90 research outputs found

    A note on bundle gerbes and infinite-dimensionality

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    Let (P,Y)(P, Y) be a bundle gerbe over a fibre bundle YMY \to M. We show that if MM is simply-connected and the fibres of YMY \to M are connected and finite-dimensional then the Dixmier-Douady class of (P,Y)(P, Y) is torsion. This corrects and extends an earlier result of the first author

    On a generalized Connes-Hochschild-Kostant-Rosenberg theorem

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    The central result here is an explicit computation of the Hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds X as their spectrum. More precisely, the Hochschild homology is identified with the space of differential forms on X, and the periodic cyclic homology with the twisted de Rham cohomology of X, thereby generalizing some fundamental results of Connes and Hochschild-Kostant-Rosenberg. The Connes-Chern character is also identified here with the twisted Chern character.Comment: 35 pages, latex2e, uses xypic. To appear in, Advances in Mathematic

    Chern character in twisted K-theory: equivariant and holomorphic cases

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    It has been argued by Witten and others that in the presence of a nontrivial B-field, D-brane charges in type IIB string theories are measured by twisted K-theory. In joint work with Bouwknegt, Carey and Murray it was proved that twisted K-theory is canonically isomorphic to bundle gerbe K-theory, whose elements are ordinary vector 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. In this paper, we study in more detail the Chern-Weil representative of the Chern character of bundle gerbe K-theory that was introduced previously, and we also extend it to the equivariant and holomorphic cases. Included is a discussion of interesting examples.Comment: 24 pages, Latex2e. To appear in CM

    Principal infinity-bundles - Presentations

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    We discuss two aspects of the presentation of the theory of principal infinity-bundles in an infinity-topos, introduced in [NSSa], in terms of categories of simplicial (pre)sheaves. First we show that over a cohesive site C and for G a presheaf of simplicial groups which is C-acyclic, G-principal infinity-bundles over any object in the infinity-topos over C are classified by hyper-Cech-cohomology with coefficients in G. Then we show that over a site C with enough points, principal infinity-bundles in the infinity-topos are presented by ordinary simplicial bundles in the sheaf topos that satisfy principality by stalkwise weak equivalences. Finally we discuss explicit details of these presentations for the discrete site (in discrete infinity-groupoids) and the smooth site (in smooth infinity-groupoids, generalizing Lie groupoids and differentiable stacks). In the companion article [NSSc] we use these presentations for constructing classes of examples of (twisted) principal infinity-bundles and for the discussion of various applications.Comment: 55 page