Chern Character for Twisted Complexes

2005

We explore the relations of twisted K-theory to twisted and untwisted classical cohomology. We construct an Atiyah-Hirzebruch spectral sequence, and describe its differentials rationally as Massey products. We define the twisted Chern character. We also discuss power operations in the twisted theory, and the role of the Koschorke classes.

2020

Viewing the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model, we construct the cyclic-homology Chern–Weil homomorphism. To realize the thus constructed Chern–Weil homomorphism as a Cartan model of the homomorphism tautologically induced by the classifying map on cohomology, we replace the unital subalgebra of coaction-invariants by its natural H-unital nilpotent extension (row extension). Although the row-extension algebra provides a drastically different model of the cyclic object, we prove that, for any row extension of any unital algebra over a commutative ring, the row-extension Hochschild complex and the usual Hochschild complex are chain homotopy equivalent. It is the discovery of an explicit homotopy formula that allows us to improve the homological quasi-isomorphism arguments of Loday and Wodzicki. We work with families of principal coactions, and instantiate our noncommutative Chern–Weil theo...

Communications in Mathematical Physics, 2003

It was argued in , that in the presence of a nontrivial Bfield, D-brane charges in type IIB string theories are classified by twisted Ktheory. In , it was proved that twisted K-theory is canonically isomorphic to bundle gerbe K-theory, whose elements are ordinary Hilbert bundles on a principal projective unitary bundle, with an action of the bundle gerbe determined by the principal projective unitary bundle. The principal projective unitary bundle is in turn determined by the twist. This paper studies in detail the Chern-Weil representative of the Chern character of bundle gerbe K-theory that was introduced in [4], extending the construction to the equivariant and the holomorphic cases. Included is a discussion of interesting examples.

Eprint Arxiv 0804 1274, 2008

In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of modules on schemes, as well as its quasi-coherent and perfect versions. We also explain how ideas from derived algebraic geometry and higher category theory can be used in order to construct a Chern character for these categorical sheaves, which is a categorified version of the Chern character for perfect complexes with values in cyclic homology. Our construction uses in an essential way the derived loop space of a scheme X, which is a derived scheme whose theory of functions is closely related to cyclic homology of X. This work can be seen as an attempt to define algebraic analogs of elliptic objects and characteristic classes for them. The present text is an overview of a work in progress and details will appear elsewhere.

Comptes Rendus Mathematique, 2004

Following the idea of Galois-type extensions and entwining structures, we define the notion of a principal extension of noncommutative algebras. We show that modules associated to such extensions via finite-dimensional corepresentations are finitely generated projective, and determine an explicit formula for the Chern character applied to the thus obtained modules.

Annales Scientifiques de l’École Normale Supérieure, 2004

In this paper, we develop twisted K-theory for stacks, where the twisted class is given by an S 1 -gerbe over the stack. General properties, including the Mayer-Vietoris property, Bott periodicity, and the product structure K i α ⊗ K j β → K i+j α+β are derived. Our approach provides a uniform framework for studying various twisted K-theories including the usual twisted K-theory of topological spaces, twisted equivariant Ktheory, and the twisted K-theory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted K-groups can be expressed by so-called "twisted vector bundles".

Journal für die reine und angewandte Mathematik (Crelles Journal), 2002

We construct for an equivariant homology theory for proper equivariant CW-complexes an equivariant Chern character, provided that certain conditions are satis®ed. This applies for instance to the sources of the assembly maps in the Farrell-Jones Conjecture with respect to the family F of ®nite subgroups and in the Baum-Connes Conjecture. Thus we get an explicit calculation in terms of group homology of Q n Z K n RG and Q n Z L n RG for a commutative ring R with Q r R, provided the Farrell-Jones Conjecture with respect to F is true, and of Q n Z K top n À C Ã r GY F Á for F RY C, provided the Baum-Connes Conjecture is true.

Journal of Geometry and Physics, 2011

We define a new kind of algebroid which fulfills a Leibniz rule, a Jacobi identity twisted by a 3-form H with values in the kernel of the anchor map, and the twist is closed under a naturally occurring exterior covariant derivative. We give examples and define three kinds of cohomologies, two via realizations as Q-structures on graded manifolds. The paper

K-Theory and Noncommutative Geometry, 2008

Twisted K-theory has its origins in the author's PhD thesis [27] and in a paper with P. Donovan [19]. The objective of this paper is to revisit the subject in the light of new developments inspired by Mathematical Physics. See for instance E. Witten [42], J. Rosenberg [37], C. Laurent-Gentoux, J.-L. Tu, P. Xu [41] and M.F. Atiyah, G. Segal [8], among many authors. We also prove some new results in the subject: a Thom isomorphism, explicit computations in the equivariant case and new cohomology operations.

