39
Twisted Ktheory and Loop groups
 Proceedings of the International Congress of Mathematicians, Vol. III (Beijing
"... Abstract. Twisted Ktheory has received much attention recently in both mathematics and physics. We describe some models of twisted Ktheory, both topological and geometric. Then we state a theorem which relates representations of loop groups to twisted equivariant Ktheory. This is joint work with ..."
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Abstract. Twisted Ktheory has received much attention recently in both mathematics and physics. We describe some models of twisted Ktheory, both topological and geometric. Then we state a theorem which relates representations of loop groups to twisted equivariant Ktheory. This is joint work with Michael Hopkins and Constantin Teleman. The loop group of a compact Lie group G is the space of smooth maps S 1 → G with multiplication defined pointwise. Loop groups have been around in topology for quite some time [Bo], and in the 1980s were extensively studied from the point of view of representation theory [Ka], [PS]. In part this was driven by the relationship to conformal field theory. The interesting representations of loop groups are projective, and with fixed projective cocycle τ there is a finite number of irreducible representations up to isomorphism. Considerations from conformal field theory [V] led to a ring structure on the abelian group R τ (G) they generate, at least for transgressed twistings. This is the Verlinde ring. For G simply connected R τ (G) is a quotient of the representation ring of G, but that is not true in general. At about this time Witten [W] introduced a threedimensional topological quantum field theory in which the Verlinde ring plays an important role. Eventually it was understood that the fundamental object in that theory is a “modular tensor category ” whose Grothendieck group is the Verlinde ring. Typically it is a category of representations of a loop group or quantum group. For the special case of a finite group G the topological field theory is specified by a certain
The resolution property for schemes and stacks
"... Abstract. We prove the equivalence of two fundamental properties of algebraic stacks: being a quotient stack in a strong sense, and the resolution property, which says that every coherent sheaf is a quotient of some vector bundle. Moreover, we prove these properties in the important special case of ..."
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Cited by 17 (0 self)
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Abstract. We prove the equivalence of two fundamental properties of algebraic stacks: being a quotient stack in a strong sense, and the resolution property, which says that every coherent sheaf is a quotient of some vector bundle. Moreover, we prove these properties in the important special case of orbifolds whose associated algebraic space is a scheme. (Mathematics Subject Classification: Primary 14A20, Secondary 14L30.) 1
On a generalized ConnesHochschildKostantRosenberg theorem
 Adv. Math
, 2006
"... Abstract. The central result here is an explicit computation of the Hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds X as their spectrum. More precisely, the Hochschild homology is identified with the space of differential fo ..."
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Cited by 14 (3 self)
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Abstract. The central result here is an explicit computation of the Hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds X as their spectrum. More precisely, the Hochschild homology is identified with the space of differential forms on X, and the periodic cyclic homology with the twisted de Rham cohomology of X, thereby generalizing some fundamental results of Connes and HochschildKostantRosenberg. The ConnesChern character is also identified here with the twisted Chern character. 1.
Supersymmetric WZW models and twisted Ktheory of SO(3)
, 2004
"... We present an encompassing treatment of D–brane charges in supersymmetric SO(3) WZW models. There are two distinct supersymmetric CFTs at each even level: the standard bosonic SO(3) modular invariant tensored with free fermions, as well as a novel twisted model. We calculate the relevant twisted K–t ..."
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Cited by 11 (3 self)
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We present an encompassing treatment of D–brane charges in supersymmetric SO(3) WZW models. There are two distinct supersymmetric CFTs at each even level: the standard bosonic SO(3) modular invariant tensored with free fermions, as well as a novel twisted model. We calculate the relevant twisted K–theories and find complete agreement with the CFT analysis of D–brane charges. The K–theoretical computation in particular elucidates some important aspects of N = 1 supersymmetric WZW models on nonsimply connected Lie groups.
Heisenberg groups and noncommutative fluxes
"... We develop a grouptheoretical approach to the formulation of generalized abelian gauge theories, such as those appearing in string theory and Mtheory. We explore several applications of this approach. First, we show that there is an uncertainty relation which obstructs simultaneous measurement of ..."
