Twisted K-theory and K-theory of bundle gerbes

Journal of High Energy Physics, 2000

In this note we propose that D-brane charges, in the presence of a topologically non-trivial B-field, are classified by the K-theory of an infinite dimensional C * -algebra. In the case of B-fields whose curvature is pure torsion our description is shown to coincide with that of Witten.

Journal of High Energy Physics, 1998

Journal of High Energy Physics, 2003

Witten has argued that charges of Type I D-branes in the presence of an H-flux, take values in twisted KO-theory. We begin with the study of real bundle gerbes and their holonomy. We then introduce the notion of real bundle gerbe KO-theory which we establish is a geometric realization of twisted KOtheory. We examine the relation with twisted K-theory, the Chern character and provide some examples. We conclude with some open problems.

International Journal of Geometric Methods in Modern Physics, 2004

In this review we show how K-theory classifies RR-charges in type II string theory and how the inclusion of the B-field modifies the general structure leading to the twisted K-groups. Our main purpose is to give an expository account of the physical relevance of K-theory. To do that, we consider different points of view: processes of tachyon condensation, cancellation of global anomalies and gauge fixings. As a field to test the proposals of K-theory, we concentrate on the study of the D6-brane, now seen as a non-abelian monopole.

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.

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.

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.

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.

2004

Twisted complex K-theory can be defined for a space X equipped with a bundle of complex projective spaces, or, equivalently, with a bundle of C * -algebras. Up to equivalence, the twisting corresponds to an element of H 3 (X; Z). We give a systematic account of the definition and basic properties of the twisted theory, emphasizing some points where it behaves differently from ordinary K-theory. (We omit, however, its relations to classical cohomology, which we shall treat in a sequel.) We develop an equivariant version of the theory for the action of a compact Lie group, proving that then the twistings are classified by the equivariant cohomology group H 3 G (X; Z). We also consider some basic examples of twisted K-theory classes, related to those appearing in the recent work of Freed-Hopkins-Teleman.

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.