Gerbes,Simplicial Forms and Invariants for Families of Foliated Bundles

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 ech-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 SU(2)-bundles.Work supported in part by the Erwin Schrödinger International Institute of Mathematical Physics, Wien, Austria and by the Statens Naturvidenskabelige Forskningsråd, DenmarkSupported in part by the European Union Network EDGE.Supported in part by Fonds zur Förderung der wissenschaftlichen Forschung, Projekt P 14195 MATAcknowledgement The results of the paper go back a few years but the presentation follows a talk given by the first author in November 2002 during the program Aspects of Foliation Theory at the Erwin Schrödinger Institute in Vienna. Both authors gratefully acknowledge the hospitality and support of the Erwin Schrödinger Institute. The second author visited Å;rhus on several occasions during the preparation of this work and would like to thank the Department of Mathematics at Aarhus University for its hospitality and support. Finally we want to thank the referee for some very useful comments in particular on the terminology of gerbes and Deligne cohomology.