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Gerbes, (twisted) \(K\)-theory, and the supersymmetric WZW model. (English) Zbl 1058.81067

Wurzbacher, Tilmann (ed.), Infinite dimensional groups and manifolds. Based on the 70th meeting of theoretical physicists and mathematicians at IRMA, Strasbourg, France, May 2004. Berlin: de Gruyter (ISBN 3-11-018186-X/pbk). IRMA Lectures in Mathematics and Theoretical Physics 5, 93-107 (2004).
Summary: The aim of this talk is to explain how symmetry breaking in a quantum field theory problem leads to a study of projective bundles, Dixmier-Douady classes, and associated gerbes. A gerbe manifests itself in different equivalent ways. Besides the cohomological description as a DD class, it can be defined in terms of a family of local line bundles or as a prolongation problem for an (infinite-dimensional) principal bundle, with the fiber consisting of (a subgroup of) projective unitaries in a Hilbert space. The prolongation aspect is directly related to the appearance of central extensions of (broken) symmetry groups. We also discuss the construction of twisted K-theory classes by families of supercharges for the supersymmetric Wess-Zumino-Witten model.
For the entire collection see [Zbl 1050.22003].

MSC:

81T50 Anomalies in quantum field theory
58J90 Applications of PDEs on manifolds
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