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Twisted equivariant Ktheory with complex coefficients
, 2008
"... Using a global version of the equivariant Chern character, we describe an effective method for computing the complexified twisted equivariant Ktheory of a space ..."
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Cited by 50 (6 self)
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Using a global version of the equivariant Chern character, we describe an effective method for computing the complexified twisted equivariant Ktheory of a space
Twisted Ktheory of differentiable stacks
 ANN. SCI. ÉCOLE NORM. SUP
, 2004
"... In this paper, we develop twisted Ktheory for stacks, where the twisted class is given by an S 1gerbe over the stack. General properties, including the Mayer–Vietoris property, Bott periodicity, and the product structure K i α ⊗K j β → Ki+j α+β are derived. Our approach provides a uniform framew ..."
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Cited by 50 (12 self)
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In this paper, we develop twisted Ktheory for stacks, where the twisted class is given by an S 1gerbe over the stack. General properties, including the Mayer–Vietoris property, Bott periodicity, and the product structure K i α ⊗K j β → Ki+j α+β are derived. Our approach provides a uniform framework for studying various twisted Ktheories including the usual twisted Ktheory of topological spaces, twisted equivariant Ktheory, and the twisted Ktheory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted Kgroups can be expressed by socalled “twisted vector bundles”. Our approach is to work on presentations of stacks, namely groupoids, and relies heavily on the machinery of Ktheory (KKtheory) of C ∗algebras.
Geometrical interpretation of dbranes in gauged wzw models
 JHEP
"... We show that one can construct Dbranes in parafermionic and WZW theories (and their orbifolds) which have very natural geometrical interpretations, and yet are not automatically included in the standard Cardy construction of Dbranes in rational conformal field theory. The relation between these th ..."
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Cited by 27 (0 self)
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We show that one can construct Dbranes in parafermionic and WZW theories (and their orbifolds) which have very natural geometrical interpretations, and yet are not automatically included in the standard Cardy construction of Dbranes in rational conformal field theory. The relation between these theories and their Tdual description leads to an analogy between these Dbranes and the familiar Abranes and Bbranes of N = 2 theories. May
Twisted Ktheory of Lie groups
"... I determine the twisted K–theory of all compact simply connected simple Lie groups. The computation reduces via the Freed–Hopkins–Teleman theorem [1] to the CFT prescription, and thus explains why it gives the correct result. Finally I analyze the exceptions noted by Bouwknegt et al [2].CONTENTS 1 ..."
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I determine the twisted K–theory of all compact simply connected simple Lie groups. The computation reduces via the Freed–Hopkins–Teleman theorem [1] to the CFT prescription, and thus explains why it gives the correct result. Finally I analyze the exceptions noted by Bouwknegt et al [2].CONTENTS 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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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
Dbranes on group manifolds and fusion rings
 JHEP
"... Abstract. In this paper we compute the charge group for symmetry preserving Dbranes ..."
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Abstract. In this paper we compute the charge group for symmetry preserving Dbranes
Ktheory in quantum field theory
 Current Develop. Math
"... Abstract. We survey three different ways in which Ktheory in all its forms enters quantum field theory. In Part 1 we give a general argument which relates topological field theory in codimension two with twisted Ktheory, and we illustrate with some finite models. Part 2 is a review of pfaffians of ..."
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Abstract. We survey three different ways in which Ktheory in all its forms enters quantum field theory. In Part 1 we give a general argument which relates topological field theory in codimension two with twisted Ktheory, and we illustrate with some finite models. Part 2 is a review of pfaffians of Dirac operators, anomalies, and the relationship to differential Ktheory. Part 3 is a geometric exposition of Dirac charge quantization, which in superstring theories also involves differential Ktheory. Parts 2 and 3 are related by the GreenSchwarz anomaly cancellation mechanism. An appendix, joint with Jerry Jenquin, treats the partition function of RaritaSchwinger fields. Grothendieck invented KTheory almost 50 years ago in the context of algebraic geometry, specifically in his generalization of the Hirzebruch RiemannRoch theorem [BS]. Shortly thereafter, Atiyah and Hirzebruch brought Grothendieck’s ideas into topology [AH], where they were applied to a variety of problems. Analysis entered after it was realized that the symbol of an elliptic operator determines an element of Ktheory. Atiyah and Singer then proved a formula for the index of such an operator (on a compact manifold) in terms of the Ktheory class of the symbol [AS1]. Subsequently, Ktheoretic ideas permeated other areas of linear analysis, algebra, noncommutative geometry, etc. One of the pleasant surprises of the past few years has been the relevance of Ktheory to superstring theory and related parts of theoretical physics. Furthermore, the story involves not only topological Ktheory, but also the Ktheory of C ∗algebras, the Ktheory of sheaves, and other forms of Ktheory. Not surprisingly, this new arena for Ktheory has inspired some developments in mathematics which are the subject of ongoing research. Our exposition here aims to explain three different ways in which topological Ktheory appears in physics, and how this physics motivates the mathematical ideas we are investigating. Part 1 concerns topological quantum field theory. Recall that an ndimensional topological theory assigns a complex number to every closed oriented nmanifold and a complex vector space to every closed oriented (n − 1)manifold. Continuing the superposition principle and ideas of locality to
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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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 over orbifolds and twisted Ktheory
"... Abstract. In this paper we construct an explicit geometric model for the group of gerbes over an orbifold X. We show how from its curvature we can obtain its characteristic class in H 3 (X) via ChernWeil theory. For an arbitrary gerbe L, a twisting L Korb(X) of the orbifold Ktheory of X is constru ..."
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Abstract. In this paper we construct an explicit geometric model for the group of gerbes over an orbifold X. We show how from its curvature we can obtain its characteristic class in H 3 (X) via ChernWeil theory. For an arbitrary gerbe L, a twisting L Korb(X) of the orbifold Ktheory of X is constructed, and shown to generalize previous twisting by Rosenberg [28], Witten [35], AtiyahSegal [2] and Bowknegt et. al. [4] in the smooth case and by AdemRuan [1] for discrete torsion on an orbifold. Contents