Twisted K-theory – old and new

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.

Journal of Topology, 2007

Using a global version of the equivariant Chern character, we describe the complexified twisted equivariant K-theory of a space with a compact Lie group action in terms of fixed-point data. We apply this to the case of a compact group acting on itself by conjugation, and relate the result to the Verlinde algebra and to the Kac numerator at q = 1. Verlinde's formula is also discussed in this context.

Journal of High Energy Physics, 2004

Recently, Diaconescu, Moore and Witten provided a nontrivial link between K-theory and M-theory, by deriving the partition function of the Ramond-Ramond fields of Type IIA string theory from an E 8 gauge theory in eleven dimensions. We give some relations between twisted K-theory and M-theory by adapting the method of [1], . In particular, we construct the twisted K-theory torus which defines the partition function, and also discuss the problem from the loop group picture, in which the Dixmier-Douady class is the Neveu-Schwarz field. In the process of doing this, we encounter some mathematics that is new to the physics literature. In particular, the eta differential form, which is the generalization of the eta invariant, arises naturally in this context. We conclude with several open problems in mathematics and string theory.

We use the spectral sequence developed by Graeme Segal in order to understand the Twisted G-Equivariant K-Theory for proper and discrete actions. We show that the second page of this spectral sequence is isomorphic to a version of Equivariant Bredon cohomology with local coefficients in twisted representations. We furthermore give an explicit description of the third differential of the spectral sequence, and we recover known results when the twisting comes from finite order elements in discrete torsion. 1 2 NOÉ BÁRCENAS, JESÚS ESPINOZA, BERNARDO URIBE, AND MARIO VELÁSQUEZ

Algebraic & Geometric Topology, 2014

We use a spectral sequence to compute twisted equivariant K-Theory groups for the classifying space of proper actions of discrete groups. We study a form of Poincaré Duality for twisted equivariant K-theory studied by Echterhoff, Emerson and Kim in the context of the Baum-Connes Conjecture with coefficients and verify it for the Group Sl 3 Z. In this work, we examine computational aspects relevant to the computation of twisted equivariant K-theory and K-homology groups for proper actions of discrete groups. Twisted K-theory was introduced by Donovan and Karoubi [DK70] assigning to a torsion element α ∈ H 3 (X, Z) abelian groups α K * (X) defined on a space by using finite dimensional matrix bundles. After the growing interest by physicists in the 1990s and 2000s, Atiyah and Segal [AS04] introduced a notion of twisted equivariant K-theory for actions of compact Lie Groups. In another direction, orbifold versions of twisted K-theory were introduced by Adem and Ruan [AR03], and progress was made to develop computational tools for Twisted Equivariant K-Theory with the construction of a spectral sequence in [BEUV13]. The paper [BEJU12] introduces Twisted equivariant K-theory for proper actions, allowing a more general class of twists, classified by the third integral Borel cohomology group H 3 (X × G EG, Z). We concentrate in the case of twistings given by discrete torsion, which is given by cocycles α ∈ Z 2 (G, S 1) representing classes in the image of the projection map H 2 (G, S 1) ∼ = → H 3 (BG, Z) → H 3 (X× G EG, Z). Under this assumption on the twist, a version of Bredon cohomology with coefficients in twisted representations can be used to approximate twisted equivariant K-Theory, by means of a spectral sequence studied in [BEUV13] and [Dwy08]. The Bredon (co)-homology groups relevant to the computation of twisted equivariant K-theory, and its homological version, twisted equivariant K-homology satisfy a Universal Coefficient Theorem, 1.13. We state it more generally for a pair of coefficient systems satisfying conditions 1.12. Theorem (Universal Coefficient Theorem). Let X be a proper, finite G-CW complex. Let M ? and M ? be a pair of functors satisfying Conditions 1.12. Then, there exists a short exact sequence of abelian groups 0 → Ext Z (H G n−1 (X, M ?), Z) → H n G (X, M ?) → Hom Z (H G n (X, M ?), Z) → 0

Journal of Geometry and Physics, 2014

This is a study of twisted K-theory on a product space T ¢ M . The twisting comes from a decomposable cup product class which applies the 1-cohomology of T and the 2-cohomology of M . In the case of a topological product, we give a concrete realization for the gerbe associated to a cup product characteristic class and use this to realize twisted K 1 -theory elements in terms of supercharge sections in a Fredholm bundle. The nontriviality of this construction is proved. Equivariant twisted K-theory and gerbes are studied in the product case as well. This part applies Lie groupoid theory. Superconnection formalism is used to provide a construction for characteristic polynomials which are used to extract information from the twisted K-theory classes.

