0 Differentiable Cohomology of Gauge Groups

There is a well-known theory of differentiable cohomology H diff (G, V ) of a Lie group G with coefficients in a topological vector space V on which G acts differentiably. This is developed by Blanc in [Bl]. It is very desirable to have a theory of differentiable cohomology for a (possibly infinite-dimensional) Lie group G, with coefficients in an arbitrary abelian Lie group A, such that the groups H l diff (G,A) have the expected interpretations. For instance, H diff (G,A) should classify the Lie group central extensions of G by A. In this paper we introduce such a theory and study various differentiable cohomology classes for finite-dimensional Lie groups and for gauge groups. We are mostly interested in the coefficient group A = C. In that case, we have the exponential exact sequence relating the differentiable cohomologies with coefficients Z, C and C. This allows us easily to compute H l diff (G,C ) for a compact Lie group G: it is isomorphic to the cohomology H (G,Z). In the case of gauge groups, we construct various differentiable cohomology classes, including the central extension of a loop group as a special case. We also prove reciprocity laws for gauge groups of differentiable manifolds with boundary embedded in a complex manifold, in the spirit of the Segal-Witten reciprocity law for loop groups. The definition of H l diff (G,A) uses simplicial sheaves. We consider the classifying space BG as a simplicial manifold, which in simplicial degree p is equal to G. Then over each manifold G we have the sheaf A of smooth A-valued functions. These sheaves organize into a simplicial sheaf A over BG. We then define H l diff (G,A) to be the degree l hypercohomology of BG with coefficients in this simplicial sheaf. Our motivation is to construct differentiable analogs of the classes in the cohomology H(Gδ,C ) of the discrete group Gδ constructed by Cheeger and Simons [Chee-S] using geodesic simplices. Similar classes have been constructed by Beilinson using his Chern classes in Deligne cohomology. We construct a differentiable cohomology class in H diff (G,C ) corresponding to a characteristic class in H(BG,Z). In fact, we construct a more powerful holomorphic class in the holomorphic group cohomology H hol (GC,C ) where GC is the complexification of G. We conjecture that these classes map to the classes in [Chee-S] under the natural map from the holomorphic cohomology of GC to the cohomology of the discrete group Gδ. In the spirit of secondary characteristic classes, we construct an extension of these differentiable cohomology classes involving differential forms of degree 0, 1, · · · , p − 1 on the various stages G of the simplicial manifold BG. Again the constructions are done holomorphically on BGC. The precise content of the construction is that it yields a class in the Deligne (hyper)-cohomology of BGC. Deligne cohomology is defined using a complex of sheaves, and the corresponding