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Gerbes, simplicial forms and invariants for families of foliated bundles

by Johan L. Dupont and Franz W. Kamber
Preprints Number 8 (August 2003)
The notion of smooth Deligne cohomology is conveniently reformulated in terms of the simplicial deRham complex. In particular the usual Chern-Weil and Chern-Simons theory is well adapted to this framework and rather easily gives rise to characteristic Deligne cohomology classes associated to families of bundles and connections. In turn this gives invariants for families of foliated bundles. The construction provides representing cocycles in the usual Čech-deRham model for smooth Deligne cohomology called 'gerbes with connection' as they generalize usual Hermitian line bundles with connection. A special case is the Quillen line bundle associated to families of flat $\operatorname{SU}(2)$-bundles.
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Former version: Version 1
Published in Communications in Mathematical Physics
The original publication is avaliable at springerlink.com