Abstract. 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. 1.

Abstract. We show that the p-dimensional noncommutative Yang–Mills model corresponding to a (p − 1)-brane allows solutions which correspond to lower branes. This may be interpreted as the Morita equivalence of noncommutative planes of various dimensions. 1.

Abstract. We establish the Thom isomorphism in twisted K-theory for any real vector bundle and develop the push-forward map in twisted K-theory for any differentiable proper map f: X → Y (not necessarily K-oriented). The push-forward map generalizes the pushforward map in ordinary K-theory for any K-oriented differentiable proper map and the Atiyah-Singer index theorem of Dirac operators on Clifford modules. For D-branes satisfying Freed-Witten’s anomaly cancellation condition in a manifold with a non-trivial B-field, we associate a canonical element in the twisted K-group to get the so-called D-brane charges. Contents

The role of the gauge invariance in noncommutative field theory is discussed. A basic introduction to noncommutative geometry and noncommutative field theory is given. Background invariant formulation of Wilson lines is proposed. Duality symmetries relating various

Topological integrals appear frequently in Lagrangian field theories. On manifolds without boundary, they can be treated in the framework of ordinary (co)homology using the formalism of Cheeger–Simons differential characters. String and D–brane theory involve field theoretic models on worldvolumes with border. On manifolds with boundary, the proper treatment of topological integrals requires a generalization of the usual differential topological set up and leads naturally to relative (co)homology and relative Cheeger–Simons differential characters. In this paper, we present a construction of relative Cheeger–Simons differential characters which is computable in principle and which contains the ordinary Cheeger–Simons differential characters as a particular case.

In this paper, we develop differential characters in twisted K-theory 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 K-theory 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 K-theory 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 D-brane charges on a compact Lie group with non-trivial twisting by a Deligne class.

Abstract. We describe a categorical framework for the classification of D-branes on noncommutative spaces using techniques from bivariant K-theory of C ∗-algebras. We present a new description of bivariant K-theory in terms of noncommutative correspondences which is nicely adapted to the study of T-duality in open string theory. We systematically use the diagram calculus for bivariant K-theory as detailed in our previous paper [12]. We explicitly work out our theory for a number of examples of noncommutative manifolds.

We analyse unstable D-brane systems in type I string theory. Generalizing the proposal in hep-th/0108085, we give a physical interpretation for real KK-theory and claim that the D-branes embedded in a product space X×Y which are made from the unstable Dpbrane system wrapped on Y are classified by a real KK-theory group KKO p−1 (X, Y). The field contents of the unstable D-brane systems are systematically described by a hidden Clifford algebra structure. We also investigate the matrix theory based on non-BPS D-instantons and show that the spectrum of D-branes in the theory is exactly what we expect in type I string theory, including stable non-BPS D-branes with Z2 charge. We explicitly construct the D-brane solutions in the framework of BSFT and analyse the physical property making use of the Clifford algebra. 1

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We review some aspects of string tachyon dynamics with special emphasis on effective actions and K-theory interpretation. 1