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Cited by 10 (3 self)
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We develop a grouptheoretical approach to the formulation of generalized abelian gauge theories, such as those appearing in string theory and Mtheory. We explore several applications of this approach. First, we show that there is an uncertainty relation which obstructs simultaneous measurement of electric and magnetic flux when torsion fluxes are included. Next we show how to define the Hilbert space of a selfdual field. The Hilbert space is Z2graded and we show that, in general, selfdual theories (including the RR fields of string theory) have fermionic sectors. We indicate how rational conformal field theories associated to the twodimensional Gaussian model generalize to (4k + 2)dimensional conformal field theories. When our ideas are applied to the RR fields of string theory we learn that it is impossible to measure the Ktheory class of a RR field. Only the reduction modulo torsion can be measured. May
On the topology of Tduality
 Rev. Math. Phys
"... We study a topological version of the Tduality relation between pairs consisting of a principal U(1)bundle equipped with a degreethree integral cohomology class. We describe the homotopy type of a classifying space for such pairs and show that it admits a selfmap which implements a Tduality tran ..."
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Cited by 10 (1 self)
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We study a topological version of the Tduality relation between pairs consisting of a principal U(1)bundle equipped with a degreethree integral cohomology class. We describe the homotopy type of a classifying space for such pairs and show that it admits a selfmap which implements a Tduality transformation. We give a simple derivation of a Tduality isomorphism for certain twisted cohomology theories. We conclude with some explicit computations of twisted Ktheory groups and
D–branes in N = 2 coset models and twisted equivariant K–theory
"... The charges of Dbranes in KazamaSuzuki coset models are analyzed. We provide the calculation of the corresponding twisted equivariant Ktheory, and in the case of Grassmannian cosets, su(n + 1)/u(n), compare this to the charge lattices that are derived from boundary conformal field theory. ..."
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Cited by 9 (4 self)
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The charges of Dbranes in KazamaSuzuki coset models are analyzed. We provide the calculation of the corresponding twisted equivariant Ktheory, and in the case of Grassmannian cosets, su(n + 1)/u(n), compare this to the charge lattices that are derived from boundary conformal field theory.
GERBES, (TWISTED) KTHEORY, AND THE SUPERSYMMETRIC WZW MODEL
, 2002
"... The aim of this talk is to explain how symmetry breaking in a quantum field theory problem leads to a study of projective bundles, DixmierDouady classes, and associated gerbes. A gerbe manifests itself in different equivalent ways. Besides the cohomological description as a DD class, it can be def ..."
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Cited by 7 (1 self)
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The aim of this talk is to explain how symmetry breaking in a quantum field theory problem leads to a study of projective bundles, DixmierDouady classes, and associated gerbes. A gerbe manifests itself in different equivalent ways. Besides the cohomological description as a DD class, it can be defined in terms of a family of local line bundles or as a prolongation problem for an (infinitedimensional) principal bundle, with the fiber consisting of (a subgroup of) projective unitaries in a Hilbert space. The prolongation aspect is directly related to the appearance of central extensions of (broken) symmetry groups. We also discuss the construction of twisted Ktheory classes by families of supercharges for the supersymmetric WessZuminoWitten model. This paper is based on a lecture at the meeting “La 70eme Rencontre entre
Extended manifolds and extended equivariant cohomology
"... Abstract. We define the category of manifolds with extended tangent bundles, we study their symmetries and we consider the analogue of equivariant cohomology for actions of Lie groups in this category. We show that when the action preserves the splitting of the extended tangent bundle, our definitio ..."
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Cited by 6 (1 self)
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Abstract. We define the category of manifolds with extended tangent bundles, we study their symmetries and we consider the analogue of equivariant cohomology for actions of Lie groups in this category. We show that when the action preserves the splitting of the extended tangent bundle, our definition of extended equivariant cohomology agrees with the twisted equivariant de Rham model of Cartan, and for this case we show that there is localization at the fixed point set, à la AtiyahBott. 1.