We generalize Hansen--Strobl's definition of $H$-twisted Courant algebroid such that the twist $H$ of the Jacobi identity is a 4-form in the kernel of the anchor map and is closed under a naturally occurring exterior covariant derivative. We give examples and define a cohomology.

Cohomological Methods in Homotopy Theory, 2001

In an earlier paper [10], we showed that for any discrete group G, equivariant K-theory for finite proper G-CW-complexes can be defined using equivariant vector bundles. This was then used to prove a version of the Atiyah-Segal completion theorem in this situation. In this paper, we continue to restrict attention to actions of discrete groups, and begin by constructing an appropriate classifying space which allows us to define K * G (X) for an arbitrary proper G-complex X. We then construct rational-valued equivariant Chern characters for such spaces, and use them to prove some more general versions of completion theorems. In fact, we construct two different types of equivariant Chern character, both of which involve Bredon cohomology with coefficients in the system G/H → R(H). The first, ch

Algebraic Topology, 2009

In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of O-modules on schemes, as well as its quasi-coherent and perfect versions. We also explain how ideas from derived algebraic geometry and higher category theory can be used in order to construct a Chern character for these categorical sheaves, which is a categorified version of the Chern character for perfect complexes with values in cyclic homology. Our construction uses in an essential way the derived loop space of a scheme X, which is a derived scheme whose theory of functions is closely related to cyclic homology of X. This work can be seen as an attempt to define algebraic analogs of elliptic objects and characteristic classes for them. The present text is an overview of a work in progress and details will appear elsewhere. Contents 1 Motivations and objectives 1 2 Categorification of homological algebra and dg-categories 6 3 Loop spaces in derived algebraic geometry 10 4 Construction of the Chern character 13 5 Final comments 15

Let K be a finite group and let G be a finite group acting on K by automorphisms. In this paper we study two different but intimately related subjects: on the one side we classify all possible multiplicative and associative structures with which one can endow the twisted G-equivariant K-theory of K, and on the other, we classify all possible monoidal structures with which one can endow the category of twisted and G-equivariant bundles over K. We achieve this classification by encoding the relevant information in the cochains of a sub double complex of the double bar resolution associated to the semidirect product K ⋊ G; we use known calculations of the cohomology of K, G and K ⋊ G to produce concrete examples of our classification. In the case in which K = G and G acts by conjugation, the multiplication map G ⋊ G → G is a homomorphism of groups and we define a shuffle homomorphism which realizes this map at the homological level. We show that the categorical information that defines the Twisted Drinfeld Double can be realized as the dual of the shuffle homomorphism applied to any 3-cocycle of G. We use the pullback of the multiplication map in cohomology to classify the possible ring structures that the Grothendieck ring of representations of the Twisted Drinfeld Double may have, and we include concrete examples of this procedure.

Differential Geometry and its Applications, 2010

We describe the Cartan and Weil models of twisted equivariant cohomology together with the Cartan homomorphism among the two, and we extend the Chern-Weil homomorphism to the twisted equivariant cohomology. We clarify that in order to have a cohomology theory, the coefficients of the twisted equivariant cohomology must be taken in the completed polynomial algebra over the dual Lie algebra of G. We recall the relation between the equivariant cohomology of exact Courant algebroids and the twisted equivariant cohomology, and we show how to endow with a generalized complex structure the finite dimensional approximations of the Borel construction M × G EG k , whenever the generalized complex manifold M possesses a Hamiltonian G action.

Let K be a finite group and let G be a finite group acting over K by automorphisms. In this paper we study two different but intimate related subjects: on the one side we classify all possible multiplicative and associative structures which one can endow the twisted G-equivariant K-theory over K, and on the other, we classify all possible monodical structures which one endow the category of twisted and G-equivariant bundles over K. We achieve this classification by encoding the relevant information in the cochains of a sub double complex of the double bar resolution associated to the semi-direct product K ⋊ G; we use known calculations of the cohomology of K, G and K ⋊ G to produce concrete examples of our classification. In the case on which K = G and G acts by conjugation, the multiplication map G ⋊ G → G is a homomorphism of groups and we define a shuffle homomorphism which realizes this map at the homological level. We show that the categorical information that defines the Twisted Drinfeld Double can be realized as the dual of the shuffle homomorphism applied to any 3-cocycle of G. We use the pullback of the multiplication map in cohomology to classify the possible ring structures that Grothendieck ring of representations of the Twisted Drinfeld Double may have, and we include concrete examples of this procedure.