state-sum models and gerbes

where G is a finite group. If we want to understand the differential geometry behind the DW-model we have to give up the finiteness of G of course. If G is a Lie group, we can ask ourselves what geometric objects correspond to smooth homomorphisms from π1(M) to G (we will not explain here what we mean by smoothness exactly). The answer is well known: principal Gbundles with flat connections. We explain this in some more detail. Let {Ui} be a covering of M by open sets such that all intersections Ui1...ip = Ui1 ∩ · · · ∩ Uip are contractible. We present a principal G-bundle, P , by its transition functions gij : Uij → G, which satisfy gji = g −1 ij and the cocycle condition gijgjkg −1 ik = 1 on Uijk. A connection, A, in P can be defined in terms of local 1-forms, Ai on Ui, with values in the Lie algebra of G, which satisfy