2011

For an integral cohomology class H of degree n+2 on a space X, we define twisted Morava K-theory K(n)(X; H) at the prime 2, as well as an integral analogue. We explore properties of this twisted cohomology theory, study a twisted Atiyah-Hirzebruch spectral sequence, and give a universal coefficient theorem (in the spirit of Khorami). We extend the construction to define twisted Morava E-theory, and provide applications to string theory and M-theory.

arXiv: Algebraic Topology, 2019

We provide a systematic approach to twisting differential KO-theory leading to a construction of the corresponding twisted differential Atiyah-Hirzebruch spectral sequence (AHSS). We relate and contrast the degree two and the degree one twists, whose description involves appropriate local systems. Along the way, we provide a complete and explicit identification of the differentials at the $E_2$ and $E_3$ pages in the topological case, which has been missing in the literature and which is needed for the general case. The corresponding differentials in the refined theory reveal an intricate interplay between topological and geometric data, the former involving the flat part and the latter requiring the construction of the twisted differential Pontrjagin character. We illustrate with examples and applications from geometry, topology and physics. For instance, quantization conditions show how to lift differential $4k$-forms to twisted differential KO-theory leading to integrality result...

Trends in Mathematics, 2006

In a previous paper , we have introduced the gauge-equivariant K-theory group K 0 G (X) of a bundle π X : X → B endowed with a continuous action of a bundle of compact Lie groups p : G → B. These groups are the natural range for the analytic index of a family of gauge-invariant elliptic operators (i.e., a family of elliptic operators invariant with respect to the action of a bundle of compact groups). In this paper, we continue our study of gaugeequivariant K-theory. In particular, we introduce and study products, which helps us establish the Thom isomorphism in gauge-equivariant K-theory. Then we construct push-forward maps and define the topological index of a gaugeinvariant family.

International Congress of Mathematicans, 2002

Twisted K-theory has received much attention recently in both mathematics and physics. We describe some models of twisted K-theory, both topological and geometric. Then we state a theorem which relates representations of loop groups to twisted equivariant K-theory. This is joint work with Michael Hopkins and Constantin Teleman.

Algebraic and Geometric Topology, 2014

Replaces Previous version. Includes comments on poincare duality for twisted equivariant in the context of proper and discrete actions and the Baum-Connes Conjecture. We use a spectral sequence proposed by C. Dwyer and previous work by Sanchez-Garcia and Soule to compute Twisted Equivariant K-theory groups of the classifying space for proper actions of Sl3(Z). After proving a Universal coefficient theorem in Bredon Cohomology with specific coefficients, we compute the twisted equivariant K-homology and state a relation to the Baum-Connes Conjecture with coefficients.

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 K-theory, 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". Our approach is to work on presentations of stacks, namely groupoids, and relies heavily on the machinery of K-theory (KK-theory) of C *-algebras.  2004 Elsevier SAS RÉSUMÉ.-Dans cet article, nous développons la K-théorie tordue pour les champs différentiables, où la torsion s'effectue par une S 1-gerbe sur le champ en question. Nous en établissons les propriétés générales telles que les suites exactes de Mayer-Vietoris, la périodicité de Bott, et le produit K i α ⊗ K j β → K i+j α+β. Notre approche fournit un cadre général permettant d'étudier diverses K-théories tordues, en particulier la K-théorie tordue usuelle des espaces topologiques, la K-théorie tordue équivariante, et la K-théorie tordue des orbifolds. Nous donnons également une définition équivalente utilisant des opérateurs de Fredholm, et nous discutons les conditions sous lesquelles les groupes de K-théorie tordue peuvent être réalisés à partir de "fibrés vectoriels tordus". Notre approche consiste à travailler sur les réalisations concrètes des champs, à savoir les groupoïdes, et s'appuie de façon importante sur les techniques de K-théorie (KK-théorie) des C *-algèbres.

Communications in …, 2002

In this note we introduce the notion of bundle gerbe K-theory and investigate the relation to twisted K-theory. We provide some examples. Possible applications of bundle gerbe K-theory to the classification of D-brane charges in nontrivial backgrounds are briefly discussed.

2022

We present a decomposition of rational twisted G-equivariant Ktheory, G a finite group, into cyclic group equivariant K-theory groups of fixed point spaces. This generalises the untwisted decomposition by Atiyah and Segal [AS89] as well as the decomposition by Adem and Ruan for twists coming from group cocycles [AR03].

Advances in Applied Clifford Algebras, 2008

The purpose of this short paper is to make the link between the fundamental work of Atiyah, Bott and Shapiro [1] and twisted K-theory as defined by P. Donovan, J. Rosenberg and the author [2] [8] . This link was implicit in the literature (for bundles over spheres as an example) but was not been explicitly defined before.