Analysis of gauge-equivariant complexes and a topological index theorem for gauge-invariant families

We continue our study of gauge equivariant K -theory. We thus study the analysis of complexes endowed with the action of a family of compact Lie groups and their index in gauge equivariant K -theory. We introduce various index functions, including an axiomatic one, and show that all index functions coincide. As an application, we prove a topological index theorem for a family D = ( D b ) b∈B of gauge-invariant elliptic operators on a G -bundle X → B , where G → B is a locally trivial bundle of compact groups, with typical fiber G . More precisely, one of our main results states that a-ind( D ) = t-ind( D ) ∈ K G 0 ( X ), that is, the equality of the analytic index and of the topological index of the family D in the gauge-equivariant K -theory groups of X . The analytic index ind a ( D ) is defined using analytic properties of the family D and is essentially the difference of the kernel and cokernel K G -classes of D . The topological index is defined purely in terms of the principal symbol of D .