Twisted K-theory and loop groups

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.

Journal of Geometry and Physics, 2009

In this paper, we develop differential twisted K-theory and define a twisted Chern character on twisted K-theory which depends on a choice of connection and curving on the twisting gerbe. We also establish the general Riemann-Roch theorem in twisted K-theory and find some applications in the study of twisted K-theory of compact simple Lie groups.

2008

In this paper, we develop differential twisted K-theory and define a twisted Chern character on twisted K-theory which depends on a degree three 3 Deligne cocycle. We also establish the general Riemann-Roch theorem in twisted K-theory and find some applications in the study of twisted K-theory of compact simple Lie groups.

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.

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].

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...

K-Theory, 2000

In this paper we study the "holomorphic K -theory" of a projective variety.

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

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.

We compare different algebraic structures in twisted equivariant K-Theory for proper actions of discrete groups. After the construction of a module structure over untwisted equivariant K-Theory, we prove a comple- tion Theorem of Atiyah-Segal type for twisted equivariant K-Theory. Using a Universal coefficient Theorem, we prove a cocompletion Theorem for Twisted Borel K-Homology for discrete Groups.

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.

Pacific Journal of Mathematics, 2013

The main result of this paper establishes an explicit ring isomorphism between the twisted orbifold K-theory ω K orb ([ * / G]) and R(D ω (G)) for any element ω ∈ Z 3 (G; S 1). We also study the relation between the twisted orbifold K-theories α K orb (ᐄ) and α K orb (ᐅ) of the orbifolds ᐄ = [ * / G] and ᐅ = [ * / G ], where G and G are different finite groups, and α ∈ Z 3 (G; S 1) and α ∈ Z 3 (G ; S 1) are different twistings. We prove that if G is an extraspecial group with prime number p as an index and order p n (for some fixed n ∈ ‫,)ގ‬ under a suitable hypothesis over the twisting α we can obtain a twisting α on the group ‫ޚ(‬ p) n such that there exists an isomorphism between the twisted K-theories α K orb ([ * / G ]) and α K orb ‫ޚ(/ * [(‬ p) n ]). Velásquez was partially supported by Colciencias through the grant Becas Generación del Bicentenario #494, Fundación Mazda para el Arte y la Ciencia, and Prof. Wolfgang Lück through his Leibniz Prize. We want to express our gratitude to Professor Bernardo Uribe for his important suggestions and ideas for our work.

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.

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