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69
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 41 (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
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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Cited by 23 (2 self)
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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
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 22 (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.
Surjectivity for Hamiltonian Gspaces in Ktheory
 T rans. Amer. Math. Soc
"... Abstract. Let G be a compact connected Lie group, and (M, ω) a Hamiltonian Gspace with proper moment map µ. We give a surjectivity result which expresses the Ktheory of the symplectic quotient M /G in terms of the equivariant Ktheory of the original manifold M, under certain technical conditions ..."
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Cited by 17 (8 self)
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Abstract. Let G be a compact connected Lie group, and (M, ω) a Hamiltonian Gspace with proper moment map µ. We give a surjectivity result which expresses the Ktheory of the symplectic quotient M /G in terms of the equivariant Ktheory of the original manifold M, under certain technical conditions on µ. This result is a natural Ktheoretic analogue of the Kirwan surjectivity theorem in symplectic geometry. The main technical tool is the Ktheoretic AtiyahBott lemma, which plays a fundamental role in the symplectic geometry of Hamiltonian Gspaces. We discuss this lemma in detail and highlight the differences between the Ktheory and rational cohomology versions of this lemma. We also introduce a Ktheoretic version of equivariant formality and prove that when the fundamental group of G is torsionfree, every compact Hamiltonian Gspace is equivariantly formal. Under these conditions, the forgetful map K ∗ G (M) → K ∗ (M) is surjective, and thus every complex vector bundle admits a stable equivariant structure. Furthermore, by considering complex line bundles, we show that every integral cohomology class in H 2 (M;�) admits an equivariant extension in H 2 G (M;�). 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 14 (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 13 (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
The ring structure for equivariant twisted Ktheory
, 2006
"... We prove, under some mild conditions, that the equivariant twisted Ktheory group of a crossed module admits a ring structure if the twisting 2cocycle is 2multiplicative. We also give an explicit construction of the transgression map T1: H ∗ (Γ•; A) → H∗−1 ((N ⋊ Γ) •; A) for any crossed module N ..."
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Cited by 13 (0 self)
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We prove, under some mild conditions, that the equivariant twisted Ktheory group of a crossed module admits a ring structure if the twisting 2cocycle is 2multiplicative. We also give an explicit construction of the transgression map T1: H ∗ (Γ•; A) → H∗−1 ((N ⋊ Γ) •; A) for any crossed module N → Γ and prove that any element in the image is ∞multiplicative. As a consequence, we prove that, under some mild conditions, for a crossed module N → Γ and any e ∈ ˇ Z3 (Γ•; S1), that (N) admits a ring structure. As an applithe equivariant twisted Ktheory group K ∗ e,Γ cation, we prove that for a compact, connected and simply connected Lie group G, the equivariant twisted Ktheory group K ∗ [c],G(G) is endowed with a canonical ring structure K i+d [c],G(G)⊗Kj+d [c],G(G) → Ki+j+d [c],G (G), where d = dim G and [c] ∈ H2 ((G⋊G) •; S1).
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 12 (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.