Results 1  10
of
38
Noncommutative correspondences, duality and Dbranes in bivariant Ktheory
, 2007
"... We describe a categorical framework for the classification of Dbranes on noncommutative spaces using techniques from bivariant Ktheory of C∗algebras. We present a new description of bivariant Ktheory in terms of noncommutative correspondences which is nicely adapted to the study of Tduality in ..."
Abstract

Cited by 15 (2 self)
 Add to MetaCart
(Show Context)
We describe a categorical framework for the classification of Dbranes on noncommutative spaces using techniques from bivariant Ktheory of C∗algebras. We present a new description of bivariant Ktheory in terms of noncommutative correspondences which is nicely adapted to the study of Tduality in open string theory. We systematically use the diagram calculus for bivariant Ktheory as detailed in our previous paper [12]. We explicitly work out our theory for a number of examples of noncommutative manifolds.
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 ..."
Abstract

Cited by 13 (0 self)
 Add to MetaCart
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).
Twists of Ktheory and TMF
 In Superstrings, geometry, topology, and C∗algebras
, 2010
"... ar ..."
(Show Context)
DIFFERENTIAL TWISTED KTHEORY AND APPLICATIONS
, 2007
"... In this paper, we develop differential characters in twisted Ktheory and use them to define a twisted Chern character. In the usual formalism the ‘twist’ is given by a degree three Čech class while we work with differential twisted Ktheory with twisting given by a degree 3 Deligne class. This res ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
(Show Context)
In this paper, we develop differential characters in twisted Ktheory and use them to define a twisted Chern character. In the usual formalism the ‘twist’ is given by a degree three Čech class while we work with differential twisted Ktheory with twisting given by a degree 3 Deligne class. This resolves an unsatisfactory dependence on choices of representatives of differential forms in the definition of the Chern character map for twisted Ktheory in the current literature. Twisted eta forms and twisted spin c structures are also defined. To show the efficacy of our point of view we use our approach to study Dbrane charges on a compact Lie group with nontrivial twisting by a Deligne class.
The index of projective families of elliptic operators: The decomposable case
"... An index theory for projective families of elliptic pseudodifferential operators is developed under two conditions. First, that the twisting, ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
(Show Context)
An index theory for projective families of elliptic pseudodifferential operators is developed under two conditions. First, that the twisting,
Topological Tduality for torus bundles with monodromy, in preparation
"... Abstract. We give a simplified definition of topological Tduality that applies to arbitrary torus bundles. The new definition does not involve Chern classes or spectral sequences, only gerbes and morphisms between them. All the familiar topological conditions for Tduals are shown to follow. We det ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
(Show Context)
Abstract. We give a simplified definition of topological Tduality that applies to arbitrary torus bundles. The new definition does not involve Chern classes or spectral sequences, only gerbes and morphisms between them. All the familiar topological conditions for Tduals are shown to follow. We determine necessary and sufficient conditions for existence of a Tdual in the case of affine torus bundles. This is general enough to include all principal torus bundles as well as torus bundles with arbitrary monodromy representations. We show that isomorphisms in twisted cohomology, twisted Ktheory and of Courant algebroids persist in this general setting. We also give an example where twisted Ktheory groups can be computed by iterating Tduality. 1.
Twisted Ktheory and finitedimensional approximation, arXiv:0803.2327. 21
"... Abstract. We provide a finitedimensional model of the twisted Kgroup twisted by any degree three integral cohomology class of a CW complex. One key to the model is Furuta’s generalized vector bundle, and the other is a finitedimensional approximation of Fredholm operators. 1. ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We provide a finitedimensional model of the twisted Kgroup twisted by any degree three integral cohomology class of a CW complex. One key to the model is Furuta’s generalized vector bundle, and the other is a finitedimensional approximation of Fredholm operators. 1.
Projective Dirac operators, twisted Ktheory and local index formula, arXiv:1008.0707
 J. Noncommutative Geom. Mathematics Department, Caltech, 1200 E. California Blvd. Pasadena, CA 91125, USA Email
"... I am very grateful to BaiLing Wang. He foresaw the possibility that the projective spin Dirac operator defined by [19] in formal sense can be realized by a certain spectral triple, and introduced his interesting research project to me in 2008. The spectral triple in his mind turned out to be the pr ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
I am very grateful to BaiLing Wang. He foresaw the possibility that the projective spin Dirac operator defined by [19] in formal sense can be realized by a certain spectral triple, and introduced his interesting research project to me in 2008. The spectral triple in his mind turned out to be the projective spectral triple constructed in this paper. Without his insight, I wouldn’t have been writing this thesis. I also wish to thank my advisor, Matilde Marcolli, for her many years of encouragement, support, and many helpful suggestions on both this research and other aspects. iv We construct a canonical noncommutative spectral triple for every oriented closed Riemannian manifold, which represents the fundamental class in the twisted Khomology of the manifold. This socalled “projective spectral triple ” is Morita equivalent to the wellknown commutative spin spectral triple provided that the manifold is spinc. We give an explicit local formula for the twisted Chern character for Ktheories twisted with torsion classes, and with this formula we show that the twisted Chern character of the projective spectral triple is identical to the Poincaré dual of the Ahat genus of the manifold.